Radiometric Temperature Measurements, Volume 42: I. Fundamentals (Experimental Methods in the Physical Sciences)

EXPERIMENTAL METHODS PHYSICAL SCIENCES IN THE Thomas Lucatorto and Al Parr, Editors in Chief Founding Editors L. MARTO...

Author: Zhuomin M. Zhang | Benjamin K. Tsai | Graham Machin

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EXPERIMENTAL METHODS PHYSICAL SCIENCES

IN THE

Thomas Lucatorto and Al Parr, Editors in Chief Founding Editors L. MARTON C. MARTON

VOLUME FOURTY TWO

EXPERIMENTAL METHODS PHYSICAL SCIENCES

IN THE

RADIOMETRIC TEMPERATURE MEASUREMENTS I. Fundamentals Editors

Zhuomin M. Zhang George W. Woodruff School of Mechanical Engineering Georgia Institute of Technology Atlanta, GA 30332-0405, USA

Benjamin K. Tsai Optical Technology Division, National Institute of Standards and Technology, Gaithersburg, MD 20899-8441, USA

Graham Machin Engineering Measurement Division National Physical Laboratory Teddington, Middlesex, TW11 0LW, UK

Amsterdam Boston Heidelberg London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo Academic Press is an imprint of Elsevier

Academic Press is an imprint of Elsevier Linacre House, Jordan Hill, Oxford OX2 8DP, UK 84 Theobald’s Road, London WC1X 8RR, UK Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA Copyright r 2010 Elsevier Inc. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://www.elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made Library of Congress Cataloging-in-Publication Data Radiometric temperature measurements / edited by Zhang, Tsai, and Machin. p. cm. Includes bibliographical references and index. ISBN 978-0-12-374021-2 1. Temperature measurements. 2. Radiation pyrometers. 3. Radiation–Measurement. I. Zhang, Zhuomin M. II. Tsai, Benjamin K. III. Machin, Graham. QC271.R336 2009 536’.5–dc22 2009026233 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-374021-2 ISSN: 1079-4042 For information on all Academic Press publications visit our web site at elsevierdirect.com Printed and bound in United States of America 10 11 12 13 14 10 9 8 7 6 5 4 3 2 1

In memory of David P. DeWitt 2 a pioneer, teacher, and friend

LIST

OF

CONTRIBUTORS

Numbers in parenthesis indicate the pages on which the author’s contributions

begins.

George P. Eppeldauer (133)

Optical Technology Division, National Institute of Standards and Technology,

Gaithersburg, MD 20899-8441, USA

Leonard M. Hanssen (181)

National Institute of Standards and Technology, 100 Bureau Drive, Gaithersburg,

MD 20899-8442, USA

Ju¨rgen Hartmann (241)

Physikalisch-Technische Bundesanstalt, Braunschweig and Berlin, Germany

Jo¨rg Hollandt (241)

Physikalisch-Technische Bundesanstalt, Braunschweig and Berlin, Germany

Boris Khlevnoy (241)

All-Russian Research Institute for Optical and Physical Measurements, Section

M-4, Moscow, Russia

Bong Jae Lee (73)

Department of Mechanical Engineering and Materials Science, University of

Pittsburgh, Pittsburgh, PA 15261-3648, USA

Yue Liu (297)

Center for Advanced Diffusion-Wave Technologies (CADIFT), Department of

Mechanical and Industrial Engineering, University of Toronto, Toronto, ON,

Canada M5S3G8; School of Mechanical Engineering, Dalian University of

Technology, Dalian, Liaoning 116024, China

Graham Machin (1, 29)

Engineering Measurement Division, National Physical Laboratory, Teddington,

Middlesex TW11 0LW, UK

Andreas Mandelis (297)

Center for Advanced Diffusion-Wave Technologies (CADIFT), Department of

Mechanical and Industrial Engineering, University of Toronto, Toronto, ON,

Canada M5S3G8

xi

xii

List of Contributors

Sergey N. Mekhontsev (181)

National Institute of Standards and Technology, 100 Bureau Drive, Gaithersburg,

MD 20899-8442, USA

Svetlana Morozova (241)

All-Russian Research Institute for Optical and Physical Measurements, Section

M-4, Moscow, Russia

Sergey Ogarev (241)

All-Russian Research Institute for Optical and Physical Measurements, Section

M-4, Moscow, Russia

Alexander V. Prokhorov (181)

National Institute of Standards and Technology, 100 Bureau Drive, Gaithersburg,

MD 20899-8442, USA

Fumihiro Sakuma (241)

National Metrology Institute of Japan, Tsukuba, Japan

Benjamin K. Tsai (29)

Optical Technology Division, National Institute of Standards and Technology,

Gaithersburg, MD 20899-8441, USA

Howard W. Yoon (133)

Optical Technology Division, National Institute of Standards and Technology,

Gaithersburg, MD 20899-8441, USA

Zhuomin M. Zhang (1, 73)

George W. Woodruff School of Mechanical Engineering, Georgia Institute of

Technology, Atlanta, GA 30332-0405, USA

VOLUMES IN SERIES EXPERIMENTAL METHODS (FORMERLY METHODS

OF

IN THE

PHYSICAL SCIENCES

EXPERIMENTAL PHYSICS)

Editors-in-Chief Thomas Lucatorto and Albert C. Parr Volume 1. Classical Methods

Edited by lmmanuel Estermann

Volume 2. Electronic Methods, Second Edition (in two parts)

Edited by E. Bleuler and R. O. Haxby

Volume 3. Molecular Physics, Second Edition (in two parts)

Edited by Dudley Williams

Volume 4. Atomic and Electron Physics - Part A:

Atomic Sources and Detectors; Part B: Free Atoms

Edited by Vernon W. Hughes and Howard L. Schultz

Volume 5. Nuclear Physics (in two parts)

Edited by Luke C. L. Yuan and Chien-Shiung Wu

Volume 6. Solid State Physics - Part A: Preparation, Structure,

Mechanical and Thermal Properties; Part B: Electrical,

Magnetic and Optical Properties

Edited by K. Lark-Horovitz and Vivian A. Johnson

Volume 7. Atomic and Electron Physics - Atomic Interactions

(in two parts)

Edited by Benjamin Bederson and Wade L. Fite

Volume 8. Problems and Solutions for Students

Edited by L. Marton and W. F. Hornyak

Volume 9. Plasma Physics (in two parts)

Edited by Hans R. Griem and Ralph H. Lovberg

Volume 10. Physical Principles of Far-Infrared Radiation

Edited by L. C. Robinson

Volume 11. Solid State Physics

Edited by R. V. Coleman

xiii

xiv

Volumes in Series

Volume 12. Astrophysics - Part A: Optical and Infrared Astronomy Edited by N. Carleton Part B: Radio Telescopes; Part C: Radio Observations Edited by M. L. Meeks Volume 13. Spectroscopy (in two parts) Edited by Dudley Williams Volume 14. Vacuum Physics and Technology Edited by G. L. Weissler and R. W. Carlson Volume 15. Quantum Electronics (in two parts) Edited by C. L. Tang Volume 16. Polymers - Part A: Molecular Structure and Dynamics; Part B: Crystal Structure and Morphology; Part C: Physical Properties Edited by R. A. Fava Volume 17. Accelerators in Atomic Physics Edited by P. Richard Volume 18. Fluid Dynamics (in two parts) Edited by R. J. Emrich Volume 19. Ultrasonics Edited by Peter D. Edmonds Volume 20. Biophysics Edited by Gerald Ehrenstein and Harold Lecar Volume 21. Solid State Physics: Nuclear Methods Edited by J. N. Mundy, S. J. Rothman, M. J. Fluss, and L. C. Smedskjaer Volume 22. Solid State Physics: Surfaces Edited by Robert L. Park and Max G. Lagally Volume 23. Neutron Scattering (in three parts) Edited by K. Skold and D. L. Price Volume 24. Geophysics - Part A: Laboratory Measurements; Part B: Field Measurements Edited by C. G. Sammis and T. L. Henyey Volume 25. Geometrical and Instrumental Optics Edited by Daniel Malacara Volume 26. Physical Optics and Light Measurements Edited by Daniel Malacara

Volumes in Series

Volume 27. Scanning Tunneling Microscopy

Edited by Joseph Stroscio and William Kaiser

Volume 28. Statistical Methods for Physical Science

Edited by John L. Stanford and Stephen B. Vardaman

Volume 29. Atomic, Molecular, and Optical Physics Part A: Charged Particles; Part B: Atoms and Molecules;

Part C: Electromagnetic Radiation

Edited by F. B. Dunning and Randall G. Hulet

Volume 30. Laser Ablation and Desorption

Edited by John C. Miller and Richard F. Haglund, Jr.

Volume 31. Vacuum Ultraviolet Spectroscopy I

Edited by J. A. R. Samson and D. L. Ederer

Volume 32. Vacuum Ultraviolet Spectroscopy II

Edited by J. A. R. Samson and D. L. Ederer

Volume 33. Cumulative Author lndex and Tables of Contents,

Volumes 1-32

Volume 34. Cumulative Subject lndex

Volume 35. Methods in the Physics of Porous Media

Edited by Po-zen Wong

Volume 36. Magnetic Imaging and its Applications to Materials

Edited by Marc De Graef and Yimei Zhu

Volume 37. Characterization of Amorphous and Crystalline Rough

Surface: Principles and Applications

Edited by Yi Ping Zhao, Gwo-Ching Wang, and Toh-Ming Lu

Volume 38. Advances in Surface Science

Edited by Hari Singh Nalwa

Volume 39. Modern Acoustical Techniques for the Measurement

of Mechanical Properties

Edited by Moises Levy, Henry E. Bass, and Richard Stern

Volume 40. Cavity-Enhanced Spectroscopies

Edited by Roger D. van Zee and J. Patrick Looney

Volume 41. Optical Radiometry

Edited by A. C. Parr, R. U. Datla, and J. L. Gardner

Volume 42. Radiometric Temperature Measurements. I. Fundamentals

Edited by Z. M. Zhang, B. K. Tsai, and G. Machin

xv

PREFACE Temperature measurement and control has played and continues to play a vital role in many scientiﬁc and technological advances. Radiometric temperature measurement, that is the measurement of temperature based on thermal radiative emission, has a long history from fundamental studies of Planckian emission, to many industrial applications including iron and steel production and materials and chemical processing, to playing a fundamental role in the realization and dissemination of successive international temperature scales. Radiation thermometry is attractive in many challenging temperature measurement situations because it is a noncontact, nonintrusive, and fast technique. Thermal radiation is governed by the fundamental physical laws established over one hundred years ago by Kirchhoff, Stefan, Boltzmann, Wien, and, in particular, Planck. These laws directly link emitted blackbody radiation, total or spectrally resolved, to the thermodynamic temperature of the emitting source. Actual practical measurements by radiation thermometry, however, are prone to a number of uncertainties associated with, for example, surface emissivity and environmental effects such as absorption by dust or smoke and reﬂected ambient radiation. While a number of books have been published on thermometry, in general, no comprehensive book devoted to radiometric temperature measurement has been published since the publication in 1988 of Theory and Practice of Radiation Thermometry, edited by D.P. DeWitt and G.D. Nutter. In recent years, there have been tremendous developments in instrumentation. For instance, infrared focal plane arrays can now produce images with a spatial resolution of order 10 mm with a temperature resolution of 0.01 K. While the expert in the ﬁeld can keep abreast of these rapidly advancing techniques through the information presented at periodic international temperature symposia and through the technical literature, it is very difﬁcult for a newcomer to ﬁnd a deﬁnitive up-to-date summary of the practice of radiation thermometry. This book aims at ﬁlling that gap by covering basic theory, measurement fundamentals, standards and calibration, and summaries of current practice of radiation thermometry in different technical ﬁelds at a level accessible to the newcomer but also comprehensive enough to provide the information needed to understand and bring to bear the latest technique to a particular radiometric temperature measurement problem. This two-volume set on Radiometric Temperature Measurement (I. Fundamentals and II. Applications) is written for those who will apply radiation xvii

xviii

Preface

thermometers in industrial practice, who will use thermometers in scientiﬁc research, who design and develop thermometers for instrument manufacturers, and who will design the thermometers to address particular measurement challenges. These volumes are more than a practice guide. We hope that by presenting the fundamental principles and pointing out the pitfalls in applying radiation thermometry in various settings, our readers will gain knowledge in: (1) the proper selection of the type of thermometer; (2) the best practice in using radiation thermometers; (3) awareness of the uncertainty sources and subsequent appropriate procedure to reduce the overall measurement uncertainty; and (4) understanding of the calibration chain and its current limitations. We have also added a large number of references at the end of each chapter as a source for those seeking a deeper or more detailed understanding. The author(s) of each chapter were chosen from a group of international scientists who are experts in the ﬁeld and specialist(s) on the subject matter covered in the chapter. It is intended that together the two volumes will form a comprehensive summary of the current practice of radiation thermometry. The ﬁrst volume concentrates on the fundamental aspects, while the second volume mainly focuses on the industrial and practical applications. In the fundamental volume, Chapter 1 provides a historical overview of radiation thermometry, explains the basic fundamentals and commonly used terms, and lists the various types of radiation thermometers. The concepts of temperature, its scale realization, calibration, traceability, measurement, uncertainty analysis, and future approaches, are extensively elaborated in Chapter 2. The basic theory on blackbody radiation, radiative properties, and the electromagnetic wave theory are discussed in Chapter 3. Chapter 4 focuses on the design and characterization of radiation thermometers. Chapter 5 addresses the theoretical and computational characterization of isothermal and nonisothermal blackbody cavities by analytical and Monte Carlo methods. In Chapter 6, radiance sources used for calibration such as ﬁxed-point blackbodies, variable temperature blackbodies, cryogenic blackbodies, high stability and other tungsten-based lamps are described. Chapter 7 is an overview of some complementary surface temperature measurement techniques, such as thermal reﬂectance, interferometry, ellipsometry, and photothermal radiometry with application examples. The volume on applications begins with a review of the state-of-the-art industrial applications of radiation thermometry, including a critique of multiwavelength thermometry (Chapter 1). Chapter 2 describes experimental characterization of blackbody cavities with an extensive survey on the measurement techniques. Chapter 3 focuses on the application of optical ﬁber thermometry for semiconductor processing, with an emphasis on rapid thermal processing and in situ calibration of lightpipe thermometers using thin-ﬁlm thermocouples. Chapter 4 reviews the

Preface

xix

state-of-the-art practice of radiation thermometry in the steel industry, highlighting speciﬁc manufacturing processes. Chapter 5 deals with thermal imaging in ﬁreﬁghting and other thermographic applications along with standards of measurement and application. Chapter 6 discusses remote sensing of earth and sea surface temperatures and reviews different instruments and their measuring capabilities. Finally, Chapter 7 covers four aspects of clinical radiation thermometry: ear thermometry, medical thermal imaging, medical pulsed photothermal radiometry, and microwave radiometry for clinical applications. This two-volume set is a tribute to David DeWitt (193422005) who has been an inspiration for us and to many others in the radiation thermometry community. In his last eight years, he dedicated his research to temperature measurement and calibration for rapid thermal processing in microelectronics manufacturing industry. He will always be remembered as a leader in the ﬁelds of radiation thermometry and heat transfer engineering. The editors sincerely thank all of the chapter authors for their outstanding contributions and hard work. We also express appreciation to Dr. Tom Lucatorto and Dr. Al Parr, the series editors, for their constant encouragement during this process and their careful review of the chapter contents. Finally, we would like to thank our families for their full support and enduring patience throughout the writing and editing of this book. Zhuomin M. Zhang Georgia Institute of Technology Benjamin K. Tsai National Institute of Standards and Technology Graham Machin National Physical Laboratory June 2009

NOMENCLATURE A An Au al B c c0 c1 c1L c2 c3 cp cv D D D d E E e f fr or ft H h _ I i J k k L Ll M M Mem m me NA

surface area (m2) projected area (m2) directional absorptance of a semitransparent material absorption coefﬁcient (m1) magnetic induction or magnetic ﬂux density (T or Wb/m2) phase velocity of electromagnetic wave (m/s) speed of light in vacuum (m/s) ﬁrst radiation constant (W m2) ﬁrst radiation constant for spectral radiance (W m2/sr) second radiation constant (m K) third radiation constant (m K) speciﬁc heat at constant pressure ( J/(kg K)) speciﬁc heat at constant volume ( J/(kg K)) electric displacement (C/m2); dynamic matrix density of states detectivity (m Hz1/2/W) diameter or ﬁlm thickness (m) electric ﬁeld vector (N/C or V/m) irradiance (W/m2); energy ( J); magnitude of electric ﬁeld (V/m) elementary charge (C) distribution function bidirectional reﬂectance (or transmittance) distribution function magnetic ﬁeld vector (A/m or C/(m s)) Planck’s constant ( J s) Planck’s constant divided by 2p, h/2p, J s Radiant intensity (W/sr) pﬃﬃﬃﬃﬃﬃﬃ 1 electric current density (A/m2) wavevector (m1) Boltzmann’s constant (J/K); also magnitude of wavevector (m1) radiance (W/(m2 sr)) spectral radiance (W/(m2 mm sr)) magnetization vector (A/m or C/(m s)) exitance (W/m2); molecular weight (kg/kmol) self-exitance or emitted exitance (W/m2) mass (kg) electron mass (kg) Avogadro constant (mol1)

xxi

xxii

n n~ P p p Q q qv R Ru Re r S S Sj T Tu Tl Tt t u V v v x; y; z

Nomenclature

real part of refractive index or refractive index; number density (m3) complex refractive index dipole moment per unit volume (C/m2); propagation matrix momentum vector (mv or _k) (kg m/s); dipole moment (C m) pressure (Pa or N/m2); magnitude of momentum (kg m/s); magnitude of dipole moment (C m) heat or radiant energy ( J) heat transfer rate (W) radiative heat ﬂux (W/m2) universal gas constant ( J/(mol K)); gas constant ( J/(kg K)); electrical resistance (O or V/A); detector responsivity (V/W or A/W) directional-hemispherical reﬂectance electrical resistivity (O m) radius or radial coordinate (m); also Fresnel reﬂection coefﬁcient Poynting vector (W/m2) entropy ( J/K); detector output signal (V or A) strength of the jth phonon oscillator temperature (K) directional-hemispherical transmittance radiance temperature (K or 1C) total radiance temperature (K or 1C) temperature (1C); time (s); Fresnel transmission coefﬁcient energy density ( J/m3) volume (m3); voltage (V) velocity (m/s) speciﬁc volume (m3/kg), also speed (m/s) coordinates (m)

Greek Symbols au b bl g dl e e0 ~ eu Z y yB yc

directional absorptivity phase shift (rad) attenuation coefﬁcient (m1) scattering rate (rad/s) radiation penetration depth (m) dielectric function or relative permittivity; also emissivity electric permittivity of vacuum (F/m or A2 s/(N m2)) complex dielectric function directional emissivity efﬁciency polar or zenith angle (rad) Brewster’s angle (rad) critical angle (rad)

xxiii

Nomenclature

k l m m0 n r ru s sl t F f O o op

extinction coefﬁcient (i.e., imaginary part of the refractive index) wavelength (m) relative magnetic permeability; also mobility (m2/(V s)) magnetic permeability in vacuum (N/A2) frequency (Hz) density (kg/m3); electric charge density (C/m3) directional-hemispherical reﬂectivity Stefan2Boltzmann constant (W/(m2 K4)); electric conductivity (A/V) scattering coefﬁcient (m1) relaxation time (s); internal transmissivity; spectral transmittance scattering phase function azimuthal angle (rad) solid angle (sr) angular frequency (rad/s); solid angle (sr); scattering albedo plasma frequency (rad/s)

Subscripts 0 1,2,3 B, b or bb e em h i L m r t l, n

vacuum medium 1,2,3 blackbody electron emitted hole incident radiance medium reﬂected total (integration over all wavelengths or frequencies) spectral property in terms of wavelength or frequency

Superscripts h p s u

hemispherical TM wave or p (parallel) polarization TE wave or s (perpendicular) polarization used to signify a directional radiative property complex conjugate

CHAPTER 1

Overview of Radiation Thermometry Zhuomin M. Zhang1 and Graham Machin2

Contents 1. Introduction 2. Basics of Radiometric Temperature Measurements 2.1. Total radiation thermometry 2.2. Spectral radiation thermometry 2.3. Radiation thermometer design considerations 3. Types of Radiation Thermometers 3.1. Classical radiation thermometry 3.2. Emissivity correction methods 3.3. Fiber-optic radiation thermometry 4. Terms Commonly Used in Radiation Thermometry 5. Summary References

1 4 4 6 9 11 11 13 18 21 24 24

1. Introduction An intuitive understanding of temperature, associated with the sensations of hot and cold, has been recognized since ancient times. However it was not until 1592, when Galileo invented the air thermoscope, that quantitative thermometry began. Extensive investigations of various types of liquid-in-glass thermometers in the 17th century led to the establishment of the Fahrenheit temperature scale (ﬁrst reported in 1724) and the Celsius (or centigrade) temperature scale (174221744). Some 100 years later in the middle of the 19th century, based on the principles of thermodynamics, Lord Kelvin (William Thomson) laid the foundation of the thermodynamic temperature scale. Today the reliable 1

2

George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405, USA Engineering Measurement Division, National Physical Laboratory, Teddington, Middlesex, TW11 0LW, UK

Experimental Methods in the Physical Sciences, Volume 42 ISSN 1079-4042, DOI 10.1016/S1079-4042(09)04201-5

r 2010 Elsevier Inc. All rights reserved.

1

2

Zhuomin M. Zhang and Graham Machin

measurement of temperature has become an essential component of nearly every ﬁeld of science and industry, including agriculture, biomedicine, chemical and materials processing, earth and atmospheric sciences, manufacturing, and remote sensing [1,2]. Since the early 1900s there have been tremendous advances in all areas of science and technology, e.g., in microelectronics, materials processing, ﬁber optics, laser, and imaging. Consequently, there has been a surge of research and development activities in temperature measurement, either by modifying earlier methods or by developing novel techniques. This continuous progress in thermometry is readily seen from the papers presented at the symposia on Temperature: Its Measurement and Control in Science and Industry. The ﬁrst symposium was held in 1919, and the two most recent ones were held in 1992 (seventh) and 2002 (eighth) [3,4]. In addition, the triennial TEMPMEKO symposia on temperature and thermal measurements in industry and science [5], organized by the Technical Committee (TC12) on Temperature and Thermal Measurements of the International Measurement Confederation (IMEKO), have also charted the developments of thermometry, since their inception in 1961. The most recent proceedings of these symposia (TEMPMEKO 2007, tenth symposium) were published as special issues in International Journal of Thermophysics (200722009) [6]. Every object emits thermal radiation when it is at a temperature above absolute zero. When its temperature approaches 1,000 K, the object begins to emit visible light. As the temperature of an object rises, it begins to glow, ﬁrstly dull red (near 900 K) progressing through light red, orange, yellow, white, blue, and so on. In previous times, metalworkers and glassblowers relied heavily on this incandescence to determine the working temperature ranges by color in order to properly temper, anneal, and/or shape the work pieces. In essence, the human eye was the ﬁrst radiation thermometer. A sound understanding of electromagnetic waves, especially thermal radiation, was developed by Maxwell, Kirchhoff, Stefan, Boltzmann, Rayleigh, Jeans, Wien, and Planck in the late 19th century and at the beginning of the 20th century. Subjective measurements by the eye began to be superseded when the ﬁrst optical pyrometer was introduced by the French chemist Le Chatelier in 1892. A United States patent was granted to Morse in 1899 for an optical pyrometer for noncontact temperature measurements. The Morse pyrometer used the human eye as the photodetector to compare the brightness temperature of the object with that emitted by a tungsten ﬁlament, and the device became known as the disappearing ﬁlament pyrometer. It was the dominant instrument for high-temperature measurements in the ﬁrst half of the 20th century [7]. Total radiation thermometers based on the Stefan2Boltzmann fourth-power law were also developed during the earlier part of the 20th century.

Overview of Radiation Thermometry

3

In the 1950s, several national laboratories began to employ photomultipliers (based on the photoelectric effect and secondary electron emission) as the physical detector for the calibration of optical pyrometers [8]. More recently, signiﬁcant advances in optoelectronics, photodetectors, and radiometers have led to the production of a wide variety of detectorbased luminous intensity scale and detector-based radiation thermometers [9]. Detector-based radiation thermometers have been the prevailing instruments for noninvasive temperature measurement since the mid1980s. These devices generally measure infrared radiation, and hence the term infrared thermometer is commonly used for radiation thermometers. Line scanners and/or infrared cameras/imagers have been used to obtain thermal images that map the surface temperature distribution of an object. For specialist applications, thermal radiation can be used to measure temperatures down to a few kelvins. Indeed, measurement of the cosmic microwave background radiation indicates that the temperature of the universe is around 2.7 K [10]. Since 1990 radiation thermometers, whose operation is based on Planck’s law of thermal radiation, have been used to establish the temperature scale above 961.781C using the freezing point of silver, gold, or copper as a reference temperature according to the International Temperature Scale of 1990 (ITS-90) [11213]. In addition, because radiation thermometry is based on a fundamental physical law, primary radiation thermometers have been used to realize the temperature scale in the range from 1,000 to 3,000 K [14]. The development of high-performance cryogenic radiometers has made it possible to accurately characterize narrowband ﬁlter radiometers for low uncertainty radiometric temperature measurements [15220]. Radiation thermometry belongs to the noninvasive class of temperature measurement techniques. It has several unique advantages such as the ability to reliably follow rapid temperature changes. In some cases, it can measure small objects or map the temperature distribution with a spatial resolution of a few micrometers. It can be used in a harsh environment where the high temperature, high pressure, corrosion, and/or contaminations would prohibit the use of other types of thermometers. Traditionally, radiation thermometers have been used in determining ﬂame and furnace temperatures for combustion and materials processing, glass, aluminum, plastics, and steel industries. More recently, radiation thermometers have also been used in semiconductor processing, asphalt and cement industry, paper industry, food industry, and advanced manufacturing such as laser welding and laser cutting. Furthermore, thermal imaging has been used for remote sensing of earth surface and sea temperatures, monitoring the thermal performance of buildings, tracking forest ﬁres, homeland security, medical diagnostics, as well as diagnostics of hot spots in microelectronic circuit boards. This two-volume set is devoted to radiometric temperature measurements. Part 1 (Vol. 42) focuses on the fundamental aspects and Part 2

4

Zhuomin M. Zhang and Graham Machin

(Vol. 43) deals with various industrial and scientiﬁc applications. This chapter provides an overview of the theory (Section 2), classiﬁcation (Section 3), and common terminology (Section 4) related to this important ﬁeld.

2. Basics of Radiometric Temperature Measurements A schematic setup for radiometric temperature measurement is illustrated in Figure 1. The source of thermal emission is the target for which the temperature is to be measured. A lens conﬁnes the ﬁeld of view and a bandpass optical ﬁlter determines the spectral region within which the thermal radiation is to be measured. An aperture is placed in front of the detector to avoid stray light being detected. The output signal (voltage or current) from the detector is ampliﬁed and measured by a signal processor. This output signal is related to the spectral radiance emitted by the surface and hence its temperature can be deduced. When a radiation thermometer is used to measure the temperature of a real surface, two issues arise. The ﬁrst is the unknown emissivity of the surface (which affects the emitted radiation from the target), and the second is the inﬂuence of the emission from and absorption by the environment (which can signiﬁcantly inﬂuence the radiation reaching the detector). These effects are shown in Figure 1. In addition, thermal or other electromagnetic radiation from nearby objects can also be reﬂected from the target to the radiometer, causing additional uncertainty. Methods, as described later, have been developed in an attempt to overcome these problems with varying degrees of success. The development of optical ﬁbers has allowed radiometric temperature measurements for surface locations otherwise inaccessible by imaging radiometers. In the rest of this section, different general techniques of radiation thermometry are described.

Filter dΩt

At Target

Figure 1

To Signal Processor

Lens Atmosphere Emis. & Abs.

Detector Assembly

Aperture

Schematic diagram of radiometric temperature measurement.

5

Overview of Radiation Thermometry

2.1. Total radiation thermometry The fundamental theory and methodology of radiation thermometry have been well established. Total or broadband radiation thermometry is based on the Stefan2Boltzmann law of thermal radiation (more details are given in Chapter 3). For a blackbody, the total energy leaving the surface per unit area (W/m2) is proportional to its temperature to the fourth power. Since emission from a blackbody is diffuse, the exitent radiance, Lb, from a blackbody at temperature T0 can be expressed as the total radiant exitance divided by p, such that, Lb ¼

sT 40 p

(1)

where s is the Stefan2Boltzmann constant (given in Appendix A). A real surface has an emissivity between 0 and 1, hence, the exitent radiance from an object at temperature T, without considering reﬂections, becomes L em ¼

sT 4 p

(2)

Here, e is the directional-total emissivity since it is assumed that the emission is conﬁned within a small solid angle (see Chapter 3 for more details). The total radiance temperature Tt (or apparent temperature) is deﬁned as the temperature at which a blackbody would emit the same radiance toward the detector as that given in Equation (2). It can be easily seen from Equations (1) and (2) that T t ¼ ðÞ1=4 T

(3)

which is always lower or at most equal to the true surface temperature. The difference between the true temperature and the radiance temperature becomes DT ¼ T ð1 1=4 Þ

(4)

Equation (4) gives the error in temperature if no emissivity correction is applied. The relative error DT/T is 33% for e ¼ 0.2, 12% for e ¼ 0.6, and 4% for e ¼ 0.85. Suppose the emissivity is estimated and a correction to the total radiance temperature is made to deduce the true surface temperature. Then, from Equation (3), we can see that a variation in corresponds to a variation in T by dT 1 d ¼ T 4

(5)

which is obtained by taking the derivative of Equation (3) with ﬁxed Tt. In Equation (5), d signiﬁes a small variation or uncertainty in the indicated quantity. When the target temperature is 1,000 K, a 4% uncertainty in

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Zhuomin M. Zhang and Graham Machin

emissivity will result in a temperature uncertainty of 10 K. It should be noted that unless the thermal environment is well controlled, the uncertainty becomes much larger than that indicated by Equation (5). For this reason, total radiation thermometers are not particularly suited to industrial applications. On the other hand, broadband devices, that approximate to total radiation thermometers, based on thermopile or pyroelectric detectors, are deployed to measure near ambient temperatures because in this measurement situation the impact of emissivity is small. In special cases, such as in the infrared ear thermometer, the measurement area is similar to a blackbody cavity, i.e., the emissivity of the tympanic membrane/ear canal combination is close to unity. The measurement with such devices is essentially based on radiative heat exchange between the source and the receiver. Reasonable accuracy (0.120.21C), for infrared ear thermometers and other broadband devices can be obtained through calibration, but issues pertaining to the conditions of use remain. A very specialized example of using a total radiation thermometer is in experiments for measuring the Stefan2Boltzmann constant and the temperature of a blackbody source with an absolute cryogenic radiometer [21] (see also Chapter 2). In this case, very high accuracy can be achieved under strictly controlled environmental conditions.

2.2. Spectral radiation thermometry The measurement equation of a bandpass spectral radiation thermometer is given by Z

lþDl=2

S l ¼ At dOt

tl RðlÞL l ðlÞdl

(6)

lDl=2

where Sl is the detector output signal (in volts or amperes), At the target area perceived by the radiometer through a solid angle dOt (referring to Figure 1), tl the spectral transmittance of the optical system (e.g., lens and ﬁlter), R(l) the detector responsivity, Ll(l) the spectral radiance that reaches the detector from the target and the environment within the optical path and ﬁeld of view, and Dl the bandwidth of the ﬁlter. It is assumed that there is no transmission outside of the bandwidth of the ﬁlter. For simplicity, the emission by and reﬂection from the detector are neglected in Equation (6). For a narrowband ﬁlter, Equation (6) can be approximated as follows S l ¼ C I L l ðlÞ

(7)

where CI is the instrument constant which is independent of the target. For a freely radiating target (i.e., with cold surroundings and negligible environment emission and absorption), the exitent spectral radiance is due

7

Overview of Radiation Thermometry

purely to emission, therefore, L l ðlÞ ¼ L em;l ðl; T Þ ¼ l L b;l ðl; T Þ

(8)

where L b;l ðl; T Þ is the blackbody spectral radiance given by Planck’s law [13]: L b;l ðl; T Þ ¼

l

5

c 1L c =lT 2 ðe

1Þ

(9)

where c 1L ¼ 2hc 20 is the ﬁrst radiation constant for spectral radiance, and c 2 ¼ hc 0 =k the second radiation constant. Note that vacuum conditions are assumed for simplicity, while in practice the effect of the refractive index of the air may have to be considered for very high-accuracy applications. In radiation thermometry, Wien’s formula L b;l ðl; T Þ ¼

c 1L 5 c 2 =lT le

(10)

is appropriate in most cases because c 2 =lT is usually much greater than unity. The radiance temperature Tl is deﬁned by equating the blackbody radiance at Tl with the exitent radiance of the object at T; hence, L b;l ðl; T l Þ ¼ l L b;l ðl; T Þ

(11)

Using Wien’s approximation given in Equation (10), the surface temperature is related to the spectral radiance temperature by 1 1 l ¼ þ ln l T T l c2

(12)

The difference between the true temperature and the radiance temperature is lT l ln l DT ¼ T c2

(13)

As an example, Figure 2 shows the temperature difference for a target temperature at 1,000 K at different wavelengths, along with that calculated from Equation (4) for a total radiation thermometer. It can be seen that as the wavelength decreases, the difference between the true temperature and the radiance temperature decreases. Furthermore, the difference becomes smaller when the emissivity is closer to unity. The uncertainty in the measured temperature due to the uncertainty of emissivity can be obtained by differentiating T with respect to el using Equation (12) for ﬁxed Tl, resulting in DT lT Dl ¼ T c 2 l

(14)

It can be seen from Equation (14) that the impact of an uncertainty in the emissivity, given the measurement conditions, on the temperature accuracy decreases as l decreases. Now according to Wien’s displacement law, the

8

Zhuomin M. Zhang and Graham Machin

500 T = 1000 K 400

λ = 0.65 μm λ = 0.95 μm

ΔT = T - Tλ (K)

λ = 1.5 μm 300

λ = 2.4 μm Total

200

100

0 0

0.2

0.4 0.6 Emissivity

0.8

1

Figure 2 The difference between the true temperature and the radiance temperature, as a function of emissivity, for both spectral and total radiation thermometers.

wavelength at which L b;l ðl; T Þ is maximum is given by lmax ¼ c 3 =T , where c3 is the third radiation constant. Since c 2 =c 3 5 (constant values are given in Appendix A), Equation (14) can be further approximated by DT l Dl T 5lmax l

(15)

The signiﬁcant advantage of using spectral over total radiation thermometry at lolmax becomes clear when the inﬂuence coefﬁcients in Equation (15) are compared with those of Equation (5). It should be noted that the error due to using Wien’s approximation rather than the exact Planck formula is 0.7% at lmax and smaller at lolmax . Hence, Wien’s approximation is acceptable for most radiometric temperature measurement situations. If the target is not a blackbody and the surrounding radiation is not negligible, L l ðlÞ in Equations (6) and (7) is the sum of the emitted (by both the target and the environment) and reﬂected spectral radiances: L l ðlÞ ¼ tenv ½l L b;l ðl; T Þ þ L r;l ðlÞ þ L em;l ðl; T env Þ

(16)

where the term tenv accounts for the transmittance through the atmosphere (see Figure 1) and L em;l ðl; T env Þ is the contribution from the atmospheric emission (which is usually negligible except in the case of hot gases or plasmas) that reaches the radiometer. The reﬂected spectral radiance, Lr,l(l), depends on the distribution of the surrounding radiation incident on the

Overview of Radiation Thermometry

9

target and the bidirectional reﬂectance distribution function (BRDF) of the target. Hence, except for well-controlled environments or in some limited cases, L l ðlÞ is difﬁcult to determine and corrections must be made to obtain the true temperature of the target.

2.3. Radiation thermometer design considerations The central problems in applying the technique of radiation thermometry are (1) determining the target emissivity, which depends on temperature, surface conditions, and angle of view and (2) reducing the inﬂuence of surrounding radiation and environmental effects such as absorption along the viewing path so that the true surface temperature can be obtained. Another important issue is the accurate determination of the instrument constant CI given in Equation (7). Neither of these issues is insurmountable, nevertheless great care and patience are required to perform radiometric temperature measurements correctly. Calibration of the radiation thermometer using a traceable blackbody source is a necessary ﬁrst step to determine the instrument constant and thus provide a sound platform for correct surface temperature measurements. Several techniques have been developed in radiation thermometry to mitigate the emissivity problem, such as the use of a reﬂecting mirror, application of a controlled reference source, dual-wavelength pyrometer, multi-spectral pyrometry, and spectrometric methods or active laser methods. An example of the laser method is to slightly modulate the surface temperature with a laser source and measure the modulated signal. This method is called photothermal radiometry [22] and can, in principle, be used to deduce surface temperature without knowledge of its emissivity. Optical ﬁber thermometry, based on a sapphire lightpipe, is another technique that has been employed in the semiconductor industry to help mitigate the effect of surface emissivity, using the ‘‘ripple technique’’ [23,24]. Details of how to apply radiation thermometry to particular temperature measurement problems will be discussed in subsequent chapters of this volume and in the companion volume devoted to such measurements [25]. In general, issues that need to be addressed when developing a radiation thermometer are the following: wavelength selection; the optical system (such as lenses, mirrors, ﬁlters, and the use of aperture and ﬁeld stops) to collect the thermal radiation and to deﬁne the ﬁeld-of-view of the target and detector area; the detector that matches the spectral band and the signal level of the emitted thermal radiance, along with the readout electronics matched to the detection system. A detailed treatment of radiation thermometer design will be given in Chapter 4. Brieﬂy, the wavelength selection should match the wavelength of radiation emitted by the target, which in turn depends upon its temperature. An attempt should be made to

10

Zhuomin M. Zhang and Graham Machin

reduce the atmospheric effect by choosing a wavelength band that avoids emission or absorption by gases such as carbon dioxide and water vapor. The bandwidth can be selected to match with the detector signal requirements and signal-to-noise ratio. The sensitivity of the emitted radiance to temperature can be obtained by differentiating Equation (11) with respect to T using Equation (10), giving dL em;l c 2 dT ¼ L em;l lT T

(17)

If at a given T and l, we can approximate L em;l as proportional to T to the mth power, i.e., L em;l ¼ aT m , we have dL em;l dT ¼m L em;l T

(18)

Comparison of Equations (17) and (18) suggests that the larger the m, the more sensitive the emitted radiance is to a temperature change of the target. Or to put it another way, a small change in temperature results in a large change in radiance. This is because the Planck curve radiance as a function of temperature has a very steep slope (and hence higher sensitivity) toward shorter wavelengths. As an example, for T ¼ 1,000 K and l ¼ 1 mm, we have mE14.4, which is much greater than the sensitivity of a total radiation thermometer (fourth power). It can be shown with the long-wavelength approximation of Planck’s law (i.e., the Rayleigh2Jeans limit) that the effective power or inﬂuence coefﬁcient m ¼ 1 occurs when l-N, hence, shorter wavelengths are generally preferred. This is consistent with the predicted reduction in temperature uncertainty (when the wavelength of measurement decreases) for a given emissivity uncertainty as prescribed by Equation (14). On the other hand, the spectral radiance predicted by the Planck function decreases rapidly with decreasing wavelength for wavelengths shorter than lmax at any given temperature, there is a trade-off in terms of the magnitude of the signal and the sensitivity of the radiant power with respect to the temperature change. In most applications, the appropriate wavelengths of operation for a radiation thermometer are from a few tenths of the peak wavelength lmax to slightly above lmax. For example, if a focal plane array is used for thermographic imaging near room temperature, the operating wavelengths should be somewhere between 3 and 12 mm, taking care to avoid atmospheric absorption bands. Examples of detectors used for thermal imaging are liquid-nitrogen-cooled InSb detectors with high sensitivity from 3 to 5 mm and MCT (a HgCdTe alloy) focal plane arrays that operate in the spectral band 8212 mm. Radiation detectors can be divided into photon detectors (nonthermal) and thermal detectors. In photon detectors, direct interaction of electromagnetic radiation (or photons) with electrons in a material may result in photoelectric, photoconductive, or photovoltaic effects. For example, in a

Overview of Radiation Thermometry

11

photoconductive process, the absorption of incoming photons causes electronic transitions from the valence band to the conduction band and, subsequently, generates free carriers (electrons and holes) that vary the conductivity of the semiconductor material. Photon detectors are more sensitive and have a faster response than thermal detectors, which rely on the temperature change of the detector material caused by the absorption of the incoming photons. Typical photon detectors that are used widely in radiation thermometers are based on silicon (Si) and indium gallium arsenide (InGaAs) photodiodes. On the other hand, thermal detectors are commonly characterized by a broadband absorption and excellent linearity. Typical thermal detectors are thermopiles, pyroelectric detectors, and bolometers [26]. The bolometer was ﬁrst invented by the American astronomer Langley in 1878 based on the resistance change of a Pt metal strip (lampblack coated) due to a temperature change associated with the absorption of the incoming thermal radiation. Modern bolometers based on superconducting materials operate in the transition region between superconducting and normal states and, hence, can be much more sensitive than those manufactured from normal metals [27]. Uncooled microbolometer arrays have recently been fabricated using microfabrication techniques and commercially applied for thermal imaging and remote sensing [28].

3. Types of Radiation Thermometers In this section the common, and many of the less common, types of noncontact thermometry methods are brieﬂy described. Classical radiation thermometry is based on single-spot devices, with some restrictions in spectral band and with a single detector. Line scanners and thermal imagers, which are extensions of the basic method, are also brieﬂy discussed. Some more specialized methods for emissivity correction are also described, such as twocolor pyrometry, multiwavelength radiometry, and polarimetry, as well as active compensation methods such as reﬂectometry, increasing the apparent emissivity using multiple reﬂections, and some modulation techniques. Finally, the use of ﬁber optics in thermometry is outlined, including lightpipes, phosphor, Fabry2Perot interferometry, and Bragg grating thermometry.

3.1. Classical radiation thermometry Classical radiation thermometers comprise essentially three components: (1) A means for collecting the radiation, either a lens that transmits in the

sensitivity band of the detector, or a mirror, or even just an aperture. (2) A means for restricting the spectral band of radiation sensed by the

detector; this could be an interference ﬁlter, a colored glass ﬁlter or the intrinsic bandpass of the detector, or a combination of the latter two.

12

Zhuomin M. Zhang and Graham Machin

(3) A detector that is sensitive to the radiation emitted by the object of

interest; its selection would be governed by the thermal emission spectrum at the temperature of the object as well as concerns for the required speed and sensitivity. Single-spot thermometers are extensively used in industry over a very wide temperature range, from process control to quality control and diagnostics (spot checks). More details are given in the companion volume [25]. These thermometers can be grouped into three types according to the temperature of the object being measured. 3.1.1. High-temperature radiation thermometers Used for temperature measurements typically above 6001C. Si photodetectors are almost universally used due to their excellent linearity with a large dynamic range, low noise, fast response, low cost and easy fabrication with large areas, and wide spectral response (for wavelengths from approximately 400 to 1,050 nm) [29]. Lens optics are commonly adopted. This type of radiation thermometer is extensively used for process control and in heavy industry, such as glass and steel manufacturing. 3.1.2. Mid-temperature radiation thermometers Used for typically temperatures from 1001C to 6001C. There exist a wider variety of technologies for detector choices since the operating wavelengths are usually between 1 and 3 mm. Examples are uncooled InGaAs or Ge photodiodes and PbS photoconductive detectors, as well as liquid nitrogencooled InSb and InAs photodetectors of high detectivity. Thermal detectors such as pyroelectric detectors and bolometers are also used. Lens optics are mostly employed and the target sizes are generally larger compared with those used by high-temperature radiation thermometers in order to obtain sufﬁcient radiant ﬂuxes. These type of detectors are commonly used for aluminum, galvanneal, and other metal processes. 3.1.3. Low-temperature radiation thermometers Typically for temperature measurements and thermal imaging from below 01C to around a few hundred degrees Celsius. Since the wavelength range of interest is from 3 to 15 mm, wideband or unﬁltered thermal detectors are frequently used, including bolometers, pyroelectric detectors, and thermopiles. Liquid nitrogen or thermoelectrically cooled MCT detectors are also common. Replica optics and mirror optics are mostly used instead of lens optics. This type of radiation thermometer ﬁnds a wide range of applications in, for example, the food, paper and plastic industries.

13

Overview of Radiation Thermometry

An extension of low-temperature radiation thermometers are thermal imagers. In recent years, scanning systems with cryogenic cooled detectors (usually InSb or MCT) have largely given way to focal plane arrays and uncooled detector arrays for thermal imaging and remote sensing [30].

3.2. Emissivity correction methods The biggest issue when trying to measure temperature by radiation thermometry is the emissivity of the surface. Various strategies have been proposed to overcome the effect of emissivity with various degrees of success. Some of these are discussed below and are classiﬁed into two types: passive methods relying only on the emitted radiation from the surface or active approaches reliant upon determining the emissivity by other independent means. 3.2.1. Dual-wavelength or ratio pyrometers The essential measurement performed by a dual-wavelength pyrometer (also called two-color pyrometer) is that of the emitted radiant ﬂux from the same surface at two different wavelengths, l1 and l2. The underlying assumption is that for the surfaces whose temperatures are being measured, the emissivity is independent of wavelength at least from l1 to l2. This is true for gray surfaces. By applying Equations (7), (8), and (11) to both wavelengths, we see that the ratio of the signals becomes r¼

S l1 C I;1 1 l52 ec2 =l2 T l52 c2 c2 ¼ exp S l2 C I;2 2 l51 ec2 =l1 T l51 l2 T l1 T

(19)

Usually C I;1 C I;2 due to the closeness of the two wavelengths. The temperature obtained from Equation (19) is called the ratio temperature: Tr ¼

1 c 2 ðl1 2 l1 Þ lnðrl51 =l52 Þ

(20)

This strategy is only successful if the emissivity of the surface is the same (which is rarely true for real surfaces) at the wavelengths of operation of the pyrometer. For surfaces whose emissivity varies linearly with wavelength, the slope of e(l) can in principle be used to correct Equation (20), but the slope itself is also difﬁcult to determine. Hence, if the surface does not exhibit a ﬂat emissivity, then it is always better to use a single-spot pyrometer and correct for the known emissivity. The beneﬁts of dual-wavelength pyrometry lie not so much in its ability to correct for emissivity, as surfaces with an emissivity which is invariant with wavelength are rare, but in the fact that it can perform reliable thermometry in situations which are unsuited to single-spot pyrometry.

14

Zhuomin M. Zhang and Graham Machin

These are where the target is smaller than the spot size, or where wavelength neutral, but absorbing gas or smoke obscures the ﬁeld of view. In fact targets that only ﬁll a few percent of the ﬁeld of view can have their temperature determined successfully by this method. More sophisticated types of dual-wavelength pyrometry have been implemented for speciﬁc situations. For instance the aluminum and steel industries present particular challenges for noncontact thermometry that can be addressed using dual-wavelength pyrometry, not in simple ratio form as described above but where an a priori known mathematical function describes the relationship of emissivity at two different wavelengths. A successful implementation of this technique to the galvanneal process is described in Ref. [31] while its application in the aluminum industry is outlined in Refs. [32,33].

3.2.2. Multiwavelength pyrometers Multiwavelength pyrometry has been, and is periodically proposed, as the solution to the emissivity problem. Suppose the emissivity can be expressed in terms of a given function of l with n unknown constants. If the radiance temperature is measured at more than n wavelengths, the n constants together with the unknown surface temperature can be determined using the measured blackbody radiance or radiance temperature at each wavelength. This is the principle of the multiwavelength (or multicolor) radiation thermometer. The least-squares method is often used if the measured radiance temperature is at more than (n+1) wavelengths. Detailed discussions have been given in the literature [34236]. Classic examples of this are given in Gebbie et al. [37] and Svet [38]. It is proposed that by using the spectral radiance at a number of wavelengths (or even the full emitted spectrum), coupled with an assumed model of emissivity, it is possible to deduce both the temperature and the surface emissivity from the radiance data alone. However it has been shown [34,35,39] that, when there is no a priori knowledge of the emissivity variation with wavelength, multiwavelength pyrometry has distinct disadvantages and generally results in much larger temperature uncertainties than those possible by much simpler methods, e.g., single-spot pyrometry with a reasonable estimate of the emissivity [35]. As with dual-wavelength pyrometry, there are some speciﬁc situations where multiwavelength pyrometry can convey advantage, provided additional information is known, in particular the functional relationship of the emissivity function with wavelength. This has been demonstrated, e.g., on measurements of SiC samples [35] and also in the aluminum process industry [38]. In both cases, a reasonable understanding of the variation of emissivity with wavelength was achievable.

Overview of Radiation Thermometry

15

In summary, multiwavelength pyrometry alone cannot solve the surface emissivity problem. For proper implementation a priori knowledge of at least the functional form of the emissivity with wavelength is required to constrain the uncertainties in temperature estimates. For some speciﬁc situations, multiwavelength pyrometry might convey a measure of advantage, while on the whole single-spot pyrometry, armed with a reasonable estimate of the surface emissivity is likely to be a simpler solution and is likely to yield lower uncertainties for the temperature estimates. 3.2.3. Fourier-transform infrared spectrometry Fourier-transform infrared (FTIR) spectrometry has been used extensively in chemical analysis, remote sensing, and the measurement of radiative properties. Since FTIR measurement is in a broad spectral region, the emissivity function can also be obtained through modeling, similar to multiwavelength pyrometry. Some dielectric materials (e.g., SiO2, Al2O3, MgO) exhibit blackbody behavior at a single wavelength, called the Christiansen wavelength. This allows determination of the sample temperature as well as emissivity by measuring the emission spectrum [40]. More discussion on the Christiansen wavelength will be given in Section 5.2 of Chapter 3. Markham et al. [41] used the FTIR emission spectra at three slightly different sample temperatures (DTE501C) to measure the surface temperature at about 8001C in a ﬂame environment. This ratio-difference method eliminates the effect of background radiation and does not require a priori knowledge of the surface emissivity, as long as the change in el is negligible at a given l. Large modiﬁcations of the sample temperature, however, may cause the emissivity to vary and/or affect the process condition. Another apparatus used a hemi-ellipsoidal mirror to measure the radiative properties as well as temperature of the surface [42]. A diffusely radiating near-blackbody source and the sample were mounted in the two foci of the hemi-ellipsoidal mirror; therefore, the radiation leaving the source was reﬂected diffusely onto the sample. This apparatus was used to measure the spectral radiative properties (within 223% uncertainty) and surface temperature (within 101C uncertainty) [42]. Hanssen and coworkers [43245] have developed an FTIR facility for characterization of materials emissivity and used the FTIR to determine surface temperatures through a separate measurement of the reﬂectance with integrating spheres. An FTIR spectrometer was also used together with optical ﬁbers to measure the temperature of a high-ﬂux solar furnace [46]. Generally speaking, using an FTIR for thermometry is more expensive than traditional radiation thermometers, and thus it is not used routinely. The only reason for using an FTIR is because the additional information such as gas emission and surface properties that may be inferred from the measurements is of value to the investigator.

16

Zhuomin M. Zhang and Graham Machin

3.2.4. Polarization pyrometry The technique known as polarization pyrometry is a possible method, for metallic surfaces, of overcoming the emissivity problem, even if it is varying due to oxidation or scale growth. The surface temperature is determined by examining the polarization of the sum of the radiation emitted by the surface and thermal radiation emitted by a blackbody reﬂected from the surface [47,48]. The method is a relatively straightforward application of Kirchhoff’s law for individual polarization. The surface is irradiated at high angle of incidence, around 701, with unpolarized blackbody radiation; a detector then measures the specularly reﬂected radiation, which is partially polarized, since the reﬂection for p polarization exhibits a dip around this angle. The radiation emitted from the surface is also partially polarized (higher for p polarization than for s polarization). It can be shown that the total radiation will not be polarized if the temperatures of the blackbody and surface are equal, so that the unknown temperature of the surface can be inferred from the blackbody temperature when the measured radiation is unpolarized. No quantitative response of the detector is needed since one only needs to ﬁnd the temperature when the detector response is independent of the analyzer (a polarizer between the sample and the detector). Experiments have shown that this technique is feasible [49,50]. However, these early versions were limited in measurement time because the temperature of the blackbody had to be the same as that of the surface, this was time-consuming to achieve. One way of reducing the measurement time would be to ﬁx the temperature of the blackbody and place a variable attenuator in front of the cavity. By changing the radiance by means of the attenuator, the null polarization position can be found rapidly [49]. More recent measurements have been made of very low emissivity metals near room temperature [50] using this technique. However, in general polarimetry has not been implemented on a wide scale. Additional discussion of the use of ellipsometry for temperature and ﬁlm thickness measurements will be presented in Chapter 7. 3.2.5. Emissivity-enhancement methods Several methods have been used to increase the apparent emissivity of the surface and, consequently, to alleviate the emissivity problem. One important method that has been used extensively over the years is the so-called ‘‘gold-cup’’ method [51,52]. In this method, a conventional single-spot radiation thermometer is used, but the surface is not imaged directly. Instead, a metallic hemisphere bowl (usually cooled) is coated with a highly reﬂective material (e.g., gold, which gives the name of this method). This hemispherical mirror is then brought close to the radiating surface and a quasi-blackbody radiation ﬁeld is generated in the region

Overview of Radiation Thermometry

17

enclosed by the ‘‘cup.’’ This radiation ﬁeld is then measured through a small hole in the cup by the detector either directly or through the intermediary of a ﬁber-optic bundle. The surface temperature can thus be inferred from the radiance temperature of the effective blackbody at the same temperature. Different shapes of reﬂecting cavity type have been modeled [53] and optimized for particular situations; e.g., aluminum sheet that has a very low surface emissivity. Another design [33] allows the temperature measurement of aluminum billets, which have to be conditioned to a particular temperature, before they are extruded. The ‘‘cup’’ in this case has a particular form that allows for some variation in the position of the billet with respect to the cup while maintaining a high and ﬁxed emissivity. In some industrial processes it is not possible to deploy an active emissivity-enhancement method. However through the use of the geometry of the situation it is still possible to greatly reduce the emissivity effects. This is done through positioning a conventional pyrometer so that it views into a wedge made by two surfaces of the same temperature (e.g., the wedge generated where steel is being coiled from a ﬂat sheet) [54] or where it measures the radiation ﬁeld between two slabs of material [55,56]. In rapid thermal processing of silicon wafers, a reﬂective shield can be used to enhance the emissivity [23], and models have shown that although the effective emissivity can be signiﬁcantly enhanced care must be taken to address the BRDF [57,58]. 3.2.6. In situ emissivity determination A straightforward method for emissivity correction is to directly measure the reﬂectance by built-in reﬂectometry. A pulsed or cw laser source can be used to measure the reﬂectance of the target and the emissivity can be inferred by the use of Kirchhoff’s law [59]. Another method, ﬁrst proposed and trialed by DeWitt and Kunz [60], involves the introduction of a small thermally induced perturbation to the thermal radiation emitted by a (hot) surface. This is done by employing the photothermal effect whereby a modulated laser beam, with sufﬁcient power is required (typically a class 4 laser) to introduce a periodic change in the emitted radiation. The perturbation is small usually of order of a degree or so on surfaces that can be in excess of 1,0001C. The basic method works as follows: two high-powered lasers operating at wavelengths l1 and l2 are modulated at frequencies o1 and o2 respectively. The critical measurement is that of the modulated thermal radiance at l1 arising from laser l2 and the modulated thermal radiance at l2 arising from laser at l1. The ratio of these two thermal radiances, from which temperature can be calculated, is effectively independent of surface emissivity, optical transmission, and reﬂected radiation [61,62].

18

Zhuomin M. Zhang and Graham Machin

Different variations of the basic technique have been developed [61265], some employing optical ﬁbers [62264] to transfer the laser beams to the measurement surface and to capture the emitted signal. Others have used mirror [64,65] instead of lens optics [61,63]. Most have been demonstrated in the laboratory and uncertainties in the technique have been typically from 31C to about 101C dependent on the surface under investigation [63265]. Some developments have gone further and been speciﬁcally developed for trials in industrial settings [62,64,65]. Typical target applications include steel production, reformer tube measurements, and the heat treatment of turbine blades. However despite this signiﬁcant activity no commercial device currently exists and there is resistance to its introduction into an industrial setting largely because of safety concerns about the use of high-powered lasers. One feature of the method is that it can be used to determine thermophysical properties such as emissivity and thermal diffusivity as well as surface temperatures [66]. This feature may be useful in the future if on-line measurement of properties for process control is required.

3.3. Fiber-optic radiation thermometry Fiber-optic radiation thermometry generally falls into two classes. The ﬁrst is termed ‘‘thermal ﬁber-optic thermometry.’’ The distinguishing feature of this type of thermometer is that an optical ﬁber or optical-ﬁber bundle is used to convey emitted thermal radiation to a detector. This is often used in very hostile environments or where access is limited. Typical applications are in aerospace engines or the steel industry. A variant on this is the use of lightpipes or incorporated blackbody methods that are often used in semiconductor processing. A second class of ﬁber-optic thermometers do not depend on emitted thermal radiation but on the temperature-sensitive properties of something else, often, but not necessarily, incorporated into the ﬁber itself. This group of techniques is termed for convenience as ‘‘nonthermal ﬁber-optic thermometry.’’ The most common of these are based on properties such as ﬂuorescence timescales or building into the ﬁber itself a temperature-sensitive element such as a Fabry2Perot interferometer or a Bragg grating. This second type of ﬁber-optic thermometer is often, but not exclusively, for relatively low-temperature applications (up to a few hundred degrees Celsius). They are generally used where electromagnetic immunity is essential such as in microwave processing of materials or in high-magnetic ﬁelds, e.g., magnetic resonance imaging. One big advantage over the ﬁrst type of thermometer is that the thermometric performance of the device is less, or even not, sensitive to the ﬁber being bent, changes in transmission losses, or even progressive optical degradation 2 provided, of course, that light can still be transmitted by the ﬁber. Not all types are

Overview of Radiation Thermometry

19

described in this section and the interested reader is referred to Refs. [2,67269] for a comprehensive outline of methods. 3.3.1. Thermal ﬁber-optic thermometry These types of thermometers are simply an extension of the classical radiation thermometry technique. The detector and wavelength-deﬁning ﬁlter are usually the same as in conventional radiation thermometry. The main difference is that instead of free space between the thermometer and the object a ﬁber-optic bundle is used to convey the thermal radiation from the object whose temperature is being determined. The ﬁber-optic bundle is nearly always heavily armored to protect it from the hostile environment in which it needs to operate. This type of thermometer is mostly used either because of accessibility issues, i.e., no clear line of sight to the target is possible, or to remove the sensitive detector and electronics from the hostile environment. One particular application is monitoring the health of turbine blades in, e.g., the combustor region of a gas turbine engine both on military and civil aircraft engines [70,71]. Here, early detection of overheating blades triggers preventative maintenance ensuring optimal and safe performance of aircraft or power generation turbines [72]. 3.3.1.1. Immersed optical ﬁber. This type of device is used in the metalprocessing industry, primarily the steel industry [73]. Here an optical ﬁber is progressively immersed into a vat of molten metal from a large reel of ﬁber. On immersion the ﬁber tip forms an equivalent cylindrical cavity with an isothermal wall of molten metal. Since the ﬁber is so thin (0.125 mm) even a small immersion creates a long cavity compared to its diameter and hence the conditions for producing a good blackbody are achieved. The ﬁber itself then conveys this blackbody radiation back to a detection system that corrects for the changing length of the ﬁber reel and determines the temperature. The original work required the use of special metal clad ﬁbers [73] but subsequent research has shown that with care more ordinary ﬁbers can be used for the same application [74]. 3.3.1.2. Lightpipe and incorporated blackbody types. Fiber optics can be used as lightpipes. In this case, single-crystal sapphire ‘‘ﬁber’’ is used. The advantage is that it can be used to very high temperatures (above 1,8001C) in a variant on this sensor the end of the ﬁber is coated with a refractory metal forming a quasi-blackbody on the tip thereby negating emissivity effects arising from the target [75]. These types of thermometers, particularly the single-crystal lightpipe types, are used extensively in the semiconductor processing industry where minimal intrusion, rapid response, and robustness against high temperatures are essential [76278].

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3.3.2. Nonthermal ﬁber-optic thermometry 3.3.2.1. Phosphor-based methods. This ‘‘noncontact’’ method is based on the use of phosphors. The technique utilizes inorganic compounds that ﬂuoresce when appropriately stimulated. The excitation usually comes from a spectrum that has higher photon energy than the corresponding ﬂuorescence. Typical excitation sources are a xenon ﬂash or a bright light emitting diode. The brightness of the ﬂuorescence decays exponentially over time and it is this ﬂuorescence lifetime (typically W1 ms), which is a strong function of temperature, and hence is exploited for thermometry. There are many different types of materials that can be used as phosphors, e.g., Cr3+-doped insulating crystals, Nd:YAG, GaAs, and so forth. The exact choice of phosphor depends on the range, sensitivity, and even the stability required by the particular thermometer. These are described in detail in Ref. [69]. The phosphor itself is often incorporated into the tip of a ﬁber, the stimulation light is conveyed to the phosphor material by the ﬁber and the resultant ﬂuorescence is then conveyed by the same ﬁber to the detection system. This type of device is often used in environments unsuitable for metal sensors such as microwave ovens or magnetic resonance imaging devices [79]. Alternatively, the object whose temperature needs to be determined can be coated with a phosphor, which is then stimulated by an external light source and the luminescence detected by an optical system. This method has been used to investigate turbine engines, internal combustion engines, and in the metal-processing industry such as the galvanneal process [80,81]. Understanding the temporal and temperature responses of the phosphors is essential to their performance as temperature measurement devices. A carefully thermostatted environment is required if this response is to be accurately determined [82]. This type of thermometry is still an active research ﬁeld, e.g., identifying new phosphors [83] and pushing the boundaries of applications such as nanoscale thermometry using ﬂuorescence [84] and in microﬂuidics [85]. 3.3.2.2. Temperature-sensitive elements intrinsic to the ﬁber. This type of sensor depends on the change of a particular property of the temperature-sensitive element with temperature. There are several different types of thermometers based on different properties; the two described below are both based on changes in dimension of the sensitive element, in both cases the sensor is incorporated within the ﬁber. (1) A Fabry2Perot interferometer employs the interference effect inside a

dielectric ﬁlm whose surface is coated by highly reﬂecting mirrors. The thickness of the dielectric is a function of temperature. In a Fabry2Perot thermometer, a Fabry2Perot etalon is essentially ﬁxed

21

Overview of Radiation Thermometry

to the end of the ﬁber. Light is sent down the ﬁber and partially reﬂects off the ﬁber2dielectric interface and partially off the dielectric2air interface. These two reﬂected beams interfere with each other and an interference pattern is detected, this can be related to the gap between the two surfaces. As the temperature of the etalon changes this gap and also the refractive index changes, the interference pattern shifts and this shift can be calibrated in terms of temperature [67,86]. (2) A Bragg grating when incorporated within the matrix of an optical ﬁber can also be used as a temperature sensor. In this type of sensor the temperature-sensitive element is a periodic change in the index of refraction in the core of the optical ﬁber. When a broadband source of light is sent down the ﬁber and the region containing the grating is exposed to a change in temperature the period of grating changes and the wavelength of light reﬂected from the grating changes [2,67,87]. Interpretation of the output of such devices can be difﬁcult as the output is sensitive to strain as well as temperature [87]. A number of this type of sensor can be distributed along a long ﬁber (length of kilometers) and interrogated using a broadband light source. The time taken for the reﬂected light to return to the detector indicates how far a particular sensor is from the detector. This type of device is particularly useful for monitoring large industrial plants.

4. Terms Commonly Used in Radiation Thermometry This section provides a glossary of radiation terms in layman language. More details can be found from, e.g., Meyer-Arendt [88], International Commission of Illumination (CIE) publication [89], American Society of Testing and Materials (ASTM) documentation [90,91]. Absorption:

Absorptance: Absorptivity:

Bidirectional reﬂectance distribution function (BRDF):

The process of converting photon energy to some other types of energy through radiation2matter interaction. See absorptivity; usually used for a semitransparent medium. Fraction of the incident radiation that is absorbed by matter. Modiﬁers: directional, hemispherical, spectral, or total. Ratio of the reﬂected spectral radiance to the spectral irradiance from a given direction.

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(Continued )

Blackbody:

Blackbody cavity:

Diffuse:

Directional: Emission:

Emissivity:

Exitance:

Exitent: Flux:

Gray surface:

Heat ﬂux: Hemispherical: Incident: Intensity: Irradiance:

The ideal emitter and absorber. The exitance is diffuse and depends on the temperature and wavelength only. See blackbody cavity. Modiﬁer for blackbody behavior or properties. An isothermal cavity that emits, through its opening, thermal radiation that closely follows the blackbody distribution. Modiﬁer referring to the directional independence of the emitted, reﬂected, or incident radiance. Modiﬁer referring to the particular direction. The process in which radiation or photon energy is released from matter due to radiative transitions. Also emittance: ratio of the radiation emitted from a surface to that emitted by a blackbody at the same temperature. Modiﬁers: directional, hemispherical, spectral, or total. Rate of radiant energy leaving a surface per unit area as a result of emission, reﬂection, or transmission (W/m2). Sometimes it is called radiosity. Modiﬁers: radiant, photon, self, spectral, total, or blackbody. Modiﬁer referring to ﬂux leaving a surface. Time rate of propagation of radiant energy (often called radiant power instead, W), photons (photon ﬂux, s1), or lumens (luminous ﬂux, lm/s). A surface whose spectral absorptivity and the emissivity are independent of wavelength over the spectral region of signiﬁcance. Rate of energy transfer across a surface per unit area (W/m2). Modiﬁer referring to all directions in the space above a surface. Modiﬁer referring to radiation striking a surface. Radiant power (ﬂux) per unit solid angle (W/sr). Modiﬁers: spectral, total. Rate at which radiant energy is incident on a surface, from all directions above the surface, per unit area (W/m2). Modiﬁers: spectral, total.

Overview of Radiation Thermometry

23

(Continued )

Kirchhoff ’s law:

Lambertian surface: Planck’s law: Radiance:

Radiant ﬂux or radiant power: Radiance temperature:

Ratio temperature:

Reﬂectance:

Reﬂectivity:

Self-exitance: Semitransparent:

Solid angle:

Spectral: Specular:

Relation between emissivity and absorptivity of a material. At thermal equilibrium, the emissivity of a body (or surface) equals its absorptivity. A surface with equal exitent radiance in all directions. Spectral distribution of blackbody exitance or radiance. Radiant power or ﬂux (W) from a surface per unit solid angle per unit area. Modiﬁers: spectral, total, incident, exitent, emitted, reﬂected, or blackbody. Time rate of propagation of radiant energy (W). Apparent temperature of a surface over a given spectral band deﬁned as the temperature of a blackbody radiator having the same spectral radiance over that spectral band. Modiﬁers: spectral, total. Temperature of a blackbody radiator having the same ratio of spectral radiance, at two ﬁxed wavelengths, as that from a surface. Fraction of the incident radiation reﬂected by matter. Modiﬁers: bidirectional, directionalhemispherical, hemispherical2hemispherical, hemispherical2directional, specular, diffuse, spectral, or total. See reﬂectance; used for reﬂection between two semi-inﬁnite media with a perfectly smooth interface. Exitance due to thermal emission from the surface; also emitted exitance. Refer to a medium that absorbs radiation volumetrically and a portion of the incident radiation can penetrate through the medium. Region subtended by an area element on the surface of a unit sphere with respect to the center of the sphere. Unit: steradian (sr). Modiﬁer referring to a monochromatic quantity. Refer to a perfectly smooth surface for which the angle of reﬂection is equal to the angle of incidence. Modiﬁer for the reﬂected component in the specular direction.

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(Continued )

Stefan2Boltzmann law: Thermal radiation: Total: Transmittance: Wien’s displacement law: Wien’s formula:

Exitance of a blackbody is proportional to its absolute temperature to the fourth power. Electromagnetic energy emitted by matter at a ﬁnite temperature. Modiﬁer referring to the integration over all wavelengths. Fraction of the incident radiative energy transmitted through a medium. Locus of the wavelength corresponding to peak emission by a blackbody. Approximate blackbody distribution function that is good at relatively short wavelengths.

5. Summary This chapter gives a general introduction to radiation thermometry, from the basic operating principles according to the fundamental laws of thermal radiation to the central problems with the practical application of the technique associated with the unknown emissivity of the surfaces and also the effect of the environment. The selection of wavelengths for a measurement is shown to be dependent on the temperature ranges. Some emissivity correction methods are summarized and typical radiation thermometers and their operating mechanisms are outlined, including traditional single detector, thermographic imagers, and various optical ﬁber thermometers. Terms commonly used in radiation thermometry are also described in straightforward language for the convenience of the reader.

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Zhuomin M. Zhang and Graham Machin

T. Rice, S. Poland, B. Childers, et al., ‘‘Fibre-Optic Temperature Sensors 2 A New Temperature Measurement Toolbox,’’ in Ref. [4], pp. 101521020. K, A. Wickersheim, ‘‘Fibre-Optic Thermometry: An Overview,’’ in Ref. [3], pp. 7112714. K. T. V. Grattan and Z. Y. Zhang, Fibre Optic Fluorescence Thermometry, Chapman & Hall, London, (1995). http://www.landinst.com/combustion/downloads/pdf/Article_Optical_Pyrometry.pdf http://www.landinst.com/combustion/downloads/pdf/turbine_temperature_analysis. pdf T. G. R. Beynon, ‘‘Radiation Thermometry Applied to the Development and Control of Gas Turbine Engines,’’ in Temperature: Its Measurement and Control in Science and Industry, edited by J. F. Schooley, American Institute of Physics, New York, Vol. 5, pp. 4712477 (1982). Y. Yamada, A. Ohsumi, Z. Yamanaka, and T. Yamada, ‘‘Temperature Measurement of Molten Metal by Immersion-type Optical Fiber Radiation Thermometer,’’ in Proceedings of TEMPMEKO ‘96, edited by P. Marcarino, Levrotto & Bella, Torino, pp. 3472352 (1997). M. dePodesta, private communication. R. R. Dils, ‘‘High Temperature Optical Fiber Thermometer,’’ J. Appl. Phys. 54, 119821201 (1983). B. K. Tsai and D. P. DeWitt, ‘‘Characterization and Calibration of Light Pipe Radiation Thermometers for Use in Rapid Thermal Processing,’’ in Ref. [4], pp. 103321038 (2003). B. K. Tsai, ‘‘A Summary of Lightpipe Radiation Thermometry Research at NIST,’’ J. Res. Natl. Inst. Stand. Technol. 111, 9230 (2006). B. Adams and C. Schietinger, ‘‘In-situ Optical Wafer Temperature Measurement,’’ in Ref. [4], pp. 108121086 (2003). T. Samulski, ‘‘Fibreoptic Thermometry: Medical and Biomedical Applications,’’ in Ref. [3], pp. 1185-1190. B. W. Noel, W. D. Turley, W. Lewis, K. W. Tobin, and D. L. Beshears, ‘‘Phosphor Thermometry on Turbine Engine Blades and Vanes,’’ in Ref. [3], pp. 124921254. S. W. Allison, M. R. Cates, D. L. Beshears, and G. T. Gillies, ‘‘Phosphor Thermometry at ORNL,’’ in Ref. [4], pp. 103321038 (2003). L. Rosso, V. C. Fernicola, and A Tiziani, ‘‘A Calibration System for Fluorescence Thermography,’’ in Ref. [4], pp. 104521050. H. Aizawa, H. Uchiyama, T. Katsumata, S. Komuro, T. Morikawa, H. Ishizawa, and E. Toba, ‘‘Fibre-Optic Thermometer Using Sensor Materials with Long Fluorescence Lifetime,’’ Meas. Sci. Technol. 15, 148421489 (2004). S. W. Allison, G. T. Gillies, A. J. Rondinone, and M. R. Cates, ‘‘Nanoscale Thermometry via the Fluorescence of YAG:Ce Phosphor Particles: Measurements from 7 to 771C,’’ Nanotechnology 14, 8592863 (2003). D. Ross and L. E. Locascio, ‘‘Fluorescence Thermometry in Microﬂuidics,’’ in Ref. [4], pp. 105121055. E. W. Saaski and J. C. Hartl, ‘‘Thin-Film Fabry Perot Temperature Sensors,’’ in Ref. [3], pp. 7312734. Y.-J. Rao, ‘‘Review of In-Fibre Bragg Grating Sensors,’’ Meas. Sci. Tecnol. 8, 3552375 (1997). J. R. Meyer-Arendt, ‘‘Radiometry and Photometry: Units and Conversion Factors,’’ Appl. Opt. 7, 208122084 (1968). CIE, International Lighting Vocabulary, Section 17.4, International Commission of Illumination (1987). ASTM, E1256-95: Standard Test Methods for Radiation Thermometers (Single Waveband Type), American Society for Testing and Materials (2007). J. C. Richmond and D. P. DeWitt, Applications of Radiation Thermometry, ASTM STP-895 (1985).

CHAPTER 2

Temperature Fundamentals Graham Machin1 and Benjamin K. Tsai2

Contents 1. Introduction 2. Primary Thermometry 2.1. Temperature: fundamental principles 2.2. Primary thermometry methods 2.3. The formal deﬁnition of the SI unit: the kelvin 2.4. Summary 3. A Short History of Temperature Scales Leading to the International Temperature Scale of 1990 3.1. Temperature scales 3.2. History of temperature scales: ITS-27 to ITS-90 3.3. Description of ITS-90 3.4. ITS-xx? 4. Scale Realisation, Calibration and Traceability 4.1. Practical realisation of the ITS-90 scale by non-contact thermometry 4.2. Calibration 4.3. Traceability 4.4. Accreditation 4.5. Establishing the validity of world metrology: the BIPM ‘Mutual Recognition Arrangement’ 5. Uncertainty Estimation in Non-Contact Temperature Scale Realisation and Dissemination 5.1. Basic principles of uncertainty analysis 5.2. Uncertainty in scale realisation above the silver point 5.3. Uncertainty in scale realisation below the silver point 5.4. Uncertainty in scale dissemination

1 2

29 30 30 32 39 40 41 41 41 43 46 47 48 51 51 53 54 55 55 57 60 62

Engineering Measurement Division, National Physical Laboratory, Teddington, Middlesex TW11 0LW, UK Optical Technology Division, National Institute of Standards and Technology, Gaithersburg, MD 20899-8441, USA

Experimental Methods in the Physical Sciences, Volume 42 ISSN 1079-4042, DOI 10.1016/S1079-4042(09)04202-7

r 2010 Elsevier Inc. All rights reserved.

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Graham Machin and Benjamin K. Tsai

6. Possible Future Approaches to High-Temperature Scale Realisation and Dissemination 6.1. The mise-en-pratique for the deﬁnition of the kelvin 6.2. Possible future non-contact thermometry scale realisation methods 6.3. A practical implementation of primary thermometry 7. Summary Acknowledgements References

63 63 65 66 66 67 67

1. Introduction In this chapter the basic principles of temperature, its deﬁnition and measurement are given. Beginning with a brief outline of the fundamental principles, the reader is introduced to the concept of thermodynamic temperature and some of the primary means of its determination. The requirement for a deﬁned scale is explained and a brief historical outline of temperature scales is given including a description of the current temperature scale, the International Temperature Scale of 1990 (ITS-90). Easy access to that scale, in the context of radiation thermometry, is the concern of the next section with the issues of calibration, traceability and accreditation addressed. This is followed by a discussion of how to determine uncertainties in temperature measurement with the emphasis ﬁrmly on non-contact thermometry. The chapter concludes with a forward look, discussing possible approaches that could lead to better non-contact thermometry scale realisation and dissemination in the future.

