UNITEXT for Physics
Giovanni Giusfredi
Physical Optics Concepts, Optical Elements, and Techniques
UNITEXT for Physics Series Editors Michele Cini, University of Rome Tor Vergata, Roma, Italy Attilio Ferrari, University of Turin, Turin, Italy Stefano Forte, University of Milan, Milan, Italy Guido Montagna, University of Pavia, Pavia, Italy Oreste Nicrosini, University of Pavia, Pavia, Italy Luca Peliti, University of Napoli, Naples, Italy Alberto Rotondi, Pavia, Italy Paolo Biscari, Politecnico di Milano, Milan, Italy Nicola Manini, University of Milan, Milan, Italy Morten Hjorth-Jensen, University of Oslo, Oslo, Norway
UNITEXT for Physics series, formerly UNITEXT Collana di Fisica e Astronomia, publishes textbooks and monographs in Physics and Astronomy, mainly in English language, characterized of a didactic style and comprehensiveness. The books published in UNITEXT for Physics series are addressed to graduate and advanced graduate students, but also to scientists and researchers as important resources for their education, knowledge and teaching.
More information about this series at http://www.springer.com/series/13351
Giovanni Giusfredi
Physical Optics Concepts, Optical Elements, and Techniques
123
Giovanni Giusfredi European Laboratory for Non-Linear Spectroscopy (LENS) Istituto Nazionale di Ottica—Consiglio Nazionale delle Ricerche (INO-CNR) Sesto Fiorentino, Italy
ISSN 2198-7882 ISSN 2198-7890 (electronic) UNITEXT for Physics ISBN 978-3-030-25278-6 ISBN 978-3-030-25279-3 (eBook) https://doi.org/10.1007/978-3-030-25279-3 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
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About the Author
Giovanni Giusfredi is currently a senior associate at INO-CNR, having retired in 2016 from INO, where he had been a lead researcher for almost 30 years. He remains actively involved in research and collaborates with colleagues at the European Laboratory for Non-linear Spectroscopy. He is also a founding member of CNR ppqSense. His research interests are wide ranging and he has coauthored more than 220 titles, including 71 articles in international journals. He is skilled and experienced in the teaching of Physical Optics and the History of Optics.
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Chapter 2 Geometrical Optics All what we see, it is seen in a rectilinear direction. Pseudo-Eukleidēs, Catoptrics, 2nd postulate
Introduction Geometrical Optics is one of the oldest of the physical sciences, but still remains the most effective approach for explaining a good part of the most common optical phenomena. It is particularly useful for tracing the propagation of light in inhomogeneous media and for describing or designing optical instruments. The emphasis of this discipline is to find the path of light rays, imagined as geometric lines along which energy flows. It is based on a few simple observations: a) light propagates in a straight line in homogeneous media and, in particular, it is possible to produce thin beams of light, similar to geometrical rays within the physically unattainable limit of an infinite subtlety; b) the laws of reflection and refraction; c) different light beams propagate without disturbing each other; d) “natural” sources are generally uncorrelated between them, for which their light beams overlap without showing interference. On the other hand, the electromagnetic field associated with visible light is characterized by very small wavelengths, on the order of 106 107 m. Therefore, the phenomena that violate the first and the last of the above observations can be observed only with accurate experiments. Indeed, the effects of diffraction or interference are almost hidden using natural sources, for which the visibility of the fringes is reduced. The diffraction phenomena appear when there are rapid changes in the amplitude of the field, such as that produced by a sharp obstacle, particularly when some dimension of the optical system, such as the diameter of an aperture, is comparable to the wavelength; or in the neighborhood of a focal point; or over long distances compared to the transverse dimension of a wave, particularly when there is a delimitation imposed upon it. Lastly, point (c) follows from the linearity of the media at the ordinary beams’ intensity. In this chapter, we will explore the consequences of such observations taken as empirical data. However, we will derive the laws of Geometrical Optics by Maxwell’s equations within the limit at which the wavelength tends to zero. We will also see that, within such a limit, the intensity can be deduced from the transverse dimension of a thin pencil of rays and that the polarization state can be associated with each ray. Therefore, in Geometrical Optics, the rays are associated with the Electronic supplementary material The online version of this chapter (https://doi.org/10.1007/ 978-3-030-25279-3_2) contains supplementary material, which is available to authorized users.
