Micro-optomechatronics

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Micro-Optomechatronics

Hiroshi Hosaka Graduate School of Frontier Sciences The University of Tokyo Tokyo, Japan

Yoshitada Katagiri NTT Microsystem Integration Laboratories Nippon Telegraph and Telephone Corporation Atsugi, Japan

Terunao Hirota Graduate School of Frontier Sciences The University of Tokyo Tokyo, Japan

Kiyoshi Itao Graduate School of Management of Science & Technology Tokyo University of Science Tokyo, Japan

Marcel Dekker

Copyright © 2005 Marcel Dekker, Inc.

New York

Kyoritsu Advanced Optoelectronic Series is credited for providing an English translation from a portion of a Japanese publication issued by Kyoritsu Shuppan (1999). Although great care has been taken to provide accurate and current information, neither the author(s) nor the publisher, nor anyone else associated with this publication, shall be liable for any loss, damage, or liability directly or indirectly caused or alleged to be caused by this book. The material contained herein is not intended to provide specific advice or recommendations for any specific situation. Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress. ISBN: 0-8247-5983-4 This book is printed on acid-free paper. Headquarters Marcel Dekker, 270 Madison Avenue, New York, NY 10016, U.S.A. tel: 212-696-9000; fax: 212-685-4540 Distribution and Customer Service Marcel Dekker, Cimarron Road, Monticello, New York 12701, U.S.A. tel: 800-228-1160; fax: 845-796-1772 World Wide Web http://www.dekker.com The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the headquarters address above. Copyright ß 2005 by Marcel Dekker. All Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Current printing (last digit): 10

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Preface

Micro-optomechatronics is a technology that fuses optics, electronics, and mechanics by the MEMS technology. This technology is used primarily for information and telecommunications equipment. This book explains the basis and the application of micro-optomechatronics. In information operation, mechanical movements are not required. Use of movement in space, however, often simplifies systems structure and increases the signalto-noise ratio of transducers remarkably over a system constructed only with solid-state components. There are many examples of information instruments that use optics, such as optical memories, optical communication devices, and optical measurement instruments. Moreover, control systems made of mechanical components and electronic circuits are necessary for precise space movement. Here, the fusion of optics, electronics, and mechanics is generated. Generally, speed and precision of motion are improved by the miniaturization of movable parts. In addition, the load is small, and the range of movement is narrow in information devices. Thus, the application of MEMS technology needs to be studied extensively. This book systematically discusses many micro-optomechatronics devices. First, all devices are classified into groups depending on the control methods of power and the position of the laser beam. Next, the devices are explained in detail according to the classification of control methods. Finally, optics and dynamics, which are the theoretical background of control methods, are discussed. This book is aimed chiefly for university students, graduate students, and research engineers in the mechanical and electronics industries. It presumes that readers will have knowledge in dynamics and electromagnetism taught in general education courses in universities. In this book, laser

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oscillation, Maxwell’s equation, the mechanics of materials, fluid dynamics, and machine dynamics are explained. A major portion of this book is an English translation of a Japanese book issued by Kyoritsu Shuppan in 1999 and the authors kindly acknowledge the Kyoritsu Advanced Optoelectronic Series for use of this material. This book also discusses the next generation of optical memory, in a section written originally for this book, because the progress of optical memory is fastest in this field and new technologies have been generated in these last four years. This book, first, explains examples of microoptomechatronics devices in information and communication systems. Then the basis of optics and dynamics are explained as it is necessary to understand the theoretical background of these devices. Chapter 1 (K. Itao) deals with the world of micro-optomechatronics. History, applications, and component technologies are explained. Chapter 2 (H. Hosaka, K. Itao, and Y. Katagiri) presents a technological outline of micro-optomechatronics. An outline of power control and position control of a laser beam, which is the performance decision factor of micro-optomechatronics, is also described. The method of both controls is classified. Details of each method are explained in the following chapters with application devices. Chapter 3 (Y. Katagiri) outlines intermittent positioning in micro-optomechatronics. This chapter details devices used in information and communication systems. In this chapter, devices that use intermittent positioning for laser beam control are also explained. The laser with tunable cavity, the pulse source laser, and an optical filter are discussed in detail. Chapter 4 (Y. Katagiri) deals with constant velocity positioning in micro-optomechatronics. The optical filter as used for optical communication systems is explained. Chapter 5 (H. Hosaka and Y. Katagiri) concerns follow-up positioning in micro-optomechatronics. Optical disk drives and their focusing and tracking servomechanisms, sampled servo systems, flying heads, and a laser sensor with a composite cavity are discussed. In Chapter 6 (Y. Katagiri) we deal with the fundamental optics of micro-optomechatronics. In this chapter and the next, basics optics and dynamics, which are useful for understanding the theoretical background of micro-optomechatronics, are described. The Maxwell equation, the wave propagation equation, and the laser oscillation are also discussed here. Chapter 7 (H. Hosaka) discusses the fundamental dynamics of microoptomechatronics. The dynamics of elastic beams, fluids, and microsized objects are also explained.

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Chapter 8 (T. Hirota and K. Itao) concerns a novel technological stream toward nano-optomechatronics. Nanotechnology and a near field optical memory are discussed and explained in detail. Hiroshi Hosaka Yoshitada Katagiri Terunao Hirota Kiyoshi Itao

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Contents

Preface

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Chapter 1

The World of Micro-Optomechatronics 1 What is Mechatronics? 2 The Trend of Innovation 3 Positioning of Micro-Optomechatronics 4 Microdynamics and Optical Technology References

1 1 5 9 10 13

Chapter 2

Technological Outline of Micro-Optomechatronics 1 Precision and Information Devices Created by Optical Technology 2 Essence of Micro-Optomechatronics Technology 3 Control of Optical Beam Intensity 4 Control of Optical Beam Position References

15

Intermittent Positioning in Micro-Optomechatronics 1 Moving Micromirrors and Their Application 2 Micromechanical Control of Cavities Based on Slide Tuning Mechanism and its Applications References

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Constant Velocity Positioning in Micro-Optomechatronics 1 Phase-Locked Loop for Constant Velocity Positioning 2 Linear Wavelength Scanning 3 Practical Examples of Linear Wavelength Scanning References

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Chapter 3

Chapter 4

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100 108 111 125

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Contents

Follow-Up Positioning in Micro-Optomechatronics 1 Follow-Up Positioning in Conventional Optical Disk 2 Follow-Up Positioning of Optical Disk Head Mounted on Flying Head 3 Displacement Sensors Based on Coupled Cavity Lasers References

127 127

Fundamental Optics of Micro-Optomechatronics 1 Fundamental Optics 2 Optical Resonators and Their Applications 3 Optics of Dielectric Thin Films 4 Extraordinary Electromagnetic Waves in Condensed Matter with Free Electrons References

161 163 182 197

Chapter 7

Fundamental Dynamics of Micro-Optomechatronics 1 Dynamics of Microsized Objects 2 Equation of Motion of the Beam 3 Fluid Dynamics around Microsized Objects 4 Movement of the Beam with Air Resistance 5 Stick–Slip Caused by Friction Force References

225 225 226 243 249 257 262

Chapter 8

Novel Technological Stream Toward Nano-Optomechatronics 1 The Coming of Nanotechnology 2 Nano-Optomechatronics for Optical Storage 3 Summary References

265 265 268 289 289

Chapter 6

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1 The World of Micro-Optomechatronics

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WHAT IS MECHATRONICS?

Almost one billion years ago, when life appeared on earth, information existed as genes in cell nuclei. Life evolved to higher forms as the genes changed. Humankind, on the top of evolution, created language. Circulation of any information to anyone was enabled by means of language. The invention of writing supported circulation of information for practical use by storing it. Information storage was dramatically improved by the epochmaking invention of paper. Modern printing technology, another invention, accomplished by Gutenberg, enabled worldwide circulation of huge amounts of information. When modern times arrived, a traffic revolution broke out as a part of the Industrial Revolution, and the circulation of information was promoted. Another revolution in communication broke out with the invention of Morse code. This was the beginning of the telecommunication era. This telegraph technology was eventually taken over by telephony, which was further improved to digital communication technology using computers. Digital communication technology integrated telegraphy and telephony into data communication technology based on the Internet Protocol. Now we stand at the multimedia age (Fig. 1). Important discoveries in the natural sciences show a concentration from 1900 to 1960, but the principal industrial inventions were achieved in the second half of the twentieth century. Japan was acknowledged as a worldwide leader of industry in the last quarter of the 20th century as Japan achieved great success in various industries, including not only the automobile, shipbuilding, and semiconductor industries but also precision machinery, providing products such as watches and cameras as well as 1

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Figure 1 The history of communications.

electronics-based products including home appliances and information equipment. Mechatronics has been also much advanced simultaneously with industrial development. Such a quarter-century is remembered as a Golden Age in Japanese history [1]. Mechatronics technology is hierarchically classified, from the point of view of function, into materials, parts of machines and electronics, equipment (devices), and systems. These elements of mechatronics are supported by fundamental technologies concerned with not only fabrication and measurement but also data processing including modern control schemes [2]. Table 1 presents how mechatronics technology supports a wide variety of industries existing today. Figure 2 is a tree-shaped diagram to show the relationship between industry and corresponding technology. This figure is from the Mechatronics Education and Research Motion, promoted by the Mechatronics Subcommittee with its chief examiner Professor Suguru Arimoto, under the supervision of the Automatic Control Research Coordination Committee of the sixteenth Science Council of Japan [3].

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Mechatronics-Related Technologies Supporting Each Industry

Information and communication industry

Consumer electronics industry Heavy electricity industry

Industrial machinery industry

Business machinery industry Medical and welfare products industry

Automobile and transportation industry Aerospace industry

Naval industry Railways industry Construction works industry Environmental industry

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Design, manufacturing, mass production techniques of semiconductors, liquid crystal, and magnetic head. Composition and mass production techniques of information input/output and storage devices. Automation technique of communication lines construction work. Wearable micromachine technology. Design, manufacturing, mass production, recycling, interface, and energy conservation design techniques of AV products. High-efficiency power generation, electric power preservation, power control and management techniques. Industrial plant, atomic reactor maintenance techniques. Radioactive waste treatment system with low environmental impact. High-speed, and high-precision machine tool techniques. Technology for making NC an open architecture. Manufacturing system integration techniques. Inverse manufacturing technique. Design and manufacturing techniques of fax, printer, and copy machine. Digitalization, systemization, and miniaturization techniques. Technology for cancer medical treatment apparatus and high-precision image processing equipment. SOR and electron beam diagnosis equipment technology. Patient transfer system. Home care medical equipment technology. Wearable information systems for physiological information monitoring. Intelligent engine technology for ultralow pollution. Recycle technology. Car safety control technology. Car navigation and intelligent transportation system. Super high-speed engine integrated control technology. Danger evasion system. Active vibration suppression technique. Fault diagnosis technology. Spatial robot remote manipulation technology. Welding and coating automation technology. Simulation technology. Attitude control and obstacle detection technology. Underwater robot technology. Technology for high-speed trains using vibration and inclination control. Collision simulation technology. Railroad track state automatic measurement system. Active and passive vibration control technology. Building construction work automation. Coating robot. Vibration estimation simulation technology. Environmental information sensing technology. Waste treatment equipment technology. Artificial environment design technology. Recycle system technology.

The World of Micro-Optomechatronics

Table 1

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Figure 2 Ref. 3.)

Chapter 1

Mechatronics-related technologies supporting each industry. (From

The technical term mechatronics was born in Japanese industry [4]. As a word for a new technology, it came to be internationally used at the beginning of the 1980s. Mechatronics pushed Japan to the top as a leading country in the supply of original high-tech products to the world. At first the word merely expressed the miniaturization of products and the unification

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of electrical and mechanical appliances; in 1990 or thereabouts, mechatronics has been understood worldwide and recognized as a new current in technology. In the same period, a special international journal, Mechatronics, focusing on the subjects of interest in these areas, began to be published by Pergamon Press in England with Professor R. W. Daniel of Oxford University as a chief editor. Four issues were published every year from 1991 to 1997, and eight have been published per year since 1998. Daniel stated in the first issue that the word mechatronics best describes the remarkable contribution of Japan to these interdisciplinary technologies by which automation and robotic conversion of the factory have been carried out to supply advanced electrical and mechanical products such as cameras, camcorders, compact disks, and CD players. Two major scientific organizations in U.S., the Institute of Electrical and Electronics Engineers (IEEE) and the American Society for Mechanical Engineers (ASME), started a program of collaboration in publishing the IEEE/ASME Transactions on Mechatronics, whose first issue appeared in March of 1996. Two Japanese professors contributed to this program; the editorial policy was drafted by Fumio Harashima, and Masayoshi Tomizuka described in the first issue how important the magazine is to provide an opportunity for scientists and engineers belonging to the two completely different scientific parties to exchange their ideas. In the earlier issues of the magazine, mechatronics was temporarily defined as ‘‘the synergetic integration of mechanical engineering with electronic and intelligent computer control in designing and manufacturing industrial products.’’ The point was that when the robotic market was just approaching ten billion U.S. dollars, the mechatronics market was estimated at ten times larger than the robotic market. This was an underestimate; if the estimation was carried out in Japan, it could be enormously enlarged by integrating the related markets concerned with automobiles and multimedia appliances.