2. Primary Thermometry 2.1. Temperature: fundamental principles The deﬁnitions of temperature and its relation to classical thermodynamics and statistical mechanics are discussed in detail in a number of works [e.g. 123], so it is not necessary to go into full details and only general remarks will be made here. Temperature is one of the seven base quantities of the International System of Units (SI) [4], along with mass, length and time, among others. Temperature is an intensive quantity, like pressure and density, not an extensive quantity, like length or mass. To illustrate, if two 1-m rods are placed end to end, their combined length is 2 m. But if a litre of water at

Temperature Fundamentals

31

201C is added to another litre of water at 201C, the temperature of the water is still 201C. That temperatures are not additive has profound implications for its measurement and the generation of a temperature scale. So what is temperature? It can simply be seen as the ‘degree of hotness’ or, more speciﬁcally the ‘potential for heat transfer’. A proper understanding of temperature as a fundamental physical quantity, and its relationship with heat, was only developed in the 19th century when the principles of classical thermodynamics (by Kelvin, Joule, and Clausius and others) and statistical mechanics (by Boltzmann, Maxwell and others) were elaborated. 2.1.1. Classical thermodynamics The zeroth law of thermodynamics [5,6] describes the existence of thermal equilibrium between two or more macroscopic systems (ensembles of particles of, for e.g., atoms or molecules of a gas, liquid or solid, etc.). If, when a system is placed in contact with a second system, there are no changes in the parameters describing the state of the two systems (such as pressure, volume, etc.), then these two systems are in thermal equilibrium with each other, there is no heat ﬂow between them, and their temperatures are the same. Thus, temperature is the parameter of state that has the same value for any systems which are in thermal equilibrium. (This property is used when the temperature of an object is measured by placing a thermometer in contact 2 bringing it into thermal equilibrium 2 with the object.) The ﬁrst law of thermodynamics is concerned with heat and work and is fundamentally a statement of the principle of the conservation of energy applied to thermodynamic systems. Further ideas about temperature come from the second law of thermodynamics, one expression of which is concerned with the efﬁciencies of heat engines. For an ideal heat engine operating in a cycle, heat Q1 is taken in from a reservoir at temperature T1, and heat Q2 is given out to a cooler reservoir at temperature T2. The work done is (Q1Q2), and the ratio of the two heats can be used to deﬁne the ratio of the ‘thermodynamic temperatures’ of the reservoirs: Q1/Q2 ¼ T1/T2. The crucial point here is that the thermodynamic temperatures, T1 and T2, are independent of the properties of any material substance, that is the working system of the ideal engine. Thus, if a working system can be found which closely approximates the ideal condition, then it could be used to derive thermodynamic temperatures (or their ratios), and all such systems would give the same results. A more fundamental mathematical, rather than physical, formulation of the second law is due to Carathe´odory [3], who showed that in the neighbourhood of a given state of a thermodynamic system, there exist states which are not accessible along purely adiabatic paths. From this, he was able to deduce the existence of a thermodynamic temperature equivalent to that deduced by Kelvin on physical grounds. It is sometimes

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stated that the expression of the second law related to heat engines is too specialised and the apparently more general treatment of Carathe´odory is to be preferred. Whether this is really the case is a moot point, and the reader is referred to Ref. [3] and references therein. But whatever the merits of Carathe´odory’s approach, the classical one based on heat engines is certainly more physically intuitive. 2.1.2. Statistical mechanics Statistical mechanics [7,8], developed in parallel with classical thermodynamics, is concerned with the statistical behaviour of large ensembles of particles and shows how for a given system the macroscopic properties are related to the kinetics of the microscopic particles of which it is composed. For a system in thermal equilibrium, the Maxwell2Boltzmann distribution function, f(E), relates the probability of occupancy for a particular state of energy, E, to the temperature: f (E)Bexp(E/kT), where k is the Boltzmann constant. It can further be shown that the temperature is a direct measure of the average kinetic energy of the constituent (noninteracting) particles, which is (1/2)kT per degree of freedom. The important point here is that a temperature could be deﬁned either through classical thermodynamics or by statistical mechanics. One of the triumphs of physics in the 19th century was that it brought the two ﬁelds into one uniﬁed theoretical framework, showing that the temperature parameters that appeared in the two formalisms are exactly equivalent. 2.1.3. Summary From the seemingly disparate disciplines of classical thermodynamics and statistical mechanics, there emerged the same unique system parameter, thermodynamic temperature, T, leading to the prospect of deriving a universally applicable temperature scale to replace the arbitrary schemes which had previously been used. The determination of thermodynamic temperature by primary thermometry and the current deﬁnition of the kelvin are the subject of the next section.

2.2. Primary thermometry methods Primary thermometry is temperature measurement where the equation of state of a well-deﬁned physical system can be explicitly written down and used, or where a rigorous physical relationship exists between temperature and the measured physical property of the sensor. For example, the equation of state of a gas or the relationship between the mean-square noise voltage across a resistor and temperature is well known and can be used for accurate thermometry, albeit with considerable difﬁculty. In contrast, neither the expansivity of mercury in glass nor the resistivity of platinum as

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Temperature Fundamentals

function of temperature is well known from ﬁrst principles, and thermometers using these properties must be calibrated before use. This section brieﬂy outlines some of the primary methods of thermometry which have been used in recent years [1,9]. 2.2.1. Constant-volume gas thermometry Constant-volume thermometry is based on the equation of state for an ideal gas pV ¼ nRT

(1)

where p is the pressure, V the ﬁxed volume, n the number of moles of gas in V, R the molar gas constant and T the thermodynamic temperature. In principle, this is a very simple method: for a known amount of gas in a known and constant volume, and given the value for R, a measurement of pressure can be used to determine the temperature. In practice, ratios of pressures are used to determine ratios of temperatures, starting from a known reference temperature such as the triple point of water. In this way, the gas density n/V can be eliminated from the equation, to ﬁrst order of accuracy. Even so the experiments are difﬁcult, for many practical reasons to do with sensing and measuring the pressure, the constancy of the volume and density, adsorption or desorption of the gas in the bulb and sensing tube, etc. A comprehensive list of factors that introduce uncertainties into constant-volume gas thermometry are given in Ref. [9] and discussed in detail in Ref. [1]. One further difﬁculty is that at ﬁnite pressures and densities, the ideal gas equation has to be modiﬁed to take into account two- and three- (and if necessary higher-order) particle interactions. This is done through the introduction of temperature-dependent virial coefﬁcients B(T) and C(T): n n 2 pV ¼ nRT 1 þ BðT Þ þ þ CðT Þ V V

(2)

Measurements of pV/n must be made at a series of gas densities, n/V, and extrapolated to zero density to obtain the value of RT. Again, the process is simpliﬁed by determining temperatures relative to a reference ﬁxed point. Until quite recently constant-volume gas thermometry has been the principal thermodynamic method over much of the range, and it formed the main basis of the ITS-90 up to about 900 K. 2.2.2. Acoustic thermometry This method is based on the relation between the speed of sound of a gas, cs, and the thermodynamic temperature c 2s ¼

gRT þ A 1 p þ A 2 p2 þ M

(3)

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Graham Machin and Benjamin K. Tsai

where M is the molar mass of the gas; g ¼ cp/cv is the ratio of speciﬁc heat capacities at constant pressure and constant volume, respectively; and A1, A2, y are the acoustic virial coefﬁcients [10]. In a real gas, the speed of sound has a small pressure dependence, written here as a virial expansion in pressure. As in constant-volume gas thermometry, this can be treated by measuring the speed of sound at several pressures and extrapolating to zero, whereupon T can be simply deduced from the above-mentioned equation. The technique is not new, but in recent years acoustic thermometry has become the most precise and widely practised form of primary thermometry at temperatures up to about 550 K [11214]. The reason for this is that all the parameters can be readily determined, and in particular the gas pressure is not required, at least to ﬁrst order of accuracy. The main challenge of the method is the accurate determination of cs. Most modern acoustic thermometry takes place in spherical or quasi-spherical resonators, a technique pioneered at the National Institute of Standards and Technology (NIST) in the 1980s [15], in which the velocity is determined from the resonant frequencies within the (quasi)-spherical enclosure. To convert the frequency into velocity one needs to know the radius, and to avoid having to measure it absolutely it is usual to calibrate the resonator at the triple point of water, whereupon only the change in the radius with temperature needs to be known or determined. In the wake of the developments in primary acoustic thermometry, the technique is likely to be a candidate for practical primary thermometry in the future, and so could be used for the direct dissemination of T instead of T90. 2.2.3. Johnson noise thermometry Johnson noise thermometry [16] is based on the temperature dependence of the mean-square noise voltage /V 2S measured in a bandwidth Df developed across a resistor Re. The temperature dependence is obtained from the well-known Nyquist equation: hV 2 i ¼ 4kT Re Df

(4)

where k is the Boltzmann constant. The method has the great attraction of being a purely electrical method of primary thermometry, which uses a probe resembling a standard resistance thermometer, and if successfully developed it could ﬁnd widespread application. The main problems with the method are (a) the very small voltages that are developed (typically mV), (b) the long measurement times needed to get a good statistical average for V 2, (c) the need to extract the signal from extraneous noise in the ampliﬁers and connections and (d) the need to know the bandwidth and gain of the ampliﬁers.

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Temperature Fundamentals

Once again, it helps in practice to determine the ratio of noise voltages from two resistors, one at the unknown temperature and the other at a known temperature, usually the triple point of water, so that the ampliﬁer bandwidth can be eliminated from the equation. Nevertheless, technical difﬁculties have so far prevented the wide application of Johnson noise thermometry except at very low or very high temperatures, where modest accuracy is acceptable, or where the environment is hostile to other forms of contact thermometry.

2.2.4. Radiation thermometry This section discusses the application of the laws of thermal radiation to primary thermometry and in view of its relevance to this book the topic will be treated in more detail than the methods so far considered. Although it was not until 1900 that Planck derived his famous equation relating the spectral radiance of thermal radiation inside an isothermal cavity to the wavelength and temperature, much of the groundwork had been laid in previous decades. Thus, it had already been shown, by applying the second law of thermodynamics to cavity radiation, that the intensity could not depend on the properties of the cavity walls, and that the total intensity (summed over all wavelengths) was proportional to T4 (the Stefan2 Boltzmann law). Wien had deduced his exponential equation for the radiant intensity at short wavelengths, and Rayleigh had deduced his equation for long wavelengths (giving rise to the troublesome ‘ultraviolet catastrophe’). Planck moulded Wien’s argument and Boltzmann’s theory into a coherent whole, though not without introducing the equally troublesome concept of quantisation of the radiation. Although Planck was not aware of Rayleigh’s work, a few months after Rayleigh’s work was published, Planck published his equation and its characteristics, which are described in Chapter 3.

2.2.5. Spectral radiometry The concepts of spectral radiometry will be discussed in detail in Chapter 4, and only an outline of the technique will be given here. The basic requirements as far as thermometry are concerned are twofold, to have a source of spectral exitance that closely resembles a blackbody radiator and to have a well-characterised absolutely calibrated spectral radiometer. The spectral exitance, M(l, T), the power emitted per unit area per unit wavelength interval, by such a blackbody (in vacuo) is given by: M ðl; T Þdl ¼

2phc 2 dl 5 expððhcÞ=ðklT ÞÞ 1 l

(5)

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Graham Machin and Benjamin K. Tsai

where k is the Boltzmann constant, h the Planck constant, c the speed of light and l the wavelength. The challenge of the technique is to measure M(l, T), using an absolutely calibrated radiometer. A radiometer is a simple device almost universally comprised of a detector, a ﬁlter to deﬁne the spectral band and a well-characterised aperture. A schematic diagram of such a device is shown in Figure 1 [17]. The measurement equation for absolute radiometry appears in the equation below: Z

I photo ¼ g

sðlÞM ðl; T Þdl

(6)

where Iphoto is the measured photocurrent, g a geometric factor determined from the two deﬁning apertures (one on the radiometer, the other deﬁning the emitting area of the blackbody) and their separation, and s(l) the absolute spectral irradiance response of the ﬁlter radiometer, determined using a cryogenic electrical substitution radiometer (ESR, see Section 2.2.6) and a reference detector. Given that Iphoto, g and s(l) are known, the equation can be solved iteratively to determine T, for any given blackbody temperature, provided it is emitting enough radiant power in the spectral range of the ﬁlter radiometer. Alternatively, an a priori curve ﬁt of Iphoto versus T has been successfully implemented in several ﬁlter radiometer applications [18221]. In recent years, this technique has been deployed widely at high temperatures, particularly above the silver point [22,23] where exceptionally good detectors based on silicon photodiodes have been used. Typical wavelengths of operation for such ﬁlter radiometers are between 400 and 1,000 nm. These wavelengths are usually deﬁned by interference ﬁlters with bandwidths typically of the order of 10 nm. The method has also been used to a limited extent at lower temperatures to around the indium point

Figure 1 Schematic of a NIST ﬁlter radiometer depicting basic elements (detector, aperture, and ﬁlter) [17].

37

Temperature Fundamentals

(B1561C), using other photon detectors such as InGaAs and longer wavelengths, typically around 2,000 nm, and bandwidths of B100 nm [24]. It is clear that in the national metrology institutes (NMIs) where signiﬁcant investment has been made, uncertainties based on absolute radiometry are now comparable with those of relative methods. However, the difﬁculties of determining and maintaining adequate knowledge of both g and s(l) are ameliorated by measuring ratios of the radiances from blackbody sources at two temperatures (although in this case, unlike in gas thermometry, the triple point of water is not available as a reference point, not being within the range of the radiometer). The ITS-90 uses ratios to a blackbody at the freezing point of silver, gold or copper to achieve a low uncertainty for scale realisation. The uncertainty of a thermodynamic scale realised from these ﬁxed points is limited by the thermodynamic uncertainty of the freezing point temperatures. Thus, the relative radiometry approach, while not giving the lowest possible uncertainties, nevertheless gives simple access to a repeatable scale without the need to know g and with only the requirement to measure s(l) by relative methods, which is quite straightforward. It should be noted that developments currently under way, discussed in Section 6, could lead to a cost-effective and simple method of thermodynamic temperature scale realisation above the silver point with uncertainties lower than the ITS-90 through the implementation of high-temperature ﬁxed points and speciﬁed interpolation equations [25].

2.2.6. Total radiometry Total radiometry utilises the Stefan2Boltzmann law for the total radiant exitance M(T) of a blackbody, which is essentially the integration of Planck’s law over all wavelengths. The total radiant exitance, M(T), for a blackbody at temperature T is given by: M ðT Þ ¼

2p5 k4 T 4 ¼ sT 4 15h3 c 2

(7)

where s¼

2p5 k4 W ¼ 5:670400 108 2 4 m K 15h3 c 2

(8)

is the Stefan2Boltzmann constant. Often the wavelength range of a ‘total’ radiation thermometer is limited by the transmission of an optical component, or the response of the detector, and the T4 law is not accurately obeyed. These instruments are more correctly called ‘broadband’ radiation thermometers, and they must be calibrated empirically.

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To implement the technique as a primary thermometer, one requires a blackbody radiator with a high emissivity over the complete wavelength range (which may extend well beyond 100 mm) and a calibrated detector with perfect response over this range, the two being separated by an aperture system of known geometry. The energy detected must all originate directly from the radiator, and none must be scattered, reﬂected or diffracted into or out of the beam. Other mechanisms of heat transfer, for example by gas transfer, must be eliminated. Corrections must be applied for any signiﬁcant departures from these conditions. It is clear that only a thermal detector is capable of meeting these requirements, and that it must itself be a blackbody absorber of radiation and consequently must be much colder than the radiator. It must also be a calorimeter, in which the energy absorbed can be measured. The aperture system is part of a radiation trap which deﬁnes the beam in such a way that the radiation transfer function (geometrical factor) can be derived. In practice, once again, this requirement can be relaxed if the apparatus is used to measure temperature ratios, in which case the geometrical factor cancels, to ﬁrst order. The radiation trap provides the interface and buffer between the hot radiator and cold detector, and it is clear that it must absorb all the radiation from the radiator (typically W95%) which is not in the geometrical beam, hence its name. The radiation trap must also include a shutter so that the radiation beam from the emitting blackbody can be shut off. This is necessary because the measurement of the energy absorbed by the detector is best determined by electrical substitution, as is now described. After it is absorbed, the energy ﬂows through a heat link to a reservoir maintained at a ﬁxed temperature. The detector itself then comes to a steady temperature which depends on the constant energy ﬂux. The energy is determined by shutting off the radiation beam and substituting electrical energy by passing a current through a heater on the detector. The detector temperature remains steady if the electrical energy exactly matches the previously measured radiated energy and the substitution is managed without losses or imperfections [26]. ESRs have been used for many years; for example, a total radiation calorimeter was used by Blevin and Brown with a radiator at the freezing point of gold and the detector at room temperature [27]. However, the thermal imperfections of ESRs which operate at ordinary ambient temperatures are such that uncertainties are likely to be W0.1%. This has been improved by a factor of 10 or more with the advent of cryogenic ESRs in the 1980s. Among the advantages of a helium-cooled detector: are high cavity adsorption with minimal back radiation, high thermal diffusivities and low heat capacities of the materials and consequently fast response, precise temperature control and uniformity, the availability of superconducting leads, ensuring that all the electrical energy is deposited in the detector, and the ability to achieve high vacuum.

Temperature Fundamentals

39

Quinn and Martin [28] were the ﬁrst to successfully apply cryogenic radiometry to thermometry. Their ESR, constructed at the National Physical Laboratory (NPL), was used for the determination of the Stefan2Boltzmann constant as well as for the measurement of temperature from 1301C to 1001C [29]. A diagram of the cryogenic radiometer is shown in Figure 2. The method is generally implemented as a thermometer

Figure 2

The cryogenic radiometer of Quinn and Martin [28].

40

Graham Machin and Benjamin K. Tsai

in relative mode, that is relative to a reference total exitance at the triple point of water Ttpw, so that M ðT Þ ¼ M ðT tpw Þ

T T tpw

4 (9)

However, the basic operation of a cryogenic ESR is the same as that operating at room temperature but with much reduced uncertainties. While total radiometry cannot in any way be considered a simple thermometry method, the techniques developed, in particular cryogenic electrical substitution radiometry, have found very widespread use in optical radiation measurements, both directly (e.g. in determining laser power) and indirectly (e.g. in absolute detector characterisation). The latter development is essential for the successful implementation of absolute spectral radiometry which was described in the previous section. 2.2.7. Other thermodynamic methods There are many other methods of thermodynamic thermometry that can be employed. These include dielectric constant gas thermometry, refractive index thermometry, nuclear orientation, nuclear magnetic resonance, Doppler broadening and others that are all well beyond the scope of this work. The interested reader is referred to Refs. [1,9] and references therein.

2.3. The formal deﬁnition of the SI unit: the kelvin The discussion so far has been concerned with the methods of thermodynamic thermometry, but in order to obtain numerical values the unit must be deﬁned. The unit speciﬁed in the SI is the kelvin, and the current deﬁnition has been in place since 1954 [4]. The formal wording, dating from 1967, is: kelvin, K: The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. It follows that the thermodynamic temperature of the triple point of water is exactly 273.16 K (i.e. Ttpw ¼ 273.16 K). The deﬁnition was clariﬁed in 2005 [4] to specify exactly the isotopic composition1 of the water that was to be used for the construction of water triple point cells. Any variation from this isotopic composition results in a temperature correction to the particular cell in question. It is common practice to express thermodynamic temperature (T), in terms of a difference from a reference temperature, T0 ¼ 273.15 K (the ice 1

At its 2005 meeting the CIPM afﬁrmed that: This deﬁnition refers to water having the isotopic composition deﬁned by the following amount of substance ratios: 0.00015576 mole of 2H per mole of 1H, 0.0003799 mole of 17O per mole of 16O, and 0.0020052 mole of 18O per mole of 16O. These isotopic ratios are that of Vienna standard mean ocean water known by the acronym V-SMOW.

Temperature Fundamentals

41

point). This difference is called the Celsius temperature (t) which is deﬁned by the relation t ¼ TT0 and whose unit is the degree Celsius (1C), which by deﬁnition has the same magnitude as the kelvin. This deﬁnition has proved robust and reliable for more than 50 years. However, the International Committee for Weights and Measures (CIPM), at its 94th meeting in 2005, adopted a recommendation on preparative steps towards redeﬁning the kilogram, ampere, kelvin and mole so that these units are linked to deﬁned values of fundamental constants [30]. In the next 10 years it is likely that the kelvin will be redeﬁned, probably along the following lines: The kelvin, the unit of thermodynamic temperature, is such that the Boltzmann constant is exactly 1.380 65XX 1023 J/K, where XX will ultimately be ﬁxed at a future evaluation of fundamental constant data [31]. There are some advantages of redeﬁning the kelvin in this way. It brings the kelvin into line with other units already deﬁned in terms of fundamental constants such as the metre. It means that thermodynamic temperature is no longer linked to a particular ﬁxed-point material as it is currently (i.e. the water triple point). Once redeﬁned any appropriate thermodynamic method could then be used to generate the quantity temperature, in principle independent of the triple point of water.

2.4. Summary In this section, some of the basic principles of thermodynamic temperature and its measurement have been elaborated and described. Some primary thermometry methods have been brieﬂy outlined. It is clear that primary thermometry is a major undertaking and not easily adaptable for practical purposes. It is largely because of the difﬁculty in using primary thermometers that a deﬁned scale was established, based on repeatable quantities that are relatively simple to measure with high precision. The formal deﬁnition of the kelvin has been given and a brief discussion concerning the proposed future redeﬁnition outlined. The progression from thermodynamic temperature to a deﬁned temperature scale is the subject of the next section.

3. A Short History of Temperature Scales Leading to the International Temperature Scale of 1990 3.1. Temperature scales Based on the above section, one can be forgiven for thinking that thermometry in any practical sense is very difﬁcult, but this is not so. The painstaking primary thermometry that has taken place over the past 100 years or so has fed into practical scales that have increased in sophistication

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Graham Machin and Benjamin K. Tsai

and range but have also increasingly approached true thermodynamic values. Thus the current deﬁned scale, the ITS-90, is estimated to depart from thermodynamic temperature by less than or about 10 mK at all temperatures below 600 K, and about 40 mK at the gold point (1,337 K). The thermometers that are used to realise and disseminate the deﬁned scale are not required to be characterised by an equation of state or a fundamental law of physics. The main requirements are that they should be precise, reproducible and practical; that is simple to use, calibrate and transport (important for scale dissemination). Typical examples of such thermometers are platinum resistance thermometers, thermocouples and radiation thermometers.

3.2. History of temperature scales: ITS-27 to ITS-90 3.2.1. The normal hydrogen scale of 1887 The origins of a deﬁned scale began soon after 1875 when the International Bureau of Weights and Measures (Bureau International des Poids et Mesures or BIPM) acquired international prototypes of the kilogram and metre and a set of national prototypes for the Member States of the Convention. Thermometers were required to accompany each prototype metre because it was important that the length be measured at the correct speciﬁed temperature. These were mercury-in-glass thermometers calibrated directly against a constant-volume gas thermometer using hydrogen as its working gas. This scale had as its ﬁxed points the melting point of ice and the boiling point of water, so it was not truly thermodynamic. It was known as the ‘normal hydrogen scale of 1887’. These developments set the scene for a more elaborate scheme to be developed. Fixed points, whose temperature would be determined by primary thermometry, would be used as anchor points on a deﬁned scale to calibrate platinum resistance thermometers over a much wider temperature range.

3.2.2. ITS-27 The International Temperature Scale of 1927 (ITS-27) was introduced to overcome the practical difﬁculties of a direct realisation of thermodynamic temperatures by gas thermometry, and to serve as a general replacement to the various national scales then in existence. The scale itself essentially split into three parts, distinguished by the sensor used to carry the scale: Between 1901C (the oxygen boiling point) and up to 6601C, a

platinum resistance thermometer of deﬁned quality was prescribed.

Temperature Fundamentals

43

Between 6601C and 1,0631C (the gold-point value at that time), a

Pt210%Rh/Pt (type S) thermocouple was to be used, calibrated at the freezing points of antimony,2 silver and gold. Above 1,0631C, an optical pyrometer was speciﬁed using Wien’s approximation for extrapolation and a deﬁned value for the second radiation constant3 c2. 3.2.3. ITS-48 and IPTS-48 The International Temperature Scale of 1948 (ITS-48) made some small adjustments to the ITS-27. The main ones were that the junction between the platinum resistance thermometer and the thermocouple was moved to the antimony point (6301C), the silver freezing temperature was increased by 0.32960.81C, Planck’s radiation law replaced Wien’s law, and an improved value of the second radiation constant was speciﬁed. The ITS-48 was subsequently modiﬁed in 1960, after the SI deﬁnition for the unit of temperature had been agreed. The scale was relabelled as the International Practical Temperature Scale of 1948 (IPTS-48), to emphasise that this was a practical scale, not a thermodynamic one. Signiﬁcant changes were as follows: the water triple point replaced the melting point of ice and the freezing point of zinc became a preferred alternative to the boiling point of sulphur as a calibration point. The restriction to using visible radiation from optical pyrometry was also removed. Temperatures were subsequently expressed in kelvin and degrees Celsius, and not degrees Centigrade as had been the case up to this point. 3.2.4. IPTS-68 and EPT-76 There were extensive changes with the introduction of the International Practical Temperature Scale of 1968 (IPTS-68). These involved changes to many of the values of the deﬁning ﬁxed points to bring the values into closer alignment with thermodynamic values. Particular ﬁxed points that were affected were the boiling point of oxygen (90.188 K), and the freezing points of zinc (419.581C), silver (961.931C) and gold (1064.431C). The lower limit of the scale was extended to 13.81 K, and six new deﬁning ﬁxed points were introduced: the triple point of equilibrium hydrogen (13.81 K), an intermediate equilibrium hydrogen point (17.042 K), the normal boiling point of equilibrium hydrogen (20.28 K), the boiling point of neon (27.102 K), the triple point of oxygen (54.361 K) and the freezing 2

3

Note that the antimony temperature was not speciﬁed as such, but determined by an standard platinum resistance thermometer (SPRT). The second radiation constant is deﬁned by c2 ¼ hc/k, where h is Planck’s constant, c the speed of light and k the Boltzmann constant. Its value is speciﬁed in the ITS-90 as 0.014388 mK which differs slightly from the most recent evaluation based on fundamental constants 0.0143775 mK.

44

Graham Machin and Benjamin K. Tsai

point of tin (231.96811C). The boiling point of sulphur was deleted. The interpolation for platinum resistance thermometers became more complex, and the second radiation constant value was adjusted. Some technical changes were introduced in an amendment in 1975, but there were no changes to the numerical values. The 1976 Provisional 0.5230 K Temperature Scale (EPT-76) was introduced to complement the IPTS-68. The EPT-76 was deﬁned in terms of temperatures assigned to 11 ﬁxed points, which were a mixture of superconducting transition points, and a number of boiling and triple points of helium (boiling), e-hydrogen (boiling and triple point) and neon (boiling and triple point) [1]. 3.2.5. The transition from previous scales to ITS-90 The ITS-90 itself was introduced on 1 January 1990, and was the fourth international temperature scale. It differed from the previous scales in that:

it extended to 0.65 K instead of 13.81 K; it was in closer agreement with thermodynamic temperatures; it had a number of overlapping ranges and sub-ranges; an interpolating gas thermometer (between 3 K and the neon triple point) was included as one of the deﬁning instruments; the type S thermocouple was eliminated and use of the platinum resistance thermometer extended to the silver freezing point; and radiation thermometry was extended downward to the silver freezing point, and made use of reference blackbody radiators of either the silver, gold or copper points to generate equivalent ITS-90 realisations. A list of the ﬁxed points of the ITS-90 and their values is given in Table 1.

3.3. Description of ITS-90 The formal text of the ITS-90 can be found in Ref. [32]. Its purpose is to ‘deﬁne procedures by which certain speciﬁed practical thermometers of required quality can be calibrated in such a way that the values of temperature obtained from them can be precise and reproducible, while at the same time closely approximating the corresponding thermodynamic values’. The ITS-90 has essentially four sections: (a) Between 0.65 K and 5.0 K, T90 is deﬁned in terms of the

vapour2pressure temperature relations of 3He and 4He. (b) Between 3.0 K and the triple point of neon (24.5561 K), T90 is deﬁned by means of a helium gas thermometer calibrated at three experimentally realisable temperatures having assigned numerical values (deﬁning ﬁxed points) and using speciﬁed interpolation procedures. (c) Between the triple point of equilibrium hydrogen (13.8033 K) and the freezing point of silver (1234.93 K), T90 is deﬁned by means of

45

Temperature Fundamentals

Table 1

The deﬁning ﬁxed points of the ITS-90.

No.

Temperature T90 (K)

Statea

Substance

t90 (1C)

1.

325

2. 3.

13.8033 17.035

270.15 to 268.15 259.3467 256.115

4.

20.27

252.88

5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

24.5561 54.3584 83.8058 234.3156 273.16 302.9146 429.7485 505.078 692.677 933.473 1234.93 1337.33 1357.77

248.5939 218.7916 189.3442 38.8344 0.01 29.7646 156.5985 231.928 419.527 660.323 961.78 1064.18 1084.62

He

V

e-H2 e-H2 (or He) e-H2 (or He) Ne O2 Ar Hg H2 O Ga In Sn Zn Al Ag Au Cu

T V (or G) V (or G) T T T T T M F F F F F F F

a V: Vapour2pressure point; T: triple point; G: gas thermometer point; M, F: melting point, freezing point (temperature at a pressure of 1,01,325 Pa, at which solid and liquid phases are in equilibrium).

platinum resistance thermometers calibrated at speciﬁed sets of deﬁning ﬁxed points and using speciﬁed interpolation procedures over deﬁned ranges or sub-ranges. (d) Above the freezing point of silver (1234.93 K), T90 is deﬁned in terms of one of three speciﬁed ﬁxed points and the Planck radiation law, in ratio form. Given the nature of radiation thermometry, for nearly all practical purposes only temperatures of 1001C and above are of relevance, that is only parts (c) and (d); these are described in more detail below. 3.3.1. The range from the triple point of equilibrium hydrogen 13.8033 K to the freezing point of silver 961.781C In this range T90 is deﬁned by means of a platinum resistance thermometer calibrated at speciﬁed sets of deﬁning ﬁxed points, and using speciﬁed mathematical procedures for interpolation at intervening temperatures [32].

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Graham Machin and Benjamin K. Tsai

Table 2 ITS-90. No.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

The sub-ranges for calibrating platinum resistance thermometers to the

Sub-range for platinum resistance thermometer

The triple point of e-H2 (13.8033 K) to the freezing point of silver (961.781C) The triple point of e-H2 (13.8033 K) to the triple point of water (273.16 K) The triple point of neon (24.5561 K) to the triple point of water (273.16 K) The triple point of oxygen (54.3584 K) to the triple point of water (273.16 K) The triple point of argon (83.8058 K) to the triple point of water (273.16 K) From 01C to the freezing point of silver (961.781C) From 01C to the freezing point of aluminium (660.3231C) From 01C to the freezing point of zinc (419.5271C) From 01C to the freezing point of tin (231.9281C) From 01C to the freezing point of indium (156.59851C) From 01C to the freezing point of gallium (29.76461C) The triple point of mercury (38.83441C) to the melting point of gallium (29.76461C)

The speciﬁc design of platinum resistance thermometer varies depending on the temperature range required; these can be capsule type, long-stem standard or high-temperature standard platinum resistance thermometers [33]. Temperatures are determined in terms of the ratio of the resistance R(T90) at the temperature T90, and the resistance R(273.16 K) at the triple point of water. This ratio is the so-called W ratio, W(T90) ¼ R(T90)/R(273.16 K). Speciﬁed functions relating W(T90) to T90 are given in the ITS-90 text [32]. The ITS-90 allows for the calibration of platinum resistance thermometers over particular sub-ranges, as detailed in Table 2. This has the advantage that the thermometer needs only to be calibrated in the range of interest for a particular user, which can reduce the cost and the overall calibration uncertainties. However, it is a source of ‘non-uniqueness’ because overlapping sub-ranges do not give identical results. 3.3.2. The range above the freezing point of silver (961.781C): Planck radiation law Above the freezing point of silver the deﬁnition of T90 is deﬁned by the following equation: L l ðT 90 Þ exp½c 2 =ðlT 90 ðXÞÞ 1 ¼ L l ½T 90 ðXÞ exp½c 2 =ðlT 90 Þ 1

(10)

Temperature Fundamentals

47

where T90(X) refers to any one of the silver {T90(Ag) ¼ 1234.93 K}, the gold {T90(Au) ¼ 1337.33 K} or the copper {T90(Cu) ¼ 1357.77 K} freezing points; Ll(T90) and Ll[T90(X)] are the spectral concentrations of the radiance of a blackbody at the wavelength (in vacuo) l, at T90 and at T90(X) respectively; and the second radiation constant is (deﬁned as) c2 ¼ 0.014388 mK. The great advantage this ratio formalism has over absolute radiometry is that the geometric factors, which are difﬁcult to measure and maintain, are eliminated and only the relative spectral response is required. The main disadvantage is that there are currently no deﬁned ﬁxed points above that of copper, which means the uncertainty in scale realisation increases with increasing temperature. Uncertainties will be discussed in more detail in Section 5. The ITS-90 extends only to 0.65 K, and there was until recently no deﬁned scale for temperatures below this point. This situation was remedied with the introduction of the Provisional Low Temperature Scale of 2000, PLTS-2000, which extends from 0.9 mK to 1 K. It is based on the melting pressure of 3He. For the details of the scale, the interested reader is referred to Rusby et al. [34].

3.4. ITS-xx? There is some discussion in the international temperature community about a possible replacement for the ITS-90, to further improve its precision and reproducibility, and its accuracy with respect to thermodynamic temperature. The possible redeﬁnition is currently known as ITS-xx. Recent thermodynamic temperature data, based mainly on absolute radiometry and acoustic gas thermometry, show how T90 departs from T, and this is plotted in Figure 3 [35]. There is of course no departure at the triple point of water because its temperature is ﬁxed by the present kelvin deﬁnition. More signiﬁcant issues arise because of the limitations of the interpolating instruments, such as the high-temperature platinum resistance thermometers used above 933 K, and the formal deﬁnition above the silver point, which currently limits the lowest uncertainties that can be achieved. However, revisions of the temperature scale are rare events because of the expense in changing procedures and documentation. In industry’s view, this expense is currently not worth the beneﬁt it might bring. In the meantime, one could implement some improvements through the device known as the mise-en-pratique (MeP-K, literally ‘put into practice’) for the deﬁnition of the kelvin [36]. The MeP-K is a broader-based and more ﬂexible document than the ITS-90, which can make supplementary speciﬁcations relating to the ITS-90 (but not affecting the numerical values), give practical guidance on the scale realisation and the estimation of uncertainties and include descriptions of primary thermometers for direct

48

Graham Machin and Benjamin K. Tsai

Steur Edsinger & Schooley Guildner & Edsinger Astrov et al, revised Kemp et al Quinn et al Fox et al Stock et al Taubert et al Noulkhow et al Goebel et al Yoon et al Fischer & Jung Moldover et al Ewing and Trusler Strouse et al Ripple et al Benedetto et al Pitre et al Edler et al Labenski et al various (T68 - T90) ITS-90, +u (k = 1) ITS-90, -u (k = 1) mean

0.15

(T - T90) / K

0.10

0.05

0.00

-0.05

-0.10 0

250

500

750

1000

1250

Temperature / K

Figure 3

Thermodynamic temperature data showing how T90 departs from T [35].

measurement of thermodynamic temperature [36]. Discussion of the MeP-K and its potential impact on the future practice of thermometry will be presented in Section 6. A natural driving force for this development is the proposed kelvin redeﬁnition (Section 2.3) [30,31,37]. The redeﬁnition will open up the possibility of realising and disseminating thermodynamic temperature directly without the need to refer to a deﬁned scale. This has the advantage that any appropriate thermodynamic method could be used to realise and disseminate a temperature scale. Of course, the thermodynamic uncertainties must be included (otherwise it becomes a new deﬁned scale), and appropriate and regular comparisons between NMIs would need to be performed to ensure that the integrity of world temperature metrology was safeguarded.

4. Scale Realisation, Calibration and Traceability The basic principle of the ITS-90 is to generate a temperature scale that is precise and simple to realise by NMIs (and also properly equipped commercial calibration laboratories) and to disseminate to the users of the scale through the process of calibration. To ensure that calibrations are performed in a reliable way such that traceability to the ITS-90 could be clearly demonstrated, the process known as accreditation has been implemented, as in other ﬁelds of metrology.

Temperature Fundamentals

49

This section deals with only the basic principles of scale realisation, calibration, traceability and accreditation, with the emphasis being on radiation thermometry. Practical details regarding non-contact thermometer design and blackbody sources suitable as calibration artefacts will be dealt with in subsequent chapters.

4.1. Practical realisation of the ITS-90 scale by non-contact thermometry Radiation thermometry can be performed from quite low temperatures (not generally less than 1001C) to the highest temperatures at which suitable sources of thermal radiation are available. Given the wide temperature range, different technologies need to be employed to construct practical thermometers. This section is concerned not so much with the technology but with the basic principles of establishing a temperature scale that can then be accessed by the user community. The discussion is in two parts, considering ﬁrst the range above the silver point and secondly the range below the silver point (where the ITS-90 itself is deﬁned by contact thermometry). 4.1.1. Above the Ag freezing point (961.781C) At temperatures above the freezing point of silver, non-contact thermometry is used to formally deﬁne the ITS-90. This deﬁnition is given in Section 3.3.2. To turn this deﬁnition into a reality, several things need to be in place. (a) A ﬁxed-point blackbody reference radiance source: This generates both T90(X)

and Ll[T90(X)] in Equation (10). The construction of such sources is discussed in Chapter 6 but this essentially comprises of a graphite crucible ﬁlled with an ingot of very high purity metal (either Ag, Au or Cu) with typical metallic impurities o10 ppm. Inserted into this ingot is a cylindrical graphite cavity that has been designed to have a very high emissivity and hence serve as a blackbody. The crucible is then heated and melted in a suitable low thermal gradient furnace and the freezing point provides the reference radiance for the starting point of the ITS-90 [18,38,39] (Figure 4). (b) A variable high-temperature radiance source: This generates Ll[(T90)], the spectral radiance at temperature T90 in Equation (10). The hightemperature source has historically been a high-stability tungsten ribbon lamp [40]. These lamps, however, were awkward to use having a small target size and low emissivity. The limitations of lamps, in more recent times, have been overcome through the use of variable high-temperature blackbody sources [39,41243], which can have

50

Figure 4

Graham Machin and Benjamin K. Tsai

Schematic of gold-point blackbody source [39].

Figure 5 Schematic of variable-temperature blackbody source for use above 1,0001C [39].

target sizes of 25 mm or more and high emissivities (W0.999). A typical example of a high temperature blackbody is shown in Figure 5. (c) A pyrometer, radiometer or radiation thermometer (these terms are used interchangeably) that can realise the deﬁnition equation [44]: This measures Ll(T90) and Ll[T90(X)] so that with the deﬁned value of T90(X), the temperature T90 of the variable-temperature source can be determined. Practical considerations mean that the target size of the radiation thermometer needs to be small to view the ﬁxed-point blackbody cavity (these typically have apertures of a few millimetres). The relative spectral response of the thermometer needs to be characterised and its output must be linear in response to radiance. These types of radiation thermometers are described in detail in Chapter 4.

Temperature Fundamentals

51

Apart from the above, the necessary peripherals and infrastructure are required to ensure that, in particular, the pyrometer is operating optimally. That means facilities for determining linearity, relative spectral response and target size should be readily available. If this is the case, a scale can be reliably generated with uncertainties approaching 1 K or less at 2,800 K. 4.1.2. Below the silver freezing point Below the silver freezing point, the ITS-90 is formally deﬁned by contact thermometry, that is platinum resistance thermometers calibrated against the deﬁning ﬁxed points. This means that an accurate representation of the scale by non-contact means should be available that takes its traceability from the formal deﬁnition. In practice, high-performance variable-temperature blackbody sources have been constructed, often based on heat pipe technology [45,46], which have as their temperature sensor either a standard platinum resistance thermometer or other demountable thermometer calibrated in terms of the ITS-90. These blackbodies generally have large apertures (W40 mm), high emissivities (W0.9995) and very low uncertainties in radiance temperature, typically 0.2 K. A different approach is to take traceability directly from the deﬁning ﬁxed points (e.g. Al, Zn, Sn, In) of the ITS-90 by constructing them as ﬁxedpoint blackbody radiators. A suitable radiation thermometer is then calibrated directly at these ﬁxed points and a simple equation is used to interpolate values of temperature at intermediate points [47249], analogous to the procedures in the ITS-90 for platinum resistance thermometers. However, due to limitations of the present detector technology, this approach is typically only suitable for temperature scale realisations at and above the indium point (1561C).

4.2. Calibration The aim of this section is not to detail the process of calibration for particular thermometers that has been treated adequately elsewhere [50], but rather to convey the general meaning of calibration. The word, calibrate/calibration, is widely used and much misunderstood. For instance, a glance at any dictionary gives an interesting variety of meanings. However, in metrology the word is carefully deﬁned. Calibration is deﬁned in the International Vocabulary of Metrology (VIM) [51] as the ‘operation that, under speciﬁed conditions, in a ﬁrst step establishes a relation between the quantity values with measurement uncertainties provided by measurement standards and corresponding indications with associated measurement uncertainties and, in a second step, uses this information to establish a relation for obtaining a measurement result from an indication’.

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Graham Machin and Benjamin K. Tsai

Note 1: A calibration may be expressed by a statement, calibration function, calibration diagram, calibration curve or calibration table. In some cases, it may consist of an additive or multiplicative correction of the indication with associated measurement uncertainty. Note 2: Calibration should not be confused either with adjustment of a measuring system, often mistakenly called ‘self-calibration’, or with veriﬁcation of calibration. Note 3: Often, the ﬁrst step alone in the above-mentioned deﬁnition is perceived as being calibration. Essentially, a calibration requires measurement standards, such as reference sources of known T90 temperature (‘quantity values’); and the process of calibration then relates this to the value indicated by the thermometer under test (the ‘corresponding indications’), in such a way that the measurement uncertainties are well understood. A comparison of the quantity values of the standard and the corresponding indication of the instrument, together with the measurement uncertainty, is the calibration result. Calibration to a reference standard (in thermometry, one that accurately represents ITS-90) is essential if consistent measurements are to be obtained, for example in a process plant.

4.3. Traceability It is possible to have a thermometer calibrated by a test laboratory, but that laboratory may not have the standards that are required to give a reliable calibration directly to ITS-90. This issue is addressed by implementing the concept of traceability. Again traceability has very speciﬁc meaning in metrology [51]. Metrological traceability is deﬁned as the ‘property of a measurement result whereby the result can be related to a reference through a documented unbroken chain of calibrations, each contributing to the measurement uncertainty’. Note 1: For this deﬁnition, a ‘reference’ can be a deﬁnition of a measurement unit through its practical realisation, or a measurement procedure including the measurement unit for a non-ordinal quantity, or a measurement standard. Note 2: Metrological traceability requires an established calibration hierarchy. Note 3: Speciﬁcation of the reference must include the time at which this reference was used in establishing the calibration hierarchy, along with any other relevant metrological information about the reference, such as when the ﬁrst calibration in the calibration hierarchy was performed. In addition, there are four further clarifying notes given in the VIM.

53

Temperature Fundamentals

The ‘stated reference’ is usually through an implementation of the deﬁnition of the measurement unit 2 this is usually only performed at the NMI. However, it is clear that a hierarchy should be established whereby the NMI holds the primary realisation of the unit which is then disseminated by calibration of appropriate high-quality artefacts to toplevel calibration laboratories. These artefacts provide the traceability to the national standard and this in turn streams to the next tier of calibration laboratories using appropriate artefacts calibrated against their traceable standards carried out under an established quality system. In such a manner, a reliable measurement infrastructure is established in the country (Figure 6). Furthermore, many NMIs voluntarily submit their quality systems to the scrutiny and approval of their regional metrology organizations (RMOs), whose member states are composed of NMIs in the region. The ﬁve major RMOs are the Inter-American Metrology System (SIM) in North and South America, European Association of National Metrology

Thermodynamic Temperature

ITS-90 as defined by CIPM (CCT)

ITS-90 set up at NMI

ITS-90 Set up at national standards laboratories

Reference Thermometers

ISO 17025 Accreditation

Secondary Calibration Laboratory

Working Standards

End Measurement Traceable to National Standards

Figure 6 Traceability. Note: CCT, Consultative Committee for Thermometry; CIPM, International Committee for Weights and Measures; ISO, International Organization for Standardization; ITS, International Temperature Scale; NMI, National Metrology Institute.

54

Graham Machin and Benjamin K. Tsai

Institutes (EURAMET), Asia Paciﬁc Metrology Programme (APMP), Euro-Asian Cooperation of National Metrological Institutions (COOMET) in Central and Eastern Europe and Cuba and Southern African Development Community Cooperation in Measurement Traceability (SADCMET). Participation in an RMO enhances the claim to metrological traceability throughout the world. Two further questions naturally arise. Firstly, how do you know that the calibration laboratories are performing their job properly? This is tested by third party accreditation (see Section 4.4). Secondly, how do you know you can rely on the national standard? Is it consistent with other national realisations in other NMIs? This is dealt with through the BIPM Mutual Recognition Arrangement (MRA) [52] (see Section 4.5).

4.4. Accreditation To ensure that the national calibration system is implemented in a rigorous manner third party accreditation for calibration and test laboratories is essential. Accreditation ensures that the calibration laboratory meets all the necessary requirements so that it can issue calibration certiﬁcates with credible traceability and uncertainty claims. The process of accreditation is carried out with reference to the international standard ISO/IEC 17025 (2005) [53]. The purpose of the standard is to ensure that the accredited laboratory has established, among other issues, the following important items:

the necessary trained personnel, adequate procedures and processes, laboratory premises appropriate for the calibrations, suitable reference standards for traceability purposes, prescribed methods for handling and processing the calibration data, including the calculation of uncertainties, equipment of sufﬁcient quality to perform the calibrations and uncertainties that have been derived in accordance with ISO Guide to the Expression of Uncertainty in Measurement [54], in a way which is traceable to the stated standard. Third party accreditation adds a level of independence which selfassessment may not have and increases the conﬁdence of the user community in the performance and results given by accredited laboratories.

4.5. Establishing the validity of world metrology: the BIPM ‘Mutual Recognition Arrangement’ For most users of metrology, the highest point of contact is the NMI of their country. In the United Kingdom this is the NPL; in the United States

Temperature Fundamentals

55

this is the NIST. Contact details of the NMIs can be found on the BIPM website (http://www.bipm.fr/). The issue then arises of how one can be conﬁdent in the equivalence of the measurement standards and calibration services set up and offered by the NMIs on a global scale. The role of the BIPM is critical here because it provides the framework for demonstrating worldwide equivalence of measurements and their traceability to the SI. In the past, uniformity of unit realisation and the level of mutual equivalence were established by bi- or multi-lateral comparisons among interested NMIs, and the corresponding bilateral declarations of equivalence. Comparisons of particular national realisations of the units were arranged and the results published. More recently, this rather piecemeal approach has been formalised under an initiative led by the BIPM known as the MRA [52]. Under the MRA, participating NMIs are obliged to undertake key comparisons of their measurement capabilities so as to determine the degree of equivalence of their individual national standards with those of other NMIs. The results of these key comparisons are accessible through the BIPM key comparison database (KCDB). One of the key outcomes of the MRA process was to ensure that mutual recognition of NMI calibration and measurement capabilities (CMCs) was achieved. Under the MRA, the degree of equivalence of national standards has begun to be demonstrated on a global scale, so users of national services can be conﬁdent that they are equivalent to those of other NMIs around the globe, to the level given in the KCDB.

5. Uncertainty Estimation in Non-Contact Temperature Scale Realisation and Dissemination Generally speaking, uncertainty in a measurement result is an estimate of the dispersion or spread of values within which the correct result (‘true value’) is expected to lie. Estimating and reporting uncertainties is crucial in sharing and publishing data, calibrating instruments, comparing data with theory and increasing conﬁdence in measurement results. Uncertainties are required in stating traceability at each link of the unbroken chain of comparisons from the workplace to the calibration laboratory and to the NMI. The Guide to the Expression of Uncertainty in Measurement (GUM) [54] sets forth the principles of estimating, combining and reporting uncertainties of measurement and calibration. In the context of this book, these uncertainty principles can be applied to primary realisations of the temperature scale and their dissemination to secondary and tertiary laboratories and then to users of these calibrated instruments in general application worldwide. Discussion for the uncertainty in scale realisations will be separated into the realisations above

56

Graham Machin and Benjamin K. Tsai

and below the silver point. Finally, the uncertainties associated with the dissemination of the scale after the realisation will be discussed in the context of determining the uncertainty of the calibration of a scale transfer instrument.

5.1. Basic principles of uncertainty analysis In general, the measurement equation reveals how the true value of the quantity being measured, y, the measurand, relates to other quantities, x1, x2, y, xn: y ¼ f ðx1 ; x2 ; . . . ; xn Þ

(11)

Each quantity could either be an independent variable, couched in an analytical function that can be written down, or be a critical variable not available in analytical form but whose uncertainty needs to be estimated by some other method. According to the GUM [54], uncertainties are classiﬁed as either Type A or Type B. Type A uncertainties are those which are due to random variations in repeated independent measurements undertaken with identical conditions. They are estimated by statistical means, such as calculating the standard deviations of a series of independent observations or least-squares ﬁtting. Assuming a normal distribution, the Type A standard uncertainty u of a speciﬁc independent variable t, equivalent to xi in Equation (11), is often evaluated as the standard deviation of the mean, ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sP N 2 ¯ i¼1 ðt t i Þ uðt i Þ ¼ NðN 1Þ

(12)

of N repeated measurement values. Type B uncertainty components arise from systematic effects in the measurement process, for example, due to instrument bias, calibration uncertainties, external inﬂuences and environmental conditions, and they are evaluated by methods other than statistical ones. All such effects need to be identiﬁed and investigated, and corrections applied where necessary. Each Type B component of uncertainty is assessed using appropriate means, which may include scientiﬁc judgment, previous measurement data and investigations, calibration certiﬁcates, experience, manufacturer’s speciﬁcations (with care), calibration and other reports, and handbooks of reference values. Often Type B components are most conveniently estimated to lie with equal probability anywhere within stated upper and lower limits. Thus the probability distribution is considered to be rectangular rather than normal, or Gaussian. In this case, if the upper and lower limits are a+ and a, the half interval is a ¼ (a+a)/2, and the standard uncertainty for this component is a/O3.

57

Temperature Fundamentals

Each standard uncertainty is the uncertainty in y due to xi. When the standard uncertainties are combined, the result represents the standard uncertainty in y due to all xi values for a coverage factor of k ¼ 1. Ignoring complicating factors such as correlations (interested readers can ﬁnd out more in Ref. [54]), the standard uncertainties are combined in quadrature to yield the ﬁnal total standard uncertainty, u2 ðyÞ ¼

n X

c 2 ðxi Þu2 ðxi Þ

(13)

i¼1

by employing the sensitivity coefﬁcients, c(xi). These coefﬁcients indicate how a particular standard uncertainty, u(xi), propagates, in this case with temperature. It is important to determine these coefﬁcients because one may know the uncertainty value at a particular temperature (e.g. the uncertainty in the realisation of the silver point) but often may not know this particular uncertainty at a different temperature. The way to calculate this is by determining the appropriate sensitivity coefﬁcient. These are generally determined by calculating the partial derivative of the measurand with respect to the quantity, xi. One speciﬁc example can be found in the scale realisation above the silver point, in which there are basically three sensitivity coefﬁcients. The ﬁrst sensitivity coefﬁcient, cðxi Þ ¼

T 90 2 T 90 ðXÞ

(14)

applies to uncertainties in the reference temperature, for example due to impurities, temperature drop, plateau identiﬁcation, repeatability and stability. The second sensitivity coefﬁcient, T 90 T 90 1 cðxi Þ ¼ l T 90 ðXÞ

(15)

applies to the inﬂuence of wavelength, reference detector, scattering and polarisation, repeatability of calibration, drift, out-of-band transmittance and interpolation and integration effects. The third sensitivity coefﬁcient, cðxi Þ ¼

lT 290 c2

(16)

involves the factors of emissivity, size-of-source effect, non-linearity, drift, ambient conditions, gain ratios, repeatability and positioning. Since all three sensitivity coefﬁcients depend on T, the combined uncertainty can be calculated for different temperatures to gauge the importance of temperature on the combined temperature uncertainty. Furthermore, with the aid of the sensitivity coefﬁcients, each standard uncertainty can be calculated and ranked to determine the most signiﬁcant factors to the uncertainty. General information about sensitivity coefﬁcients are given in

58 Table 3

Graham Machin and Benjamin K. Tsai

Representative conﬁdence levels for a normal distribution.

Coverage factor k

Conﬁdence level (%)

1.000 1.645 1.960 2.000 2.576 3.000

68.27 90.00 95.00 95.45 99.00 99.73

Ref. [54], while examples of how to calculate the sensitivity coefﬁcients for scale realisations are given in Refs. [55257]. The ﬁnal step is to report the expanded uncertainty, U ¼ kuðyÞ

(17)

where k is the coverage factor. A coverage factor of k ¼ 1 corresponds to one standard deviation or a conﬁdence level of 68.27% for a normal distribution. It is common to use a coverage factor k ¼ 2 in industry and calibration. Other coverage factors and the respective conﬁdence levels for a normal distribution are shown in Table 3.

5.2. Uncertainty in scale realisation above the silver point Scale realisations above the silver point are different from those below the silver point due to the different methods entailed and prescribed by the ITS-90. Above the freezing point of silver, the temperature scale is determined by reference to one of three freezing points (silver, gold or copper) through the ratios of spectral radiances as prescribed by Equation (10). Although the ITS-90 does not explicitly prescribe a particular wavelength, typically a wavelength between 650 and 655 nm is used. In addition, even though the ITS-90 also does not recommend any speciﬁc method of maintaining the scale through Equation (10), there are three general schemes of realising the temperature scale above the silver point: Scheme 1: Establishing a series of current and radiance temperature values for the thermometer, with a reference tungsten strip lamp current corresponding to the radiance temperature of the ﬁxed point. Scheme 2: Measuring signal ratios between the source at T90 and the reference lamp at the ﬁxed-point temperature, with a reference lamp current corresponding to the radiance temperature of the ﬁxed point. Scheme 3: Establishing a series of temperatures as a function of the output signals of the thermometer, with the ﬁxed-point calibration maintained on the thermometer.

Temperature Fundamentals

59

The ITS-90 simply deﬁnes the scale equation in terms of Planck’s law, referring to the Supplementary Information for ‘practical details’. In general, one needs an instrument with good linearity and known spectral response, which can measure radiances in the required temperature range at a suitable chosen wavelength. Given a calibration at one of the speciﬁed ﬁxed points, Scheme 3 can be applied to derive radiance temperatures. There are alternatives, such as building up the scale based on lamps by radiance doubling. Schemes 1 and 2 are variants of Scheme 3, although the usage of the lamp is based on convenience and avoids overusing the ﬁxed point. Although these three schemes result in different uncertainty budgets, there are four basic common uncertainty components related to the three basic operations of scale realisations: 1. Fixed-point calibration in determining a value of Ll[T90(X)] in Equation

(10). 2. Spectral characterisation of the thermometer to ﬁx the value of l. 3. Determination of the output signal (this is related to quantities such as

linearity and drift). 4. Use of a tungsten strip lamp as a radiance source at some point in the

primary scale realisation process. Typical uncertainties for the scale realisation above the silver point are shown in Figure 7 [56]. A brief summary of the uncertainty components for each of these basic operations will be given here, but the reader is referred to Refs. [55,56] for more details. In the ﬁxed-point calibration operation, the presence of impurities may affect the uncertainty. Uncertainties also arise in the graphite cavity effective emissivity due to uncertainties in the wall emissivity and other factors. Furthermore, there is uncertainty in the temperature drop at the graphite crucible cavity bottom between the metal surface and inner cavity wall, mainly caused by a lack of knowledge of the thermal conductivity of graphite. The temperature uncertainty is also affected by the plateau identiﬁcation and the repeatability of the freezing curves. Overall the uncertainties associated with the ﬁxed point are small (o50 mK) compared to some of those considered below. Various factors which affect the uncertainty in the spectral responsivity include wavelength accuracy, reference detector signal-to-noise ratios and responsivity, scattering and stray light effects, polarisation of calibration sources, calibration repeatability due to alignment and ambient temperature issues, drift due to ﬁlter ageing, ﬁlter out-of-band transmittance, and integration and interpolation issues. Overall, the dominant uncertainty components in determining the wavelength are the uncertainty in the measurement system used to determine the transmission proﬁle of the interference ﬁlter and the

60

Graham Machin and Benjamin K. Tsai

Figure 7 Uncertainty estimates for three different schemes used for realising the ITS-90 above the silver point: Scheme 1: The ﬁxed-point radiance is transferred to a reference tungsten strip lamp, and T90 values are then established and maintained on the lamp by measurement of radiance ratios. Scheme 2: The ﬁxed-point radiance is transferred to a reference tungsten strip lamp, and T90 temperatures of any radiance source are determined by the ITS-90 deﬁning equation and signal ratios between the source and the reference lamp. Scheme 3: The ﬁxed-point calibration is maintained on a radiation thermometer, and T90 values are established as a function of output signals of the thermometer. The step in uncertainty in Scheme 1 is caused by the necessity, above 1,7001C, to use a gas-ﬁlled tungsten strip lamp (which is inherently more unstable) instead of an evacuated one [56].

uncertain secular or temporal drift associated with the performance of the interference ﬁlter. The uncertainty in the output signal is inﬂuenced by the size-of-source effect4 due to different ambient surroundings and source and thermometer characteristics, detector non-linearity, drift of ampliﬁer and of the detector sensitivity, ambient conditions of temperature and humidity, gain ratios, signal noise, detector short-term stability and the voltmeter resolution. The estimation of non-linearity is potentially the largest source of uncertainty in this category. Finally, where a lamp is used additional components of uncertainty consist of the lamp current, drift and stability, base and ambient temperature, positioning and polynomial ﬁtting. The secular or temporal stability of the 4

The size of source effect is the stray light scattered into or out of the measurement spot by imperfections in the optical system; these could be diffraction, dust on the front lens, imperfectly formed lenses, etc.

Temperature Fundamentals

61

lamp is the biggest uncertainty component. In total, at least 23 independent sources of uncertainty are identiﬁed and need to be combined to determine the ﬁnal uncertainty value. The analysis shows that the dominant contributions to the overall uncertainty are those associated with determining the operating wavelength of the pyrometer, the linearity of the detector and the drift associated with the interference ﬁlter used to ﬁx the pyrometers operating wavelength [55,56]. Overall typical standard uncertainties in scale realisation could be of order 1 K at 2,500 K. The best that could be achieved is about a factor of 5 better, but in practice this is very difﬁcult to attain.

5.3. Uncertainty in scale realisation below the silver point For scale realisations below the silver point, the ITS-90 is still applicable, but two different approaches have been used for scale realisation, both of which have already been outlined in Section 4.1.2. One relies on using variable-temperature blackbody radiators (the VTBB approach) which employ contact sensors calibrated directly traceable to ITS-90. The other approach uses an interpolating radiation thermometer calibrated against a series of ITS-90 ﬁxed-point blackbody radiators (the FPBB approach). Both approaches have different sources of uncertainty [57,58]. However, the VTBB approach is the more general method, since the FPBB approach, due to limitations in detector technology, can really only be used down to the indium point, whereas the VTBB approach can easily go to 501C or lower. For the VTBB approach, 18 uncertainty components have been identiﬁed. The ﬁrst eight components deal with the blackbody source. The uncertainty due to calibration temperature is composed of uncertainties from the reference thermometer calibration, the calibrated thermometer stability and the measured temperature uncertainty. The uncertainty due to isothermal blackbody emissivity is affected by the intrinsic emissivity of the cavity wall, geometrical factors and machining imperfections. For VTBBs that cannot be guaranteed to be isothermal, the uncertainty due to the nonisothermal blackbody emissivity can be estimated by a modiﬁed summation method or Monte Carlo technique. Depending on the temperature and wavelength, the uncertainty due to reﬂected ambient radiation is signiﬁcant for longer wavelengths and lower temperatures. The uncertainty due to radiant heat exchange through the aperture can be attributed to the temperature drop at the cavity backwall between the inner cavity wall and the metal surface (in the FPBB approach), or the contact sensor (in the VTBB approach). Convection losses are due to ventilation from the ambient air, especially for larger apertures. The uncertainty due to cavity bottom uniformity can be caused by blackbody cavity design features, homogeneity of the backwall coating material, conductive and convective heat ﬂows, and

62

Graham Machin and Benjamin K. Tsai

radiation loss through the cavity aperture. The inﬂuence of ambient conditions on the temperature uncertainty comes from the noise of the blackbody radiance temperature. The next seven uncertainties are associated with the radiation thermometer. The size-of-source effect occurs because light can be scattered into the detector solid angle and scattered out of the solid angle. This effect can be signiﬁcant when the radiation thermometer is calibrated using one source and then used to measure a source of a different size or uniformity. The uncertainty due to detector non-linearity arises from the properties of the detector or the electronic circuit and can be deﬁned as the relative difference between the measured power and the actual power. The uncertainty due to the reference temperature is more pronounced at longer wavelengths and temperatures closer to ambient. The uncertainty due to ambient temperature includes changes in background radiation, changes in detector responsivity due to detector temperature changes, temperature dependence of the ampliﬁer and temperature shifts in the ﬁlter wavelength. Atmospheric absorption by gases, such as water vapour and carbon dioxide, can contribute to the total uncertainty. The uncertainty due to gain ratios can be assessed from three sources: short-term stability of a single gain, stability of a single gain and determination of each gain ratio. The contribution of the noise uncertainty can be signiﬁcant when the temperature measurements are near the lower limit of the radiation thermometer. Three other uncertainties include those due to interpolation errors, drift and the unknown temperature. Interpolation errors arise from the exact form of the polynomial used to interpolate the temperatures between the calibration points. The drift can be caused by many factors including changes in the window transmission, mirror reﬂectance, ﬁlter transmission, detector sensitivity and mean wavelength. The in-use uncertainty occurs when a calibrated radiation thermometer is used to measure an unknown temperature and can include non-linearity, gain ratio ﬂuctuations and use of different blackbody sources. In summary for the VTBB approach, the major components of uncertainty are the cavity backwall uniformity effect, convection, nonisothermal blackbody emissivity, cavity backwall heat exchange at higher temperatures and non-linearity at lower temperatures. When the FPBB approach is used, there are 15 uncertainty components. Thirteen of these components are common with the VTBB approach. Ten components are those associated with the radiation thermometer or other effects. Three components are the same as those in the VTBB approach: isothermal blackbody emissivity, reﬂected ambient radiation and cavity backwall heat exchange. The remaining two components are associated with the ﬁxed points and are due to the effects of impurities and plateau identiﬁcation. The major uncertainty components

Temperature Fundamentals

63

for the FPBB approach are those due to plateau identiﬁcation, noise and cavity backwall heat exchange (at higher temperatures). In general, the FPBB approach results in an overall lower uncertainty than the VTBB approach. However, the VTBB is more ﬂexible because it allows more choice of temperatures and temperature measurement procedures. In reality, a combination of the two approaches is ideal in providing adequate cross-checks between the two approaches and yields the optimal combination of ﬂexibility and lower uncertainties.

5.4. Uncertainty in scale dissemination Temperature scale dissemination is the process of calibrating a transfer artefact (a radiation thermometer or a blackbody source) against a primary standard (e.g. a cryogenic radiometer or a well-characterized blackbody) and transferring the scale to a blackbody source or to another radiation thermometer using the calibrated artefact to make accurate temperature measurements. Appropriate scale dissemination via calibrated radiation thermometers or blackbodies is important because accuracy depends on achieving proper scale dissemination and traceability to the national/ primary standard is at stake. A proper uncertainty analysis is critical in listing all factors that contribute to the overall uncertainty, in conﬁrming the propagation of the uncertainty from the primary standard to the test item, and in identifying the major uncertainty components in a calibration process. Two examples from radiation thermometry are given in the following two paragraphs to illustrate the uncertainty propagation for dissemination of the temperature scale from a primary standard [39]. A sample budget for calibration of a radiation thermometer above the silver point is shown in Table 4. The ﬁrst ﬁve rows of uncertainty values are expressed for a coverage factor of k ¼ 1, along with the uncertainty type (Type A or Type B). Using a coverage factor of k ¼ 1, the combined standard uncertainty is shown for the ﬁve temperatures in the next row. Finally, the expanded uncertainty coverage factor of k ¼ 2 is shown in the last row. The ﬁve components of uncertainty for this dissemination of the temperature scale are the calibrated VTBB temperature, the mean effective wavelength of the transfer radiation thermometer, the blackbody uniformity, the temperature of the test radiation thermometer and the disparity between the 1990 NIST radiance temperature scale5 relative to the thermodynamic temperature scale. In general, the dominant uncertainty component in disseminating the scale to the secondary radiation thermometer is from the uncertainty in the test radiation thermometer itself. 5

The 1990 NIST radiance temperature scale has equivalent temperature values to the ITS-90, because its basis is a gold-point blackbody source that has exactly the same temperature value as the assigned ITS-90 value.

64 Table 4 point.

Graham Machin and Benjamin K. Tsai

Sample budget for calibration of a radiation thermometer above the silver

Source of uncertainty

Type

Standard uncertainties (k ¼ 1) (1C) Temperature (1C) 900

1,100 1,500 1,900 2,700

B

0.1

0.1

0.2

0.3

0.5

B

0.0

0.0

0.0

0.1

0.1

B A

0.1 0.4

0.1 0.3

0.1 0.4

0.1 0.3

0.1 0.6

B

0.1

0.1

0.2

0.3

0.5

Combined standard uncertainty Coverage factor, k ¼ 2

0.4

0.4

0.5

0.5

1.0

Expanded uncertainty of test with respect to SI units

0.9

0.8

1.0

1.1

2.0

1. Calibration of the variabletemperature blackbody relative to the 1990 NIST Radiance Temperature Scalea 2. Mean effective wavelength measurement for the NIST Photoelectric Pyrometer 3. Blackbody uniformity 4. Test thermometer measured valuea 5. 1990 NIST Radiance Temperature Scale relative to the Thermodynamic Temperature Scale

a

Please refer to page Footnote 5.

Table 5 Sample budget for calibration of a radiation thermometer below the silver point. Factor

1001C

2001C

4001C

6001C

8001C

Reference blackbody temperature Reference blackbody emissivity Ambient background effect Test thermometer measured value

0.00 0.01 0.02 0.20

0.04 0.01 0.02 0.20

0.06 0.00 0.00 0.15

0.04 0.01 0.00 0.15

0.06 0.01 0.00 0.40

Expanded uncertainty, k ¼ 2

0.20

0.20

0.16

0.16

0.40

Another sample budget is the calibration of a radiation thermometer below the silver point in Table 5. The four uncertainty components in this budget are the reference blackbody temperature, the reference blackbody emissivity, the ambient background effect and the test thermometer

Temperature Fundamentals

65

measured value. The major component of uncertainty at all temperatures is the uncertainty due to the test thermometer measured value.

6. Possible Future Approaches to High-Temperature Scale Realisation and Dissemination 6.1. The mise-en-pratique for the deﬁnition of the kelvin The MeP-K has already been discussed in Section 3.4 [36]. Essentially, it allows a more ﬂexible approach to temperature scale realisation and dissemination. Currently, the core of the MeP-K is the ITS-90, but it is envisaged that over time a number of thermodynamic temperature measurement approaches will be included as improvements in techniques and technologies become available. In the long term, incremental improvements can be made to temperature measurement and dissemination without changing ITS-90. This is particularly desirable from the point of view of the user community who will gain all the beneﬁts of improved temperature measurement but none of the disadvantages that would be caused by a complete redeﬁnition. It is likely that the ﬁrst improvement which potentially leads to T, instead of T90 being realised and disseminated, is the application of primary radiation thermometry above the silver point. Currently, the scale in this temperature range in almost universal use is the ITS-90 (Section 3.3.2), that is, Planck’s law in ratio form referenced to a freezing point blackbody of Ag, Au or Cu. However, recent improvements in absolute radiometry [23] and developments of high-temperature ﬁxed points [25] have set the scene to move to a more fundamental basis for temperature scale realisation at these elevated temperatures.

6.2. Possible future non-contact thermometry scale realisation methods There are three possible routes for non-contact thermometry at high temperatures described in the following subsections.

6.2.1. ITS-90 [no change] The ﬁrst is to continue to realise and disseminate ITS-90. This has the advantage that it is relatively simple to establish and maintain. However, the disadvantage is that the uncertainties in realisation propagate upwards approximately as (T/Tref)2, as described in Section 5.2. Clearly, an

66

Graham Machin and Benjamin K. Tsai

improved result would be obtained if a new thermodynamic value of the reference temperature Tref is used. 6.2.2. Absolute primary radiometry The second route is to use absolute radiometry. It is certainly possible to implement this with lower uncertainties than are currently possible with ITS-90 [23,59]. This approach could be followed by a number of laboratories that have made signiﬁcant investment in primary radiometry facilities and could lead to a factor of 325 decrease in the uncertainty of scale realisation, as well as disseminating T rather than T90. The main disadvantage is that the high cost of establishing and maintaining primary radiometry facilities militates against the widespread use of this approach. In addition, regular checking of key measured parameters, which is often quite an involved process, such as redetermination of the ﬁlter radiometer absolute spectral response, is required to ensure continuing reliable scale realisation. 6.2.3. High-temperature ﬁxed-point interpolation A third approach to scale realisation could be followed where hightemperature ﬁxed points of known thermodynamic temperature could be used to realise and disseminate a thermodynamic scale with similar uncertainties to absolute radiometry. High-temperature ﬁxed points are based on the eutectic reaction of metal (or metal carbide) and carbon. The eutectic melting point is a unique and well-deﬁned point in the material phase diagram which provides a reproducible ﬁxed point. Eutectic ﬁxed points above the Cu point are the subject of intense research, the aim of which is that in the near future they could be used as routine temperature standards [60]. It has been shown [61,62] that if the temperature of these new hightemperature ﬁxed points can be established with sufﬁcient precision, then a thermodynamic temperature scale can be realised by interpolation between a series of these ﬁxed points. Alternatively, their temperatures can be deﬁned by convention with zero uncertainty (as are the current ITS-90 ﬁxed points) and a conventional scale close to thermodynamic could be realised with low uncertainties.

6.3. A practical implementation of primary thermometry The MeP-K can be used to allow alternative approaches to realising and ultimately disseminating high temperatures with uncertainties lower than are currently possible. An addendum to the MeP-K [36] will be formulated that proposes the direct dissemination of T at high temperatures by absolute radiometry. An annex to the same document would recommend the use of

Temperature Fundamentals

67

interpolation between high-temperature ﬁxed points whose temperatures have been determined a priori by absolute radiometry. It can therefore be seen that the MeP-K is a very ﬂexible document with ITS-90 remaining intact while allowing for improvements to temperature measurement to be disseminated to the user community in an invisible way. It is clear that through proper deployment of the MeP-K many of the perceived shortcomings of the ITS-90 can be adequately ameliorated.

7. Summary Temperature, an intensive quantity of measurement, is one of the seven base units of the SI. Thermodynamic temperature can be determined through a variety of primary methods, including constant-volume gas thermometry, acoustic thermometry and radiation thermometry. Since realisation of a primary temperature scale is problematic, from 1927 a succession of internationally agreed practical temperature scales have been established, whose values are as close to thermodynamic temperature as the current state of primary temperature measurement would allow. The current temperature scale, the ITS-90, deﬁnes the temperature scale from 0.65 K up to and beyond 1357.77 K by applying vapour pressure thermometry, interpolating gas thermometry, platinum resistance thermometry and radiation thermometry. One of the main topics of this chapter is the establishment of a reliable non-contact thermometry scale. Above the silver point, Planck’s law is used in ratio form referenced to a deﬁning ﬁxed point of Ag, Au or Cu. Below the silver point, the link to ITS-90 is established either through calibrated temperature sensors and well- designed blackbody radiators or through an interpolating radiation thermometer calibrated against blackbody cavities constructed from the deﬁning ﬁxed-point materials. Any temperature scale must be easily accessible, disseminated in terms of calibration processes, traceable to the SI units and amenable to accreditation to international standards. To be valid, meaningful and practical, temperature measurements must be accompanied by a thorough uncertainty analysis, which appropriately estimates the spread in measurements. Uncertainty analyses are well established for scale realisations below and above the silver point. The future of radiation thermometry may lead to one of three possible scenarios in the temperature scale: continuation of the ITS-90 with basically no change, transition to absolute primary radiometry or application of hightemperature ﬁxed-point blackbodies. To facilitate this change, it is expected that the MeP-K will be applied to allow a smooth transition and ensure equivalence of the various allowable alternative methods.

68

Graham Machin and Benjamin K. Tsai

ACKNOWLEDGEMENTS The authors gratefully acknowledge Richard Rusby, Emma Woolliams and Charles Gibson for reviewing the manuscript. The authors also thank Joachim Fischer for allowing the use of his ﬁgures in this chapter. Extracts from the International Vocabulary of Metrology 2 Basic and General Concepts and Associated Terms (VIM), 3rd edition, JCGM 200:2008 (www.bipm.org/ en/publications/guides/vim.html) are published with permission of the Director of the International Bureau of Weights and Measures (BIPM).

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Temperature Fundamentals

[18] [19]

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69

H. W. Yoon, D. W. Allen, C. E. Gibson, et al., ‘‘Thermodynamic-Temperature Determinations of the Ag and Au Freezing Temperatures using a Detector-Based Radiation Thermometer,’’ Appl. Opt. 46, 287022880 (2007). B. K. Tsai, B. C. Johnson, and R. D. Saunders, ‘‘Evaluation of the Radiation Characteristics of a High-Temperature Blackbody,’’ in Proceedings of the SPIE, edited by M. Strojnik and B. F. Andresen, The International Society for Optical Engineering, San Diego, CA, pp. 5142523 (1995). D. W. Allen, S. W. Brown, G. P. Eppeldauer, et al., ‘‘The Realisation and the Dissemination of the Detector-Based Kelvin,’’ in Proceedings of the 9th International Symposium on Temperature and Thermal Measurements in Industry and Science, edited by D. Zvizdic, L. G. Bermanec, T. Stasic, T. Veliki, LPM/FSB, Zagreb, pp. 57270 (2004). D. W. Allen, G. P. Eppeldauer, C. E. Gibson, and H. W. Yoon, ‘‘Comparison of the NIST Radiance Temperature Scale with the Detector-Based Radiance Temperature Scale from 1200 K to 2800 K,’’ in Proceedings of the 9th International Symposium on Temperature and Thermal Measurements in Industry and Science, edited by D. Zvizdic, L. G. Bermanec, T. Stasic, and T. Veliki, LPM/FSB, Zagreb, pp. 1272132 (2004). K. Anhalt, J. Hartmann, D. H. Lowe, G. Machin, M. Sadli, and Y. Yamada, ‘‘Thermodynamic Temperature Determinations of Co2C, Pd2C, Pt2C and Ru2C Eutectic Fixed-Point Cells,’’ Metrologia 43, S782S83 (2006). H. Yoon, ‘‘The Realization and the Dissemination of Thermodynamic Temperature Scales,’’ Metrologia 43, S222S26 (2006). J. Hartmann, D. R. Taubert, and J. Fischer, ‘‘Measurement of T-T90 down to Zinc Point Temperatures with Absolute Filter Radiometry,’’ in Proceedings of the 8th International Symposium on Temperature and Thermal Measurements in Industry and Science, edited by B. Fellmuth, J. Seidel, and G. Scholz, VDE Verlag GMBH, Berlin, pp. 3772382 (2002). E. Woolliams, G. Machin, D. H. Lowe, and R. Winkler, ‘‘Metal (Carbide)2Carbon Eutectics for Thermometry and Radiometry: A Review of the First Seven Years,’’ Metrologia 43, R112R25 (2006). N. P. Fox and J. P. Rice, ‘‘Absolute Radiometers,’’ in Optical Radiometry, edited by A. C. Parr, R. U. Datla, and J. L. Gardner, Elsevier Academic Press, New York (2005). W. R. Blevin and W. J. Brown, ‘‘A Precise Measurement of the Stefan2Boltzmann Constant,’’ Metrologia 7, 15229 (1971). T. J. Quinn and J. E. Martin, ‘‘A Radiometric Determination of the Stefan2Boltzmann Constant and Thermodynamic Temperatures between 401C and +1001C,’’ Philos. Trans. R. Soc. London, Ser. A 316, 852189 (1985). J. E. Martin, T. J. Quinn, and B. Chu, ‘‘Further Measurements of Thermodynamic Temperature Using a Total Radiation Thermometer: The Range 21301C to +601C,’’ Metrologia 25, 1072112 (1988). I. M. Mills, P. J. Mohr, T. J. Quinn, B. N. Taylor, and E. R. Williams, ‘‘Redeﬁnition of the Kilogram, Ampere, Kelvin, and Mole: A Proposed Approach to Implementing CIPM Recommendation 1 (CI-2005),’’ Metrologia 43, 2272246 (2006). J. Fischer, S. Gerasimov, K. D. Hill, et al., ‘‘Preparative Steps Towards the New Deﬁnition of the Kelvin in Terms of the Boltzmann Constant,’’ Int. J. Thermophys. 28, 175321765 (2007). H. Preston-Thomas, ‘‘The International Temperature Scale of 1990 (ITS-90),’’ Metrologia 27, 3210 and 107 (1990). H. Preston-Thomas, P. Bloembergen, and T. J. Quinn, Supplementary Information for the International Temperature Scale of 1990, BIPM, Se`vres, http://www.bipm.org/ (1990).