© Springer Nature Switzerland AG 2019 G. Giusfredi, Physical Optics, UNITEXT for Physics, https://doi.org/10.1007/978-3-030-25279-3_2
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Chapter 3 Interference The law is, that “wherever two portion of the same light arrive at the eye by different routes, either exactly or very nearly in the same direction, the light becomes most intense when the difference of the routes is any multiple of a certain length, and least intense in the intermediate state of the interfering portions; and this length is different for light of different color”. Thomas Young [Young 1802, p.387]
Introduction The most impressive evidence of the wave nature of light is given by the bright and dark bands that, under appropriate conditions, are formed in the overlapping zone of two or more beams of light, and which become observable, for example, by interposing a diffuser screen in such a zone. These bands are called interference fringes, and the phenomenon itself is known as interference. Historically, the first documented observations of interference fringes were made independently by Robert Boyle and Robert Hooke, who noted the colored fringes produced by thin films. Hooke, in particular, made a systematic study of them and observed the ones that are improperly called “Newton’s rings”. Newton himself realized the periodic nature of Hooke’s rings, but he gave an explanation of compromise between wave ideas and his corpuscular theory of light that was completely wrong [Landsberg 1979, p. 123]. Finally, Thomas Young, in 1802, correctly interpreted the phenomenon as interference between the reflections from the interfaces, coming to the measurement of the wavelength. To further demonstrate the wave nature of light, Young performed his famous experiment, published in 1807, in which he observed the interference produced by the light coming from two slits [Young 1807]. However, his principle of interference was not accepted until Augustin Jean Fresnel removed the various objections to the wave theory, demonstrating the validity of his diffraction theory. In the following sections, the theory of interference is developed from the equations of electromagnetism in the case of a linear and isotropic medium. In this context, interference finds its justification in the sum of the fields of the various waves and in the principle of superposition. We will first analyze the cases of twowave interference, the conditions for their observation and the interferometers, with some examples of their use, such as in the examination of the optical quality of the surfaces, in the refractive index measurements, and in stellar interferometry. Next, we will study the most significant cases of interference with many waves: the Fabry-Perot interferometer, as a prototype of resonant cavity, and the dielectric multilayers with their applications. Electronic supplementary material The online version of this chapter (https://doi.org/10.1007/ 978-3-030-25279-3_3) contains supplementary material, which is available to authorized users.
© Springer Nature Switzerland AG 2019 G. Giusfredi, Physical Optics, UNITEXT for Physics, https://doi.org/10.1007/978-3-030-25279-3_3
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Chapter 4 Diffraction Lumen propagatur seu diffunditur non solum Directe, Refracte, ac Reflexe, sed etiam alio quodam quarto modo, DIFFRACTE. Propositio I, De Lumine, P. Francesco Maria Grimaldi
Introduction Diffraction, whose name was introduced by Grimaldi in 1665, when he first discovered it and described its effects, has been conveniently defined by Sommerfeld (1949), paraphrasing the Grimaldi’s expression, as «any deviation of the light rays from rectilinear paths which cannot be interpreted as reflection or refraction». For example, if an opaque object is placed between a point source and a screen, the shadow thrown by the object does not have an edge as sharp as the one predicted by Geometrical Optics. In fact, careful observation of the shadow edge reveals that a bit of light goes into the shaded area, while darkened fringes appear in the illuminated area. On the other hand, there is also a similarity between the diffraction produced by a diffractive body and the refraction or reflection from a surface: both effects are due to a sudden discontinuity of the medium and, indeed, the diffraction fringes can be attenuated by the apodization of the obstacle edges, which consists of a gradual variation of their opacity. The phenomenon of the diffraction fringes is quite complex, and there have historically been deep disputes about their origins. Today, we can say that there is both an electromagnetic contribution from the edge of diffractive objects, and a “geometric” contribution due to their form; in this chapter, we will only deal with cases in which this second contribution is relevant and the other can be neglected. Many diffraction studies are devoted to the effects of objects placed in the path of radiation. The term diffraction, however, is also used to indicate a particular method of calculation of the wave field by means of surface or line integrals, as an alternative to methods of propagation through the integration of the wave equations in three-dimensional space. After that of Grimaldi, the first important contribution to the diffraction theory was given by Fresnel, who perfected the Huygens model by introducing Young’s interference principle in place of the envelope of wavelets. Fresnel was thus able to calculate diffraction figures with remarkable precision. Even Young presented a theory of diffraction, based on the visual observation that an aperture edge appears illuminated. Therefore, he suggested that the diffraction fringes were produced by the interference between the radiation transmitted directly from the aperture and a wave coming from the edge, which is now called a boundary-diffracted wave. An equivalent to Fresnel’s theory was then demonstrated by Maggi and Rubinowicz Electronic supplementary material The online version of this chapter (https://doi.org/10.1007/ 978-3-030-25279-3_4) contains supplementary material, which is available to authorized users.