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THE TREND OF INNOVATION

Looking back over the progress in physics, we realize that classical physics based on Newtonian mechanics had a great impact on the Industrial Revolution, which started at the beginning of the second half of the 18th century, and which until today has been influential in subsequently developed technology and industry. Newtonian mechanics shows the best applicability in the macroworld, in which objects of interest are visible to the naked eye. Invention was carried out based on mechanics, and novel mechanical products were launched; human power was substituted for by

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mechanical power generated by steam engines large enough to drive automobiles, ships, and other general machines. In short, Newtonian mechanics has developed heavy industry. However, we are now standing at the turning point and reconsidering heavy industry, which has caused global environmental destruction owing to its heavy consumption of natural resources to meet great demand. On the other hand, quantum mechanics, being the core of modern physics established in the first half of the twentieth century, is the driving force of the high-tech revolution in the twenty-first century and the second half of the twentieth. Quantum mechanics was first developed to describe phenomena in the supersmall world of atoms and atomic nuclei; then it was applied to explaining the behaviors of electrons in semiconductors. The high-tech revolution in the twentieth century in a wide variety of fields—including information, electronics, biology, new materials, and micromachines— came from semiconductor technology (Table 2) [5]. Figure 3 shows a scheme for the development of miniaturization. For promoting miniaturization, further study must be carried out on integrated circuits, integrated mechanisms, and integrated intelligence. The realization of new features with high-efficiency and high-level functionality becomes possible through the implementation of numerous microscopic artifacts by using these techniques. Using these basic technologies and adopting the latest computer technologies such as image analysis and structure analysis software, mechatronics is improving toward new technologies that boost the added value of artificial products expressed by the terms system integration and system synthesis. Today, Japan is seeing the rapid aging of the populace, and technology is strongly required to serve the medical needs and the welfare of the aged. Although the situation is different in each country, the problems are similar. Developing countries are promoting rapid industrialization and will soon overtake developed countries. If we continue consuming energy, the day is not so far distant when environmental issues will become major global problems. The provision of energy and food will become increasingly important. In the twenty-first century, science and technology will be asked to contribute to the care of the elderly, to the general welfare, and to the terrestrial environment; thus a technology that saves resources and energy will become more important. Many companies will have to collaborate on the industrialization of such technology. Mechatronics is fundamental and will be useful for realizing the goals of technology as mentioned above. Many Japanese companies have experienced the industrialization of mechatronics and related technology for last quarter

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Progress of Sciences and Transformation of Industrial Structure

Sciences

Classical physics Newtonian mechanics (end of 17th century) Thermodynamics, electromagnetics, inorganic chemistry

Main subject

The macroworld (size visible by the human eye) Energy innovation (muscle substitution)

Impact on technology Application field in the industry

Industry keyword Influence on earth environment Impact on industrial history

Thermomotor, iron manufacture, shipbuilding, automobile, chemical industry at initial stage Large and heavy Energy/resources consuming (severe) Supported the industrial revolution started in the second half of the 18th century

Modern physics Quantum mechanics (beginning of the 20th century) Nuclear physics, organic chemistry, molecular biology The microworld (size not visible by the human eye) Information innovation (cranial nerves system substitution) Electronics, atomic energy, new materials, petrochemistry, biotechnology

The World of Micro-Optomechatronics

Table 2

Small and light Energy/resources sparing (kind) Supported the high-tech revolution starting in the second half of the 20th century and extending through the 21st century

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Figure 3 Expansion of miniaturization technology.

of the twentieth century and hence will contribute to the collaboration with other countries. Computers are devices for data processing, and they carry out communication between humans and machines. All sorts of multimedia appliances are used to store, edit, and produce sound, images, and pictures. Nevertheless, it is important to be substantial in the real world where humans and machines perform versatile works for manufacturing in factories, and for medical treatment and rehabilitation in hospitals and related facilities, environmental activities, and various domestic duties. Such substantial activities are achieved by mechatronics. Thereby mechatronics links the virtual (computer) world and the real world. In the twenty-first century, mechatronics will have to be extended to a technology that unifies computer and human daily activity. In other words, human-oriented mechatronics should positively contribute to many problems in medical care, human welfare, and elderly care. Furthermore, through similar unification with the natural world, nature-oriented mechatronics will be accomplished. It will contribute to improving the earthly environment and eliminating various problems in not only environmental conservation procedures but also sensor monitoring systems investigating various natural phenomena including organic reactions in the human body. A new technological evolution is now coming out for the conservation of natural resources and the saving of energy.

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The World of Micro-Optomechatronics

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9

POSITIONING OF MICRO-OPTOMECHATRONICS [6]

It is said that the origin of machinery control technology corresponds to the invention of a governor by James Watt, at the end of the eighteenth century. Moreover, there is the growth of the automobile, aircraft, and shipbuilding industries from the beginning of the twentieth century, and in 1957 an artificial satellite was launched, a milestone in the history of machine control. Furthermore, at this time, Harrison of MIT realized a high-precision ruling engine using electric control, and basic research on numerically controlled machine tools started. Given the history above, let us follow up with the germination of new technology related to mechatronics. As shown in Table 3, at the beginning of the 1960s, the process of automation started, and the second half of the 1960s saw the period of mechanical automation using electric control technology. Furthermore, entering the 1970s, the era of the combination of electrical and mechanical elements using IC and LSI, namely the mechatronics era, started, and in the second half of the 1970s, the use of the microprocessor met the era of real mechatronics, combining mechanics, electronics, and information. In this period, the laser diode was invented in cc1962 and made a continuous oscillation at room temperature in 1970.

Table 3

Period of New Technologies Germination

Period

Progress of technology and signs of new period

1960– 1965–

Process automation in chemical industry and heavy machinery industry Period of mechanical automation by the introduction of electric control technology Period of combination of mechanics and electronics by the introduction of IC and LSI electronic technology (initial stage of mechatronics) Period of combination of mechanics, electronics, and information by the introduction of microprocessor (mechatronics) Period of combination of mechanics, electronics, information, and optics by the introduction of laser diode (optical mechatronics) Period of combination of electronics, physics, mechanics, information, and optics by the introduction of micromachining (micromechatronics) Period of combination of optics, chemistry, physics, electronics, mechanics, and information realizing the synthesis of information and energy (micro-optomechatronics) Period of synthesis of nanomachine, nanocontrol, and nanosensing (nanomechatronics) Period of imitation of living organism (nanobiomechatronics)

1970– 1975– 1980– 1985– 1990–

1995– 2000–

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Then at the beginning of 1980, optical mechatronics technology, combining mechanics, electronics, information, and optics, put out its buds. But even though these technologies were put together, true fusion was still far away. In the late 1980s, through the application of micromachining technology for semiconductors to machine elements, research on microsized sensors and actuators became vigorous, giving the possibility of realizing various sensors and microactuators. Then the era of micromechatronics arrived, which sought the miniaturization of mechatronics systems to their limits and synthesized all functions on a chip. Turning to the 1990s, the time was heading for the period of microoptomechatronics, which was born from micromechatronics technology with light. Furthermore, in the second half of the 1990s, we entered a period of nanomechatronics, where nanomachines of molecular and atomic size took the main parts in cooperation with nanocontrol and nanosensing. Further, it will grow into imitation technology for living things and precise arrangements such as DNA’s helix structure and muscle mechanism, and the nanobiomechatronics period will come along eventually. In addition to the progress of these advanced component technologies, image processing, control theory, and other computer application technologies have started to integrate. A system integration, a horizontal development, is next pursued, and, with new functional devices developed, new manufacturing technology is continuously being invented centered on the industrial world.

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MICRODYNAMICS AND OPTICAL TECHNOLOGY

There are many artificial and organic systems implementing high-level functions by using microscopic movement, such as insects’ movement, the lymph flow of animals’ semicircular canals, eyeball microscopic motion, the motion of the ink-jet printer’s ink particle, atomic force and scanning electron microscopes, and very high density memory probe motion. In short, using not only the solid-state elements of semiconductors but also microscopic movements, machine systems often and drastically increase their performance. In Fig. 4, mechatronics technology used in information systems is classified into three categories; microscopic energy, micromechanisms, and micromovement measurement and control; and concrete technological themes are illustrated. First of all, as for microscopic kinetic energy technology, (1) understanding of energy flow, (2) energy supply, and (3) energy

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Figure 4 Main items of microscopic motion systems technology.

transformation are probably the main techniques. Regarding (1), as to equipment size from centimeter to micrometer, it is necessary to explain energy loss caused by airflow resistance, structural damping, and supporting point loss. Furthermore, it is necessary to elucidate energy loss caused by the interference of element cantilevers used in comb actuators that have several hundred microactuators in them. Regarding (2), it is necessary to investigate the wireless driving method of microcantilevers by laser light and electromagnetic waves and to investigate the microgeneration mechanism using oscillators or rotors. Finally, concerning (3), an efficient energy conversion method using resonant vibration is important. Next, related to micromechanisms, the following research is necessary: (1) structural design, (2) the development of the actuator, and (3) microdynamics data accumulation. Regarding (1), there are various mechanisms based on the microcantilever: the V-groove sliding mechanism, the microrotation mechanism, the inchworm mechanism, and the microhinge mechanism. Regarding (2), a great number of actuators for microscopic movement using piezoelectric elements, electrostatics, electromagnetism, or laser beams are promoted. Considering (3), it will be necessary to accumulate experimental and theoretical data of tribology and stick-slip that appear in the positioning of microsized movement where the inertial force is negligible, such as in positioning of optical fibers and also the data of microtapping that appears in the AFM (atomic force microscope) and the SNOM (scanning near field optical microscope).

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Lastly, about micromovement measurement and control, various techniques such as (1) microscopic oscillation elucidation, (2) microsensor development, (3) surface shape observation, (4) microbody recognition control, and (5) system evaluation are essential. Regarding (1), researches on biorhythm, microscopic oscillation, insects’ movement measurement, and transient observation technique of microscopic force coming from static friction to kinetic friction in micromotion are important. Considering (2), the development of sensor elements using microcantilevers and sensing systems for miniature three-dimensional position measurement system due to geomagnetism, gravity, acceleration, and Coriolis force are important. Considering (3), importance is put on the development of the threedimensional surface shape measurement method using a three-dimensional electronic beam measuring instrument or a scanning electron microscope. As for (4), the tracking method of a microscopic object for recognition and image processing is necessary. Considering (5), the evaluation method of mechanical characteristics of microscopic object is important. Figure 5 shows the classification of micromotion observed in information and precision systems: continuous, intermittent, and passive movements. If we take out the major phenomena dominating micromotion from there, the resonance phenomenon, the stick-slip phenomenon, the static friction and kinetic friction mixture phenomenon, and the tapping phenomenon (the microscopic collision phenomenon) appear. The microscopic vibration theory constitutes the basis of the above microdynamics technologies. In nature, we can see the microoscillation

Figure 5 Micromotion and dynamics.

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phenomenon in many places: the movement of celestial bodies, atomic and molecular oscillations, pendulum movement, and tide flow. In living things, microscopic vibration exists in birds’ twitter, hummingbirds’ hovering, the heartbeat, eardrum vibration, and the subtle oscillation of skin. Edison and Bell used microscopic vibration phenomena in the gramophone and the telephone, and it became the roots of information home appliances. Finally, in recent years, information-sensing equipment and precision information equipment that use microscopic vibration are developed in great numbers. As an example of the former, there are the microscopic telephones, microphones, and microearphones in the acoustical vibration field, the piezo ink-jet printer and the microscanner in the vision field, the odor sensor by crystal oscillator in the smelling field, and the contact sensor by oscillator and vibrator in the mobile telephone in the field of touching. Also, as examples of the latter, there are the SPM (scanning microscope), ultrasonic sensors, vibration transportation devices, and micropower generators [7]. In this way, together with information systems’ miniaturization, machinery became organized on microvibrations, as if it were imitating living beings. When optical technology joined microdynamics technology, optical micromechatronics technology was born. The following chapters will explain the world of the unification of microdynamics and optical technology in detail.

REFERENCES 1. 2. 3.

4. 5. 6.

7.

Itao, K. Mechatronics of Electronics, Information and Communication; Institute of Electronics, Information and Communication: Corona, 1992. in Japanese. Itao, K. Technological portrait of opto-mechatronics. Mechanical Design 1992, 36, 10. in Japanese. Takano, M.; Arimoto, S.; Futagawa, A.; Kosuge, K.; Itao, K.; Kurosaki, Y. Proposal to Mechatronics Education and Research. Automatic Control Research Coordination Committee Report, The Sixteenth Science Council of Japan, Also presented in Itao, K. Mechatronics systems’ locus. Journal of the Japan Society for Precision Engineering 1999, 65, 1. in Japanese. Mori, T. Technical appearance of mechatronics. Journal of the Japan Society for Precision Engineering 1991, 57 (12), 2089. in Japanese. Mituhashi, T. High-technology and Japanese Economy. Iwanami: Iwanami Shinsho, 1992; 24pp. in Japanese. Itao, K. The development of optical micromachine technology. Optical micromachines. Journal of Japan Society of Applied Physics 1998, 67 (6). in Japanese. Itao, K. Information Microsystems—Microvibrations Theory. Asakura, 1999; in Japanese.