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R. Rusby, M. Durieux, A. L. Reesink, et al., ‘‘The Provisional Low Temperature Scale from 0.9 mK to 1 K, PLTS-2000,’’ J. Low Temp. Phys. 126, 6332642 (2002). J. Fischer, M. DePodesta, K. Hill, et al., Working Group 4 Report to CCT, BIPM Document CCT/08-13, http://www.bipm.org/ (2008). Consultative Committee for Thermometry (CCT), The Mise-en-Pratique for the Deﬁnition of the Kelvin, http://www.bipm.org/utils/en/pdf/MeP_K.pdf (2006). The International Committee for Weights and Measures (CIPM), Recommendation of the International Committee for Weights and Measures: Preparative Steps Towards New Deﬁnitions of the Kilogram, the Ampere, the Kelvin, and the Mole in Terms of Fundamental Constants, http://www.bipm.org/cc/CIPM/Allowed/94/CIPM-Recom1CI-2005EN.pdf (2005). H. C. McEvoy, G. Machin, R. Friedrich, J. Hartmann, and J. Hollandt, ‘‘Comparison of the New NPL Primary Standard Ag Fixed-Point Blackbody Source with the Primary Standards Fixed Point of PTB,’’ in Temperature: Its Measurement and Control in Science and Industry, edited by D. C. Ripple, B. C. Johnson, C. W. Meyer, et al., AIP, New York, pp. 9092914 (2002). C. E. Gibson, B. K. Tsai, and A. C. Parr, Radiance Temperature Calibrations, NIST Special Publication 250-43 (1998). R. D. Lee, H. J. Kostkowski, P. R. Chandler, T. P. Jones, J. Tapping, and H. Kunz, ‘‘Intercomparison of the IPTS 68 Above 1064 C by Four National Laboratories,’’ in Temperature: Its Measurement and Control in Science and Industry, edited by H. H. Plumb, ISA, New York, pp. 3772393 (1972). V. I. Sapritsky, B. B. Khlevnoy, V. B. Khromchenko, et al., ‘‘Blackbody Sources for the Range 100 K to 3500 K for Precision Measurements in Radiometry and Radiation Thermometry,’’ in Temperature: Its Measurement and Control in Science and Industry, edited by D. C. Ripple, B. C. Johnson, C. W. Meyer, et al., AIP, New York, pp. 6192624 (2002). C. E. Brookley and W. E. Llewellyn, ‘‘Determination of Blackbody Radiance at Temperatures Above 23001C,’’ in Temperature: Its Measurement and Control in Science and Industry, edited by J. F. Schooley, AIP, New York, pp. 119521199 (1992). Y. Yamada, N. Sasajima, H. Gomi, and T. Sugai, ‘‘High Temperature Furnace Systems for Realizing Metal-Carbon Eutectic Fixed Points,’’ in Temperature: Its Measurement and Control in Science and Industry, edited by D. C. Ripple, B. C. Johnson, C. W. Meyer, et al., AIP, New York (2002). J. Fischer, G. Neuer, E. Schreiber, and R. Thomas, ‘‘Metrological Characterisation of a New Transfer-Standard Radiation Thermometer,’’ in Proceedings of the 8th International Symposium on Temperature and Thermal Measurements in Industry and Science, edited by B. Fellmuth, J. Seidel, and G. Scholz, VDE Verlag GMBH, Berlin, pp. 8012806 (2002). J. Fischer, ‘‘Developments in Infrared Radiation Thermometry,’’ in Proceedings of the 7th International Symposium on Temperature and Thermal Measurements in Industry and Science, edited by J. F. Dubbeldam and M. J. de Groot, VSL, Delft, pp. 27-34 (1999). B. Chu and G. Machin, ‘‘The Upgraded NPL Blackbody Calibration Facility,’’ in Proceedings of the 7th International Symposium on Temperature and Thermal Measurements in Industry and Science, edited by J. F. Dubbeldam and M. J. de Groot, VSL, Delft, pp. 5432548 (1999). T. Ricolﬁ, M. Battuello, F. Girard, G. Machin, H. McEvoy, S. Ugur, and A. Diril, ‘‘Radiation Temperature Scales between the Indium and Silver Points Realised at IMGC, NPL and UME using a Fixed-Point Calibration Technique,’’ Meas. Sci. Technol. 13, 209022093 (2002). F. Sakuma and S. Hattori, ‘‘Establishing a Practical Temperature Standard by Using a Narrow-Band Radiation Thermometer with a Silicon Detector,’’ in Temperature: Its

Temperature Fundamentals

[49]

[50] [51] [52]

[53] [54]

[55] [56]

[57] [58] [59]

[60] [61]

[62]

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Measurement and Control in Science and Industry, edited by J. F. Schooley, AIP, New York, pp. 4212427 (1982). B. Chu, H. McEvoy, and G. Machin, ‘‘Use of an InGaAs Radiation Thermometer to Verify the Accuracy of the NPL Blackbody Reference Sources from 1561C to 6001C,’’ in Temperature: Its Measurement and Control in Science and Industry, edited by D. C. Ripple, B. C. Johnson, C. W. Meyer, et al., AIP, New York (2002). J. V. Nicholas and D. R. White, Traceable Temperatures, Wiley, New York (2001). ISO/IEC, International Vocabulary of Metrology 2 Basic and General Concepts and Associated Terms (VIM), 3rd ed., JCGM 200:2008, ISO (2007) (www.bipm.org/en/ publications/guides/vim.html). The International Committee for Weights and Measures (CIPM), ‘‘Mutual Recognition of National Measurement Standards and of Calibration and Measurement Certiﬁcates Issued by National Metrology Institutes,’’ http://www.bipm.org/ utils/en/pdf/mra_2003.pdf (2003). International Organization for Standardization (ISO)/International Electrotechnical Commission (IEC), General Requirements for the Competence of Testing and Calibration Laboratories, ISO/IEC 17025:2005 (2005). Working Group 1 of the Joint Committee for Guides in Metrology (JCGM/WG 1), Evaluation of Measurement Data 2 Guide to the Expression of Uncertainty in Measurement, JCGM 100:2008 (2008). (Available at: http://www.bipm.org/utils/common/ documents/jcgm/JCGM_100_2008_E.pdf) J. Fischer, M. Battuello, M. Sadli, et al., Uncertainty Budgets for Realisation of Scales by Radiation Thermometry, BIPM Document CCT/03-03, http://www.bipm.org/ (2003). J. Fischer, M. Battuello, M. Sadli, et al., ‘‘Uncertainty Budgets for Realization of ITS90 by Radiation Thermometry,’’ in Temperature: Its Measurement and Control in Science and Industry, edited by D. C. Ripple, B. C. Johnson, C. W. Meyer, et al., AIP, New York, pp. 6312638 (2002). J. Fischer, P. Saunders, M. Sadli, et al., Uncertainty Budgets for Calibration of Radiation Thermometers below the Silver Point, BIPM Document, Open Access, CCT-WG5/ docs-03, http://www.bipm.org/ (2008). P. Saunders, J. Fischer, M. Sadli, et al., ‘‘Uncertainty Budgets for Calibration of Radiation Thermometers below the Silver Point,’’ Int. J. Thermophys. 28, 106621083 (2008). H. J. Jung, ‘‘On the Determination of the Thermodynamic Temperature of High Temperature Blackbodies via ITS-90 or Alternative Methods,’’ in Proceedings of the 7th International Symposium on Temperature and Thermal Measurements in Industry and Science, edited by P. Marcarino, Levrotto, and Bella, Torino, pp. 235-244 (1997). G. Machin, P. Bloembergen, J. Hartmann, M. Sadli, and Y. Yamada, ‘‘A Concerted International Project to Establish High Temperature Fixed Points for Primary Thermometry,’’ Int. J. Thermophys. 28, 197621982 (2007). P. Bloembergen, Y. Yamada, N. Yamamoto, and J. Hartmann, ‘‘Realizing the HighTemperature Part of a Future ITS with the Aid of Eutectic Metal-Carbon Fixed Points,’’ in Temperature: Its Measurement and Control in Science and Industry, edited by D. C. Ripple, B. C. Johnson, C. W. Meyer, et al., IP, New York, pp. 2912296 (2002). P. Saunders and D. R. White, ‘‘Interpolation Errors for Radiation Thermometry,’’ Metrologia 41, 41246 (2004).

CHAPTER 3

Theory of Thermal Radiation and Radiative Properties Zhuomin M. Zhang1 and Bong Jae Lee2

Contents 1. Introduction 2. Thermodynamics of Blackbody Radiation 2.1. Radiance, heat ﬂux, irradiance, and exitance 2.2. Blackbody cavity and radiation pressure 2.3. Entropy and the Stefan2Boltzmann law 2.4. Planck’s law of blackbody radiation 3. Radiative Properties of Materials 3.1. Emissivity 3.2. Absorptivity 3.3. Kirchhoff’s law 3.4. Reﬂectivity 3.5. Effective or apparent emissivity 3.6. Atmospheric transmission effects 3.7. Semitransparent materials 4. Electromagnetic Wave Theory 4.1. Macroscopic Maxwell’s equation 4.2. Wave equation and wave propagation 4.3. Fresnel reﬂection and transmission coefﬁcients 4.4. Reﬂectivity and emissivity 5. Optical Properties 5.1. The Drude model for a conductor 5.2. The Lorentz oscillator model for a nonconductor 5.3. Optical constants of a semiconductor 6. Radiative Properties of Layered Structures 6.1. The ray-tracing method 6.2. Thin-ﬁlm optics

1

2

73 76 76 78 79 81 84 84 88 89 90 93 95 98 98 98 100 103 106 109 110 112 116 119 119 121

George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405, USA Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA 15261-3648, USA

Experimental Methods in the Physical Sciences, Volume 42 ISSN 1079-4042, DOI 10.1016/S1079-4042(09)04203-9

r 2010 Elsevier Inc. All rights reserved.

73

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Zhuomin M. Zhang and Bong Jae Lee

6.3. Radiative properties of multilayer structures 6.4. Thin ﬁlms on a thick substrate 7. Summary Acknowledgement References

123 125 128 129 129

1. Introduction Understanding the fundamentals of thermal emission and the radiative properties of materials is not only important to the establishment of radiation thermometry but also essential to the proper selection and application of radiometric temperature measurement techniques to industrial processes. This chapter describes the basic theory of thermal radiation and electrodynamics, as well as the deﬁnition and prediction of optical and radiative properties of solid materials. The nature of light has been a perplexing subject for ancient philosophers as well as for modern scientists. While earlier wave theory developed by Huygens in 1678 was able to explain light reﬂection and refraction, it was incomplete and was largely supplanted by the corpuscular theory of light formulated by Newton in 1704. However, Young’s doubleslit experiment in 1801 demonstrated interference phenomenon and hence the wave nature of light, as proposed by Huygens, reemerged as a description of light. Fresnel (178821827) further advanced the wave theory to explain polarization and interference effects. The discoveries of infrared (heat) radiation by Herschel in 1800 and ultraviolet (chemical) radiation by Ritter in 1801 broadened the understanding of the optical spectrum. Built on several separate relations, Maxwell in 1873 established the electromagnetic-wave theory, which was further conﬁrmed by Hertz’s discovery of radio waves due to electrical vibration. The Michelson2Morley experiment in 1887 demonstrated that light waves can travel in vacuum without a medium with a speed independent of direction. Electromagnetic waves cover a broad range of frequencies, and are transverse waves, as illustrated in Figure 1 for a linearly polarized plane wave propagating along the y-axis. Here, k is the wavevector that points in the direction of propagation and l is the wavelength. The electric ﬁeld vector E oscillates parallel to the z-axis and the magnetic ﬁeld vector H oscillates parallel to the x-axis. Note that E, H, and k form a right-handed triplet. The surfaces of constant phase are parallel to the x2z plane. The speed of propagation is related to the wavelength lm (in the medium) and frequency n by: c ¼ lm n

(1)

The frequency is independent of the medium in which radiation propagates, but the speed of light and the wavelength are inversely proportional to the

75

Theory of Thermal Radiation and Radiative Properties

z E

k

y x

H

Figure 1

Illustration of the propagation of a plane wave of wavelength l.

Frequency, (Hz) 1022

1020

1018

1016

1014

1012

1010

108

106

Thermal Radiation -ray

x-ray

Optical Radiation

Micro-wave Radio-wave

Visible UV 10−6

10−4

10−2

IR 100

102

104

106

108

Wavelength, (μm)

Figure 2 The electromagnetic-wave spectrum. The visible wavelengths are from about 380 to 750 nm. Thermal radiation covers the wavelength region from 0.1 to 1,000 mm.

refractive index n of the medium. Therefore, we have lm ¼ l/n and c ¼ c0/n, where l is used as the wavelength in vacuum and c0 is the speed of light in vacuum. The refractive index of air is about 0.03% greater than that of vacuum, and thus, it is often approximated to 1. It should be noted that c0 is a fundamental constant, which is ﬁxed to an exact value [1]; see Appendix A for fundamental and other physical constants. In the 1983 version of the International System (SI) of Units [2], the unit of length (meter) is deﬁned based on the speed of light in vacuum and the unit of time (second). In practice, monochromatic radiation refers to radiation that is conﬁned to a very narrow spectral band, such as that from lasers or spectral lines emitted by gases. Polychromatic or broadband radiation, such as that emitted from a furnace or a heated plate, can be viewed as a superposition of monochromatic radiation. The electromagnetic spectrum covers a large spectral range from g-rays to radio waves, as shown in Figure 2. The visible spectrum covers the wavelength region from about 380 to 750 nm.

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Zhuomin M. Zhang and Bong Jae Lee

Thermal radiation often refers to the spectral region at wavelengths from approximately 0.1 to 1,000 mm, which covers most of the radiation emitted by objects at temperatures from about 20 to 15,000 K. In this deﬁnition, thermal radiation includes part of the ultraviolet (UV) region and the entire visible (VIS) and infrared (IR) regions. On contrary, the term optical radiation is used rather loosely and often is taken to include UV, VIS, and IR spectral regions, from extreme-deep UV (l ¼ 10 nm) to very far-IR (l ¼ 1,000 mm). Nevertheless, the wave approach to the fundamental nature of light is only one side of the story. In 1900, Planck introduced the concept of energy quanta in order to derive the law of thermal radiation (a more detailed discussion on thermal radiation will be presented in the next section). In 1905, Einstein used the concept of energy quantization to explain the phenomenon of photoelectric emission. Later in 1924, de Broglie hypothesized that a physical particle could also exhibit wave nature. Wave2particle duality became an established foundation of quantum mechanics and more broadly modern physics. According to this theory, radiation is made of discrete energy quanta, called photons. The energy of an individual photon is proportional to its frequency, i.e., E ¼ hn

(2)

where h is the Planck constant, and the frequency n is in Hertz. Although photons are massless, they possess momentum in addition to energy. The momentum of each photon is given as: p¼

hn h ¼ c lm

(3)

In vector form, p ¼ _k, where _ ¼ h=2p is the reduced Planck constant and the magnitude of the wavevector is k ¼ 2p/lm. More details on the mechanism of radiation emission, as well as how radiation propagates inside a medium and how it interacts with materials, will be presented in the remainder of this chapter. Section 2 discusses the thermodynamic theory of blackbody radiation. Section 3 deﬁnes radiative properties of materials and the relationship between different properties. Section 4 elaborates on the electromagnetic-wave theory and how to apply the macroscopic Maxwell equations to describe wave propagation inside a medium as well as wave reﬂection and refraction at the interface between different media. Section 5 addresses models of the frequency-dependent dielectric functions of various solid materials in the optical frequency region. Section 6 presents methods to obtain radiative properties of layered structures. A brief summary is given in Section 7.

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Theory of Thermal Radiation and Radiative Properties

2. Thermodynamics of Blackbody Radiation 2.1. Radiance, heat ﬂux, irradiance, and exitance The spectral radiance is deﬁned as the radiant power dq from a pencil cone, as shown in Figure 3, from a surface area dA in the direction s^ (deﬁned by the zenith angle y and the azimuthal angle f), per unit wavelength interval, per unit solid angle, and per unit area projected onto the direction s^. Hence the spectral radiance is L l ðl; y; fÞ ¼

dq dA cosy dO dl

(4)

where dO is the element solid angle, and the unit of Ll is W/(m2 sr mm). Note that q is also called radiant ﬂux, denoted by the symbol F in some chapters of this book. Using the spherical polar coordinates shown in Figure 4, by deﬁnition, we have dO ¼

dAn rdy r siny df ¼ ¼ siny dydf r2 r2

(5)

The total radiance is the integral of the spectral radiance over all wavelengths, i.e., Z

1

L¼

L l ðlÞdl

(6)

0

It should be noted that in most thermal radiation books, L and Ll are called intensity and spectral intensity, respectively [3,4]. However, the proper metrological terminology for these expressions is spectral or total radiance as given in Equation (4) or (6), respectively. This is the convention that is observed here and throughout this book. z

sˆ

θ

dΩ dA

y

φ x

Figure 3

Illustration of radiance and the solid angle.

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Zhuomin M. Zhang and Bong Jae Lee

z θ

dθ dAn

r dΩ dA

y dφ φ

x

Figure 4

Spherical coordinates showing the solid angle dO ¼ sin hdhd/.

The radiative heat ﬂux is the rate of energy ﬂow per unit area across an element area dA, W/m2, i.e., q00 ¼

dq ¼ dA

Z

2p

Z

p=2

L cosy siny dydf f¼0

(7)

y¼0

Note that dA can be either a real surface or an imaginary surface. For a vacuum enclosure, dA can be any surface element inside the volume. Assuming the enclosure is at thermal equilibrium (i.e., with isothermal walls at steady state), photon energies exist uniformly throughout the volume. Because photons travel with a speed of c0 in all directions, it can be shown from kinetic theory that [5,6] q00 ¼

uc 0 4

and

q00l ¼

ul c 0 4

(8)

where u and ul are the total (with units of J/m3) and spectral energy densities (with units of J/m3 mm), respectively. The radiative heat ﬂux that leaves a surface is called exitance M, which includes emitted, reﬂected, and transmitted radiant power toward the hemisphere above the surface. If the radiance from a surface is independent of the direction, the surface is said to be diffuse and from Equation (7), one obtains, M ¼ pL

and

M l ¼ pL l

(9)

where M is the total exitance and Ml is the spectral exitance. In many books on heat transfer, the radiative heat ﬂux leaving the surface solely by emission is called the emissive power and the exitance is also called radiosity. The terms self-exitance (emission from a surface neglecting any reﬂected and transmitted radiation) and exitance (outgoing radiative heat ﬂux from a surface) are observed throughout this book.

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Theory of Thermal Radiation and Radiative Properties

The irradiance is the radiative heat ﬂux incident upon a surface from all directions above the surface. The symbols E and El are used, respectively, for the total and spectral irradiance. Equation (7) can be used to calculate irradiance if the incoming radiance is prescribed as a function of the polar and azimuthal angles of incidence.

2.2. Blackbody cavity and radiation pressure The enclosure concept helped greatly in establishing the laws of blackbody radiation. In 1862, Kirchhoff coined the term ‘‘black body,’’ which is spelled as one word (blackbody) in current practice. A blackbody absorbs all radiation impinged onto it and emits the maximum amount of radiation at any given temperature. Theoretically, a blackbody can be realized by a vacuum enclosure whose walls are kept at a constant temperature, as shown in Figure 5(a). The nature of radiation inside the blackbody cavity is independent of the properties of the wall materials. Since radiation carries energy and momentum, the absorption and reﬂection of radiation by an object will result in a force, which is termed as radiation pressure. Consider radiation in a thermal-equilibrium enclosure, the momentum exchange is related to the energy exchange divided by the speed of light. Hence, the radiation pressure p (in pascal or N/m2) on a perfectly reﬂecting wall can be written as: 2 p¼ c0

Z

2p

f¼0

Z

p=2

Lcos2 y siny dydf

(10a)

y¼0

Here, the factor 2 arises because the reﬂected ﬂux also contributes to the radiation pressure, and the additional cos y is needed since only the momentum component in the direction normal to the surface contributes to the radiation pressure. Because L is independent of the directions for a L (, T )

U, V, S, P L (, T ) L(T) LT()

L (, T ) T

(a)

Figure 5

T (b)

Illustration of (a) an isothermal enclosure and (b) a blackbody cavity.

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Zhuomin M. Zhang and Bong Jae Lee

blackbody cavity, Equation (10a) can be reduced to p¼

4pL u ¼ 3c 0 3

(10b)

where we have utilized Equations (8) and (9). In terms of the spectral properties, it can be shown that pl ¼ ul/3. In a blackbody cavity, p is independent of the properties of the wall and is uniform inside the medium. The same is true for the thermodynamic temperature T. Furthermore, the spectral distribution Ll should depend on l and T only. When a small hole is opened in a large blackbody cavity, as shown in Figure 5(b), the emitted radiation may be considered as diffuse and closely follows the blackbody distribution. For this to be true, the aperture should be small compared to the dimensions of the enclosure, while at the same time the diameter of the aperture must be much greater than the characteristic wavelength to avoid diffraction effects. In order for the cavity to behave like a blackbody, its smallest dimension should be much greater than the characteristic wavelength of thermal radiation [6].

2.3. Entropy and the Stefan2Boltzmann law Boltzmann in 1884 conceived a thought experiment involving radiation as the working ﬂuid in a piston-cylinder arrangement during a reversible process. The ﬁrst law of thermodynamics can be written as: dU ¼ dQ pdV

(11)

where U is the internal energy, V the volume of the cylinder or blackbody cavity, and dQ the inﬁnitesimal amount of heat transferred into the cylinder. In a reversible process, dQ ¼ TdS, so that the entropy change is dS ¼

dU þ pdV T

(12a)

Note that dU ¼ d(uV) ¼ Vdu + udV and p ¼ u/3. Thus, we have dS ¼

V du 4u dT þ dV T dT 3T

(12b)

The coefﬁcients on the right-hand side (RHS) are nothing but (qS/qT )V and (qS/qV )T, respectively. The derivative of (V/T )(du/dT ) with respect to V gives (q2S/qVqT ) ¼ (1/T )(du/dT ); and the derivative of (4u/3T ) with respect to T is (q2S/qTqV ) ¼ (4/3T )(du/dT )(4u/3T 2 ). Because (q2S/qVqT ) ¼ (q2S/qTqV ), one obtains du 4u ¼ dT T

(13)

Theory of Thermal Radiation and Radiative Properties

81

It can be seen that u is proportional to T to the fourth power or upT4. In terms of the radiance and exitance, it can be shown by applying Equations (8) and (9) that Lb ¼

sT 4 p

and

M b ¼ sT 4

(14)

where s is a fundamental constant called the Stefan2Boltzmann constant, and the subscript ‘‘b’’ denotes blackbody. This is the Stefan2Boltzmann law, ﬁrst postulated by Stefan in 1879 based on his analysis of previously measured data and later derived by Boltzmann in 1884 from thermodynamics. Boltzmann’s thought experiment also allowed him to conceptually design a perfect Carnot engine with radiation as the working ﬂuid. Subsequent analysis showed that the temperature that describes the radiation ﬁeld was the same as the thermodynamic temperature based on the Carnot cycle. Furthermore, it is consistent with the deﬁnition of thermodynamic temperature for an equilibrium system according to 1 ¼ T

@S @U

(15) V

Note that the entropy of blackbody radiation can be expressed as: S¼

16sT 3 V 3c 0

(16)

2.4. Planck’s law of blackbody radiation By treating the radiation ﬁeld as standing waves, Wien derived a formula for the spectral distribution of blackbody radiation in 1896, called Wien’s formula, which is applicable to the short-wavelength region of the blackbody spectrum but deviates toward long wavelengths. In the subsequent years, Planck attempted to rigorously derive the blackbody distribution and ﬁnally succeeded in 1900, by using statistical entropy to obtain the mean energy of an oscillator. The most important contribution of this work was the counter-intuitive assumption of discrete energies (hv) that the oscillator could possess. Also in 1900, Rayleigh used the equipartition theorem to show that the blackbody emission should be proportional directly to temperature but inversely to the fourth power of wavelength. Jeans derived a more complete expression in 1905. The Rayleigh2Jeans formula agreed with experiments at sufﬁciently high temperatures and long wavelengths, where Wien’s formula failed, but disagreed with experiments at short wavelengths. In 1924, Bose used quantum statistics (independent particles) to derive Planck’s law. The theory of quantum statistics was further advanced by Einstein to form Bose2Einstein statistics. The following derivation of

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Zhuomin M. Zhang and Bong Jae Lee

Planck’s law is based on the blackbody cavity shown in Figure 5(a) using quantum statistics. Under thermodynamic equilibrium, the spectral radiance Ll(l,T) must be a function of l and T only. Let nndn be the number of particles (photons) per unit volume in the frequency interval between n and n + dn, then nn dn ¼ f BE ðnÞDðnÞdn

(17a)

Here, fBE(n) is the Bose2Einstein distribution given by [6] f BE ðnÞ ¼

1 ehn=kT 1

(17b)

where k is the Boltzmann constant, and D(n) is the density of states. The number of states per unit volume in the momentum space with a spherical shell from p to p + dp is equal to the volume of the spherical shell divided by Planck’s constant to the third power. Therefore, DðnÞdn ¼

4pp2 dp 4pn2 dn ¼ c 30 h3

(17c)

8phn3 1Þ

(17d)

The spectral energy density is un ¼ 2hnnn ¼

c 30 ðehn=kT

where the factor of 2 arises from the two polarizations or polarization degeneracy. Note that Ln ¼ unc0/4p, we have the spectral radiance for a blackbody: L b;n ðn; T Þ ¼

2hn3 c 20 ðehn=kT 1Þ

(18)

This is Planck’s law of blackbody radiation. As Lb,ldl ¼ Lb,ndn the spectral radiation of blackbody in terms of wavelength can be written as: L b;l ðl; T Þ ¼

l

5

c 1L c =lT 2 ðe

1Þ

(19)

where c 1L ¼ 2hc 20 is the ﬁrst radiation constant for spectral radiance, and c2 ¼ hc0/k is the second radiation constant. Strictly speaking, the cavity must be evacuated, because any medium except for vacuum is dispersive and the speed of light depends on the frequency. For a gas or dielectric medium whose refractive index n can be assumed constant in the thermal radiation spectrum, Equation (19) can be modiﬁed by multiplying a factor n2 while treating l as the wavelength in vacuum [4,6], viz. L b;l ðl; T Þ ¼

n2 c 1L l5 ðec2 =lT 1Þ

(20a)

83

Theory of Thermal Radiation and Radiative Properties

Alternatively, if the wavelength is taken as that in the medium lm, Planck’s law becomes L b;lm ðlm ; T Þ ¼

c 1L 5 c 2 =ðnlm T Þ 2 n lm ½e

(20b)

1

since L b;lm ðlm ; T Þdlm ¼ L b;l ðl; T Þdl [4]. As shown in Figure 6 for blackbody radiation in vacuum, the Planck distribution has a peak and approaches zero at extremely short or long wavelengths. By setting the derivative of the Planck function to zero, we obtain the important Wien displacement law: lmax T ¼ c 3

(21)

where lmax is the wavelength corresponding to the peak radiance, and c3 is known as the third radiation constant. It can be seen from Figure 6 and Equation (21) that the maximum emission peak shifts toward shorter wavelengths as the temperature increases. Historically, Wien derived Equation (21) in 1893 by a thought experiment, similar to what Boltzmann did in deriving Equation (14). In the long wavelength limit when c2/lT{1, since ex1Ex for x{1, the RHS of Equation (19) approaches L b;l ðl; T Þ

c 1L T c 2 l4

(22)

109 Spectral Radiance (W m−2 μm−1 sr−1)

10000 K 107

Lb, (, T) =

6000 K

c1L 5

(ec2/T−1)

105

2000 K 103

1000 K 101

300 K

10-1 10-3 10-5 10-1

T = 50 K

100

101

102

Wavelength (μm)

Figure 6

Planck’s blackbody distribution at different temperatures.

103

84

Zhuomin M. Zhang and Bong Jae Lee

which is the Rayleigh2Jeans formula. When plotted on a log2log scale for a given temperature, Equation (22) is a straight line with a slope of 4. This is indeed the case as can be seen from Figure 6 at wavelengths much longer than lmax. In the limit of extremely long wavelengths, the spectral radiance is proportional to temperature. The Rayleigh2Jeans formula fails as the wavelength is reduced to near the emission peak of the blackbody curve and in particular in the short-wavelength region, where the Rayleigh2 Jeans equation predicts inﬁnite energy density at zero wavelength. This is clearly a failure of the theory to accurately reﬂect the underlying physical processes and thus is termed the ‘‘ultraviolet catastrophe.’’ At short wavelengths, when c2/lTc1, the RHS of Equation (19) can be approximated by: L b;l ðl; T Þ

c 1L 5 c 2 =lT le

(23)

This is called Wien’s formula, which gives approximately the correct result all the way to l ¼ lmax, where the relative error is 0.7%. Hence, for many industrial applications, Wien’s formula is an excellent approximation. Note that Wien’s formula always underpredicts the spectral radiance. When lWlmax, the error becomes large, as shown in Figure 7. At very long wavelengths, Wien’s formula asymptotically approaches c1Ll5 and becomes independent of temperature; this is physically incorrect. Furthermore, Equation (19) can be integrated over the whole spectrum to obtain the Stefan2Boltzmann law, Equation (14), using the relation Z

1 0

ex

x3 p4 dx ¼ 1 15

(24)

It can be shown that, for blackbody emission, nearly 25% of the energy is at lolmax and W95% of energy falls in the spectral region between 0.5lmax and 50lmax. Once the spectral radiance is known, integration over the wavelength band, the solid angle, and area gives the total power. As an example, consider two coaxial disks, each with a diameter of 1 cm, facing each other and separated at the distance of 1 m. One disk is the source, while the other is the detector. Assuming that both are blackbodies, and supposing that a narrow-band ﬁlter is placed in between so as to allow only radiation from 1,200 nmrlr1,220 nm to pass through without loss (i.e., transmittance t ¼ 100%), the goal is to estimate the total radiant power received by the cold detector (i.e., negligible emission) from the source at a temperature of 1,400 K. Since the dimensions of the surfaces are much smaller than the distance, we can assume all the radiation is from the normal direction. The solid angle subtended by the source as viewed from the detector is dO ¼ As/ L2 ¼ 7.85 105 sr. Neglecting the effect of the ﬁlter, the radiant power

85

Theory of Thermal Radiation and Radiative Properties

Percentage Error of Wien's Formula (%)

101

100

10-1

10-2

λmaxT

10-3 0

1000

2000

3000

4000

5000

6000

λT (μm-K)

Figure 7

Relative error of Wien’s formula compared with Planck’s law.

reaching the detector is qwithout filter ¼ LðT ÞdO cosy dA ¼

sT 4 dO cosy dAd ¼ 428 mW p

(25)

where cos y ¼ 1. With the ﬁlter, we have qwith filter ¼ tL l ðl; T ÞdO cosy dAdl W ð7:85 105 srÞ ¼ 9:39 103 sr m2 mm ð7:85 105 m2 Þ ð2:0 102 mmÞ ¼ 1:16 mW

ð26Þ

In this example only a small fraction of the total radiation is transmitted through the narrow-band ﬁlter.

3. Radiative Properties of Materials 3.1. Emissivity For real materials, the rate of emitted energy per unit area cannot exceed that of a blackbody at the same temperature. The ratio between the spectral radiance emitted by a real surface to that of a blackbody at the same

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Zhuomin M. Zhang and Bong Jae Lee

temperature is called the directional spectral emissivity, 0l ðl; y; f; T Þ

L em;l ðl; y; f; T Þ L b;l ðl; T Þ

(27)

Here, subscript ‘‘em’’ is used for the emitted. Note that in general the emissivity also depends on the surface temperature. It can be viewed as the ratio of the emitted energy between the surface and that of a blackbody at the same temperature per unit time, per unit solid angle, per unit wavelength interval, and per unit projected area toward the direction deﬁned by the zenith angle y and azimuthal angle f. The directional total emissivity is deﬁned based on the total radiance as follows: 0t ðy; f; T Þ

L em ðy; f; T Þ p ¼ L b ðT Þ sT 4

Z

1 0

0l L b;l ðl; T Þdl

(28)

The ratio of the exitance deﬁned in Equation (9) emitted from an object to that from a blackbody at the same temperature is called the hemispherical spectral emissivity, hl ðl; T Þ

M em;l ðl; T Þ 1 ¼ pL b;l p

Z

2p

Z

0

p=2

0l cosy siny dydf

(29)

0

where Mem,l is due to emission only and is called self-exitance. Similarly, the hemispherical total emissivity is deﬁned as: ht ðT Þ

M em ðT Þ p ¼ 4 sT sT 4

Z

1

hl L b;l ðl; T Þdl

(30a)

0

which can also be expressed as: ht ðT Þ

1 p

Z 0

2p

Z 0

p=2

0t cosy siny dydf

(30b)

In this chapter, the prime sign signiﬁes a directional property, or directional-hemispherical property, and superscript ‘‘h’’ designates a hemispherical property for clarity. In other chapters of this book and in many other publications, the prime or superscript is often omitted and readers need to distinguish directional from hemispherical properties from the context. Many industrial materials have spectrally selective surfaces for which the emissivity is a complicated function of temperature, wavelength, the direction of emission, polarization, as well as surface conditions, such as ﬁnishing, roughness, contamination, coatings, etc. For perfectly smooth surfaces, the emissivity can be predicted from the electromagnetic-wave theory, as will be discussed in the following sections. Based on the deﬁnition of blackbody and emissivity, the value of emissivity of real surfaces is between 0 and 1. The range of the normal total emissivities for a

Theory of Thermal Radiation and Radiative Properties

87

Normal Total Emissivities at 25 °C

Plants, water, skin, wall paints, and anodized finishes 0.88 – 0.98 Graphites, minerals, soil, and glasses 0.76 – 0.95 Oxides and ceramics 0.4 – 0.8 Oxidized metals 0.25 – 0.68 Unpolished metals 0.1 – 0.4 Polished metals 0.04 – 0.13 Highly polished metals, foils or films 0.02 – 0.07

Figure 8 Ranges of the normal total emissivity of common materials at room temperature [3,7].

variety of materials near room temperature is shown in Figure 8. Generally speaking, conductors have a higher reﬂectance and lower emissivity in the VIS and IR. Pure metals usually have a very low emissivity. The nature of surface ﬁnish and oxidization can greatly affect its emissivity. Some spectrally selective materials that appear highly reﬂecting in the visible may have a high emissivity (such as white paint and paper) in the mid-IR spectrum. It is important to bear in mind that at near room temperature, the most important spectral region is from 5 to 50 mm. A comprehensive compilation of radiative properties of solids is given in Ref. [7]. Additional information on the emissivity of real materials can be found from Refs. [3,4,8]. For metallic materials or electrical conductors, the emissivity averaged over two polarizations is nearly constant for y less than about 401, increases to a maximum near 801, and ﬁnally drops to zero at the grazing angle. For nonconductors, the polarization-averaged emissivity is nearly constant for y less than about 701 and decreases sharply to zero when y approaches 901. The hemispherical emissivity is higher (r30%) than the normal emissivity for conductors and slightly lower (r5%) than the normal emissivity for nonconductors. Note the assumption of diffuse (i.e., angle independent), emission is a good ﬁrst-order approximation in many engineering applications. In these cases, the hemispherical emissivity may be assumed equal to the normal emissivity. This assumption is not applicable for structured surfaces for which the emissivity may strongly depend on the direction. For a diffuse emitter, the emissivity is independent of the viewing direction; thus, 0l ¼ hl

and

0t ¼ ht

(31)

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Zhuomin M. Zhang and Bong Jae Lee

A gray or grey surface is one such that its emissivity and absorptivity are independent of wavelength. The following relations between the emissivities exist for a gray surface: 0l ¼ 0t

and

hl ¼ ht

(32)

In the literature, both emissivity and emittance are used interchangeably. While it has been suggested that emissivity should be used for perfectly smooth surfaces and emittance should be used for real materials [4], most people do not make such a distinction and tend to use emissivity for all sorts of materials. For semitransparent materials, reﬂectance, transmittance, and absorptance are most frequently used. Absorptivity is often used interchangeably with absorptance. Reﬂectivity is commonly used as the fraction of energy reﬂected at the interface between two semi-inﬁnite media. Emissivity determination is one of the central problems in radiation thermometry as discussed in Chapter 1. The measurement and modeling of emissivity become extremely important in many applications measuring surface temperature using radiometric methods, such as rapid thermal processing (RTP) of semiconductors and the steel and glass industries. Further discussion on theoretical modeling of emissivity for opaque and semitransparent materials will be given later in this chapter.

3.2. Absorptivity The directional spectral absorptivity depends on the temperature, as well as the direction and wavelength of the incident radiation. Unless the incident radiance is extremely intense to cause nonlinear absorption, a0l is considered as an inherent material property that is independent of the environment. The directional spectral absorptivity is the ratio of the absorbed spectral radiance to that incident on the surface of the object: a0l ðl; y; fÞ

L abs;l ðl; y; fÞ L i;l ðl; y; fÞ

(33)

In general, a0l ðl; y; fÞ also depends on the temperature of the object. The hemispherical spectral absorptivity is deﬁned based on the ratio of the absorbed irradiance to the irradiance coming onto the surface from all directions in the upper hemisphere; therefore, ahl ðlÞ

E abs;l ðlÞ El ðlÞ

(34a)

Theory of Thermal Radiation and Radiative Properties

89

where Z

2p

Z

p=2

L i;l ðl; y; fÞ cosy siny dydf

El ðlÞ ¼ 0

(34b)

0

and Z

2p

Z

p=2

a0l ðl; y; fÞL i;l ðl; y; fÞ cosy siny dydf

Eabs;l ðlÞ ¼ 0

0

(34c)

In general, ahl ðlÞ depends on the angular distribution of the radiation incident onto the surface (i.e., incoming radiation). If the incoming radiation is diffuse (or isotropic), then the radiance is independent of the direction. In this case, we have ahl ðlÞ

1 p

Z 0

2p

Z 0

p=2

a0l ðl; y; fÞ cosy siny dydf

(35)

The directional total absorptivity a0t is the fraction of absorbed total radiance, but it is not used very often. The hemispherical total absorptivity is deﬁned according to aht

R1 R1 h Eabs;l ðlÞdl a E l ðlÞdl E abs ¼ R0 1 l ¼ 0R 1 E E ðlÞdl l 0 0 E l ðlÞdl

(36)

It should be mentioned that only a0l is an inherent property of the materials, whereas a0t , ahl , and aht are not, because they also depend on the directional distribution and/or spectral distribution of the incident radiation. Note that all the above deﬁnitions can be tied to the ratio of energy absorbed to that of the incident energy, as described in Ref. [4].

3.3. Kirchhoff’s law Consider an object placed in a blackbody enclosure, such as the one shown in Figure 5(a). Under thermodynamic equilibrium, the emitted radiant power from the object is ht As M b ðT s Þ ¼ ht As pL b ðT s Þ, where As and Ts are the surface area and the temperature, respectively, and Lb is the blackbody radiance. On contrary, the absorbed power by the object is aht As EðT w Þ, where Tw is the wall temperature and E(Tw) ¼ pLb(Tw) is the irradiance on the object. Noting that Ts ¼ Tw and, under equilibrium, the absorbed power must equal to the emitted power, we have aht ¼ ht

(37a)

90

Zhuomin M. Zhang and Bong Jae Lee

It can be shown that the equality also applies to directional and spectral properties [4]. Thus, under thermal equilibrium, we also have a0t ¼ 0t ;

ahl ¼ hl ;

and

a0l ¼ 0l

(37b)

The equalities given in Equations (37a) and (37b) are for thermal equilibrium and are generally not applicable when the incoming radiation is not from an environment that is in thermal equilibrium with the object. This is because the absorptivities depend on the incoming radiation from surroundings, except for a0l which is an inherent property of the material that is independent of the condition of the incoming radiation. Hence, the following equation holds for most practical conditions, a0l 0l

(38)

This equation known as Kirchhoff’s law states that the directional spectral absorptivity is equal to the directional spectral emissivity [325]. The emissivity deﬁnition is solely based on the spontaneous emission from the surface and is an inherent property of the material that does not depend on the surroundings. However, the absorptivity is deﬁned based on the net absorbed energy by treating stimulated or induced emission as negative absorption. Under these deﬁnitions, Kirchhoff’s law given in Equation (38) is always valid in terms of the directional spectral absorptivity and emissivity for any given polarization. The only assumptions are (a) the material under consideration is at a uniform temperature, at least within several penetration depths near its surface and (b) the external electromagnetic ﬁeld is not too strong to alter the material’s inherent properties, as in a nonlinear interaction. For a diffuse surface, ahl ¼ a0l and hl ¼ 0l ; hence, we have ahl ¼ hl . If a surface is gray, then a0t ¼ a0l and 0t ¼ 0l ; thus we have a0t ¼ 0t . In practice, a surface may be considered ‘‘gray’’ if the emissivity is independent of the wavelength over the spectral regions of importance. If a surface is diffuse-gray, then there is no spectral and directional dependence of the absorptivity or emissivity. This means that Equations (37a) and (37b) hold regardless of whether the surface is in an isothermal enclosure or not. Kirchhoff’s law has important applications in determination of the radiative properties. For example, the emissivity can be measured indirectly by measuring the absorptivity, or reﬂectivity as discussed below.

3.4. Reﬂectivity Let us ﬁrst consider an opaque surface, where radiation cannot penetrate very deep inside the material and there is no radiation transmission. For a mirror-like perfectly smooth surface, the reﬂected rays are in the specular direction, as shown in Figure 9(a). This requires that the root-mean-square (rms) surface roughness be less than one-hundredth of the wavelength

91

Theory of Thermal Radiation and Radiative Properties

Specular reflection

Diffuse reflection

BRDF from rough surface

(a)

(b)

(c)

Figure 9 Schematics of surface reﬂection: (a) specular or mirror-like, (b) diffuse, and (c) bidirectional reﬂectance distribution function (BRDF) from a rough surface.

[9211]. Another ideal surface is a perfectly diffuse surface, as shown in Figure 9(b), where the reﬂected radiance is independent of the direction. Such an ideal surface is more difﬁcult to realize even with surface roughness. In practice, there exists a bidirectional distribution of the reﬂected radiance from a rough surface, as shown in Figure 9(c). Consider radiation incident on a surface within an element solid angle dO with a spectral radiance Li,l at an incidence angle y. The incoming radiant power is given by qi,l ¼ Li,ldO cos ydAdl. The fraction of this power absorbed by the material is determined by the directional spectral absorptivity a0l , while the remaining is reﬂected toward the hemisphere. The directional-hemispherical spectral reﬂectivity r0l is deﬁned as the ratio of the reﬂected energy over the hemisphere to the incident energy from a given direction. Hence, r0l ðl; T ; y; fÞ ¼ 1 a0l ðl; T ; y; fÞ

(39a)

Similar relations can be drawn for the directional-hemispherical total reﬂectivity and the directional total absorptivity, r0t ðT ; y; fÞ ¼ 1 a0t ðT ; y; fÞ. By using Kirchhoff’s law, it can be seen that for an opaque surface, we have 0l ðl; T ; y; fÞ ¼ 1 r0l ðl; T ; y; fÞ

(39b)

This equation provides a powerful indirect means for determining the directional spectral emissivity. Spectrophotometers, such as dispersive instruments with gratings or ﬁlters and Fourier transform infrared (FTIR) spectrometers, often equipped with a suitable reﬂectance accessory or an integrating sphere, can be used to measure the spectral reﬂectivity [12219]. Lasers are also commonly used for reﬂectivity measurements at single or multiple wavelengths. In general, it is necessary to employ two directions to describe the reﬂection by real surfaces. The bidirectional reﬂectance distribution function (BRDF), also known as bidirectional reﬂectance, is a fundamental radiative property that describes the redistribution of reﬂected energy.

92

Zhuomin M. Zhang and Bong Jae Lee

z sˆi

dΩi

dΩr

i

sˆr

r i r x

Figure 10 Geometry of the incident and reﬂected beams.

BRDF is deﬁned as a ratio of the reﬂected spectral radiance to the incident spectral irradiance, according to the geometry depicted in Figure 10 for the incident and reﬂected beams [4,19221]: f r ðl; s^i ; s^r Þ ¼ f r ðl; yi ; fi ; yr ; fr Þ ¼

dL r L i cosyi dOi

(40)

where Li is the spectral radiance of the incident radiation within the solid angle dOi in the direction s^i , and dLr is the spectral radiance of reﬂected light in the direction s^r . An important principle of the BRDF is reciprocity [4], which states that the BRDF remains the same when the incidence and reﬂection directions are interchanged. The reciprocity principle can be expressed as: f r ðl; yi ; fi ; yr ; fr Þ ¼ f r ðl; yr ; fr ; yi ; fi Þ

(41)

The directional-hemispherical properties can be obtained from integration of BRDF over the reﬂecting hemisphere. The directional-hemispherical spectral reﬂectance can be expressed as: R0l ðl; yi ; fi Þ

Z ¼ 2p

f r ðl; s^i ; s^r Þ cosyr dOr

(42)

where 2p indicates the solid angle of the upper hemisphere. Here, R0l is used instead of r0l , which is reserved for smooth opaque surfaces. For an opaque surface, R0l ¼ r0l , and thus Equation (42) is applicable to any surface. It should be noted that the hemispherical-directional reﬂectance depends on the angular distribution of the incoming radiance. When the incoming radiance is independent of the direction, the hemisphericaldirectional spectral reﬂectance is equal to the directional-hemispherical

Theory of Thermal Radiation and Radiative Properties

93

spectral reﬂectance. For a diffuse surface, the reﬂected radiance is uniform in all directions and BRDF is related to the directional-hemispherical reﬂectance by f r ¼ R0l =p. For a specular surface, BRDF is a delta function around the specular reﬂection angle [6,9]; in this case, the specular reﬂectance is the same as the directional-hemispherical reﬂectance, or simply the reﬂectance denoted by r0l . While the BRDF of a rough surface can be predicted by solving Maxwell’s equations with the knowledge of surface roughness and optical properties of each medium, this is usually impractical due to the difﬁculties in determining the actual surface topography as well as in obtaining the solution of complete 3D vector equations at the boundary. Hence, approximation methods, such as the Rayleigh2Rice perturbation theory, the Kirchhoff approximation, and the geometric optics approximation, are often employed since the rigorous electromagnetic solution requires tremendous memory space and CPU time, and boundary conditions are very rarely known with sufﬁcient precision. Notice that these approximations are only applicable with certain ranges of roughness parameters and the wavelength [9,11]. Monte Carlo models have been developed to simulate semiconductor, metal, and coated microrough surfaces [21,22] based on geometric optics in which a rough surface can be viewed as consisting of microfacets. The slope or orientation of microfacet is important in modeling the BRDF of rough surfaces. Two surface realization methods have been used: one is the perturbation method that incorporated the slope distribution obtained by analyzing the topographic data measured with an atomic force microscope (AFM); the other is directly applied to the surface topographic data from an AFM. The results agree reasonably well with experiments and demonstrate the importance of incorporating the anisotropic slope distribution in the BRDF modeling [22]. A typical setup for BRDF measurements is shown in Figure 11. The laser beam passes through a polarizer and a beam splitter: one beam goes to the reference detector that monitors the laser power and the other reaches the sample. The sample and signal detector can be rotated independently. The BRDF measurement equation is [20] f r ¼ CI

Vs V m cosyr DOr

(43)

where Vs and Vm are outputs of the signal and reference detectors, respectively, and DOr is the reﬂection solid angle. The instrument constant CI compensates the beam-splitter ratio and the difference in the responsivities of the two detectors. In essence, Vs is proportional to the power reaching the detector reﬂected from the sample and Vm is proportional to the power that is incident on the sample. Note that the

94

Zhuomin M. Zhang and Bong Jae Lee

Goniometric Table

BeamPolarizer splitter

Signal Detector

Position Controller

Laser Computer Sample Reference Detector Pre-amp

Lock-in

Figure 11 Schematic of the experimental setup for measuring BRDF [20].

measured BRDF values are always ﬁnite because the solid angle in the experiment is ﬁnite. Measurements and modeling of BRDF and directional-hemispherical reﬂectance are important for many industrial applications. One of the examples is for temperature measurement in RTP [23]. Radiation thermometers are commonly used to monitor the semiconductor wafer temperature during RTP [24]. The emissivity of the wafer must be determined in advance to correct the thermometer reading. The uncertainty of the radiometric temperature measurement largely depends on how accurately the emissivity of the silicon wafer is known. Furthermore, chamber reﬂections must be considered in order to determine the radiant ﬂux reaching the radiometer through the optical ﬁber or lightpipe [25]. Since the back surface of the silicon wafer is usually rough, knowledge of the BRDF is crucial to the emissivity modeling or heat transfer analysis in an RTP chamber. For this reason, there have been a number of studies that dealt with the modeling and measurements of BRDF of rough silicon wafers with or without thin-ﬁlm coatings [21,22,26228].

3.5. Effective or apparent emissivity One of the methods to improve the accuracy of temperature measurement is to enhance the emissivity through the introduction of a cavity on the workpiece, usually by drilling a hole, as illustrated in Figure 12. Suppose the temperature of the workpiece is T, the area of the opening is A1, and the wall area of the cavity is A2. Assume all surfaces are diffuse with a spectral emissivity of el (which is independent of the direction). Without the cavity, the power emitted by the surface in the wavelength interval dl is elpLb,l(l,T)A1dl, as in the case of Figure 12(a). There will be more power

95

Theory of Thermal Radiation and Radiative Properties

Detector

Detector

A1,

T

A2,

T (a)

(b)

Figure 12 Illustration of effective emissivity: (a) the detector views a ﬂat workpiece of area A1, and (b) the detector views a cavity on the workpiece with a surface area of A2.

emitted from the opening with the cavity as in the case of Figure 12(b). One can trace the multiple reﬂections to ﬁnd out the power emitted by the wall (surface A2) that escapes through the opening; i.e., "

# A1 A1 A1 A1 A1 2 2 ð1 l Þ þ l pL b;l ðl; T ÞA2 dl þ 1 1 ð1 l Þ þ A2 A2 A2 A2 A2 (44a)

For a blackbody of surface area A1, the radiant power emitted toward the hemisphere is qb ¼ pL b;l ðl; T ÞA1 dl

(44b)

The ratio of Equation (44a) to Equation (44b) deﬁnes the effective emissivity, which is eff ¼

l 1 ð1 A1 =A2 Þð1 l Þ

(45)

It can be seen that eeff -1 as A1/A2-0, as expected. More complicated analysis of the effective emissivity of cavities can be performed using the radiation network method or a Monte Carlo method [3,4,29].

3.6. Atmospheric transmission effects A major challenge in practicing radiation thermometry is the environmental absorption and scattering by the medium in the optical path. If the medium is at high temperature, emission from the gases must also be considered. This section brieﬂy reviews the equations governing the radiative transfer in the atmosphere without considering wave effects. Usually, gases (such as Ar, N2, and O2) do not absorb or emit radiation in the VIS and near-IR region unless their temperature is extremely high. Polar gases, such as carbon

96

Zhuomin M. Zhang and Bong Jae Lee

dioxide, water vapor, and hydrocarbon gases, emit and absorb radiation in speciﬁc wavelength bands over a wide range of temperatures. Gas molecules and aerosols in the atmosphere, pulverized coal in a combustor, soot in ﬂames, dust, and other particulates also scatter radiation. As shown in Figure 13, at steady state, the spectral radiance L l ðl; x; s^Þ also depends on the location or spatial coordinate x and directional vector s^. As the ray travels in the medium, the radiance is attenuated by absorption and scattering. If the emission of the medium is not negligible, the radiance is enhanced by emission. Furthermore, rays from all directions can be scattered toward the traveling direction, which is called in-scattering. The macroscopic description of the spectral radiance is known as the radiative transfer equation (RTE) [4]: dL l sl ¼ al L b;l ðl; T Þ bl L l þ dx 4p

Z

L l ðl; x; s^0 ÞFl ðs^0 ; s^ÞdO0

(46)

4p

where al and sl are the absorption and scattering coefﬁcients, respectively; bl ¼ al + sl is the attenuation (or extinction) coefﬁcient; and Fl ð^s0 ; s^Þ is the scattering phase function (Fl1 for isotropic scattering). Note that ˆ L(, + d, s)

dξ

(a) Attenuation by absorption: dL = −aL(, , ˆs)

ˆ L(, , s) ˆ L(, + d, s)

dξ

(b) Enhancement by emission: dL = aLb,(, T)

T ˆ L(, , s)

out-scattering

dξ

ˆ L(, , s)

ˆ L(, + d, s)

(c) Effect of scattering

in-scattering

Figure 13 Schematic of radiative transfer in a participating medium: (a) attenuation by absorption, (b) enhancement by emission, and (c) effect of scattering.

Theory of Thermal Radiation and Radiative Properties

97

Lb,l(l,T) in Equation (46) should be obtained from Equation (20a) to include the effect of the refractive index while l is the vacuum wavelength. The RHS of Equation (46) is composed of three terms: (1) the ﬁrst accounts for the contribution of emission, which depends on the local gas temperature T(x); (2) the second is the attenuation by absorption and scattering in the considered volume; and (3) the third is the contribution of in-scattering from all directions (integration over the solid angle 4p) to the direction s^. If only absorption is considered, Equation (46) reduces to dLl/dx ¼ alLl or L l ¼ L l;0 expðal xÞ

(47)

where Ll,0 is the radiance at x ¼ 0. The radiance will exponentially decay inside the medium. Furthermore, the radiation penetration depth or photon mean free path is deﬁned as dl ¼ a1 l . At a distance x ¼ dl, the radiance is attenuated by e1. In general, Equation (46) is an integro-differential equation, subject to often complicated boundary conditions. The integration of the spectral intensity over all wavelengths and all directions gives the radiative heat ﬂux. Unless the temperature ﬁeld is prescribed, the RTE is coupled with the heat conduction or convection equation to completely describe the heat ﬂow for a particular set of boundary conditions. Solutions of Equation (46) are rarely needed in radiometric temperature measurements. However, the characteristics of the absorption and scattering of materials should be known in order to estimate the impact of the environment. Gas radiation is more important for radiation temperature measurements in combustion furnaces and in remote sensing of ﬁres, as will be discussed in other chapters of this book. Scattering by individual gas molecules is usually negligible because their diameters are much smaller than the wavelengths of interest. When radiation travels through a gas, some of the energy may be absorbed. The absorption of radiant energy (or photons) raises the energy levels of individual molecules within the gas. Conversely, gas molecules may spontaneously lower their energy levels and emit photons. These changes in energy levels are called radiative transitions. The most important transitions for radiation thermometry are bound2bound transitions between vibrational energy levels coupled with rotational transitions, because they are prevalent in the near-IR. The photon energy must be exactly the same as the difference between two energy levels in order for the photon to be absorbed or emitted; therefore, the quantization of the energy levels results in discrete spectral lines for absorption and emission. The rotational lines superimposed on a vibrational line give a band of closely spaced spectral lines, called the vibration2rotation spectrum. A large number of quantum states in polyatomic gases can produce spectral lines that are very closely

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Zhuomin M. Zhang and Bong Jae Lee

spaced to each other; some lines may overlap as a consequence of broadening. The results are called vibration2rotation bands. Band models are, therefore, developed to facilitate the computation of the total radiative properties. More elaborated discussions on the absorption and emission mechanisms as well as radiative properties of gases can be found from Refs. [4,30232]. Small particles are important for radiative transfer in the open atmosphere and in combustion chambers. Soot particles ranging from 5 to 100 nm in size are formed during combustion processes and emit thermal radiation (often stronger than the emission from combustion gases) in a continuous spectrum over the IR and some VIS regions. In pulverized coal combustion furnaces, coal, char, and ﬂy ash particles are important as well as soot particles. Particles can also scatter electromagnetic wave or photons, causing a change in the direction of propagation. In the early 20th century, Mie developed a theory for scattering by a spherical particle based on Maxwell’s equations, known as the Mie scattering theory, which can be used to predict the scattering phase function [30]. In the case when the particle sizes are small compared with the wavelength the formulation reduces to the simple expression obtained by Rayleigh, known as Rayleigh scattering. In the Rayleigh scattering limit, the wavelength of the scattered light remains unchanged and the scattering efﬁciency is inversely proportional to l4. The absorption efﬁciency, however, is proportional to the inverse of l. The scattering phase function is nearly isotropic. Soot particles usually fall in the Rayleigh limit. For spheres whose diameters are much greater than the wavelength, geometric optics can be applied. If the scattering by one particle is affected by the presence of other particles, the scattering is called dependent scattering, which requires more complicated treatments to obtain the scattering phase function [4,30].

3.7. Semitransparent materials If a material is not opaque, the bidirectional transmittance distribution function or BTDF is deﬁned as: f t ðl; s^i ; s^r Þ ¼

dL t L i cosyi dOi

(48)

which is similar to the BRDF given in Equation (40), with the replacement of the reﬂected radiance by the transmitted radiance. The directionalhemispherical spectral transmittance can be calculated by integrating the BTDF over the hemisphere: T 0l ¼

Z 2p

f t ðl; s^i ; s^t Þ cosyt dOt

(49)

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99

From energy conservation, the directional spectral absorptance of a semitransparent material is given by: A0l ¼ 1 R0l T 0l

(50)

Semitransparent materials are important as optical lenses, windows, and ﬁlters for radiation thermometer applications. Further discussion on how to calculate radiative properties of multilayer ﬁlms will be given in Section 6. The transmittance of a ﬁlter over a spectral band needs to be deﬁned based on the radiance distribution and the wavelength interval, i.e., R l2

T 0filter

T 0l L l dl ¼ lR1 l2 l1 L l dl

(51)

4. Electromagnetic Wave Theory 4.1. Macroscopic Maxwell’s equation Microscopically, a medium consists of molecules or atoms that are made of electrons and nuclei. In general, nuclei are in random motion due to thermal excitation, and electrons are in orbitals around nuclei. Therefore, the microscopic electromagnetic ﬁelds generated by these charged particles ﬂuctuate, over distances on the order of 1010 m, with periods of near 1013 s for nuclear vibrations and 1017 s for electron orbital motion [33]. However, all the microscopic ﬂuctuations are averaged out in most of measuring devices, resulting in reasonably smooth and slowly varying macroscopic quantities. The macroscopic electromagnetic ﬁelds are described by macroscopic Maxwell’s equations given as follows: rE¼

@B @t

rH¼Jþ

@D @t

(52)

(53)

rD¼r

(54)

rB¼0

(55)

Here, B is the magnetic induction ﬁeld, D the electric displacement vector, J the electric current density in units (A/m2), and r the charge density in the medium. Equation (52) is Faraday’s law of induction, which shows the interdependence of electric and magnetic phenomena. It states that a

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Zhuomin M. Zhang and Bong Jae Lee

time-varying magnetic ﬁeld produces an electric ﬁeld in a coil; in other words, any closed electric ﬁeld line generates a time-varying magnetic ﬁeld. Equation (53) is Ampere’s law with Maxwell’s correction [33,34]. It states that through any closed magnetic ﬁeld line, there is an electric current density or an electric displacement or both. Gauss’s law given in Equation (54) describes how the divergence of electric ﬁeld is affected by the charge. On contrary, Equation (55) is the Gauss’s law of magnetism, which states that the divergence of magnetic ﬁeld is zero and implies that there exist no magnetic monopoles. Note that D ¼ e0E + P, where e0 is the electric permittivity of vacuum, and P is the polarization vector that is the dipole moment in a unit volume. Polarization is related to the response of permanent or induced dipole moment to the electric ﬁeld, see Refs. [35,36] for a detailed discussion. Because P ¼ e0(e1)E, where e is the relative permittivity or dielectric function of the medium, the electric displacement vector can be expressed by the electric ﬁeld for a linear isotropic medium as D ¼ ee0E. Note that e is a function of frequency for a dispersive medium and becomes a dyadic tensor for an anisotropic medium. Furthermore, from Ohm’s law, J ¼ sE, where s is the electric conductivity. At optical frequencies, both e and s are complex variables and the distinction between a conductor and a dielectric diminishes [6]. The magnetic induction is related to the magnetic ﬁeld H by B ¼ m0(H + M), where m0 is the magnetic permeability of vacuum and M is the magnetization vector, that is, magnetic moment in unit volume. With the relation M ¼ (m1)H, where m is the relative magnetic permeability, the magnetic induction can be expressed by the magnetic ﬁeld for a linear isotropic medium as B ¼ mm0H. Magnetic materials, whose relative permeability is different from unity, rarely exist in nature at optical frequencies. Metamaterials with nanostructures can possess a negative m at some optical frequencies, resulting in certain unique optical and thermal radiative properties for emerging applications, such as superlens that can provide images with resolution below the diffraction limit, invisible cloaking, enhanced transmittance, and coherent thermal emission [37,38].

4.2. Wave equation and wave propagation Let us consider the situation of free space for simplicity, such that r ¼ 0, s ¼ 0, e ¼ 1, and m ¼ 1. After substituting the constitutive relations into Maxwell’s equations and then combining Equations (52) and (53) in terms of the electric ﬁeld, we obtain r 2 E ¼ m0 0

@2 E @t 2

(56)

Theory of Thermal Radiation and Radiative Properties

101

where the vector identity r (r E) ¼ r(r E)r2E ¼ r2E has been used. Equation (56) is the wave equation, whose solution is a function of all four variables x, y, z, and t. A special kind of solution for the wave equation is a plane wave, whose time dependence is harmonic (i.e., sin ot or cos ot) with the angular frequency o. In the one-dimensional case, the solutions of the wave equation are linear combinations of sin(kx7ot) and cosðkx otÞ, where k ¼ o/c0 ¼ 2p/l. By extending the above solution to three dimensions, a monochromatic plane wave solution of the wave equation in Equation (56) can be written as: E ¼ E0 eiðkrotÞ

(57)

where E0 is the amplitude vector, r ¼ x^x þ y^y þ z^z the position vector, and k ¼ kx x^ þ ky y^ þ kz z^ the wavevector. The physical electric ﬁeld is Re(E). Since any time-dependent function can be expressed as a Fourier expansion of time-harmonic components, Equation (57) can be integrated over all frequencies to obtain the total electric ﬁeld at given time and position. The surface perpendicular to the wave propagation direction is called a wavefront and can be determined by k r ¼ constant. In vacuum, the wavefront travels in the k direction at the speed c0 ¼

o 1 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ k m0 0

(58)

which is called phase speed. The magnetic ﬁeld can be obtained from the electric ﬁeld using Maxwell’s equations. In vacuum, both the electric and magnetic ﬁelds are perpendicular to the wavevector k and satisfy the following relations: k E ¼ om0 H

(59a)

k H ¼ o0 E

(59b)

Consequently, E, H, and k are orthogonal to each other and form a righthanded triplet, as depicted in Figure 1. The discussion of electromagnetic wave propagation in free space can be extended to any medium with a relative permittivity e and permeability m. In the case of a lossless medium, both e and m are real. The refractive index of the medium is deﬁned as: n¼

pﬃﬃﬃﬃﬃ m

(60)

The phase velocity is reduced by a factor n; thus, the phase speed in the medium becomes: c¼

c0 n

(61)

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Zhuomin M. Zhang and Bong Jae Lee

and the wavelength in the medium pﬃﬃ becomes lm ¼ l/n. For nonmagnetic materials, it is obvious that n ¼ . Because the phase speed is inversely proportional to the refractive index, electromagnetic waves of different frequencies will travel at different phase speeds in a dispersive medium. The wave group is a superposition of waves traveling in different speeds and, therefore, will continuously change shape as it propagates through the medium. The group amplitude is modulated as a wave with its own front, called group front, which travels at its own velocity. The modulation of the group amplitude forms an envelope or wave packet. The velocity at which the wave packet travels is called the group velocity, given by vg ¼

@o k @k k

(62)

in an isotropic medium [33,39]. Generally speaking, the group velocity is considered as the velocity of energy transport. In some cases, the refractive index of a material can be less than unity depending on the frequency region, resulting in the phase velocity greater than c0 at that frequency. However, the wave packet or energy of the polychromatic wave propagates at the group velocity, which is always less than c0. Metamaterials may possess a negative refractive index and thus a negative phase velocity [6]; but the group velocity is always positive [40]. If the conductivity s is included, the wave equation for a nonmagnetic material becomes: r2 E ¼ m0 s

@E @2 E þ m0 0 2 @t @t

(63)

Substituting Equation (57) into Equation (63), it can be shown that k2 ¼ iom0 s þ o2 m0 0

(64)

suggesting that the wavevector should be complex. The complex dielectric function can be deﬁned as: ~ ¼ 0 þ i00 ¼ þ i

s o0

(65)

Accordingly, the complex refractive index n~ ¼ n þ ik is related to the complex dielectric function by ~ ¼ ðn þ ikÞ2 . The imaginary part k of the complex refractive index is called the extinction coefﬁcient. In addition, the refractive index and the extinction coefﬁcient are called the optical constants. It can be seen from Equations (64) and (65) that k2 ¼ ~o2 m0 0 or k ¼ n~ o=c 0 . It should be noted that some texts use ~ ¼ 0 i00 for the dielectric function and n~ ¼ n ik for the complex refractive index. In this case, the choice of time-harmonic term is exp(iot), such that Equation (57) must be rewritten as E ¼ E0ei(otk r) [6].

Theory of Thermal Radiation and Radiative Properties

103

Equation (57) represents a plane wave whose electric ﬁeld vector is parallel to E0. Such a wave is said to be linearly polarized since the oscillation direction does not change with time. Mathematically, two orthogonal linearly polarized waves can be superimposed to represent ^ that an arbitrarily polarized wave. Consider two unit vectors, a^ and b, are real and perpendicular to each other. The electric ﬁeld vector can be expressed as: ^ iðkrotÞ E ¼ ðE1 a^ þ E2 bÞe

(66)

where E1 and E2 are complex quantities, which may be expressed as E1 ¼ |E1|eif1 and E2 ¼ |E2|eif2. It can be shown that ReðEÞ ¼ jE1 j cos y1 a^ þ jE2 j cos y2 b^

(67)

where y1 ¼ k r ot + f1 and y2 ¼ k r ot + f2. The simplest case is when y1 ¼ y2, that is E1 and E2 are in the same phase. In this case, the electric ﬁeld is linearly polarized. If the phases of E1 and E2 are different, the electromagnetic wave is generally elliptically polarized. For the special case when |E1| ¼ |E2|, the wave becomes circularly polarized. Since thermally emitted light is randomly polarized, polarization effects are usually not considered in radiative heat transfer. This assumption, however, may be invalid for structured surfaces or upon reﬂection from a surface due to the effect of polarization. The energy density of the electromagnetic ﬁelds in a lossless medium is given by: 1 u ¼ ðE D þ B HÞ 2

(68a)

The time-averaged energy density for the time-harmonic ﬁelds becomes: 1 u ¼ ðE D þ B H Þ 4

(68b)

where denotes the complex conjugate. Furthermore, the vector S ¼ E H represents the energy ﬂux and is called the Poynting vector, which indicates the instantaneous energy ﬂux. In reality, the time-averaged Poynting vector S is employed to describe the radiant power through a unit area. The time-averaged Poynting vector can be expressed in terms of the complex electric and magnetic ﬁelds as: 1 S ¼ ReðE H Þ 2

(69)

In an absorbing medium, it can be shown that the amplitude of either the electric or magnetic ﬁeld decays exponentially according to exp(2pkx/l), where x is the coordinate along the direction of propagation. Hence, the

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Zhuomin M. Zhang and Bong Jae Lee

Poynting vector decays proportionally to exp(4pkx/l) ¼ exp(alx), where al ¼ 4pk/l is the absorption coefﬁcient.

4.3. Fresnel reﬂection and transmission coefﬁcients The solution of the wave equation considered so far is for waves traveling in an unbounded medium. However, radiation is often incident from one medium to another. Consider a plane wave at an angular frequency o incident from medium 1 to medium 2, which are separated by a smooth interface, as shown in Figure 14. Each medium is assumed to be homogenous and isotropic, so that e and m describe the material response to external electric and magnetic ﬁelds, respectively. It is a common practice to analyze the wave propagation for two separate polarizations. In the case of s polarization as shown in Figure 14(a), the electric ﬁeld is perpendicular to the x2z plane. This is also called a transverse electric (TE) wave, for which the incident electric ﬁeld can be expressed as: Ei ¼ y^ E i eiðk1x xþk1z zÞ

(70)

where Ei is the amplitude of the incident wave at the origin. The timeharmonic term exp(iot) is omitted for simplicity. Note that k21x þ k21z ¼ k21 ¼ m1 1 o2 =c 20 . Boundary conditions require that the wavevector of the reﬂected wave to be kr ¼ k1x x^ k1z z^ . Hence, the reﬂected electric ﬁeld can be expressed as Er ¼ y^ E r eiðk1x xk1z zÞ

(71)

where Er is the amplitude of the reﬂected wave. Similarly, the transmitted electric ﬁeld is given by: Et ¼ y^ E t eiðk2x xþk2z zÞ

Figure 14 Reﬂection and transmission of a linearly polarized plane wave at the interface between two semi-inﬁnite media: (a) s polarization; (b) p polarization.

(72)

Theory of Thermal Radiation and Radiative Properties

105

where Et is the amplitude of the transmitted wave. It should be noted that k22x þ k22z ¼ k22 ¼ m2 2 o2 =c 20 . The electric ﬁeld in medium 1 is a superposition of the incidence and the reﬂected ﬁelds. At the interface (z ¼ 0), boundary conditions require that the tangential component of the electric ﬁeld (Ey) must be continuous. Thus, we have k1x ¼ k2x ¼ kx

(73)

Ei þ Er ¼ Et

(74)

and

Note that Equation (73) results in Snell’s law, that is, n1sin y1 ¼ n2sin y2 for two dielectrics. The magnetic ﬁeld can be obtained using Maxwell’s equation, H ¼ (iomm0)1r E. The requirement of the tangential component of magnetic ﬁeld (Hx) to be continuous at the boundary yields the following equation: k1z k2z ðEi E r Þ ¼ Et m1 m2

(75)

The Fresnel reﬂection and transmission coefﬁcients for s polarization are deﬁned according to the ratios of the electric ﬁelds as r s12 ¼ E r =Ei and t s12 ¼ Et =E i , respectively. From Equations (74) and (75), it can be shown that [6,39] r s12 ¼

Er k1z =m1 k2z =m2 ¼ Ei k1z =m1 þ k2z =m2

(76)

t s12 ¼

Et 2k1z =m1 ¼ Ei k1z =m1 þ k2z =m2

(77)

and

For p polarization or transverse magnetic (TM) waves, the magnetic ﬁeld is in the y-direction, as shown in Figure 14(b). The Fresnel reﬂection and transmission coefﬁcients are deﬁned based on the ratios of the magnetic ﬁelds. The same procedure can be applied to determine the magnetic ﬁelds and then the electric ﬁelds in both media. After the boundary conditions are implemented, we obtain p

r 12 ¼

H r k1z =1 k2z =2 ¼ H i k1z =1 þ k2z =2

(78)

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Zhuomin M. Zhang and Bong Jae Lee

and p

t 12 ¼

Ht 2k1z =1 ¼ H i k1z =1 þ k2z =2

(79)

For nonmagnetic materials, the Fresnel reﬂection and transmission coefﬁcients reduce to

r 12 ¼

and t 12

8 n~ 1 cosy1 n~ 2 cosy2 > > < n~ cosy þ n~ cosy ;

for s polarization

n~ 2 cosy1 n~ 1 cosy2 > > ; : n~ 2 cosy1 þ n~ 1 cosy2

for p polarization

1

1

2

2

8 2~n1 cosy1 > > < n~ cosy þ n~ cosy ; 1 1 2 2 ¼ 2~ n cosy > 2 1 > ; : n~ 2 cosy1 þ n~ 1 cosy2

(80)

for s polarization (81) for p polarization

Note that the Fresnel coefﬁcients for s and p polarizations are different even at normal incidence because they are deﬁned based on the electric and magnetic ﬁelds, respectively. Ellipsometry is a major technique for optical property measurements and material diagnostics, especially in the UV, VIS, and near-IR spectral regions [41]. Ellipsometers utilize the polarization effects to determine the ratio of the reﬂection coefﬁcients for p polarization and s polarization: rp ¼ tanðcÞeiD rs

(82)

where c and D are deﬁned as ellipsometric angles (or parameters). For incidence on the surface of a semi-inﬁnite medium, measurements of c and D allow the determination of n and k because the reﬂection coefﬁcients are related to optical constants by Equation (80). For a dielectric ﬁlm on an opaque substrate, ellipsometers can be used to measure the refractive index and the ﬁlm thickness, if the refractive index of the substrate is previously determined. These quantities can be used to deduce surface temperature as will be discussed in Chapter 7.

4.4. Reﬂectivity and emissivity The reﬂectivity at the interface between two media can be expressed by the time-averaged Poynting vector. For s polarization, referring to Figure 14(a), the z component of the incident Poynting vector at the interface is 1 k1z S 1z ¼ Re ðE þ E ÞðE E Þ i r i r om0 m 1 2

(83)

Theory of Thermal Radiation and Radiative Properties

107

The reﬂected wave and the incident wave are in general coupled such that the energy ﬂow cannot be separated into a reﬂected ﬂux and an incident ﬂux [42]. Under the assumption that medium 1 is nonabsorbing and k1z is real, Equation (83) can be rewritten as [6]: S 1z ¼ S iz S rz

(84)

where S iz ¼

k1z k1z jE i j2 and S rz ¼ jE r j2 2om0 m1 2om0 m1

(85)

The spectral reﬂectivity r0l at the interface is deﬁned as the ratio of the reﬂected ﬂux to the incident ﬂux. Hence, r0l ¼

S rz jEr j2 ¼ S iz jEi j2

(86a)

Alternatively, Equation (86a) can be expressed as: r0l ¼ jr s12 j2

(86b)

where r12 is the Fresnel reﬂection coefﬁcient given in Equation (76). For p polarization, the same procedure can be followed to show that r0l ¼

jH r j2 p ¼ jr 12 j2 jH i j2

(87)

From Equation (80), it can be shown that for normal incidence from air (n1E1) to a medium with n~ 2 ¼ n2 þ ik2 , the reﬂectivity is independent of polarization and can be expressed as: r0l;n ¼

ðn2 1Þ2 þ k22 ðn2 þ 1Þ2 þ k22

(88)

The emissivity of a surface can be calculated from Equation (39b), that is, subtracting the reﬂectivity from unity. However, the reﬂectivity should be averaged over the two polarizations calculated from Equations (86) and (87) for incidence from air to the medium. Knowledge of the optical constants of materials is important for the prediction of their radiative properties. Figure 15(a) shows the reﬂectivity as a function of the incidence angle when radiation is incident from air (n1 ¼ 1) to a dielectric medium with n2 ¼ 1.5 for both polarizations. By carefully examining Equation (80), it can be seen that the reﬂection coefﬁcient can be zero for p polarization p only. The angle at which r 12 ¼ 0 is called the Brewster angle (also known as

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Zhuomin M. Zhang and Bong Jae Lee

1 (a) n1 = 1.0, n2 = 1.5

Reflectivity

0.8

s polarization p polarization

0.6

0.4 θB = 56.3°

0.2

0

0

30

60

90

Angle of Incidence (deg)

1

θc = 41.8°

0.8

Reflectivity

s polarization p polarization

0.6

0.4

θB = 35.6°

0.2

0

(b) n1 = 1.5, n2 = 1.0

0

30

60

90

Angle of Incidence (deg)

Figure 15 Reﬂectivity at the interface between air and a prism (n ¼ 1.5) for both polarizations: (a) radiation incident from air; (b) radiation incident from the prism.

the polarizing angle): yB ¼ tan1(n2/n1), which is 56.31 when n1 ¼ 1 and n2 ¼ 1.5. At the Brewster angle, the propagating directions of the reﬂected and transmitted waves form a right angle (i.e., y1 + y2 ¼ p/2). If the medium is absorbing, the reﬂectivity for p polarization will show a minimum, which is greater than zero [39]. The Brewster angle can be employed to build polarizers and to achieve nearly perfect transmission for p-polarized laser light.