© Springer Nature Switzerland AG 2019 G. Giusfredi, Physical Optics, UNITEXT for Physics, https://doi.org/10.1007/978-3-030-25279-3_4
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Chapter 5 Fourier’s Optics Introduction We have seen that, in Fresnel’s approximation the propagation of a wave between two parallel planes can be expressed in terms of a Fourier’s transform. Here, we examine an alternative technique, relying on the fact that the field present on the first plane can be represented by its spectrum in plane waves, for which, in a homogeneous space, one can determine their propagation in a simple way. By recombining these waves, one can therefore easily rebuild the field on the second plane with an inverse transform. This fact has two important applications, the first concerns the mathematical and numerical techniques that can be used to calculate the diffracted field, and the second is, in a certain sense, opposite to the first and concerns the processing of signals by optical means. In particular, we will briefly discuss some topics that make use of the Fourier’s transform, including sampling theorems and the numerical techniques for the calculation of diffraction, the formation of images and analysis of the quality of optical systems, the theory of coherence and some of its applications, spatial filtering and finally diffraction gratings.
5.1 Mathematical preliminaries 5.1.1 Some special, frequently used functions In optics one often refers to the concept of a point source, when, for example, its dimensions are very small compared to the resolving power of the instrument and then the source is assimilated to a point with a finite emissive power, but also of infinite intensity. Other times, then, an extended source is assimilated into a continuous collection of point sources, with an overall finite intensity. Mathematically, the concept of a point source is rendered by Dirac’s function, which is not an ordinary function, but rather belongs to the category of generalized functions, endowed with particular discontinuity, which have meaning only within an integral. A solid discussion of these functions requires the theory of distributions, but, here, we limit ourselves to understanding them as the limit of a sequence of ordinary functions with “good behavior”. Some functions of this type and others that are particularly useful in optics [Goodman 1996] are as follows. Electronic supplementary material The online version of this chapter (https://doi.org/10.1007/ 978-3-030-25279-3_5) contains supplementary material, which is available to authorized users.
© Springer Nature Switzerland AG 2019 G. Giusfredi, Physical Optics, UNITEXT for Physics, https://doi.org/10.1007/978-3-030-25279-3_5
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Chapter 6 Propagation of laser beams in linear media The Gaussian, Bessel, and Bessel-Gauss approaches Introduction If the medium is homogeneous, linear, isotropic, and non-dispersive, any electromagnetic wave can, in principle, be decomposed into plane waves, which are a simple and effective basis for complex wave equation solutions (1.4.12). But, in real cases, we have to deal with electromagnetic beams of limited size, and the decomposition in plane waves is not always the most appropriate. Consider, for example, the case of a monochromatic, and therefore continuous (CW, that is, continuous wave, single-frequency) laser beam; at every point in the space, its electric field oscillates sinusoidally over time. The spatial trend in this field is not equally simple: at first sight, a collimated laser beam is a genuinely good representation of a pencil of parallel geometric rays between them, but, with more careful observation, we find that the beam tends to expand due to the diffraction. As we have already seen, the solution to diffraction problems is difficult to obtain in general. However, if we limit ourselves to considering waves whose spectrum of wave vectors from the development in plane waves remains contained within a narrow cone centered around a direction of predominant propagation, let’s say, the z-axis, the wave equations can be simplified in a less drastic way than for Geometrical Optics, while maintaining (in many cases) an excellent representation of the diffraction (when, along the path of the beam, there are no diffracting objects). The equation that is obtained is called the paraxial wave equation, and its characteristic solutions constitute a basis for replacing the plane waves. The advantage of this treatment consists in the fact that, in many practical cases, it is sufficient to consider a finite number of such solutions, instead of the infinite or very large number that would be required for a plane wave development. One class of characteristic solutions of the paraxial equation is that of Gaussian beams, which constitute a series of transverse modes and are particularly suitable for treating laser beams. Their field decays as exp(–r2/w2), where r is the distance from the beam axis and w is a parameter that depends on the position along the axis. A different approach to the problem of diffraction derives from the observation that, for a linear medium, the propagation between two planes orthogonal to the zaxis can be seen as the response (output) of a linear system to an excitation input. In particular, as we have seen above, the “classical” treatment of diffraction conElectronic supplementary material The online version of this chapter (https://doi.org/10.1007/ 978-3-030-25279-3_6) contains supplementary material, which is available to authorized users.
© Springer Nature Switzerland AG 2019 G. Giusfredi, Physical Optics, UNITEXT for Physics, https://doi.org/10.1007/978-3-030-25279-3_6
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