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2 Technological Outline of Micro-Optomechatronics

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PRECISION AND INFORMATION DEVICES CREATED BY OPTICAL TECHNOLOGY

To explain relationships between fundamental characteristics of optical and mechanical functions, optical devices are classified by energy and information effects in Fig. 1. Energy effects [1] are divided into radiation pressure chemical changes, heating effects, and optoelectric (OE) conversion. The heating effects are divided into magnetism change, phase change, swellings, and melting. Information effects are based on the characteristic properties of waves and usually use propagating light. In recent years, near field light localized at dielectric surfaces has also come into use. Propagating light is characterized by traveling in straight lines, interference, diffraction, reflection, refraction, polarization, and resonance. Many devices of microoptomechatronics are realized based on such properties. In the first application field, there are communication devices. The optical magnetism relay [2] and the optical distortion relay [3] use the energy effect, and the optical fiber switch [4], the waveguide switch [4], the optical MDF (main distributing frame), the wavelength tunable laser, and the optical disk filter use the information effect. In the second application field, there are information memories. Data recording is carried out on magneto-optical, phase-change, and rewritable compact disks by using the energy effects. Data reproduction, tracking, and focusing are carried out for all kinds of disks based on the information effect of light. In the third application field, there is input/output equipment. Digital micromirror devices (DMD) [3], laser printers, blurring-free VTRs, autofocus cameras, and scanners work based on the information effect. A photophone [5] uses the energy effect. In the fourth application field, there are measurement apparatus. They include 15

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Figure 1

Chapter 2

Basic characteristics and application for micro-optomechatronics of light.

an optical fiber gyroscope [6], the optical tiltmeter [7], the CCL sensor, the SNOM (scanning near field optical microscope) [8], and the microencoder [3], all of which are based on the information effect. The optical thermooscillator [4] use the energy effect. In the fifth application field, there are processing, handling, and other power-oriented equipment. These applications include those for the microworld, such as optical tweezers, optical grippers, optical distortion actuators, laser processing machines, and the optical molding machine; all of these use the energy effect. Most devices that use the information effect are already commercialized. Commercialized devices based on the energy effect include optical disk recording, optical molding, and laser processing equipment. Noncontact motion drives are prosperous in the technology of the research level that uses the energy effect. Because the driving force by optical energy is very small, objects to be manipulated are limited to minute ones. So it is applied mostly to information devices, for example, the movement of relay electrodes and the handling of optical parts. The actual controlling technique of optical beams is explained in the following sections.

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Technological Outline of Micro-Optomechatronics

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ESSENCE OF MICRO-OPTOMECHATRONICS TECHNOLOGY

Why are mechanical technologies used for information processing that essentially needs no objective movement? The answer is simple; the system is simplified and an S/N ratio is improved remarkably by using space movement compared with systems consisting only of solid-state elements. In optical micromechatronics, how to control the intensity and position of the optical beam is essential. In short, optical mechatronics technology can be defined as the precise control technology for optical beams. Each of the control technologies is explained in the following sections.

2.1

Intensity Control of Optical Beam

In optical micromechatronics, many functions are realized by controlling the intensity of optical beams both temporally and spatially as shown in Table 1. In the time domain, it is most useful to control the optical strength by using a small semiconductor laser. Because the semiconductor laser emits photons by converting input electrical energy to optical energy, the output power can be controlled easily and quickly (at a maximal frequency of several gigahertz) by modulating the input current. So this method is widely used for data coding in information and communication devices. In microoptomechatronics, this method is also used for driving optical-thermo oscillator and photophones. By using the property of coherent short optical pulses, it is possible to achieve extremely high intensity. Many light wave components whose frequencies are precisely controlled can be concentrated

Table 1

Classification of Optical Beam Intensity Control

Control domain Method

Application

Time domain

Space domain

Control of pouring current of semiconductor laser Mode synchronization Optical thermo-oscillator (bending moment excitation) Microrelay Photophone (sound wave excitation) Material processing by pulse light source

Lens (positioning, forming)

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Diffraction Optical tweezers Optical disk (pit recording) Hologram (exposure)

18

Chapter 2

in the time domain to generate a sharp beat waveform. This method of generating short pulses is called mode-locking. Since most materials melt in high-intensity light, this method can be used for material processing and basic experiments for nuclear fusion. In the space domain, the intensity of an optical beam is controlled by making use of spatial nonuniformity of refractive index. Beam convergence by lens is a major example of it and is used for the generation of pits on optical disks. Optical tweezers that trap minute objects are achieved by using the intensity gradient formed near the focal point of a lens. Light intensity control is also carried out through making a diffraction pattern. For example, a diffraction pattern can be designed to have the lens function that concentrates optical energy of the plane wave homogeneously distributed in space to a desired point. A typical example of this diffraction pattern formation is holography, which is used for optical memories and displays. 2.2

Position Control of an Optical Beam

The technology for optical beam position control is classified by accuracy and method as shown in Table 2. Accuracy of positioning is classified into three categories by aspects of light. In the first category, optical power is used; accuracy is defined by the size of the receiving and emitting elements (around 1 mm). In the second category, optical interference is used; required accuracy is several tenths of a wavelength (around 0.1 mm). The focusing servomechanism of optical disks uses a wave property of light, but because it does not use interference directly, the required accuracy is a little low and is about the length of a wavelength (around 1 mm). In the third category, optical phase or near field light is used, or loss and accuracy are strictly specified; accuracy is requested to much less than 1/10 of wavelength. There are three methods in positioning; intermittent, continuous, and follow-up. In intermittent positioning, the object is positioned from point to point; it is used in tuning laser wavelengths and assembly processes. A route between the target points is arbitrary. In micro-optomechatronics, in order to position a minute object with high accuracy, actuators with high resolution are needed. Also in order to reduce a positioning time, movable parts should be as light as possible. Moreover, compensating for the friction force is necessary, because this force becomes dominant in minute objects. In Chap. 7, positioning under large friction force is explained. Continuous positioning moves an object under the condition providing a moving position and/or speed that have been determined in

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Positioning system Intermittent Positioning accuracy

Usage of optical power (>1 mm)

Usage of optical interference (>0.1 mm) Usage of optical phase (<0.1 mm)

Photophone DMD Optical tiltmeter Waveguide switch Magneto-optical actuator Optical distortion actuator Optical fiber Waveguide Connector (radial direction) Optical disk stamper Connector (axial direction) Wavelength tunable laser Optical gyroscope, disk filter

Continuous

Follow-up

Scanner

Autofocused camera Blurring prevention VTR

Encoder Optical disk rotation

Optical disk focus servo Optical disk track servo Optical disk slider Optical tweezers CCL sensor SNOM tapping

Disk filter

Technological Outline of Micro-Optomechatronics

Table 2 Positioning Method and Accuracy at Micro-Optomechatronics

19

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20

Chapter 2

advance. Since inertial force and viscous resistance affects the motion, positioning becomes more difficult than with intermittent positioning. Typical examples are seeking control of optical disk heads and rotation control of a disk-shaped optical filter. The simplest continuous control schemes refer to a constant velocity control. The disk filter is a typical example: wavelength accuracy of around 0.01 nm is achieved by rotating a disk at a constant speed. Since the rotating speed will be changed by friction or external disturbance force, it is necessary to compensate for these disturbances to maintain a constant velocity. Such high-performance control is accomplished by measuring the transient rotation angle of the disk using a sensor to generate a phase error signal from a clock signal as a reference, and by controlling the motor torque so as to reduce the error signal to zero in a negative feedback loop. The follow-up positioning is a kind of continuous positioning and makes the second object follow at a constant distance to the first object, which moves in an unknown manner. Although the absolute position of the second object is needed to detect it in the usual continuous positioning, the relative position of the first and the second object is needed to detect in the follow-up positioning. A typical example is the tracking control of optical disks. It is necessary to make a laser beam follow the pit with a diameter of about 1 mm and with a moving speed of about 10 m/s with an accuracy of about 0.1 mm. Section 4 explains the dynamic analysis of continuous positioning and follow-up positioning.

3

CONTROL OF OPTICAL BEAM INTENSITY

In this section a technology concerning light intensity control and its applications is described among the optical beam control technologies. 3.1

Usage of Optical Radiation Pressure

When light is applied to an object, a part (or all) of the momentum of the photon is transferred to the object. Consequently, the object receives force from the photon according to the law of conservation of momentum. Such force is called the radiation pressure. Radiation pressure has been detected by experiment since old times. Figure 2 shows the experimental system by Stimmer in 1964 [10]. The rotating moment was generated by irradiating the high-power laser beam to rotation mirrors hung by wire in a vacuum chamber. The torque is obtained by detecting the angle of rotation of the mirror by this moment from the angle of reflection of the laser.

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21

Figure 2 Measuring method of a photon. (From Ref. 10.)

The radiation pressure is analyzed by the same method as that for solid particles. Let us study the case in which light with intensity I is vertically applied to a high-reflectivity mirror that is at a standstill, as shown in Fig. 2. The change in momentum becomes 2h
/c when a photon reflects by colliding with the mirror. Because the number of photons colliding with the mirror at time intervals dt is I dt/h
, the impulse the mirror receives is F t ¼

I h
2It t 2 ¼ h
c c

ð1Þ

When the irradiation area of light on the surface of mirror is assumed to be S, the radiation pressure P becomes P¼

2I cS

ð2Þ

Let us consider the case in which light 1 mm in wavelength, having an intensity of 10 mW, is applied to the area of 1 mm2. The optical beam corresponds to a flow of photons of 5
1016 per second. Carrying out the calculation according to the above equation, we estimate the radiation pressure as 60 N/m2, and we also estimate the force that acts on the entire irradiation area as 60 pN. It is almost equal to gravity acting on an object with one side length of about 7 mm and a density of 2 g/cm2. The laser manipulation technology is a method to catch and handle small particles in a space by using the radiation pressure. The fundamental concept was proposed by Ashkin. It is now being actively used as optical tweezers in the fields of biology, chemistry, and physics to

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22

Chapter 2

perform trapping and transferring of the living body cell and the microcapsule. When the optical power is P ¼ 0.3 (W), the radiation pressure F is estimated as F¼

P ¼ 109 c

ðNÞ

ð3Þ

where c is the speed of light in a vacuum. Thus the pressure of light is extremely small. However, if the object is very small, this force can become more significant. Figure 3 explains the principle of how the radiation pressure acts on a minute transparent object. Radiation pressure is generated at every point the light is refracted. The total force is obtained by integrating the pressures at all the points. (This is shown by an arrow in the figure.) The strength of the acting power shows strong dependence on the size and shape of the particle, and the refractive index difference between the particle and the surrounding material. Hence optimizing the acting power according to the particle of interest, we can freely manipulate the particle in a free space. For instance, the radiation pressure can float a particle made of transparent dielectric material in the air, eliminating the influence of the gravity [11]. In the instrument suspending an object in the air as shown in Fig. 4, the particle diameter is 20 mm and the power of the Ar laser is 150 mW. The vertical position of the particle is measured and fed back to an electric

Figure 3 Principle of radiation pressure. (From Ref. 11.)

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23

Figure 4 Floating of minute ball by radiation pressure.

optical modulator, and the laser power is adjusted to generate an optimal pressure that balances with gravity. Moreover, moving the object in water is easily achieved because the radiation pressure can be remotely applied to an object of interest whenever the medium is transparent. Thereby the radiation pressure is used for the cell operation. It is also possible to rotate a small object using the spin momentum of photons. Photons in a pure circularly polarized state have the spin momentum h/p, whose signature depends on the direction of the polarization. Consider an absorbable object. If the object acquires photons in a circularly polarized state, it receives the momentum of photons by the angular momentum conservation theorem (see Fig. 5a). This momentum transfer is a driving force to rotate the object. Such rotary actuation is also available for some particular transparent objects. They are anisotropic optical media exhibiting dichroism. These materials exhibit two refractive indices according to the polarization of light and hence give a phase difference between the two transmitted lights with different polarization. As the linearly polarized light consists of two circularly polarized lights with different directions, the light transmitted through such an anisotropic object has a phase shift due to the dichroism. This phase shift is dependent on the transmission length. Hence all kinds of

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Chapter 2

Figure 5 Rotation of object using spin momentum of photons. (a) Rotation of object by absorption of circularly polarized light. (b) Rotation of object by change of polarization direction.

polarization states can be realized by changing the length. In the particular case that the length is adjusted to give a phase shift corresponding to a pure circularly polarized state (phase shift: d ¼ p/2), a linearly polarized light is transformed into a circularly polarized light while transmitted through the object. This means that the object receives the recoil angular momentum from the light by this polarization transformation (see Fig. 5b). This angular momentum is also a source of the driving force for rotary actuation. This mechanism is generally extended to the rotary actuation based on angular momentum transformation.

3.2 3.2.1

Usage of Photothermal Conversion Vibration Excitation by Photothermal Stress (Optical Oscillating Sensor)

When the thermal effect of optical energy (absorption) is used, optical energy can be converted to mechanical energy. This type of generation of mechanical energy is called photothermal drive. This enables the noncontact drive of a minute object. We explain the vibration excitation mechanism of the cantilever by the photothermal effect. When light is irradiated onto the side of a cantilever, a thermal expansion is brought out in the irradiated part by temperature

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25

Figure 6 Mechanism of bending moment generation in a beam by photothermal conversion.