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Theory of Thermal Radiation and Radiative Properties

In Figure 15(b), the reﬂectivity is plotted when the radiation is incident from a dielectric medium to air. In this case, the Brewster angle is yB ¼ 35.61. When the incidence angle is greater than a certain value, the reﬂectivity becomes unity for both polarizations. This phenomenon is called total internal reﬂection, which occurs when the radiation is incident from an optically denser medium to an optically rarer medium. The onset angle of total internal reﬂection is called the critical angle, given by yc ¼ sin1(n2/n1) according to Snell’s law. Total internal reﬂection is important for optical ﬁber or lightpipe thermometers [24,25] and near-ﬁeld radiative transfer due to photon tunneling [43]. Surface roughness can modify the emissivity. Figure 16 shows the emissivity of silicon at the wavelength l ¼ 635 nm near room temperature. It should be noted that silicon wafers of approximately 400-mm thickness are opaque at this wavelength due to strong band-gap absorption. The emissivity data were deduced from the directional-hemispherical reﬂectance measured with an integrating sphere by Lee et al. [18]. Except at the emission angle of 401, the measurement results for the smooth surface agree well with those predicted using the optical constants from Refs. [44,45]. The rough surface has an rms roughness of 630 nm and exhibits an anisotropic slope distribution (sample no. 2 in Ref. [18]). While the emissivity is a strong function of polarization, the polarization-averaged emissivity does not change signiﬁcantly up to the Brewster angle. At smaller angles, the emissivity is enhanced by surface roughness. However, the 1 Silicon λ = 635 nm

Emissivity

0.8

0.6 Theory, p 0.4

Theory, s Smooth, p Smooth, s

0.2

Rough, p

Rough, s 0 0

30

60

90

Angle of Emission (deg)

Figure 16 Comparison of calculated and measured emissivity of silicon versus the angle of emission for both polarizations at the wavelength of 635 nm.

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Zhuomin M. Zhang and Bong Jae Lee

emissivity of the rough surface is lower for p polarization at large angles; this is due to the polarization mixing caused by the randomly oriented microfacets. The polarization-averaged emissivity is enhanced by 0.0520.06 due to anisotropic surface roughness. This example illustrates that emissivity at oblique incidence is polarization dependent. Furthermore, the emissivity variation due to surface roughness must be compensated in order for the radiation thermometer to provide accurate measurements of surface temperatures.

5. Optical Properties Several simple dielectric function models are described to account for free-electron absorption, phonon vibrations, and interband absorption. These models are convenient for prediction of the VIS and IR optical constants of conductors, dielectrics, and semiconductors. In the case of nonmagnetic materials, the complex dielectric function is related to the complex refractive index by: ~ðoÞ ¼ 0 þ i00 ¼ ðn þ ikÞ2

(89)

It should be mentioned that the real and imaginary parts of the dielectric function are intrinsically related based on the Kramers2Kronig dispersion relation, which is a consequence of the causality and very useful in determining the frequency-dependent dielectric function of real materials [30,33].

5.1. The Drude model for a conductor In a conductor, electrons in the outermost orbits are ‘‘free’’ to move in accordance with the external electric ﬁeld. The dielectric function of a conductor can be modeled by considering the electron movement under the electric ﬁeld. From Newton’s second law, the equation of motion for an electron can be expressed by assuming the damping force to be proportional to its velocity as: me ðx€ þ g_xÞ ¼ eE

(90)

where me is the electron mass, e the electron charge, and g the strength of the damping and called the scattering rate. The damping term is mainly due to the collision of electrons with lattice vibrations, and the inverse of the scattering rate is called the relaxation time. The electric ﬁeld E in Equation (90) is assumed to be macroscopic ﬁeld by neglecting the local ﬁeld effects [30,36]. For a harmonic excitation, that is, E ¼ E0eiot, the solution of

Theory of Thermal Radiation and Radiative Properties

111

Equation (90) is x¼

e=me E oðo þ igÞ

(91)

Note that the electric current density J is related to the electron velocity by: ~ J ¼ ne e_x ¼ sE

(92)

where ne is the number density of electrons and s~ the complex conductivity. From Equation (92), the conductivity can be expressed in terms of the frequency and scattering rate as: ~ sðoÞ ¼

ne e2 =me s0 g ¼ g io g io

(93)

where s0 ¼ nee2/(gme) is the dc conductivity. The complex dielectric function of a metal can be calculated by using Equation (65) as follows: ~ ðoÞ ¼ 1

o2p s0 g=0 ¼ 1 o2 þ iog o2 þ iog

(94)

where eN accounts for high-frequency contributions and op is the plasma frequency given by: rﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃ s0 g ne e2 ¼ op ¼ 0 me 0

(95)

The value of eN is usually on the order of unity and has little inﬂuence on the optical constants of conductors in the IR region. The plasma frequency is in the UV region for most of the metals. When ooop, the real part of the dielectric function is negative such that nok. At very low frequencies, the real part of the dielectric function is much smaller than the imaginary part and, therefore, nEk. Metals become highly reﬂective in the VIS and IR region. The values of scattering rate and plasma frequency for several metals can be found in Ref. [46]. Figure 17 shows the normal reﬂectivity calculated from Equation (88) of gold, using the Drude model with parameters of eN ¼ 1, op ¼ 7.25 104 cm1, and g ¼ 2.16 102 cm1 [46], as well as using the experimentally determined optical constants from Ref. [44]. In the IR wavelengths, the reﬂectivity is very high and the emissivity is below 0.023 at lW1 mm. In this region, the reﬂectivity predicted by the Drude model agrees well with that calculated from optical constants obtained from experiments. However, there exist signiﬁcant deviations in the VIS and UV regions. The simple Drude model signiﬁcantly overpredicts the reﬂectivity and the resulting

112

Zhuomin M. Zhang and Bong Jae Lee

1

0.8

Reflectivity

Au

0.6

Drude model [46]

0.4

Tabulated data [44]

0.2

0 0. 1

1

10

Wavelength (μm)

Figure 17 Normal reﬂectivity of gold calculated from the simple Drude model and from the experimentally determined optical constants.

reﬂectivity exhibits a sharp edge at the plasma frequency. There exist complicated interband transitions in the VIS region. The interband transition at wavelength around 500 nm causes signiﬁcant absorption when the photon energy is higher than the band-gap energy. Etchegoin et al. [47] used a critical-point model to include interband transitions and obtained good agreement of the optical constants of gold at wavelengths from 200 to 1,000 nm. Although discrepancies exist in the shorter wavelength region, the Drude model is easy to apply and in the absence of measurement data provides insight on the dielectric function of conductors in the IR and microwave spectral regions. The measured optical constants of a large number of metals are tabulated in Ref. [44]. While the data vary from sample to sample as well as between measurements performed by different groups, the tabulated data should be considered as more reliable than those predicted by the Drude model.

5.2. The Lorentz oscillator model for a nonconductor The electrons in a nonconductor are bound to molecules and cannot move freely. In contrast to free electrons, a bound charge experiences a binding force, which can be viewed due to an imaginary spring with a resonance frequency o0. The binding force can be expressed as mo20 x, where x represents the displacement of a bound charge and m is the mass associated with the oscillator. Consequently, the equation of motion for an oscillator

113

Theory of Thermal Radiation and Radiative Properties

with a positive charge of e, subject to spring-damping forces, can be expressed as follows [30]: mð€x þ g_x þ o20 xÞ ¼ eE

(96)

where g is the damping coefﬁcient or scattering rate. With the assumption of harmonic electric ﬁeld, exp(iot), the solution of Equation (96) is x¼

e=m E o20 o2 iog

(97)

Note that the dipole moment p due to a bound charge is p ¼ ex ¼

o20

e2 =m E o2 iog

(98)

If the number density of the oscillator is n, then the polarization (dipole moment per unit volume) is P ¼ np. There exist different kinds of oscillators in a real material, such as bound electrons or lattice ions. From the response of a single-charge oscillator to a time-harmonic electric ﬁeld, we can extend the response function to a collection of oscillators. Assume that there are N types of oscillators. The mass, resonance frequency, scattering rate, and number density for the jth oscillator are mj, oj, gj, and nj, respectively. Then, the dielectric function can be expressed as a superposition of the contributions from each type of oscillators. Therefore, ~ ðoÞ ¼ 1 þ

N X

o2p;j

j

o2j o2 iogj

(99)

where o2p;j ¼ nj e2 =ðmj 0 Þ and op, j may be viewed as the plasma frequency of the jth oscillator. In the derivation of the above equation, the constitutive relation P ¼ 0 ð~ 1ÞE has been used. As in the Drude model, eN accounts for high-frequency contributions. The Lorentz model was ﬁrst derived based on the classical theory at the beginning of the 20th century. Later, it was conﬁrmed by quantum mechanics. The resonance frequency corresponds to the transverse optical phonon mode. Since the parameters for the Lorentz model are more difﬁcult to be modeled as compared to those for the Drude model, they are usually determined from ﬁtting the measured reﬂectance. Figure 18 shows the dielectric function, optical constants, and normal emissivity of c-axis crystalline Al2O3 (sapphire) based on the Lorentz parameters obtained by Barker [48] for ordinary rays in which the electric ﬁeld is perpendicular to c-axis. There are four oscillators located at wavelengths of 15.7, 17.6, 22.6, and 26.0 mm, with

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Zhuomin M. Zhang and Bong Jae Lee

300

Dielectric Function

(a)

ε′

200

ε′′ 100

0

Al2O3

-100 0

5

10

15

20

25

30

35

Wavelength (μm)

14 12

Optical Constants

(b) 10

n

8

κ

6 4 2 0 0

5

10

15

20

25

30

35

Wavelength (μm) 1 (c)

Emissivity

0.8 0.6 0.4

Opaque 0.2 0

0.5 mm

0

5

10

15

20

25

30

35

Wavelength (μm)

Figure 18 (a) Dielectric function, (b) optical constants, and (c) normal emissivity of Al2O3.

Theory of Thermal Radiation and Radiative Properties

115

corresponding wavenumbers of 635, 569, 442, and 385 cm1. There is a sharp increase in 00 ¼ Imð~Þ at almost the resonance frequency. The real part of the dielectric function 0 experiences oscillation across the resonance frequency. Regarding the dielectric function, the normal dispersion is associated with an increase in 0 with frequency, and the anomalous dispersion is the reverse. Normal dispersion can be found everywhere except in the vicinity of the resonance frequency oj. In addition, in the region where anomalous dispersion occurs, the imaginary part of the dielectric function becomes comparable or larger than the real part of the dielectric function. Therefore, the material becomes highly reﬂective near the resonance frequency, and the radiation inside the material is rapidly attenuated or dissipated. The spectral region with a large imaginary part of the dielectric function is also called the region of resonance absorption. Because phonon absorption only affects a relatively narrow band, the extinction coefﬁcient is rather small for lo10 mm and lW28 mm. The normal emissivity plotted in Figure 18(c) is calculated for a semiinﬁnite (opaque) medium, using method discussed previously, and for a 0.5-mm-thick ﬁlm, using the formula that will be detailed in Section 6.1. The emissivity oscillates with wavelength due to the variation of optical constants. However, a resonance peak in k does not necessarily correspond to a maximum emissivity. This is because a large k will result in a large reﬂectance, especially when kcn (i.e., when 0 o0), and consequently, a low emissivity. Furthermore, there exists a wavelength at which the emissivity becomes unity, that is, at l ¼ 9.5 mm for sapphire. This wavelength is called the Christiansen wavelength at which n ¼ 1 and k{1 so that the reﬂectance is nearly zero [30,49]. On contrary, the photon penetration depth dl ¼ 1/al ¼ l/(4pk) is less than a few tens of micrometers, small enough for a slab with a thickness of 100 mm to be considered as opaque at this wavelength. Hence, the emissivity will be almost 1 and the material behaves like a blackbody at the Christiansen wavelength. This phenomenon is called the Christiansen effect. Because of this unique characteristic, the Christiansen effect has been used to calibrate radiation thermometers and to determine IR emissivity of dielectric materials [49252]. Interestingly, the Christiansen wavelength depends little on the angle of incidence and temperature [50,52]. In addition, it is nearly the same for both the ordinary and extraordinary rays for a uniaxial crystal [44]. Table 1 lists the Christiansen wavelength for a number of insulators, mostly taken from Ref. [44], except that of Si3N4 is from Ref. [49], those of LaAlO3 and LaGaO3 are from Ref. [53], and that of ZrO2 is from Ref. [54]. In some polar semiconductors, such as GaAs, InSb, and ZnSe, the Lorentz model can also describe the contribution of the phonon vibration to the dielectric function in the IR. At the wavelength corresponding to n ¼ 1, due to free-carrier contributions, the extinction coefﬁcient is too large to obtain a near-zero reﬂectance or close-to-unity emissivity.

116 Table 1

Zhuomin M. Zhang and Bong Jae Lee

Christiansen wavelength of several materials [44,49,53,54].

Name

Chemical formula

Christiansen wavelength (mm)

Aluminum oxide Aluminum oxynitride Barium ﬂuoride Barium titanate Beryllium oxide Calcium ﬂuoride Cesium bromide Lanthanum aluminate Lanthanum gallate Lithium tantalate Magnesium ﬂuoride Magnesium oxide Potassium bromide Silicon carbide (6 H) Silicon dioxide Silicon nitride Strontium titanate Titanium dioxide (rutile) Yttrium oxide Zirconia

Al2O3 AlON BaF2 BaTiO3 BeO CaF2 CsBr LaAlO3 LaGaO3 LiTaO3 MgF2 MgO KBr SiC SiO2 Si3O4 SrTiO3 TiO2 Y2O3 ZrO2

9.5 9.5 20.9 12.8 8.0 16.0 78.0 12.5 14.0 10.7 12.6 11.8 50.0 10.0 7.4 8.3 11.5 11.8 14.5 13.0

5.3. Optical constants of a semiconductor The optical constants of a semiconductor largely depend on its crystalline structure, temperature, and impurity levels. In this section, emphasis is given to single-crystal silicon because of its common use in the semiconductor industry. There have been numerous studies on the optical and radiative properties of silicon materials. The fundamental physics has been well established regarding the crystalline structure, electron and phonon dispersion, and scattering mechanisms [55]. The absorption processes associated with the band-gap absorption, free-carrier absorption, and lattice absorption are well understood. Model expressions, which relate the optical constants or sometimes the dielectric function to the wavelength and temperature, have been developed to cover certain spectral and temperature ranges with good accuracy. However, even though numerous functional expressions for the optical constants of silicon exist, none of them covers the entire range of wavelengths and temperatures of interest. Several functional expressions for optical constants of lightly doped silicon can be found in the review paper by Timans [23]. Measurement data of the optical constants of lightly doped silicon (doping concentration less than 1015 cm3) over wide wavelength regions

Theory of Thermal Radiation and Radiative Properties

117

at room temperature can be found in the handbook edited by Palik [44]. However, few experimental data exist at high temperatures. Sato [56] was the ﬁrst to comprehensively study the emissivity of silicon wafers from direct measurements as well as from the reﬂectance and transmittance measurements in the temperature range between room temperature and 8001C. The refractive index values based on his experimental data are consistently higher than those obtained from recent studies. In the following section, several functional expressions are selected to illustrate the optical constants of silicon in the VIS and near-IR region at different temperatures. Figure 19(a) shows the refractive index of silicon at 251C and 8001C, in the wavelength region from 0.5 to 1.5 mm. The refractive index decreases slightly as the wavelength increases. At room temperature, the refractive index values based on each expression agree well with the data compiled in Ref. [44]. Since no experimental data exist at wavelengths between 0.84 and 1.1 mm, the expressions of Jellison and Modine [57] and Li [58] are separately extrapolated to this wavelength region, which is between the two vertical dash-dotted lines. The refractive index obtained from these expressions differs in the extrapolated region, and the difference becomes larger at higher temperatures. A difference in the refractive index may cause an error in the prediction of radiative properties and in the temperature measurement by radiation thermometers. The absorption coefﬁcient al of lightly doped silicon is plotted for silicon in Figure 19(b) at 251C and 8001C, in the wavelength region from 0.5 to 1.5 mm. The squares represent the values converted from the extinction coefﬁcient in Ref. [44] and the circles indicate the data by Saritas and McKell [59]. Since no theoretical functions exist in the wavelength region between 0.84 and 1.1 mm, the Jellison and Modine expression [57] is extrapolated to longer wavelengths, while the expression of Timans [60] coupled with the free-carrier absorption formula [61] is extrapolated to shorter wavelengths. The agreement between the calculated extinction coefﬁcient values and the measured data from both Palik’s handbook [44] and Saritas and McKell [59] is very good at room temperature. However, as in the case of the refractive index, the disagreement between the expressions increases with temperature. Although it is not shown in the ﬁgure, lattice absorption occurs in the wavelength range between 6 and 25 mm. It should be noted that the effect of lattice absorption is negligible in most RTP applications compared to the band-gap absorption as well as free-carrier absorption. For heavily doped Si at room temperature, ionized donors or acceptors produce free carriers, which can greatly enhance the absorption in the IR region. The Drude free-electron model has been used to describe the freecarrier absorption of doped Si in many studies [23,62265]. Fu and Zhang [64] combined the interband absorption, lattice vibration, and free-electron

118

Zhuomin M. Zhang and Bong Jae Lee

4.4 Jellison-Modine [57] Li [58]

4.2

Refractive Index

Palik [44] 4.0

3.8 800°C

3.6

25°C

(a) 3.4 0.5

0.7

0.9

1.1

1.3

1.5

Wavelength (μm) 105

Jellison-Modine [57] Timans [60] Palik [44] Saritas et al. [59]

Absorption Coefficient (cm−1)

800°C 104

103

25°C

102

101

(b) 100 0.5

0.7

0.9

1.1

1.3

1.5

Wavelength (μm)

Figure 19 Optical constants of lightly doped silicon at selected temperatures [45]: (a) refractive index; (b) absorption coefﬁcient.

absorption to model the dielectric function of silicon from low to high doping concentrations at temperatures from 300 to 1,000 K. More recently, Basu et al. [65] compared different models for the mobility and carrier concentration of heavily doped silicon near room temperature.

Theory of Thermal Radiation and Radiative Properties

119

An improved Drude model was developed for doping concentration exceeding 1018 cm3 and validated by reﬂectance and transmittance measurements in the wavelength region from 2 to 20 mm with an FTIR spectrometer [65].

6. Radiative Properties of Layered Structures The radiative properties of surfaces can be strongly affected by thinﬁlm coatings [6]. If the multiply reﬂected waves interfere with each other, the radiative properties for a nondispersive medium exhibit evenly spaced fringes when plotted in terms of the wavenumber (1/l). In this case, the multiply reﬂected waves are coherent and the ﬁlm is said to be ‘‘thin.’’ On contrary, a ‘‘thick’’ layer refers to the case when the effect of interference between multiply reﬂected waves can be neglected in calculating the radiative properties. In other words, the waves are incoherent. In general, when the thickness is comparable to the wavelength, interference effects become important. However, this does not guarantee complete coherence in reality because of the nature of the source and imperfect surfaces (e.g., roughness and nonparallelism), resulting in partially coherent waves. Partial coherence theory considers the intermediate regime between the coherent limit and incoherent limit [66,67]. In the following, the radiative properties of ﬁlms are discussed by using the ray-tracing method for thick ﬁlms and thin-ﬁlm optics for thin ﬁlms. Because all radiative properties are presented on spectral basis, the subscript l is omitted for simplicity in notation. Furthermore, discussions are limited to the case of optically smooth surfaces.

6.1. The ray-tracing method Figure 20 shows a thick ﬁlm or slab placed between two dielectrics (media 1 and 3). The refractive index and the extinction coefﬁcient of the ﬁlm are n2 and k2, respectively, and the thickness is d2cn2l. It should be noted that if the extinction coefﬁcient is comparable to the refractive index, the transmittance of a thick ﬁlm will be negligible and the ﬁlm can be regarded as a semi-inﬁnite medium. Therefore, for a thick ﬁlm, it is usually appropriate to assume that the layer is slightly absorbing, that is, k2{n2. The refractive index of medium 1 or 3 is indicated by n1 and n3, respectively. In this ﬁgure, rij represents the reﬂectivity at the interface between medium i and medium j (i, j ¼ 1, 2, or 3) and can be calculated for each polarization using Equation (86b) or (87).

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Zhuomin M. Zhang and Bong Jae Lee

n1

1

12

(1 − 21)

21

(1 − 12) 2

n2 , 2

d2

23

n3

3

(1 − 23)

Figure 20 Schematic of the ray-tracing method for calculating the reﬂectance and transmittance by a thick ﬁlm.

By tracing the energy of all reﬂected rays, the directional-hemispherical spectral reﬂectance of a thick ﬁlm can be expressed as: R0l ¼ r12 þ r23 ð1 r12 Þð1 r21 Þt2i þ ð1 r12 Þð1 r21 Þr21 r223 t4i þ ¼ r12 þ

r23 ð1 r12 Þ2 t2i 1 r12 r23 t2i

ð100Þ

where ti is the internal transmissivity given by: 4pk2 d 2 a2 d 2 ¼ exp ti ¼ exp l cos y2 cos y2

(101)

Here, y2 is the angle of refraction inside the slab. In general, the angle of refraction is complex due to the absorption. For a slightly absorbing medium with k2{1, however, y2 can be determined using Snell’s law by neglecting absorption [68]. It should be noted that the absorption by the layer is accounted by the internal transmissivity. Similarly, the directionalhemispherical spectral transmittance can be obtained as: T 0l ¼

ð1 r12 Þð1 r23 Þti 1 r12 r23 t2i

(102)

For randomly polarized incidence, the radiative properties should be averaged over s and p polarizations. The directional spectral absorptance can be obtained from A0l ¼ 1 R0l T 0l and it is equal to the directional spectral emittance (or emissivity) according to Kirchhoff’s law. In the case of emissivity of a slab in air, we obtain 0l ¼

ð1 rÞð1 ti Þ 1 rti

(103)

121

Theory of Thermal Radiation and Radiative Properties

where r is the reﬂectivity at the air2medium interface. This equation has been used to calculate the normal emissivity of a semitransparent Al2O3 slab (see Figure 18).

6.2. Thin-ﬁlm optics When the thickness of each layer is comparable or less than the wavelength of incident radiation, the wave interference effects inside a layer become important. Figure 21 shows a thin ﬁlm with thickness d2. Here, rij and tij represent the Fresnel reﬂection and transmission coefﬁcient, respectively, at the interface between medium i and medium j (i, j ¼ 1, 2, or 3). In order to account for wave interference effects, the amplitude and phase of electric ﬁeld (for s polarization) or magnetic ﬁeld (for p polarization) need to be taken into account. After superposition of multiply reﬂected waves considering phase shifts upon traveling in the layer, the reﬂection and transmission coefﬁcients of the ﬁlm can be expressed as: r ¼ r 12 þ

t 12 t 21 r 23 ei2k2z d2 1 r 21 r 23 ei2k2z d2

(104)

and t¼

t 12 t 23 eik2z d2 1 r 21 r 23 ei2k2z d2

(105)

The above equations are known as Airy’s formulae [39,69]. Based on the reﬂection and transmission coefﬁcients, the directional-hemispherical spectral reﬂectance and transmittance can be calculated as: R0l ¼ jrj2 1, 1

1

(106)

t21

r12

x z

t12

r21

2

d2

2, 2 r23

3, 3

3

t23

Figure 21 Illustration of the Fresnel reﬂection and transmission coefﬁcients at the boundaries of a thin ﬁlm.

122

Zhuomin M. Zhang and Bong Jae Lee

1

Transmittance

0.8 0.6 0.4 Thin-film optics

0.2

Free spectral range

Ray-tracing method 0 1000

1200

1400

1600

1800

2000

Wavenumber (cm−1)

Figure 22

Normal transmittance of a 30-mm-thick silicon wafer.

8 Reðk3z =m3 Þ 2 > > jtj ; > < Reðk1z =m1 Þ 0 Tl ¼ > Reðk3z =3 Þ 2 > > : Reðk = Þ jtj ; 1z 1

for s polarization (107) for p polarization

Again, A0l ¼ 1 R0l T 0l and Kirchhoff’s law states that 0l ¼ A0l . For randomly polarized incidence, the radiative properties should be separately calculated based on Fresnel coefﬁcients for s and p polarizations and then averaged over two polarizations. Figure 22 shows the transmittance of a silicon wafer calculated based on either the thin-ﬁlm optics (solid line) or the ray-tracing method (dashed line). The ﬁrst and last media are assumed to be air (n ¼ 1). The optical constants of silicon are taken from the tabulated values [44] and the wafer thickness is set to be 30 mm in the calculation. When wave interference in the ﬁlm is important, the resulting transmittance oscillates periodically as a function of wavenumber. The period of oscillation, that is, the separation between two consecutive interference maxima (or minima), is called the free spectral range Dn (in cm1) that is given by: Dn ¼

1 2n2 d2 cosy2

(108)

where n2 is the refractive index of silicon and d2 is its thickness. In contrast, the ray-tracing method (or geometric optics) completely ignores the interference effects and results in no fringe. In many practical applications, partial coherence must be taken into consideration to predict the radiative properties. In reality, partial coherence is often caused by the limited resolution of the instrument, beam divergence, as well as nonparallelism and surface roughness of the sample [6,9,10].

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Theory of Thermal Radiation and Radiative Properties

1 = 1, 1 = 1

1 x

2, 2

2

d2

N−1, N−1

N−1

dN−1

N, N

N z

Figure 23 structure.

Geometry for calculating the radiative properties of a multilayer

6.3. Radiative properties of multilayer structures The matrix formulation provides a convenient way to calculate the radiative properties of multilayer structures [6,69,70]. For the jth medium (refer to Figure 23), the relative electric permittivity is ej and the relative magnetic permeability is mj. The thickness of the jth layer is denoted by dj. It is assumed that e1 ¼ 1 and m1 ¼ 1, that is, the top semi-inﬁnite medium is air or vacuum. The electromagnetic wave is incident from air at an angle of incidence y1 and is reﬂected from or transmitted through the following layers. For s-polarized waves, the y component of electric ﬁeld in the jth medium can be expressed as a summation of forward and backward waves in the z-direction: ( ik z A1 e 1z þ B1 ek1z z eiðkx xotÞ ; j ¼ 1 Ey ðx; zÞ ¼ ikjz ðzzj1 Þ þ Bj ekjz ðzzj1 Þ eiðkx xotÞ ; Aj e

j ¼ 2; 3; . . . ; N

(109)

where Aj and Bj are the amplitudes of forward and backward waves in the jth layer, respectively, z1 ¼ 0 and zj ¼ zj1 + dj( j ¼ 2, 3, y, N1), and kx and kjz are the parallel and perpendicular components of the wavevector, respectively. Notice that kx is conserved throughoutqthe layers due to the ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

phase-matching condition, and kjz is given by kjz ¼ mj j o2 =c 20 k2x . The magnetic ﬁeld can be obtained from the electric ﬁeld using Maxwell’s equation. From the boundary conditions at the interface, ﬁeld amplitudes

124

Zhuomin M. Zhang and Bong Jae Lee

of adjacent layers can be related as: Aj Bj

! ¼

ðPj D1 j Djþ1 Þ

Ajþ1 Bjþ1

! ; j ¼ 1; 2; . . . ; N 1

(110)

Here, Pj is the propagation matrix given by: P1 ¼ I ¼

1 0

0 1

(111a)

and Pj ¼

eikjz dj

0

0

eikjz dj

! ;

j ¼ 2; 3; . . . ; N 1

(111b)

In addition, Dj is the dynamical matrix given as: Dj ¼

!

1

1

kjz =mj

kjz =mj

;

j ¼ 1; 2; . . . ; N

(112)

By applying Equation (110) to all layers, we obtain A1 B1

! ¼

N1 Y

Pj D1 j Djþ1

j¼1

AN BN

! (113)

Consequently, the spectral radiative properties of the N-layer system are B1 B 1 A21

(114)

ReðkNz =mN Þ AN A N Reðk1z =m1 Þ A21

(115)

R0l ¼

and T 0l ¼

For p-polarized waves, the same procedure can be applied using the magnetic ﬁeld with mj’s being replaced by ej’s. A more detailed derivation suitable for both nonmagnetic and magnetic materials can be found from Ref. [70]. Figure 24 shows the normal spectral transmittance of a one-dimensional multilayer structure as a wavelength-selective ﬁlter with a center wavelength at 550 nm. The structure is composed of alternating SiC (nHE2.69) and SiO2 (nLE1.47) thin ﬁlms. In order to suppress the

125

Theory of Thermal Radiation and Radiative Properties

1

M1

0.6

0.4

M2

Transmittance

0.8

0.2

0 0.3

0.4

0.5

0.6

0.7

0.8

0.9

Wavelength (μm)

Figure 24 Normal transmittance of a multilayer structure for a spectrally selective ﬁlter, which is constructed by two sets of four unit cells of SiC/SiO2 ﬁlms with different periods.

transmission above and below the center wavelength, two sets of SiC/SiO2 photonic crystal are used with different periods. The ﬁlter performance is optimized by varying the thickness of SiC and SiO2 layers. For the top structure M1, the thicknesses of SiC and SiO2 are set to be 66 and 121 nm, respectively. For the bottom structure M2, the SiC and SiO2 thicknesses are set to be 33 and 61 nm, respectively. Due to the wave interference effects, the ﬁlter provides higher transmittance at wavelengths between 500 and 600 nm. Hence, the SiC/SiO2 structure can be used for ﬁlter radiometers [71].

6.4. Thin ﬁlms on a thick substrate In many cases, one needs to consider structures of thin ﬁlms coated on a thick substrate, where waves inside the ﬁlms must be treated as coherent but waves in the substrate can be treated as incoherent. Figure 25 shows the geometry of a thick substrate with thin-ﬁlm coatings on both sides. Here, rta is the reﬂectance of the top multilayer ﬁlms for rays incident from air, assuming that the substrate extends to inﬁnity; whereas rts is the reﬂectance for rays incident from the substrate to the top ﬁlms. Note that the transmittance tt of the top multilayer ﬁlm is the same regardless of whether the ray is incident from air or the substrate, when absorption inside the substrate is negligibly small. Similarly, rbs and tb are the reﬂectance and transmittance, respectively, for the multilayer ﬁlms at the bottom of

126

Zhuomin M. Zhang and Bong Jae Lee

ta

1

ds

s

t

t

Air

ts

bs

b

1

Substrate

Air

Figure 25 Schematic of thin-ﬁlm coatings on both sides of a thick substrate.

the substrate for rays incident from the substrate. The transfer matrix formulation discussed in the previous section can be separately applied to calculate the reﬂectance and transmittance of the thin ﬁlms at the top and bottom, by treating the substrate as a nonabsorbing dielectric. Nevertheless, the absorption of the substrate can be accounted for by introducing the internal transmissvity ti, as done in the ray-tracing method discussed previously. Consequently, the radiative properties of the substrate with thin-ﬁlm coatings in the semitransparent region can be expressed as [23,45]: R0l ¼ rta þ

t2i t2t rbs 1 t2i rts rbs

(116)

ti tt tb 1 t2i rts rbs

(117)

0l ¼ 1 R0l T 0l

(118)

T 0l ¼

and

Lee and Zhang [72] developed a software tool, named Rad-Pro for Radiative Properties, using VBA programming in Excel. It allows users to easily calculate and plot the directional, spectral, and temperature dependence of the radiative properties of silicon wafers with various thin-ﬁlm coatings. In Rad-Pro, users can choose to use the formulation for coherent and incoherent, as well as other options, such as the polarization state of incident radiation and the option that treats the silicon substrate as a semi-inﬁnite medium (opaque). This software tool and user

127

Theory of Thermal Radiation and Radiative Properties

1 Thin films coated on Si substrates 0.8 Reflectance

(v) Curves: Calculation Marks: Measurement

0.6

(i)

0.4

(iv)

0.2

(ii) (iii) 0 0.5

0.6

0.7

0.8

0.9

1

Wavelength (μm)

Figure 26 Near-normal reﬂectance of silicon substrates with coatings [45]: (i) bare substrate with 4-nm native SiO2, (ii) 128-nm Si3N4 ﬁlm, (iii) 306-nm SiO2 ﬁlm, (iv) 868-nm SiO2 ﬁlm, and (v) 50-nm polysilicon and 170-nm SiO2 ﬁlms.

manual are downloadable free of charge at the following website: www.me.gatech.edu/Bzzhang. A few examples are given below to illustrate the effect of thin ﬁlms on the radiative properties. Figure 26 compares the calculated reﬂectance with that measured at near-normal incidence, in the spectral region from 0.5 to 1 mm, using a calibrated spectrophotometer at the National Institute of Standards and Technology [45]. In this region, the silicon wafer is opaque. Excellent agreement exists between the predicted and the measured values. The reﬂectance with one layer coating never exceeds that of the bare silicon. For the nitride-coated wafer, there exists a reﬂectance minimum of o0.003 at l ¼ 1 mm. For the oxide-coated wafer, interference effects can be clearly seen and the period decreases with increasing ﬁlm thickness. The variation is the largest in the reﬂectance spectra for the silicon wafer with a 50 nm polysilicon ﬁlm on top of a 170 nm silicon-dioxide coating. Figure 27 shows the calculated normal emissivity for a 700-mm-thick silicon wafer at different temperatures at wavelengths from 0.5 to 5 mm. The silicon wafer is assumed to be lightly doped and polished on both sides. The optical constants of polysilicon are assumed to be the same as those of silicon. Note that silicon wafers are semitransparent at wavelengths longer than the band-gap wavelength lg. Hence, the emissivity depends on the thickness of the wafer. At 251C, free-carrier absorption is negligibly small so that the emissivity at wavelengths longer than lgE1.1 mm is almost zero. At 5001C, thermally excited carriers can result in signiﬁcant absorption and subsequently a large emissivity beyond 1.1 mm. With multilayer coatings,

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1 300 nm SiO2, 25 °C 300 nm SiO2, 500 °C

0.8

Emissivity

800 nm SiO2, 500 °C 0.6

0.4

0.2 Polysilicon/SiO2 0 0.5 0.6

0.8

1

3

5

Wavelength (μm)

Figure 27 Calculated normal emissivity of coated silicon of 700-mm thickness at different temperatures. In the case of the polysilicon and SiO2 coatings, the thicknesses are 100 and 300 nm, respectively.

the emissivity can be close to unity at certain wavelengths and very small at some other wavelengths. The correction of radiometric temperature measurement caused by the uncertainty in emissivity posts a signiﬁcant challenge in lightpipe thermometers used in RTP industry. Additional discussion on the application of lightpipe thermometry in semiconductor processing can be found from Chapter 3 in the companion volume, Vol. 43 of this series.

7. Summary Beginning with a discussion of the historical perspectives in the development of the thermal radiation theory, this chapter describes the thermodynamics foundation of the Stefan2Boltzmann law and the Planck law of blackbody radiation. Deﬁnitions of the radiative properties and their relationships are then presented, for example, emissivity, absorptivity, and reﬂectivity for opaque materials, as well as the reﬂectance, transmittance, and absorptance for semitransparent materials. Brief discussions on effective emissivity and radiative transfer in participating media are also given. Maxwell’s electromagnetic wave theory is introduced for the prediction of reﬂection and transmission at the interfaces as well as in multilayer structures. Furthermore, frequency-dependent dielectric function models are described for metals, dielectric materials or insulators, and semiconductors, in the VIS and IR regions. The fundamental concepts and basic

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theories described in this chapter are essential for the practice of radiation thermometry described in subsequent chapters.

ACKNOWLEDGEMENT Partial support was provided by the National Science Foundation (CBET-0500113, 0828701).

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P. J. Timans, ‘‘Emissivity of Silicon at Elevated Temperatures,’’ J. Appl. Phys. 74, 635326364 (1993). P. Vandenabeele and K. Maex, ‘‘Inﬂuence of Temperature and Backside Roughness on the Emissivity of Si Wafers during Rapid Thermal Processing,’’ J. Appl. Phys. 72, 586725875 (1992). P. J. Hesketh, J. N. Zemel, and B. Gebhart, ‘‘Polarized Spectral Emittance from Periodic Micromachined Surfaces. 2. Doped Silicon-Angular Variation,’’ Phys. Rev. B 37, 10803210813 (1988). J. P. Hebb, ‘‘Pattern Effects in Rapid Thermal Processing’’, Ph.D. Dissertation, Massachusetts Institute of Technology, Cambridge, MA (1997). C. J. Fu and Z. M. Zhang, ‘‘Nanoscale Radiation Heat Transfer for Silicon at Different Doping Levels,’’ Int. J. Heat Mass Transfer 49, 170321718 (2006). S. Basu, B. J. Lee, and Z. M. Zhang, ‘‘Infrared Radiative Properties of Heavily Doped Silicon at Room Temperature,’’ J. Heat Transfer (accepted). G. Chen and C. L. Tien, ‘‘Partial Coherence Theory of Thin-Film Radiative Properties,’’ J. Heat Transfer 114, 6362643 (1992). K. Fu, P.-F. Hsu, and Z. M. Zhang, ‘‘Uniﬁed Analytical Formulation of Thin-Film Radiative Properties Including Partial Coherence,’’ Appl. Opt. 45, 6532661 (2006). Z. Zhang, ‘‘Optical Properties of a Slightly Absorbing Film for Oblique Incidence,’’ Appl. Opt. 38, 2052207 (1999). P. Yeh, Optical Waves in Layered Media, Wiley, New York (1988). C. J. Fu, Z. M. Zhang, and D. B. Tanner, ‘‘Energy Transmission by Photon Tunneling in Multilayer Structures Including Negative Index Materials,’’ J. Heat Transfer 127, 104621052 (2005). M. Durak and F. Samadov, ‘‘Realization of a Filter Radiometer-Based Irradiance Scale with High Accuracy in the Region from 286 nm to 901 nm,’’ Metrologia 41, 4012406 (2004). B. J. Lee and Z. M. Zhang, ‘‘Rad-Pro: Effective Software for Modeling Radiative Properties in Rapid Thermal Processing,’’ in Proceedings of the 13th IEEE International Conference on Advanced Thermal Processing of Semiconductors-RTP 2005, Santa Barbara, CA, pp. 2752281 (2005).

[61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72]

CHAPTER 4

Radiation Thermometer Designs Howard W. Yoon and George P. Eppeldauer

Contents 1. 2. 3. 4.

Introduction Design Considerations Basic Optical Designs Component Characterizations 4.1. Photodiode selection 4.2. Preampliﬁer design 4.3. Preampliﬁer gain calibration 4.4. Spectral ﬁlters 5. System-Level Characterizations 5.1. Spectral responsivity measurements 5.2. Size-of-source effect 5.3. Characterization of linearity 6. Radiation Thermometry below the Silver Point 6.1. Design considerations for an InGaAs detector-based thermometer 6.2. Detector optimization 6.3. Temperature-dependent detector performance 6.4. Optical performance 6.5. Blackbody measurements and interpolation functions 6.6. Uncertainties 7. Calibration of Radiation Thermometers Using ITS-90 8. Thermodynamic Temperature Measurements Using Radiometric Techniques 8.1. Radiance responsivity calibration of AP1 8.2. Summary of systematic corrections to AP1 8.3. Uncertainty estimates of thermodynamic temperature Using AP1 and similar instruments 9. Summary References

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Optical Technology Division, National Institute of Standards and Technology, Gaithersburg, MD 20899-8441, USA Experimental Methods in the Physical Sciences, Volume 42 ISSN 1079-4042, DOI 10.1016/S1079-4042(09)04204-0

r 2010 Elsevier Inc. All rights reserved.

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1. Introduction Since the publication of the book ‘‘Theory and Practice of Radiation Thermometry’’ [1] in 1988, the designs of radiation thermometers have undergone drastic changes. With the exception of few industries, disappearing ﬁlament pyrometers are not commonly used. The improvements in detector performance have led to an expansion of detector-based radiation thermometry in which the temporally stable response of these improved detectors allows the calibration to be maintained. Also, the improvements in micro-bolometer array performance along with lower costs have led to inexpensive hand-held thermal imaging cameras for nondestructive testing and thermal diagnostic uses. Although thermal imagers can be calibrated for use as transfer standards, the lowest measurement uncertainties are obtained using single spot-radiation thermometers that are capable of viewing small-aperture ﬁxed-point blackbodies (FPBBs). In this chapter, the instrument design and characterization of spot-radiation thermometers capable of transferring temperature scales with the lowest uncertainties will be discussed. Radiation thermometers can be calibrated using the International Temperature Scale of 1990 (ITS-90) [2] or using absolute radiometry for thermodynamic temperature determinations, although the latter is still rare. The chapter is also limited to radiation thermometers that operate in the visible [3] to the near-infrared [4] wavelength regions. The choice of the detector, optimization of the detector for stability and linearity, and characterization of radiation thermometers for size-of-source effect (SSE) will be discussed.

2. Design Considerations The primary consideration in the design of a radiation thermometer is the desired range of operational temperatures. Figure 1 shows the Planck radiances for blackbodies ranging in temperatures from the freezing point of indium, or 430 K, to 3,000 K. Due to the nonlinear relationship between radiance and temperature, the central wavelength of the radiation thermometer should be at shorter wavelengths than the peak wavelength of the Planck radiances for the temperature of interest. For the range of temperature from 1501C to 9621C (Ag freezing temperature) and for temperatures greater than 9621C, the center wavelengths of commonly used radiation thermometers are at 1,550 and 650 nm, respectively. The identiﬁcation of the temperature range of use is also important since the ITS-90 temperature traceability is different for the two temperature ranges. If the temperature range is limited to those ranges then, for the highest

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107 106

Spectral Radiance [W/(cm2 sr cm)]

105 104 103 102 101 100 10-1

In Sn Zn Al Ag Au T3000K

10-2 10-3 10-4 10-5 10-6

500

1000

1500

2000

2500

3000

Wavelength [nm]

Figure 1 The Planck radiances for blackbody temperature ranging from the freezing point of In to 3,000 K. The center wavelengths of commonly used radiation thermometers are indicated by the two lines at 650 and 1,550 nm.

performance thermometers, these considerations restrict the types of detector and optical designs to those using Si or InGaAs diodes with regular glass optics. It is important to have low-uncertainty photocurrent measurement systems or transimpedance current-to-voltage ampliﬁers since the proper design of the ampliﬁers is critical in obtaining linear, temporally stable photocurrent measurements for the lowest uncertainties. The following sections discuss the optical designs, the choice of detectors, the choice of spectral ﬁlter, the current-to-voltage ampliﬁer, and characterizations of the spectral responsivity, linearity, and SSE of radiation thermometers. Finally, calibrations of radiation thermometers for measurements of ITS-90 and thermodynamic temperatures are considered.

3. Basic Optical Designs Radiation thermometers typically have either a reﬂective optic or a refractive lens to collect the radiation and to deﬁne the angular ﬁeld of view (AOV) and the target size. As shown in Figure 2, the simplest optical setup

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Blackbody

Field Stop

Lens

Preamplifier

FOVAOV Aperture Stop

Filter Detector

Figure 2 The schematic of a simple radiation thermometer showing the blackbody source, the objective lens, ﬁeld stop, spectral ﬁlter, detector, and the preampliﬁer. In this design, the aperture stop is in front of the lens before the ﬁeld stop.

Blackbody

Lens

Field Stop

Aperture Stop

Filter

Preamplifier

Detector

Figure 3 The schematic of a simple radiation thermometer with a Lyot stop showing the blackbody source, the objective lens, ﬁeld stop, spectral ﬁlter, detector, and the preampliﬁer. The aperture stop is behind the ﬁeld stop to act as a Lyot stop.

consists of an objective lens that focuses the radiation onto a ﬁeld stop which limits the area of collection or target size. If the analysis is limited to geometric optics, the AOV is simply the angle subtended by the diameter of the aperture stop and the object distance. The target area is the area of the ﬁeld stop multiplied by the magniﬁcation factor of the object distance over the image distance. A spectral ﬁlter is usually placed between the collected radiation and the detector to limit the spectral range of the radiation transmitted. The spectrally selected radiation is incident on the photodiode producing the photoinduced current, which is converted to a voltage using a preampliﬁer. Each of these items in turn will be discussed further in this chapter. Another variation of the radiation thermometer is to have additional optics, as shown in Figure 3, to collimate or refocus the radiation with additional lenses after the ﬁeld stop. These additional lenses enable the placement of the aperture stop after the ﬁeld stop. In this design, the collimated radiation is transmitted through a large area of the spectral ﬁlter to reduce possible effects due to spatial inhomogeneities of the ﬁlter. The impact of this design on the resulting SSE will be discussed later in this chapter. Due to the developments in optical modeling software, the above simple designs are easily modeled using commercial software to determine, among many other optical properties, imaging quality such as the Strehl ratio, the effect of aberrations, and chromatic focal shift. It is also possible to design custom lenses for the desired characteristics to improve the imaging

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quality or to reduce the chromatic focal shift. It is recommended that such modeling optical software be used in trying to design a new radiation thermometer or to understand the operations of an existing radiation thermometer. It is also possible to model the scatter and the diffraction from the internal elements of the radiation thermometer with more sophisticated versions of the optical modeling software.

4. Component Characterizations 4.1. Photodiode selection Photodiodes are photovoltaic detectors that usually have advantageous electronic and radiometric characteristics. Single element Si and unextended InGaAs photodiodes are the best choices for the 20021,650 nm wavelength range. As these photodiodes are made up of different materials, they have different energy bands. Incident photons, with energy hn greater than the energy band gap, create electron2hole pairs in the semiconductor at or near the junction. Typical spectral radiant power responsivity characteristics of Si, Ge and InGaAs photodiodes are shown in Figure 4. The nonuniformity of spatial responsivity of these devices has to be matched to the uncertainty requirement of a radiant power responsivity 1

Responsivity [A/W]

0.8

Si 1227 photodiode Si 1337 Ge InGaAs

0.6

0.4

0.2

0

400

800

1200

1600

Wavelength [nm]

Figure 4 Spectral power responsivities of frequently used detectors between 200 and 1,800 nm.

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measurement. Across the whole active surface, the spatial nonuniformity of responsivity (ratio maximum-to-minimum) of high-quality, large-area silicon photodiodes is between 0.3% and 0.5%, and that of InGaAs photodiodes is 0.5% or higher. Ge photodiodes have nonuniformities of about 1% or higher. Selection of photodiodes with good spatial uniformity of responsivity may be necessary if radiant power responsivity measurements are needed with reduced relative uncertainties. For broadband applications, the spectral dependence of the spatial uniformity should be tested. Several photodiode radiometric characteristics that need to be known are temperature dependence of responsivity (if temperature control is not applied), linearity [5], and stability. The temperature-dependent responsivities of four photodiodes in regular use for radiometry applications are shown in Figure 5. At low irradiance levels, the linearity of silicon photodiodes has been veriﬁed in an extremely wide dynamic range of 11 decades. An example over six decades is shown in Figure 6. The linearities of InGaAs diodes have also been demonstrated, and the range is shown to be limited by the low shunt resistance of InGaAs diodes when they are not cooled to below ambient temperatures. Detector electronic characteristics that can inﬂuence radiation thermometer performances are shunt resistance, parallel (junction) capacitance, and noise of the detector and the attached measuring circuit. The shunt resistance determines the voltage gain for the input noise and drift. The

Si and Ge responsivity change %/°C

0.020

InGaAs

3

0.015 2

Ge

0.010

Si 1227 Si 1337

0.005

1

0

0.000

InGaAs responsivity change %/°C

4

0.025

-0.005 400

600

800 1000 1200 1400 1600 1800 Wavelength [nm]

Figure 5 The temperature-dependent responsivities of the four photodiodes shown in Figure 4.

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0.6

Relative error / %

0.4 0.2 0 -0.2 -0.4 -0.6 0.01

0.1

1

10

100

1,000

10,000

Detector current / nA

Figure 6 The nonlinearity in a silicon detector determined using the combinatorial method. Three separate sets of 120 measurements are plotted.

parallel detector capacitance has to be known to determine the gain equations of the photocurrent measuring electronic circuit [6]. For different photodiode impedances, different circuit solutions are needed to obtain the lowest measurement uncertainty and lowest noise [7]. The noise ﬂoor determines the sensitivity and dynamic range of photocurrent measurements. Photodiodes are usually linear down to the noise ﬂoor dominated by the circuit components of the current-to-voltage converter. The noise ﬂoor for Si photodiodes is in the 1015 A range. Recently, thermoelectrically cooled regular InGaAs photodiodes can be obtained commercially with shunt resistances equal to that of high-quality silicon photodiodes. Since all noise components in the detector circuit for these InGaAs photodiodes are similar to silicon photodiodes, silicon-type performance can be achieved using these cooled InGaAs detectors [8]. These siliconperformance InGaAs detectors can be used in radiation thermometers where high sensitivity is needed.

4.2. Preampliﬁer design Since the photocurrent generated in the photodiode is converted to a voltage by a current-to-voltage preampliﬁer, it must be matched to the characteristics of the detector if high measurement performance is needed [9]. The simplest version of a widely used photocurrent meter is shown in Figure 7. The photocurrent of photodiode P (that has a shunt resistance RS and a parallel capacitance CJ) is measured by a short-circuit current meter. The current-to-voltage conversion in the current meter is realized by an operational ampliﬁer (OA) and a feedback resistor R that has a parallel (stray) capacitance of C.

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C R P OA V

Figure 7 Simpliﬁed circuit diagram of a photocurrent meter.

The load resistance (input resistance of the current meter) RI for the photodiode is [10]: RI ¼

R A

(1)

where A is the open loop gain of the OA. For linear operation, the requirement for short-circuit photocurrent measurements is RI{RS. If the nonlinearity requirement is lower than 0.01%, RI must be four orders of magnitude smaller than the shunt resistance of the detector. For lowuncertainty photocurrent measurements, the photodiodes have to be selected for high shunt resistance, typically 225 GO for a Si photodiode for radiometric applications. After selecting the photodiode, the other passive components of the feedback can be calculated. Usually, R is changed in decade increments to cover a wide dynamic signal range. The feedback resistors R of the OA can change from about 1 kO to 100 GO. A photocurrent meter (which is an analog control loop) can operate in the linear regime when these feedback resistors are used. At R ¼ 1012 O, the current-to-voltage conversion becomes nonlinear. Using R ¼ 1011 O, the settling time of the output voltage V can be 2 min. With the right design, RS and R should be similar to each other and R ¼ 10 GO is usually enough to perform very low noise in the output voltage V of the current meter. At the OA output, the signal-produced voltage V has to be always much greater than the superimposed total noise voltage originating from the noise components of the photodiode current meter. The signal-tonoise ratio at the current meter output has to satisfy the uncertainty requirement of a measurement. The signal-to-noise ratio also depends on the noise pickup. To avoid noise pickup, the wires between the photodiode pins and the OA inputs must be short, rigid (moving free), and electrically shielded. Best results can be obtained if the photodiode and the preampliﬁer are combined in an electrically shielded (metal) measuring head. The power supply should be located in a separate shielded box which is connected to the measuring head with a shielded cable. The shielding to the measuring head should propagate from the electric power-line outlet

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Radiation Thermometer Designs

(using a three-pin line connector) through the shield of the cables. In order to avoid unwanted signal pickup through ground loops, the common (ground) lead of the preampliﬁer circuit should be combined with the electrical shield in one point only, which should be the 0 V output pin of the power supply (typically 715 V). The high-resistance components in the preampliﬁer, including the printed circuit board (PCB) shown in Figure 8, should be kept clean to eliminate shunt effects such as leakage currents. These issues are more fully explored in Ref. [9]. PCB C1 R1 C2 R2 C3 R3 C4 R4 C5 R5 R6 R7

4

2 photodiode

3

1 OA 8

7

6

5 104 V/A 1010 V/A rotary switch

+15V 0V

A H E

Figure 8

-15V

P1

B shield D

BNC out

Offset zero

Circuit diagram of a simple photocurrent meter.

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Howard W. Yoon and George P. Eppeldauer

4.3. Preampliﬁer gain calibration The radiation thermometer measurements over the temperatures of interest could result in signal changes that span more than six decades. Although the diodes themselves might be linear, the resulting signal is a combined output from the diode and the preampliﬁer, and thus the linearity of the preampliﬁer gains must be assessed. It would be difﬁcult to compensate the radiation thermometer for large nonlinearities, even if the thermometer was used only as an interpolating instrument, since the nonlinearities would be a part of the interpolating function. The dc signal gain (V/IP) of a preampliﬁer must be known or calibrated against high accuracy standards, such as resistor or current-source standards. In order to avoid tedious and limited accuracy signal-gain calibrations, the feedback resistors (R) should be purchased with a 0.01% uncertainty (tolerance) of their decade nominal values. Such resistors are commercially available from 1 kO to 100 MO with a resistance temperature coefﬁcient of 10 ppm/1C. The R ¼ 1 GO signal gain can be calibrated against the R ¼ 100 MO signal gain by measuring the same constant (stable) input current (or optical radiation when the detector is connected to the preampliﬁer input). In order to utilize the 0.01% uncertainty of resistors R, a beam shutter should be used at the input of the radiometer. The output offset voltage of the preampliﬁer can be canceled out from the measurement by subtracting the dark reading (shutter closed) from the signal plus dark reading (shutter open) at the output of the preampliﬁer (or the following digital voltmeter).

4.4. Spectral ﬁlters Radiation thermometers that operate in the visible or the near-infrared wavelength region are typically constructed using spectral ﬁlters to restrict the incident radiation on the diode to a selected wavelength region. The spectral ﬁlters can be constructed using interference layers deposited on glass substrates or using broadband absorbing glasses [11]. The interference ﬁlters are constructed using multilayer thin ﬁlm deposition that can be much more susceptible to temperature and humidity changes than the broadband absorbing glass ﬁlters. Since the spectral transmittance of the interference ﬁlters is critically dependent on the ﬁlm thickness, any change in the ﬁlm thickness due to temperature or humidity-related expansion of the thin ﬁlms would lead to an unwanted change in the spectral transmittance. The increasing use of ‘‘hard’’ or ion-assisted deposited thin ﬁlms has reduced temperature and humidity-related changes [12]. For the most stable operation, the spectral ﬁlters should be temperature stabilized. If the thermal link to a heat sink is an issue, then the ﬁlters could be stabilized at slightly higher than the ambient temperatures to reduce the electrical power usage and the thermal load.

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Radiation Thermometer Designs

5. System-Level Characterizations Once the radiation thermometer is constructed from the individual components consisting of the lenses, detector, and preampliﬁer into a system, system-level characterizations are needed. The desired uncertainties of these system-level characterizations depend on the temperature range and the desired use of the radiation thermometer. For the realization of ITS-90 below 9621C, the spectral characterization of the radiation thermometer should be performed to conﬁrm that the spectral responsivity is not seriously affected by overlap with any of the atmospheric absorption lines. The atmospheric transmittance for a 1-m path at sea level and 296 K temperature is shown in Figure 9. Since the temperature calibration of the radiation thermometer is performed using a set of FPBBs or a variabletemperature blackbody (VTBB), a low-uncertainty spectral responsivity characterization is not critical. The nonlinearity characterizations are also not critical since the interpolations using the blackbody source account for nonlinearities in the radiation thermometer. For the realization of the ITS90 above 9621C, the spectral responsivity measurements and the linearity characterizations are critical since both measurements are used along with 1.0

Transmittance

0.8

0.6

0.4

0.2

0.0 1000

1500

2000

2500

3000

3500

4000

4500

5000

Wavelength, nm

Figure 9 ditions.

The atmospheric transmittances for 1 m path at sea level and 296 K con-

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Howard W. Yoon and George P. Eppeldauer

one of the Ag-, Au-, or Cu FPBBs to extrapolate the temperature scale to higher temperatures. The SSE characterizations are still needed in both cases since the radiation thermometer is used with sources of different apertures.

5.1. Spectral responsivity measurements 5.1.1. Measurement geometry The calibration of detector spectral responsivity requires the use of standard detector. Although thermopile or pyroelectric detectors could be used to derive the scale, modern radiometric realizations are performed using cryogenic, electrical substitution radiometers. Thermopiles and pyroelectric detectors have low responsivities, which result in high noise levels at the typical output from a monochromator. Since the primary standard cryogenic radiometers measure radiant power, the calibration transfer starts from power mode typically using a Si-trap detector. The radiant power mode measurement geometry is illustrated for a laser beam in Figure 10. The uniformity of the spatial responsivity of detectors, calibrated in underﬁlled mode, has to be sufﬁciently high to keep the relative standard uncertainty low for power measurements. Measurement of stabilized laser beams is commonly applied in responsivity calibrations where the reference detector is a cryogenic radiometer or a multiple-input-reﬂection Si-trap detector standard. In spectral radiant power responsivity measurements, where broadband sources are used with either narrow-band ﬁlters or monochromators, the incident beam is usually convergent onto the trap detectors. These calibrated trap detectors are then used in either laser- or monochromatorbased spectral response measurement facilities. Since only a few national measurement institutes (NMIs) have laser-based facilities, the more commonly used monochromator-based facility will be ﬁrst described. The laser-based facility will be described later in the section on realization of thermodynamic temperatures. The details of the responsivity calibrations can be found in Ref. [13]. 5.1.2. Monochromator-based determination of spectral responsivity At most NMIs, monochromator-based measurement setups are used for determining spectral responsivity. Best results are obtained if similar type Power in laser beam detector aperture

Figure 10 Measurement geometry of radiant power measurement.

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Radiation Thermometer Designs

test and working standard detectors are used that have high sensitivity in a limited wavelength range. A typical monochromator-based responsivity measurement setup is shown in Figure 11. A broadband lamp is imaged onto the input slit of the monochromator using a lens or mirror imaging system. Mirrors, unlike lenses, have no internal reﬂections and are preferable for wide spectral range applications because they show no chromatic aberrations. Spherical mirrors are most commonly used for imaging, operated at near-normal angles of incidence to reduce off-axis aberrations. Spherical aberration is usually negligible for slow (i.e., f/8 or greater) mirrors. Double monochromators are used in high accuracy applications where high stray light rejection is needed. Single monochromators have higher throughput but lower performance in terms of spectral stray light. The dispersing elements are either prisms or gratings. The dispersion of prisms is highly nonlinear. Therefore, when a constant spectral bandpass is needed, elaborate cams have to be used for both the wavelength and the slit-width drives. It is difﬁcult to ﬁnd prism materials that have high enough transmission and dispersion in a wide wavelength range. Prism monochromators are more complex, hence, more expensive than grating versions. However, the use of grating-based monochromators is not without limitations. Gratings have anomalies that render their spectral efﬁciency highly irregular, and they need order sorting (cut-on) ﬁlters. The output slit is imaged at the surface of the detectors. A light shutter can be used to subtract the output offset voltage of the detector2ampliﬁer from the output signal. A monochromator-based setup can be used for irradiance mode calibrations if the output beam of the monochromator is collimated and spatially uniform. For radiance responsivity calibrations in the case of radiation thermometers, an integrating sphere with a large enough exit port lamp

Detector stage Standard

monochromator filter

mirror mirrors Shutter

Test detector

Figure 11 Spectral responsivity measurement setup using a monochromator.

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Howard W. Yoon and George P. Eppeldauer

can be used, which is irradiated by the monochromatic output beam from the monochromator. In this case, the exit port radiance of the sphere can be determined against an irradiance mode detector standard. If the signal from the integrating sphere is not sufﬁcient, then the exit slit or the opening of the monochromator can be directly imaged by the radiation thermometer. The output from the monochromator will not be spatially or angularly uniform nor will it be truly monochromatic due to the ﬁnite opening of the exit slits.

5.2. Size-of-source effect In many ﬁelds of radiometry, the unknown radiances from a deﬁned target area are determined by comparisons to reference radiance sources that can have different sizes and shapes of radiating surfaces. In photometry, such measurements determine the luminance of the target; and in radiation thermometry, the radiance temperature of the target is measured. Because the requirements for the uncertainties in the temperature determinations are so stringent, the need for correcting the contributions of the surroundings is critical in radiation thermometry. In practice, radiation thermometers are used to assign the radiance temperatures of either tungsten-strip lamps or large-area VTBBs in comparison to those of FPBBs that have typically small, 3- to 6-mm diameter openings to increase their temperature uniformities/effective emissivities [14]. These radiance comparisons can be between sources of different diameters and shapes, and thus it is important to characterize the systematic corrections or the SSE, which are necessary when comparing radiance sources of different diameters [15,16]. For comparison of FPBBs with the lowest uncertainties, the systematic corrections for the SSE can be one of the dominant terms in the total uncertainty budget [17]. Although SSE has been extensively studied with application to radiation thermometry [18], the effort to reduce out-of-ﬁeld scatter has a much longer history in the development of coronagraphs [19]. The problem of measuring the corona, or the electromagnetic activity of the sun, is similar to the problem in the rejection of stray radiation from blackbody sources in that the intense radiation from the solar disk must be rejected in order to study the corona. A coronagraph, or a telescope dedicated to the imaging of the corona, must create an artiﬁcial eclipse within the telescope, and imaging the sun onto an opaque target can achieve good rejection of the direct solar radiation. The placement of the aperture stop behind the ﬁeld stop is attributed to Lyot in the development of the coronagraph [20]. Furthermore, the characterization of the radiation thermometer to reduce the SSE when viewing sources with different source sizes is analogous to the characterizations required to assess the radiometric rejection properties of the coronagraph.

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Radiation Thermometer Designs

In radiation thermometry, the SSE has been measured using a uniform variable-diameter radiance source with a central obscuration (‘‘indirect method’’) [21] or without the central obscuration (‘‘direct method’’). Comparison studies have also been performed using blackbody cavities and radiance sources to study the effect of volume radiation of the blackbody on the SSE [22]. All three methods have been shown to be equivalent [23]. These studies have shown that SSE measured with a source diameter of 50 mm can be reduced to o1 102 in industrial radiation thermometers and to o1 103 in reference radiation thermometers. Even with large SSE, systematic corrections can be made, but these corrections require knowledge of the spatial distribution of the source radiance, and the spatial uniformity can also be dependent on time-varying environmental conditions. The reduction of the SSE to o5 105 would eliminate the need for measurement of the spatial uniformity, and the SSE could then be considered as one of the uncertainty components in the total uncertainty budget.

5.2.1. Measurements of SSE Investigation of the SSE could be performed using a light-tight optical bench as shown in Figure 12. The light-tight box around the base plate is constructed of anodized, extruded aluminum. The walls of the light-tight box are formed of opaque-black plastic and covered with an internal black applique´ [24]. A circular opening of 64 mm diameter with a sharp, beveled edge is machined on the front plate to act as the front bafﬂe, and the objective lenses are placed on a 6-cm long V-groove mount placed behind this opening. The V-groove optical mount is designed for quick interchange of objective lenses for testing purposes. An adjustable iris with a sharp, well-deﬁned inner edge is placed between the front bafﬂe and the objective lens, and is used to reduce the aperture to 44 mm diameter so that Collimating lens Objective Lens

f(c) i(o)

Front baffle

Adjustable Iris

Photodiode Field Stop

i(c) Lyot stop

Figure 12 The schematic of the radiation thermometer used for the investigation of SSE. The objective lens is placed on a V-groove mount for quick and reproducible changes of the lens.

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Howard W. Yoon and George P. Eppeldauer

the ground-glass edge of the objective lens is not illuminated. The side of the iris facing into the radiometer is also covered with the black applique´, to reduce the back reﬂectance onto the objective lens. The 1- and 0.6-mm diameter ﬁeld stops are fabricated from a bimetal BeCu/Ni substrate with black antireﬂective coated and metal-reﬂective sides. The ﬁeld stop is tilted about 101 from the optical axis to avoid back reﬂections toward the objective lens. The direct reﬂection from the ﬁeld stop is directed toward the front corner of the radiometer, and since the entire inside of the radiometer is covered with low-reﬂective black covering, the direct reﬂectance from the ﬁeld stop is suppressed. The metallic side of the ﬁeld stop is placed toward the object side for use in possible alignments with laser illumination and to visually align sources. The ﬁeld stop, the aperture stop, the collimating lens, and the detector are all placed on movable holders on an optical bench so that the physical conﬁguration could be quickly changed. A variable-aperture iris could also be placed between the ﬁeld stop and the detector to act as a limiting aperture stop or a Lyot stop. A quasi-monochromatic radiance source of 50 mm diameter is used for these studies. The source utilizes 36 LEDs placed on a 50-mm diameter circuit board and can be temperature stabilized using a thermoelectric (TE) cooler. Since the light emitting diode (LED) source is nearly monochromatic and the spectral radiance is centered at 660 nm, spectral ﬁlters are not used. The spatial uniformity of the LED source is measured to be varying by 72% over the central 40-mm region with the edge radiance falling to about 60% of the central radiance. The temporal stability of the LED source was monitored using a photodiode attached to the side of the LED source. The low-reﬂectance central obscuration was constructed with a hard polymer shaped into a conical-cavity shape to reduce the back reﬂectance [25]. The 2-mm central obscuration was glued into place in the centered hole in the front ground-glass diffuser plate. The effective diameter of the LED source was varied using a variable-aperture steel iris, which could be adjusted from 2 to 50 mm diameter. The physical separation of the iris to the front diffuser was minimized to reduce the back reﬂections from the rear of the iris into the central obscuration. Although an LED source was used for these measurements, typically, integrating sphere sources (ISS) are used with a central obscuration. The central obscurations are constructed using opaque materials formed into a small blackbody to reduce the reﬂectance, and placed on transparent windows to be placed over the opening of the ISS. The central obscurations are made with larger diameters than the target size to avoid scattering from the edges of the obscuration. Due to the limited sizes of integrating spheres, it is difﬁcult to measure large W30 cm diameter SSE. Alternatively, if the radiation thermometer is stable, a small integrating sphere or another Lambertian source could be translated in the x and y planes to simulate a

149

Radiation Thermometer Designs

much larger diameter source. Studies have shown the equivalence between the direct, the indirect, and the translation methods [22]. 5.2.2. SSE determination The SSE contribution of different objective lenses and changing optical designs were tested using the indirect method by plotting the ratios of signals, sðd; d0 Þ ¼

nðL; dÞ nðL; d0 Þ nðLÞ

(2)

where d is the diameter of the uniform radiance source, d0 the diameter of the central obscuration, and L the radiance of the source. The signal measured while viewing the unobstructed radiance source is n(L); the signal measured while viewing the central obscuration with the diameter of the radiance source, d, is n(L, d ); and the signal measured with the diameter of the variable aperture at the diameter of the central obscuration, d0, is n(L, d0). If stray radiation during the measurements is suppressed, then the signal n(L, d0) should be equal to the signal with the radiance source turned off. Optimally, the signal from monitor diode attached to the side of the source should be used to determine if the radiance of the source changed for any reason during the measurements. The output from ISSs could depend on the size of the opening. The measurement of the SSE could also be performed using the direct method without the central obscuration, sðdÞ ¼

nðdÞ nðref Þ nðdÞ

(3)

where n(d ) is the signal when viewing the center of the source and n(ref ) the signal at the largest diameter of the source. In both types of measurements, the radiance of the source should be constant, and monitor detectors on the source should be used to determine whether the source has remained stable with changes in the output diameter of the opening. The spectral radiance from the ISS is often dependent on the opening diameter since the ratio of the area of the opening versus the total internal area changes [26]. 5.2.3. Importance of the Lyot stop The SSE measurements shown in Figure 13 are performed with the Lyot stop (or glare stop). The placement of the Lyot stop is determined by the focal lengths of the objective and the collimating lenses in Figure 12. The purpose of the Lyot stop is twofold. The Lyot stop is used to block radiation originating from internal scatter away from the objective lens; it is also used to block the diffracted radiation from the edge of the objective lens. The

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Howard W. Yoon and George P. Eppeldauer

2.5×10-4 500 mm and 300 mm fl 500 mm and 500 mm fl GRIN 200 mm fl 300 mm fl achromat 200 mm fl achromat 250 mm fl achromat

SSE σ ( d, 2 mm)

2.0×10-4

1.5×10-4

1.0×10-4

5.0×10-5

0.0 0

10

20 30 40 Source Diameter (d ) [mm]

50

Figure 13 The measured SSE for different objective lenses. All the measurements were performed with the Lyot stop. The lenses with smaller number of elements have lower SSE from the reduced surfaces for scattering with a strong dependence on the surface ﬁnish.

Lyot stop then becomes the limiting aperture stop in the radiation thermometer, and should be placed at the image distance of the collimating lens with the distance from the collimating lens to the objective as the object distance, 1 1 1 þ ¼ iðoÞ þ f ðcÞ iðcÞ f ðcÞ

(4)

where i(o) is the image distance of the objective lens or the distance from the objective lens to the ﬁeld stop, f (c) the focal length of the collimating lens, and i(c) the image distance or the Lyot stop distance from the collimating lens. Placement of the aperture stop at any other position after the collimating lens would not be as effective in blocking the diffracted radiation from the objective lens and would not be considered a Lyot stop. If multiple-element objective lenses are used, then there would be a corresponding increase in the number of stops since the scattering from the individual surfaces in the objective lens system would appear as volume scatterers. The importance of using a Lyot stop is shown in Figure 14. The SSE was measured using the same 250-mm focal length (ﬂ) achromat lens with the Lyot stop and also without the collimating lens and the Lyot stop. For

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Radiation Thermometer Designs

1.6×10-3 without Lyot stop with Lyot stop

1.4×10-3

SSE σ ( d, 2 mm)

1.2×10-3 1.0×10-3 8.0×10-4 6.0×10-4 4.0×10-4 2.0×10-4 0.0 0

10

20

30

40

50

Source Diameter (d) [mm]

Figure 14 SSE comparison of 250 mm focal lens, PAC088, with and without the Lyot stop. The SSE increases by 20 at 50 mm diameter without the Lyot stop.

measurements without the Lyot stop, the Si diode was placed directly behind the ﬁeld stop. The SSE increases rapidly as the diameter of the source is increased resulting in a 20-fold increase in the SSE at 50 mm as compared to the measurements with the Lyot stop. The diameter of the Lyot stop depends on the focal lengths of the objective and the collimating lenses since the de-magniﬁcation factor, M, is M¼

iðoÞ þ f ðcÞ iðcÞ

(5)

Thus, for a 200-mm focal length objective lens with a 50-mm focal length collimating lens, the de-magniﬁcation factor, M, is 6.66. In this case, a 6mm diameter Lyot stop was effective in reducing the SSE. This corresponded to a 6 mm 6.66D40-mm diameter aperture stop at the plane of the lens leading to a reduction of the effective objective lens diameter by about 4 mm. Further reductions in the Lyot stop diameter would lead to lower SSE but at the expense of decreasing the throughput of the radiometer. 5.2.4. Understanding the source of SSE An analogy of the indirect method of measuring SSE and dark-ﬁeld imaging in microscopy can be made to understand the contribution of

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scatter from the objective lens. In dark-ﬁeld imaging as illustrated in Figure 15 (left), the annular stop leads to illumination of the sample at oblique angles. The direct illumination is not collected by the microscope objective, and only the scattered light from the sample is collected by the microscope objective. In the SSE measurements as shown in Figure 15 (right), the annular stop is replaced by the radiance source with a central obscuration, and the sample is replaced by the objective lens of the radiation thermometer. If there were perfect imaging with no scattering, then no radiation would be collected by the objective just as no radiation would be scattered into the ﬁeld stop of the radiation thermometer. If the scattering within the radiation thermometer is reduced, then any radiation through the ﬁeld stop is primarily from the scattering from the objective lens. Any defects or particulate contamination on the objective lens acts as a secondary source at the plane of the objective lens. Increasing the diameter of the SSE test source increases the radiation incident on the defects on the objective lens or the irradiance (W/cm2). The increase in the irradiance on any particle leads to a corresponding increase in the radiance of the particle resulting in an increase in the SSE. Thus, it is critical to reduce the irradiance on the objective lens by the use of front bafﬂes and lowreﬂectance inner walls on the radiometer. Since the defects on the objective Scattered radiation

Microscope objective

Sample

Annular stop

Direct radiation

Field Stop

Radiometer Objective Lens

SSE Source

Condenser lens

Figure 15 (Left) Measurement schematic of dark-ﬁeld microscopy. The situation is analogous to the ‘‘indirect’’ method of SSE determination with the objective lens as the sample and the source being the annular stop and the radiation from the condenser lens. (Right) Optical schematic of the SSE measurement. Only the scattered radiation is collected by the ﬁeld stop showing that the SSE determinations measure the scattered transmittance of the objective lens.

Radiation Thermometer Designs

153

lens act to collect radiation, it is important to shield the objective lens from stray radiation as much as possible. Since any surface contamination can lead to an increase the scatter, prior to the SSE measurements, all lenses should be cleaned. In the experimental arrangement shown in Figure 12, the SSE was found to be critically dependent on the cleanliness of the lens with an increase in the SSE by about 10% or more with few dust particles on the surface of the objective lens. Since cleaning residues also scatter radiation, any cleaning marks also lead to an increase in the SSE. The scattering from large particulate contamination on the objective lens and from defects on/in the objective lens is difﬁcult to describe and may depend on the size, the shape, and the material of the scatterer. Small particles scatter primarily in the forward direction as described by Rayleigh scattering [27], and since the scattering has a 1/l4 wavelength dependence, one would expect that reducing the SSE would be more difﬁcult at the shorter wavelengths as well. If the scattering is more isotropic than forwarddirected, then the use of a long image distance could be beneﬁcial in reducing the contribution of objective lens scatter. Due to the contribution of scatter from the objective lens in increasing the SSE, any windows placed between the objective lens and the blackbody also lead to an increase in the SSE. The imperfections and the particulate contaminations on the window surfaces act to scatter radiation into the ﬁeld stop. Thus, if any changes are made to the optical path from the source to the radiation thermometer, the SSE should be measured prior to the use of the modiﬁed radiation thermometer. The placement of the aperture stop is also important in reducing the SSE. If the aperture stop is placed in front of the objective lens, then the diffracted radiation from the edge of the aperture stop cannot be removed. Just as the radiance of the objective lens defects increases as the incident irradiance is increased from the larger diameter of the SSE source, the amount of diffracted radiation increases from the greater irradiance incident at the edge of the aperture stop. If the limiting aperture is placed in front of the objective lens, then the SSE is expected to increase as the aperture stop diameter is decreased since the diffracted edge is pulled in closer to the center. However, if a Lyot stop is used, then the SSE should decrease as the Lyot stop diameter is decreased since more of the diffraction from the edge of the objective lens is removed. Thus, the placement of the aperture stop is important in understanding the dependence of the SSE on the aperture stop diameter. 5.2.5. Importance of optical performance Poor imaging performance of the objective lens can also lead to an increase in the SSE. The main considerations for optical performance can be divided into chromatic aberrations and other optical aberrations such as spherical

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Howard W. Yoon and George P. Eppeldauer

aberrations, coma, and astigmatism, among others. The chromatic aberrations are considered separately since the radiometer can be refocused to minimize chromatic aberrations while the other types of optical aberrations require a redesign of the lens. Most commercial achromatic doublets are optimized for inﬁnite conjugate and can provide diffractionlimited performance under inﬁnite-conjugate operation. These lenses are not diffraction limited with 1:1 or 2:1 ﬁnite-conjugate ratios, but custom, achromatic lens designs can be optimized for diffraction-limited performance at 1:1 conjugate ratios or any other conjugate ratios. The calculated optical performances of the lens combinations evaluated for SSE in Figure 13 are shown in Table 1. Due to the limitations of the modeling program that was used, the Gradient Index (GRIN) lens was not evaluated. The imaging performance is evaluated using Optical Software for Layout and Optimization (OSLO) from Lambda Research although any other optical modeling program can be used. The calculations are performed at 650 nm with a 1-mm diameter object size and 44-mm diameter aperture top at the entrance of the objective lens system. The imaging at the focal plane of the ﬁeld stop is quantiﬁed by comparing the diffraction-limited radius with the geometric-root-mean-squared (RMS) radius and by the Strehl ratio. If the ratio of the diffraction-limited radius and the geometric-RMS radius is o1, then the system can be considered to be diffraction limited with other sources of optical aberrations well controlled. The Strehl ratio is a measure of the point-spread function and is equal to 1.00 for a perfect imaging system. In Table 1, the optical imaging quality of the doubled achromat objectives is evident from the near unity Strehl ratio and the diffractionlimited performance. Doubled achromats with matching or mismatched focal lengths are known for diffraction-limited imaging performance; however, due to the inter-reﬂections between the lenses the SSE is increased by about a factor of four over that of a single lens as evidenced in Figure 13. The measurements in Figure 13 for the double achromats were performed with the same lenses used for the single achromat measurements Table 1 SSE.