Figure 7 Energy flow in photothermal drive.

increase, as shown in Fig. 6. The bending moment is generated by this thermal expansion, and the cantilever deflects. When an optical irradiation is stopped, the temperature of the irradiation part decreases rapidly by thermal diffusion and the moment disappears. When the light is irradiated in a row of pulses, moment is generated at every arrival of the pulse. A similar effect as the mechanical forced vibration is achieved. Figure 7 shows the conversion process and the dissipation process from optical energy to the vibration energy. The essence of the bending moment generation is nonuniformity of the temperature distribution in the direction of the thickness of the oscillator, and such uniformity is generated by the optical irradiation.

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26

3.2.2

Chapter 2

Sound Wave Generation by Photothermal Expansion (Photophone)

The photophone is an apparatus that converts optical signals (amplitude modulation) into sound waves. Figure 8 shows the outline of its structure. Thermal energy exists in the process that converts optical energy into sound energy. An optical signal is irradiated from the optical fiber to the medium of the absorption cell. The medium increases its temperature and heats surrounding air after absorbing optical energy. The heated air expands and a resulting air-density wave propagates in the converter. This wave is a longitudinal wave and so is recognized as sound. The size of the converter was r1 ¼ 0.4 mm, r2 ¼9 mm and 85 cm in length in Ashkin’s experimental apparatus. When converting an optical signal to a sound signal, the conversion efficiency was almost flat in the range from 300 to 3300 Hz. This bandwidth includes almost all that of the human voice. The research was started with a prototype photophone by Alexander Graham bell at the end of 19th century and was further studied by Ashkin in the Bell laboratory. An optical speaker still being studied is based on the same principle. Carbon fibers are packed into a ventricle in the air, and the heat alternatively causes expansion and compression of the air in the ventricle according to the optical signal. The sound impedance of the air in free space is matched with that of the horn. Various experimental investigations on such mechanisms have been carried out to find an optimal structure of the horn and the heat absorption material. Typical examples are shown in Fig. 9.

Figure 8 Structure of photophone. (From Ref. 5.)

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27

Figure 9 Example of experiment of photophone. (From Ref. 5.)

3.2.3

Magnetic Circuit Control (Microrelay)

This section presents a micromagnetic actuator controlling magnetic force by using the temperature-dependent magnetic phase transition phenomenon of ferromagnetism. The operation principle is that the equivalent gap length is changed by heating a magnet around the gap with a laser; the magnetism decreases with increasing the temperature and completely vanishes at a temperature higher than the Curie point as shown in Fig. 10. Its advantages over conventional electrostatic and electromagnetic actuators are as follows. (1) It is easy to provide the self-latching function. (2) In submillimeter size, it has larger power than that of electrostatic actuators. (3) The structure is simpler than that of the electromagnetic type, and manufacturing is easier. (4) The wireless operation control is possible because it uses light. On the other hand, its defects are as follows. (1) Displacement is mainly in binary motion, and multistep control is

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Chapter 2

Figure 10

Principle of thermomagnetization microrelay. (From Ref. 12.)

difficult. (2) The moving frequency is several tens of Hz at most because heat is used. A microsize relay (1.5 mm
200 mm) is developed as an application. Switch operation at 10 ms was realized by heating it with a 30 mW laser light. 3.2.4

Writing in Recording Film (Optical Disk)

A typical structure of an optical disk is shown in Fig. 11. The disk has a sandwich structure, a recording film lies between two transparent substrates. The inside of the disk has grooves to guide the optical beam. The recording film is protected and the reading error with dusts and other defects is prevented by the transparent substrates. Because the laser beam converges to a point on the recording layer with a lens, the optical beam diameter corresponds to about 1 mm at the disk surface. The size of the dust is much smaller than the diameter, and its shadow is reduced greatly on the recording layer. Then the probability of recording error caused by blocking of laser light by the dusts or other defects becomes very small. For recording, the laser forms microscopic pits on the recording layer along the guide grooves with a heating energy of approximately 1 nJ. Figure 12 describes typical data recording and reproducing methods for write-once and rewritable disks. In the write-once type, the recording process is as follows: the recording film is locally evaporated by laser heating, and uneven pits are generated. In the reproducing process, a change of power in reflected light is detected. Also, there are methods that can make flat pits with different reflectivity or transparency by thermally transforming the recording material. In the rewritable type, there is a magneto-optical method of which the recording process is as follows: a laser beam is irradiated to the recoding layer, the layer temperature rises up to about

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Technological Outline of Micro-Optomechatronics

Figure 11

Structure of the optical disk.

Figure 12

Major recording methods.

29

200 C partially, and at the same time a magnetic field is externally applied. Then the magnetization direction of the heated area is reversed. In the magneto-optical method, data codes are reproduced as follows: a laser beam is applied to the recording layer, the polarization angle of the beam is

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30

Chapter 2

rotated by the magnetic Kerr effect, which is the interaction between light and magnetism, a change of reflected light power is detected by a photodiode, and the direction of the magnetization of the recording layer is detected. Another read/write method of a rewritable disk is the usage of the crystal-amorphous reversible change of chalcogenide recording film. When a high-power laser is exposed to the recording layer for a short time, the exposed part is rapidly melted and cooled and then enters the amorphous state. When a low-power laser is exposed to the recoding layer for a long time, the recording material is melted and cooled slowly and then returns to the original crystal state. Data are reproduced by detecting the difference of reflectivity between crystal-amorphous states. Reversible phase change is possible more than 106 times by using a recording material made of chalcogenide.

4 4.1

CONTROL OF OPTICAL BEAM POSITION Continuous Positioning Control

In rotational movements of a disk filter, an optical disk, and the polygon mirror of a laser beam printer, the space information memorized in the medium is transformed to temporal signals. Since the accuracy that changes space information into time information depends on the accuracy of the medium movement, it is necessary to make a rotation speed into a fixed value strictly. For this purpose, the angle of the rotating object is detected and motor control is performed so that the difference of this angle and the reference signal created externally becomes zero. The typical structure of a constant-speed rotation system is shown in Fig. 13. A direct-current servomotor generates rotational movement. Rotation of the motor is measured by the rotary encoder. This signal and an external electric signal are compared, and synchronous rotation is realized by controlling the rotation of the motor so that the signal difference becomes zero. A control block diagram is shown in Fig. 14. The rotation angle x of a moving object is measured, the difference between this angle signal and the reference signal r is calculated, and the driving force Cðx_  r_Þ þ Kðx  rÞ of a motor is generated so that the angle deviation and the speed deviation, which is the differential of the angle, become zero. If disturbance, such as friction, is written as F, the movement equation is given by Mx€ ¼ F þ Kðr  xÞ þ Cð_r  x_ Þ

r ¼ vt

ð4Þ

This equation is equal to that of the mechanism that the position x of mass M is controlled by the spring K and the dashpot C to the target position

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Technological Outline of Micro-Optomechatronics

Figure 13

Constant-speed rotation mechanism.

Figure 14

Block diagram of continuous positioning.

31

r ¼ vt, which moves in proportion to time t (Fig. 15). The dynamic model of the head seek control in the optical disk, where the object follows a variable speed target, is also given by this model, although the relationship between r and t is not so simple as a linear relation. In the actual microoptomechatronics systems, characteristics of the motors and the control circuits are more complicated. But the essential of positioning control is the same. 4.2

Classification of Follow-Up Positioning Control

Many types of follow-up positioning control are used in micro-optomechatronics. Its classification is shown in Fig. 16. The follow-up control is

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32

Chapter 2

Figure 15

Dynamic model of continuous positioning.

Figure 16

Follow-up positioning in micro-optomicromechatronics.

divided into active and passive controls. In active control, an object is driven by an actuator such as a moving coil, the object position is measured by a sensor, and the sensor signal is fed back to the actuator. In passive control, a natural constraining force such as air bearing force is generated between the controlled object and the target object, and the object position is controlled without sensors. Furthermore, there are out-of-plane (the direction of distance) and in-plane (the right and left direction) controls as moving directions. In active control, there are continuous and discrete controls as the methods of data processing. 4.3

Active Continuous Follow-Up Positioning

A typical example of the follow-up, out-of-plane positioning is the focus servo of the optical disk, which positions the focus of a laser beam in the

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33

axial direction. In an optical disk, it is necessary to position the focus point of the laser on the disk surface, which vibrates about 0.5 mm, and a permissible error is about 1 mm. For this purpose, a gap between the focus point and the disk surface is detected from the reflected reproduction beam, and the objective lens is driven so that the gap deviation becomes zero. There are various methods in carrying out the focus error detection, and a typical method is the astigmatic method. The electromagnetic actuator shown in Fig. 17 is built in an optical head and an objective lens can be driven up and down. If the current proportional to the gap length is added to the coil, the focal point is always positioned on the disk surface. A typical example of the follow-up, in-plane positioning, which positions the focus of a laser beam in the vertical direction of the beam axis, is track servo in an optical disk. In the optical disk drive, it is necessary to make a laser beam follow pits that vibrate about 70 mm in plane with an accuracy of about 0.1 mm. For this purpose, a position error is detected from the reflected recording/reproduction beam, and the objective lens is driven so that the error becomes zero. There are various methods for detecting in-plane focus position errors, and a typical method is the push–pull method. Details of an astigmatic method and the push–pull method are explained in Chap. 5.

Figure 17

Focusing control mechanism. (From Ref. 13.)

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Chapter 2

Figure 18

Block diagram of active continuous follow-up positioning.

We describe the dynamic model of active, continuous, follow-up positioning. The model is common to in-plane and out-of-plane controls. First, the position difference x  r and velocity difference d(x  r)/dt between a target position and an object position are created by a sensor and a circuit. Next, with an amplifier, a difference signal of displacement is multiplied by K and a velocity difference signal is multiplied by C, and the sum of them is applied to an actuator. Then the actuator force becomes larger as the position difference and the velocity difference become larger, and the position and velocity of the controlled object become closer to those of the target object. To simplify the analysis, the actuator and the controlled object are modeled by a rigid body of 1 degree of freedom. In order to restrain their movement in other directions than the follow-up direction, some supporting mechanisms (e.g. a parallel leaf spring) are necessary, and their spring constant is set to k. Moreover, let M be the mass of the actuator and the moving object. Then the block diagram of a control system becomes as shown in Fig. 18. Moreover, if a displacement sensor (the push–pull method or the astigmatic method) does not have an error and its gain is set to 1, the equation of motion is given by Mx€ þ kx ¼ Kðr  xÞ þ Cð_r  x_ Þ

ð5Þ

This equation is the same as that of 1-degree-of-freedom spring-massdashpot system as shown in Fig. 19. That is, displacement feedback becomes equivalent to a spring, and velocity feedback becomes equivalent to a dashpot. Moreover, if k is set to zero, this model becomes the same dynamic model as that of the continuous positioning system shown in the previous section. 4.4

Passive Continuous Follow-Up Positioning

A method of positioning a laser beam in the axis direction without using a sensor and an actuator is described. A typical method is the use of a flying

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Figure 19

Dynamic model of active continuous follow-up positioning.

Figure 20

Conceptual figure of flying slider. (From Ref. 13.)

35

slider. Compared with the active method, the structure of the control system is simple, and the mass of a movable part becomes small. A conceptual figure of the flying slider is shown in Fig. 20. At first, the slider is pushed onto the disk surface by a spring suspension. When a disk rotates, surrounding air is dragged and the air is stuffed into the tapered

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Chapter 2

Figure 21

Dynamic model of passive follow-up positioning.

part of the slider. Then the air pressure increases, a support spring deflects, and the slider goes up from the disk surface. Since the air pressure increases as the gap between the slider and the disk decreases, the air film works as a spring. Also, air pressure depends on the relative velocity of the slider against the disk in the vertical direction, and the air film also works as a viscous damper. If the spring constant and the damping coefficient of an air film are denoted as K and C, and the spring constant of the support spring is denoted as k, the passive positioning system becomes a 1-degree-of-freedom system, as shown in Fig. 21. This model is dynamically the same as that of the continuous active follow-up positioning systems. 4.5

Dynamic Characteristic of Positioning Control Systems

As stated above, continuous and follow-up positioning systems result in the same 1-degree-of-freedom system. In the following, a positioning performance is analyzed by using the system of Figs. 15 and 19. First, constant speed positioning is studied. The positioning error in this case is generated by the disturbance force acting on the object. If a coordinate system moves with r in Fig. 16, the spring and the dashpot are fixed to the ground, and the positioning error is given by the displacement x when disturbance acts on M. To simplify the analysis, disturbance is assumed as a sine wave, and the steady-state displacement is calculated. Then the positioning error is given by Mx€ þ Cx_ þ ðK þ kÞx ¼ F sin !t

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ð6Þ

Technological Outline of Micro-Optomechatronics

37

Amplitude B of x at steady state is given as B¼

F 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 Kþk  1  ð!=!1 Þ2 þ ð2ð!=!1 ÞÞ2

ð7Þ

The result is shown in Fig. 22. The horizontal axis shows the relative frequency of disturbance !/! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1, the vertical axis shows relative amplitude B/F, and !1 ¼ ðK þ kÞ=M,  ¼ C=2M!1 . Let us study the conditions needed to improve the performance of constant speed positioning. First, since the response amplitude for direct current disturbance (! ¼ 0) is 1/(Kþk), K is desirable as large as possible. Next, , i.e., C, is desirable as large as possible in order to reduce the amplitude at the resonant frequency. These show that the movement of the object becomes stable by using an actuator with large generating power since K and C correspond to the output force of the actuator. Moreover, in the rotation systems such as an optical disk, disturbance vibration is known to be larger for lower frequency. In this case, it is better to raise the resonance frequency, and for this purpose M should be as small as possible. It turns out that a small and powerful actuator is required to generate a stable movement. Moreover, it is found that the displacement at resonant frequency cannot be suppressed when the velocity feedback is not used (C ¼ 0), and that the displacement feedback is not enough to obtain sufficient accuracy.

Figure 22 Positioning error amplitude produced by sinusoidal disturbance force. (From Ref. 13.)

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38

Chapter 2

Next, active and passive follow-up controls are studied. In this case, it is necessary to put the displacement x of the controlled object into agreement with the position r of the target object in Fig. 19. To simplify the analysis, a case where the target object vibrates by r ¼ A sin !t is studied. Then the equation of motion is given by Mx€ þ kx ¼ Kðr  xÞ þ Cð_r  x_ Þ

ð8Þ

Since the required value is the relative displacement of x and r, the equation is rewritten by using it as an unknown parameter as Me€ þ Ce_ þ ðk þ KÞe ¼ Kr,

e¼xr

The amplitude E of e at steady state is given as   2   ð!=!1 Þ2  !22 =!1  ffiA E ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 1  ð!=!1 Þ2 þð2ð!=!1 ÞÞ2

ð9Þ

ð10Þ

Figure 23 shows the relationship between input-and-output amplitude ratio and input frequency obtained by the above formula. Here, the gain takes the peak value at the resonance frequency of the 1-degree-of-freedom system. rffiffiffiffiffiffiffiffiffiffiffiffi Kþk ð11Þ !1 ¼ M

Figure 23 Positioning error amplitude produced by sinusoidal movement of target position. (From Ref. 13.)

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Technological Outline of Micro-Optomechatronics

The gain takes the minimum value at frequency rffiffiffiffiffi k !2 ¼ M

39

ð12Þ

This agrees with the resonance frequency decided by the support spring k and the objective mass M. At this frequency, since support spring force and inertia force balance, force does not act on K and C. Therefore, the spring K does not deflect, and the distance between the controlled object and the target object is kept constant. Let us study the conditions needed to improve the performance of follow-up positioning. First, since the relative displacement at ! ¼ 0 is k/(Kþk), K and k should be as large as and as small as possible, respectively. Next, since the peak height at the resonant frequency is 1/2, C should be as large as possible. Next, since the resonant frequency moves to right and the graph goes down as K/M becomes large, M should be as small as possible. That is, the generated force by the actuator or the air film (K and C ) should be as large as possible, the additional spring (k) should be as small as possible, and the mass of the moving part (M) should be as small as possible. These are the same as in the case of continuous positioning. As mentioned above, to realize high precision positioning in microoptomechatronics, microactuator technology with large generating power is indispensable. 4.6 4.6.1

Follow-Up Discrete Control In-Plane Movement

The way to position the focus of a laser beam, based on the discrete position information, in the direction vertical to the beam axis is described. The typical one is a sampled servo in the optical disk. As described already, in the optical disk, it is necessary to make a laser beam follow the pit, with an accuracy of about 0.1 mm, which vibrates in the disk plane at the amplitude of about 70 mm. The sampled servo detects a position depending on marks written discretely on the data recording surface. Since the position information is recorded separated from the data signal spatially, and since the track error signal does not leak to the data signal and the focus error signal, high S/N data reproduction and highaccuracy lens positioning can be realized in the sampled servo system. The composition of the optical system is simple compared with the continuous servo system (push–pull method), so this system is promising as a means of tracking in the near field optical disk in future, too. Moreover, there is the advantage that the position detection error by the movement of the objective

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40

Chapter 2

lens and the inclination of the disk hardly appears. However, the position information cannot be obtained without the servo mark, so the objective lens must move with inertia between the servo marks. By the occurrence of the disturbance between marks, the track positioning error, the decrease of the S/N ratio of the reproduced signal, the miserasure of the neighboring track data can occur. If we increase the number of servo marks to suppress this, the memory capacity of the data decreases. The balance of amounts of data and servo marks is important. Details of the sampled servo are explained in Chap. 5. 4.6.2

Out-of-Plane Movement

The way to position a light source, based on discrete position information, in the direction of the beam axis is explained. The typical one is the tapping mode of scanning near field optical microscopy (SNOM). In SNOM, a minute opening of several tens of nm in diameter is made at the end of the optical fiber, and the optical property of the object surface is measured with a resolution of about 10 nm by irradiating the light from the optical fiber to the measured object. Therefore, the distance between the light source and the measured object must be made equal to or less than several tens of nm. There are three methods to make the probe follow the object surface: contact mode, noncontact mode, and tapping mode. The comparison of each method is shown in Table 3. The contact mode slides the probe on the object surface. It is the simplest, but to make the probe follow the unevenness of the object surface, it is necessary to press the probe by a certain amount of force. Thus, it is not suitable for the scan of a flexible object such as a living creature. In the noncontact mode, the system measures the bending quantity of the probe caused by the attractive force

Table 3

Scanning Methods of Probe Microscope (DI Inc. Catalog) Mode

Contact AFM

Non-contact AFM Tapping AFM

Force detection mode

Horizontal resolution Pressing force to the sample Friction force Soft sample observation Stability of observation

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Static

Dynamic

Dynamic

Large Large

Small Nothing

Small Nothing

Technological Outline of Micro-Optomechatronics

41

between the object and the probe and controls a gap between them to make the bending quantity constant. There is another method of measuring a resonance frequency change by vibrating the probe. There is no damage to the object, but it is easily influenced by an external disturbance because the detected attractive force is very small. The tapping method uses the amplitude decrease that occurs when the probe’s tip touches the object, while forcing the probe to vibrate at the first natural frequency. The system makes controls that add constant vibrational force to the probe and moves the measured object up and down so that the vibration amplitude of the probe is constant. Because the vibration amplitude is easy to detect even for a small change, it can detect minute contact force, so that the damage to the object is small. In the tapping mode, the position information is obtained only at the moment of collision, so it becomes a discrete sampling control. Moreover, in the transition period from the contact to the noncontact condition, the amplitude does not recover at once but gradually according to A ¼ 1  ð1  rÞe!n t A0

r¼

A1 A0

ð13Þ

where A, A0, and A1 are transient, steady-state, and initial amplitudes, and , !n, and t are damping ratio, angular natural frequency, and time, respectively. By the above formula, the vibration cycle N, which is necessary for amplitude offset to decrease to 1/e of the initial offset, is given by N

1 2

ð14Þ

So the system has a time delay given above for the measurement of distance. In present SNOM, the resonance of the scanning mechanism also becomes the limit factor of the control performance. As for the tapping mode, it is shown that the bigger the amplitude and the smaller the probe radius, the smaller the influence of the surface absorption layer is [14]. Also, for applying to the memory, the analysis of the characteristic of the movement in the case of scanning an object surface at high speed is studied [15].

REFERENCES 1.

Shimoda, K. The dynamics action of the laser light. Journal of the Japan Society for Precision Engineering 1992, 58, 406. in Japanese.

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Chapter 2

2.

Hashimoto, E.; Uenishi, Y.; Watanabe, A. The research of the heat magnetization control mechanism. The Micro Mechanical Relay Made by Ni Plating. NTT R&D 1995, 44 (10), 1099–1104. in Japanese. Igarashi, I.; Esashi, M.; Fujita, H. Micro Opto Mechatronics Handbook; Asakura Shoten, 1997; in Japanese. Shimokawa, F.; Sato, M.; Nagaoka, S. Micro mechanical optical switch. NTT R&D. 1999, 48 (9), 665–673. in Japanese. Kleinman, D.A.; Nelson, D.F. The photophone—an optical telephone receiver. J. Acoust. Soc. Am. 1976, 59 (6), 1482–1494. Ooetsu, T. The optical fiber sensor, Ohmsha. 1986, in Japanese. Nagato, R.; Hosaka, H.; Itao K.; Sakamoto K. A two-DOF inclination sensor based on a shellfish balancing organ. 10th Int. Conf. Solid-State Sensors Actuators, Transducers ’99, Digest of Tech. Papers, 2, 1999; pp. 1554–1557. Ootsu, M.; Kawata, S. Near Field Nanophotonics Handbook. Optronics, 1997; in Japanese. Suzuki, Y.; Kondou, Y.; Hanada, T.; Katagiri, Y.; Itao, K. The Design method of the light and heat drive optical fiber vibrator. Journal of the Japan Society For Precision Engineering 1997, 63 (8), 1101–1106. in Japanese. Stimmer, M.; Slawsky, Z.I. Torsion pendulum photometer, The Rev. Sci. Instruments 1964, 35 (3), 311–313. Ashkin, A. et al. Optical levitation high vacuum. Appl. Phys. Lett. 1976, 28 (6), 333. Higurashi, E.; Ohguchi, O.; Tamamura, T.; Ukita H.; Sawada, R. Optically induced rotation of dissymmetrically shaped fluorinated polyamide microobjects in optical traps. J. Appl. Phys. 1998, 82 (6), 2773–2779. Ono, K.; Tagawa, N.; Nakayama, M.; Ichihara, J.; Yoshimura, S. Memories and printers. Ohmsha 1995, in Japanese. Nakajima; H.; Kuroda, S.; Takahashi, H.; Osumi, H.; Hosaka, H.; Itao, K. Micro tapping control on wet surface, Micromechatronics 2000, 44 (4), 35–47. in Japanese. Aoyagi, S.; Kitada, H.; Kusuda, Y.; Matsumoto, Y.; Takano, M.; Hosaka, H.; Itao, K. Tapping motion analysis of a cantilever based on elastic vibration theory under the condition that impulsive force and moment are given on its tip analysis method in the case that modal functions are not orthogonal. J. Japan Soc. Precision Engineering 2002, 68 (8), 1030–1036. in Japanese.

3. 4. 5. 6. 7.

8. 9.

10. 11. 12.

13. 14.

15.

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3 Intermittent Positioning in Micro-Optomechatronics Intermittent positioning is defined as a control technique that moves and holds an object to a desired position. The technique is used in microoptomechatronics for optical components typically including mirrors and gratings so that optical properties specified by wavelength, optical phase, etc., should satisfy these requirements. Since reconfigurations of systems are frequently requested on demand in practical applications, a quick control process is essential. Small inertial and compact monolithic moving mechanisms achieved by micromachining are advantageous for this purpose. Such devices include recent micromechanical devices that miniaturize optical systems similar to those of the conventional systems, e.g., optical switches having electrically or magnetically changeable optical paths, optical filters having changeable resonant cavities, and wavelength-tunable lasers including filters to determine oscillation wavelengths and optical gain media. In this chapter, fundamentals of intermittent positioning are described using two practical applications; one involves micromechanically tunable modelocked semiconductor lasers as tunable optical pulse sources; the other has wavelength-tunable disk-shaped optical bandpass filters. Their practical implementations are also mentioned to show how the technique is practically used. Control of optical properties using optical components is based on the aspect of light as a wave typically featured by interference and diffraction. These optical components include an optical resonator such as a Fabry– Perot resonator consisting of two face-to-face mirrors. Optical resonators can offer not only wavelength selectivity but also various other optical parameters such as group delay and group velocity. These parameters are directly tuned by controlling the length of resonators. There are two approaches to the control. One uses miniaturized mechanical systems with improved kinetic properties based on inertial effects [1–8]. These systems are typically formed with micro moving mirrors. Various kinds of moving 43

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44

Chapter 3

mirrors have been presented. Figure 1 shows two typical micro moving mirrors, both of which are driven by electrostatic forces. One uses a membrane, which is suspended by beams on a substrate to form a small air gap (Fig. 1a). Electrostatic force is generated between the membrane and the substrate. The surface of the membrane as a mirror surface is vertically moved by deforming the membrane. The other uses an interdigit structure: a beam suspended on a substrate has a digit facing a ground electrode spaced by a small gap (Fig. 1b). A capacitance is formed between the digit and the electrode, and this generates an electrostatic force to bend the beam. The sidewall of the beam acting as a mirror surface is horizontally displaced by the bending. These mirrors can be monolithically fabricated by LSIcompatible processing techniques, so we can easily construct tunable microoptical resonators with lengths of several micrometers to control optical parameters including wavelength, group delay, and group velocity. Their small inertia, which provides a mechanical resonant frequency much higher than that of typical mechanical fluctuations, promises quick and stable controllability. The latter approach is based on a slide-tuning mechanism [24,25]. A typical mechanism uses a wedge-shaped structure with no moving part (see Fig. 2). The collimated light beam effectively forms an optical cavity at which the beam passes through the structure. The cavity length is changed according to the position of the beam. Since the wedge structure provides a gradual slope, it is easy precisely to obtain a required cavity length. If such a changeable structure is formed in a conventional bandpass filter configuration consisting of a half-wave resonant cavity sandwiched by two quarter-wave high-reflectivity coatings (HRCs), the transmission center wavelength can be controlled according to the position of a collimated light beam. This tunable device offers high mechanical stability: the mechanical fluctuation, which may affect the slight displacement of the beam, has a negligible influence on the cavity. Although there remains the problem of

Figure 1 Schematic structure of micro moving mirrors. (a) Mirror using membrane. (b) Mirror using sidewall of suspension beam.