The calculated imaging performance of the lens combinations tested for

Achromat focal Diffraction limited Geometric RMS Ratio geometric/ length (mm) radius (mm) radius (mm) diffraction radii

Strehl ratio

500 300 300 250 200

0.990 0.921 0.158 0.112 0.071

0.00872 0.00532 0.01070 0.00886 0.00589

0.00122 0.00211 0.04410 0.05040 0.04880

0.14 0.40 4.12 5.69 8.29

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Radiation Thermometer Designs

so that the SSE should be not subject to the differences in the quality of the lens surfaces. Although the optical performances of the single achromats used at ﬁnite conjugate is far from diffraction limited, the SSE was found to be lowest with the single achromats. At longer wavelengths, the optical performance of the single achromats improves. Table 2 shows the optical modeling of the PAC088 lens at selected wavelengths where radiation thermometers can be constructed. The imaging performance as given by the Strehl ratio increases from 0.112 at 650 nm to 0.310 at 2,250 nm, demonstrating that radiometers designed in the visible wavelengths can also be used in near-infrared wavelength region with improved performance. There are other considerations in the near infrared such as the effect of the antireﬂective coating on the lens and the lens transmittance. If a single achromat with diffraction-limited optical performance is desired, then custom lenses can be designed. In Table 3, an example of an optimized lens design found using OSLO is given. Although many different glass types can be chosen for the optimization of optical performance, the most common achromat Schott glass combinations are chosen for this example. The optical performance of the optimized achromat is listed in Table 4 showing the nearly diffraction-limited performance at the wavelengths of interest. With such lenses, if the surface scatter is controlled then low SSE with good imaging is possible. Furthermore, with the choice of ﬂint-glass and crown-glass elements in the achromat, the objective lens Table 2

The optical performance of the PAC088 lens at different wavelengths.

Wavelength (nm)

Diffraction limited spot radius (mm)

Geometric Ratio diffraction/ spot radius (mm) geometric radii

Strehl ratio

900 1,550 2,250

0.0124 0.0219 0.0336

0.0527 0.0545 0.0556

0.144 0.243 0.310

4.25 2.49 1.65

Note: The optical performance improves at longer wavelengths.

Table 3 The lens parameters for the 1:1 imaging optimized lens with a focal length of 200 mm at 650 nm. Surface

Radius of curvature (mm)

Center thickness (mm)

Material

R1 R2 R3

102.70 73.52 317.68

10.0 4.0 2

BK7 SF5 2

Note: The optimized lens is constructed using common achromat glass types.

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Optical performance of the 1:1 optimized achromat lens from Table 3.

Wavelength (nm)

Diffraction limited Geometric spot spot diameter (mm) diameter (mm)

Ratio diffraction/ geometric spot diameters

Strehl ratio

650 900 1,550 2,250

0.00589 0.00816 0.0144 0.0219

1.02 0.74 0.56 0.31

0.808 0.791 0.851 0.955

0.00598 0.00601 0.00811 0.00684

Note: The lens is found to be diffraction limited in performance from 650 to 2,250 nm; the lens performances are evaluated at the focus.

can be made to have small chromatic focal shifts at 650 and 900 nm avoiding the need to refocus at the two wavelengths. Even with the best optical designs, the fabrication of objective lenses for the lowest scatter could be difﬁcult since many different vendors with varying fabrication capabilities exist. For instance, a contribution to SSE can arise from the scattering from the glue between the elements of the achromat. Although the surface ﬁnish of the lens can be speciﬁed by a scratch-and-dig criterion, the glue scatter cannot be speciﬁed using the surface ﬁnish criterion and would be dependent on the control of the manufacturing process. The increased scatter from the surface and interface imperfections from a vendor’s lens could be visually observed with a bright light source. Higher scattering is immediately observed with a greater SSE. If the radiation thermometer is to be used for only radiance ratios of blackbody sources, then optical performance is not so critical. Since there is very little spatial nonuniformity in the cavities of FPBBs or standard-quality blackbodies, achieving diffraction-limited performance can be sacriﬁced for low-scatter criterion. Lens-based optical systems perform optimally for onaxis designs with worse optical performance as the imaging goes off-axis. Since radiation thermometers are designed for on-axis performance with small o3-mm diameter target areas, the optical aberrations are minimized. If sources with greatly different spatial extents are to be compared, then an assessment of the optical performance of the point-spread function or the Strehl ratio should be determined.

5.2.6. Impact of SSE The sources of SSE in radiation thermometers are due to the internal design as well as the scatter properties of the objective lens system. If the inside of the radiometer is not coated with low-reﬂective material, then secondary sources could be present from the internal reﬂections, which can be reﬂected back into the objective lens. To control the reﬂections within the

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Radiation Thermometer Designs

radiometer, the internal dimensions should be such that the multiple reﬂections can be controlled and reduced. The sensitivity of the radiometers to SSE can be converted to temperature using the derivative of the Wien approximation, DL c 2 DT ¼ l T2 L

(6)

where L is the spectral radiance, T the temperature, c2 the second radiation constant, and l the center wavelength of the radiation thermometer. The Wien approximation is valid since measurements with the Si-diode-based radiation thermometers are all performed at a shorter wavelength region from the peak of the Planck radiance for temperatures r3,000 K. The corrections for SSE are converted to temperatures using Equation (6). From the use of Equation (6), these measurements demonstrating SSE of 3 105 with a 25-mm diameter source indicate that SSE effects of o2 mK at 1,337 K and o9 mK at 3,000 K with a radiation thermometer centered at 650 nm can be achieved.

5.3. Characterization of linearity 5.3.1. Importance of linearity Radiation thermometers are often used to assign the spectral radiance or radiance temperatures of sources using ratios of signals that can differ by ﬁve decades or more. For performing ratios between such sources with low uncertainties, the linearity of the detectors used in the transfer radiometers needs to be characterized. The linearity characterizations are not so important if the radiation thermometer is used in a temperature range where interpolations are possible. The linearity characterization becomes especially critical at temperatures above 9621C, where high temperatures are determined using extrapolations from lower temperatures or signals. 5.3.2. Physical causes of nonlinearity The linearity of photodiodes with respect to continuous radiation is mainly inﬂuenced by two effects: ﬁrstly, the local minority carrier distribution within the semiconducting material; secondly, the current path through the device. The ﬁrst effect depends on the penetration depth of the optical radiation and may result in a wavelength-dependent nonlinearity, if deep electron or hole traps within the semiconductor material affect the minority carrier diffusion length (particularly in lower grade silicon detectors). Second, as a photodiode is never an ideal diode, ohmic resistances take place along the current path. Generally at high irradiance levels, the path and terminal resistance connected in series to the junction of the diode determine the photocurrent (i.e., short-circuit output current). In addition,

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nonlinearities can occur if the photodiode is overﬁlled since any incident radiation outside the active area of the photodiode could be trapped and not be converted to photocurrent. The nonlinearity with overﬁlling of the photodiode is especially problematic with InGaAs diodes [28]. 5.3.3. Measurement of linearity There are essentially three recognized methods for assessing the linearity of optical and infrared detectors [29]: the superposition method [30,31], the attenuation method [32], and the differential or ac method [33]. Both the superposition and the differential methods utilize linearity, but the superposition method, ﬁrst proposed by Coslovi and Righini [30], most clearly exploits the same principles as the combinatorial approach. The linear scale factors are typically determined in separate comparisons with standard sources, such as lamps or blackbodies. The superposition method is most commonly used in the form known as ﬂux doubling [34]. The result, a, can be obtained from any measurement of a rational attribute (quantity), A, of an artifact X. If the linearity test is carried out with two test artifacts X with attributes of the same magnitude, then 2AðXÞ Að2XÞ ¼ 2xðaÞ xð2aÞ

(7)

This shows that the instrument error at 2a can be found in terms of the two measurements and the nonlinearity at a. Equation (7) is applied recursively so that the nonlinearity is determined at 2a, 4a, 8a, and so on, with each expressed in terms of the sequence of measurements and the nonlinearity at a, the lowest radiance measurement. Thus, the nonlinearity is determined at a sequence of exponentially spaced points, except for one piece of information. It turns out that the error in the ﬁrst point is required only for absolute radiometry [35]. Thus, the ﬂux-doubling method enables measurement of nonlinearity and an additional calibration point is required to anchor the detector response absolutely. This doubling technique is commonly utilized in radiation thermometry by alternatively introducing neutral density ﬁlters to change the signal and then also by adjusting the source to perform the measurement at another signal level. In this way, the entire signal range of over ﬁve decades could be tested with a set of neutral density ﬁlters and a VTBB source. Unfortunately, the ﬂux-doubling method produces sparse, exponentially spaced data with cumulating uncertainties. It also, unlike overdetermined systems, has no surplus of data for demonstrating the validity of the nonlinearity model or for estimating uncertainties. A version of the superposition method that sums light over two paths, as with the ﬂux-doubling method, but with three ﬁlter wheels inserted into

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Radiation Thermometer Designs

Beamsplitter

Detector

Mirror

Filter wheel 3 Filter wheel 2

Filter wheel 1

Mirror Beamsplitter

Figure 16

Stabilised source

Simpliﬁed schematic diagram of the detector linearity testing facility.

the system [36], is shown in Figure 16. Filter wheels 1 and 2 both have ﬁve controlled positions as indicated by indices i and j, respectively, with i ¼ 0 and j ¼ 0 corresponding to an opaque shuttered position. Filter wheel 3 has six controlled positions indicated by index k, with k ¼ 0 being a shuttered position. The total ﬂux through the two paths is additive so that, acalc ði; j; kÞ ¼ f1 ði; kÞ þ f2 ðj; kÞ

(8)

For a single path there are 30 possible ﬁlter combinations, of which 10 are nominally shuttered. Thus, there are a total of 20 nonzero combinations of ﬁlters, or 20 different ﬂux levels, for each path. The shuttered positions are used to compensate for the dark current of the detector, so are not included among the valid combinations. With all possible combinations of i, j, and k, there are 120 nonzero ﬂux levels. The system is a realization of the combinatorial method with 120 measurements and 40 unknown parameters. A least-squares ﬁt is used to ﬁt the 40 artifact parameters and simultaneously determine the distribution of the nonlinearities. For the best solid-state detectors, no correction equation is necessary so the leastsquares minimization function is s2 ¼

1 XXX ½ameas ði; j; kÞ acalc ði; j; kÞ2 80 i j k

(9)

Standard uncertainties of 0.02% have been achieved for silicon detectors, and 0.04% for InGaAs detectors.

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6. Radiation Thermometry below the Silver Point Accurate determination of radiance temperatures from 1501C to 1,0001C is important for the calibration of radiation thermometers that are used in turn for process control or to transfer the calibration to other radiation thermometers. In this temperature range, one can establish a scale on radiation thermometers from the ITS-90 either by using the ITS-90 FPBBs from the In-point to the Ag-point with an interpolation equation or by using a VTBB whose temperatures are determined by contact platinum resistance thermometers (PRT) [37]. In both approaches, the FPBB or the VTBB emissivity has to be measured or modeled so that the lowest uncertainties in the transfer to the radiation thermometer can be achieved. International comparisons of scales in this temperature range have revealed that the lowest uncertainties in radiance temperatures can be achieved using the ﬁxed-point interpolation technique instead of using a VTBB [38]. Uncertainties from the use of FPBBs are possibly lower since the temperatures are well known and high spatial temperature uniformity of the cavity results in a high-emissivity blackbody. VTBBs are expected to have larger uncertainties arising from difﬁculties in achieving uniform temperature distributions within the cavity and from differences in temperature between the contact thermometer and that of the radiating surface of the blackbody, thus resulting in larger uncertainties in the emissivity of the blackbody. Although the method of calibration using ﬁxed points is well established, commercial radiation thermometers that are temporally stable and able to view the small cavity openings of the FPBBs are not readily available. Furthermore, custom InGaAs radiation thermometers constructed by the NMIs have conﬂicting designs [39,40]. Although reduction of SSE is a critical design feature due to the need to compare small-area FPBB cavities to large VTBB openings, implementation of the Lyot stop has been lacking. Furthermore, in the past, methods to decrease the noise-equivalent current of the radiation thermometer have not been extensively explored. A generalized design for an InGaAs-based radiation thermometer capable of measurements from the In-point to the Ag-point is described. This device also incorporates a Lyot stop while utilizing a low-scatter objective lens. The construction utilizes carbon2epoxy rods for structural stability with temporal stability of the responsivity increased by the use of a room-temperaturestabilized ﬁlter and cooled detector. The detector is contained in a hermetically sealed package to decrease the detector temperature below ambient temperatures in order to increase the shunt resistance for low-noise operation.

6.1. Design considerations for an InGaAs detector-based thermometer The schematic of the InGaAs radiation thermometer is shown in Figure 17. The internal elements are mounted on four 15-mm diameter

AMPLIFIER

DIODE

EYEPIECE

LYOT STOP

LENS

FIELD STOP

MAIN LENS

SHUTTER

Radiation Thermometer Designs

DETECTOR LENS

FILTER LASER and SPLITTER

Figure 17 The schematic of the InGaAs-based radiation thermometer with the optical components labeled. The outer diameter of the main body is 20.3 cm with a total length of 75 cm, excluding the front bafﬂe. For alignment, the eyepiece is used to view the image on the tilted ﬁeld stop. A laser can be made incident on the ﬁeld stop for additional alignment.

161

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Howard W. Yoon and George P. Eppeldauer

graphite2epoxy rods. The graphite2epoxy rods have a coefﬁcient of thermal expansion of 0.5 106 K1, which is smaller than that of fused silica [41]. The rods are attached to the front plate holding the objective lens, and various separations between the optical elements can be smoothly adjusted by sliding along the rods. The angled ﬁeld stop is constructed from polished stainless steel with a beveled 3-mm diameter hole. The ﬁeld stop is angled to reduce the back reﬂection of the image onto the objective lens, thus reducing the SSE. The specular reﬂection from the ﬁeld stop is directed to a ﬂat mirror for viewing through the telescope objective. The radiation through the ﬁeld stop is collimated by the 70-mm focal length achromat and the plane of the objective lens is focused onto the Lyot stop. The presence of the Lyot stop is critical in reducing the SSE. The roomtemperature-stabilized interference ﬁlter is constructed using hard, ionassisted deposited layers at a center wavelength of 1,550 nm. The full-width at half-maximum (FWHM) bandpass of the ﬁlter is 100 nm. Due to the size of the four-stage TE cooler, the maximum size of the regular InGaAs detector is limited to a diameter of 3 mm. The detector lens focuses the 3mm diameter object at the ﬁeld stop onto a 1.5-mm diameter region on the detector, leading to an underﬁlled detector. The outer case of the radiation thermometer is ﬂoating with respect to the optical elements attached to the graphite rods and is only ﬁxed at the middle supporting plate. The case is constructed from anodized aluminum tubing. The use of the Al tube facilitates rapid thermal equilibration between the laboratory temperature and the internal temperatures. Additionally, the heat-sink plate for the InGaAs detector is thermally connected to the external tubing with copper-braided contacts that work to dissipate the heat generated from the use of the TE cooler. With only passive cooling of the heat sink, the detector was cooled to 701C.

6.2. Detector optimization The most important aspect of the radiation thermometer design is the selection of the detector. Due to the linearity and the wavelength region of the spectral power responsivity, a regular InGaAs diode was chosen. Since the small target diameter and the f/x limit the throughput of the thermometer, the preampliﬁer should have a dynamic range from 105 to 1010 V/A for measurements of blackbodies from the In-point to the Agpoint. In a transimpedance ampliﬁer circuit, the shunt resistance of the input diode should have a value greater than or equal to the feedback resistance for linear, low-noise ampliﬁcation of the current. The measurement of the shunt resistances of both 1- and 3-mm diameter InGaAs diodes in Figure 18 shows that the shunt resistances have a semilogarithmic relationship to temperature. The shunt resistances also increase as a linear function of the detector area, as expected for defects

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Radiation Thermometer Designs

1011 IGA 3 mm IGA 1 mm IGA 3 mm

Shunt Resistance, Ω

1010

109

108

107

106

-40

-20

0

20

40

Temperature, °C

Figure 18 The temperature-dependent shunt resistances of 1- and 3-mm diameter InGaAs (IGA) diodes. The shunt resistance of the diode should be comparable to the feedback resistance of the preampliﬁer circuit for low-noise, linear operation.

with a uniform surface density. Figure 18 shows that for 1-mm diameter diodes, shunt resistances W1010 O are achieved at detector temperatures of 201C, which is possible with two-stage TE cooling. For 3-mm diameter diodes, detector temperatures of at least 401C are needed to achieve W1010 O shunt resistance. Unless the diode material quality is improved, larger diodes are expected to require greater cooling to obtain the same shunt resistances.

6.3. Temperature-dependent detector performance The importance of the ability to cool the detector to below ambient temperature is shown in Figure 19. The equivalent dark current and its standard deviation at a preampliﬁer gain of 1010 V/A are shown. The standard deviations are calculated from 10 measurements made during the period of 10 power-line cycles that correspond to an electrical bandwidth of 0.13 Hz. The equivalent dark current, calculated from the output offset voltage and the current-to-voltage conversion, is only weakly dependent on the detector temperature with a decrease from 3 pA to about 1 pA as the detector is cooled from room temperature to 701C. The change in the equivalent dark current with decreasing temperature indicates that stabilizing the detector temperature is important in avoiding drift in the output signal, especially at the In-point where the net photocurrent is only

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Howard W. Yoon and George P. Eppeldauer

3.5

100

3.0

2.0 10 1.5

St. Dev., fA

Dark Current, pA

2.5

1.0 0.5 0.0 -80

-60

-40

-20

0

20

1 40

Detector Temperature, °C

Figure 19 The dark current (left axis) and the standard deviation (right axis) of the dark current measured with a bandwidth of 0.3 Hz as a function of the 3-mm diameter InGaAs detector temperature. The standard deviation of the dark current of the InGaAs detector at 701C nearly approaches Si-type detector performance.

about 23 pA. With cooling of the detector, the standard deviation of the dark current (the noise-equivalent current) changes much more, from about 100 to 2.4 fA. Although the detector was cooled to lower temperatures than 701C, the heat generated from the TE coolers could not be readily dissipated, which leads to self heating of the detector. We observed a slow drift of the equivalent dark current when the temperature of the detector was set to lower temperatures that precluded stable measurements of the noise-equivalent dark current. These measurements indicate that close to Si-type performance of a few femtoamps noiseequivalent current [6] is possible with four-stage TE cooling. This lownoise performance is especially important at the In-point where any decrease in the noise-equivalent power directly leads to a decrease in the noise-equivalent temperatures.

6.4. Optical performance The InGaAs radiation thermometer utilizes a commercial 50-mm diameter achromat with a focal length of 200 mm with nearly 1:1 ﬁnite-conjugate imaging. Since commercial achromatic lenses are designed for inﬁniteconjugate imaging, the imaging performance suffers as shown by the Strehl ratio in Table 4. The Strehl ratio increases as the wavelength increases which indicates that imaging performance is better at 1,550 nm than at 500 nm. Furthermore, achromatic lenses generally have two wavelengths

165

Radiation Thermometer Designs

where the image distances are equal for a set object distance with the chromatic focal shift depending on the lens material and thickness. Since the radiation thermometer is designed to transfer a temperature scale from one blackbody to another without any desired spatial structure within the cavity, the need for imaging performance is much reduced. Although the alignment of the radiation thermometer can be performed in the visible wavelength, the object distances at 1,550 nm and at 500 nm differ by 25 mm for the same image distance. Since the internal focus is difﬁcult to adjust, the object distance can be offset by this difference to account for chromatic aberrations.

6.5. Blackbody measurements and interpolation functions The InGaAs radiation thermometer was calibrated using the ITS-90 FPBBs at the freezing temperatures of In, Sn, Zn, Al, and Ag. The freezing-point cavities have internal openings of 6 mm diameter which is larger than the 4.5-mm spot size of the radiation thermometer, and thus the opening overﬁlls the radiation thermometer. The freezing-point cavities are inserted into furnaces with heat-pipe liners to ensure the spatial uniformity during the melts and freezes. The freeze duration lasts for about 30 min, and these measured net currents shown in Table 5 are the results of averaging many measurements. The measured photocurrent, i, at the ﬁve ﬁxed points is ﬁtted using a modiﬁed-Planck function [42], i¼

A expðc 2 =ðBT þ CÞÞ 1

(10)

where T is the temperature in kelvin, c2 the second radiation constant, and A, B, and C are the ﬁtting parameters. The nonlinear minimization to determine the ﬁtting parameters can be performed using Excel or another Table 5 The gains and the photocurrents at the ﬁxed-point temperatures measured using a variable-temperature blackbody set at the respective temperatures. Fixed point

Temperature (1C)

Gain (V/A)

Signal (A)

NETD (1C)

In Sn Zn Al Ag

156.5985 231.928 419.527 660.323 961.78

1010 109 107 106 105

2.26251E-11 5.36748E-10 7.45110E-08 2.23358E-06 2.63766E-05

0.0050 0.0011 0.0015 0.0009 0.0013

Note: The standard deviations of the photocurrents at the respective temperatures are converted to noise-equivalent temperature difference (NETD).

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Howard W. Yoon and George P. Eppeldauer

1E-4 0.005

1E-5

0.004

1E-7 0.003 1E-8 0.002

1E-9 1E-10 1E-11

NETD, K

Photocurrent, A

1E-6

0.001 0

200

400

600

800

1000

Temperature, °C

Figure 20 The measured photocurrents of the InGaAs radiation thermometer at various ﬁxed-point temperatures. The interpolations are performed using the photocurrent measurements. The NETD is determined from the standard deviation of the measurements.

ﬁtting program. The ﬁtted parameters are shown in Figure 20 along with the photocurrents that were measured at the respective temperatures. The residuals of the ﬁt shown can indicate whether the measurements at any ﬁxed point are in agreement with the overall ﬁtted curve. The residuals shown as percent differences from the ﬁtted curve indicate that the Snpoint measurements are off from the rest of the freezing points that have much smaller residuals. A major advantage of the modiﬁed-Planck function is that the function can be inverted to T¼

1 c2 C B lnððA=iÞ þ 1Þ

(11)

which gives the relationship between the measured photocurrent and the temperature.

6.6. Uncertainties The uncertainties of the InGaAs thermometer described above can be broadly classiﬁed into those related to the FPBBs, characterizations of the radiation thermometer, and the use of the radiation thermometer. In Table 6, the standard uncertainties are listed as either temperature or percent uncertainties. The standard uncertainties in Table 6 are converted to temperature uncertainties using a sensitivity equation similar to Equation (6).

167

Radiation Thermometer Designs

Table 6 source.

Table of standard uncertainty components listed according to their

Blackbody

Radiation thermometer (RT)

RT use

Description

FPBB scheme (%) (k ¼ 1)

Impurities Plateau identiﬁcation Blackbody emissivity, isothermal Cavity bottom heat exchange Size-of-source effect Nonlinearity Atmospheric absorption Gain ratios Noise Interpolation error Drift

6.5 mK 10 mK 0.02 35 mK 0.005 0.002 0.01 0.002 0.001 0.03 0.05

Note: Items are stated in either temperature or in percents.

7. Calibration of Radiation Thermometers Using ITS-90 The temperatures above the freezing temperature of Ag are determined using Planck radiance ratios to the Ag, Au, or Cu FPBBs [43]. The extrapolations to higher temperatures from the ﬁxed point are performed from the relative spectral responsivity, S(l). The radiance ratios are determined using integrated Planck radiances, L, convolved with the spectral responsivity [44], R SðlÞ Lðl; T 1 Þ dl Lðlm ; T 1 Þ ¼R Lðlm ; T 0 Þ SðlÞ Lðl; T 0 Þ dl

(12)

where lm is the mean effective wavelength. Since the Planck radiances change as a function of temperature and materials can have spectrally dependent emissivities, the spectral responsivities are reduced to an effective wavelength [45], le, with R ð1=lÞSðlÞ Lðl; T Þ dl 1 ¼ R le SðlÞ Lðl; T Þ dl

(13)

The concept of an effective wavelength enables the use of Planck radiances at a single wavelength to replace the spectral responsivity of the radiation thermometer with a delta function spectral width although the radiation thermometer has a ﬁnite spectral width. Since the radiance ratios in

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Howard W. Yoon and George P. Eppeldauer

Equation (12) are performed at different temperatures, the radiance ratio is determined at a mean effective wavelength, 1 1 1 1 ¼ þ lm 2 lm ðT 0 Þ lm ðT 1 Þ

(14)

In practice, the measured photocurrents at the two temperatures are compared using i1 Lðlm ; T 1 Þ ¼ i0 Lðlm ; T 0 Þ

(15)

where the unknown temperature, T1, is obtained from the ratios of Planck radiances at the mean effective wavelength that matches the measured photocurrent ratios. Note that with the widespread use of computers, the use of effective wavelength is reducing since the integrals can be easily evaluated using spreadsheet programs. In addition, and again, since the wavelength integrals can be easily evaluated, the implementation of ITS-90 can be performed without the use of the mean effective wavelength concept. Since only the relative spectral responsivity is obtained from the typical monochromator-based calibrations, the absolute throughput of the radiation thermometer, tabs, needs to be determined from the measurements using one of the Ag, Au, or Cu ﬁxed points. The measurement equation, Z

i0 ¼ tabs

S rel ðlÞ Lðl; T 0 Þ dl

(16)

is used to determine the absolute throughput. In Equation (15), the photocurrent, i0, is measured, and the spectral radiance is known from the given temperature, T0. Once the absolute throughput is determined, the radiation thermometer is used to measure the temperature of a blackbody using Z

i1 ¼ tabs

S rel ðlÞ Lðl; T 1 Þ dl

(17)

where i1 is the measured photocurrent, and the temperature, T1, can be determined by iteratively changing the integrand until the calculated photocurrents match the measured photocurrent.

8. Thermodynamic Temperature Measurements Using Radiometric Techniques The measurements of temperature can be classiﬁed into primary and secondary thermometry [46]. Primary thermometry is deﬁned as

Radiation Thermometer Designs

169

determining thermodynamic temperatures using equations of state and physical constants without any adjustable temperature-dependent parameters. Since primary thermometers are not easily transportable and are difﬁcult to use, secondary thermometers are used as transfer devices. Because of the difﬁculties in maintaining primary thermometers for routine calibrations, the ITS-90 assigns ﬁxed values for the phase-transition temperatures of various pure materials based upon the best thermodynamic temperature determinations available at the time of specifying the scale. Secondary interpolating thermometers are based upon these ﬁxed points and the assigned temperatures are ultimately traceable to primary thermometers through the ﬁxed points. Some examples of secondary thermometers are standard PRTs and thermocouples. Another type of primary thermometer is the total and the spectral radiometer, calibrated using electrical substitution radiometry and used in conjunction with a blackbody. Such total radiometers in the past have determined the Stefan2Boltzmann constant using the wavelength integrated Planck’s radiance law to a relative combined uncertainty of 0.013% [47]. If only a narrow spectral region is selected by the use of spectral ﬁlters, then the spectral radiometer can be used to determine the temperatures of blackbodies with the use of Planck’s radiation law within the spectrally selected region. In the past, detector-based radiance temperature measurements have been performed using ﬁlter radiometers calibrated for spectral irradiance responsivity [48]. These ﬁlter radiometers with precision apertures are typically used in nonimaging applications to measure the temperatures of blackbodies, but such optical ﬂux techniques are difﬁcult to use to measure the FPBBs with sub-centimeter diameter cavity openings. The nonimaging radiometers are used to measure spectral irradiance using double-aperture ﬂux transfer techniques, and the small, o1 cm internal apertures of FPBBs make it difﬁcult to deﬁne a source aperture. Such techniques can also result in large uncertainties in the diffraction corrections needed from the use of the small deﬁning apertures. However, small-aperture blackbodies can be measured using imaging radiometers in which the ﬁeld stop deﬁnes the area of measurement. In the past, absolute imaging radiometers or pyrometers have been constructed from ﬁlter radiometers using a lens with a measured transmittance [49], but similar component-based calibrations cannot be used with low ﬁnal uncertainties if a number of optical elements such as lenses and ﬁlters are introduced into the optical path. In this section, the NIST Absolute Pyrometer (AP1) is described in which the radiation thermometer is calibrated at the system level. The AP1 is designed and constructed at NIST and is calibrated for radiance responsivity as a system using a cryogenic, electrical substitution radiometer-based method. The performance of the system is demonstrated by measuring the Ag- and Au-point thermodynamic temperatures, and these

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Howard W. Yoon and George P. Eppeldauer

are compared to the previous evaluations. Through use of a device such as the AP1, the thermodynamic temperatures can be directly disseminated in this range using detector-based calibrations with comparable or lower uncertainties than that of ITS-90.

8.1. Radiance responsivity calibration of AP1 The AP1 was calibrated for absolute radiance responsivity using the NIST Spectral Irradiance and Radiance responsivity Calibrations using Uniform Sources (SIRCUS) facility as illustrated in the sequence of measurements in Figure 21(a)2(d). The optical power of the power-stabilized and collimated laser source is measured using a cryogenic electrical substitution radiometer, the NIST Primary Optical Power Radiometer (POWR) in Figure 21(a). Since the POWR is difﬁcult to use on a daily basis, the power responsivity is transferred to six-element Si-trap detectors in Figure 21(b). From the knowledge of the spatial uniformity and the area of the precision, deﬁning aperture on the trap detectors, the detector-based spectral irradiance responsivity is determined. The laser is input into an integrating sphere to obtain a spatially uniform and Lambertian source in Figure 21(c),

(a)

Electrical Substitution Radiometer

Laser

Si-trap Detector

(b) Precision Aperture

Precision Aperture Si-trap Detector

(c) d

(d)

Integrating Sphere

Radiation Thermometer

Figure 21 The procedure for detector-based spectral radiance responsivity calibrations with the use of the electrical substitution radiometer. (a) Measurement of the spectral power of the laser with the cryogenic electrical substitution radiometer. (b) Determination of the spectral power responsivity of the Si-trap detector. (c) Measurement of the spectral irradiance of the ISS. (d) Determination of the radiance responsivity of the radiation thermometer as a system.

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and the spectral irradiance of the source is determined using the Si-trap detector. The spatial uniformity at the exit aperture of the integrating sphere is mapped to determine the uncertainties due to the nonuniformity of the sphere. The power-stabilized laser is transmitted through a ﬁber which is agitated in a water-ﬁlled ultrasonic bath. The rapid vibrations in the ultrasonic bath remove the laser speckle in the sphere output since the integration times of the detectors are much longer than the time duration of the speckle pattern. A small-diameter (38.1 mm) integrating sphere is used for increased throughput and the 5-mm diameter exit port of the ISS is chosen to approximate the 6-mm diameter opening of the FPBB cavities. With precision apertures on both the output port of the integrating sphere and the entrance of the Si-trap detector combined with the knowledge of the distance separating the sphere and the trap detector, the monochromatic spectral radiance of the integrating sphere can be assigned from the detector-calibrated irradiance responsivity of the trap detector. The aperture area and the distance measurement device are traceable to length standards. The AP1 is then calibrated for spectral radiance responsivity by substitution with the trap detector as shown in Figure 21(d). Once the monochromatic, spectral radiance of the ISS is known, any radiometer or radiation thermometer can be calibrated at the system level. The procedure is repeated at other wavelengths until the full spectrum of the radiance responsivity is obtained. The detector-based radiance responsivity of the AP1 is shown along with the spectral radiance of the gold-point blackbody in Figure 22. The interference ﬁlter used in the AP1 shows better suppression than 1 107 in the out-of-band region relative to the peak value of the responsivity. The FWHM of the radiance responsivity is 10 nm, with the centroid wavelength of 649.115 nm. The total expanded uncertainty of the radiance responsivity at a single wavelength is 0.15% (k ¼ 2). The wavelength spacing between calibrations was decreased near the peak of the spectral responsivity since interference effects from inter-reﬂections can lead to oscillations in the spectral responsivity which are wavelength dependent. The sources of the interference effects are from the remaining coherence of the laser through the ISS combined with the inter-reﬂections between optical elements in the radiation thermometer. Although effort was made to reduce the interference effects by designing the optical ﬁlter with a 51 tilt of one of the surfaces from the optical axis, the interference effects are evident as o1% peak oscillations of the responsivity. Since the radiation thermometer is used to measure the radiance from a blackbody, which is an incoherent source, the convolved radiance does not exhibit the interference effects, and the interference effects in the radiance responsivity depend on the spatial coherence of the incident radiation. The increase in the uncertainties due to the interference effects can be minimized by performing measurements at closely spaced wavelength intervals.

172

10-4

10-4 1337.33 K

10-6

10-6

10-8

10-8

10-10

10-10

10-12

10-12

10-14 400

500

600

700

800

900

1000

Spectral Radiance, L [W/(cm2sr) nm]

Radiance Responsivity,SL [A/W/(cm2sr)]

Howard W. Yoon and George P. Eppeldauer

10-14

Wavelength [nm]

Figure 22 The radiance responsivity of the AP1 measured in the SIRCUS facility plotted with the Planck radiance at 1337.33 K. The center wavelength is 650 nm with a FWHM of 10 nm. The out-of-band suppression of o1 107 is achieved in the wavelength region from 700 to 1,000 nm.

The thermodynamic temperatures of blackbodies are found from the radiance responsivity and Planck’s law by the use of Z ic ¼

S L Lðl; T Þ dl

(18)

where ic is the calculated photocurrent, SL the absolute radiance responsivity, and L(l, T ) the spectral radiance given by the Planck radiance law. The Planck radiance law is Lðl; T Þ ¼

c 1L 1 n2 l5 expðc 2 =ðnlT ÞÞ 1

(19)

where c1L and c2 are the ﬁrst and second radiation constants, T is the thermodynamic temperature, l the wavelength of the radiation in air, and n the refractive index. For these calculations, the Committee on Data for Science and Technology (CODATA) values [50] for the radiation constants and the refractive index of air of n ¼ 1.00029 were utilized. There are no adjustable parameters in the measurement equation relating the radiance responsivity of the AP1 to the thermodynamic temperature, and Equation (19) can be used to generate a curve of photocurrent to blackbody temperatures. The one-to-one relationship is shown in Figure 23. The photodiode used in the AP1 is operated in the photovoltaic mode without any bias voltages, and this type of photodiode

173

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10-5

Photocurrent [A]

10-6 10-7 10-8 Au 10-9 10-10 10-11 1000

1500

2000

2500

3000

Temperature [K]

Figure 23 The calculated photocurrent as a function of the blackbody temperature. The calculation is performed with no adjustable parameters.

in the photovoltaic mode has been shown to be linear over a wide dynamic range [51]. The range of calculated photocurrents of the AP1 is from 1 1011 A at the Ag-point to 1 106 A near 3,000 K which is in the linear regime for photodiodes. The plot in Figure 23 also indicates the need for low-noise current-to-voltage ampliﬁers to measure the low photocurrents near the Ag and Au freezing temperatures. Other corrections such as the SSE, nonlinearity of the preampliﬁer, and the emissivity of the blackbody will be applied as corrections to the measured photocurrents before comparison to the calculated photocurrents in Figure 23. The relationship between photocurrent and blackbody temperatures can be ﬁtted with an interpolation function [52] to facilitate conversions from signals to temperatures.

8.2. Summary of systematic corrections to AP1 The correction factors listing the dominant systematic corrections to the AP1 signals are listed in Table 7. The AP1 signals were corrected prior to the application of Equation (18) by using s i ¼ G im

(20)

where im is the measured photocurrent, i the corrected photocurrent, G the gain correction, e the emissivity, and s the SSE correction. The corrected photocurrent can then be directly compared to the calculated

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Table 7 Systematic correction factors to the AP1 signal. Type of correction

Gain, G

Emissivity, e

SSE, s

Signal correction factor Uncertainty (k ¼ 2)

1.0017 0.022%

0.99964 0.02%

0.99963 0.02%

Table 8 The gain ratios of the preampliﬁer measured using a 1 W white LED source with a Si detector. Gains compared 6

5

10 /10 107/106 108/107 109/108 1010/109

Gain ratios

Measurement uncertainty (k ¼ 2)

9.998 10.001 10.000 10.000 9.957

0.002 0.002 0.002 0.002 0.002

photocurrents in Figure 23. The gain correction factor is the normalization factor of gain 1010 to gain 108 as shown in Table 8 since the responsivity calibrations were performed at gain 108 while the Ag- and Au-freezing temperatures were measured at gain 1010. The SSE correction has the effect of reducing the measured signal since the responsivity calibration was used with a 5-mm diameter opening of the ISS with no other source of radiation outside the opening. The emissivity of blackbodies also reduces the radiance and thus compensation can be made using Equation (20).

8.3. Uncertainty estimates of thermodynamic temperature Using AP1 and similar instruments The discussion of the uncertainties of the measurements must start from the initial realization of the cryogenic electrical substitution radiometer in the NIST SIRCUS facility, whose component standard uncertainties are listed in Table 9. The two largest standard uncertainties are due to the spatial and the angular uniformity outputs of the integrating sphere and the long-term stability of the trap detectors. Since the windows in the multielement diode trap detectors are removed to avoid multiple reﬂections between the windows and the diode surfaces, the power responsivities of the diodes can change due to progressive contamination or surface damage. The ﬁnal total uncertainty leads to the uncertainties in Table 10. The individual uncertainties are listed in Table 10 as a percent of the signals and then converted to temperature uncertainties. The dominant

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Table 9 The total uncertainty of the spectral radiance responsivity calibrations in the NIST SIRCUS facility broken down into the component uncertainties.

1 2 3 4 5 6 7

SIRCUS uncertainty components (k ¼ 1)

Type

Uncertainty (%)

Trap responsivity Aperture areas Distance Sphere spatial and angular uniformity Ampliﬁer gain Temperature coefﬁcient of trap Temporal stability of trap Combined standard uncertainty (k ¼ 1) Expanded total uncertainty (k ¼ 2)

B B B B A B B

0.025 0.004 0.010 0.050 0.010 0.003 0.050 0.076 0.153

Note: Trap refers to the Si-diode trap detector used for the realization.

Table 10 Uncertainty components for the Ag and Au freezing temperature measurements.

1 2 3 4 5 6 7

Signal uncertainty component (%) (k ¼ 2)

Ag

Au

Spectral radiance responsivity Temporal stability of responsivity Plateau identiﬁcation Emissivity Nonlinearity Dark current drift Size-of-source effect Total uncertainty in signal Temperature uncertainty (K) (k ¼ 2)

0.15 0.03 0.01 0.02 0.022 0.03 0.02 0.16 0.110

0.15 0.03 0.01 0.02 0.022 0.03 0.02 0.16 0.129

contribution to the temperature uncertainties is that of the radiance responsivity which is 0.15% (k ¼ 2). Since the center wavelength of the AP1 is in the wavelength and temperature regions where the Wien approximation is applicable, the uncertainty of the temperature from the uncertainty of radiance is found from the derivative of the Wien approximation and is given in Equation (6). Using the AP1, the Ag- and Au-freezing temperatures are in agreement with the measurements of Fox et al., who also used detector-based ﬁlter radiometry to measure the Ag- and Au-freezing temperatures, 1235.009 K (70.088 K) and 1337.330 K (70.098 K), respectively [53]. The differences of the AP1 measurements with the earlier results by Fox et al. are within the combined uncertainties. The gold freezing temperature is also in agreement with the detector-based measurements of Mielenz et al. [54].

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All these results are in agreement with the ITS-90 temperatures. The ITS-90 temperature assignments of the respective freezing temperature are primarily from the ratio pyrometry work of Fischer and Jung [55] with radiance ratios taken from a lower temperature blackbody at 730 K. The thermodynamic temperature of the reference blackbody was assigned from two conﬂicting constant-volume gas thermometry measurements [56] with differences of 30 mK at 730 K. Since at that time the differences between the two gas thermometry results could not be resolved, a compromise of 15 mK between the two results was agreed at 730 K, or half the value of the 30 mK difference. Since the uncertainties in temperature increase as DT a ¼

T 2a DT b T 2b

(21)

where DTa and DTb are the uncertainties of the blackbody at temperatures Ta and Tb; the resulting thermodynamic temperature offset at 1,337 K resulting from ratio pyrometry is expected to be 50 mK. In addition to the thermodynamic temperature uncertainties, the uncertainties in realizing a temperature scale using ﬁxed points include uncertainties due to material purity, emissivity of the cavity, furnacetemperature uniformity, and furnace operation. Even if the ﬁxed point is ﬁlled initially with ultrapure (99.999%) metal, the metal in the blackbody can become contaminated due to operation at high temperatures, and any contamination depresses the freezing temperature. To have reproducible and stable freeze plateaus, the furnace should have a uniform temperature distribution. The electrical power heating of the furnace also has to be optimized to achieve reproducible and ﬂat freeze plateaus. The dependence of the freezing temperature on the impurities and the operational parameters can lead to realization uncertainties of 50 mK (k ¼ 2) [3], and temperature scale comparisons can require side-by-side operation of the respective ﬁxed points to achieve lower temperature uncertainties.

9. Summary The transition to detector-based radiance temperatures would lead to better agreement with thermodynamic temperatures and lead to direct dissemination of thermodynamic temperatures. This transition would be in accordance with the movement to having deﬁned quantities or ﬁxed values for the fundamental constants as the basis for the SI units instead of ﬁxed artifacts as has already occurred for the candela in 1979. The dissemination of detector-based temperatures would also result in having a common basis, the electrical watt and length units, for photometric, radiometric, and temperature scales possibly leading to resolution of any inconsistencies in

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the national scales. Currently very few laboratories around the world can achieve competitive uncertainties with the current ITS-90 deﬁnition above the silver point. However, in the future as primary radiometry facilities improve, more laboratories will be able to realize the disseminate T at high temperatures, with lower uncertainties than the current ITS-90 allows. To allow for this eventuality, the mise-en-pratique for the deﬁnition of the kelvin [57] will facilitate the realization of T alongside ITS-90 where it is advantageous to do so.

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D. K. Walker, K. J. Coakley, and J. D. Splett, ‘‘Non-linear Modelling of Tunnel Diode Detectors,’’ in Geoscience and Remote Sensing Symposium, IEEE International, Anchorage Alaska, pp. 396923972 (2004). K. D. Mielenz and K. L. Eckerle, ‘‘Spectrophotometer Linearity Testing Using the Double-Aperture Method,’’ Appl. Opt. 11, 229422303 (1972). P. Saunders and D. R. White, ‘‘Propagation of Uncertainty due to Non-linearity in Radiation Thermometers,’’ Int. J. Thermophys. 28, 209822110 (2007). H. W. Yoon, J. J. Butler, T. C. Larason, and G. P. Eppeldauer, ‘‘Linearity of InGaAs Diodes,’’ Metrologia 40, S1542S158 (2003). T. Ricolﬁ, M. Battuello, F. Girard, G. Machin, H. McEvoy, S. Ugur, and A. Diril, ‘‘Radiation Temperature Scales between the Indium and Silver Points Realized at IMGC, NPL and UME Using a Fixed-Point Calibration Technique,’’ Meas. Sci. Technol. 13, 209022093 (2002). F. Girard and T. Ricolﬁ, ‘‘Analysis of Two Methods for Calibrating IR Thermometers in the Middle-temperature Range,’’ in Proceedings of TEMPEMKO 2004, Ninth International Symposium on Temperature and Thermal Measurements in Industry and Science, edited by D. Zvizdic, Laboratory for Process Measurement, Croatia, pp. 827-832 (2005). T. Ricolﬁ and F. Girard, ‘‘A Transfer Standard Infrared Thermometer for the Temperature Range 1501C to 10001C,’’ in Proceedings of TEMPEMKO 1999, Seventh International Symposium on Temperature and Thermal Measurements in Industry and Science, edited by J. F. Dubbledam and M. J. de Groot, Vol. II, IMEKO/Nmi Van Swinden Laboratorium, Delft, The Netherlands, pp. 5932598 (1999). B. Gutschwager, S. Schiller, J. Hartmann, and J. Hollandt, ‘‘A Lens-Free InGaAs Transfer Radiation Thermometer with an Extended Calibration Range above 9601C,’’ in Proceedings of TEMPEMKO 2004, Ninth International Symposium on Temperature and Thermal Measurements in Industry and Science, edited by D. Zvizdic, Laboratory for Process Measurement, Croatia, pp. 6052610 (2005). G. Romeo and G. Frulla, ‘‘Analytical and Experimental Results of the Coefﬁcient of Thermal Expansion of High-Modulus Graphite2Epoxy Materials,’’ J. Compos. Mater. 29, 75127650 (1995). F. Sakuma and M. Kobayashi, ‘‘Interpolation Equations of Scales of Radiation Thermometers,’’ in Proceedings of TEMPEMKO 1996, Sixth International Symposium on Temperature and Thermal Measurements in Industry and Science, edited by P. Marcarino, Levrotto and Bella, Turin, pp. 3052310 (1997). H. Preston-Thomas and T. J. Quinn, Supplementary Information for the International Temperature Scale of 1990, Se`vres, Bureau International des Poids et Mesures (1990). J. L. Gardner, ‘‘Effective Wavelength for Multicolor/Pyrometry,’’ Appl. Opt. 19, 308823091 (1980). P. Saunders, ‘‘Uncertainty Arising from the Use of the Mean Effective Wavelength in Realizing ITS-90,’’ in Temperature: Its Measurement and Control in Science and Industry, edited by D. C. Ripple, B. C. Johnson, C. W. Meyer, et al., AIP, New York, pp. 6392644 (2002). T. J. Quinn, Temperature, Academic Press, San Diego, pp. 20-22 (1990). T. J. Quinn and J. E. Martin, ‘‘A Radiometric Determination of the Stefan2Boltzmann Constant and Thermodynamic Temperatures between 401C and +1001C,’’ Philos. Trans. R. Soc. 316, 852189 (1985). H. W. Yoon and C. E. Gibson, ‘‘Determination of Radiance Temperatures Using Detectors Calibrated for Absolute Spectral Power Response,’’ in Proceedings of TEMPEMKO 1999, Seventh International Symposium on Temperature and Thermal Measurements in Industry and Science, edited by J. F. Dubbledam and M. J. de Groot,

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Vol. II, IMEKO/Nmi Van Swinden Laboratorium, Delft, The Netherlands, pp. 7372742 (1999). N. P. Fox, J. E. Martin, and D. H. Nettleton, ‘‘Radiometric Aspects of an Experiment to Determine the Melting/Freezing Temperature of Gold,’’ Metrologia 28, 2212227 (1991). P. J. Mohr and B. N. Taylor, ‘‘CODATA Recommended Values of the Fundamental Physical Constants: 2002,’’ Rev. Mod. Phys. 77, 12107 (2005). G. P. Eppeldauer and J. E. Hardis, ‘‘Fourteen-Decade Photocurrent Measurements with Large-Area Silicon Photodiodes at Room Temperature,’’ Appl. Opt. 30, 309123099 (1991). F. Sakuma and S. Hattori, ‘‘Establishing a Practical Temperature Standard by Using a Narrow-Band Radiation Thermometer with a Silicon Detector,’’ in Temperature: Its Measurement and Control in Science and Industry, edited by J. F. Schooley, AIP, New York, Vol. 5, pp. 4212427 (1982). N. P. Fox, J. E. Martin, and D. H. Nettleton, ‘‘Absolute Spectral Radiometric Determination of the Thermodynamic Temperatures of the Melting/Freezing Points of Gold, Silver and Aluminium,’’ Metrologia 28, 3572374 (1991). K. D. Mielenz, R. D. Saunders, and J. B. Shumaker, ‘‘Spectroradiometric Determination of the Freezing Temperature of Gold,’’ J. Res. NIST 95, 49267 (1990). J. Fischer and H. J. Jung, ‘‘Determination of the Thermodynamic Temperatures of the Freezing Points of Silver and Gold by Near-Infrared Pyrometry,’’ Metrologia 26, 2452252 (1989). R. L. Rusby, R. P. Hudson, M. Durieux, J. F. Schooley, P. P. M. Steur, and C. A. Swenson, ‘‘Thermodynamic Basis of the ITS-90,’’ Metrologia 28, 9218 (1991). G. Machin, P. Bloembergen, J. Hartmann, M. Sadli, and Y. Yamada, ‘‘A Concerted International Project to Establish High Temperature Fixed Points for Primary Thermometry,’’ Int. J. Thermophys. 28, 197621982 (2007).

CHAPTER 5

Calculation of the Radiation Characteristics of Blackbody Radiation Sources$ Alexander V. Prokhorov, Leonard M. Hanssen and Sergey N. Mekhontsev

Contents 1. Introduction 2. Deﬁnitions of Principal Quantities 3. Deterministic Methods 3.1. Approximate expressions for isothermal diffuse cavities 3.2. Method of integral equations for diffuse cavities 3.3. Integrated effective emissivity of diffuse cavities 3.4. Series technique for multiple reﬂections in cavities 3.5. Cavities with diffusely emitting and specularly reﬂecting walls 3.6. Specular2diffuse cavities 3.7. Method of successive reﬂections 4. Monte Carlo Method 4.1. Stochastic approach to radiative heat transfer 4.2. Ray tracing 4.3. Stochastic models for optical properties 4.4. Monte Carlo modeling of specular2diffuse blackbodies 4.5. Modeling of direction-dependent optical characteristics 5. Numerical Comparison of Results Obtained by Various Methods 6. Conclusions Acknowledgement References

181 182 187 187 189 199 201 204 205 205 208 208 210 211 212 227 231 233 234 234

$

Certain commercial software is identiﬁed in this paper in order to specify the computational procedure adequately. Such identiﬁcation is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the software identiﬁed is necessarily the best available for the purpose.

National Institute of Standards and Technology, 100 Bureau Drive, Gaithersburg, MD 20899-8442, USA Experimental Methods in the Physical Sciences, Volume 42 ISSN 1079-4042, DOI 10.1016/S1079-4042(09)04205-2

r 2010 Elsevier Inc. All rights reserved.

181

182

Alexander V. Prokhorov et al.

1. Introduction A source of optical radiation whose radiation characteristics can be calculated on the basis of fundamental physical laws makes possible the calibration of radiation thermometers in absolute units of temperature. Theoretically, a perfect blackbody is the most suitable object for this purpose. At the same time, a perfect blackbody is an ‘‘ideal’’ object, that is, not more than a physical abstraction that does not exist in the real world. It is well known that the perfect blackbody conditions are realized inside a closed isothermal cavity with opaque walls (see, for instance, Refs. [124]). However, the blackbody radiation of a closed cavity is inaccessible and unobservable. Thus, instead of using a completely enclosed cavity, one can approach the ideal by leaving only a small opening in the cavity. The radiation escaping the cavity through the opening will then very closely mimic blackbody radiation. From this point forward, we shall use the term blackbody for a real radiator whose radiation properties approach those of a perfect blackbody. This term is equivalent to the following terms found frequently within the technical literature: blackbody radiation source, blackbody reference source, blackbody calibration source, blackbody simulator, artiﬁcial blackbody, imperfect blackbody, laboratory blackbody, etc. In order to employ a blackbody as a standard reference source for radiation thermometry, it is necessary to know how large the differences are between the radiation characteristics of the blackbody 2 for a given geometry and cavity wall material 2 and those of a perfect blackbody. Radiometry, radiation thermometry, optical remote sensing, and other areas of modern science and technology require determination of the effective emissivity of blackbody calibration sources with a relative uncertainty of 0.01% or less. Similar requirements of accuracy are necessary for radiance temperature and other quantities employed in the design, characterization, and calibration of blackbody radiators. Due to the difﬁculty of performing accurate measurements, computational methods have, until recently, remained the primary mode of investigation of blackbody radiation characteristics, for those deviating only slightly from a perfect blackbody. This chapter is devoted to the computational characterization of radiation emitted by blackbodies. We shall emphasize work of the last two decades (1990s and 2000s), while still including earlier milestone work. The interested reader may refer to an excellent survey [5] for information regarding work reported prior to 1982. Useful information can also be found in additional overviews in Refs. [628]. Section 2 describes the terminology we use and provides deﬁnitions of principal quantities used in this chapter. Conventional, deterministic methods for calculation of the radiation characteristics of isothermal and nonisothermal

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cavities are considered in Section 3. The applications of these methods are primarily determined by the model of radiation characteristics (diffuse, specular, etc.) adopted for the cavity walls. Section 4 describes the application of the stochastic (Monte Carlo) ray-tracing method to computer modeling of blackbody radiators. This computational method has become prevalent in the last decades of the 20th century due to its great generality, ﬂexibility, and a number of other advantages. In Section 5, a comparison of some results obtained by various methods is described.

2. Definitions of Principal Quantities The quantitative measure for the difference in the radiation characteristics between an artiﬁcial blackbody and a perfect one is the effective emissivity (sometimes, the terms apparent emissivity or cavity’s emissivity are used; we shall consider all these terms as interchangeable). The qualiﬁer effective is used due to the effect produced by multiple reﬂections. Unlike in the case of a ﬂat sample, the outgoing radiation of an element of the cavity wall consists not only of its own thermal radiation, but also of radiation falling from other surface elements and reﬂected by the element under consideration. Generally speaking, the effective emissivity is the ratio of a radiometric quantity (usually, radiance or spectral radiance) that characterizes the blackbody at a certain temperature to the same quantity of a perfect blackbody having the same temperature. Real-world cavities are always nonisothermal. Temperature nonuniformity might signiﬁcantly change the cavity radiation characteristics. The effective emissivity of a nonisothermal cavity is a function of a temperature assigned to the perfect blackbody in the effective emissivity deﬁnition (often referred to as the reference temperature) and might be less than or greater than unity, depending on the assigned reference temperature. Let us consider an isothermal cavity with a small opening. The radiation escaping the cavity through this opening differs little from the radiation of a perfect blackbody. Due to the geometrical invariance of the spectral radiance along rays, the spectral radiance of the opening is equal to that of the cavity internal surface projected by the rays passed through the opening in opposite directions. In its turn, the outgoing radiation ﬂux from the cavity wall toward the opening consists of two parts: the ﬂux of the surface element’s own thermal emission and the ﬂux reﬂected by this element. The ﬁrst part depends only on the emissivity and temperature of the cavity wall and does not depend upon the presence of the rest of the cavity. The reﬂected ﬂux depends on the surface element reﬂectance and on the radiation ﬂux falling onto this element from the rest of the cavity. The opening affects the reﬂected ﬂux, because the radiant ﬂux that should arrive from the wall of the closed cavity is absent if the opening is present.

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Moreover, in the case of a cavity with the opening, the radiation ﬂux propagated from the opening to the rest of the cavity is also absent. Accordingly, the irradiation of a target area from the rest of the cavity is somewhat lower than that in the case of a closed cavity. The net result is that the radiation ﬂux exiting from the isothermal cavity is greater than that of a ﬂat sample having the same temperature and made of the same material as the cavity, but less than the radiation ﬂux of a perfect blackbody at the same temperature. The increase of the radiation ﬂux of a blackbody in comparison with a ﬂat sample is known as the cavity effect. Now, we shall deﬁne the most important quantities that characterize blackbody radiation sources. Unless otherwise speciﬁed, we shall assume that the environment is nonrefracting, nonabsorbing, nonscattering, and nonemitting (i.e., vacuum at 0 K), or that these effects are negligible. We will also assume that the optical properties of the cavity wall do not depend on temperature. The effective emissivity of a blackbody may be deﬁned in a variety of ways depending on the variables one considers. First, let us consider radiation emitted from a point on the blackbody wall at a particular wavelength, l, with coordinates speciﬁed by the vector n, and the direction in which the radiation is emitted is given by the vector x. Under these conditions, the effective emissivity, ee, is deﬁned by the following equation: e ðl; n; x; T 0 Þ ¼

L l ðl; n; xÞ L l;bb ðl; T 0 Þ

(1)

where Ll and Ll,bb are the spectral radiances (in W/(m3sr) of the blackbody being considered and a perfect blackbody at a reference temperature T0. In Ref. [5], the deﬁnition of spectral directional effective emissivity is formulated using the spectral exitance Ml for which, however, the dependence on direction is indicated. This represents a confusion of terminology. The spectral exitance can be employed in deﬁnitions only for blackbodies with Lambertian (diffusely emitting and reﬂecting) internal walls; the rigorous deﬁnition for a blackbody with an arbitrary angular distribution of emitted and reﬂected radiation requires the use of radiance. The denominator in Equation (1) is deﬁned according to Planck’s law: L l;bb ðl; T 0 Þ ¼

c1 pl ½expðc 2 =ðlT 0 ÞÞ 1 5

(2)

where c1 and c2 are the ﬁrst and second radiation constants, respectively [9]. This quantity of effective emissivity, which deﬁnes local, directional, and spectral properties, is the primary radiation characteristic of a blackbody cavity. Other kinds of effective emissivities can be obtained by averaging over the spatial, angular, and spectral domains. For example, integration over a ﬁnite spectral range [l1, l2] results in the following equation for the

Calculation of the Radiation Characteristics of Blackbody Radiation Sources

185

bandlimited effective emissivity: R l2 e ðl; l1 ; l2 ; n; x; T 0 Þ ¼ R l2 l1

L l ðl; n; xÞ dl

l1

L l;bb ðl; n; x; T 0 Þ dl

(3)

Similarly, integration over the entire spectrum of radiation reduces the equation to a ratio of radiances. Application of the Stefan2Boltzmann law further simpliﬁes the equation to: e ðn; x; T 0 Þ ¼

pLðn; xÞ sT 40

(4)

where L is the radiance of the cavity wall and s is the Stefan2Boltzmann constant [9]. One can also integrate with respect to the solid angle, over a hemisphere. This transforms the term for spectral radiance, Ll, to spectral radiant exitance, Ml. Thus, the spectral hemispherical effective emissivity at a particular wavelength, l, is then deﬁned as e;h ðl; n; T 0 Þ ¼

M l ðl; nÞ M l ðl; nÞ ¼ M l;bb ðl; T 0 Þ pL l;bb ðl; T 0 Þ

(5)

Further integration over a bandwidth of radiation or the complete spectrum results in bandlimited and total hemispherical effective emissivities: R l2

e;h ðl; n; T 0 Þ ¼

l1 M l ðl; nÞ dl R l2 l1 M l;bb ðl; nÞ dl

e;h ðn; T 0 Þ ¼

R l2 ¼ p

l1

M l ðl; nÞ dl

l1

L l;bb ðl; nÞ dl

R l2

M ðnÞ M ðnÞ ¼ M bb ðT 0 Þ sT 40

(6)

(7)

In some situations it is useful to know the spectral integrated effective emissivity. This quantity, ee,c, is the ratio of the spectral radiation ﬂux falling onto the detector from a blackbody, Fl, to the spectral radiation ﬂux, Fl,bb, from a perfectly black surface that replaces the blackbody aperture and has the temperature T0: e;c ðl; T 0 Þ ¼

Fl ðlÞ Fl;bb ðl; T 0 Þ

(8)

Integration over a bandwidth of radiation or the complete spectrum results in bandlimited and total integrated effective emissivities: R l2 e;c ðl; l1 ; l2 ; T 0 Þ ¼ R l2 l1

l1

Fl ðlÞ dl

Fl;bb ðl; T 0 Þ dl

(9)

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e;c ðT 0 Þ ¼

F Fbb ðT 0 Þ

(10)

A common calibration method, which does not utilize optical elements, is composed of a blackbody with a circular aperture of radius, Ra, and a circular detector of radius, Rd. The aperture and detector are coaxial, lying on two parallel planes separated by a distance, Hd. In the limiting case where Rd ¼ Ra and Hd-N, we refer to the quantity ee,n, the normal effective emissivity. If Rd ¼ Ra and Hd-0, then we refer to, the hemispherical effective emissivity, ee,h. The latter characterizes the overall radiative heat loss of a cavity through its aperture. Depending on the particular viewing conditions that are used for various types of radiation thermometers, pyrometers, radiometers, etc., one can deﬁne the appropriate types of effective emissivities by averaging the local directional effective emissivity over a visible part of the cavity internal surface and a suitable solid angle. All the previous deﬁnitions have been developed for a nonradiating background environment. However, real environments have temperatures greater than absolute zero. Therefore, thermal radiation from the surrounding environment will irradiate the aperture of a blackbody cavity and can fall onto the detector after multiple reﬂections inside the cavity. Because the spectral, spatial, and angular distributions of background radiation are hard to predict, one usually considers the simplest case of isotropic blackbody radiation corresponding to the background temperature, Tbg. Assuming that the detector does not distort the isotropy of the background radiation, the effect of the background radiation on the spectral local directional effective emissivity of a nonisothermal blackbody can be taken into account by the second term in the equation: e ðl; n; x; T 0 ; T bg Þ ¼ e ðl; n; x; T 0 Þ þ ½1 e ðl; n; xÞ

expðc 2 =ðlT 0 ÞÞ 1 expðc 2 =ðlT bg ÞÞ 1

ð11Þ

Here ee(l, n, x, T0, Tbg) is the spectral effective emissivity of a nonisothermal blackbody taking into account the background radiation; ee(l, n, x, T0) does not include this correction; ee(l, n, x) is the spectral effective emissivity of an isothermal blackbody. By analogy, the bandlimited and total effective emissivities of a nonisothermal blackbody taking into account the background radiation can be deﬁned by the equations: e ðl; l1 ; l2 ; n; x; T 0 ; T bg Þ

R l2

¼ e ðl; l1 ; l2 ; n; x; T 0 Þ þ ½1 e ðl; l1 ; l2 ; n; xÞ Rll1 2 l1

L l;bb ðl; T bg Þ dl L l;bb ðl; T 0 Þ dl

ð12Þ

187

Calculation of the Radiation Characteristics of Blackbody Radiation Sources

and

T bg e ðn; x; T 0 ; T bg Þ ¼ e ðn; x; T 0 Þ þ ½1 e ðn; x; T 0 Þ T0

4 (13)

Another important quantity is the radiance temperature, TS, that is deﬁned as the temperature of a perfect blackbody, for which the spectral radiance at the speciﬁed wavelength l has the same value as for the thermal radiator under consideration. For an imperfect blackbody having the spectral effective emissivity ee, the radiance temperature can be computed as T S ðl; n; xÞ ¼ c 2

expðc 2 =ðlT 0 ÞÞ 1 l ln 1 þ e ðl; n; x; T 0 Þ

1 (14)

Rather than identifying the temperature associated with a particular wavelength, we can also identify the temperature at which the radiance emitted from a perfect blackbody is equal to the radiance emitted from a blackbody radiator. This temperature is referred to as the radiation temperature, TR: T R ðn; xÞ ¼ T 0

p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 e ðn; x; T 0 Þ

(15)

The equations deﬁning the radiance and radiation temperatures for different viewing conditions can be written by analogy to the equivalent types of effective emissivity. In this section, we have considered only a portion of the characteristics describing the radiation of blackbody radiators that appear in published works. There has been no uniﬁed system presented in literature for terminology used in this section that is similar to the nomenclature developed by Nicodemus et al. [10] for material optical properties. Deﬁnitions concerned with the viewing conditions using an optical system are completely absent. This can sometimes lead to serious difﬁculties when comparing the results obtained by independent researchers. At the same time, we have not attempted to establish a comprehensive terminology for the radiation characteristics of blackbodies, but rather to provide useful terms and clarity about their meaning. In the following sections, we shall consider the principal methods used for calculating the radiation characteristics that we have deﬁned above.

3. Deterministic Methods 3.1. Approximate expressions for isothermal diffuse cavities The ﬁrst and simplest approximate analytical expressions were derived for an isothermal cavity whose internal surface emits and reﬂects optical radiation according to Lambert’s law (diffuse cavity approximation).

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If Kirchhoff’s law holds for effective characteristics of a cavity (see Refs. [11,12] for detailed discussions), then to calculate the normal effective emissivity ee,n of a cavity, it is sufﬁcient to compute its normal effective absorptance ae,n (or normal effective reﬂectance, re,n, since ae,n + re,n ¼ 1 for opaque cavity walls). In other words, e;n ¼ ae;n ¼ 1 re;n

(16)

for a beam of radiation falling into the cavity perpendicularly to its aperture. Ribaud [13] considered only the ﬁrst reﬂection; assuming that all radiation is absorbed by the cavity after a single reﬂection, he obtained: e;n ¼ 1 r

O p

(17)

where r is the reﬂectance of a cavity wall and O the solid angle subtended by the cavity aperture from the center of the area irradiated by an inﬁnitely thin incident ray. Gouffe´ [14] considered a cavity of arbitrary shape, with an aperture area, s, and a total surface area, S (aperture included). Gouffe´ assumed that the reﬂected radiation is uniformly distributed over the cavity walls from each reﬂection. Summing the radiation ﬂuxes escaping the cavity following each reﬂection, and using the expression for an inﬁnite geometric progression, he obtained the following formula for the effective emissivity: e ¼

½1 þ ð1 Þððs=SÞ ðO=pÞÞ ð1 ðs=SÞÞ þ ðs=SÞ

(18)

where e ¼ 1r. Equation (18) gives the exact value of the effective emissivity for the case of an isothermal diffuse spherical cavity because the assumption that the reﬂected radiation is uniformly distributed over the cavity walls from each reﬂection is correct for a spherical cavity. For a sphere, s/S ¼ O/p and Equation (18) can be rewritten as: e ¼

ð1 ðs=SÞÞ þ ðs=SÞ

(19)

Treuenfels [15] represented the effective emissivity of a diffuse cavity in the form: e ¼ 1

1 X

f k rk

(20)

k¼1

where fk is the fraction of the radiation ﬂux, reﬂected k times before emerging from the cavity and 1 X k¼1

fk ¼ 1

(21)

Calculation of the Radiation Characteristics of Blackbody Radiation Sources

189

Treuenfels made an assumption that the ratio of successive fractions, b ¼ fk+1/fk, is constant and always less than 1, from which he derived the following form for the effective emissivity: e ¼

þ f 1 ð1 Þ

(22)

Equation (22) is the equivalent of Equation (19), if s/S is substituted for f1. For a variety of simple geometric shapes, f1 can be evaluated using R 2 F dA f 1 ¼ RA xs A F xs dA

(23)

where A ¼ S s is the cavity radiating surface area and Fxs is the diffuse view factor [3,4] from a surface element dA at a position n to the aperture s that is equal to the ratio of the radiation ﬂux diffusely emitted by dA into the hemisphere to its fraction falling onto the aperture (other terms used are angle factor [1], conﬁguration factor [2], and form factor [16]). Using analytical integration, Treuenfels obtained expressions for f1 for several simple geometries such as a spherical cavity, and inﬁnite grooves of cylindrical, triangular, and rectangular proﬁles. Despite the improved sophistication of Treuenfels’ method as compared to Gouffe´’s method, it does not provide an assured increase of the accuracy for calculation of the effective emissivity of an arbitrarily shaped cavity. In addition, neither of these methods takes into account the real temperature distribution over cavity’s radiating surface.

3.2. Method of integral equations for diffuse cavities The internal walls of diffuse cavities emit and reﬂect optical radiation according to Lambert’s cosine law. Let us consider a diffuse cavity (see Figure 1). For simplicity, we shall assume the radiative characteristics to be temperature independent and uniform over the entire cavity surface. The element dA of a cavity surface A at the point n has the temperature T(n). The outgoing radiation of the element dA consists of its own thermal radiation and the radiation falling onto the element dA from the remainder of a surface A. The surface density (or radiosity) B of the outgoing radiation ﬂux can be expressed as [1]: BðnÞ ¼ sT 4 ðnÞ þ ð1 ÞEðnÞ

(24)

where E(n) is the irradiance of the element dA produced by the rest of the cavity’s internal surface.

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Blackbody cavity Aperture

x

dA′

ξ′

z y

ξ

θ′

d

n

θ

n′

dA

Figure 1 Drawing for derivation of the equation for effective emissivity. n and nu are the normals to the surface elements dA and dAu, respectively.

It is deﬁned as Z

Bðn0 Þ dF dAdA0 ðn; n0 Þ

EðxÞ ¼

(25)

A

where the integration occurs over the entire cavity’s radiating surface A and dF dAdA0 ¼

cosy cosy0 0 dA pd 2

(26)

is the differential view factor between elements dA at n and dAu at nu(see Figure 1). By substituting Equation (25) into Equation (24), we obtain: Z

4

Bðn0 Þ dF dAdA0 ðn; n0 Þ

BðnÞ ¼ sT ðnÞ þ ð1 Þ

(27)

A

The local hemispherical effective emissivity of a cavity wall is deﬁned as e;h ðn; T 0 Þ ¼

BðnÞ sT 40

(28)

For a cavity with perfectly diffuse walls, the local hemispherical effective emissivity is equal to the local directional effective emissivity; therefore we can omit the subscript ‘‘h’’ in the left-hand side of Equation (28). After substituting Equation (27) into Equation (28), we obtain:

T ðnÞ e ðn; T 0 Þ ¼ T0

4

Z

e ðn0 ; T 0 ÞKðn; n0 Þ dA0

þ ð1 Þ

(29)

A

where Kðn; n0 Þ ¼ dF dAdA0 ðn; n0 Þ=dA0 is the kernel of the linear integral equation, in which the unknown function ee(n, T0) is placed inside the integral sign. For an isothermal cavity, Equation (29) can be rewritten in the form: Z e ðnÞ ¼ þ ð1 Þ A

e ðn0 ÞKðn; n0 Þ dA0

(30)

Calculation of the Radiation Characteristics of Blackbody Radiation Sources

191

By introducing the spectral radiosity Bl ðl; nÞ ¼

c 1 ðlÞ

þ ½1 ðlÞEl ðl; nÞ l5 expðc 2 =ðlT ðnÞÞÞ 1

(31)

which is the sum of the spectral exitance of thermal radiation, emitted from a diffuse surface element with a spectral emissivity e(l), and the reﬂected portion of the incident spectral irradiance El. One can write the analogs of Equations (29) and (30) for the spectral effective emissivity of a nonisothermal and an isothermal cavity: e ðl; n; T 0 Þ ¼ ðlÞ

expðc 2 =ðlT 0 ÞÞ 1 þ ½1 ðlÞ expðc 2 =ðlT ðnÞÞÞ 1

Z

e ðl; n0 ; T 0 ÞKðn; n0 Þ dA0 A

(32) Z e ðl; nÞ ¼ ðlÞ þ ½1 ðlÞ

e ðl; n0 ÞKðn; n0 Þ dA0

(33)

A

Equations (32) and (33) must be solved for every value of wavelength l. Mathematically, the equations presented above all have the form of a Fredholm’s integral equation of second kind: Z

b

Kðn; n0 Þe ðn0 Þ dn0 ¼ LðnÞ

e ðnÞ r

(34)

a

where r ¼ 1 e, a r n, nurb, and 8 for an isothermal blackbody > > > > > > < ðlÞ expðc 2 =ðlT 0 ÞÞ 1 for the spectral effective emissivity expðc 2 =ðlT ðnÞÞÞ 1 LðnÞ ¼ > > > T ðnÞ 4 > > for the total effective emissivity > : T0

(35)

A variety of methods, analytical and numerical, have been developed (e.g., see Refs. [17219]) for solving equations of the form of Equation (34). Only two geometrical conﬁgurations are known to have exact analytical solutions. The ﬁrst is a diffuse concave cylindrical groove. Its radiative properties were comprehensively studied in Ref. [20]. The second is a spherical cavity [21]. The sphere has a notable property: the view factor between two surface elements, inﬁnitesimal or ﬁnite, does not depend on their positions. Thus, dF dAdA0 ¼ dF AdA0 ¼

dA0 4pR2

(36)

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Alexander V. Prokhorov et al.

and F dAA0 ¼ F AA0 ¼

A0 4pR2

(37)

where R is the sphere radius. After substitution of Equation (36) into Equation (27) and simple transformations, for isothermal spherical cavity, we obtain e ¼

1 ð1=2Þð1 Þð1 þ cosfÞ

(38)

where f is the opening half-angle. It is easy to show that Equation (38) is equivalent to Equation (19) (Gouffe´ formula). The closed-form solution for the nonisothermal diffuse spherical cavity can be found in Ref. [22]. In particular, for axisymmetrical distributions of temperature T(x) and the wall’s emissivity e(l, x), the spectral radiosity can be expressed as Rx ½1 ðl; xÞ 0 0 ðl; x0 ÞL l;bb ðl; T ðx0 ÞÞ dx0 Rx Bl ðl; xÞ ¼ ðl; xÞL l;bb ðl; T ðxÞÞ þ (39) 2R 0 0 ½1 ðl; x0 Þ dx0

All the geometrical parameters are shown in Figure 2. By dividing both sides of Equation (39) by Ll,bb(l, T0), we can obtain the expression for the local effective emissivity. A number of approximate methods for solving Equations (29) and (32) have been developed in precomputer era. However, they are of historical interest only due to the signiﬁcant difﬁculty of analytical integration and the unsatisfactory accuracy of the results. We shall consider three numerical methods that, in principle, allow us to obtain exact solutions, assuming no limitations on computing time, s

s″ R

0

x0

x

dx

Lλ(s) s′

Figure 2

Geometry of a nonisothermal spherical cavity (from Ref. [22]).