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Intermittent Positioning in Micro-Optomechatronics

45

Figure 2 Slide tuning mechanism using wedge-shaped cavity.

how to reduce the positioning time, it will be removed by using a conventional high-speed intermittent positioning technique.

1

MOVING MICROMIRRORS AND THEIR APPLICATION

1.1 1.1.1

Characterization of Micro Moving Mirrors Variation of Moving Mechanisms

Detailed kinetic performances are described for the micro moving mirror with interdigit structures. The fundamental micro moving mirrors are realized by a suspended metal beam with a comb actuator at its center as shown in Fig. 3a, and they can be displaced by bending the beam as shown in Fig. 3b. The electrostatic force produced by the comb actuator is used for the bending, but it is only effective when the interdigit electrode gap of the actuator is in the range of several micrometers. Hence this simple moving mechanism must be modified for practical implementation. First of all, the number of the digits is increased to enhance such an intrinsically small electrostatic force (Fig. 3c). Then a configuration with two combs is used to realize a push–pull operation, because the force is unidirectional (Fig. 3d). Finally, a folded beam is used to reduce the stiffness to achieve a maximal distortion under the limited drive force (Fig. 3e). 1.1.2

Static Performance

The distortion of the beam is numerically estimated by the condition that the electrostatic force produced by the comb drives should be balanced with the corresponding elasticity caused by the distortion.

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Chapter 3

Figure 3 Micro moving mirror driven by comb actuators. (a) Fundamental structure. (b) Displacement mechanism. (c) Force enhancement by comb structures. (d) Push–pull mechanism. (e) Folded beam mechanism.

Figure 4 Configuration of comb actuators.

Electrostatic Force. Let us find a driving force generated by electrostatic comb actuators. Consider two combs forming an interdigit structure spaced by d acting as a capacitor, as shown in Fig. 4. Suppose that a comb is DC-biased by V to remain stationary under an external force F. When the comb is displaced by dx along the x-axis against the external force, the potential energy is changed by the external force as W ¼ F x

ð1Þ

This displacement changes the electrostatic energy of the capacitance. Assuming that this energy change is given by the change transfer dQ, we can

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Intermittent Positioning in Micro-Optomechatronics

47

express the change as 1 We ¼ V Q 2

ð2Þ

The electric system supplies the energy necessary for the charge transfer as Ws ¼ V Q

ð3Þ

The energy change of the capacitor must be equal to the sum of the potential energy change and the supply of the system based on the energy conservation theorem, We ¼ W þ Ws

ð4Þ

Hence we obtain 1 W ¼  V Q 2

ð5Þ

The charge stored in the capacitance changes by the charge transfer, and this change is considered to come from the capacitance change under a constant voltage as Q ¼ V C

ð6Þ

Therefore we can relate the electrostatic parameters to the external force as 1 W ¼  V2 C 2

ð7Þ

Finally we obtain the attractive force generated between the two combs as F¼

@W 1 2 @C "0 2 ¼ V ¼ V @x 2 @x 2d

ð8Þ

where "0 is the permittance of vacuum. This equation indicates that the magnitude of the force increases in proportion to the squared applied voltage. It is also obvious that the force is unidirectional because of the squared voltage. The direction agrees with the one along which x increases. The estimated force corresponds to a single capacitor formed by the combs. The digit of the combs has two capacitors, so the net force magnitude per digit equals to double the force for a single capacitor. Assuming N, the number of the digits of the comb, the total force is enhanced to 2NF. This enhancement is enough for compensating the small electrostatic force. Dynamics of Microstructures. Consider the distortion of the beam under the force generated by the comb actuators as shown in Fig. 5.

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Chapter 3

Figure 5 Calculation model of distorted beam.

We will carry out calculations using a centralized load approximation method. Suppose that the beam L in length with two fixed edges gives a distortion in the direction along the y-axis at an arbitrary position apart from a fixed edge by x. This distortion is related to the moment Mx, and the corresponding distortion angle  as Z y Mx ¼ ¼ dx þ 1 ð9Þ x EIz Z y Mx 0 ¼ dx dx þ 1 þ 2 ð10Þ ¼ x EIz where E is Young’s coefficient and Iz the moment of the beam in the direction of the fast axis. The moment is given for a rectangular beam w in width and h in height Iz ¼

hw3 12

ð11Þ

Suppose that a concentrated force W is applied to the center of the beam. It is enough to consider the hemiregion 0  x  L/2. on the left hand of the beam, taking account of symmetry. The moment Mx at an arbitrary position in the hemiregion is expressed using a moment generated at the fixed edge, 1 Mx ¼ xW þ M 2

ð12Þ

Therefore the distortion angle at this position is obtained using Eq. (9) as ¼

1W 2 M x  x þ 1 4 EIz EIz

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ð13Þ

Intermittent Positioning in Micro-Optomechatronics

49

Since the distortion angle becomes zero at a fixed edge, we obtain 1 ¼ 2 ¼ 0. Hence the distortion y is given by integrating Eq. (10) to obtain y¼

1 W 3 1M 2 x  x þ y1 12 EIz 2 EIz

ð14Þ

where y1 ¼ 0 because of the condition of fixed edges. As the distortion angle at the center should be zero, we obtain M¼

LW 8

ð15Þ

Hence we can obtain a distortion of the beam at an arbitrary position as y¼

1 W 2 x ð4x  3LÞ 48 EIz

ð16Þ

The displacement of the micro moving mirror given at the center of the beam is expressed as y¼

1 WL3 192 EIz

ð17Þ

This equation indicates that the displacement at the center of the beam decreases in order to the third power with miniaturizing the moving mechanism, assuming a constant driving force. This means that smaller moving mechanisms need larger driving forces. This is a shortcoming for micromechanisms that drive moving parts. Hence it is reasonable for enhancing the driving force. However, the enhancement becomes difficult on reducing the device size. To eliminate this problem, a folded beam structure has been proposed to construct micro moving mechanisms, as shown in Fig. 6a. When a local force G is applied to the center of the folded beam, the distortion is given as the superposition of the distortions of each beam element, as shown in Fig. 6b. For the case of the folded beam shown in the figure, the distortion is given by y¼

Wð2L31 þ L32 Þ : 192EIz

ð18Þ

Electrically Controllable Distortion. The electrostatic force generated by the comb electrodes was obtained according to Eq. (8). Since this equation is combined with Eq. (18), we can find that the micro moving

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Chapter 3

Figure 6 Modeling of micro moving mirror with folded beams. (a) Schematic structure. (b) Calculation model of distorted triple-folded beam.

mirror is displaced according to the squared voltage as y¼N

"0 L3 V2 384dEIz

ð19Þ

where N is the number of the capacitances formed between the comb electrodes. Since the monolithic fabrication techniques can achieve microstructures with a large number of the comb electrodes, the electrostatic driving force for the micromirror is rather larger than we consider. Consequently, micro moving mechanisms driven by electrostatic forces can be practical. Experimental Demonstration. Static performance of the moving micromirrors can be readily clarified by the combination of theory and experiment. Experiments were performed with an Ni micro moving mirror with three folded beams fabricated on a silicon substrate. The specification of the mirror is shown in Table 1. We can readily calculate the linear dependence of the displacement of the mirror on the squared voltage by using the parameters shown in the table. We can also experimentally estimate the linear dependence. Figure 7 compares the measured dependence with the calculated one. It is obvious that they show good agreement. This means that the displacement of the mirror can be numerically estimated if the detailed structural parameters of the moving mirrors are clarified in advance. 1.1.3

Frequency Response of Micro Mirrors

To minimize the positioning time, it is necessary to clarify the dynamic performance of the moving mirrors such as the indicial responses, which are

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Intermittent Positioning in Micro-Optomechatronics Table 1

51

Specification of Comb Actuator

Density () Young’s constant (E) Beam width (w) Beam thickness (h) L1 beam length L2 beam length

8900 kg/m3 170 GPa 7.5 mm 20 mm 1200 mm 1300 mm

Figure 7 Static displacement characteristics of micro moving mirror.

characterized by parameters including a rise-up time, and settling time. These parameters can be theoretically derived from the specifications of the micro mirrors, but the theoretical derivation usually needs complicated calculation procedures. These parameters, however, can be readily estimated from measured frequency responses of the mirror. Figure 8 shows a frequency response measured at the center of the beam. The first and second resonant oscillations are clearly detected (The second oscillation gives a minimal gain for the frequency response because of the characteristic spatial mode having a knot at the center of the beam.) The first resonant frequency of around 1 kHz is relatively high compared with those of conventional mechanical systems, and the flat phase region indicates the potential highspeed controllability. To be more quantitative, we estimate that the first resonant oscillation gives a damping factor  of 0.031. The rise-up time Tr, defined as the time while the distortion reaches 90% from 10% of the required value, and the

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Chapter 3

Figure 8 Dynamic property of micro moving mirror.

settling time Ts, defined as the time while the deviation of the distortion from the required value becomes less than 10%. These parameters are given with the damping factor as Tr ffi

1:02 2f1

ð20Þ

Ts ffi

2:30 2f1 

ð21Þ

The micromirror having the above specifications gives Tr ¼ 0.2 ms and Ts ¼ 14 ms. The indicial response of the micromirror featured by these parameters verifies the micro-optomechatronic control scheme based on the effect of reducing the inertial for high-speed applications. 1.2 1.2.1

Micromechanically Tunable Mode-Locked Lasers [20] Figures of Merits

Small low-driving power optical-pulse sources are desired in various practical fields including optical communication systems and measurement. The requirements of practical pulse sources include sufficient tunability in repetition rate to suit the application systems. However, while conventional monolithic mode-locked lasers with a fixed cavity length offer excellent stability, they suffer from instability in mode-locked oscillation as the discrepancy between the natural repetition rate and the electrical signal frequency increases. This problem will remain unresolved until the natural repetition rate is changed to reduce the discrepancy. External-cavity lasers

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53

Figure 9 Micromechanically tunable mode-locked laser diode for repetition rate control.

with a mechanically changeable cavity length are suitable for repetition-ratetunable mode-locked operation. The problem is that the current configuration using separate components, such as an actuator for displacing the external mirror and lenses for generating a collimated beam in the external cavity, has poor mechanical stability and large size, limiting the maximum possible repetition rate. Hence a simple external-cavity laser configuration consisting of a laser diode and a moving external mirror located very close to the laser facet has been proposed and demonstrated for repetition-rate tunable mode-locked lasers. The micromechanical tuning mechanism is used together with a multisegment laser diode for realizing such a tunable external-cavity laser configuration (see Fig. 9). An essential element of the tuning mechanism is a monolithic micro moving mirror controlled by electrostatic comb drives. The moving mirror uses a side wall of a triple-fold-beam thin-film spring suspended over a substrate and separated by a narrow air gap. A differential (push–pull) manner of mirror displacement is achieved using two comb drives. These comb drives consist of interdigitated fingers and stationary electrodes attached to the substrate. A typical example of such a mechanism is fabricated with Ni films 20 mm thick on a silicon substrate. This micro moving mirror is placed facing the laser facet with an antireflection (AR) coating. The mirror is displaced along the optical axis of the laser diode. This displacement changes the cavity length and therefore changes the repetition rate of the pulse train from the laser. The detailed mechanism of mode-locked pulse generation is explained in the following section.

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Chapter 3

Figure 10 Schematic structure of optical pulse source using micromechanically tunable mode-locked semiconductor laser.

The microminiaturized configuration enables a compact packaging as used for conventional monolithic semiconductor lasers, as shown in Fig. 10. 1.2.2

Principle of Mode-Locking in Semiconductor Lasers

Temporal Aspect of Mode-Locked State [9]. Mode-locking is the coupling of the eigenmodes of semiconductor lasers with the same mode spacing and the same phase. Such a mode-locked state is represented using a field component for the mth eigenmode: Em ¼ m cos½ð!0 þ 2mfÞt þ  The frequency f is close to the mode spacing as c fffi 2nL

ð22Þ

ð23Þ

The coupled modes perform as a single mode, which is called supermode. An interesting effect is that the light power pulsates in such a mode-locked state. The repetition rate of the pulses corresponds to the frequency f. As the number to be coupled increases, the pulse becomes sharper (Fig. 11). Major Schemes for Mode-Locking. To achieve mode-locking, it is necessary to provide a mechanism of encouraging mode coupling for semiconductor lasers. There are three major methods to achieve modelocking in semiconductor lasers: active mode-locking, passive mode-locking, and hybrid mode-locking. A schematic configuration to accomplish these methods is shown in Fig. 12. Detailed explanations are described for each method as follows.

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55

Figure 11 Various temporal waveforms of mode-locked pulses as functions of the number of modes (N).

Figure 12 Mode-locking schemes. (a) Active mode-locking. (b) Passive modelocking. (c) Hybrid mode-locking.

Active Mode-Locking [10–13]. Multisegment laser diodes including a DBR segment for limiting undesirable spectral broadening are generally used for mode-locking (see Fig. 12a). The active mode-locking is achieved by modulating an injection current to the end segment of the multisegment lasers. The current modulation in semiconductor lasers usually implies two types of modulation: intensity modulation and phase modulation. Assuming

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56

Chapter 3

a real sinusoidal function as the electric field component of light with an optical frequency
0 for simplicity, the optical intensity modulation at a frequency f is expressed using an amplitude form: EðtÞ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ cosð2
0 tÞ 1 þ m cosð2ftÞ ¼ cosð2
0 tÞ   1 m2 m3
1 þ m cosð2ftÞ þ 2 cos2 ð2ftÞ þ 3 cos3 ð2ftÞ þ    2 2!2 3!2 ð24Þ

where m is the modulation depth. We find the nth cos(2p
0t) cos(2pnft), which is readily expanded into cos[2p(
0  nf )t]. Here the amplitude of the electric field to be unity for simplicity. The phase modulation is, on expressed in a similar form:

order term as two terms as is also assumed the other hand,

EðtÞ ¼ cos½2
0 t þ
 sinð2ftÞ ¼ cosð2
0 tÞ cos½
 sinð2ftÞ  sinð2
0 tÞ sin½
 sinð2ftÞ 1 X ¼ J0 ð
Þ cosð2
0 Þ þ Jn ð
Þ n¼1


½cosð2
0 þ 2nfÞ þ ð1Þn cosð2
0 þ 2nfÞ

ð25Þ

where Jn(z) is the Bessel function. Hence these expressions show that both types of modulation produce a series of optical sidebands at every modulation frequency around the original optical carrier at
¼
0. Let us consider the effect of such modulation on oscillation of multisegment lasers. As shown in Fig. 13, the laser oscillates in a multiplemode state in the free-running condition, which provides a primary mode accompanied with some subordinate modes spaced by a round-trip frequency of the laser. When the laser is modulated at a frequency f very close to the round-trip frequency, the subordinate modes are pulled into the nearest modulation-sideband frequency and therefore contribute to the growth of locked modes spaced by f. The state of these locked modes offers a unique supermode that exhibits a pulsed temporal waveform; consequently the active mode-locking is achieved. Passive Mode-Locking [14–15]. Another approach is a passive mode-locking scheme using a saturable absorber (SA) that can be readily achieved by applying a reverse bias to the semiconductor laser media.

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Intermittent Positioning in Micro-Optomechatronics

57

Figure 13 Spectrum of mode-locked pulses compared with that of multiple modes in the free-running condition.

Optical pulses can pass through the saturable absorber owing to their extremely high peak power. The absorber, on the other hand, is optically modulated at every passage of the pulses. This modulation accelerates the mode coupling. A multisegment laser diode is used for accomplishing the passive mode-locking. The laser consists of multiple segments, each of which is electrically separated by narrow grooves (see Fig. 12b). An end segment of the laser is usually used as the SA by simply applying a reverse voltage, while the other segments injected a DC current for laser oscillation. Although the original seed of the mode-locking cannot be specified, any pair of emissions in the laser can produce optical sidebands in the SA region, and this leads to mode-locking. Passive mode-locking has remarkable features compared with active mode-locking. No electrical modulation at a repetition frequency is required for the locking owing to the modulation effect in the SA. The repetition rate is only dependent on the effective cavity length of the laser. Hence ultrafast optical pulses can be expected at a frequency of even higher than hundreds of gigahertz, if laser oscillation is achieved in such a short cavity with a limited optical gain. This drawback is readily eliminated by a harmonic mode-locking scheme. Harmonic mode-locking is a state in which multiple pulses circulate in the laser cavity, so that the repetition rate is multiplied by the number of circulating pulses. Hence high-repetition-rate pulse generation is expected even for lasers with a longer cavity length by using the harmonic mode-locking scheme. This can be readily achieved by making up the segment configuration of the multisegment laser. Typical examples include colliding-pulse mode-locked (CPM) lasers (see Fig. 14a) [14]. The CPM lasers have an SA segment in the center of the cavity and allow two pulses circulating. Since these pulses collide at the SA segment, the effect of

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Chapter 3

Figure 14 Colliding-pulse mode-locked lasers. (a) Original colliding-pulse modelocked (CPM) laser. (b) Harmonic colliding-pulse mode-locked (HCPM) laser. (c) SH trance and corresponding optical spectrum of 6th HCPM laser.

the SA is emphasized. The concept of the CPM can be also extended to harmonic CPM operation based on simultaneous pulse collisions in all SA segments (see Fig. 14b). Figure 14c typically shows a temporal SH trace and the corresponding optical spectrum of the harmonic CPM lasers having a repetition rate of about 192 GHz. Hybrid Mode-Locking. Hybrid mode-locking is a combination of the active and passive mode-locking schemes as shown in Fig. 12c. We can generate electrically synchronized ultrashort optical pulses using the SA effect [22]. Such a locking scheme is accomplished by adding an RF signal to the reverse voltage for the SA segment of the multisegment laser diode. When a reverse bias voltage is applied to the SA, the potential energy of the carrier electrons in the semiconductor medium is changed. This change generally shifts the absorption coefficient versus optical frequency curve   !. Hence the light-absorption characteristics of the SA are strongly dependent on the reverse bias voltage. The absorption coefficient, derived from the imaginary part of the permittivity is related to the refractive index regarded as the real part according to the Kramers–Kronig relations: Z1 0 0 1 ð! Þ 00 d!0 ð26Þ ð!Þ ¼  PV 0  1 !  !

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Intermittent Positioning in Micro-Optomechatronics

Figure 15

59

Transmission characteristics of saturable absorbers.

1 ð!Þ ¼ PV  0

Z

1

00 ð!Þ d!0 0 1 !  !

ð27Þ

with ð!Þ ¼ 0 ð!Þ þ i 00 ð!Þ

ð28Þ

The change of the refractive index much affects the absorption efficiency of semiconductors, so the transmittance is changed according to the reverse bias voltage as shown in Fig. 15. Hence if an RF signal is applied to the SA with appropriate DC biasing, we can achieve an intensity modulation simultaneously satisfying the condition of passive mode-locking. 1.2.3

Tuning Mechanism for Mode-Locked Pulses

Let us consider an external-cavity laser configuration for repetition-rate tunable, passively mode-locked semiconductor lasers as shown in Fig. 16. The laser cavity consists of the semiconductor laser and the external cavity formed between the laser facet and the reflection surface of the external micro moving mirror. The mode-locked pulses circulate in the externalcavity laser passing through the laser facet on the mirror side whose reflectivity is reduced by antireflection coating. The effective laser-cavity length is given by Leff ¼ nL þ h

ð29Þ

where h is the external-cavity length, L the internal-cavity length of the original semiconductor laser, and n the effective refractive index of the laser waveguide. The external cavity is usually filled with air, so the refractive

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Chapter 3

Figure 16 External cavity laser configuration for repetition-rate-tunable modelocked semiconductor lasers.

index is estimated as 1.0. Assuming that the external-cavity length is much smaller than the repetition rate, this is thus given by

c F0 h ¼ ffi F0 1  F¼ ð30Þ 2ðnL þ hÞ 1 þ ðh=nLÞ nL where F0 is the repetition rate when h ¼ 0. This condition means that the external mirror is attached to the laser facet. From this equation, we can readily derive the fact that the repetition rate is linearly controlled by the linear displacement of the mirror as
F ¼ F0


h nL

ð31Þ

Since the external-cavity length is much smaller than the internal-cavity length, the repetition rate F is almost constant even though the length is changed. However, as the internal-cavity length becomes much shorter, a slight change in the length becomes more effective for tuning the repetition rate; nevertheless the maximal external-cavity length is several micrometers. 1.2.4

Example of Micromechanically Tunable Mode-Locked Laser

A micromechanical tunable mode-locked laser has been constructed by combining the micro moving mirror as mentioned above with a multisegment laser. The original semiconductor laser is a multisegment laser consisting of seven segments with a total length of 1950 mm fabricated on an InGaAsP/InP strained quantum-well laser substrate. The reflectivity of the facet on the gain segment side is reduced by an AR coating with a residual reflectivity of around 2%. Each segment of the laser can be

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61

Figure 17 Autocorrelation trace and corresponding optical spectrum of modelocked pulses at a rate of 22 GHz.

independently biased in either the forward or the reverse direction to optimize the mode-locking condition. The stable passive mode-locking has been achieved under optimal reverse biasing of the edge-segment with a length of 190 mm as a saturable absorber while biasing the other segments forward. The constructed laser therefore produces mode-locked pulses with a tunable repetition rate at around 22 GHz. The mode-locking is confirmed by an optical spectrum and a corresponding second-harmonic generation (SHG) autocorrelation trace of the light output as shown in Fig. 17. The spectral full width at half-maximum of 0.56 nm and the pulse width of 3.9 ps derived from the SHG trace width of 5.9 ps assuming a hyperbolic-secant pulse shape gives the time-bandwidth product of 0.31. This means that the generated pulses are nearly transform-limited (TL). A more direct observation is performed with a streak camera that can transiently detect optical pulses. Figure 18 shows a typical temporal trace of the streak camera image measured for the pulses generated from the above mode-locked laser. This measurement scheme is however inadequate to investigate the repetition-rate tunability, because the change in the rate caused by the mirror displacement is too small to detect by the camera. Detecting the pulses with a photodiode is suitable to the above purpose. The detected signal (mode-locked signal) represents the transient pulse timing, the

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62

Chapter 3

Figure 18 Direct temporal measurement of mode-locked pulses at around 22 GHz by using a synchroscannedstreak camera.

Figure 19

Mode-locked signals of micromechanically tunable mode-locked laser.

repetition rate corresponds to the peak frequency, and the timing jitter corresponds to the phase noise of the mode-locked signal. This scheme is applied to the investigation of the tunability. Figure 19 shows typical power spectra of the mode-locked signals for various comb-drive voltages. These spectra show that the repetition rate is surely changed according to the comb-drive voltage corresponding to the mirror displacement. Figure 20 shows the peak frequency as a function of the squared voltage. The relationship exhibits an undulation with a period of half the wavelength coming from the residual reflectivity of the AR-coated facet. Such undulation is undesirable for practical use, but it can be much reduced by improving the AR coating. The mean repetition-rate change produced by displacing the mirror provides a linear relationship (shown by the solid line) with a change rate of 4.8 MHz/mm. This verifies the tuning concept.

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Intermittent Positioning in Micro-Optomechatronics

Figure 20

1.2.5

63

Measured repetition rate vs. squared comb-drive voltage.

Application to Optical Transmission Systems [21]

Optical Pulse Sources in Transmission Systems. Long-span optical transmission systems are performed with repeaters to maintain high-quality signals while suppressing propagation loss, waveform distortion, and timingjitter accumulation while transmitting signals (Fig. 21a). High-quality modelocked optical pulses can be used for return-to-zero (RZ)-formatted transmission. Hence mode-locked lasers can be applied to the repeaters for transmission systems using such RZ-formatted signals (Fig. 21b). Requirements. Large-capacity transmission systems employ timedivision multiplexing. All clocks installed in the systems must be synchronized with a master oscillator to discriminate multiplexed signals. Synchronizing the mode-locked lasers is therefore particularly important for practical implementation. The pulse timing is usually required to be as stable as that of electrical oscillators. It is assumed that an accuracy of 1 Hz is necessary for 20 Gbps data transmission with suppressing an error rate of less than 109. The 20 GHz mode-locked laser with an effective refractive index of 3.2 has a cavity length of L ¼ c/(2nf ) ¼ 2343.75 mm, and the accuracy of the length to satisfy the above requirement is nominally given by L ¼ L

f c ¼ f  0:0001 nm f 2nf2

ð32Þ

The mode-locked lasers are formed by cleaving with an accuracy of 5 mm. This accuracy produces a repetition-rate error of  20 MHz for 20 GHz

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64

Chapter 3

Figure 21 Configuration of time-division-multiplexing-based transmission systems. (a) Concept of long-span transmission with optical amplifiers and repeaters. (b) Schematic diagram of repeater.

mode-locked lasers. Moreover, the temperature dependence of the cavity length due to thermal expansion may produce a repetition-rate change of 20 kHz per degree, assuming an extremely small expansion coefficient of 106. Hence the micromechanical cavity-length control scheme by itself is insufficient for achieving the required accuracy. On the other hand, the electrical control of the refractive index, which provides only 0.2 0.3% change in the laser-cavity length corresponding to the repetition-rate change of 2 3 MHz nominally achieves fine tuning with a resolution of less than 1 Hz, but its tunable range is intrinsically limited by the physical properties of the semiconductor media. PLL Stabilization Supported by Micro Moving Mirror [16–19]. We find that cooperation of these two control schemes enables generation of mode-locked pulses that satisfy the required condition with compensating the uncertainty of the cavity length depending on the cleaving accuracy. Figure 22 shows a diagram of this composite control scheme. The micro moving mirror (MM) of the mode-locked laser diode (MLLD) with a control range of 10 mm is used for coarse repetition-rate control. The uncertainty caused by the cleaving is compensated for by this control. Further fine repetition-rate control is performed by the electrical method based on the dependence of the refractive index on the electrical

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65

Figure 22 Schematic diagram of PLL-stabilizing passively mode-locked semiconductor lasers using cavity-length control via micro moving mirror.

control parameters including the reverse bias voltage. This control compensates for not only the slight discrepancy between the required length and the mechanically controlled cavity length but also the cavity length fluctuations caused by thermal expansion in the lower frequency range and carrier-density fluctuation in the higher frequency range from DC to several gigahertz. This electrical compensation is easily applied to synchronizing modelocked pulses to an electrical signal using a phase-locked loop (PLL) technique (Fig. 23). The conventional PLL uses a negative feedback loop consisting of a voltage-controlled oscillator (VCO), a phase comparator, an amplifier, and an appropriate loop filter. The signal generated from a freerunning VCO is compared with a master clock by the comparator to produce a phase-error signal. This signal is negatively applied to the frequency-control element of the VCO via the amplifier and the loop filter. Minimizing the phase-error signal under the optimal loop gain, the VCO is synchronized to the clock. In the PLL circuit for optical pulse sources, the passively mode-locked laser corresponds to the VCO. The electrical repetition-rate control element is the saturable absorber: the repetition rate is controlled by the reverse voltage in the range that maintains the stable mode-locking. The range is often very small according to the kinds of mode-locked lasers, but the PLL scheme can be always available if the laser-cavity length is optimized by the micro moving mirror in advance. Figure 24 confirms the PLL synchronization: the free-running mode-locked laser exhibits a broad mode-locked signal corresponding to large phase noises, but the PLL-synchronized laser exhibits a sharp spike at the clock frequency, meaning that the phase noise is much reduced.