Calculation of the Radiation Characteristics of Blackbody Radiation Sources

193

rounding errors, etc. The ﬁrst method is the method of iterations (or the method of successive approximations). Let us rewrite Equation (34) in the form Z

b

K ðx; x0 Þe ðx0 Þ dx0 þ jðxÞ

e ðxÞ ¼ r

(40)

a

By substitution into the right-hand side of Equation (40) of an arbitrary continuous trial function ð0Þ e ðxÞ instead of ee(x), we will obtain ð1Þ e ðxÞ

Z

b 0 0 K ðx; x0 Þð0Þ e ðx Þ dx þ jðxÞ

¼r a

(41)

Successive repetition of this process results in ðnÞ e ðxÞ ¼ r

Z a

b

Kðx; x0 Þðn1Þ ðx0 Þ dx0 þ jðxÞ e

(42)

where ðnÞ e ðxÞ; n ¼ 1; 2; . . . are successive approximations to the solution of Equation (30). Note that Equation (42) reduces to Equation (40) in the limit that ðnÞ ðxÞ ! e ðxÞ for n-N. Usually, the integration is performed e ð1Þ numerically. Assuming that ð0Þ e ¼ 0 and e ¼ , we can transform Equation (43) to the form [23]: ðn1Þ ðnÞ ðxÞ þ r e ðxÞ ¼ e

Z a

b

Kðx; x0 Þ½ðn1Þ ðx0 Þ ðn2Þ ðx0 Þ dx0 e e

(43)

Numerical integration is performed with the help of one of the quadrature formulas. Although Equation (43) requires the use of the results of two preceding iterations for each step, convergence of the iteration process itself is improved. The second method is the series method. We assume the solution of Equation (34) to be of the form e ðxÞ ¼ LðxÞ þ

1 X

rk Ek ðxÞ

(44)

k¼1

After substitution of Equation (44) into Equation (34) and collecting the coefﬁcients of equal powers of r, we obtain the recurrent system of equations for the functions Ek(x): Rb

Kðx; x0 ÞLðx0 Þ dx0 Rb E 2 ðxÞ ¼ a Kðx; x0 ÞE1 ðx0 Þ dx0 ¼ a K 2 ðx; x0 ÞLðx0 Þ dx0 Rb Rb E 3 ðxÞ ¼ a Kðx; x0 ÞE2 ðx0 Þ dx0 ¼ a K 3 ðx; x0 ÞLðx0 Þ dx0 Rb

Ek ðxÞ ¼

Rb a

E1 ðxÞ ¼

a

::: 0

0

Kðx; x ÞE k1 ðx Þ dx0 ¼

Rb a

K k ðx; x0 ÞLðx0 Þ dx0

(45)

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Alexander V. Prokhorov et al.

where 0

0

K 1 ðx; x Þ ¼ Kðx; x Þ;

Z

0

b

Kðx; x00 ÞK k1 ðx00 ; x0 Þ dx00 ;

K k ðx; x Þ ¼

k ¼ 2; 3; . . .

a

(46)

are the iterated kernels of the integral equation. Rewriting Equation (44) in terms of Equations (45) and (46) results in the Neumann series: e ðxÞ ¼ LðxÞ þ

1 X

rk

Z

b

K k ðx; x0 ÞLðx0 Þ dx0

(47)

a

k¼1

Sydnor in Ref. [24] notes that the series representation, in the special case of wavelength-dependent surface emissivity, offers a signiﬁcant saving in computation time for the evaluation of a large number of monochromatic values ee(l, x). The third approach to solving Equation (34) is the quadrature method. In this method, the linear integral equation is reduced to the system of linear algebraic equations using one of the quadrature formulas (usually, Simpson’s or the trapezoidal rule) for the integral calculations. Generally, the quadrature formula can be written in the form: Z

b

zðxÞ dx ¼

n X

a

Ai zðxi Þ þ Dn ½z

(48)

i¼1

where xi (i ¼ 1, 2, y, n) are the nodes of the quadrature on the interval [a, Pn b], AiZ0 are the weighting factors that do not depend upon z(x), i¼1 Ai ¼ b a, and Dn[z] is the remainder term. If the remainder term is negligible, one can rewrite Equation (34) as the system of n linear algebraic equations with the n variables ei ¼ e(xi) such that e;k r

n X

Ai Kðxk ; xi Þe;i ¼ Lk ;

k ¼ 1; 2; . . . ; n

(49)

i¼1

where Lk ¼ Lðxk Þ; K ki ¼ Kðxk ; xi Þ. Equation (49) can be written in an equivalent form: n X ½dik rAi K ki e;i ¼ Lk ;

k ¼ 1; 2; . . . ; n

(50)

i¼1

where dik is the Kronecker delta, deﬁned as (

dik ¼

1; 0;

when i ¼ k when iak

(51)

Calculation of the Radiation Characteristics of Blackbody Radiation Sources

195

Equation (50) is solvable by standard methods [25]. Integral equations of the theory of radiative heat transfer have been intensively applied to diffuse cavities of simple shapes since the early 1960s. For example, Sparrow and coworkers computed the distributions of local effective emissivity for isothermal cylindrical [26] and conical [27] cavities without lids using a digital computer and method of successive approximations. Alfano [28,29] obtained analogous results for cylinders with and without a diaphragm (lid). 3.2.1. Zonal approximation method When solving the integral equations discussed above inaccuracies in the calculation of the local effective emissivity can occur in cases where singularities exist. In particular, the geometry of the cavity may create singular points at the apex of a cone or at a junction of cavity walls. At some of these points, the view factors have an indeterminate form that can be evaluated with the help of L’Hoˆpital’s rule. Kelly [30], for example, obtained the expression for the local effective emissivity of a vertex in an isothermal diffuse conical cavity: e ð0Þ ¼

þ ð1 Þ sin3 ðb=2Þ

(52)

where b is the angle at the cone vertex. In other cases, e.g., for points lying on a junction line between the bottom and lateral wall of cylindrical cavity, the view factor is indeﬁnite and the radiosity has a discontinuity. More importantly, the kernel in Equation (39) contains the absolute values of the difference of two homonymous coordinates (as |xxu| or |xx|) and, correspondingly, has discontinuities in the slope (cusps) at x ¼ xu or x ¼ x. Direct application of the quadrature formulas to these points leads to inaccuracies that can only be reduced by greatly increasing the number, n, of quadrature nodes. Because the volume of computations for the iterative solution of an integral equation is proportional to n2, the computational task grows rapidly. Bedford and Ma [31] demonstrated that these inaccuracies cannot be completely eliminated by reasonable increasing the number of nodes in quadrature formula. To resolve this problem they proposed using the method of zonal approximation, which essentially transforms the integral in Equation (34) into a summation. Following Hottel and Saroﬁm [4], Bedford and Ma proposed to use the zonal approximation for the integral of the form: Z

L

e ðxÞ dF xx ¼ 0

n Z X i¼1

xi

xiþ1

e ðxÞ dF xx

(53)

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where x1 ¼ x1 ¼ 0, xn+1 ¼ xn+1 ¼ L. Assuming ee(x) to be a slowly varying function of x, one can write: Z

L

e ðxÞ dF xx ¼ 0

n 1X ½e ðxiþ1 Þ þ e ðxi Þ 2 i¼1

Z

xiþ1

dF xx

(54)

xi

The integral in Equation (54) may be expressed by the difference of diffuse view factors F xxi and F xxiþ1 between an inﬁnitesimal ring at x and two circular disks at xi and xi+1. Bedford and Ma [31234] applied the zonal approximation to diffuse isothermal and nonisothermal conical, cylindrical, cylindro-conical, double conical, and cylindro-inner-conical cavities, with and without diaphragms. Note that zonal approximation is not the only method that provides a correct solution for the Fredholm’s integral equations of the second kind with discontinuous kernels. In 1971, Yamamoto [35] used integration by parts to avoid the singularity in the equation kernel and computed the distributions of the local effective emissivity for isothermal and nonisothermal diffuse cylindrical cavities. Chandos and Chandos [36] applied R L the method described in Ref. [37] in order to evaluate the integral 0 e ðxÞFðx; xÞ dx, where F(x, x) has a discontinuous slope at x ¼ x. They used a uniform grid x0, x1, y, xn on the interval [0, L] for the quadrature method. In the interval [xj1, xj], ee(x) is replaced by the linear interpolation ððxj xÞe ðxj1 Þ þ ðx xj1 Þ e ðxj ÞÞ=ðxj xj1 Þ. The integral is approximated by the sum: Z

n X

L

e ðxÞFðx; xÞ dx ﬃ 0

½aj e ðxj1 Þ þ bj e ðxj Þ

(55)

j¼1

where aj ¼

1 xj xj1

Z

xj

ðxj xÞFðx; xÞ dx

(56)

ðx xj1 ÞFðx; xÞ dx

(57)

xj1

and bj ¼

1 xj xj1

Z

xj

xj1

This method was applied to the integral equations for conical, cylindrical, and extended conical (i.e., biconical) isothermal cavities and shown to have very good accuracy. 3.2.2. Cavities with screened walls A geometrical shape widely used for blackbody cavities is the cylindro-innerconical cavity (see Figure 3). A cylindro-inner-conical cavity is not a convex

197

Calculation of the Radiation Characteristics of Blackbody Radiation Sources

x

r=R

r

r = R1 2α

r=0 y

x = y = R/tanα

x=y=0

Figure 3

x = y = 2R/tanα

x=L

Cylindro-inner-cone (from Ref. [34]).

cavity. The internal conical bottom can partially obstruct the radiation exchange between cylindrical walls, resulting in partial screening of the internal surface. Thus, there are regions inside the cylindro-inner-cone that cannot ‘‘view’’ one another due to screening by the reentrant cone base. Bedford et al. studied this cavity in Ref. [34]. To simplify the analysis, they imposed the condition LZ2R/tan a that ensures the visibility of the entire bottom from a point on the lid and vice versa. One can write the system of integral equations for e ðxÞ ¼ e ðx; l; T x ; T 0 Þ; e ðyÞ ¼ e ðy; l; T y ; T 0 Þ; and e ðrÞ ¼ e ðr; l; T r ; T 0 Þ, and then apply the zonal approximation method as described above: ( n1 r X e ðxÞ ¼ LðxÞ þ ½e ðxiþ1 Þ þ e ðxi ÞjF x;xiþ1 F x;xi j 2 i¼1 þ

n X ½e ðyjþ1 Þ þ e ðyj ÞðF x;yjþ1 F x;yj Þ j¼1

) n X ½e ðr kþ1 Þ þ e ðr k ÞðF x;r kþ1 F x;yk Þ þ k¼m

( n r X ½e ðxiþ1 Þ þ e ðxi ÞjF y;xiþ1 F y;xi j e ðyÞ ¼ LðyÞ þ 2 i¼1 ) n X þ ½e ðr kþ1 Þ þ e ðr k ÞðF y;r kþ1 F y;r k Þ

(58)

k¼m

( n1 r X ½e ðxiþ1 Þ þ e ðxi ÞðF r;xiþ1 F r;xi Þ e ðrÞ ¼ LðrÞ þ 2 i¼1 ) n X þ ½e ðr jþ1 Þ þ e ðr j ÞðF y;r jþ1 F y;r j Þ j¼1

Even if the condition LZ2R/tana is fulﬁlled, mutual screening can take place for view factors dF y;xiþ1 ;xi ; dF x;yjþ1 ;yj , and dF x;xiþ1 ;xi .

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xi+1 xi y

v

u

w

Figure 4 For derivation of the view factor dF y;xiþ1 ;xi with screening taken into account (from Ref. [32]).

Let us consider as an example the view factor dF y;xiþ1 ;xi that describes the radiant transfer from the element dAy on the conical bottom and an incomplete circular band on the cylindrical wall. Let us assume that the element dAy of the conical surface has the coordinates (y, Ry tan a, 0) as shown in Figure 4. When xi+1o2R/tan a, dF y;xiþ1 ;xi can be expressed as the absolute value of the difference in view factors between dAy and circular segments deﬁned by xi and xi+1: dF y;xiþ1 ;xi ¼ jF y;xi F y;xiþ1 j

(59)

The view factor dF y;xi can be evaluated using Stokes theorem which converts the double integral over the area of the segment at xi to the sum of three single contour integrals around the boundary of the segment (see Ref. [1], pp. 1282136). For the local coordinate system, (u, v, w) chosen in such a way that one of the three integrals becomes zero, and the other two correspond to each part of the two-part boundary:

F y;xi

( "pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ #) a2i b2 sina 2ð2R2 ai Þ 1 fðxi Þ tan fðxi Þ þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2ﬃ tan ¼ ai þ b 2p 2 a2i b 1 R sinfðxi Þ (60) þ tan1 p ðxi yÞ seca ( "pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ #) a2i b2 ðxi yÞ cosa 2ai fðxi Þ fðxi Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2ﬃ tan1 tan þ 2pðR y tanaÞ 2 ai þ b a2i b

Calculation of the Radiation Characteristics of Blackbody Radiation Sources

199

where xi tana fðxi Þ ¼ cos1 1 R ai ¼ R2 þ ðxi yÞ2 þ ðR y tanaÞ2 b ¼ 2RðR y tanaÞ

(61)

If xiW2R/tan a, there is no screening, and the circular segments become complete circular disks. Then the expression for dF y;xi reduces to F y;xi

" # " # sina ai 2R2 ðxi yÞ cosa ai ¼ 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2ﬃ þ 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2ﬃ 2 2ðR y tanaÞ a2i b a2i b

(62)

To derive the remaining view factors in Equations (58) and treat their singularities, the same approach as described in Refs. [31233] is applied. Meier et al. [38] examined a spherical cavity with a cylindrical extension. Such a cavity has the mutually screened regions of the internal surface: if the radius of the cylinder is less than that of the sphere, the spherical part of the cavity may not be completely visible from a point on its cylindrical part. They composed a system of the integral equations and derived the necessary expressions for view factors. However, the authors of Ref. [38] were apparently unacquainted with Ref. [34], since the solution for the effective emissivity was only determined in ﬁrst approximation of a power series expansion. Nevertheless, since the authors of Ref. [38] only examined the quality of a cylindrical2spherical cavity with an internal coating of very high emissivity (eZ0.95), the ﬁrst-approximation solution was sufﬁcient.

3.3. Integrated effective emissivity of diffuse cavities The method of integral equations for diffuse cavities enables the calculation of the distributions of the local effective emissivity over the internal surface of a cavity. In practice, however, it is necessary to know the values of effective emissivity that correspond to the speciﬁc arrangement of the cavity and the detector. These values can be obtained by integrating the distribution of the local effective emissivity over the area of visibility. Let us consider two diffuse cavities of arbitrary shape, the same depth L, and reference temperature T0, with no internal screening, and a coaxial circular detector of radius Rd, spaced at a distance H from the cavity aperture (Figure 5). The principal distinction between the two systems depicted in Figure 5 is the difference in the visibility of the internal cavity surface from any point on the detector. For a cavity without a lid, the entire internal surface can be observed from any point on the detector. Thus, the

200

Alexander V. Prokhorov et al.

Cavity without lid Detector Rd

L

H

Rd Detector Cavity with lid

Figure 5

Cavity-detector systems with (lower) and without (upper) vignetting.

radiation ﬂux falling onto the detector is equal to Z

L

e ðx; T 0 ÞF xd ðL þ H xÞ dAx

F ¼ M bb ðT 0 Þ

(63)

0

where Mbb is the radiant exitance of a blackbody (in W/m2) and Fxd(x) is the view factor between the annular element dAx of the cavity surface placed at a distance x from the bottom vertex and the detector. According to the integrated emissivity deﬁnition (10), RL

e ðx; T 0 ÞF xd ðL þ H xÞ dAx RL 0 F xd ðL þ H xÞ dAx Z L 1 ¼ e ðx; T 0 ÞF xd ðL þ H xÞ dAx F dd ðHÞ 0

e;c ðH; T 0 Þ ¼

0

(64)

where Fdd is the view factor from the cavity aperture to the detector (view factor between two coaxial disks). Sparrow and Heinisch [39] computed the hemispherical (H ¼ 0) effective emissivity and examined the dependence of the integrated effective emissivity on H and Rd for a cylindrical cavity without a lid. They found that the distance between the detector and the cavity aperture plays an important role on the normal effective emissivity results, and that the detector radius is only of secondary importance. Furthermore, the normal effective emissivity exceeds the hemispherical effective emissivity for cavities having a depth-to-radius ratio greater than unity. Alfano and Sarno¨ [40] expanded the analysis to bafﬂed cylindrical cavities but assumed that the detector was inﬁnitesimal.

Calculation of the Radiation Characteristics of Blackbody Radiation Sources

201

For a bafﬂed cavity, there are three zones on its internal surface (see lower part of Figure 5): one that is visible from every point on the detector; the other that is invisible from any point on the detector; and the third that is a zone of penumbra (shaded area in Figure 5) where the visibility depends on the positions of both the location on the cavity and the location on the detector. The size of the penumbra region decreases when H increases. Thus, the inﬂuence of the vignetting effect on the integrated effective emissivity can be neglected for an inﬁnitely distant detector (i.e., for the normal effective emissivity) because all of the rays falling onto detector are parallel to the cavity axis. A double integration with variable integral terms was applied in Ref. [41] to obtain rigorous solutions for the bafﬂed conical and cylindrical cavities. This method was generalized in Ref. [42] for the precise evaluation of the integrated effective emissivity of any practical cavity with circular symmetry. An alternative approach to the view factors calculation based on the Stokes’s theorem and contour integration was proposed in Ref. [43]. In this approach, the triple numerical integration is replaced by a single integration, which, in turn, is replaced by a zonal summing approximation. The authors of Ref. [43] considered a diffuse frustum cone with a reentrant cone base and an optional stop of radius R0 between the cavity aperture and the detector. The authors indicated that the computing time was substantially reduced and the precision of the integrated emissivity calculation was increased as compared to the conventional formulation.

3.4. Series technique for multiple reﬂections in cavities 3.4.1. Diffuse cavities Ohwada in Ref. [23] described a method for calculating the effective emissivity of diffuse cavities that is based on iterative solution of the integral equations of the form of Equation (40). The main equation of this method is like Equation (43) which links an nth approximation to those of orders n1 and n2. Ohwada referred to her method as a series technique (see Refs. [44,45]). In the numerical solution of the integral equations, the zonal approximation method [31] was employed for view factors. Ohwada applied this technique to the calculation of the local effective emissivities of isothermal cylindrical [46], conical [44], cylindro-conical and double conical cavities [45], as well as their integrated effective emissivities [47]. In Refs. [48,49], this technique was extended to the spectral, total, and bandlimited effective emissivities of nonisothermal cavities of various geometrical shapes. Ohwada and Sakate also calculated effective emissivities for a cylindrical cavity with a longitudinal strip opening [50].

202

Alexander V. Prokhorov et al.

(a)

(b)

Detector h

P

r

N R N0

γ l

θ y

Q

x

Figure 6 (a) Schematic illustration of incident and reﬂected rays, and angle c. (b) Geometry of the cavity and the detector (from Ref. [51], with alterations).

3.4.2. Non-Lambertian cavities Ohwada in Ref. [51] attempted to extend the series technique to cavities with non-Lambertian internal walls. The positions Q and P in Figure 6(a) are on the cavity wall and point R is on the cavity wall or on the cavity opening. Area elements located at P, Q, and R are denoted by DP, DQ, and DR, respectively. Radiation leaves DP, falls on DQ, and after undergoing a reﬂection, falls on DR. The number of the area elements on the cavity wall is given by k, while the total number of area elements on the cavity wall and opening is given by m. The direction from Q to R is designated CQR. The radiance of DQ in the direction CQR is designated L(Q, CQR). Then L n ðQ; CQR Þ ¼ L n1 ðQ; CQR Þ þ

k X

f ðCQP ; CQR Þ½L n1 ðQP ; CPQ Þ

P¼1

(65)

L n2 ðQP ; CPQ Þ cosðfQP ÞDoQP

where L 0 ðQ; CQR Þ ¼ 0 L 1 ðQ; CQR Þ ¼ 0 L B ðT Þ for

Q ¼ 1; 2; . . . ; k;

(66) R ¼ 1; 2; . . . ; k; . . . ; m (67)

where CQP is the direction from Q to P, fQP the angle between the normal QN0 to DQ and the line connecting Q and P, DoQP the solid angle subtended by P when viewed from a point Q, L the radiance of the cavity wall at temperature T, and e0 is the surface emissivity, assumed to be independent of the direction for simplicity.

Calculation of the Radiation Characteristics of Blackbody Radiation Sources

203

To simulate a non-Lambertian surface, the angle g between the normal QN0 and the line QN, bisecting +PQR, is introduced. The bidirectional reﬂectance distribution function (BRDF), in accordance with its deﬁnition [10] is expressed as follows: f ðCQP ; CQR Þ ¼ P m R¼1

r0 g½gðCQP ; CQR Þ

(68)

g½gðCQP ; CQR Þ cosðfQR ÞDoQR

where fQR is the reﬂection angle, r0 ¼ 1e0 the directional-hemispherical reﬂectance, and g½gðCQP ; CQR Þ ¼ exp½ag2 ðCQP ; CQR Þ

(69)

Such a Gaussian dependence of the BRDF on the angle g can be used for approximate description of optical properties of randomly rough surfaces. When a ¼ 0, the BRDF is Lambertian. When a-N, the reﬂection is specular. The integrated effective emissivity for the detector arrangement shown in Figure 6(b) is calculated according to the following equation: PP

e;c ¼

LðQ; CQR Þ cosðfQR ÞDoQR DQL PP L B ðT Þ cosðfQR ÞDoQR DQL

Q R

(70)

Q R

where L(T ) is the radiance of a black body at temperature T of the cavity surface. The summations are carried out over the ranges of Q and R. Radiation leaving DQ in the direction of CQR arrives at the detector. The dependences of ee,c upon various geometrical parameters was computed and compared with that for a Lambertian reﬂection [51]. However, results of calculations performed for a closed cavity (perfect blackbody) have a deviation in effective emissivity from unity of up to 0.01. In Ref. [51], Ohwada supposed that the calculation accuracy might be improved by subdividing the cavity surface into a ﬁner mesh and increasing the number of signiﬁcant digits for computing the quantities related to the geometric factors. In her subsequent paper [52], Ohwada concluded that the reason for the poor accuracy in her calculation was due to an intrinsic contradiction in the model related to the optical properties of the cavity wall. The integration of the BRDF containing the Gaussian factor as in Equation (69) over the entire hemispherical range of viewing directions results in a directional-hemispherical reﬂectance that depends on incident angle. According to Kirchhoff’s law, the emissivity of such a surface must also depend on the emission angle. However, in Ref. [51], the angular independence of r0 and e0 was assumed. This was improved in Ref. [52] by using the reciprocity rule and considering r0 and e0 as functions of the incident angle for several more

204

Alexander V. Prokhorov et al.

realistic models of the BRDF. These improvements were implemented and the deviations of effective emissivity from unity for a closed cavity were reduced to values less than the truncation error (about 104); the hemispherical, normal, and integrated emissivities for a coaxial detector, and for a detector with optical system consisting of a lens and two stops were computed, and the inﬂuence of the surface condition (i.e., of the parameter a) on these values was investigated.

3.5. Cavities with diffusely emitting and specularly reﬂecting walls In the infrared spectral range, when the surface roughness becomes negligible in comparison with the wavelength of the incident radiation, the diffuse model cannot adequately describe the optical properties of surfaces such as polished and oxidized metals, and other materials with signiﬁcant surface (i.e., not volumetric) reﬂection. For this case, a model that incorporates diffuse thermal emission and specular reﬂection is more suitable. The calculation of the effective emissivity for these cavities is greatly simpliﬁed. The problem is reduced to ray tracing and multiplication of the radiance by r, after each reﬂection. According to Kirchhoff’s law, the local directional effective emissivity is given by e ðx; y; fÞ ¼ 1 re ðaperture; y; f; 2pÞ

(71)

where re ðaperture; y; f; 2pÞ ¼ rnðy;fÞ is the effective reﬂectivity of the cavity and n(y, f) is the number of reﬂections undergone by a ray falling from the direction (y, f) before it is reﬂected out of the aperture in any direction. The most common geometry for a cavity with specularly reﬂecting walls is a cone. For a ray entering a cone at an oblique angle c and striking the surface at an angle g to the generatrix of the cone, the number of reﬂections occurring before the ray ﬁnally leaves the cone is [53] n ¼ Ent

p 2g þ 2a cosc 2a cosc

(72)

where a is the cone apex half-angle and Ent(x) represents the greatest integer which is less than or equal to x. Equation (72) shows that it is preferable to view a specular cone at an oblique angle rather than axially. The practical limit to n is set by the width of the beam, since c varies over the width due to the curvature of a conical surface. The presence of even a small fraction of diffuse reﬂection can signiﬁcantly decrease the effective emissivity of a specularly reﬂecting conical cavity. The analytical approach becomes increasingly complicated, if one or more of the surfaces inside a cavity reﬂect thermal radiation diffusely. In Ref. [54], a cylindro-inner-conical cavity with a specularly reﬂecting ﬂat lid

Calculation of the Radiation Characteristics of Blackbody Radiation Sources

205

is considered. The authors used Stokes theorem to reduce the double integrals to a single contour integral around a ﬁnite area on the specular lid and the exchange factors developed in Ref. [55]. The general form of the exchange factor is dExi ;xj ¼

1 X

ðrm Þn df n

(73)

n¼0

where xi is the position of an emitting element dAi, xj the position of a receiving element dAj, rm the specular reﬂectance of the lid, and dfn the fraction of diffusely distributed radiative ﬂux from dAi that arrives at dAj with n intervening ideal (i.e., with rm ¼ 1) specular reﬂections. The authors of Ref. [54] found that the distributions of the effective emissivity over the internal surface of a cavity with a specular lid are very close to those in the similar purely diffuse cavity (see Ref. [34]).

3.6. Specular2diffuse cavities According to the specular2diffuse (SD) model of reﬂection (e.g., see Chapter 5 of Ref. [1]), the directional-hemispherical reﬂectance r is represented by the sum of diffuse and specular components rd and rs, respectively. It is supposed that r, rd, and rs do not depend on incident angle. The differential exchange factor dEdA,dAu, which replaces the view factor in diffuse2specular enclosures, is deﬁned as the sum of fractions of diffuse radiation leaving dA that arrives at dAu both directly and after all possible specular reﬂections. Analytical derivation of the exchange factors can be a very laborious process. Hence, the integral equations method is rarely applied to specular2diffuse cavities. However, Kowsary and Mahan in Ref. [56] used the integral equation method for calculating the effective emissivity of a specular2diffuse spherical cavity with an isothermal surface. They calculated the hemispherical effective emissivity of the specular2diffuse spherical cavity against the opening half-angle for various values of the ratio rs/r.

3.7. Method of successive reﬂections De Vos [57] considered a cavity (see Figure 7) with opaque walls and one opening. One can express the spectral radiation ﬂux emitted by a surface element dw in the direction Wow within the solid angle dOow and for the temperature T, as ow I B dw cosWow dOow

(74)

where IB is the radiant intensity of a blackbody at a temperature T, ow the hemispherical emissivity of dw at temperature T in the direction Wow , and dOow the solid angle subtended by do when viewed from dw.

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Alexander V. Prokhorov et al.

dΩwn

dn

Inw dw

θwn θwo do

dΩwo Iwo

Figure 7 Blackbody cavity with one opening, dw and dn are the elements of the cavity inner surface, do is an element of the cavity opening (from Ref. [57]).

The spectral radiation ﬂux coming from a given surface element dn and reﬂected by dw into the solid angle dOow is given by o I wn dOnw dw cosWnw r no w dOw

(75)

where I wn is the radiant intensity of dn in the direction of dw at a temperature T, dOnw the solid angle subtended by dn when viewed from dw, dOow the solid angle subtended by do when viewed from dw, and r no w the partial reﬂectivity of dw at temperature T for the direction from dn to do. The partial reﬂectivity, r ab x (the quantity used by De Vos), of the surface element dx is the fraction of the radiation reﬂected in the direction b per unit solid angle when the radiation is incident from the direction a. From this deﬁnition, it follows that the directional-hemispherical reﬂectivity for radiation falling from the direction a is a

Z

r an dOn

r ¼

(76)

2p

where dOn is the solid angle in a direction n and ran is the partial reﬂectivity in this direction. The integration is performed over the hemispherical solid angle. The Helmholtz’ theorem of reciprocity can be written for the partial reﬂectivities: r ab cosWa ¼ r ba cosWb

(77)

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Calculation of the Radiation Characteristics of Blackbody Radiation Sources

The relationship linking the De Vos’s partial reﬂectivity with the BRDF (see Ref. [10]) is r ab ¼ f r ða; bÞ cosWb

(78)

The ﬁrst-order approximation for I wn is I wn

¼ IB 1

X

r wh n

! dOhn

kn wn

(79)

h

where wn is the emissivity of dn at temperature Tn, in the direction of dw, kn ¼

I B ðT Þ I B ðT n Þ I B ðT Þ

(80)

The second-order approximation for I ow is given by I ow ¼ I B 1

X h

h r oh w dOw

XZ

h on n r wh n dOn r w dOw

Z

n kn wn r on w dOw

! (81)

h

The integration is to be extended over the entire surface (except the opening). In the third-order approximation, terms containing the product of three partial reﬂectivities and three solid angles must be added to Equation (81). In principle, the method described enables the calculation of the directional effective emissivity for cavities whose walls have arbitrary BRDFs (or partial reﬂectivities). Usually, however, less complicated and more idealized models of reﬂection are employed (e.g., purely diffuse, purely specular, or mixed, SD models). Campanaro and Ricolﬁ in Ref. [58] used De Vos’s method for calculating the effective emissivity of an isothermal specular2diffuse sphere. Using a third-order approximation, they showed that the effective emissivity decreases as the specular component is increased. The same authors in Ref. [59] applied De Vos’s technique to the computation of the total effective emissivity of a nonisothermal spherical cavity with a diffusely emitting and reﬂecting internal surface. An analytical solution was derived for the general temperature distribution case. Assuming a linear temperature gradient along the axis normal to the opening, the authors obtained numerical dependences for various values of the ratio of the opening radius to the sphere radius. Quinn [60] applied De Vos’s method to an isothermal diffuse cylindrical cavity with and without a diaphragm. For the open-ended cylinder, Quinn obtained a second-order approximation for the effective emissivity of the bottom center: e ¼ 1

r r2 I 2 D2

(82)

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Alexander V. Prokhorov et al.

where D is the length of the cylinder of unit radius, and 1 I2 ¼ p

Z 0

D

Dx 2x cosf 1 2cosf df dx (83) 2 2 2 tan x x þ 4cos2f p=2 ½1 þ ðD xÞ

Z

p=2

For the lidded cylinder, Quinn found that e ¼ 1

ra2 ra2 I 02 D2

(84)

where a is the dimensionless radius of the aperture, I 02 ¼ 2

Z

D 0

ð1 þ

xðD xÞ dx þ ðD xÞ2 2

x2 Þ2 ½1

(85)

The integrals I2 and Iu2 were evaluated numerically, for D ranging from 4 to 20 and tabulated. Quinn and Martin [61] investigated the radiation heat transfer from a blackbody cavity radiator to an absolute cavity detector. Both cavities have the shape of a cylindro-inner cone and coated inside with black paint. The paint has a specular component of reﬂection that increases with wavelength and incident angle. A second-order approximation was used. Terms of higher order were taken into account by summation of all reﬂections starting from the third, but were found to be negligible. In Ref. [62], Berry applied a second-order approximation of De Vos’s method to the isothermal diffuse cylinder-inner-cone (cylinder with a reentrant cone). Redgrove and Berry [63] used De Vos’s approach to compute the effective emissivity of the same cavity but with a specular2diffuse reﬂection from the internal surface. They analyzed the dependences of the effective emissivity on the geometrical parameters of a cavity, the emissivity of the wall, and the ratio rs/r. A correction term for nonisothermal conditions to the total effective emissivity was also derived. The difﬁculty of analytical integration, as well as the accumulation of truncation errors due to the numerical calculation of nested integrals, restricted the number of reﬂections that could be considered to two. This limitation can lead to signiﬁcant errors in calculating the effective emissivity for a cavity with even moderately low values of wall emissivity.

4. Monte Carlo Method 4.1. Stochastic approach to radiative heat transfer The Monte Carlo method is a class of numerical techniques based on the implementation of a stochastic model to an object under consideration in

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209

order to evaluate its characteristics on the basis of the law of large numbers. Several excellent monographs [64269] cover the basics of various Monte Carlo methods including time-saving algorithms, methods of convergence acceleration, etc. Because of their reliance on repeated computation and random or pseudorandom numbers, Monte Carlo methods are best suited to calculation by a computer and are used when it is impossible or impractical to compute an exact result with a deterministic algorithm. Physicists working on the Manhattan Project at Los Alamos National Laboratory originally coined the term Monte Carlo in the 1940s [70]. It is named after the famous gambling casino in Monaco because the method is analogous to the random and repetitive processes common in gambling. Initially, the Monte Carlo method was applied to studying neutron interactions with matter. Monte Carlo’s probabilistic treatment of the radiation2matter interaction effects is well suited to solving computational problems concerning atmosphere optics [71], radiative heat exchange [72,73], computer graphics [74277], remote sensing, infrared scene and target simulation, etc. When we consider optical radiation transfer problems within the framework of geometric optics, the application of the Monte Carlo method is a conceptual process performed by a computer in which a particle (sometimes referred to as a bunch, or bundle of particles or even of rays) starts from a point on a surface and moves away from the surface in a random direction. Frequently, the term photons is used for such particles; however, real photons have nothing physically in common with the particles of such a model. The number of particles moving in each of a ﬁnite (but very large) number of directions is deﬁned by a solid angle and each is weighted to match the angular distribution of radiation ﬂux. After a sufﬁciently large number of particles have been traced from the source to the receiver by this process (the limit is determined by computer performance and the investigator’s accuracy requirements), the process is terminated and the fraction of particles reaching a receiver from the source is determined. Since the energy content of each particle is known, the ﬂux falling on the receiving surface can easily be determined. In real situations, an intermediate surface may interact with the radiation resulting in partial reﬂection, absorption, or transmission by that surface. Any reﬂected radiation can propagate according to its own distribution of reﬂected radiance where it can be partially reﬂected again. In general, there are an inﬁnite number of multiple reﬂections between two mutually irradiating surfaces, and the Monte Carlo approach easily accommodates this situation. In a computer stochastic model, the actual distributions of radiation ﬂuxes over spatial and angular domains are replaced with probability density functions (PDF) for the appropriate random variates. Effective computer modeling of these random variates is the most important component of the

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Monte Carlo method. Modeling an arbitrary PDF is usually based on a pseudorandom number generator (PRNG) 2 a computer program that produces a sequence of ﬂoating-point numbers uniformly distributed on the [0, 1) interval (see Ref. [78]). Often the correctness and the precision of the modeling depend on the quality of these random sequences. A PRNG designed for critical modeling programs must undergo a thorough examination of periodicity, multidimensional equidistribution, various statistical tests of distribution uniformity, etc. [79]. Recent investigations show that most widely used PRNG have serious defects [80]. On the basis of their own experience, the authors of Ref. [81] recommend the Mersenne-Twister PRNG [82], which provides an extremely large period of 21993721 and up to 623-dimensional equidistribution for 32-bit accuracy. The Mersenne-Twister PRNG is freely distributed software. Recent versions of the code for its algorithm are available on the web (see Ref. [83]). The computational problem that is usually solved by the Monte Carlo method concerns the evaluation of the mean value of a random variate. The random uncertainty of such an estimate decreases as N 1/2 (N is the number of random trials) and does not depend on the dimension of the problem solved. For radiative transfer inside cavities, the dimension of the estimated random variate is the maximum number of consecutive reﬂections that a ray undergoes until it escapes the cavity. It is sufﬁcient to compare this with the method of successive reﬂections, described earlier where the precision of the results decreases due to accumulations of the truncation errors at the computation of nested integrals, to understand the advantages of the Monte Carlo method.

4.2. Ray tracing Instead of a particle or a bunch, or bundle of particles, we shall refer hereinafter to the rays traced inside a cavity in keeping with the terminology used in ray or geometric optics. Each of the n surfaces that form a cavity can be described in terms of the Cartesian coordinate system as Fi ðx; y; zÞ ¼ 0;

i ¼ 1; 2; . . . ; n

(86)

The parametric equations of a ray in Cartesian coordinates are: 8 0 > < x ¼ x þ ox t y ¼ y0 þ o y t > : z ¼ z0 þ o t z

(87)

where (xu, yu, zu) and (x, y, z) are the coordinates of the start and end points of a ray, respectively, (ox, oy, oz) are the coordinates of a unit vector x of

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a ray direction, and t is a time-like parameter. First Equations (86) and (87) are solved together to obtain the parameters ti for each i. Once a minimal positive t is found, then we can determine the coordinates (x, y, z) using Equation (87). There are several books devoted to ray tracing in computer graphics [75,77,84]. They contain efﬁcient algorithms for determining the intersection points of a ray with the surfaces of common cavity structures.

4.3. Stochastic models for optical properties The optical properties of a surface can be exhaustively described by the spectral BRDF [10]: f ðl; yi ; fi ; yr ; fr Þ ¼

dL l;r ðl; yr ; fr Þ dEl;i ðl; yi ; fi Þ

(88)

where Ll,r is the spectral radiance of reﬂected radiation, El,i the spectral irradiance (in W/m3) from incident radiation, and (yi, fi) and (yr, fr) are the directions of incidence and observation, respectively, deﬁned by their spherical coordinates. In the case of unpolarized optical radiation, the BRDF is a function of ﬁve arguments. Usually, experimental data exist only for several wavelengths, and angular scanning is performed in the plane of incidence. This is why it is often impossible to directly use the measured BRDF in the Monte Carlo algorithms. Therefore, it is necessary to have a reliable model of the BRDF that satisﬁes the experimental data and predicts physically plausible values for the BRDF for the complete range of arguments where there is a lack of experimental data. The model of BRDF must obey the energy conservation law: Z rðl; yi ; fi ; 2pÞ ¼

2p fr ¼0

Z

p=2

yr ¼0

f ðl; yi ; fi ; yr ; fr Þ sinyr cosyr dyr dfr 1

(89)

where r(l, yi, fi, 2p) is the spectral directional-hemispherical reﬂectance. The spectral directional emissivity of an opaque body is equal to ðl; yi ; fi Þ ¼ 1 rðl; yi ; fi ; 2pÞ

(90)

The model of BRDF should also be consistent with the reciprocity principle: f ðl; yi ; fi ; yr ; fr Þ ¼ f ðl; yr ; fr ; yi ; fi Þ

(91)

If the angular distribution of the emitted or reﬂected radiant intensity is known, it is possible to construct a simple stochastic model by generating random rays, which are uniformly distributed in a hemisphere, and assigning to each ray a statistical weight proportional to the radiant intensity

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in that direction. However, such a model is computationally inefﬁcient, especially for BRDF with signiﬁcant nonuniformities. For more effective modeling, the stochastic model of BRDF, which is incorporated into a Monte Carlo ray-tracing algorithm, should provide importance sampling [85]. In other words, random directions of reﬂection (yr, fr) should be generated from a general population that has a PDF equal to f(l, yi, fi, yr, fr) cos(yr).

4.4. Monte Carlo modeling of specular2diffuse blackbodies 4.4.1. Early Monte Carlo studies Polgar and Howell [86] applied the Monte Carlo method to the stochastic modeling of the reﬂective properties of a diffuse conical cavity irradiated by collimated radiation. Corlett [87] described in detail a modeling algorithm for the thermal radiation exchange among opaque specular2diffuse surfaces separated by vacuum. The algorithm developed was applied to the calculation of radiative heat transfer from one surface to another in a closed specular2diffuse cavity. Random numbers u uniformly distributed on the segment (0, 1] were employed in the following cases: to choose between reﬂection and absorption: absorption, if uaoa, where a is a surface absorptance, and reﬂection otherwise; (ii) to choose between specular and diffuse reﬂection: specular, if usors/r, where rs is the specular component of the directional-hemispherical reﬂectance r, and diffuse otherwise; (iii) to generate a direction (y, f) of diffuse reﬂection: in the local spherical coordinate system, (i)

cosy ¼

pﬃﬃﬃﬃﬃ uy ;

f ¼ 2puf

Toor and Viskanta [88] investigated the precision of their Monte Carlo calculations against known analytical solutions for specular2diffuse surfaces. They also considered directional-dependent specular reﬂection, which is predicted from Fresnel’s equations in terms of the complex index of refraction n~ ðlÞ ¼ nðlÞ ikðlÞ. In addition, they calculated the BRDF f ðl; yi ; fi ; yr ; Þ, which was approximated by f ðl; yi ; fi ; yr ; Þ rðl; yi Þf 1 ðl; yi ; fi ; yr ; Þ

(92)

where f 1 ðl; yi ; fi ; yr Þ is Beckmann’s reﬂection distribution function for a perfect conductor (Beckmann’s solution [89] is based on Kirchhoff’s approximation of diffraction integrals; it satisﬁes reciprocity but violates energy conservation due to neglecting multiple reﬂections). The function fN also depends upon parameters of surface roughness. To implement the importance sampling (i.e., to generate the random directions of reﬂection according to the BRDF), Toor and Viskanta

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employed the inverse transform of cumulative distribution function. For polar and azimuthal angles, cumulative distribution function can be written as R yr R 2p

0 0 f ðl; yi ; fi ; yr ; fr Þ cosyr sinyr dfr dyr Ryr ðl; yi ; fi Þ ¼ R p=2 R 2p 0 0 f ðl; yi ; fi ; yr ; fr Þ cosyr sinyr dfr dyr

(93)

and R p=2 R fr Rfr ðl; yi ; fi Þ ¼ R0p=2 R02p 0

0

f ðl; yi ; fi ; yr ; fr Þ cosyr sinyr dfr dyr

(94)

f ðl; yi ; fi ; yr ; fr Þ cosyr sinyr dfr dyr

The values of y versus Ry and f versus Rf were calculated and stored in computer memory for each angle of incidence yi with the increments of 51. From this table the direction of the reﬂection (yr, fr), corresponding to a given angle of incidence yi, was obtained by interpolation. Although Refs. [86288] are not directly devoted to the radiative properties of blackbody radiators, they showed the applicability of the Monte Carlo method to the calculation of the effective emissivity of blackbody radiators and demonstrated the possibilities of using the stochastic approach to the modeling of radiation heat transfer. 4.4.2. Hemispherical effective emissivity of diffuse cavities Sparrow and coworkers in a series of publications [90292], proposed a method of variance reduction in the Monte Carlo modeling of the radiative properties of a diffuse cavity which they called energy partitioning. They applied it to isothermal bafﬂed conical and cylindrical cavities. The essence of the method is the following. A random point on a radiating surface of a cavity emits a ray. Let F represent the fraction of the radiation emitted at that point, which passes directly out of the aperture, that is, the diffuse view factor of the aperture from that point. The energy content E of the ith ray is partitioned into two portions. One portion, FiE, passes directly out of the aperture and is tallied. The other portion, (1Fi)E, remains within the cavity and is carried by the ray along a straight path. Before the trajectory of the ray is traced, a random number Ra, having a value between 0 and 1, is selected to determine whether or not the ray will be absorbed at its intersection with the cavity wall or the bafﬂe. Absorption takes place if Rara ¼ e. If the ray is absorbed, its trajectory ends, and the next ray is traced. On the other hand, if the ray is not absorbed, its point of intersection is determined by constructing a straight-line trajectory based on the random angles y and f of a local spherical coordinate system. At the point of intersection, a second partitioning may take place. Let Fi1 denote the view factor of the aperture as seen from the point of impingement

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(note that Fi1 ¼ 0 if the point of intersection is on the bafﬂe). Then, the two partitioned portions are Fi1(1Fi)Eand (1Fi1)(1Fi)E. The ﬁrst of these passes out of the aperture and is tallied. Whereas the second remains in the cavity and continues its life cycle, experiencing successive partitionings until absorption occurs. Any ray released from a location on the cavity wall will contribute at least one tally (and, perhaps, many tallies) to the determination of the cavity radiant exitance Eout. The total hemispherical effective emissivity is equal to " # N N X Eout ðAw =Aa Þ X e ¼ ¼ Fi þ Gi N sT 4 Aa i¼1 i¼1

(95)

where Gi ¼ 0, if no reﬂections occur; Gi ¼ (1Fi)Fi1 for one reﬂection; Gi ¼ (1Fi)Fi1 + (1Fi)(1Fi1)Fi2 for two reﬂections; and so forth. To avoid a direct hit of the second portions into the aperture, random directions of diffuse emission and reﬂection were not generated into the entire hemisphere, but into the hemisphere not including the conical solid angle subtended by the aperture. We omit the appropriate trigonometric transformations but note that there is a more simple way: to ignore the traces of the second portion if it hits the aperture, and to generate a new direction. The cases of emitting and nonemitting bafﬂes were considered. Hemispherical effective emissivities of conical and cylindrical cavities were computed for various values of wall emissivity and geometrical parameters. The convergence of the computational process was studied. Although the energy partitioning method requires time-consuming calculations, the stochastic process converges faster than a conventional Monte Carlo technique. The main drawback of the algorithm described above is its poor applicability to the calculation of the directional effective emissivity, because if a detector of ﬁnite size is placed in front of the cavity aperture, the probability of a ray hitting the detector, as well as the view factor for detector, tends to zero as the distance between the aperture and the detector tends to inﬁnity. 4.4.3. Effective emissivity of isothermal specular2diffuse cavities According to the generalized Kirchhoff’s law [11,12] applied to the isothermal opaque cavity: e ¼ ae ¼ 1 r e

(96)

where ee, ae, and re are the effective emissivity, the effective absorptivity, and the effective reﬂectivity, respectively. Consequently, we can consider the reﬂection of radiation by the cavity, instead of the emission, and replace the conditions of cavity observation with those of cavity irradiation. In this case, rays start at the point of observation,

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215

proceed in the direction of observation, pass through the cavity aperture, and repeatedly reﬂect off the cavity walls until either they escape the cavity through the aperture or the energy carried by the ray becomes negligible. Such a method is referred to as backward (or inverse) ray tracing. The reversibility of ray trajectories is possible because of the reciprocity principle (see, for instance, Refs. [93,94]) and allows us to evaluate the directional effective emissivity along an inﬁnitely thin ray. Many Monte Carlo algorithms for radiation transfer are based on backward ray tracing. One such algorithm was developed by Ono in Refs. [95,96] for calculating the directional effective emissivities of isothermal specular2diffuse cavities. He used a serial expression for the directional effective reﬂectivity of a cylindro-conical cavity: rD e ¼

Z

A

dO10 r D;1 0 þ

Z

dO10 r D;1 0

W

Z

A

dO21 r 0;2 1 þ

(97)

j

where r i is the partial reﬂectivity (see Section 3.7 and Ref. [57]), and the ith term fi is given by Z

fi ¼

W

dO10 r D;1 0

Z

W

dO21 r 0;2 1

Z

Z

::: W

A

dOii1 r i2;i i1

(98)

Equation (98) expresses fi as the fraction of radiation that enters the cavity at the location x0 from a direction D and undergoes i reﬂections in the cavity according to the partial reﬂectivity before passing out of the aperture. Thus, the effective reﬂectivity rD e is given by rD e ¼

1 X

fi

(99)

i¼1

For the uniform SD model, reﬂectance does not depend on the j directions of incidence and viewing, so r i ¼ r. If we set r ¼ 1 (no absorption) in the nested integrals in Equation (97), we can calculate rD e for an arbitrary r by rD e ¼

1 X

F ir i

(100)

i¼1

where Fi is the value fi in Equation (98) for r ¼ 1. For a diffuse surface, F1 is the view factor from the location x0 to the aperture and F2 is equivalent to I2 of Quinn’s formula in Ref. [60], where x0 is the center of the plane bottom of a diffuse cylindrical cavity. The standard deviation of the effective reﬂectivity calculated according to Ono’s method is proportional to N1/2 such that sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 1X Dr F i r 2i N i¼1

(101)

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Ono in Refs. [95,96] investigated the normal effective emissivity of isothermal cylindrical and cylindro-conical cavities with specular2diffuse walls using ray tracing and computing the fractions Fi. In the Figure 8, the normal effective emissivity of the cylindro-conical cavity with the depthto-diameter ratio L/d ¼ 3 and rs/r ¼ 0.3 are depicted as functions of the apex angle y for various values of wall reﬂectance r. Figure 8 shows that the effective emissivity of a cylindro-conical cavity with specular2diffuse walls depends strongly on the conical bottom apex angle. In fact, the normal effective emissivity of a specular2diffuse cylindro-conical cavity is determined by the number of specular reﬂections of a ray prior to escape from a cavity. The dependence of the effective emissivity on y becomes smoother as the specular fraction of the reﬂectance decreases. The same method was applied by Ono et al. [97] to a cylindrical cavity with a lateral hole. In overviews [6] and [98], Ono described his method in detail. Prokhorov et al. [99] described a different algorithm for the numerical modeling of specular2diffuse cavities. They also used the SD model and introduced the diffusity D ¼ rd/r, which does not depend on 1.00 r = 0.1 0.2 0.99

ε

0.3

0.96

0.4

0.97

0.96

30

60

90

120

150

180

θ°

Figure 8 Normal effective emissivities as a function of the apex angle h of a cylindro-conical cavity with L/d ¼ 3 and qs/q ¼ 0.3, for various surface reﬂectivities r. The case of a cavity with a purely diffuse surface is shown by thin lines (from Ref. [95]).

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incidence angle. The method of statistical weights was used to accelerate the convergence of the computational process. The method of statistical weights consists of the following. Before the ﬁrst interaction with the wall, a statistical weight w equal to unity is assigned to the ray. After each reﬂection, the statistical weight is multiplied by the wall reﬂectance r to obtain a new weight. The effective emissivity is evaluated as the ratio of the sum of the statistical weights of rays escaping the cavity to those launched into the cavity. Ray tracing of a single ray terminates when its statistical weight becomes less than some small, prescribed value, or when the ray leaves the cavity through the aperture after a specular reﬂection. If the ray hits the aperture after a diffuse reﬂection, the last direction of reﬂection is ignored, and a new direction of diffuse reﬂection is generated. There exists a strict proof [100] of the fact that the variance of the result obtained by the method of statistical weights is always less than that obtained by the conventional method, which uses the termination of ray trajectories after the absorption event. In the method of statistical weights, each ray, having a statistical weight greater than a predeﬁned threshold value, makes a contribution to the result. The algorithm described in Ref. [99] also includes an analytical calculation of the view factors to the aperture from every point of a diffuse reﬂection (an analog of the energy partitioning method in Refs. [90292]). To save computing time (e.g., when we need to compute the spectral effective emissivity for a large number of wavelengths and the corresponding spectral emissivities of the cavity walls), the method of dependent trials is used. According to it, a set of emissivity values is assigned to the cavity walls, and a corresponding set of statistical weights is assigned to each ray. By transforming the statistical weights at each point of reﬂection, we use a single trajectory to process a variety of spectral data. This algorithm was applied to the calculation of the effective emissivities of an isothermal specular2diffuse cavity with a conical bottom, a cylindrical middle part, and a tapered conical diaphragm. Steinfeld [101] used the Monte Carlo method to examine the absorption of a spherical cavity with a specular2diffuse internal surface. If one considers a spherical cavity as a radiation source, then the effective absorptance calculated in this article corresponds to the hemispherical effective emissivity of the source. Chu et al. [102] applied the Monte Carlo technique to a cylindrical cavity with an inner-conical bottom and reported good agreement with the results deduced in Ref. [34]. Prokhorov and Hanssen applied backward ray tracing and the method of statistical weights to a cylindrical cavity with a ﬂat diaphragm and a ﬂat inclined bottom [103] (see Figure 9). It was assumed that the cavity internal walls are uniform and gray. Various conditions of observation were studied. The following estimator

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y

y Rc β

Ra

z

x

x z H

Figure 9

Cylindrical cavity with an inclined bottom (from Ref. [103]).

for the local directional effective emissivity was obtained: e ðn; xÞ ¼

N mX i ðn;xÞ X rk1 N i¼1 k¼1

(102)

where N is the number of rays traced and mi the number of reﬂections of the ith ray. The case when the aperture is observed along an inﬁnitely thin ray parallel to the cavity axis corresponds to the local normal effective emissivity. The distribution of the local normal effective emissivity ee,n(x, y) across the aperture is not necessarily uniform. For a cavity with specular walls, the local normal effective emissivity is e;n ðx; yÞ ¼ 1 ð1 Þmðx;yÞ

(103)

where m(x, y) is the number of successive reﬂections (prior to escaping the cavity) of a ray that enters into the cavity parallel to the cavity axis and intersects the aperture plane at a point with coordinates (x, y). The YZ plane is the only plane of symmetry for a cavity with an inclined bottom. When the bottom forms an angle of 301 with the cylinder at y ¼ Rc this angle is equal to 1501 at y ¼ Rc. Thus for a purely specular cavity, the rays entering into the cavity aperture through points with coordinates (x, y) and (x, y) after their ﬁrst reﬂections from the bottom, fall on the cylindrical wall at different angles. Their further trajectories will also be different, so that the total number of reﬂections varies. The points on the aperture that correspond to the same number of reﬂections form continuous zones of unusual shape. The zones may have irregular structure due to the three-dimensional and nonaxisymmetric nature of the cavity. The shape of these zones can change when H or b is varied. In Figure 10, a map of the zones of the aperture, which correspond to a certain number of reﬂections for backward traced rays, is depicted for a specular cavity without a diaphragm, Rc ¼ Ra ¼ 1, H ¼ 8, and b ¼ 301.

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1

β = 30° 0.8 0.6 0.4

y

0.2 5

0

6

-0.2

9 8

-0.4 7

-0.6 -0.8 -1 0

0.2

0.4

0.6

0.8

1

x

Figure 10 Distribution of the number of reﬂections over the aperture of a specular cavity with Rc ¼ Ra ¼ 1, K ¼ 8, and b ¼ 301 (from Ref. [103]).

Numbers on the map indicate the number of reﬂections for every zone. Because the distribution is symmetrical, only the right half is shown. Three-dimensional views of the distributions of the local normal effective emissivity ee,n(x, y), for the cavities with Ra ¼ Rc ¼ 1, H ¼ 8, b ¼ 301, e ¼ 0.7, and D ¼ 0, 0.1, 0.5, and 1 are depicted in Figure 11. For a cavity with purely specular walls, the distribution looks like a set of plateaus, or ﬂat terraces having different heights and often of complicated shape. The presence of a relatively small diffuse component leads to a signiﬁcant decrease in the distribution’s step heights. For instance, at D ¼ 0.5, the relief created by the distribution of ee,n(x, y) becomes almost indistinct. At D ¼ 1 (purely diffuse walls), the distribution assumes a very smooth convex form with the minimum at (x ¼ 0, y ¼ 1). To compute the average normal effective emissivity ee,n, the same algorithm is used as for the local normal effective emissivity ee,n(x, y), except that the rays hit the aperture at points that are uniformly distributed over the circular aperture area Sa.

220

0.999 0.999

0.998

0.998

0.997

0.997 1

0.996 1 0.8

0.8 0.6

0.6

0.4

0.4 0.2 y

0 -0.2

0.8 -0.4

0.4

-0.6

D=0

-0.8

0.2

y

0.6 x

0 -0.2

0.8 -0.4

0.4

-0.6

D = 0.1

-0.8

-1 0

0.996 0.994 0.992 0.990 1

0.2

1

0.2

1

0.6 x

-1 0

0.995 0.990 0.985 1

0.8 0.4 y D = 0.5

0.8 0.6

0.2 0 -0.2

0.8 0.6 0.4 x 0.2

-0.4 -0.6 -0.8

-1 0

1

0.4 0.2 y D=1

0 -0.2

0.8 -0.4

0.4

-0.6 -0.8

0.2

0.6 x

-1 0

Figure 11 Distribution of the local normal effective emissivity over the cavity aperture for Ra ¼ Rc ¼ 1, H ¼ 8, b ¼ 301, e ¼ 0.7, and three values of diffusity, D. Because of symmetry, only the right halves of the distributions are shown (from Ref. [103]).

1

Alexander V. Prokhorov et al.

0.6

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221

Integrated Effective Emissivity, εe

1.000

0.995

0.990

0.985

D=0 D = 0.25 D = 0.5 D = 0.75 D=1

0.980

0.975 0

2

4

6

8 Hd

10

12

14

Figure 12 The integrated effective emissivity of a cavity with a diaphragm as a function of the distance Hd between the cavity aperture and the detector, for several values of diffusity, D; b ¼ 301, Rc ¼ 1, Rd ¼ Ra ¼ 0.5, H ¼ 4, and e ¼ 0.7 (from Ref. [103]).

The algorithm for calculation of the integrated effective emissivity that allows taking into account the vignetting effect is also described. The integrated effective emissivity of a cavity having Rc ¼ 1, Ra ¼ 0.5, H ¼ 4, b ¼ 301, e ¼ 0.7, as a function of the distance Hd between the cavity aperture and a coaxial circular detector of radius Rd are plotted in Figure 12 for several values of diffusity, D. Prokhorov et al. [104,105] applied the Monte Carlo method to ﬂat radiators with isothermal concentric grooves of triangular and trapezoidal proﬁles (Figure 13). They compared the results obtained for an SD model of reﬂection using the STEEP3 [106] code, based on the Monte Carlo algorithm [107], with that obtained for Fresnelian2Lambertian (FL) model, whose BRDF is expressed by the equation f FL ðl; yi ; fi ; yr ; fr Þ ¼ d

rðlÞ dðyr yi Þdðfr fi pÞ þ ð1 dÞFðnl ; kl ; yi Þ p sinyr cosyi (104)

where the ﬁrst term is the diffuse (Lambertian) component, d is the Lambertian BRDF weighting coefﬁcient, d the Dirac’s delta-function, yi the incidence angle, nl the ratio of the spectral refractive indices above and

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Alexander V. Prokhorov et al.

β

z p

ft fb r

Figure 13 Geometrical model of the radiator: p is the pitch, ft and fb are the widths of the ﬂat areas at the top and bottom, and b is the included angle (from Ref. [104]).

below the surface, kl the spectral extinction coefﬁcient, and F is the Fresnelian reﬂectance for unpolarized radiation. In general, the following conclusion can be made. Although the FL model is more realistic, it requires sufﬁcient experimental data to ﬁt model parameters. The simpler SD model can be applied when the necessary experimental data are absent. For V-grooved structures, the SD model produces slightly lower values of effective emissivity than the FL model for comparable parameters. Zhang and Dai [108,109] also used the Monte Carlo ray-tracing technique to calculate the effective emissivity of an infrared blackbody source with a V-grooved bottom, which was assumed to be diffuse and isothermal. Ishii et al. [110] applied a commercially available ray-tracing program, OptiCAD (supplied by OptiCAD Co.) [111] to isothermal cylindro-conical cavities with a grooved cylindrical surface. The SD model of reﬂection was adopted, following a similar method of calculation as that of Ono [95]. The Monte Carlo method is successfully used for numerical modeling of blackbodies with a reﬂecting cavity. Quinn and Martin [112] ﬁrst proposed this blackbody design and Usadi [113] systematically described it. Prokhorov and Martin [114] modeled the radiative heat transfer from such a blackbody of a very sophisticated shape into the cryogenic radiometer. Bidirectional ray tracing was performed with rays propagating ﬁrst into the radiator and then into the radiometer cavity. Many authors [1152119] have used the Monte Carlo-based software to predict the radiation characteristics of such blackbodies in the design stage. 4.4.4. Effective emissivity of nonisothermal specular2diffuse cavities The Monte Carlo method has also been applied to the calculation of the radiation characteristics for specular2diffuse cavities that are not isothermal. Sapritsky and Prokhorov [120] proposed a Monte Carlo algorithm to calculate the spectral and total effective emissivity of nonisothermal cavities based on the reciprocity principle and backward ray tracing. The spectral,

Calculation of the Radiation Characteristics of Blackbody Radiation Sources

223

local effective emissivity can be written in the form: e ðl; n; x; T x ; T 0 Þ ¼ e ðl; n; xÞ þ De ðl; n; x; T x ; T 0 Þ

(105)

where ee(l, n, x) is the local spectral effective emissivity of an isothermal cavity, n and x are the position and direction vectors, respectively, and Dee(l, n, x, Tx, T0) is the correction term for nonisothermal conditions. According to the generalized Kirchhoff’s law for isothermal cavities (see Refs. [11,12]): e ðl; n; xÞ ¼ 1 re ðl; n; xÞ

(106)

where re(l, n, x) is the directional-hemispherical spectral effective reﬂectivity for the direction of radiation incident into the cavity, which coincides with the direction of the observations x. The authors considered the interval l1rlrl2 where the diffusity D ¼ rd(l)/r(l) is constant. Initially, each incident ray is assigned a statistical weight of unity, which is multiplied by r(l) following every reﬂection. Specular or diffuse reﬂection is selected by means of a pseudorandom number uD. If the next pseudorandom number uDoD, the reﬂection is considered to be diffuse. Otherwise, it is specular. After each diffuse reﬂection, the statistical weight is reduced to take into account the radiation loss through the cavity aperture. This is done by applying the diffuse view factor Fa(n) between an element of the wall area at the point of reﬂection n and the cavity aperture. If, after diffuse reﬂection, a particle escapes from the cavity through the aperture, the last direction of reﬂection is ignored and the computation begins for another ray. For a ﬁxed number of ray trajectories N, it is simple to show that j1 mi N X Y 1X j e ðl; n; xÞ ¼ 1 r ðlÞF a ðnj Þ ½1 Fðnk Þ N i¼1 j¼1 k¼1

(107)

where mi is the number of reﬂections in the ith trajectory. For a specular reﬂection, Fa(n) ¼ 1 if a ray escapes the cavity, otherwise Fa(n) ¼ 0. The maximum number of reﬂections in the trajectory can be estimated from the relation

lnðdmin Þ m ¼ Ent lnð1 min Þ

(108)

where d is the permissible component of the calculation uncertainty due to the interruption of a ray trajectory, emin ¼ Min e(l) at l1rlrl2. The spectral directional effective emissivity of a nonisothermal cavity can be calculated using the reciprocity theorem, which permits substitution of the direction of incident radiation into the cavity for the direction of the observation. Each time a ray is reﬂected from the wall, the spectral radiance of a blackbody is calculated at the cavity wall temperature for that reﬂection

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point in accordance with Planck’s law. The weighted summation of these values along the ray trajectory makes it possible to evaluate the spectral radiance at the point n in the direction x. Correction for the nonisothermal case is made using the expression De ðl; n; x; T ref Þ ¼

mi N X X ðlÞ rj1 ðlÞ½L e ðl; T j Þ L e ðl; T ref Þ (109) NL e ðl; T ref Þ i¼1 j¼1

where Tj, j ¼ 1, 2y, mi, are the cavity wall temperatures at the jth reﬂection point. Sapritsky and Prokhorov in Ref. [121] expanded the applicability of the algorithm [120] and computer code to a cavity having a conical bottom, a conical middle part, and a conical diaphragm. In Ref. [121], the problem of choosing a reference temperature was also discussed. The following hypothesis was suggested: ‘‘For each nonisothermal cavity, there is a characteristic reference temperature (called the optimal reference temperature), at which the spectral effective emissivity coincides with that value for the same isothermal cavity.’’ Later, this hypothesis was disproved and in fact we can conclude that such a temperature can be found only as a rootmean-square approximation. Along with the growth of the performance of personal computers, the algorithm that is described by Equations (107) and (109) has undergone changes. Because cavities with screened walls were included, the algorithm for calculation of the aperture’s view factors was eliminated since an increase in the number of rays traced is more effective than the computation of the aperture’s view factors. In Ref. [107], the estimator of el,e(l, Tref) for nonisothermal cavities is l;e ðl; T ref Þ ¼

mj j1 N X Y j ðlÞ expðc 2 =ðlT ref ÞÞ 1 X r ðlÞ expðc 2 =ðlT j ÞÞ 1 k¼1 k N i¼1 j¼1

(110)

Ballico [122] considered the effective emissivity and radiance temperatures of graphite tube blackbody furnace that is observed by a singlewavelength (0.65 mm) pyrometer with a single-lens objective. From a given element of the blackbody surface, only radiation that (i) lies within the acceptance cone formed by the aperture and source and (ii) passes through the detector image will be detected. Ballico computed the vignetting effect by the zonal method, using the Lambertian approximation and a series of simpliﬁcations. If the radiating surface of a cavity is non-Lambertian, the radiance of rays passing through the intersection of the acceptance cone and the detector image varies with the direction. To account for this effect, the Monte Carlo ray-tracing method was applied. The simple SD model assumes that directional-hemispherical reﬂectance does not depend on the incidence angle. This assumption is too crude

Calculation of the Radiation Characteristics of Blackbody Radiation Sources

225

for the problem considered, so Ballico [122] adopted the FL model. The BRDF can be written as (there is an obvious misprint in Ref. [122], in Equation (21) for BRDF; we provide the corrected formula from Ref. [10] in Ballico’s designation): rðyi ; fi ; yr ; fr Þ ¼ ð1 sÞ

rðlÞ dðyr yi Þdðfr fi pÞ þ sFðyi ; yr ; nÞ p sinyr cosyi

(111)

where s is specularity degree and d the Dirac’s delta-function,

2

2 1 ncosyi cosw

1

ncosw cosyi

Fðyi ; yr ; nÞ ¼

þ 2 ncosyi þ cosw 2 ncosw þ cosyi

(112)

sin w ¼ (sin yi)/n and n is the complex refractive index of wall’s material. After ray tracing inside the nonisothermal cavity, the rays are sent through the detection optics, and followed through multiple surface reﬂections. The solid angle-weighted average of the surface radiances is calculated where the rays are eventually absorbed. The effective emissivity is given by the ratio of the detected power to that received by the pyrometer from the same radiator, but having perfectly black walls. The radiance temperature of a high-temperature graphite tube blackbody was computed through its effective emissivity. The vignetting effect for a pyrometer was also estimated. The Monte Carlo model employed in Ref. [122] has one free parameter, the degree of specularity s of the surface. A value of s was ﬁtted using the best agreement between the computed and the measured radiance temperatures as the criterion. Hartmann and coworkers [1232125] described another Monte Carlo ray-tracing algorithm for computing the spectral effective emissivity of diffuse isothermal and nonisothermal cavities. They also used inverse ray tracing and considered ‘‘photons’’, which enter the cavity through the aperture. Once the photon reaches the cavity wall, a random number generator is used to determine the next process of the photon. With a probability of 1 w ðlÞ, where w ðlÞ is the local emissivity of the cavity wall, the photon is reﬂected from the surface in a randomly distributed direction. The photon is traced until it is either absorbed or reemitted. The effective emissivity eiso for an isothermal cavity is equal to the ratio of absorbed photons to the photons sent into the cavity. It depends only on the geometry of the cavity and the local emissivity w ðlÞ of the cavity walls. For the effective emissivity enoniso, of a nonisothermal cavity, every absorbed photon is weighted by the spectral radiance according to Planck’s law at a temperature T at the location where the photon is absorbed. The spectral radiance L(l, T) assigned to a single absorbed photon is connected to the spectral radiance Ls(l, T) of a blackbody at the same temperature by Lðl; T Þ ¼ w ðlÞL s ðl; T Þ

(113)

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For the slightly different temperature TDT in a nonisothermal cavity, Equation (113) can be rewritten using Wien’s approximation for Planck’s law as

c 2 DT L s ðl; T Þ Lðl; T DT Þ ¼ w ðlÞ 1 lT 2

(114)

or, in equivalent form, Lðl; T DT Þ ¼ w ðlÞDT ;w ðl; DT ÞL s ðl; T Þ

(115)

where DT ;w ðl; DT Þ is the change in local emissivity due to a temperature difference DT. Thus, for the entire cavity: L cav;iso ¼

1X FðOn Þw ðlÞL s ðl; T Þ N n

(116)

and L cav;noniso ¼

1X FðOn Þw ðlÞDT ;w ðl; DT n ÞL s ðl; T Þ N n

(117)

The emissivity of a nonisothermal blackbody is noniso

1X c 2 DT n ¼ FðOn Þw ðlÞ 1 N n lT 2

(118)

where T is the reference temperature (usually, the temperature of the cavity bottom center). Murthy et al. [126] applied the Monte Carlo method to computer modeling of heat-ﬂux sensor calibration using a high-temperature blackbody and the cooled sensor placed inside the blackbody to achieve higher heat-ﬂux levels in a nearly hemispherical irradiation environment. The blackbody radiator and an inserted heat-ﬂux sensor differ widely in temperature. In addition, the sensor’s surface partially reﬂects the radiation falling on it. The computations make possible the determination of the optimum distance from the sensor to the cavity bottom in order to attain the maximal value of the effective emissivity and hence the maximal radiative heat ﬂux. Ishii et al. [127] performed ray tracing of blackbody cavities that serve as the standard sources for calibration of infrared ear thermometers. Including the effect of the background radiation, they obtained the following expression for the spectral effective emissivity: " # kn N X 1X k1 kn e ðlÞ ¼ aðlÞr ðlÞRl;b ðl; T ðn; kÞ; T 0 Þ þ r ðlÞRl;b ðl; T s ; T 0 Þ N n¼1 k¼1

(119)

Calculation of the Radiation Characteristics of Blackbody Radiation Sources

227

where a(l) is the directional absorptance and r(l) the directionalhemispherical reﬂectance of the surface given by r(l) ¼ 1a(l) for an opaque body. N is the total number of incident ray bundles, kn the total number of reﬂections of the nth ray bundle inside the cavity, T(n, k) the temperature at the location x(n, k), for which an nth ray bundle reaches the cavity inner surface at the kth impact, Rl,b(l, T(n, k), T0) the blackbody spectral radiance ratio at the wavelength l between the local temperature T(n, k) and the reference temperature T0 and Ts is the temperature of the surroundings. The angular dependence of the effective emissivity of nonisothermal cavities, with the shape of cylindro-cone and a cylinder with an inclined bottom, which are suitable for the calibration of ear thermometers with wide view angles, was investigated.

4.5. Modeling of direction-dependent optical characteristics The SD model of reﬂection is sufﬁciently powerful to calculate the effective emissivity of blackbody cavities in most practical cases. However, for some situations and materials, the SD model cannot ensure the precision necessary to satisfy modern metrological requirements. One of the major advantages of the Monte Carlo method is the possibility to model optical radiation transfer between surfaces with directionally dependent optical characteristics, including random rough surfaces. To model optical radiation transfer between non-Lambertian surfaces using the Monte Carlo method, it is necessary to generate random directions of reﬂection using a distribution of the probability density that is determined by the BRDF of the reﬂecting surface. Since measurements of BRDF are usually carried out for a limited range of angles of incidence and reﬂection, it is necessary to have a model of BRDF that reproduces the measurement results with sufﬁcient accuracy, has physically plausible behavior within the entire domain of its deﬁnition, and has a directionalhemispherical reﬂectance that depends on the incident angle in accordance with experimental data. Zaworski et al. [128] computed the spatial distribution of radiation passed through a rectangular gap with rough walls. The polar and azimuthal spherical coordinates for the reﬂection lobe were considered as Gaussian variates with respect to the specular direction. Standard deviations of Gaussian distributions were ﬁtted to experimental data. However, a signiﬁcant discrepancy between the computed and measured resulting distributions of radiation passed through the gap was obtained. The authors suggested that the discrepancy is due to a lack of measured values of the bidirectional reﬂectance at large incident angles and deﬁciencies of the model adopted. A better approach to modeling a random rough surface must have a physical basis. Micro-facet based models represent a rough surface as a

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collection of planar mirror-like facets randomly tilted with respect to the plane of the horizon. The size of the micro-facets must be small enough as compared to an irradiated area for statistics methods to be applicable. Torrance and Sparrow [129] proposed a one-dimensional micro-facet model of a rough surface consisting of randomly oriented equilateral V-grooves with random Gaussian slopes. Only the ﬁrst reﬂection is considered and the analytical expression is derived for a geometric attenuation factor, allowing for masking and shadowing effects (partial illumination of a facet shadowed by an adjacent one and the partial visibility of a facet occluded by another). Although this model is very simple, it predicts the peak shift in the BRDF at large incidence angles toward larger reﬂection angles. These off-specular peaks are observable experimentally for all rough materials. The Torrance2Sparrow model does have an intrinsic defect: its directional-hemispherical reﬂectance depends on incidence angle even when the facet’s reﬂectance is equal to unity. However, because of its simplicity, the use of the Torrance2Sparrow model continues to spread in the ﬁelds of computer graphics, remote sensing, and radiation heat transfer. Zhou et al. [130] applied a modiﬁed Ward’s model of reﬂection [131] to Monte Carlo modeling of the effective emissivity of a rough silicon wafer. Further modiﬁcations to the model may be necessary to obtain better agreement of the incidence angle dependence of the directionalhemispherical reﬂectance with experimental data. Prokhorov et al. [132] and Hanssen and Prokhorov [133] described a new model of reﬂection from rough surfaces, referred to as the TETRA BRDF model that is also based on micro-facet theory and developed within the framework of geometric optics. It employs the generation of random tetrahedral pits (see Figure 14), whose walls reﬂect each ray according to Fresnel’s law. Several arrangement types for the projection of the lower tetrahedron vertex onto the tangent plane were considered: (R) a random point uniformly distributed inside the base triangle, (G) the base triangle centroid (center of gravity), (V) one of the vertices of the base triangle, and (C) the circumcircle center. The depth h of the tetrahedral pit is considered as a random variate distributed according to the two-parameter Weibull probability density deﬁned as " # b b1 h b pðhÞ ¼ b h exp ; Z Z

h40

(120)

where bW0 is the shape parameter and ZW0 is the scale parameter of the distribution. An incident ray undergoes one or several reﬂections from the tetrahedron’s walls, and then continues to participate in the radiation transfer on the macro-level. If a large number of rays are aimed at the same

229

Calculation of the Radiation Characteristics of Blackbody Radiation Sources

Lower vertex projection

Incident ray

Surface tangent plans Reflected ray x1, y1 l1 x0 , y0

x2, y2

l3

l2 x3, y3 Lower vertex

Figure 14 Tetrahedral pit with a triangular base in the surface tangent plane (from Ref. [132]).

point of a surface along the same direction and the rays scattered by the random tetrahedrons are registered, one can construct a resultant BRDF. This procedural model of reﬂection has as a prototype the twodimensional model proposed by Torrance and Sparrow. The in-plane sections of the TETRA-G, TETRA-V, TETRA-R, and TETRA-C BRDFs generated for b ¼ 2, Z ¼ 0.1, and incident angles of 01, 301, and 601 are shown in Figure 15. Three-dimensional plots of the TETRA-G BRDF in spherical coordinates for b ¼ 2, nl ¼ 2.5, kl ¼ 2.0, and l ¼ 10.6 mm and for three incidence angles are presented in Figure 16, where all BRDF maxima are normalized to unity. The local directional spectral effective emissivity of a cavity having an arbitrary temperature distribution over a radiating surface can be computed by the following equation: e ðl; T ref ; n0 ; x0 Þ ¼

j1 mi N X Y 1 rðl; n0 ; x0 Þ X L l;bb ðl; T ij Þ rðl; nik ; xik Þ (121) NL l;bb ðl; T ref Þ i¼1 j¼1 k¼0

where xk is the direction of the incidence of the ith ray onto the kth point, nk of reﬂection; n0 and x0 are the viewing point and the viewing direction, respectively, r is the directional-hemispherical reﬂectance, and Tij the temperature at the point of the jth reﬂection of the ith trajectory. As an example, the distributions of the local normal effective emissivity across the aperture were computed for a conical cavity with an apex angle b ¼ 301, without a diaphragm and a cylindrical cavity with a diaphragm, and with a radius of the bottom Rc ¼ 1, an aperture radius Ra ¼ 0.5, and a length L ¼ 4. Both cavities are assumed to be isothermal. The results of the calculations are presented in Figures 17 and 18. It was assumed that the

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TETRA-G

TETRA-V

3.0

14.0

2.5

12.0 10.0 BRDF, sr-1

BRDF, sr-1

2.0 1.5 1.0

6.0 4.0

0.5 0.0 -90

8.0

2.0

-60

-30

0

30

60

0.0 -90 -60 -30

90

0

30

60

90

Viewing Angle, degree

Viewing Angle, degree

TETRA-R

TETRA-C

5.0 35 30

4.0 BRDF, sr-1

BRDF, sr-1

25

3.0

2.0

20 15 10

1.0 5

0.0 -90 -60 -30

0 0

30

60

Viewing Angle, degree

90

-90 -60 -30

0

30

60

90

Viewing Angle, degree

Figure 15 In-plane TETRA-G, TETRA-V, TETRA-R, and TETRA-C BRDFs generated for b ¼ 2, g ¼ 0.1, and incident angles of 01, 301, and 601 (from Ref. [132]).

reﬂection from a tetrahedron’s facet obeys Fresnel’s law and that nl ¼ 2.5 and kl ¼ 2.0. For comparison, calculations were also performed for various diffusities D within the framework of the conventional SD model, in which the reﬂectance r was chosen to be numerically equal to the Fresnelian reﬂectance for normal incidence. For each surface, the directional-hemispherical reﬂectance for a set of incident angles is computed prior to ray tracing, then, during ray tracing, interpolation is used.

Calculation of the Radiation Characteristics of Blackbody Radiation Sources

η

θi = 0º

θi = 30º

231

θi = 60º

0.05

0.1

0.15

0.2

Figure 16 3D plots of the TETRA-G BRDF in spherical coordinates for three incident angles; b ¼ 2, nk ¼ 2.5, kk ¼ 2.0, and k ¼ 10.6 mm. All BRDF maxima are normalized to unity (from Ref. [132]).

Figures 17 and 18 show signiﬁcant differences in the distributions obtained for the two models. None of the dependences computed for the TETRA model can be approximated by the dependences computed for the specular2diffuse one. In addition, the calculations performed for a cylindrical cavity (Figure 18) using the TETRA model predict a signiﬁcantly greater nonuniformity of the distribution of normal effective emissivity over the cavity aperture than that for the SD model.

5. Numerical Comparison of Results Obtained by Various Methods Several computational techniques of the effective emissivity are in current use. McEvoy et al. [134] compared calculations of the effective emissivity performed with the help of several programs. The ﬁrst was a

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1

Local Normal Effective Emissivity

0.95 η= 0 η = 0.05 η = 0.1 η = 0.15 η = 0.2 D=0 D = 0.2 D = 0.4 D = 0.6 D = 0.8 D=1

0.9

0.85

0.8

0.75

0.7 0

0.2

0.4

0.6

0.8

1

R

Figure 17 Distributions of the local normal effective emissivity across the aperture of an isothermal conical cavity with an apex angle of 301 computed for the TETRA and specular--diffuse models of reﬂection plane (from Ref. [132]).

program written at the National Physical Laboratory (UK) [63] to calculate the emissivity of an isothermal cylindrical cavity with a reentrant cone on the back wall by de Vos’s method [57]. A correction term for nonisothermal conditions was also calculated. The emissivity was also calculated using a different program, the commercially available, modeling program, STEEP3 [106]. For a graphite wall with an emissivity e ¼ 0.92, both programs gave an estimate for the total emissivity of the cavity, of 0.99998. This corresponds to a temperature correction at the Ag melting point of 2 mK at l ¼ 906 nm. To assess the dependence of the cavity emissivity on the value chosen for the wall emissivity, the calculation was repeated for graphite walls with an emissivity of 0.81. In this case, the estimated emissivity of the cavity was 0.99996, a temperature correction at the Ag melting point of 4 mK at l ¼ 906 nm. To provide further veriﬁcation of the validity of the software, the emissivity of the cavity was calculated at the Physikalisch-Technische Bundesanstalt (Germany) using their emissivity software [1232125]. It calculates emissivities of isothermal and nonisothermal cavities based on a Monte Carlo ray-tracing method. Assuming an isothermal cavity with wall emissivities of 0.92 and 0.81 the cavity emissivities were found to be 0.99999 and 0.99997, respectively. The results from all three programs

233

Calculation of the Radiation Characteristics of Blackbody Radiation Sources

1

0.95

Local Normal Effective Emissivity

0.9 η =0 η = 0.05 η = 0.1

0.85

η = 0.15 η = 0.2

0.8

D= 0 D = 0.2 D = 0.4

0.75

D = 0.6 D = 0.8 D= 1

0.7

0.65

0.6 0

0.1

0.2

0.3

0.4

0.5

R

Figure 18 Distributions of the local normal effective emissivity across the aperture of an isothermal cylindrical cavity with Rc ¼ 1, Ra ¼ 0.5, and H ¼ 4 computed for the TETRA and specular--diffuse models of the reﬂection (from Ref. [132]).

agreed to within 1 105 (which corresponds to the difference in radiance temperature of 1 mK at 950 nm and 0.7 mK at 650 nm), providing conﬁdence in the validity of the software. The time has perhaps arrived to conduct a more comprehensive intercomparison of various software tools being used for the calculation of effective emissivity with an aim to deduce their actual accuracy. Clearly, the experimental veriﬁcation must remain the ultima ratio in determining a preference of one computational method over another.

6. Conclusions Calibration of radiation thermometers is performed using blackbody radiation sources. Radiation thermometry metrology is impossible without

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a reliable determination of the radiation characteristics of such sources. Of primary importance is the effective emissivity. In most instances, calculation is the only way to determine the effective emissivity. In order to achieve the necessary accuracy, even the most advanced computational method must be based on experimental data of temperature and optical properties of the cavity’s walls. In this chapter, we have considered both conventional (deterministic) and stochastic (Monte Carlo ray tracing) methods for the effective emissivity computational determination and made an attempt to show the advantages of the stochastic methods. However, this is not to say that the deterministic, especially, analytic (i.e., formula-based) methods must be excluded from consideration. They allow investigation of the most general dependences and trends, despite their limited accuracy or applicability. Issues such as the numerical modeling of the temperature distributions over a cavity’s radiating surface, polarization of the radiation emitted by a blackbody cavity (which can be important especially for cavities whose walls have a signiﬁcant specular reﬂection), the scattering and refraction of a cavity’s radiation, etc., remain beyond the scope of this chapter and require additional discussion.

ACKNOWLEDGEMENT The authors wish to acknowledge the considerable effort of Catherine Cooksey in helping to edit this Chapter and thereby making the content more accessible to its readers.

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CHAPTER 6

Blackbody and other Calibration Sources Ju¨rgen Hartmann1, Jo¨rg Hollandt1, Boris Khlevnoy2, Svetlana Morozova2, Sergey Ogarev2 and Fumihiro Sakuma3

Contents 1. Introduction 1.1. History of thermal radiation and blackbody radiators 1.2. Technical realization of blackbody radiation 2. Fixed-Point Blackbody Radiators 3. Variable Temperature Blackbodies for Temperatures up to 1,0001C 3.1. Heat-pipe blackbody cavities 3.2. Practical blackbody cavities 3.3. Flat plate radiators 4. Cryogenic/Vacuum Blackbodies 4.1. Special requirements for low-temperature blackbody sources 4.2. Low-temperature blackbody designs in different low-temperature radiometric facilities 5. Variable Temperature Blackbody Sources above 1,0001C 5.1. High-temperature furnaces that operate in air to B1,8001C 5.2. High-temperature furnaces with resistively heated carbon cavities 5.3. Temperature gradients 6. Lamp Sources 7. Summary Acknowledgements References

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Physikalisch-Technische Bundesanstalt, Braunschweig and Berlin, Germany All-Russian Research Institute for Optical and Physical Measurements, Section M-4, Moscow, Russia National Metrology Institute of Japan, Tsukuba, Japan

Experimental Methods in the Physical Sciences, Volume 42 ISSN 1079-4042, DOI 10.1016/S1079-4042(09)04206-4

r 2010 Elsevier Inc. All rights reserved.

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1. Introduction The calibration of optical instruments for measuring thermal radiation can be performed in one of the two ways: either the absolute spectral responsivity of a detector with respect to a primary detector standard is determined, or the detectors (or more generally detector-based systems) are calibrated through using known sources of spectral radiation. The absolute determination of detector spectral responsivity requires sophisticated measurement equipment and methods, available only in a limited number of national metrology institutes (NMIs), therefore nearly all calibrations of instruments used for the measurement of thermal radiation are performed using accurately known sources of spectral radiation. In principle two sources of calculated spectral radiation are available, either synchrotron sources calculable using the Schwinger equation or blackbody sources calculable using the Planck function [1]. In practice, due to the major investment required to acquire a synchrotron source, only the application of blackbody radiation offers a practical solution for instrument calibration at nearly all NMIs and calibration laboratories [2]. This chapter is therefore solely concerned with the history, theory, design and application of blackbody radiators for radiation thermometry applications.

1.1. History of thermal radiation and blackbody radiators Thermal radiation is generated and emitted by any object in thermodynamic equilibrium above absolute zero. The origin of thermal radiation is the thermal movement of charged particles. Oppositely charged particles moving with respect to each other forming simple dipoles. The electromagnetic radiation emitted by these dipoles is called thermal radiation. At high temperatures (above B800 K) this emitted thermal radiation can be observed by the human eye. At the beginning of the industrial revolution, besides candles, light through the combustion of gas was the only reliable source of optical radiation. The spectral radiance or luminance of such a source could not be calculated analytically. To produce a reliable source of optical radiation a detailed empirical procedure was required. However by the mid-1800s it was already known that an absolute radiation source should exist. This was arrived through general thermodynamic considerations by Gustav Kirchhoff [3]. The radiation emitted by such an absolute radiation source should not depend on the optical properties of the surface of the source or the material used, but only on its temperature. Additionally, Kirchhoff’s considerations led him to conclude that for such a source its emissivity would equal its absorptivity. A perfect thermal emitter should, therefore, also be a perfect absorber and is therefore called a blackbody, because it

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absorbs all incoming radiation and consequently appears black (at least at temperatures below B800 K where self emission is too far in the infrared to be visible with the naked eye). Kirchhoff restricted himself to theoretical considerations, not seeking a practical realization of such a blackbody. However, he argued that it was scientiﬁcally and technologically important to ﬁnd the analytical representation of the spectral radiance emitted by such a blackbody. Kirchhoff’s theoretical investigations inspired a number of famous scientists, like Wilhelm Wien, Max Planck, Lord Rayleigh, James Jeans and others, to seek an analytic solution for the spectral radiance emitted by a blackbody [4]. Besides this theoretical interest there was also a strong practical reason for solving this problem. In the last 20 years of the 19th century electrical lighting began to replace gas lighting. Werner von Siemens wanted a standard source of optical radiation, to show that his electrical streetlighting was superior to the more traditional gas lighting. To initiate the development of such a standard radiation source (and other relevant measurement techniques) he decided to fund the foundation of an independent government institution dedicated to measurement. Mainly due to the efforts of von Siemens and Herman von Helmholtz (who later became its ﬁrst president) the Physikalisch-Technische Reichsanstalt (PTR) was established in 1887 (this was renamed the Physikalisch-Technische Bundesanstalt (PTB) in 1950). One of the ﬁrst tasks of the PTR was the practical investigation of blackbody radiation. This required that practical blackbody sources be constructed. Wien and Lummer [5] were the ﬁrst to use cavity radiators as technical approximations for blackbody sources. Such radiators were then used at the end of the 19th century for fundamental investigations into blackbody radiation between 90 and 1,800 K. Figure 1 shows the radiometry laboratory of the PTR around the year 1900. Typical blackbodies for realizing temperatures between 90 and 1,000 K are shown in Figure 2 [7]. For the temperatures up to about 1,000 K wellstirred liquid saltpetre was used (left side of Figure 2), while for the lowest temperatures down to 90 K liquid nitrogen was used as cooling agent (right side of Figure 2). For investigation of thermal radiation at temperatures up to 1,800 K a new type of blackbody was developed at the PTR, the so-called electrically heated blackbody [8]. In this blackbody the radiating cavity was formed from a thin platinum layer heated by direct electrical current (Figure 3). For the highest temperatures, up to 2,600 K, a blackbody shown schematically in Figure 4 was constructed [9]. This blackbody consisted of a graphite tube heated by an electrical current 2 a concept still in use today (see Section 5). The intention of Wien was to experimentally verify his law of blackbody radiation, which he had discovered in 1896 [10]. The ﬁrst series of experiments for temperatures up to about 1,000 K did demonstrate close agreement between his theory and experiment [11]. However at higher

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Temperature Radiometry laboratory of the PTR around the year 1900 [6].

Figure 2 Typical blackbodies for realizing temperatures up to 1,000 K (left) and down to 90 K (right) [7].

temperatures signiﬁcant differences began to surface between Wien’s theory and the experimental data especially in the long wavelength range [12]. The results of these experiments are shown in Figure 5. Wien’s approach accurately characterized the spectral radiance emitted from a blackbody at low temperatures and short wavelengths, but failed to

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Figure 3

Picture of the electrically heated blackbody [8].

Figure 4

Electrically heated graphite blackbody for temperatures up to 2,600 K [9].

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E 140

130

120

110

1646° (1653°)

100 Schwarzer Körper. 90

80 Experiment 70

60 1460° 50

Theory

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30 1259° 20 1095° 10

998° 904° 723° 0

1

2

3

4

5

6

λ

Figure 5 Comparison of experimental with the theoretical predicted data using Wien’s law of radiation [12].

do so at longer wavelengths and higher temperatures. In this region another analytical representation, obtained by Rayleigh and Jeans [13], gives excellent agreement with experiments. However their formalism failed at low temperatures and short wavelengths, leading to inﬁnite spectral

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radiance at the shortest wavelengths, popularly known as the ultra-violet catastrophe. By 1899 two representations for the spectral radiance emitted by a blackbody were known, accurately describing the spectrum for short (Wien [14]) or long wavelengths (Rayleigh and Jeans [13]). It was Max Planck who discovered an empirical representation, which interpolated between both solutions and which also ﬁtted exactly the experimental data for all wavelengths and all temperatures. However, Planck was not satisﬁed by this empirical approach desiring a sound theoretical basis. In October 1900 he succeeded in doing so but only by making a revolutionary assumption. The assumption was that the energy of thermal radiation was not distributed continuously, but is formed of small ﬁnite packets of energy [15]. This implied that a change in the energy of light or thermal radiation was discrete and not continuous as was expected from classical electromagnetic theory contrary to an implicit assumption underlying classical physics that of ‘natura non facit saltus’, i.e. nature does not make jumps. Planck’s fundamental assumption has become famous in the history of physics initiating the beginning of quantum mechanics. The formula developed by Planck is [15217] L b;l ¼

2hc 2 1 5 l expðhc=klT Þ 1

(1)

with the spectral radiance Lb,l of a blackbody (indicated by the subscript b) in vacuum, wavelength l, speed of light c, temperature T and Boltzmann’s constant k. Equation (1) forms the basis for the use of blackbody radiators as standard sources for thermal radiation. For radiometry applications (e.g. lighting technologies) hightemperature blackbodies constructed from electrically heated graphite components are utilized (Section 5). For lower temperatures, particularly important for industrial radiation thermometry (Chapter 8), blackbodies with cavities formed by heat-pipes and covering the temperature range from 601C up to 1,0001C are utilized (Section 4). For lower temperatures, such as for space applications, blackbodies operating at cryogenic temperatures are required. This dictates the use of special operating environments requiring a vacuum or dry gas to prevent icing and also requires that the whole optical path be cooled to liquid nitrogen temperatures to suppress the background radiation on the measured signal (Section 3). Finally the International Temperature Scale of 1990 (ITS-90) [18] speciﬁes that blackbodies with radiating cavities immersed in freezing metals, of Ag, Au or Cu, so-called ﬁxed-point radiators, serve as primary standards for radiation temperature at the deﬁning ﬁxed points of the scale (Section 2).

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1.2. Technical realization of blackbody radiation To make use of the equation derived by Planck (Equation (1)) a near ideal blackbody has to be realized. Already Wien and Lummer [5] had found in 1895 that a cavity uniform in temperature having a small opening in comparison to its inner surface area, forms a source of nearly ideal blackbody radiation. Quantifying the small departures of practical blackbody cavities from the ideal situation is one of the key problems of radiation thermometry and radiometry. This departure from ideality is characterized by the emissivity of the cavity and is described in the next section. 1.2.1. Blackbody emissivity To quantify how close a practical blackbody is to an ideal blackbody the concept of emissivity has been introduced. Emissivity is a measure of the deviation of the ability of a real surface to emit thermal radiation in comparison to an ideal blackbody, i.e. an emissivity of unity describes an ideal blackbody. The emissivity of a real surface is in principle a function of wavelength, temperature and angle of view. Emissivity is measured by determining the ratio of the spectral radiance of the source/surface with respect to the spectral radiance of an ideal blackbody at the same temperature. However, if the emissivity is close to unity, as in the case of a practical blackbody, a low uncertainty measurement of that emissivity is very difﬁcult. Only a few results of such measurements are reported in literature [19]. In such cases it is more usual to calculate the emissivity by a variety of mathematical models usually either based on Monte Carlo simulations or by the multiple reﬂection method [20]. Recently software has become available that uses multiple ray tracing combined with the Monte Carlo method and can be used to investigate the emissivity of isothermal blackbodies with a complicated geometry [21]. A simple method for estimating the emissivity of a cavity radiator, ecavity, with a wall emissivity ewall, length l and radius r was given by Bauer and Bischoff in 1970 [22] and is given as cavity ¼ 1

1 wall 1 : wall 1 þ ðl 2 =r 2 Þ

(2)

Using this simple formula the increase of emissivity by drilling a hole into a solid body can be calculated. In Figure 6 this calculation is shown for a range of values of wall emissivities and ratios of length to radius of a cavity. Equation (2) only takes into account the ratio of the radius to the length of the cavity, neglecting the form of the bottom or the temperature distribution or other additional effects, which might inﬂuence the cavity emissivity. More sophisticated methods for estimating cavity emissivity have been described in the literature [16,17,23], and an outline of these are given

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Blackbody and other Calibration Sources

effective emissivity

1

0.95 wall emissivity 0.9

0.9 0.75 0.5 0.3 0.1

0.85

0.8 0

5

10

15

20

25

ratio l/r

Figure 6

Enhancement in emissivity by geometrical effects according to Equation (2).

Photon

Figure 7 Schematic representation of a cavity radiator for generation of blackbody radiation.

below. According to Kirchhoff’s general thermodynamic considerations the thermal radiation generated by enclosed isothermal walls is blackbody radiation, in this case the construction material of the cavity thus formed is unimportant. To sample this radiation a small hole must be made in the cavity as shown in Figure 7. Due to geometric constraints when a photon enters the cavity it will undergo several reﬂections at the cavity walls and the probability for the escape of this photon is low, therefore the cavity has a high absorption. The only optical radiation emitted from this cavity is that generated by the cavity walls. As long as the internal walls of the cavity are isothermal the radiation emitted by this hole will be a close approximation to blackbody radiation.

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The approximation improves when the area of the opening gets smaller in comparison to the inner surface area of the enclosure. An important complication that must be considered is that of cavity non-isothermality. This is particularly an issue for large aperture cavities compared to their length such as those used for remote sensing instrument calibration. The large aperture (Section 5) area leads to a non-isothermal temperature distribution on the cavity walls due to the signiﬁcant radiation loss through the large aperture. This has a strong adverse inﬂuence on the emissivity of the blackbody and has to be considered in high-accuracy metrological measurements. Bedford and co-workers [24] addressed this topic, adapting the series method for use with non-isothermal blackbodies. A different approach, by Prokhorov [25], determines the emissivity of a non-isothermal blackbody by the Monte Carlo method; this is considered further in Section 5. Here a method is described for calculating the emissivity of isothermal and then non-isothermal blackbody cavities, based on the Monte Carlo method. In this approach the inverse ray tracing method is applied and photons are considered which enter the cavity through the aperture instead of photons emitted by the blackbody [2,26]. A photon is launched into the cavity at a certain angle and its path through the cavity is calculated through geometrical considerations. Only emission angles permitted by the experimental situation are considered. Once the photon reaches the cavity wall, a random number generator is used to determine the subsequent path of the photon. Diffuse reﬂection according to Lambert’s cosine law for the inner surface of the blackbody is assumed. With a probability of 1ew(l) the photon is reﬂected from the surface in a randomly distributed direction, with ew(l) being the local emissivity of the cavity wall. The path of the photon is tracked until it is either absorbed in the walls of or escapes from the cavity. The isothermal emissivity eiso is simply the ratio of absorbed photons to photons launched into the cavity. This emissivity, eiso, depends only on the geometry of the cavity and the local emissivity ew(l) of the cavity walls, i.e. the spectral radiance Lcav,iso(l,T) emitted by an isothermal cavity is L cav;iso ðl; T Þ ¼ iso L b;l ðl; T Þ

(3)

For the emissivity eDT(l), which results from a non-isothermal temperature distribution of cavity walls, every absorbed photon is weighted with the spectral radiance according to Planck’s law with the temperature T of the location where the photon is absorbed. The spectral radiance Ll(l,T) assigned to a single absorbed photon is connected to the spectral radiance Lb,l(l,T) of a blackbody at the same temperature by L l ðl; T Þ ¼ w ðlÞL b;l ðl; T Þ

(4)

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Blackbody and other Calibration Sources

For the slightly different temperatures TDT in a non-isothermal cavity, Equation (4) can be rewritten in linear approximation as

@L b;l ðl; T Þ DT L l ðl; T DT Þ ¼ w ðlÞL b;l ðl; T DT Þ ¼ w ðlÞ L b;l ðl; T Þ @T (5)

Using Wien’s approximation for Planck’s law, the derivative is @L b;l ðl; T Þ c2 ¼ L b;l ðl; T Þ @T lT 2

(6)

so that Equation (5) becomes approximately

h i c2 L l ðl; T DT Þ ¼ w ðlÞ 1 2 DT L b;l ðl; T Þ lT

(7)

Equation (7) can be written as L l ðl; T DT Þ ¼ w ðlÞDT ;w ðl; DT ÞL b;l ðl; T Þ

(8)

with eDT,w(l,DT) specifying the change in local emissivity due to a temperature difference DT. Equation (8) describes the spectral radiance contribution to the overall emitted spectral radiance by one photon emitted from a location with temperature TDT [2]. The emissivity eDT describes the transition from an isothermal cavity to a non-isothermal cavity deﬁned by DT ¼

L cav; noniso L cav; iso

(9)

1.2.2. Consideration of the effective refractive index of blackbody radiation In optical measurements the medium refractive index has to be considered. Especially where the optical radiation passes through different media, as also refraction at the contact surface between the two media has then to be accounted for. In laboratory conditions, on leaving the cavity the medium the radiation passes through is ambient air. The formulas for the index of refraction of air derived by Edle´n and modiﬁed by others can be used to calculate the refractive index with low uncertainty [27231]. For standard reference laboratory conditions the refractive index of ambient air is shown in Figure 8 [28]. However, the situation is more complex when dealing with the measurement of blackbody radiation generated in a cavity radiator. Due to the different temperatures inside and outside the blackbody and due to the purging of the cavity with argon or other inert gases, the index of refraction of the media inside and outside the blackbody are, in general, different. Usually the index of refraction of ambient air is taken into account in the optical path between the opening of the cavity and the

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1,000300

refractive index of air

1,000295 1,000290 1,000285 1,000280 1,000275 1,000270 1,000265 200

700

1200

1700

2200

wavelength / nm

Figure 8 Refractive index of air at 1,013 hPa, 0.03% CO2 content, 50% relative humidity at 231C.

detecting instrument. However, this accounts only for the optical radiation outside the cavity, neglecting all effects originating inside the cavity. For the generation of the blackbody radiation inside the cavity the index of refraction of the medium inside the cavity has to be taken into account [16] and this might well differ signiﬁcantly from the value of the ambient medium [2]. Including the index of refraction Equation (1) can be rewritten as [2,16] L b;l ¼

2hc 2 =n2 l5 exp½hc=ðk nlT Þ 1

(10)

Equation (10) describes the spectral radiance at the exit aperture of a cavity radiator containing a medium with refractive index n inside as a function of the wavelength l inside this medium. The radiation emitted by the blackbody cavity has to travel the distance between the aperture and the detector through the ambient medium, usually air at ambient conditions. This will cause a wavelength shift according to the index of refraction of the ambient medium, namb. To account for this, Equation (10) is rewritten in terms of vacuum wavelength l0 ¼ nl and accounting for the wavelength shift in the ambient medium requires replacement of l0 by nambl resulting in L b;l ¼

2hn3 c 2 =n5amb l5 exp½hc=ðk namb lT Þ 1

(11)

Comparing Equation (11) with Equation (10) reveals that the effect of including the transmission of the radiation through the ambient medium

253

Blackbody and other Calibration Sources

with refractive index namb introduced the factor (n/namb)3, changing the effective spectral radiance Lb,l by this factor [2]. Another effect that has to be accounted for is the reﬂection occurring at the contact area between the media inside of the blackbody cavity, the hot gas and the outside media, usually ambient air. For perpendicular incidence the corresponding reﬂectivity R is namb n 2 R¼ namb þ n

(12)

In Equation (12) it is assumed that the transition between the two media inside and outside the blackbody is abrupt, resulting in an upper limit for the expected reﬂectivity. Consideration of Equation (12) leads to the ﬁnal formula for the spectral radiance emitted by a blackbody radiator at temperature T with a medium of refractive index n into a medium of refractive index namb L b;l ¼ aðnamb ; nÞ

2hc 2 =n2amb l5 exp½hc=ðknamb lT Þ 1

(13)

with the factor "

# namb n 2 n3 aðnamb ; nÞ ¼ 1 n3amb namb þ n

which accounts for the effect of the two different media inside and outside the blackbody cavity. For a proper consideration of the effect of refractive index the factor a(namb, n) has to be taken into account, when calculating the spectral radiance emitted by a blackbody radiator according to Planck’s law. The error neglecting the different refractive indices might be tolerable in some photometric experiments. However, it has to be taken into account for accurate radiance temperature measurements and absolute radiometry currently reaching uncertainties of a few tenths of a degree for temperatures around 3,273 K [2,32234]. This is especially important for the determination of the thermodynamic phase transition temperatures of the novel metal2carbon and metal2carbide2carbon eutectics where an uncertainty in temperature of about 0.1 K is required and the abovementioned effect has at least to be considered in the uncertainty budget [2,35]. Once the emissivity of the blackbody is known and, for the very highest accuracy work, the inﬂuence of the index of refraction is considered, blackbodies are calculable sources for thermal radiation. These are very widely used in thermometry, radiometry and photometry. Various types of blackbodies have been developed as standard sources of thermal radiation.

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The overview given in the following section, is restricted to the temperature range usually covered by commercial radiation thermometers and important for radiometry, photometry and radiation thermometry, i.e. from 100 up to 3,273 K. It is of necessity selective and the reader should refer to the references for a more comprehensive overview.

2. Fixed-Point Blackbody Radiators The vast majority of ﬁxed-point blackbodies realized in NMIs are the ITS-90 deﬁning ﬁxed points for scale realization above the silver point, namely blackbody sources of Ag, Au or Cu. To achieve the best realization of such ﬁxed points the blackbody has to have a very high emissivity, typically 0.99999, be realized in a very low thermal gradient environment and constructed from the purest metals and crucible material. The material of construction of such ﬁxed points is graphite which fortuitously has an intrinsically high surface emissivity. The most important factors for a good ﬁxed-point blackbody are a high emissivity of the cavity and high purity of the metallic ingot. A high emissivity of the ﬁxed-point blackbody is realized by a large ratio of cavity length to cavity diameter and the use of graphite as crucible material, which already has an intrinsically high emissivity. Maintaining the high purity of the used metal ingot requires great care during the fabrication of the ﬁxed-point blackbody. In addition and especially for the high melting ﬁxed points, special precautions with respect to robustness of the cells have to be taken into consideration. The best method to establish the reliability of a ﬁxed-point blackbody is through comparison with another ﬁxed-point blackbody [36]. For the ITS-90 ﬁxed-point blackbodies the proﬁle of the freezing plateau, such as the ﬂatness and the depth and recovery of the super cooling, give important information about the soundness of the crucible, the purity of the ﬁxedpoint material and crucible-furnace conﬁguration. A typical example of such a realization is described below. The ﬁxed-point blackbody in Figure 9 uses a single zone heater and a sodium heat-pipe to ensure very low-temperature gradients along the ﬁxed-point crucible. High emissivity is achieved through the large dimensions of the cavity (100 mm in length and 16 mm in diameter), with an aperture of 3 mm in diameter. A large amount of ﬁxed-point material is necessary: about 3.5 kg of 99.999% pure gold. The large amount of material, together with the sodium heat-pipe surrounding the ﬁxed point, results in long freezing plateaux. A typical plateau is plotted in Figure 10, showing that the plateau time is about 90 min. Such a ﬁxed point is typical of those used in NMIs for their primary scale realization. Besides primary realizations of the ITS-90 ﬁxed-point blackbody sources, secondary ﬁxed-point sources can be used for routine calibration of

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Blackbody and other Calibration Sources

Graphite-crucible

Water Cooled Housing

Sodium Heat-pipe

Argon Gas Inlet Heating Zone

Figure 9

Fixed Point Metal (Au)

Aluminia Tube

Insulation

Design of the PTB gold ﬁxed-point blackbody [34].

1340.0 1339.5

temperature / K

1339.0 1338.5 1338.0 1337.5 1337.0 1336.5 1336.0 1335.5 1335.0 0

50

100

150

200

250

300

350

400

time / min

Figure 10 Typical melting and freezing plateau obtained from the gold ﬁxed point shown in Figure 9.

radiation thermometers, and also scale realization, particularly below the silver point. In these cases smaller ﬁxed points are generally deployed (though with larger apertures than typical primary ﬁxed-point blackbody cavities), which are of lower cost, are small and easy to operate but still have reasonable uncertainties [37]. These ﬁxed points cover a wider range of temperatures than those of the ITS-90 deﬁning ﬁxed points for the scale above the silver point, and are used to calibrate radiation thermometers in conjunction with a suitable interpolation equation. Table 1 gives the speciﬁcation and Figure 11 a schematic diagram of typical practical blackbody furnaces. The one shown in Figure 11 was developed by the NMIJ.

256 Table 1

Ju¨rgen Hartmann et al.

Speciﬁcations of practical-type ﬁxed-point blackbodies.

Speciﬁcations

Copper

Freezing temperature1084.621C (1357.77 K) Maximum power 700 W consumption Aperture size 3 mm Cavity emissivity 0.999870.0001 Cavity shape Amount of metal Purity of metal Plateau duration Operation Aperture diameter Size of the furnace Temperature control Gas purge

Silver

Aluminium

Zinc

961.781C (1234.93 K) 600 W

660.3231C (933.473 K) 500 W

419.5271C (692.677 K) 400 W

6 mm 0.999570.00025 with f6 mm cavity aperture 0.99970.0005 without f6 mm cavity aperture

Cylinder cone 26 cm3 Better than 99.999% or 99.9999% 10 min75 min Switch of heater current or temperature setting 6 mm (stainless steel) Diameter 10 cm Length 25 cm R-type thermocouple Argon gas 200 mL/min

250 mm Stainless Steel Capsule Heater

100 mmφ

Stainless Steel Radiation Shield Alumina Tube

Ar Gas

ThermoCouple Firebrick Insulator Ceramic Wool

Alumina Cement Graphite Crucible Aperture

Figure 11 Design of practical-type ﬁxed-point blackbody designed by NMIJ.

The graphite crucible contains about 26 cm3 metal of 99.999%2 99.9999% purity. For temperatures above 2001C, argon gas is introduced to prevent oxidation of the metal and graphite. The temperature is controlled by an R-type thermocouple. A stainless steel aperture of 6 mm in diameter

Blackbody and other Calibration Sources

Figure 12

Schematic of the ﬁxed-point crucible design.

Figure 13

Photograph of the ﬁxed-point blackbody furnace.

257

is set in front of the cavity. Figure 12 shows the ﬁxed-point design. The construction material is graphite. The ﬁxed point is composed of four parts: a cavity, a cylinder, a lid and an aperture, screwed together. The cavity is a cylinder of 10 mm in diameter and 45 mm in length with a 1201 (full angle) cone at its base. The ﬁxed points shown in Figure 12 are operated in furnaces shown in Figure 13.

Ju¨rgen Hartmann et al.

12

0.02

10

0.01

8

0.00

6

-0.01

4

-0.02

2

-0.03

0

-0.04

-2

-0.05

-4

-0.06

-6 0

5

10

15

Temperature difference (K)

Temperature difference (K)

258

-0.07 20

Time (min)

Figure 14 Freezing curve of aluminium. The data shown by squares is the same plateau expanded 200 times in scale.

A typical freezing curve for such a ﬁxed point is shown in Figure 14. As can be seen good performance is conﬁrmed by the sharp return out of the undercool and a ﬂat freezing plateau. Until quite recently no reliable high-temperature ﬁxed-point blackbody sources were available. However this situation is in the process of changing and signiﬁcant worldwide effort has been spent on the development and characterization of high-temperature ﬁxed points, with freezing and melting points above that of copper (1,0851C). Since 1999 metal2carbon and metal2carbide2carbon eutectic and peritectic materials have shown their potential for such high-temperature ﬁxed points [38]. Since then these ﬁxed points have been investigated in several international projects [2,39241]. As a result of these investigations a number of high-temperature ﬁxed points at elevated temperatures are available and summarized in Table 2 [42]. Typical cell design for such high-temperature ﬁxed points are shown in Figures 15 and 16 [2,43]. These high-temperature ﬁxed points will have a signiﬁcant impact in both primary and secondary thermometry. More details can be found in Section 6 of Chapter 2.

3. Variable Temperature Blackbodies for Temperatures up to 1,0001C The temperature range from about 601C up to 1,0001C is most relevant for industrial purposes as it covers, e.g. the applicable temperature

259

Blackbody and other Calibration Sources

Table 2 High temperature ﬁxed points and their approximate phase transition temperatures. Eutectic

Approximate temperature (K)

Approximate temperature (1C)

Metal2carbon Fe-C Co-C Ni-C Pd-C Rh2C Pt-C Ru-C Ir-C Re-C

1,426 1,597 1,602 1,765 1,930 2,011 2,226 2,565 2,747

1,153 1,324 1,329 1,492 1,657 1,738 1,953 2,292 2,474

Metal2carbide2carbon B4C—C d(MoC)2C TiC-C ZrC-C HfC2C

2,659 2,856 3,032 3,155 3,458

2,386 2,583 2,759 2,882 3,185

Metal2carbide2carbon peritectic Mn7C32C 1,604 Cr3C22C 2,099 WC2C 3,022

1,331 1,826 2,749

range for the plastic, aluminium and part of the glass industries as well as food industry. Depending on the application, blackbodies of different accuracy and design can be used. The task of such blackbodies is to characterize and calibrate radiation thermometers and thermal imagers, serve as reference sources for the measurement of emissivity and the calibration of blackbodies by use of a high-quality transfer radiation thermometer. The most sophisticated ones are heat-pipe blackbodies, where the radiating cavity is made from a heat-pipe ensuring the best temperature homogeneity across the length of the cavity. More practical types are generally either liquid-bath blackbodies, where the cavity is immersed in a well-thermalized liquid bath or, at a higher temperature, electrically heated cavities usually placed in a furnace.

3.1. Heat-pipe blackbody cavities The variable temperature blackbodies with lowest uncertainties for the temperature range from 601C up to 1,0001C are heat-pipe blackbodies.

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Figure 15 Typical design for metal--carbon eutectic ﬁxed-point cells with dimensions in millimetres [43].

Figure 16

Alternative design of metal--carbon eutectic ﬁxed point [2].

261

Blackbody and other Calibration Sources

The cavity of this type of blackbody is formed of a heat-pipe. A heat-pipe is a doubled walled tube ﬁlled with a small amount of a ‘ﬂuid’ to enhance the heat transfer. The pressure inside the tube is adjusted so that the liquid and the gaseous phase of the ﬂuid are in equilibrium [44]. The effect of the heat-pipe is as follows. At a hot spot in the cavity an amount of the ﬂuid evaporates lowering the temperature due to the latent heat of vaporization. At cold spots in the cavity part of the gaseous phase condenses and releases its latent heat of condensation at this location. Due to this effect the temperature homogeneity of blackbody cavities formed by heat-pipes is extremely good. However, due to pressure constraints, heat-pipes with particular working ﬂuids only work over a restricted temperature range. Therefore, for blackbody applications, typically four different heat-pipe ﬂuids for the temperature range 601C up to 1,0001C are used, details are given in Table 3 [34,45,46]. A typical design of an ammonia heat-pipe blackbody is shown in Figure 17. The cavity of this blackbody has a high emissivity coating with an emissivity of about 0.95 [47]. This coating combined with a large cavity results in an emissivity of 0.9999. For temperatures below about 151C (approximately the dew point) and, in particular, below the freezing point Table 3 Heat-pipe working ﬂuids and typical operating temperature ranges used for the construction of blackbody cavities. Heat-pipe working ﬂuid

Temperature range (1C)

Ammonia Water Caesium Sodium

60 to 50 502270 2502650 50021,000 Cooling element

Housing

N2 purging Aperture 60 mm PRT

525 mm

Heat-pipe

Heating elements

Figure 17 Cross-section of the ammonia heat-pipe blackbody used at the PTB [2].

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Ju¨rgen Hartmann et al.

of water the cavity is purged with dry nitrogen to prevent condensation of water and ice forming in the cavity. For the adjacent temperature range from 501C to 2701C the heat-pipe ﬂuid used is water. The design is similar to the ammonia heat-pipe blackbody shown in Figure 17, but has a smaller cavity of 420 mm in length. Due to the higher temperatures a different high emissivity coating is used with an emissivity of about 0.9. The cavity has an effective emissivity of 0.9998. For the temperature above 2501C another blackbody design shown in Figure 18 is implemented. In this case the intrinsic emissivity of the cavity material which is sandblasted and oxidized Inconels forms the emitting surface with an emissivity of approximately 0.75. Together with the geometrical dimensions shown in Figure 18 an emissivity of 0.9996 is obtained. In this temperature range two heat-pipe materials are applied, these are caesium (2701C up to 6501C) and sodium (5001C up to 9621C). The details of these blackbodies are summarized in Table 4. These blackbodies are used in the low- and medium-temperature calibration facility of the PTB shown schematically in Figure 19. Traceability of the radiation temperature to the ITS-90 for the four blackbodies is given by standard platinum resistance thermometers (SPRT) calibrated directly against the national realization of ITS-90. These PRTs are inserted into the heat-pipe close to the bottom of the cavity as indicated in Figure 19. Optimized controlling of the temperature in conjunction with the high thermal mass of the blackbodies yields a temperature stability of about 710 mK. When radiation thermometers or thermal imagers are calibrated using the facility outlined in Figure 19 they are ﬁxed on a computer controlled xyz translation stage, which has variable ranges of over Pt 100 or thermocouple

water cooled housing

bifilar heating

200 mm 368 mm

ø 41 mm

ø 92 mm

SPRT 950 mm heat-pipe

Ceramic insulation

Figure 18 Cross-section of the heat-pipe blackbodies for temperature above 2501C up to 9621C [2].

Details of the heat-pipe blackbodies.

Blackbody

Temperature range (1C)

Heat-pipe material

Length of cavity (mm)

Diameter of cavity (mm)

Wall emissivity

Blackbody emissivity

NH3 H2O Cs Na

60 to +50 50 to 270 270 to 650 500 to 962

Steel Titanium Inconel 600 Inconel 600

525 420 368 368

60 60 41 41

0.965 0.88 0.75 0.75

0.999970.00006 0.999870.00015 0.999670.00017 0.999670.00017

Blackbody and other Calibration Sources

Table 4

263

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ammonia heat pipe bb -60 °C to 50 °C

water heat pipe bb 50 °C to 270 °C

SPRT

SPRT

caesium heat pipe bb sodium heat pipe bb 270 °C to 650 °C 500 °C to 962 °C SPRT

SPRT

y 600 mm 3000 mm x

Radiation thermometer or thermography system to be calibrated in terms of T90

Figure 19 A schematic diagram of the low- and medium-temperature calibration facility of the PTB.

3,000, 600 and 200 mm, respectively, with at least 50 mm accuracy and 10 mm reproducibility. These types of blackbodies and calibration facilities are similar to those that exist in other leading NMIs, such as NPL and VSL [45,46,48].

3.2. Practical blackbody cavities The above-mentioned types of blackbodies are for specialized use in NMIs. For more practical use in calibration laboratories or for manufacturers of radiation thermometers more practical blackbodies are available from different suppliers.1 These are essentially blackbody cavities in electrically heated furnace assemblies. A detailed description of all practical blackbody types would go beyond the scope of this book and the reader is referred to the information given by the respective manufacturer. A completely different design for heating the radiating cavity is implemented in liquid-bath blackbody radiators. Bath blackbodies apply a simple technical solution to obtain a very low-temperature gradient, where the blackbody cavity is immersed in a liquid. The usual range covered by conventional liquid-bath blackbodies is typically from 401C up to about 4501C. A schematic drawing of a sophisticated bath blackbody, optimized for lowest uncertainties in radiance temperature is shown in Figure 20. 1

For example, Land Instruments, Mikron, Heitronics, Mester, E-O Industries and Chino.

Blackbody and other Calibration Sources

Figure 20

265

Schematic drawing of a sophisticated liquid-bath blackbody [49].

Typically the cylindrical cavities are made from copper, or other metals and are completely immersed in the liquid. For the cavity in Figure 20 the optimum form of the bottom of the cavity was found to be a ﬂat bottom tilted at 301 with respect to the optical axis. Using a high surface emissivity coating of 0.95 the cavity has an emissivity of about 0.9997. Dry thermalized N2 gas is used to prevent icing and condensation within the cavity. The overall standard uncertainties in radiation temperature reached with such a liquid-bath design range from about 100 mK at ambient up to 220 mK at about 2501C [49]. Other types of liquid-bath blackbodies are described, e.g. in Ref. [48]. A special application of liquid-bath blackbodies is the calibration of infrared ear thermometers for the measurement of human body temperatures. Typical examples of such blackbodies are described and evaluated in Refs. [50] and [51] and given in Figure 21. In the case of the blackbody described in Ref. [52] its straightforward but efﬁcient design ensures the traceability of clinical thermometry to the ITS-90 via the calibrated PRT, which senses the temperature of the stirred water bath. Due to the high emissivity of the cavity and the high thermal conductivity of the copper, the temperature of the cavity is extremely close to the temperature of the water bath.

3.3. Flat plate radiators Conventional blackbody radiators by necessity suffer from the fact that the diameter of the radiating cavity has to be small with respect to the length to

266

2,122

9,012 26,977

1

108,631 115,331

105 mm

20 mm

198,194

2

3

5 mm

110 mm

50° 4

φ 20

φ 60

120°

wall thickness 1.5 mm

φ 36.1°

30°

200

Figure 21 Typical example for liquid-bath blackbodies used for the calibration of infrared ear thermometers. Left: Geometry of the liquid-bath blackbody used in Europe and described in the EN 12470-5 [52]. (1: support, 2: copper cavity coated by a black paint, 3: aperture, 4: O-ring). Middle: Geometry of the cavity (dimensions in millimetres) of the liquid-bath blackbody used in Japan and described in JIS [53]. Right: Blackbody used in the United States according to the ASTM Standard E1965-98 [54].

Ju¨rgen Hartmann et al.

39 mm

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obtain a high emissivity. However, low-temperature radiation thermometers and in particular thermal imaging systems, have a large ﬁeld of view requiring a large radiating area of the radiation source. A practical solution to the characterization of such infrared systems are ﬂat plate radiators. Here a plate, having a high and known emissivity is uniformly heated and used as a calculable radiator. However, the emissivity of the surface of such a plate will be signiﬁcantly lower than one, and their use requires careful consideration of the environmental conditions (e.g. reﬂected background radiation). In particular at higher temperatures convection losses across the surface will alter the surface temperature and distribution. To enhance the surface emissivity, special black velvet coatings have been developed, which yield emissivities of about 0.95. For further increase of the emissivity, the surface can be structured to enhance the emissivity by geometrical effects to about 0.97. It should be noted that it is difﬁcult to obtain a homogeneous temperature distribution across the radiating surface for these devices, which signiﬁcantly inﬂuences the effective emissivity of the ﬂat plate radiator. In general the emissivity of ﬂat plate radiators is critical for the overall uncertainty in realized radiance temperature. There are a wide variety of commercially available ﬂat plate radiators. Specialized ﬂat plate radiators are sometimes built for speciﬁc applications. For example, a unique ﬂat plate radiation source [55] was designed for ground-based pre-ﬂight radiometric calibration of space-based instrumentation in the 2.5215 mm range (Figure 22). The radiator consists of a black painted heated radiation surface and a heat exchanger linked with the radiation surface with supports serving as poorly conducting heat paths. The emissivity of the source estimated by means of Monte Carlo simulation software [56] on the basis of the known emissivity of NEXTEL Velvet 82111 black paint [47] is 0.9670.02. A visual representation of the ﬂat plate uniformity where sample scans obtained by means of a thermal imager under ambient conditions is shown in Figure 23.

4. Cryogenic/Vacuum Blackbodies Measurements of the characteristics of various radiation sources in the infrared spectral range is of great importance in science and technology. A major problem in using the infrared spectral range is absorption in the atmosphere, e.g. by water vapour or carbon dioxide, leading to large variations in the signal depending on the concentration of these gases [17,23]. In order to solve this problem, cryogenic-vacuum systems including low-temperature blackbody radiators and infrared radiometers have been developed at metrological centres [55,57260].

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Figure 22 Flat plate reference radiator for pre-ﬂight calibration of space-based instrumentation [55].

4.1. Special requirements for low-temperature blackbody sources The measurement of thermal radiation emitted by blackbodies with temperatures below room temperatures and in particular well below 01C requires specialized conditions and facilities. This is because the measurement of the thermal radiation emitted by such low-temperature blackbodies is strongly contaminated by the surrounding background/ambient thermal radiation, particularly if the background is at a temperature signiﬁcantly in excess of that of the blackbody. Additionally, a blackbody colder than 01C is prone to icing in a normal laboratory environment. To overcome these problems, facilities using low-temperature blackbody radiation sources are operated under vacuum in a cooled environment. A typical example of such a facility is shown in Figure 24 [60] and similar equipment has also been developed at the NIST [61]. Operating the blackbody and the detector under vacuum has another practical advantage. Due to the behaviour of Planck’s law of radiation, the maximum of the emitted spectral radiance shifts towards longer wavelengths with decreasing temperatures. Therefore, detectors with spectral responsivities in the near and mid-infrared region of the spectrum

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Figure 23 Spatial distribution of radiation temperature across the aperture of cavity shown in Figure 22 by means of a thermal imager placed at a distance of 3 m from the blackbody radiating surface. Ambient environmental temperature was 311C. (left) Aperture scan at T ¼ 72.21C, (right) aperture scan at T ¼ 28.81C.

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LN2 blackbody as radiometric zero Opto-mechanical unit Ga blackbody

LN2 cooled beam line

In blackbody Vacuum detector chamber

Detetor under test

Vacuum IR standard radiation thermometer VIRST

Figure 24 Reduced background facility of the PTB. The optical path is cooled by liquid nitrogen and the whole facility can be evacuated [60].

need to be used. For such wavelengths, i.e. between 1 and 14 mm, under laboratory conditions, absorption bands of water vapour and carbon dioxide will signiﬁcantly affect the atmospheric transmission, but in vacuum these effects are negated.

4.2. Low-temperature blackbody designs in different low-temperature radiometric facilities In the following section some examples for blackbodies operating in low background (radiance) facilities are presented. The Medium Background Modular Radiometric System (MB MRS) shown in Figure 25, contains two Thermal Reference Sources, a gallium melting point at 29.76461C and a variable temperature blackbody. The optical path consists of optical bafﬂes, a chopper, a shutter and a reference cavity (to act as a radiometric zero) and is cooled with liquid nitrogen. In between the bafﬂes and apertures there is a germanium lens, which serves as a window to separate the vacuum part of the facility (left side in Figure 25) from the non-vacuum (right side Figure 25). The design of the thermal sources permits their use under vacuum. The ﬁxed-point blackbody can be operated in vacuum or air and is used for the realization of spectral radiance or irradiance at the gallium point, the design is shown in Figure 26. The radiating cavity is made of Teﬂon, coated with ChemGlaze [62] (specular black), and includes two calibrated capsule

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Reference Cooled Cavity

Ga Fixed-Point Blackbody

Cooled Baffle Cooled Shutter Chopper LN2

Filter Radiometer

Precision Apertures

Figure 25 The Medium Background Modular Radiometric System (MB MRS).

Figure 26 Schematic diagram of the gallium ﬁxed-point blackbody.

PRTs one for control and the other for temperature measurement. The solid angle of the radiation emitted by the gallium ﬁxed-point blackbody is limited by a precision optical aperture. The gallium cell is inserted into a thick-walled copper block, whose temperature is controlled by a liquid ﬂowing through a copper tubing heat exchanger. This provides the temperature control necessary to controllably melt the gallium. A schematic diagram of the variable temperature blackbody is given in Figure 27. This blackbody can be inserted into the same heat exchanger as that used for the gallium point and can operate from 201C to +1001C. The temperature of the blackbody is controlled by a high stability (70.02 K) ﬂuid circulator. The internal diameter of the radiating cavity is 40 mm, the length of the cavity with a conical bottom is 250 mm and the apex angle of the cavity bottom is 751. The cavity is internally coated with the black paint ChemGlaze Z302 [62] yielding a normal effective emissivity from 1 to 14 mm of at least 0.9999. Another example of a variable temperature cavity for use under vacuum at cryogenic temperatures is the vacuum variable low-temperature

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Figure 27 The variable temperature blackbody.

blackbody (VTBB100) developed for the low background calibration facility at PTB [59,60,63]. The VTBB100 was constructed as a highly stable reference source for calibration of blackbody sources (by comparison) in the temperature range from 100 up to 450 K under medium-vacuum conditions (103 Pa) and a medium-background environment (liquidnitrogen-cooled cryoshroud). A schematic of the VTBB100 is shown in Figure 28. The VTBB100 represents a long radiator cavity made of oxygen-free copper with a 20 mm diameter aperture. The aperture is cooled by a heat link to a reservoir of liquid nitrogen in the vacuum chamber in which the blackbody is installed. The vacuum enclosure allows one to work at temperatures down to 100 K. The required homogeneity of temperature along the radiator cavity is determined by the design of the cavity, and is adjusted by a precision control system. The radiator control system limits the temperature gradient to 10 mK in the operating temperature range 100 up to 450 K. The blackbody BB1000 is simply an extended area blackbody for medium temperatures (near ambient up to 1,000 K), which operates under cryogenic conditions, facilitating measurements, with reduced background radiation, it can be operated under vacuum or with special gas environments. The BB1000 radiator design has two special design features of note: (1) a simple ﬂat bottom for uniform heat removal; and (2) use of reﬂecting lateral walls to reduce the size of the zone with high radiation losses. However, the usage of a ﬂat bottom leads to a decrease of the effective emissivity in comparison to a conical bottom, especially for coatings with a signiﬁcant specular component of reﬂection. It is well known that coverage with V-shaped or rectangular grooves [64266] can increase the emissivity of ﬂat surfaces. However, in this design there was potential for temperature gradients along grooves if they had been introduced and it is this problem

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Figure 28 Schematic drawing of the blackbody VTBB100: 1, body of the VTBB100; 2, second radiation screen; 3, ﬁrst radiation screen; 4, cryoshroud; 5, cavity; 6, cryoshroud heat-exchanger; 7, cryoshroud heater (HT); 8, ring type crossbar on the cavity; 9, heat-link; 10, electrical connector; 11, ﬂange for low background facility connection; H1-2, H2a-3, H3a-4, heaters of the VTBB100 ﬁrst zone, second and third zones accordingly.

that is avoided by using a ﬂat bottom. The conﬁguration of lateral walls is similar to the one proposed in Ref. [67] and was successfully applied to develop blackbodies in the low- and medium-temperature range [66]. A schematic of the blackbody BB1000 showing the grooved sidewalls is shown in Figure 29. For the ground-based validation and calibration of spaceborne instrumentation low-temperature blackbodies with large radiating areas are required to cover the ﬁeld of view of such instruments. One particular blackbody developed by VNIIOFI [55,68,69] is the precision large-area low-temperature blackbody known as the BB100-V1 for the calibration of spaceborne radiometers and thermometers within the temperature range 2402350 K from 1.5 to 15 mm [70272]. A schematic diagram of the blackbody is shown in Figure 30. The blackbody is a cylindrical copper cavity, painted with NEXTEL Velvet 821-11 black coating [47]. It is heated/cooled by liquid circulation in a copper coil tube heat exchanger around the blackbody cavity. The radiating cavity of the blackbody is comprised of a 120 mm diameter cylinder, 200 mm long with a 100 mm diameter aperture, yielding an emissivity of 0.997. This high emissivity is achieved by the use of the high emissivity coating, ensuring low-temperature gradients, and careful design

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Additional heater

Differential thermocouple

Main heater

Thermocouples

Cold Reflector Au Coated Al

Inlet LN2

2.0 1

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32

30°

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Case Outlet LN2

Nickel Multi-Foil Insulation (15 micron Foil, Ceramic Spacer, 20 − 25 Layers)

Vacuum superinsulation

MINCO PRT−100Sensor (with Backup unit)

SDL Unit T=100 - 150 K

Figure 29 Schematic drawing of the blackbody BB1000 for operation at high temperatures in a vacuum environment.

Figure 30

Cross-section of the large-area low-temperature blackbody BB100-V1.

of the radiating back wall, incorporating V-grooves. The thermometry of the cavity temperature, the temperature gradient and control is performed using ﬁve 100 O PRTs. The reference thermometer can be removed periodically to allow for recalibration. Beside the facilities described above others, like those of the NIST, have been developed to undertake calibrations required by space agencies such as

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NASA [61,73]. The advantage of such facilities is the suppression of background thermal radiation by performing the measurements in a cooled environment under vacuum or an inert gas atmosphere.

5. Variable Temperature Blackbody Sources above 1,0001C For high-temperature applications above the gold point, heat-pipe blackbodies cannot generally be used due to lack of suitable materials from which to manufacture the heat-pipe. Consequently, special hightemperature blackbodies of different designs have been constructed. Only few materials can be used for cavities in this temperature range. The most commonly used material, in particular for the highest temperatures, is graphite. Different types of graphite are used for this purpose: pure carbon tubes, CC reinforced tubes and tubes made of pyrolytic graphite [2]. Also graphite blackbody cavities can be made from silicon carbide or (less commonly) alumina for use at temperatures below 1,8001C. The radiating cavity is heated by two different methods. These are either direct resistive heating, as is usually (but not exclusively) the case for graphite cavities, or indirect heating with the heaters being somewhat removed from the emitting cavity. At these temperatures the thermal gradient along the sidewall of the cavity is often large and will signiﬁcantly affect the effective emissivity. The measurement of such gradients and determining its effect on the emissivity of one particular furnace will be described. Examples of these blackbodies are given below.

5.1. High-temperature furnaces that operate in air to B1,8001C For the medium temperature range from 6001C up to 1,5001C blackbodies with resistively or radiative heated SiC cavities are, due to simplicity of design and operation, widely used in calibration laboratories. While normal SiC can only be operated to B1,5001C, sintered SiC can be operated at even higher temperature. The emissivity of SiC is sufﬁciently high to reach effective emissivities of the blackbody cavity of about 0.995 even for cavities with openings of about 50 mm in diameter. The cavity temperature of such furnaces is usually measured by thermocouples located at the bottom of the cavity. Calibrating these thermocouples in accordance to the ITS-90 ensures the required traceability. Due to the oxidation resistance of SiC such blackbodies can be operated without a window and without purging of an inert gas, additionally due to the low thermal mass no watercooling is necessary, enabling stand-alone operation.

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5.2. High-temperature furnaces with resistively heated carbon cavities The material best suited for high-temperature blackbodies is graphite, it has a sufﬁciently low electrical resistance enabling an efﬁcient conversion of electrical power to thermal energy and its high thermal conductivity enables it to achieve good temperature uniformity along the walls. In addition, the mechanical stability of the carbon is excellent even for temperatures up to 2,800 K. The main disadvantages of graphite-based cavities is their requirement to operate in an inert gas atmosphere and that the cavities have a limited lifetime when operated at the highest temperatures due to sublimation (e.g. at and above 2,800 K the typical cavity lifetime is about 1002200 h, after which the cavity has to be replaced). Three different types of graphite cavities are used. The ﬁrst uses a resistively heated graphite cavity, the second uses a graphite cavity heated indirectly by carbon2carbon composite heaters surrounding the cavity and the third type uses resistive heating but is constructed from pyrolytic graphite rings. An example for the ﬁrst type of blackbody is shown in Figure 31. This type of blackbody consists of a gently tapered graphite tube surrounded by graphite felt insulating material wrapped with two layers of graphite foils and inserted into a silica tube. The whole assembly is purged with argon gas. The blackbody and its insulation are surrounded by an aluminium water-cooled jacket (see Figure 32). The blackbody is clamped onto graphite-composite rings, which are in turn ﬁrmly ﬁxed in water-

Figure 31 Resistively heated carbon tube-based blackbody for the operation without window. (Photo courtesy: Howard Yoon, NIST.)

Blackbody and other Calibration Sources

Figure 32 Assembly (left) of the blackbody cavity [74] and schematic (right) of the insulation [75,76] of the ﬁrst type of furnace (in this version a brass water jacket is used).

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cooled copper rings as shown in Figure 32. During operation, the blackbody is resistively heated by a voltage applied across the copper electrical contacts. The second type of radiance source, where the blackbody is radiatively heated by three surrounding resistively heated carbon2carbon composite heating elements is shown in Figure 33. The advantage of using composite material as heating elements is that they can be quite thin (e.g. 1 mm), enabling a high-voltage low-current operation, resulting in lower power requirements. However, the disadvantage of high-voltage operation is that arcing can occur during operation, this problem can be reduced by adding a few percent N2 to the Ar purging gas outside of the furnace tube, enabling operation temperature up to 2,800 K [77]. Another example of such a furnace is shown in Figure 34, this time a single zone furnace. As can be seen from Figure 34 the graphite cavity is radiatively heated by a semi-rigid C2C composite heating element. The slots are cut into the material to increase its electrical resistance. This simple geometry enables a quasi-homogeneous temperature distribution over a length of about 100 mm, with a 27 mm internal diameter. This furnace can reach temperatures as high as 3,100 K and is purged by argon or helium. When measurements are required the window in front of the cavity can be removed to enable direct access to nearly undisturbed and ideal blackbody radiation. This furnace was been developed by the NMIJ and is currently manufactured by Chino [42]. This type of furnace is mainly used for hightemperature ﬁxed-point measurements [42] and details can be found in Ref. [77]. The third type of furnace is quite different in design. The blackbody cavity is formed by a stack of pyrolytic graphite rings as shown in Figure 35. Pyrolytic graphite has a layered structure with a very high thermal and electrical conductivity longitudinal to the layers but very low thermal and

CC-composite furnace tube (φ50×800 mm)

graphite radiation insulators shields

crucible

control fiber-optic radiation thermometers radiation thermometer

purge unit reference fiber-optic center zone radiation 200 mm thermometer rear zone front zone 100 mm 100 mm

CC-composite heater elements

Figure 33 Reinforced C/C carbon tube radiative heated by rigid CC sheet heating element for the operation with and without window. In this ﬁgure a hightemperature ﬁxed point is installed in the furnace [42].

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Figure 34 Picture of the interior of the high-temperature furnace manufactured by Chino. The heating element around the supporting furnace tube can clearly be seen.

Figure 35 Inner cavity of a high-temperature blackbody (HTBB) formed by pyrolytic graphite rings.

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electrical conductivity in the perpendicular direction. Pyrolytic graphite has two advantages: (a) its layered structure ensures good electrical heating and homogeneous temperature distribution and (b) the pyrolytic graphite is extremely inert with respect to sublimation, enabling long operation at high temperatures. A design of a high temperature blackbody and furnace utilizing this approach is shown schematically in Figure 36. The pyrolytic graphite rings are clamped together by a spring from the backside and directly heated by an electrical current. This can be seen to the left of the schematic part of Figure 36. The diameter of the radiating cavity is about 37 mm and its length is about 200 mm. Applying an electrical current of about 650 A temperatures of 3,200 K can be attained [79,80] and through a modiﬁed design 3,500 K can be achieved [78]. These blackbodies can also be used without a window when purging with inert argon gas avoiding uncertain window transmission corrections. Due to the large inner diameter of the cavity, this blackbody can be used for radiometry and photometry, however to use it in this mode a detailed radiometric

Figure 36 Cross-section (top) and photograph (bottom) of a high-temperature blackbody made from pyrolytic rings [78].

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characterization is necessary to determine the blackbody cavity emissivity [2]. A different concept for heating is realized with the BB22p blackbody shown in Figure 37 [81284]. From Figure 37 it can be seen that the cavity is designed as a one-sided coaxial radiator, i.e. both electrodes are located on the same side of the housing, the electrical current goes through the inner tube, which forms the cavity, and then through the outer one that is coaxial to the cavity. The front end of the cavity does not contact any water-cooled parts and the outer tube acts as a heat shield, improving the cavity temperature uniformity. At either end of the cavity there are two pyrolytic graphite rings, which separate the cavity from the cooler massive graphite parts of the furnace. The operating temperature range of this cavity is 1,000 up to 3,000 K. It has a length of 130 mm with 22 mm diameter, with an opening of 15 mm in diameter and an effective emissivity of W0.999.

Figure 37 Coaxial design of the one-side based blackbody BB22p. 1, Spring unit; 2, water-cooled copper electrodes; 3, cavity; 4, pyrographite end rings; 5, outer tube.

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5.3. Temperature gradients One main effect inﬂuencing the emissivity of a high temperature cavity is the temperature distribution along the cavity walls [2]. To determine the cavity wall temperature of the blackbody cavity shown in Figure 36 a radiation thermometer was tilted by 51 with respect to the symmetry axis of the cavity and moved with respect to the blackbody keeping the cavity wall in the focal plane of the pyrometer for every measurement point according to Figure 38. Figure 39 shows the measured radiance temperatures of the cavity wall for several cavity temperatures from 1337 up to 3,200 K. All measured temperature distributions show a distinct temperature rise in temperature away from the bottom of the cavity, followed by a temperature drop towards the cavity opening. The qualitative explanation

blackbody cavity

radiation thermometer

Figure 38 Geometry of the temperature measurement of the cavity walls.

(Tcavity wall -Tbottom) / K

25

0

1337 K 1500 K 1800 K 2300 K

-25

2600 K 2800 K 3200 K -50 -150

-100

-50

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50

100

150

distance from cavity bottom / mm

Figure 39 Measured temperature distribution along the cavity walls of the hightemperature blackbody. Negative and positive distances of the bottom correspond to a measurement on the left side (negative values) and right side (positive values) of the cavity walls.

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for this behaviour is that the bottom is most efﬁciently cooled by radiation losses through the aperture as is also the case for the area near the opening. The wall temperatures measured by the radiation thermometer were effective radiance temperatures, which were determined by the temperature of the wall in the ﬁeld-of-view of the radiation thermometer and the reﬂected radiative power contributing from other parts of the cavity. To determine the true wall temperature at each location a ray-tracing Monte Carlo simulation programme based on the theory described in Refs. [2,26] was developed. These calculated temperature distributions were used to determine the overall cavity emissivity of the blackbody, the so-called effective emissivity, at different temperatures. The obtained effective emissivities are presented as a function of wavelength in Figure 40. Taking the temperature of the bottom of the cavity as a reference temperature, the temperature increase near the bottom of the cavity results in effective emissivities higher than one, rapidly increasing for decreasing wavelength. The results presented in Figure 40 clearly indicate that the nonisothermal distribution of the temperature of the cavity walls critically affects the effective emissivity and, with it, the apparent temperature of the blackbody. Typically blackbody cavities are used as reference radiation sources and their temperature is either determined by contact thermometry or by radiometric methods. If the temperature is determined using a contact sensor then an assumption of isothermality is generally made. However for many blackbodies, particularly the ones described in this section, the cavities are non-isothermal, and the achieved radiance temperature is then wavelength dependent. For example, in the case above determining the 1.030 1.025

1337

1.020

1500 1800

1.015 ε

2300 1.010

2600

1.005

2800

1.000

3200

0.995 0

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Figure 40 Calculated effective emissivities including the non-isothermal behaviour of the wall temperature for different blackbody temperatures as a function of wavelength. The temperature of the cavity bottom was taken as reference temperature.

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blackbody temperature of 1337 K by absolute ﬁlter radiometry, at an effective wavelength of 676 nm, results in an error of about 0.8% in spectral radiance when observing the blackbody at a wavelength of 200 nm. Beside the wavelength dependence non-isothermal cavities are also prone to additional uncertainties depending on the ﬁeld-of-view and position of the viewing of the radiation thermometer. Figure 41 shows the resulting error in temperature, when the cavity temperature has been radiometrically obtained at 676 nm and then used another wavelength. The resulting temperature error may be as large as 0.5 K, clearly indicating that a correction with respect to the nonisothermal temperature of the cavity walls is essential for use of the hightemperature blackbody studied here as a high-accuracy radiation source in the visible and UV spectral range. However, it is clear that all hightemperature cavities will suffer from this effect, usually much more than the above described high-quality blackbody, and it is necessary to carefully investigate their temperature uniformity to obtain the most precise results. One word of caution: special care should be taken when the temperature of a non-isothermal blackbody is measured with a multiwavelength ratio radiation thermometer. Such a device only works properly, if the emissivity of the blackbody does not vary with wavelength, which is only the case for an isothermal blackbody cavity.

6. Lamp Sources Tungsten strip lamps have been and are still used as reference sources of thermal radiation and radiance temperature. The main advantage of

(Tλ-Tλo) / K

0.6 0.5

1337

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Figure 41 Resulting temperature correction using the cavity temperature obtained at a wavelength k0 ¼ 676 nm at different wavelengths k.

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tungsten strip lamps is their simplicity [85]. Such lamps only require a stable power supply with electrical currents up to about 30 A, depending on the desired radiation temperature. As the tungsten strip surface is not a blackbody radiator, the emissivity of tungsten has to be considered. The emissivity of tungsten is plotted according to Ref. [86] in Figure 42. Due to the relatively low emissivity of the tungsten only radiation temperatures around 2,600 K, at a wavelength of about 650 nm, can be attained. At such radiance temperatures the tungsten strip already is at a temperature of about 3,000 K. The accuracy of tungsten strip lamps critically depends on the transmission of the window, which has to be cleaned prior to use. The cleaning procedure of the tungsten strip lamps is an important source of uncertainty [87,88]. For reproducible measurements cleaning the silica window with a tissue moistened with pure alcohol and subsequent drying with another tissue is recommended. The effect of cleaning the window has been investigated during the observation of the freezing of the gold ﬁxed point (Figure 43) and large changes in radiance can result as shown in Figure 44. It is therefore recommended that after cleaning the lamp be left for about 20 min before use. This time is required to allow the unavoidable thin layer of water to reabsorb into the lamp window. Due to the low emissivity and the necessary window, reﬂections need to be avoided between the optical instrument used for measuring the spectral radiance and the tungsten strip or the silica window. In order to detect any inter-reﬂections between the lamp and the radiation thermometer, the spectral radiance has to be measured as a function of the vertical and the horizontal angular position of the tungsten strip lamp with respect to 0.5

1200 K 1400 K 1600 K 1800 K 2000 K 2200 K 2400 K

emissivity of tungsten

0.45 0.4 0.35 0.3 0.25 0.2

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Figure 42 Emissivity of tungsten as a function of wavelength for several temperatures [86].

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1.007

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iAu /iC520

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Figure 43 Freezing plateau of the gold freezing point recorded as ratio of the signal of the gold ﬁxed point, iAu, and the tungsten strip lamp No. C520, iC520.

1.0045 1.004

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100 mK 1.0035 1.003 1.0025 1.002 980

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1020

1040 time / min

1060

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Figure 44 Inﬂuence of cleaning the window of the tungsten strip lamp No. C520 with a dry tissue, inducing a signal change corresponding to a change in temperature of about 100 mK.

the optical axis. In Figure 45 the normalized photocurrent obtained for such an investigation is plotted as a function of the angular position of tungsten strip lamp P98. Note that this problem is unique to each tungsten strip lamp and has to be measured on an individual basis. As expected the inter-reﬂection maximum of the photocurrent can clearly be seen at a horizontal angle of 01. For the calibration of such a lamp an orientation has to be used, which lies in a ﬂat region of the photocurrent signal.

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1.001

normalized signal

0.999 0.997 0.995 0.993 0.991 0.989 0.987 7.0

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Figure 45 Normalized photocurrent (at 650 nm) as a function of the angular position of the lamp P98 [2].

Calibrations of variable temperature tungsten strip lamps may be performed by increasing or decreasing the ﬁlament current. As there is evidence that prolonged heating changes the crystal structure of the tungsten ﬁlament [2,16] experiments have been performed operating a tungsten strip lamp with decreasing and increasing electrical current [2,87]. These experiments showed indeed a signiﬁcant difference in obtained radiance temperature at the same ﬁlament currents, when reaching these currents from opposite directions. So care has to be taken to use the lamp exactly the way as it has been calibrated. Additionally, the polarity of the lamps must not be changed, because this will cause large irreversible changes of the emitted radiance [16]. Tungsten strip lamps have a low emissivity and as described above suffer from inter-reﬂections. Quinn in 1967 [89] proposed an ingenious solution to these problems. Fold the tungsten strip into a long tube, and put a bundle of tungsten wires in the middle, this results in a blackbody cavity, having signiﬁcantly higher emissivity compared to the bare tungsten strip surface. Due to mechanical and power requirement limitations only tubes of 1 mm in diameter and about 20 mm long can be realized. Due to the thicker tungsten ﬁlament signiﬁcantly higher currents are required to reach the same temperature of the tungsten ﬁlament. However, due to the signiﬁcantly higher emissivity, which is around 0.95 and to ﬁrst order not dependent on wavelength, the obtained radiance temperature is nearly the same as the real temperature of the tungsten. In Figure 46 a schematic of the ﬁlament of a blackbody lamp is shown.

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45 mm

2 mm Bundle of fine tunsten wire a

Tantalum aperture Ø1 mm

5 mm Tungsten 0.025 mm thick b Tungsten 0.04 mm thick

joints are pressed tight where shown c

Figure 46 and [89].

Schematic of the ﬁlament of a blackbody lamp according to Refs. [16]

The disadvantage of all the above-mentioned lamps is the small radiating area with signiﬁcant variation of the spectral radiance across the emitted surface. Figure 47 shows the relative variation of the signal across the ﬁlament of an Osram WI 17 lamp at a radiance temperature of 2,300 K at 650 nm wavelength. To reduce the signiﬁcant variation of the spectral radiance to about 20% of that across the ﬁlament integrating spheres can be used resulting in a more homogeneous spectral radiance distribution as can be seen in Figure 48. Such integrating sphere sources are used as reference sources for measuring devices with large measuring ﬁeld and are used, e.g. in ﬂuorometry [90]. However this is only required for detectors with a measuring ﬁeld larger than the width of the tungsten ribbon of usual tungsten strip lamps. Such integrating sphere sources have the drawback of lower spectral radiance. Tungsten strip lamps are practical transfer sources for the calibration of radiation thermometers and other radiation sources. Although they are quite easy to operate, they suffer from the variation of emissivity with wavelength and temperature and from sensitivity to cleaning procedures. Due to the progress in developing high-temperature metal carbon eutectics, these novel high-temperature ﬁxed points could well replace the lamp sources for the calibration of radiation thermometers in the temperature range above 1,0001C (this is outlined in more detail in Section 6 of Chapter 2).

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1.15 1.05 0.95 20 15

1

x

mm

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5

-1 0

Figure 47 Variation of the spectral radiance across the ﬁlament of an OSRAM WI 17/G tungsten strip lamp [34].

Figure 48 Spectral radiance distribution of an integrating sphere source [34].

7. Summary Sources of thermal radiation are required to investigate and calibrate radiation thermometric devices for temperatures from about 1701C up to

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over 3,0001C. To cover this large temperature range a variety of types and concepts of thermal radiation sources is used. The most sophisticated one 2 the concept of blackbody radiators 2 has already been introduced in the late 19th century and its development is described in the historical overview of thermal radiation at the beginning of this chapter. Starting from this very beginning a wide survey from the most sophisticate blackbody radiation sources used in top-of-the-art laboratories of the National Metrology laboratories to the ﬂat plate radiators and lamps used in industrial level calibrations laboratories is given in this chapter. Also provided is an outline of the important features and how to characterize such thermal radiation sources. The chapter is completed with a description of the methods of calculating the emissivity of blackbody sources and the inﬂuence of the index of refraction on the thermal radiation emitted by blackbodies.

ACKNOWLEDGEMENTS J. Hollandt and J. Hartmann wish to acknowledge fruitful discussion with K. Anhalt, B. Gutschwager and D. Taubert. The authors thank Elzbieta Kosubek and Beate PruXeit for their valuable support in producing the manuscript.

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CHAPTER 7

Laser Optical and Photothermal Thermometry of Solids and Thin Films Yue Liu1,2 and Andreas Mandelis1

Contents 1. Introduction 2. Interferometric Thermometry 2.1. Physical principle 2.2. Applications 3. Ellipsometric Thermometry 3.1. Theoretical and experimental principles of ellipsometry 3.2. Applications 4. Photothermal Radiometric Thermometry 4.1. Principle of temperature measurement by photothermal radiometry 4.2. Applications 5. Thermoreﬂectance Thermometry 5.1. Physical principle 5.2. Instrumentation and applications 6. Summary References

297 299 299 302 304 305 308 311 313 314 323 323 324 331 331

1. Introduction Laser optical and photothermal thermometry discussed in this chapter is distinct from the conventional radiation thermometry introduced in previous chapters. In this thermometric category, the laser is used either as a probe or as an extra heat source to obtain temperature information instead

1

2

Center for Advanced Diffusion-Wave Technologies (CADIFT), Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, ON, Canada M5S3G8 School of Mechanical Engineering, Dalian University of Technology, Dalian, Liaoning 116024, China

Experimental Methods in the Physical Sciences, Volume 42 ISSN 1079-4042, DOI 10.1016/S1079-4042(09)04207-6

r 2010 Elsevier Inc. All rights reserved.

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of directly measuring the emitted radiation from a sample to calculate the temperature. Laser optical and photothermal thermometry of solids is a new branch in thermometry, which was spurred by the increasing demand for quality control in microtechnology during the past decade. Many optical thermometry methods [1,2] have been developed that take advantage of various optical phenomena. Research into engineering applications of optical thermometry have been shown to encompass the processing of metals, semiconductors, and insulators involved in the production of integrated circuits, plasma processes, laser and electron beam treatments, and ﬁlm growth. Each optical thermometric method described in this chapter involves the remote measurement of temperature-dependent optical parameters of a material, followed by the determination of the temperature from the value of the known temperature-dependent parameter(s). Temperature-dependent parameters of solids, e.g., can be the refractive index, the electronic bandgap, excited-state populations, optical phonon energy, and so on. To determine the temperature, the solid is exposed to an optical beam (usually a laser beam) that is used to measure one or more of the following parameters: the reﬂection and/or transmission coefﬁcients at a ﬁxed wavelength, the reﬂection and transmission spectra, the polarization states of the reﬂected light, and the diffraction angle or a dynamic radiation feature with respect to small temperature ﬂuctuations induced by pulsed or modulated irradiation onto the surface. Laser optical thermometry techniques developed for solids mainly include: interference thermometry, ellipsometry, photothermal radiometric thermometry, and thermoreﬂectance thermometry. These techniques will be discussed in this chapter. As a new direction in thermometry, optical thermometry has been developed with the purpose of overcoming the difﬁculties attributed to traditional measurement techniques under speciﬁc conditions. Most notably, optical beams are insensitive to electrical interference, vibration, deformation, or chemical activity in a medium while having certain characteristic features such as wavelength, polarization, propagation direction, and intensity modulation to allow the reﬂected beam to be distinguished from optical interference and background noise. Laser thermometry is a noncontact technique capable of determining temperature where it is difﬁcult or impossible to perform contact thermometry. In addition surface temperature distributions can be measured by beam scanning. The technique can be applied over a wide range of temperatures: the lower limits are cryogenic and upper limits can be as high as the material’s melting point. Although many laser thermometry methods have distinct advantages and have been developed and are quite widely used in engineering research, there still exist disadvantages that limit their widespread use. Most

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notably, laser thermometry methods are narrowly specialized in application in contrast to radiation thermometry methods which have widespread applicability. Many laser methods provide measurements for a restricted set of materials, and sometimes there are additional speciﬁcations for the specimen shape and surface properties. For example, laser interference thermometry applied to semiconductors and insulators requires the sample to be in the geometry of a plane-parallel plate, transparent to the probe beam, and having a reasonably smooth surface. Laser thermometry methods also usually require a complicated optical system. Additionally, the determination of temperature from laser thermometry often requires prior knowledge of physical properties of the sample (e.g., thermal expansion coefﬁcient, thermoreﬂectance coefﬁcient, and refractive index) including their dependence on temperature. Usually the temperature dependence of these parameters is not well known and thus requires that the material of interest be studied or calibrated prior to the implementation of any temperature measurement. For this reason the use of laser thermometry in industrial process monitoring and control is quite limited. A summary of the advantages and limitations of laser thermometry compared with traditional thermometry of solids is shown in Table 1 [1].

2. Interferometric Thermometry The interferometric thermometry technique is one of several major laser thermometry methods. First suggested by Hacman in 1968 [3], this technique was used to measure the temperature of a glass substrate where the thermal expansion is the dominant effect. This noncontact optical technique utilizes interferometry to determine temperature changes from the thermal expansion and refractive index changes of a transparent substrate. The transparent substrate has a known thickness and the front and back faces are polished and are approximately parallel. Temperature changes are measured by counting oscillations (fringes) in the reﬂectance signal.

2.1. Physical principle When an incident laser beam is directed onto a transparent plate with two parallel plane surfaces, interference due to multiple reﬂections in the transparent plates appears and can be observed as light fringes as shown in Figure 1. Figure 2 shows the schematic refraction and reﬂection of light at a three-layer sample, where y1, y2, and y3 represent incident angle in medium 1 and refracted angles in media 2 and 3, respectively. ni and

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Advantages and limitations of solid surface thermometry methods [1].

Method

Advantages

Contact thermometry

Simple signal generation Measurements can be made on any surface

Drawbacks

Radiation thermometry

Universal method Noncontact measurement Wide temperature range Temperature patterns can be recorded

Laser thermometry

High noise immunity Noncontact measurement Wide temperature range High sensitivity High spatial resolution Temperature patterns can be recorded The response time of certain methods is shorter than 1 ps

Some signals are affected by electrical interference Difﬁcult to establish reliable thermal contact Effects of detector on temperature in contact region Difﬁculties to apply to moving objects Measurement delay Restricted working life in corrosive media Lack of data on emissivity for many materials in the sensitive spectral ranges for IR detectors Inﬂuenced by external radiation Sensitive to surface details Cannot be used at low temperatures Narrow specialization for each method (i.e., often application speciﬁc) Sensitive to surface properties Complicated optical schemes in some methods Calibration for each new material is needed in most methods

L represent refractive indices of the three media and the thickness of medium 2, respectively. For a transparent plate, media 1 and 3 can be air. The difference between the optical paths of the reﬂected beams from top and bottom is d ¼ 2n2 L cos y2 . When the temperature changes, the optical

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Laser

301

Detecting Spot

Transparent Plate sample

Figure 1

Laser interference and the resulting light fringes.

Medium 1

θ1

n1 Medium 2 n2 Medium 3 n3

Figure 2

L

θ2 θ3

Schematic of refraction and reﬂection of light at a ﬁnitely thick sample.

path changes, which causes the interference fringes to shift. The variation of optical path d with respect to the temperature T is given by dd ¼ dT

dL dn2 cos y2 ¼ ð2n2 La þ 2LbÞ cos y2 ¼ 2Lg cos y2 2n2 þ 2L dT dT

(1)

where a is the linear thermal expansion coefﬁcient of the plate, b the refractive index gradient with respect to temperature, and g a total coefﬁcient of combining both a and b: g ¼ n2a+b. Both a and b can be obtained from a handbook or through calibration. When the incident angle and temperature measurement range are relatively small, the variation of the angle of refraction caused by temperature changes can be neglected. Therefore the change in the optical length difference between reﬂected

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beams shown in Figure 2 with respect to temperature variation DT is given by Dd ¼ 2Lg cos y2 DT

(2)

For each increase Dd by one wavelength l, one fringe passes through the center of the light spot. If a total number of N fringes pass through the center during a particular heating process, the temperature rise DT can be calculated as DT ¼ NM

(3)

where M¼

l l ¼ 2Lg cos yt 2Lg½1 ðsin yi =nÞ2 1=2

(3a)

The substrate temperature is determined by adding the initial temperature T0 to the temperature change DT. Therefore, the entire history of the reﬂectance signal must be obtained during heating or cooling, and the initial direction of temperature change must be known. If the measurement is interrupted and fringes are lost, the absolute temperature can only be recovered by bringing the wafer to a known temperature. For example, to measure the temperature of a 5-mm thick glass plate made of common optical glass using a He2Ne laser (l ¼ 632.8 nm), the calibrated value of M ¼ 5.5 (K) is shown in Equation (3a) in the temperature range of 300 K to 350 K. Thus, if the initial temperature T0 is 300 K, the number of shifted interference fringes caused by temperature change (increase) is N, then the temperature can be calculated from T ¼ (300+5.5N) K. From Equation (3), we can also see that the measurement resolution of this method depends on the resolutions of N and M. N can be obtained by counting fringe numbers, which are integers or tracking interference phase changes (fraction of a fringe), which can be more precise being noninteger and hence yielding higher resolution. M will be a constant once sample thickness, material, and incident angle are determined. The spatial resolution depends on the size of incident beam.

2.2. Applications The technique of interferometric thermometry can only be used when substrate materials are transparent with respect to the probe beam. Materials which have been investigated with this technique include a number of optically transparent dielectrics [326], where a He2Ne laser emitting at 632.8 nm was used and thermal expansion was the dominant effect. IR lasers at 1.15 [729] and 1.5 mm [10,11] are widely used in temperature monitoring of semiconductor processes (Si, GaAs, InP, InGaAsP), such as

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plasma etching, ﬁlm deposition, and annealing. For dielectric material temperature measurements, thermal expansion dominates the optical path variation. However, for semiconductor interferometric temperature measurements, both the thermal expansion effect and the temperature dependence of the refractive index, contribute to the interference phenomena. The temperature dependence of refractive index is caused by the temperature dependence of the energy gap Eg, to which the refractive index is directly related. Therefore, both thermal expansion and refractive index must be taken into account when temperature is determined by interferometric thermometry. Usually, the values of the thermal expansion coefﬁcient are about an order of magnitude smaller than the temperature coefﬁcient of the refractive index as shown in Table 2 [7211]. Thus, interferometry is about a factor of 10 more sensitive to a change in the refractive index than to a change in thickness. In principle, the temperature could be derived from Equations (1)2(3). However, the linear approximation to the temperature dependence of the refractive index n and a small value thermal expansion coefﬁcient leads to an unacceptably large uncertainty in temperature determination over a large temperature variation DT. Consequently, either more accurate expressions for n and a must be used or a priori calibration with respect to another thermometry technique, such as use of a calibrated thermocouple, must be performed. The similar technique of laser interferometry (using two beams: reference beam and object beam) has also been widely used for mass and heat transfer investigations. Various interferometric techniques, such as holographic interferometry [12215], shearing interferometry [16], and phase shift interferometry [17] have been used to investigate the temperature ﬁeld around a single vapor bubble. In these cases, the method of conﬁguration and data processing are different with the technique for solid surface we mainly introduced. Details of these techniques are not discussed in this section. The phase difference between the reference beam Table 2

Properties of semiconductors [7211].

Material

Thermal expansion coefﬁcient (106 K1)

Temperature coefﬁcient of refractive index (105 K1) at 1.15 mm

Temperature coefﬁcient of refractive index (105 K1) at 1.53 mm

Temperature coefﬁcient of refractive index (105 K1) at 3.39 mm

Si Ge GaAs CdTe In P

5.5 2.44 7.86 10 4.5

3.9B4.7 6.9 5.1B5.7

5.15

2.7

5.95

7 3.4 10 10 2.7

6.65

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Yue Liu and Andreas Mandelis

and object beam can be determined from the interferogram in different ways: counting the fringes and interpolating between maxima and minima, using Fourier transformation, Hilbert transformation, or in case of phase shift interferometry by combining at least three images. As a noninvasive temperature measurement technique, interferometric thermometry has many advantages over thermocouple and conventional pyrometer measurements. It is remote, therefore, minimally disturbing the processing environment and is insensitive to harsh environments. It provides a direct measurement of substrate temperature unlike a thermocouple which may not be in thermal equilibrium with the sample. Another advantage is that it measures wide temperature ranges, i.e., in principle from cryogenic to relatively high temperatures, limited only by the thermal and chemical stability of the substrate and the requirement that the material remain transparent at the probe wavelength l. Since only relative intensities are measured, any changes of window transmission do not affect the temperature measurement results. This technique can provide high spatial resolution and temperature mapping with the use of scanning equipment or an expanded beam and large detector arrays. Usually, this technique can only be used to measure the temperature of specimens which has both sides well polished. However, this is not a serious limitation in many applications. Under certain conditions, the technique can also be extended to standard semiconductor wafers polished to an optical ﬁnish on one side [18]. Although the technique has been widely used in temperature monitoring for semiconductor processes, the requirements of high-quality surface conditions (two parallel polished surfaces) and transparency of the tested material with respect to the probe beam limit its usage to nonabsorbing media.

3. Ellipsometric Thermometry Ellipsometry is a versatile and powerful optical technique for the investigation of the dielectric properties (complex refractive index or dielectric function) of thin ﬁlms. The name ‘‘ellipsometry’’ stems from the fact that the most general state of light polarization is elliptic. The technique has been known for almost a century. A surge of development of this technique appeared in the 1960s and 1970s with the advent of the invention and development of lasers. Today, it has many applications in different ﬁelds, from semiconductor physics to microelectronics and biology, from basic research to industrial applications. Ellipsometry is a very sensitive measurement technique and provides unrivalled capabilities for thin-ﬁlm metrology. Ellipsometry is also becoming more interesting to researchers in other disciplines such as biology and medicine [19221].

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As an optical technique, ellipsometry measures the change in the polarization state of light following reﬂection from a surface. Singlewavelength ellipsometry gives two measurable quantities: C, the change in amplitude ratio, and D, the change in phase difference between the parallel and perpendicular polarization components of the surface-reﬂected beam of an incident polarized plane wave. Ellipsometry can yield information about layers that are thinner than the wavelength of the probing light, even down to a single atomic layer. Ellipsometry can probe the complex refractive index or the dielectric function tensor, which gives access to fundamental physical parameters and is related to a variety of sample properties, including morphology, crystal quality, chemical composition, or electrical conductivity. It is commonly used with excellent accuracy to characterize ﬁlm thickness for single layers or complex multilayer stacks ranging from a few angstroms to several micrometers. This technique is most commonly used for thin ﬁlm optical property measurements. The measured values of C and D are sensitive to the optical properties of the surface layer and of the substrate. The relation between C and D and the optical properties of the single ﬁlm/substrate system can be expressed in terms of the complex reﬂection coefﬁcient, r: tan C expðiDÞ ¼ rðn1 ; n2 ; n3 ; L; y1 ; lÞ

(4)

where n1, n2, n3 are the complex refractive indices of the ambient, ﬁlm, and substrate, respectively. L is the thickness of the ﬁlm, y1 the angle between the incident beam and the normal to the sample surface, and l the wavelength of the incident light. Utilizing the Fresnel equations and the knowledge of the optical properties of the sample, single-wavelength ellipsometry can be used to measure the thickness and refractive index of the coating ﬁlms grown on substrates or to measure the properties of the substrate itself. Repeating the same measurements with different substrate temperatures, one can obtain the temperature dependence of the real and imaginary parts of the refractive index of the sample. Once the temperature dependence of the refractive index is known, an inverse process can be performed, leading to the measurement of the refractive index dependence on temperature, thus deducing the temperature. Ellipsometry was ﬁrst used by Ibrahim and Bashara for temperature measurement of oxide-coated silicon substrates in the early 1970s [22,23]. Most reports of temperature measurements by ellipsometry were reported in the late 1980s to the mid-1990s [24227].

3.1. Theoretical and experimental principles of ellipsometry When a plane wave impinges onto a boundary between two homogeneous media of different optical properties, it is split into two waves: a transmitted wave entering the second medium and a reﬂected wave propagating back

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into the original medium. As is well known, the incident wave can be considered as the superposition of two polarization components parallel (denoted by p) and perpendicular (denoted by s) to the plane deﬁned by the incident and reﬂected beams. When the light is directed obliquely onto a coated substrate, reﬂection and refraction occur at both interfaces of air2ﬁlm and ﬁlm2substrate. The Fresnel reﬂection coefﬁcients at each of the interfaces for each polarization are given in Chapter 3. Subscripts 1, 2, and 3 refer to air, thin layer, and substrate, respectively; y1 is the incident angle of the light with respect to the normal to the interface; and y2, y3 are refraction angles of the light at the ﬁlm and substrate interfaces, respectively, shown in Figure 2. The two interfaces yield a combined reﬂection coefﬁcient which can be obtained using the Fabry2Perot equations [28]: Rp ¼

r 1p þ r 2p ei2d 1 þ r 1p r 2p ei2d

(5)

Rs ¼

r 1s þ r 2s ei2d 1 þ r 1s r 2s ei2d

(6)

4pdn2 cos y2 l

(7)

with 2d ¼

Combining the above equations yields the expression for the reﬂection coefﬁcient of the coated ﬁlm for each polarization. The ratio of the two reﬂection coefﬁcients can be split into amplitude and phase, thereby deﬁning the ellipsometric parameters C and D: Rp tan Ce ¼ ¼ Rs iD

r 1p þ r 2p ei2d 1 þ r 1s r 2s ei2d 1 þ r 1p r 2p ei2d r 1s þ r 2s ei2d

(8)

From Equation (8), we can see that both Cand D are functions of n1, n2, n3, d, l, and y1 (y2 and y3 can be expressed in terms of y1). Under typical measurement conditions, the wavelength l, incident angle y1, and substrate refractive index n3 are usually known and are constant. Therefore, C and D can be considered functions of ﬁlm thickness d and refractive index n2. If one can measure C and D experimentally, one can derive n2 and d. This is the principle of ellipsometric measurements. In order to measure C and D, more details must be discussed. Equation (8) can also be expressed as: G ¼ tan CeiD ¼

Rp jE rp jjEis j i½ðfrp frs Þðfip fis Þ e ¼ Rs jE ip jjErs j

(9)

Here E represents the magnitude of the electric ﬁeld vector of the optical radiation, subscripts r and i denote incident and reﬂected beam,

Laser Optical and Photothermal Thermometry of Solids and Thin Films

307

respectively, and f is the phase of each polarization component with respect to the interface. Comparing the right-hand side of Equation (9) with the left-hand side, one obtains tan C ¼

jRp j jErp jjEis j ¼ jRs j jE ip jjErs j

D ¼ ðfrp frs Þ ðfip fis Þ

(10)

(11)

From Equations (10) and (11) we see that C is related to amplitudes of incident and reﬂected beams of both polarizations, while D is related to the phase changes of incident and reﬂected beams of both polarizations. In order to simplify the experiment and measure C and D directly, two requirements must be satisﬁed. (1) The amplitudes of both p and s polarizations are equal, i.e., |Eip|/|Eis| ¼ 1. Then Equation (10) is simpliﬁed tan C ¼

jE rp j jE rs j

(12)

(2) The reﬂected beam must be linearly polarized, i.e., yrp yrs ¼ 0 or p. Equation (11) is then simpliﬁed: D ¼ ðyip yis Þ or

D ¼ p ðyip yis Þ

(13)

In practice, it is not difﬁcult to satisfy these two requirements experimentally by choosing an appropriate light source, introducing a polarizer and a quarter-wave plate on the optical path of the incident beam, and a polarizer (usually called ‘‘analyzer’’) in the path of the reﬂected beam. A schematic drawing of an ellipsometry setup is shown in Figure 3. It consists of a laser (usually a 632.8 nm He2Ne laser), a polarizer, and a quarter-wave plate that provide a state of polarization, which can be varied from linearly, to elliptically, to circularly polarized light by varying the angle of the polarizer. The beam is reﬂected off the sample of interest and then passed through the analyzer. The operator adjusts the angles of polarizer and analyzer until a signal minimum is attained. This minimum signal is detected if the light reﬂected by the sample is linearly polarized, while the analyzer is set so that only light with polarization perpendicular to the incoming polarization is allowed to pass. The angle of the analyzer is therefore related to the direction of polarization of the reﬂected light, if the null condition is satisﬁed. In order to obtain linearly polarized light after reﬂection, the polarizer must provide an optical retardation between the two incoming polarizations which exactly compensate for the optical retardation caused by the polarization-dependent reﬂections at each dielectric interface. Since the amplitudes of both polarizations can be set

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Yue Liu and Andreas Mandelis

Detector

Laser Polarizer Analyzer λ/4 plate

Film Sample Substrate

Figure 3

Schematic of a typical ellipsometric setup.

equal, the ratio of the amplitudes after reﬂection equals the tangent of the angle of the analyzer with respect to the normal. Although in principle one can calculate n2 and d from C and D if other parameters are known, it is hard to directly derive the relationship between (n2, d) and (C, D). With the development of computer and numerical simulation techniques, one can obtain n2 and d much easier and faster numerically than analytically, thus extending the applications of ellipsometry signiﬁcantly. In summary, ellipsometry is an effective method for refractive index and ﬁlm thickness measurements. However, in order to perform temperature measurements using this method, we must know the relationship between temperature and refractive index and two measured ellipsometric values. Repeating the same measurements with different substrate temperatures, one can obtain the temperature dependence of the real and imaginary part of the refractive index of the ﬁlm. Once the temperature dependence of the refractive index is known, the temperature itself can be obtained through an inverse process.

3.2. Applications Ellipsometric thermometry has been applied to measuring temperatures in semiconductor processes such as Si plasma etching and rapid thermal oxidation (RTO). Kroesen et al. [25] introduced an ellipsometric technique to in situ temperature measurements of Si wafers with or without oxide during a plasma deposition process. The experiments concerning oxide ﬁlms were performed with ﬁlm thickness values that place the measurement angle D around 0 or 3601. This occurs at C values near 701 using a He2Ne

309

Laser Optical and Photothermal Thermometry of Solids and Thin Films

0 on

−400 100

(a)

plasma off 4.5

(b)

80

4.0 Psi (°)

Temperature (°C)

VDC (V)

laser, a SiO2 refractive index 1.52, and incident angle 74.141. In this region, C hardly changes with ﬁlm thickness, whereas D is very sensitive: onedegree increase in D corresponds to a thickness change of roughly 0.2 nm. The angle C is sensitive to the refractive index of bulk Si, it is therefore convenient to perform measurements of the changes of refractive index and of ﬁlm thickness independently, a feature which leads to temperature measurement. Figure 4 shows the evolution with time of several parameters while the plasma is switched off. Figure 4(a) shows the dc self-bias of the electrode as a function of time. Figure 4(b) shows the time evolution of C and of substrate temperature. Full oxide etch-off occurs at time te, after which the Si substrate is exposed and the value of C becomes a strong function of the real part of the refractive index of silicon. This is demonstrated in Figure 4(c). The measurement values of C and D are converted into ﬁlm thickness values in the case of measurements on oxide

60 Psi

3.5

40 T 100

1.0

0.1

10

0

400

800

Psi (°)

Temperature (°C)

(c)

1200

Time (s)

Figure 4 (a) The dc self-bias of the electrode as a function of time. This signal is recoded in order to monitor exactly when the plasma is switched on and off. (b) The substrate temperature as measured by a Luxtron TM ﬂuoroptic probe and the ellipsometric angle W as a function of time. At time te, the oxide etch end point is reached. (c) Logarithmic plots of the temperature data displayed in (b) and of the difference CðtÞ Cð1Þ. (From Ref. [25] with permission).

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ﬁlms, and into refractive indices in the case of a bare silicon substrate. Both ﬁlm thickness and Si refractive index are plotted versus temperature which is measured simultaneously with a ﬂuoroptic probe shown in Figure 5. From Figure 5, one can extract linear thermal expansion coefﬁcient of Si oxide: (1/l)(dl/dT) ¼ 9.9 106 K1, and temperature derivative of silicon refractive index at 632.8 nm: (dn/dT) ¼ 4.52 104 K1, which is in excellent agreement with literature values [29]. Using these measured temperature coefﬁcients, noncontact temperature measurements can be performed in this situation using in situ ellipsometry. As discussed in Section 1, ellipsometry can measure temperature and ﬁlm thickness simultaneously. Therefore, it can be used to monitor the temperature and coating thickness during ﬁlm growth on a substrate. The utility of single-wavelength ellipsometry in the monitoring and control of RTO of crystalline silicon in atmospheric pressure has been studied [26]. As a basis for ellipsometric measurements, knowledge of the temperature dependence of the optical properties of materials under investigation

0.8 Thickness Change (nm)

(a)

SiO2

0.6 1223

951

0.4 679 0.2

407

0.0 (b)

Si

Re(n)

3.90

3.89

3.88 20

40

60

80

100

Temperature (°C)

Figure 5 (a) The thickness change of SiO2 ﬁlms on silicon plotted versus the temperature change. The four lines represent four different ﬁlm thickness values: 407, 679, 951, and 1,223 nm. (b) The refractive index of silicon as a function of substrate temperature. (From Ref. [25] with permission).

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Laser Optical and Photothermal Thermometry of Solids and Thin Films

(e.g., silicon substrate and oxide ﬁlm) is essential. A least-squares ﬁt of the form nSi ¼

X

ai T i

(14)

bi T i

(15)

i

kSi ¼

X i

to the well-known temperature-dependent data of Si has been performed [30]. Here T is the wafer absolute temperature. The ﬁt yielded the coefﬁcients ai and bi for the real (nSi) and the imaginary (kSi) parts shown in Table 3. The temperature dependence of the refractive index of the oxide was found to be [30] nox ðT Þ ¼ 1:46474 þ 1:22142 105 T

(16)

Using the foregoing data [29], the change of the ellipsometric parameters C and D with temperature can be calculated for various oxide thicknesses. A least-squares ﬁt to these data yields polynomial expressions for temperature and oxide thickness as a function of C and D. A single ellipsometric measurement (a pair of C and D values) thus provides simultaneous measurements of temperature and oxide ﬁlm thickness. The ability to measure the temperature and ﬁlm thickness allows active control of the ﬁlm growth during oxidation in a RTO system such as the one shown in Figure 6. A typical experimental result of Si oxidation is shown in Figure 7. The process starts at point A corresponding to C and D values for the native oxide at room temperature. The labels show the ﬂow of the oxidation process indicated by increasing temperature (A2B), ﬁlm growth (B2C), and decreasing temperature (C2D) after the expected thickness has been reached. In summary, ellipsometric thermometry is a powerful optical temperature measurement method. It can measure both temperature and Table 3 Coefﬁcients for the least-squares ﬁt to the temperature-dependent data of van der Meulen and Hien [30] for nSi (ai ) and kSi (bi ), Equations (14) and (15). I

ai

bi

0 1 2 3 4 5 6

3.8602 0.000413765 8.11133E-8 1.3766E-10

0.0196281 5.13093E-5 3.7556E-11 1.366666E-11 6.03972E-13 1.00054E-15 4.95804E-19

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Yue Liu and Andreas Mandelis

Tungsten Halogen Lamp Heater Quartz Window Substrate Rotating Analyzer

Compensator

70°

Light Source

Shutter PTM Polarizer

A/D

Computer a2, b2 Ψ, Δ T, L

Figure 6 Schematic diagram of RTO experimental setup showing vacuum chamber, ellipsometer, and tungsten-halogen lamp heater. The arrows indicate the computer control of shutter and active feedback control of the lamp power based on the ellipsometry measurements. (From Ref. [26] with permission).

Δ (degrees)

λ = 6328 Å ϕ = 70°

LOX = 0 Å

180 A

170

B

T = 0°C

T = 1100°C

160 D C

TSET = 950°C

150

LSET = 100 Å 140

8

9

10

LOX = 120 Å 11

12

13

14

15

16

17

Ψ (degrees)

Figure 7 Process ﬂow diagram of measured ellipsometric angles W and D. The solid lines are the results of calculations performed using the temperature dependence data [29]. The horizontal lines are lines of constant temperature placed every 2201C and vertical lines are lines of constant thickness placed every 20 Å. (From Ref. [26] with permission).

Laser Optical and Photothermal Thermometry of Solids and Thin Films

313

ﬁlm thickness during a ﬁlm growth process. It has been used for the simultaneous measurement of temperature and ﬁlm thickness in the fast oxidation process of Si. However, the need of a priori knowledge of the temperature dependence of material optical properties limits the usage of this technique for temperature measuring applications.

4. Photothermal Radiometric Thermometry Conventional IR pyrometry techniques have been widely employed for temperature measurement in various industrial processes [31] as discussed extensively in this book. However, there are two factors critically limiting the accuracy of IR pyrometry. One is the need to know the emissivity of the sample, and made more complicated in cases in which emissivity changes during an industrial process. The other is the effect of reﬂected radiation both from hot ambient conditions and/or other hot objects near the surface being measured, for instance, a furnace or a plasma chamber. The former issue has been addressed extensively in the past although there still remain some challenges. Details are discussed in other sections of this book. To minimize the contributions of ambient radiation, a photothermal radiometry (PTR) technique introduced to measure temperature [32237]. In this technique, one modulated or pulsed laser is used to generate a temperature ﬂuctuation. Through measurements of the ﬂuctuating radiation at two different wavelengths and knowledge of the relationship between the emissivities at the two measured wavelengths (including the simplifying assumption that the emissivities are equal), one can derive the temperature. This technique will be discussed in this section. Another major branch of laser PTR, laser absorption radiation thermometry, also called emissivityfree active radiation thermometry, was ﬁrst proposed by Dewitt and Kunz [38]. These methods use more than one laser and measure emissive radiation in no less than two wavelengths in order to derive the temperature without the assumption of equal emissivities at two wavelengths. The details of these techniques are discussed in Chapter 4 in Vol. 43 of this series. PTR is widely used as a means of measuring thermophysical properties such as diffusivity, and also as a tool for characterizing surfaces and detecting subsurface defects and contact resistances. The basic idea is to create a small perturbation of the sample temperature using an external heating laser source. This small temperature perturbation is superposed on the original sample surface temperature and creates a corresponding change in the thermal radiation signal from the sample. Instead of measuring the direct thermal emission from the sample, the perturbed thermal radiation signal, which is not affected by the ambient radiation under appropriate conditions, is detected and used to determine the surface temperature.

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4.1. Principle of temperature measurement by photothermal radiometry The signal detected by an infrared sensor imaging a surface is the sum of radiation directly emitted from the sample surface and reﬂected by the sample from ambient radiation: S l ¼ kl tl ½l I bl ðT 0 Þ þ I al

(17)

where kl is a system constant depending on wavelength (l), detector spectral bandwidth (Dl), and geometry of the detection system; tl the transmissivity of the system; el the emissivity of the surface; I bl ðT 0 Þ the intensity of the blackbody emission; and Ial the intensity of the reﬂected ambient radiation. From Equation (17) it is seen that both emitted radiation and reﬂected ambient radiation contribute to the dc component of the signal that is delivered to an infrared detector. If a modulated or a pulsed laser beam impinges on the detected surface, the surface temperature becomes T ðr; t Þ ¼ T 0 þ DT ðr; t Þ

(18)

Furthermore, if DT ðr; tÞ is much smaller than T0, we may expand the blackbody radiation intensity in a Taylor series: I bl ½T ðr; tÞ ¼ I bl ðT 0 Þ þ

@I bl DT ðr; tÞ @T T ¼T 0

(19)

From Equation (19) it is evident that the dynamic component of the signal is proportional to ð@I bl =@T Þ, which becomes the signal based on the photothermal effect [32]. The dc component can be easily discarded and the ac component easily detected by dynamic instruments such as a lock-in ampliﬁers, boxcar integrators, oscilloscopes, etc. The temperature variation is proportional to the absorbed power per unit surface area. When one increases the laser power for a given beam size, DT ðr; tÞ increases. If DT ðr; tÞ is too large, the expansion given in Equation (18) is not sufﬁciently accurate and higher order terms have to be included. In practice, in order to avoid the complexity of nonlinear temperature dependencies, it is necessary to work with only a small increase in surface temperature. This is always possible either by reducing the incident power or by increasing the illuminated area.

4.2. Applications As mentioned above, small temperature changes can be obtained by illuminating a surface with either a modulated CW [32235] or a pulsed laser [36,37]. Therefore, there are various ways of generating photothermal signals and deducing the temperature from features of the responses, such as

Laser Optical and Photothermal Thermometry of Solids and Thin Films

315

decay time of the transient or amplitude of the modulated signal. In this section, details of photothermal thermometry, including signal detection and temperature measurement, will be presented. In order to obtain a modulated photothermal signal, usually a chopped laser beam is impinged on the surface of the tested sample to periodically modulate the surface temperature. A typical experimental setup with modulated PTR system for temperature measurement is shown in Figure 8. In the case of low laser power, the photothermal signal Sl captured by the detector and demodulated by the lock-in ampliﬁer depends on three unknown parameters: surface temperature (T0), emissivity (el), and the dynamic (harmonic or transient) component of the surface temperature (DT) induced by the laser beam. Theoretically, DT can be derived from considerations in terms of laser power, laser beam size and proﬁle, and thermophysical properties of the interrogated material. In practice, it is hard to know the above-mentioned parameters accurately, so as to determine DT with sufﬁcient accuracy. In order to obtain an accurate temperature, DT must be removed from the expression of the temperature (T0). For this purpose, only the region where the signals are proportional to the absorbed power at low level will be considered. The signal can be written in the form Z @I bl S l ðtÞ ¼ kl tl l DT ðr; tÞdS @T T ¼T 0 DS

(20)

dS is the element of the irradiated area. It is important to point out that the signal comes exclusively from this irradiated region. Therefore, the spatial resolution of the measurement must be controlled with great accuracy. To eliminate DT, two measurements at two different wavelengths, l1 and l2, MICROSCOPE OBJECTIVE

WIRE

LENS

FILTER INSB DETECTOR

ARGON LASER

CHOPPER POWER SUPPLY THERMOCOUPLE THERMOMETER

COMPUTER

LOCK-IN AMPLIFIER

Figure 8 A typical modulated photothermal radiometry system for temperature measurement. (From Ref. [34] with permission).

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Yue Liu and Andreas Mandelis

are performed, and the ratio of two signals is taken. This procedure also allows all the geometrical terms of Equation (20) to be eliminated. The ratio is independent of the dynamic surface temperature, but does depend on the value of emissivity at wavelengths l1 and l2. In many cases, Wien’s law will provide a very good approximation. The surface temperature can then be written in closed form: T0 ¼

c 2 ðð1=l2 Þ ð1=l1 ÞÞ lnððS l1 =S l2 Þðl2 =l1 Þðkl2 tl2 =kl1 tl1 Þðl61 =l62 ÞÞ

(21)

where the ratio kl2 tl2 =kl1 tl1 is a system constant that can be calibrated by using a blackbody emitter. c2 the second radiation constant. Equation (21) is applicable to both modulated and pulsed PTR methods. At this stage, we have to eliminate the unknown ratio l1 =l2 . It must be pointed out at this stage we are facing the classical problem of the unknown emissivity. In order to eliminate the unknown ratio, we either make the rarely valid assumption that l1 ¼ l2 or, preferably measure the ratio. Now, let us turn to the problem of wavelength selection. In order to choose appropriate wavelengths, several factors must be considered. Studies [31,32] show that the sensitivity of the signal as a function of temperature increases when the difference between the two wavelengths increases. Therefore, if two wavelengths are chosen to be very close, they will result in a very weak dependence of the measured quantity on the temperature. It is known that @I bl =@T is an increasing function of temperature for given l [31]. It turns out that highest sensitivity is achieved by choosing one wavelength with a very high or very low value of @I bl =@T (where the dependence on T is very weak), and the other wavelength in a region where the dependence of @I bl =@T on T is maximum. Obviously, the accuracy of the measurement requires an acceptable signal-to-noise ratio (SNR). Thus, one must seek a wavelength in the vicinity of the maximum of the ﬁrst derivative of Planck’s function with respect to T. In order to estimate the error due to the graybody assumption, one might choose both wavelengths close to each other. A ﬁnal requirement must be taken into account, namely the absorption bands of air in the IR spectrum. Metallic ﬁne wire and semiconductor temperature measurements have been studied with the modulated photothermal technique. Figure 9 shows a typical PTR temperature result from a 127-mm nickel2aluminum wire at two different wavelengths. Equation (21) is used to calculate the theoretical temperature, which is shown in Figure 9(b) as a solid line. The major source of the discrepancy between the theoretical calculation curve and experimental curves is attributed to unequal emissivities of the wire at the two wavelengths. Another possibility for the observed discrepancy is that the laser beam introduces a large dc temperature rise. In order to compensate for the emissivity variation, calibration must be carried out [34].

10-2

PRT SIGNAL (V)

3.2 μm 4.5 μm

10-3

10-4

10-5 300

400

500

600

700

800

900

TEMPERATURE (K)

SIGNAL RATIO G=S(3.2 μm)/S(4.5 μm)

(a) 3.0 2.5

20 Hz 500 Hz CALIBRATION

2.0 1.5 1.0

EQUAL EMISSIVITY ASSUMPTION

0.5 0.0 400

500

600

700

800

900

800

900

SAMPLE TEMPERATURE (K) (b) 900 PTR TEMPERATURE (K)

20 HZ 800

500 HZ THEORITICAL

700

600

500

400 400

500

600

700

THERMOCOUPLE TEMPERATURE (K) (C)

Figure 9 (a) Photothermal signals for a 127-mm nickel--aluminum wire at two different wavelengths with the laser modulation frequency at 20 Hz. (b) Ratio of the photothermal signals for the 127-mm nickel--aluminum wire. (c) Comparison thermocouple and photothermal radiometry temperature readings for the 127-mm nickel--aluminum wire. (From Ref. [34] with permission).

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A comparison of the thermocouple temperature reading and the photothermal reading based on the calibration curve is shown in Figure 9(c). The maximum temperature difference between the two methods is about 122%. Although PTR has been successfully used to measure the surface temperature of metals, additional considerations must be taken for the temperature measurement of semiconductors due to the electron2hole pair generation by the laser. It should be noted that photothermal radiometric thermometry is not really suitable for temperature measurements of some semiconductors such as highly doped silicon samples between 300 and 900 K, a range in which their emissivities depend strongly on temperature. The illuminating laser creates electron2hole pairs if its photon energy is larger than the bandgap. Therefore it gives rise to a carrier density oscillation (a carrier density wave). The blackbody emissive intensity depends only on temperature, however, the emissivity of semiconductors depends on both temperature and carrier density. The modulated laser creates modulation in both temperature and carrier density. In order to accurately determine the sample temperature based on the measured ac signals, either one of the two following requirements have to be satisﬁed. (1) The relationship between the free photo-excited carrier density wave and emissivity must be well known. Usually, this is not the case. (2) The signal component due to the carrier density variation must be much smaller than that due to temperature variation. If the second requirement is satisﬁed, the problem can be simpliﬁed to one similar to a temperature measurement of a metal as discussed earlier. Research [33] demonstrates that PTR may be applied to heavily doped semiconductors, if the effect of electron2hole pairs can be minimized by careful adjustment of the experimental conditions. These conditions, in turn, limit the application of this technique in semiconductor temperature measurements. Temperature measurements have also been performed using pulsed PTR. Unlike the modulated PTR effect where temperature and radiation from the sample surface change harmonically, the temperature and radiation of the pulsed PTR effect rise to a peak temperature and then decay. As an example, if the surface of a sample is illuminated with a laser pulse with ﬁxed power and duration, the temperature of the sample, initially at temperature T0, rises locally to a peak temperature T ¼ T0+DT and then decays. To simplify the discussion, it can be assumed that the cooling of the sample follows an exponential decay given by t T ðtÞ ¼ T 0 þ DT exp t

(22)

where t is the time measured from the end of the excitation pulse and t the time constant determined by the thermophysical properties of the sample. The decay of the radiation emitted from the surface of the sample can be

319

Laser Optical and Photothermal Thermometry of Solids and Thin Films

expressed as [31]: Iðl; T 0 ; DT ; t; t; Þ ¼

c 1l 1 5 =ðl½T þ DT expðt=tÞÞg 1 expfc l 2 0

(23)

The dynamic component of the sample radiation is given by DIðl; T 0 ; DT ; t; t; Þ ¼ Iðl; T 0 ; DT ; t; t; Þ Iðl; T 0 ; Þ

(24)

The temporal behavior of the dynamic component of the photothermal signal is shown in Figure 10. A schematic of the pulsed PTR technique is shown in the inset. The laser heats the target from an initial time t0 to a time tPEAK. As the surface of the target is heated, the emitted radiation increases. Some of the radiation is transferred by the optical system to an ac-coupled IR detector. The detector signal increases from S ¼ 0 at the time t0 to a maximum value SPEAK at time tPEAK. When the laser is turned off, the temperature decays exponentially and the signal decays back to zero. The analysis of the signal in order to measure the temperature is similar to analyses introduced for surface temperature measurements [32234] using modulated PTR. However some differences arise due to different thermal behavior between transient and stationary harmonic heating, such as signal detection and temperature derivation from transient signals. Two major signal testing analysis methods have been developed to deduce the temperature by use of various features of the detected signals. One method SIGNAL

TARGET

PULSED LASER SPEAK DETECTOR

SIGNAL

S1 = α1 SPEAK

1> α1 > α2 S2 = α2 SPEAK

0 t1

td DECAY TIME

TIME t2

EXCITATION LASER PULSE

t0

tPEAK

TIME

Figure 10 Deﬁnition of the photothermal signal measuring parameters. The inset is a schematic of the photothermal radiometry technique. (From Ref. [37] with permission).

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Yue Liu and Andreas Mandelis

calculates the temperature from the amplitude of the pulsed PTR signals obtained after a ﬁxed delay time [36], the other one calculates the temperature from the shape of the decay signal [37]. To deduce the temperature from the detected pulsed photothermal decay signal, heating time, delay, and sampling window have to be carefully chosen. Heating time is the pulse width. In some cases, it may not be controllable, as heating lasers have ﬁxed pulse width. The delay is the time between the end of the pulse and the beginning of the signal acquisition. Sampling window is integration time of signal capture. In calculating the temperature from pulsed photothermal signals, one faces the same problems discussed earlier in conjunction with modulated photothermal signals, such as unknown emissivity, wavelength selection, etc. Analyzing the ratio of photothermal signals at two wavelengths [34] is the commonest method to calculate the temperature. Table 4 gives some experimental results using pulsed PTR at 3.99 and 4.56 mm excitation wavelength, assuming l1 ¼ l2 . It must be emphasized that, for the results presented in Table 4, the reﬂected ﬂux is as large as the emitted ﬂux and much larger than the pulsed emitted signal. The reference temperature was measured by a K-type thermocouple with 1% accuracy. The fact that no systematic error appears in Table 4 shows that the technique can effectively minimize the effect of ambient radiation on the signals, though the assumption of the equivalence of emissivity at two wavelengths restricts the use of this method, unless the ratio is explicitly known. Since the amplitude of the photothermal signal depends on the target emissivity and on geometric factors, variations in these parameters increase the inaccuracy of the temperature measurements. Another method has been developed [37], in which the target temperature is obtained from the shape of the photothermal signal decay curve, and not from the signal amplitude. In this method, the decay time of the photothermal signal can be chosen to characterize the photothermal curve shape. The decay time is a parameter that can be deﬁned in several ways. One typical deﬁnition is Table 4 Experimental temperature measurements for an oxidized iron sample at wavelengths 3.99 and 4.56 mm with a 150-ms sampling width, a 100-ms delay, a 4-mm diameter laser beam, a 5-mJ pulse energy, and an average of more than 300 shots. Tref (K)

T0 (K)

792 610 507 389

79679 61978 51378 39979

Note: The reference temperatures were measured with a K-type thermocouple with an accuracy of 1% [36].

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Laser Optical and Photothermal Thermometry of Solids and Thin Films

SPECTRAL RESPONSE. (%)

based on choosing two signal levels, S1 and S2, where S1 ¼ a1SPEAK, and S2 ¼ a2SPEAK, a1,a2 are two numbers, 1Wa1Wa2W0. S1 and S2 occur at times t1 and t2, respectively. The decay time, td ¼ t2t1, is deﬁned as the difference between t2 and t1. Figure 10 shows the temporal behavior of the pulsed photothermal signal and the deﬁnition of the signal measuring parameters. A set of measurements with the experimental setup shown in Figure 11 was carried out. The results are shown in Figure 12. Two typical experimental photothermal signals measured at target temperatures of 21 and 541C, are shown in the inset of Figure 12. The decay time td was deﬁned as the time between t1 for which a1 ¼ 0.8, and t2 for which a2 ¼ 0.2. The decay time td was measured as a function of the blackbody target temperature T0. The laser pulse energy was 33 mJ and the pulse duration was 2.5 ms. As can be seen, td is proportional to T0 for both InSb and HgCdTe infrared detectors. These results show the feasibility of the method for temperature measurements. An uncertainty of approximately 751C is the limit of the experimental system. The graph generated by OSCILLOSCOPE CH - 1

60

PC

CH - 2 AMPLIFIERS

40 HgCdTe

20

InSb

0 4

8 12 WAVELENGTH (μm)

16

REJECTION FILTER MIRROR FOCUSING SYSTEM

CO2 LASER LENS

SILVER HALIDE OPTICAL FIBERS DELIVERY FIBER

THERMOCOUPLE TEMPERATURE CONTROLLER

SENSING FIBER TARGET H HEATER

Figure 11 Experimental setup of photothermal radiometry for fast temperature measurements. The overall optical spectral response of the system is shown in the inset. HgCdTe and InSb are infrared detectors; CH1 and CH2 are two channels of the oscilloscope. (From Ref. [37] with permission).

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8

HgCdTe 6

5

HgCdTe

InSb

4

SIGNAL (mVolts)

HgCdTe or InSb DECAY TIME (ms)

7

54°C

90 60 30 0

21°C 0

4 Time (ms)

8

3 40

80

120

160

BLACKBODY TARGET TEMPERATURE (°C)

Figure 12 Decay time as measured by the HgCdTe and InSb detectors (in this case a1 ¼ 0.8, and a2 ¼ 0.2), versus the temperature T of a blackbody target. Laser pulse parameters: energy of 33 mJ, pulse duration of 2.5 ms. The inset shows two typical photothermal signals measured by HgCdTe detector at a target temperature of 211C and the other at 541C. (From Ref. [37] with permission).

this measurement can further be used as a calibration graph for this type of sample. A calibration curve will be needed for each new type of sample. Studies of possible dependence on target geometry and emissivity have shown that this method is independent of these two factors [37]. Therefore, the calibration can be done at a convenient location and then used for temperature measurement at a difﬁcult location. In addition, this method has all the aforementioned advantages of photothermal radiometric temperature measurements (modulated and pulsed methods). In summary, photothermal radiometric thermometry can effectively ﬁlter out ambient contributions associated with conventional dc radiometric thermometry by analyzing the dynamic photothermal signal. Since the transient photothermal signal is generated by very small temperature variations, it is much smaller than the dc component of the radiation signal. Therefore, this technique may be limited by the ability of the measurement system to detect the very weak intensity of the dynamic radiation. This type

Laser Optical and Photothermal Thermometry of Solids and Thin Films

323

of thermometry under certain conditions, such as radiation measurements emitted from samples inside plasmas and radiation heating vacuum enclosures, is difﬁcult because unknown emissivities at different wavelengths complicate signal interpretation. All these factors limit the usage of this technique and measurement range. The only exception is that of emissivity-free photothermal thermometry which can be used in a wide variety of situations, including where there is a large background component and that the emissivity values are unknown and different at two wavelengths. These are discussed in detail in Chapter 10. In order to perform reliable temperature measurements, all factors, such as detection subsystem (detector, optical geometry, ﬁlters, and signal processor), laser operating mode (modulated or pulsed), wavelengths selection, calibration method, temperature calculation method, etc. must be carefully considered.

5. Thermoreflectance Thermometry This thermometric technique is based on the principle of laser thermoreﬂectance, i.e., the change of the refractive index of a material (and thus of its reﬂectivity) with temperature [29,39241]. The reﬂectivity of most dielectrics and metals in the visible spectral range is known to change proportionally with temperature [42245]. By taking into account the surface condition of a material as well as the incident beam wavelength, angle, and polarization, a change in temperature can be related to the change in reﬂectivity. The most important issue with such a temperature measurement technique is to ﬁnd the relationship between reﬂectance, R, and temperature, T, variations, i.e., the thermoreﬂectance coefﬁcient (1/R) (dR/dT).

5.1. Physical principle The physical mechanism of thermoreﬂectance largely remains complicated and not completely understood in detail. The complex index of refraction for most metals and dielectrics depends weakly on temperature, resulting in a dependence of reﬂectivity, R, on temperature, T, according to the Fresnel formula of vacuum2material interfaces for normal incidence: RðT Þ ¼

½nðT Þ 12 þ k2 ðT Þ ½nðT Þ þ 12 þ k2 ðT Þ

(25)

where n and k are the real and imaginary indices of refraction, respectively. The real and imaginary parts of the complex dielectric function e of a solid can be expressed as combinations of n and k: 1 ¼ n 2 k2

and

2 ¼ 2nk

(26)

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Yue Liu and Andreas Mandelis

Since e ¼ e1+ie2 ¼ (n + ik)2 commonly, k is referred to as the extinction coefﬁcient. For small temperature changes resulting in small reﬂectivity changes, thermoreﬂectance is linearized through a Taylor series expansion [46248] as shown below: RðT Þ RðT 0 Þ þ

dR ðT T 0 Þ dT

(27)

where T0 is the ambient temperature. Thus, the change in the reﬂectance is approximately proportional to the temperature change, i.e., dR DR ¼ RðT Þ R0 ’ DT dT T ¼T 0

(28)

The fractional reﬂectivity change from its value R0 measured at the reference temperature, T0, can be written as: DR 1 dR ¼ DT k DT R0 R0 dT T ¼T 0

(29)

where the term (1/R0)(dR/dT) is commonly referred to as the thermoreﬂectance coefﬁcient, k (K1). This coefﬁcient can be treated as a constant by holding constant parameters such as surface conditions, wavelength, polarization, and incident angle. This is because the temperature change is usually small when thermoreﬂectance technique is applied.

5.2. Instrumentation and applications The thermoreﬂectance technique has been introduced to measure thermal properties of semiconductors [49], thermal diffusivities of metal ﬁlms [50], semiconductor devices [46], and steel foil at low temperatures [47]. Since the early 1990s, many research groups have applied the thermoreﬂectance technique as a means of measuring the temperature of homogeneous materials such as metals [51254], semiconductors [48,55257], and metal alloys [58]. As a noninvasive optical technique, thermoreﬂectance thermometry has also been explored for temperature measurements and imaging of some devices such as laser diodes [59262], transistors [63265], integrated circuits [66], and a variety of test devices [67271]. The method offers direct access to the active area which may even be covered with a transparent encapsulation layer. In the visible or near-UV range, excellent spatial resolution can be attained with proper choice of optical components. Knowledge of the temperature dependence of reﬂectance and of the thermoreﬂectance coefﬁcient of probed materials are key issues toward enabling thermoreﬂectance temperature measurements. Unfortunately, the value of k is only known for a few bulk materials at certain wavelengths. As a consequence, k of every new tested sample or material must be

325

Laser Optical and Photothermal Thermometry of Solids and Thin Films

MIR1

MC

PL

CBS

PCBS

QWP

MIR2

Incident angle Φ = 0 TC SAMPLE FL 2

FL1 Ref (ω) PD 1

PD 2

HP

He-Ne 632.8

LIA 1

LIA 2

PC

Figure 13 Experimental setup for thermoreﬂectance coefﬁcient measurement. MIR, mirror; PL, polarizer; MC, mechanical chopper; CBS, cube beam splitter; PCBS, polarizing cube beam splitter; QWP, quarter-wave plate; FL, focusing lens; PD, photodiode; HP, heater plate; LIA, lock-in ampliﬁer; TC, thermocouple.

obtained a priori through measurement. Figure 13 shows a typical setup for thermoreﬂectance coefﬁcient measurements [58]. The setup consists of a 10-mW 632.8-nm linearly polarized He2Ne laser serving as the source. The emitted beam is mechanically chopped at a certain frequency and passes through a visible light polarizer to ensure linear polarization. The incident beam is then split into two beams using a cube beam splitter, and the 901 reﬂected beam is focused onto a silicon photodiode connected to one of two lock-in ampliﬁers referenced to the chopper frequency. This reﬂected beam is used as a reference beam to monitor laser power ﬂuctuations. The forward transmitted beam continues on through a polarizing cube beam splitter (PCBS) aligned to the polarization of the incident beam, thus allowing for the majority of the incident beam to pass. The beam then traverses a 632 nm quarter-wave retardation plate which circularly polarizes the light. The circularly polarized beam is directed normal to the sample surface where it is reﬂected with the opposite direction of circular polarization. By optimizing the optical axis orientation of the quarter-wave plate (QWP), the reﬂected beam, upon passing again through the QWP, re-emerges linearly polarized perpendicular to the polarization of the incident beam. As a result, the returning linearly polarized beam is redirected by the polarizing beam splitter to the second

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Yue Liu and Andreas Mandelis

silicon photodiode connected to the second lock-in ampliﬁer which is also referenced to the chopper frequency. The signals obtained by both lock-in ampliﬁers are recorded by a computer. In order to normalize the reﬂected signal so as to remove the effect of ﬂuctuations in laser power, the reﬂected signal is normalized by the reference signal. Upon comparison of the normalized reﬂected signal with temperature values measured with a thermocouple, one can obtain the thermoreﬂectance coefﬁcient of the material. Figure 14 demonstrates a typical measurement of a TiN sample. The thermoreﬂectance coefﬁcient of TiN calculated with Equation (29) and experimental data shown in Figure 14 is 3.12 104 K1 for probing wavelength of 632.8 nm. Studies have shown that the thermoreﬂectance coefﬁcient is related to many factors, such as wavelength and polarization of probe light source, optical system of thermoreﬂectance measurement, incident angle, signal collection circuitry, etc. In order to obtain the temperature of a measured sample accurately, all those factors must be taken into consideration. Thermoreﬂectance coefﬁcients of some materials obtained with similar setups by different research groups are shown in Table 5. By combining thermoreﬂectance thermometry with an automatic sample scanning system, one can perform temperature mapping and

140 Reflected Signal, Rm(T) Temperature

-3.10

120

-3.12 100

-3.14 -3.16

80

-3.18 60

-3.20 -3.22

Temperature (°C)

Normalized Amplitude, Rm(T) (a.u)

-3.08

40

-3.24 reference region

20

-3.26 0

100

200

300

400

500

600

700

Time (sec)

Figure 14 Transient temperature and anti-correlated normalized reﬂectance amplitude of a TiN sample. The reference signal baseline is highlighted. Mean reference values were found to be To ¼ 251C, Ro ¼ 3.24 within the ﬁrst 250 s. Rm(T) is the reﬂectance signal normalized with the laser power. Rm(T) is plotted versus time in order to facilitate visual comparison of the thermocouple and reﬂectance transients.

Laser Optical and Photothermal Thermometry of Solids and Thin Films

Table 5

327

Thermoreﬂectance coefﬁcients of certain materials.

Material

Probing wavelength (nm)

Thermoreﬂectance coefﬁcient (K1)

TiN [58] GaAs [62] GaAs [62] GaAs [70] AlGaAs [62,69] Polysilicon [66] Sijunction [66] Al (99.5%)Cu (0.5%) [53] Si [43,49,57] Au [52] TiW [49]

632.8 632.8 532 457 632.8 536 536 632.8 632.8 632.8 632.8

3.12 104 1.5 104710% 2.4 104710% 2.5 104 3.8 104710% 1.4970.04 103 6.474.9 104 (2.5570.15) 105 1.2B1.6 104 2.6 104 9.2 106

distribution measurements under the name of thermoreﬂectance thermography (TRTG) [64,71]. Figure 15 shows a typical setup for TRTG [72]. A MOSFET device with a smaller channel was activated and scanned with this setup. Figure 16 gives scanning results of the area of 50 15 mm which includes the channel as well as the gate polysilicon region depicted in the bottom left half of the ﬁgure. To produce the absolute temperature rise value plotted in Figure 16 the measured reﬂectivity ﬁeld was scaled with the experimentally obtained thermoreﬂectance coefﬁcient. The results prove the method’s capability of capturing the transient behavior of the absolute surface temperature ﬁeld. Thermoreﬂectance can also be used for temperature measurements during coating growth processes by properly interpreting the reﬂectance signals. A reﬂectance contribution of the growing coating can be approximated using a ﬁrst-order Taylor series expansion for the reﬂectivity-temperature dependence. As a result, Equation (29) can be generalized to include the ﬁrst-order dependence on thickness as well. Since the reﬂectivity is no longer that of a bulk conducting material (substrate), but instead, of a coating-substrate two-layer system, the total fractional reﬂectance, DR/R0 is given by the interferometric thermoreﬂectance [58]: DRðL; T Þ RðT ; LÞ RðT 0 ; L 0 Þ 1 @R 1 @R DL þ DT ¼ ¼ R0 R0 R0 @L R0 @T aðL; T ÞDL þ kðL; T ÞDT

(30)

where a(L,T) (m1) can be obtained formally by differentiating the total reﬂectance with respect to the thin-ﬁlm thickness, L. Equation (30) gives

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Zoom/Camera Assembly

Ar-Ion Laser, 488 nm

Oscilloscope

Beam Splitter

Collimators

PC Computer

Lens Beam Splitter

Amp

PD

λ/4 Plate Mirror

Drain Pulser z y

Objective Lens

x DUT

Thermochuck or TE Controller

Ext. Trigger

D i g i t a l C o n t r o l

Thermochuck or TE

Figure 15 Schematic of the TRTG experimental setup (thermoreﬂectance coefﬁcient calibration and scanning). DUT, device under test; PD, photodiode; Thermochuck, temptronic, 8-in, liquid stabilized thermal chuck; TE, thermoelectric device. (From Ref. [64] with permission of IEEE).

the complete theoretical reﬂectance variation when both coating thickness and temperature change during a coating deposition process. Figure 17 gives an example of comparison of the theoretical total reﬂectance variation (DR) by combining k(L)DT, and a(L)DL, together with one normalized interferometric thermoreﬂectance measurement during a TiN coating process. As expected, the interference pattern occurs at thicknesses o0.3 mm and it is essentially the same for all temperatures, T, since the signal variation caused by the temperature (thermoreﬂectance) is much smaller than the variation caused by coating thickness (interference) and can thus be neglected. It is an implicit assumption of the method that the optical properties of the thin layer are slowly varying functions of temperature and hence the interference effect dominates. This also validates the experimental assumption that a can be treated as a function independent of temperature, a ¼ a(L). Therefore, when the thickness is less than 0.3 mm, we cannot obtain reliable temperatures. However, when the coating thickness is large enough (LW0.3 mm) to be optically opaque to the incident beam, the interferometric effect tends to zero asymptotically and the signal variation is related solely to the thermoreﬂectance effect.

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Laser Optical and Photothermal Thermometry of Solids and Thin Films

50 Drain

Source

T, °C 20 18 16 14 12 10 8 6 4 2 0

40

Gate Y, μm

30

20

10

0 0

5

10

15

X, μm

Figure 16 Scan area and corresponding temperature contours on an activated MOSFET device at the time of peak temperature (channel dimensions are 15 5 mm). (From Ref. [64] with permission of IEEE).

In this range of thicknesses the temperature can be precisely obtained. At thicknesses LW0.3 mm, the signal switches between two theoretical isotherms with DT ¼ 2001C and DT ¼ 3001C [72]. This means that, compared to the onset temperature of the coating process (usually around 35024001C), the temperature increased approx. 20023001C during an entire coating process. This temperature determination is in agreement with the thermocouple temperature readings taken during this period of coating growth. This technique can be used to approximate surface temperatures at different stages of coating growth using the measured reﬂectance signal. According to the example shown in Figure 17 at the thickness of 1 mm, the experimental curve coincides with the theoretical isotherm at 2201C. This can be taken to be the coating surface temperature when the thickness is 1 mm. The reﬂectance experimental curve further reveals relatively large temperature ﬂuctuations between 2001C and 3001C during the coating growth process from a thickness of 0.3 to 2.5 mm, which is also in general agreement with thermocouple measurements. Since the response of reﬂectance to changes in temperature is extremely fast, it is also possible to measure a moving sample with a fast data acquisition technique. Figure 18 shows data from a rotating sample

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0.06 0.04 ΔR (Theoretical) Normalized in-situ measurement

0.02

-0.02

ΔT=0°C ΔT=10°C ΔT=20°C ΔT=50°C

-0.04

ΔT=100°C

ΔR (a.u.)

0.00

-0.06 ΔT=200°C ΔT=220°C ΔT=240°C ΔT=260°C ΔT=280°C ΔT=300°C

ΔT=220°C

-0.08 -0.10 -0.12 0.0

0.7

1.4

2.1

2.8

3.5

Thickness (μm)

Figure 17 Comparison of theoretical DR based on Equation (8) for various thicknesses and temperatures under real-time in situ TiN-coating process monitoring.

Figure 18 Transient measurement results of rotating sample with sampling rate 1.53 kHz and rotating speed 4.43 s/cycle.

Laser Optical and Photothermal Thermometry of Solids and Thin Films

331

temperature measurement using a suitably constructed thermoreﬂectance system [72]. This study shows that thermoreﬂectance thermometry can be used for moving sample temperature measurements. Many research results have shown that thermoreﬂectance thermometry can be used to measure surface temperatures quickly, remotely, and with a high degree of sensitivity and spatial resolution. Combined with a scanning system this technique can be used to obtain the temperature distribution of solid areas. Since laser beams are to a large extent immune to hostile ambient conditions compared to other thermometry techniques, such as during the aforementioned plasma deposition process, thermoreﬂectance thermometry appears to be the most suitable technique for measuring the surface temperature of parts undergoing plasma coating processes by properly interpreting reﬂectance signals. Combined with a fast data acquisition technique, thermoreﬂectance can be used to measure the temperature of moving samples.

6. Summary In this chapter, we have introduced four major laser optical and photothermal thermometric techniques. Compared with the conventional radiation thermometry, laser optical thermometry, usually uses a laser either as a probe or as an extra heat source to obtain temperature, instead of directly measuring the radiation from a sample to calculate the temperature. As a probe, a laser beam can carry temperature information when reﬂected from a surface under investigation. As a heat source, it can cause temperature and emitted radiation variations. The analysis of the emitted radiation can yield temperature information. As a new branch of thermometry, which has developed from the increasing demand, during the past decade, for improved quality control in microtechnology, they all have some common features compared with other conventional thermometry shown in Table 6. However, all of these methods have their own special features and limitations. Laser thermometry methods are narrowly specialized in contrast to conventional IR thermometry methods. Many laser methods provide measurements for a restricted set of materials, and sometimes there are additional speciﬁcations for the specimen shape and surface properties. Normally, laser thermometry methods require complicated optical systems. Additionally, determination of temperature from laser thermometry requires prior knowledge of physical properties of the sample including their dependence on temperature. Usually the temperature dependence of these parameters is not well known and thus it must be determined for the material of interest prior to the implementation of any temperature measurement. For this reason the use of laser thermometry in

332

Table 6

Comparison of laser optical thermometer with several conventional thermometers. Thermocouple

Resistance temperature devices (RTD) (Pt100)

Thermistor

Radiation thermometry

Optical thermometer

Operating range Accuracy

200 to 2,0001C Low, 11C common Medium Fast

250 to 8501C Very high, 0.031C common High Slow

100 to 3001C High, 0.11C common Low Medium

BSeveral thousands Low

oMelting point Low High Fast

High Low Medium

Medium High High

Low Medium Low

Low Fast for high temperature, slow for low temperature Medium Low High

Low

Low

Low

Medium

High

Linearity Thermal response

Noise problems Long-term stability Cost of measuring instrument Suitability to plasma coating process

Low Medium High

Yue Liu and Andreas Mandelis

Parameters

Laser Optical and Photothermal Thermometry of Solids and Thin Films

333

engineering materials monitoring and in controlled industrial applications is quite limited, but it has been growing steadily in recent years as new and improved techniques are being developed.

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[67] [68] [69] [70] [71] [72]

APPENDIX A: FUNDAMENTAL CONSTANTS

AND

OTHER PHYSICAL

Quantity

Symbol Value

Uncertainty

Avogadro constant Boltzmann constant Electrical permittivity (vacuum) Electron mass Elementary charge First radiation constant First radiation constant for spectral radiance Magnetic permeability (vacuum) Molar gas constant Planck constant Planck constant over 2p Second radiation constant Second radiation constant (ITS-90) Speed of light in vacuum Stefan2Boltzmann constant Third radiation constant

NA k e0

6.02214179 1023 mol 1 1.3806504 10 23 J/K 8.854187817 10 12 F/m

3.0 1016 mol 1 2.4 10 29 J/K Exact

me e c1 c1L

9.10938215 10 31 kg 1.602176487 10 19 C 3.74177118 10 16 W m2 1.191042759 10 16 W m2/sr

4.5 10 4.0 10 1.9 10 5.9 10

m0

4p 10

R h _ c2 c2

8.314472 J/(mol K) 6.62606896 10 34 J s 1.054571628 10 34 J s 1.4387752 10 2 m K 1.4388 10 2 m K

1.5 10 3.3 10 5.3 10 2.5 10 Exact

c0 s c3

299,792,458 m/s 5.670400 10 8 W/(m2 K4) 2.8977685 10 3 m K

Exact 4.0 10 5.1 10

7

N/A2

38

kg C 23 W m2 24 W m2/sr 27

Exact 5

J/(mol K) Js 42 Js 8 mK 41

13 9

W/(m2 K4) mK

Source: http://physics.nist.gov/cuu/index.html

337

SUBJECT INDEX Absorption coefﬁcient, 117–118 Absorptivity

directional, 88, 89

hemispherical, 88

spectral, 88

total, 89

Accreditation, 48, 54

Acoustic thermometry, 33–34

Acoustic virial coefﬁcients, 34

Adiabatic, 31

Adsorption, 33, 38

Airy’s formulae, 121

American Society of Testing and Materials

(ASTM), 21

Ampere’s law, 100

Ampliﬁer, 34, 60, 135, 162, 315

Aperture, 4, 9, 11, 36, 80, 148, 169, 250, 256,

270, 283

Asia Paciﬁc Metrology Programme

(APMP), 54

Bandwidth, 6, 10

Bidirectional reﬂectance distribution function

(BRDF), 9, 21, 91–94, 203, 207, 211,

227–231

Bidirectional transmittance distribution function

(BTDF), 98

Blackbody

ﬁxed-point, 49, 50, 61, 67, 134, 254–256,

258, 270

also see ﬁxed-point blackbody

heat-pipe, 260, 261–264, 261t, 261f, 263t,

liquid-bath, 264–265, 265f

perfect, 182, 187

quasi-, 16, 19

reference, 49, 64, 176, 182, 270f

variable temperature, 50f, 258–267, 271, 272f,

275

Blackbody cavity, see cavity

Blackbody radiation, 77–85

Blackbody lamp, 287–288

Boiling point of

equilibrium hydrogen (or e-hydrogen), 44

helium, 44

neon, 43

oxygen, 43

sulphur, 43

water, 42

Boltzmann constant, 32, 41, 82, 247, 337

Brewster angle, 107–109

Broad-band measurement, 144, 145

Bureau International des Poids et Mesures

(BIPM), 42

Calibration, 51–52 Calibration and Measurement Capabilities (CMC), 55

Calorimeter, 38

Cavity

bafﬂe, 201, 213

diffuse, 187–195

isothermal, 183–184, 187–189

non-Lambertian, 202–204

nonisothermal, 183, 223, 224,

225

specular-diffuse, 205

spherical, 191–192, 192f, 199, 217

Celsius, see Temperature Scale

Centigrade, see Temperature Scale

Christiansen wavelength, 15, 115,

116t

Complex refractive index, 102

Constant volume gas thermometry,

33

Coverage factor, 57, 58, 63, 58t

Critical angle, 109

Crucible, 49, 256, 257f

Current meter, 139–140

Current-to-voltage converter, 139

Dark current, 163–164, 164f

Desorption, 33

Detector, characteristic of

linearity, 11, 12, 157–159, 159f

shunt resistance, 138, 162, 163f

stability, 60, 160, 174

standard, 144, 145

uniformity, 38

Detector, type of

bolometer, 11

InGaAs, 139, 162

InSb, 10, 12, 322f

MCT (HgCdTe), 10, 321, 322f

nonthermal, 10

optical, 135

photomultiplier, 3

photon, 10–11

pyroelectric, 6, 144

Si photodiode, 140

thermal, 11, 12

339

340 Dielectric function of

conductors, 110

nonconductors, 113

semiconductor, 110, 116, 118, 128

Diffusity, 216, 221, 223

Doppler broadening, 40

Effective wavelength, 63, 167–168

Electrical substitution radiometer (ESR), 36, 170,

170f

Ellipsometry, 106, 304–313

Electromagnetic wave, 99–110

Electromagnetic ﬁeld, 99, 103

Emissivity or emittance

apparent, 94–95

directional, 86

effective, 94–95, 95f

hemispherical, 86, 87

isothermal blackbody, 61, 186

nonisothermal blackbody, 186, 226

spectral, 86

total, 86

Emissivity-enhancement method, 16–17

Emitter

blackbody, 316

diffuse, 87

Energy partitioning, 213, 214

Equation of state, 32

Euro-Asian Cooperation of National

Metrological Institutions (COOMET), 54

European Association of National Metrology

Institutes (EURAMET), 53, 54

Eutectic (ﬁxed-point), 66

Exchange factor, 205

Exitance

emitted, 77

reﬂected, 77

self-, 78, 86

spectral, 35, 78, 184

total, 78

Extinction coefﬁcient, 102, 116, 324

Expansivity, 32

Extensive quantity, 30

Fabry-Perot interferometer (or interferometry),

11, 18, 20–21

Feedback resistor, 140, 142

Field-of-view, 9, 283, 284

Fixed-point blackbody

eutectic ﬁxed-point, 66, 260f

see also boiling point

see also freezing point

see also triple point

Fourier-transform infrared (FTIR) spectrometry,

15, 91

Freezing point

of aluminum, 46t

of antimony, 46t

Subject Index

of copper, 3

of gold, 3, 38

of silver, 3, 44

of tin, 46t

of zinc, 43

Free spectral range, 122

Fresnel’s equation, 212

Fresnel’s reﬂection coefﬁcient, 104–106

Fresnel’s transmission coefﬁcient, 104–106

Gaussian probability distribution, 226

Gauss’s law, 100

Geometrical factor, 38

Gray (or grey) surface, 13, 22, 88, 90, 217

Gold-cup method, 16

Group velocity, 102

Guide to the Expression of Uncertainty in

Measurement (GUM), 55

Heat engine, 31–32

Heat transfer, 38, 94, 208–210

Heat pipe, 259–264

Hydrogen, 42, 45–46

Ideal gas, 33

Ideal gas equation, 33

Index of refraction, see refractive index

Infrared camera, 3

Internal transmissivity, 120

Intensity, 22, 35, 77, 97, 318

Intensive quantity, 30

Inter-American Metrology System (SIM), 53

Interference, 20, 21, 119, 122, 127

Interference ﬁlter, 11, 36, 142

Intermediate equilibrium hydrogen point, 43

International Bureau of Weights and Measures

(BIPM), 42

International Commission of Illumination (CIE),

21

International Committee for Weights and

Measures (CIPM), 41

International System of Units (SI), 30

International Temperature Scale; see Temperature

Scale

International Vocabulary of Metrology (VIM), 51

Interpolation, 44, 57, 62, 67, 143, 160,

165–166

Irradiance, 22, 36, 77, 79, 189, 211, 270

ISO/IEC 17025 (2005), 54

Isotopic composition, 40

ITS-xx, see Temperature Scale

Kelvin, see Temperature Scale

Kelvin redeﬁnition, see Temperature Scale

Key Comparison Database (KCDB), 55

Kinetic energy, 32

Kirchhoff ’s law, 89–90, 214, 242

Subject Index

Lambertian, 23, 202–204, 221

Lamp

blackbody, 287, 288f

tungsten strip, 284–288, 286f

Line scanner, 3, 11

Linearity of response, 51, 59, 157–159

Los Alamos National Laboratory, 209

Lyot stop, 136f, 149–151, 150f

Maxwell’s equations, 93, 98, 99

Maxwell-Boltzmann distribution function, 32

Mean effective wavelength, 63, 64t, 167–168

Mean square noise voltage, 32, 34

Measurand, 56, 57

Melting point of

gallium, 46t, 270

ice, 42

Member States of the Convention, 42

Method of iterations, 193

Method of statistical weights, 217

Method of successive reﬂections, 205–208

Micro-facet model of a rough surface, 228

Mise-en-pratique (MeP-K), 47, 65

Modiﬁed Planck function, 165, 166

Molar gas constant, 33, 337

Monte Carlo, 208–231

Mutual Recognition Arrangement (MRA),

54–55

Narrow-band ﬁlter, 84, 85, 144

National Institute of Standards and Technology

(NIST), 34, 127

National Metrology Institutes (NMIs), 37, 242

National Physical Laboratory (NPL), 39, 232

Noise, 34–35, 62, 138, 139, 140

Noise-equivalent-current, 160, 164

Normal probability distribution, 56

Nuclear magnetic resonance, 40

Nuclear orientation, 40

Nyquist equation, 34

Ohm’s law, 100

Operational ampliﬁer, 139–140

Optical constants, 102, 106, 110, 114f, 116, 118f

Optical properties, 110–118

Partial reﬂectivity, 206–207

Perfect absorber, 242

Phase velocity, 101–102

Phosphor, 20

Photocurrent, 36, 139–140, 165, 165t, 166, 168,

172, 173, 173f, 286, 287f

Photodetector, 3, 12

Photodiode, 137–139

Photon, 76, 78

Photon detector, 10–11

341 Photothermal radiometric thermometry,

313–323

Physikalisch-Technische Bundesanstalt (PTB),

243

Planck’s constant, 82

Planck’s law, 3, 7, 10, 23, 37, 65, 67, 81–85, 85f

Platinum resistance thermometer, 42, 43, 44,

45–46, 46t, 51, 160

Polarization, 16, 100, 103

Poynting vector, 103–104, 106

Preampliﬁer, 136f, 139–142, 162, 173t

Probability density function (PDF), 209

Prototype, 42

Pseudorandom number generator (PRNG), 210

Pyrometer (or pyrometry)

disappearing ﬁlament, 2, 134

dual-wavelength, 13–14

multi-wavelength, 14–15

optical, 2, 3, 43

polarization, 16

ratio, 13–14

two-color, 13–14

Radiance

exitent, 5, 6, 7

reﬂected, 91, 93

spectral, 77

total, 77

Radiance ratios, 156, 167–168

Radiant power, 10, 22, 23, 36, 77, 84, 95, 103,

144

Radiation

background, 3, 15, 62, 186

blackbody, 77–85

optical, 76, 157, 182, 211, 227, 243, 249

thermal, 76, 242–248

Radiation constant

ﬁrst, 184, 337

ﬁrst y for spectral radiance, 7, 82

second, 7, 43, 47, 82, 157, 165, 184,

316

third, 83

Radiation detector, see detector

Radiation penetration depth, 97

Radiation pressure, 79–80

Radiation thermometer

broad-band (also broadband), 5

classical, 11–12

detector-based, 3

ﬁber-optic, 18–21

multi-wavelength, 14–15

ratio, 13–14

spectral, 6–9

total, 5–6

Radiation thermometry, see radiation

thermometer

Radiation transfer function, 38

Radiation trap, 38

342 Radiative properties

for multilayer structures, 123–125

for thick ﬁlms, 119

for thin ﬁlms, 119

see absorptivity

see emissivity

see reﬂectance

see transmittance

Radiative transfer equation (RTE), 96

Radiative transition, 97

Radiator

Blackbody, 23, 35, 38, 187, 253, 285

ﬂat plate, 267

Radiometer (or radiometry)

Absolute, 36, 37, 65, 66

absolute cryogenic - (ACR), 6

cryogenic, 3, 39, 39f, 144

detector-based, 3, 169, 175

electrical substitution - (ESR), 36, 38, 40, 170,

174

ﬁlter, 36, 36f, 125, 169

photothermal, 313–323

relative, 37

spectral, 35–37

total, 37–40

Radiometer calibration, 48–49, 51–52

Radiometric temperature measurement,

4–11, 4f

Radiometry, see radiometer

Radiosity, 78

Rayleigh scattering, 98, 153

Rayleigh-Jeans formula, 81, 84

Rad-Pro software, 126

Ray tracing, 119–121

Reciprocity, 92, 206, 211, 215, 222

Rectangular probability distribution, 56

Reﬂectance or reﬂectivity

bidirectional

directional-hemispherical, 92–94, 109, 203,

205, 212, 224, 229

effective, 188

Refractive lenses, 135

Refractive index, 75, 97, 101, 102, 106, 110,

117, 118f, 119, 251–254, 252f, 303

Regional Metrology Organization (RMO), 53

Repeatability, 59

Responsivity

integrated, 167

radiance, 171–173

relative spectral, 167, 168

spectral, 59, 137, 137f, 143, 144–146

Resonator, 34

Scale dissemination, 63–65

Scale realization, 254, 255

Scale transfer, 56

Scattered light, 98, 152

Scattering, 97–98

Subject Index

Scattering coefﬁcient, 96

Self-calibration, 52

Sensitivity coefﬁcient, 57–58

Short-circuit current, 139

Signal gain, 142

Signal-to-noise ratio, 10, 140, 316

Size-of-source effect, 60, 62, 146–147

Snell’s law, 105

Solid angle, 23, 62, 77, 77f, 78f, 83, 92, 205–206,

270

Source

LED, 148

monochromator, 144–146, 145f

tungsten strip lamp, 58, 59, 285, 286, 286f,

287

Source radiance, 147

Southern African Development Community

Cooperation in Measurement Traceability

(SADCMET), 54

Spectral exitance, 35, 77, 183

Spectral ﬁlters, 142

Spectral irradiance response, 36

Spectral response, 144–146

Specularity, 225

Speed of light, 36, 73, 74, 79, 82

Speed of sound, 33, 34

Specular-diffuse model of reﬂection, 205, 222,

227

Spot-radiation thermometer, 134

Stability, 60, 62, 148, 262, 276, 304

Standard deviation of the mean, 56

Standard mean ocean water (SMOW), 40

Standard platinum resistance thermometer, 46,

51, 262

Standard uncertainty, 56–57, 167t

Statistical mechanics, 32

Stray light, 59

STEEP3, 221, 232

Stefan-Boltzmann constant, 5, 6, 39, 81,

169, 185

Stefan-Boltzmann law, 5, 24, 37, 80–81

Stokes theorem, 198, 205

Strehl ratio, 154, 155, 156

Superconducting transition, 44

Temperature

radiance, 5, 7, 8f, 23, 146, 157, 160, 187, 224,

225, 282, 283, 285

radiation, 97, 187, 262, 265, 269f

ratio, 13, 23

reference, 3, 33, 57, 62, 183, 224, 227, 283,

320

thermodynamic, 31, 40, 41, 42, 47, 48f, 66,

167–176

Temperature scale

1976 Provisional 0.5 K to 30 K Temperature

Scale (EPT-76), 44

Celsius, 1

343

Subject Index

Centigrade, 1

Fahrenheit, 1

International Practical Temperature Scale of

1948 (IPTS-48), 43

International Practical Temperature Scale of

1968 (IPTS-68), 43–44

International Temperature Scale of 1927

(ITS-27), 42–43

International Temperature Scale of 1948

(ITS-48), 43

International Temperature Scale of 1990

(ITS-90), 44–45, 45t

ITS-xx, 47–48, 48f

Kelvin, 40–41

Kelvin redeﬁnition, 48

Normal hydrogen scale of 1887, 42

Provisional Low Temperature Scale of 2000

(PLTS-2000), 47

Thermal equilibrium, 31, 32, 79, 90

Thermal image (or imaging), 3, 11, 13, 259,

262

Thermocouple, 43, 44, 304, 318

Thermoelectric cooling, 12, 139, 148

Thermoreﬂectance, 323–331

Thermoreﬂectance coefﬁcient, 324, 326,

327t Thermodynamic temperature, see temperature Thermodynamics ﬁrst law of, 31, 80

second law of, 31, 35

zeroth law of, 31

Thermometer (or thermometry)

Clinical, 265

Ellipsometric, 304–313

gas (or ideal gas), 33

infrared ear (or IR ear), 6, 226, 265, 266f

interferometric, 299–304

Johnson noise, 35–36

liquid-in-glass, 1

mercury-in-glass, 42

non-contact, 49, 55–65

photothermal radiometric, 313–323

primary, 30–41

radiation, see radiation thermometry

resistance, 34, 42, 43, 45–46, 46t, 51, 160

secondary, 168–169 thermocouple, 43, 44, 304, 318

Thermoscope, 1

Thin-ﬁlm optics, 121–123

Torrance-Sparrow model of reﬂection, 228–229

Total internal reﬂection, 109

Traceability, 52–54, 53f

Transmittance

Bidirectional, 98

directional-hemispherical, 121

Triple point of

Argon, 46t

e-hydrogen, 43–46, 46t

helium, 44

hydrogen, 43–46, 46t

mercury, 46t

neon, 46t

oxygen, 43, 46t

water, 40, 46t, 47

Type S (Pt-10%Rh/Pt) thermocouple, 43, 44

Ultraviolet catastrophe, 35, 84

Uncertainty

Measurement, 248

temperature, 5–6, 10, 14, 59, 61, 175, 175t,

176

Type A, 56

Type B, 56

Vacuo, 35, 47

V-groove, 147, 222, 228

View factor, 195, 198, 198f, 214, 215

Ward’s model of reﬂection, 228

Wave equation, 100–104

Wavefront, 101

Wavevector, 74, 76, 101, 102

Wien’s approximation, 7, 8, 251

Wien’s displacement law, 7–8, 24

Wien’s formula, 7, 81, 84, 85f

Zeroth law, see Thermodynamics Zonal method, 195–196

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