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66

Chapter 3

Figure 23 A synchronization scheme for passively mode-locked lasers using an optical phase-locked loop circuit.

Figure 24 Synchronization of passively mode-locked laser. (a) Free-running. (b) Synchronized.

Effects of the repetition-rate tuning performance on the mode-locked operation are typically shown for stabilizing the passively mode-locked pulses (see Fig. 25). Although the pulses are synchronized to an electrical signal optimized for the natural repetition frequency of the laser by using a

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67

Figure 25 Effect of micromechanical cavity-length adjustment on synchronization of mode-locked lasers.

phase-locked loop, the synchronization is readily broken by even slightly changing the signal frequency. However, the synchronization is reestablished at the new electrical signal frequency by adjusting the mirror position so as to reduce the frequency discrepancy. 1.3

Tunable Micromechanical Filter for Optical Pulse Shaping

Optical filters offer a wide variety of function including not only wavelength selectivity but also controllability over optical parameters such as group delay and group velocity. We can control light waves using such versatile properties of optical filters. Pulse shaping is a typical example, accomplished by adjusting a group-velocity dispersion (GVD) of the filter through which optical pulses are transmitted. We here focus on pulse-width narrowing which is particularly important in long-distance optical transmission systems based on time-division multiplexing (TDM). To accomplish the narrowing, we must characterize coherent optical pulses generated from mode-locked lasers and clarify their transmission characteristics through optical fibers. From these investigations we will find that chirped optical pulses can be either narrowed or broadened according to the magnitude and signature of the GVD of a filter to be transmitted. Hence we can optimize the filter for the narrowing. This section describes detailed narrowing mechanisms and presents an experimental demonstration using a micromechanically tunable interferometer as the filter.

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68

1.3.1

Chapter 3

Fourier Analysis for Optical Pulses

Optical short pulses are characterized in the frequency and time domains based on the principle of the Fourier transform. We can readily understand the effect of this transform by considering the fact that the repetition rate T corresponding to the pulse spacing in the time domain is reversely related to the repetition frequency of the pulses corresponding to the mode spacing F in the frequency domain. It is obvious that these parameters satisfy the relation FT ¼ 1. Figure 26 schematically explains the above relation. Similarly, the temporal profile of the pulse train is related to the spectral distribution. We here introduce a parameter of width at 3 dB reduction from the peak that quantitatively estimates the magnitude of distribution. Based on the Fourier analysis, we will find that the pulse width
t corresponding to the temporal distribution and the spectral width

satisfy the following inequality relation:
t

C

ð33Þ

where C is a parameter dependent on each pulse profile. This means that the minimum pulse width is limited by the spectral distribution. Of course, excessive spectral broadening does not directly contribute to the pulse-width narrowing: in this case, the parameter C becomes very large. The time– spectral width product can also take a minimal value C. Such pulses are regarded as transform limit (TL). 1.3.2

Characterization of TL Pulses

We can evaluate the optical pulses more quantitatively by performing the Fourier transform for assumed temporal pulse profiles. Let us take a

Figure 26 Characterization of optical short pulses. (a) Spectral profile. (b) Temporal waveform.

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69

representation for the electric field component of optical pulses for simplicity as EðtÞ ¼ AðtÞ exp½iðtÞ expði!0 tÞ

ð34Þ

where A(t) is the envelope of the temporal pulse waveform and !0 the angular optical frequency of the central mode of the pulses. Let us consider particular waveforms, which include Gaussian and hyperbolic secant profiles. The Gaussian profile, which is convenient for generally describing mode-locked pulses, is expressed assuming (t) ¼ 0, " # 1 t 2 ð35Þ expði!0 tÞ EðtÞ ¼ exp  2 ag with
t ag ¼ pffiffiffiffiffiffiffiffiffiffiffiffi 2 loge 2 where
t is the pulse width defined at the 3 dB power reduction from the peak power. The Fourier transform of the above field component is given by  Z 2 1 1 ag 1 EðtÞ expði!tÞ dt ¼ pffiffiffi exp  ag ð!  !0 Þ ð36Þ Eð!Þ ¼ 2 1 2 2  Hence the power spectral density jE(!)j2 gives a spectral width as pffiffiffiffiffiffiffiffiffiffiffiffi 2 loge 2
! ¼ ag

ð37Þ

Hence, we find the time–frequency product for the Gaussian pulses,
t

¼

2 loge 2 ¼ 0:4412 

ð38Þ

A similar evaluation can be done for the hyperbolic secant profile, which is often given by the pulses propagating in optically nonlinear media such as the optical soliton. The profile is expressed as EðtÞ ¼ sec h½as t exp½iðtÞ expði!0 tÞ ¼

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2 exp½iðtÞ expði!0 tÞ exp½as t þ exp½as t

ð39Þ

70

Chapter 3

with pffiffiffi  2 loge 1 þ 2 as ¼
t The Fourier transform of the hyperbolic secant profile is given by   ð!  !0 Þ Eð!Þ ¼ sec h as 2as This equation gives the spectral width as pffiffiffi  2as loge 1 þ 2

¼ 2

ð40Þ

ð41Þ

Hence the time-spectrum-width product is obtained, pffiffiffi!2  2 loge 1 þ 2
t

¼ ¼ 0:3148 

ð42Þ

Figure 27 numerically shows the relation between pulse width and corresponding spectral width for both the transform-limited Gaussian and the hyperbolic secant pulses. Since such time-spectrum-width product is readily measured using a conventional optical spectrum analyzer and autocorrelator, it is easy to distinguish whether optical pulses are transform-limited or chirped.

Figure 27 Relationship between spectral width and pulse width for transformlimited pulses.

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1.3.3

71

Propagation Characteristics

Basic Formalism. Let us consider how optical pulses propagate in a medium. We can generally use a Fourier transform technique for such consideration. The technique evaluates the effect of the medium to obtain the field component of the transmitted pulses in the frequency domain. This estimation is mathematically very simple, compared with a similar calculation procedure performed in the time domain. As can be seen in Fig. 28, the time-dependent field component E(t) is related to the frequency-dependent component, which can be readily given as the Fourier transform E(!). Since the effect of the medium on the pulse can be represented as a phase shift (!), the field component of the pulse transmitted through the medium is simply given as E(!) exp[i(!)]. Hence this component is transferred to the conventional time-dependent field by reverse Fourier transform. Even though with the time-domain analysis it is difficult to estimate the effect of dispersive media on pulse propagation, we can easily carry out such an analysis using the frequency-domain technique. The technique is more quantitatively explained. The frequencydependent field component is the Fourier transform of the temporal field component of the initial pulses, Eð!Þ ¼

1 2

Z

1

Aðt0 Þ exp½iðt0 Þ exp½it0  dt0 1

with  ¼ !  !0

Figure 28

Concept of spectral domain analysis based on Fourier transform.

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ð43Þ

72

Chapter 3

We assume that the phase shift imposed by the medium (!) is approximated, neglecting higher-order (n > 2) terms, as 1 1 ð!Þ ¼ 0 þ ð!  !0 Þ_ þ ð!  !0 Þ2 € ¼ 0 þ _ þ 2 € 2 2

ð44Þ

with 0 ¼ ð!0 Þ,



@ _ ¼ @! !¼!0

2

@  € ¼ @!2 !¼!0

Taking account of the effect of the above phase shift, the field component of the transmitted pulses is simply expressed in the frequency domain as Z1 1 expði0 Þ Aðt0 Þ exp½iðt0 Þ Eð!Þ exp½ið!Þ ¼ 2 1 

  1 2€ 0 0 _ ð45Þ
exp i   t exp i  dt 2 Hence the field component in the time domain is readily derived from the inverse Fourier transform of the above equation as Z1 Eout ðtÞ ¼ Eð!Þ exp½ið!Þ expði!tÞ d! 1 Z1 1 expði0 Þ ¼ Aðt0 Þ exp½iðt0 Þ 2 1 Z 1  

  1 exp i _  t0 þ t exp i2 € d dt0 ð46Þ
2 1 The integral with  is simplified as sffiffiffiffiffiffi   1 2€ 2 i=4 i 2 e

Fð Þ  exp i  exp½i  d ¼ exp  2 € 2€ 1 Z

1

ð47Þ

with

¼ t þ _  t0 Hence we obtain  1  Eout ðtÞ ¼ pffiffiffiffiffiffiffiffiffi exp i!0 t þ 0 þ 4 2€  Z1  i  0 0 2 _
exp iðt Þ  Aðt0 Þ dt0 tþt 2€ 1

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ð48Þ

Intermittent Positioning in Micro-Optomechatronics

73

Impulse Response. The simplest case for discussing the effect of a medium with chromatic dispersion on the pulse propagation often deals with an ideal pulse with a waveform exhibiting a delta function. The effect is regarded as the optical impulse response for the medium. We assume the initial pulse waveform as AðtÞ ¼ ðtÞ

ð49Þ

The output pulse waveform as the impulse response is therefore represented as  1  Eout ðtÞ ¼ pffiffiffiffiffiffiffiffiffi exp i!0 t þ 0 þ 4 2€  Z1  i  0 0 2 _
exp iðt Þ  ðt0 Þ dt0 tþt 2€ 1   2 1  i  _ ¼ pffiffiffiffiffiffiffiffiffi exp i!0 t þ 0 þ ð50Þ exp  tþ 4 2€ 2€ This equation can be simplified to     1  i Eout ðTÞ ¼ pffiffiffiffiffiffiffiffiffi exp i!0 T  _ þ 0 þ exp  T2 4 2€ 2€

ð51Þ

with T  t þ _ We can discuss the effect of the dispersive medium on the pulse propagation. The first-order phase dispersion term gives the time delay corresponding to the group delay g. The second-order phase dispersion term implies phase modulation. Such phase modulation provides the temporal change of the optical frequency of the pulses. Taking account of the definition of transient angular optical frequency as ðTÞ ¼ ½!0 þ
!ðTÞT the change from the mean angular frequency !0 is evaluated as

@ T2 T
!ðTÞ ¼  ¼ @T 2€ €

ð52Þ

ð53Þ

This means that the optical frequency is linearly changed by the secondorder phase dispersion. Such change is called linear chirp. This linear chirp will be necessary for pulse-width narrowing as described in following sections.

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74

Chapter 3

Propagation Characteristics of Practical Pulses. Pulses with No Phase Modulation: Let us consider the propagation characteristics of optical pulses with more practical profiles. We first deal with Gaussian pulses with no phase modulation, EðtÞ ¼ expði!0 t  s t2 Þ

with

s¼

1 2a2

ð54Þ

The field component of the transmitted pulses is nominally represented as  Z  1  1 i 2 Eout ðtÞ ¼ pffiffiffiffiffiffiffiffiffi exp i!0 t þ 0 þ exp  ðT  t0 Þ 4 1 2€ 2€ 02 0
expðst Þ dt ð55Þ The Fourier transform of the above equation is readily obtained as Z 1 1 Eout ð!Þ ¼ Eout ðt0 Þ expði!t0 Þ dt0 2 1 Z 1 1 1 ¼ pffiffiffiffiffiffiffiffiffi ei0 þi=4 Xðt0 Þ exp½ið!  !0 Þt0  dt0 2 € 1 2

ð56Þ

with  2   i  exp  exp st02 dt0 t þ _  t0 2€ 1

Z XðtÞ ¼

1

The Fourier transform of X(t) having a convolution form is given by   Z Z 1   1 1 i 2 exp  ð  t0 Þ exp st02 dt0 exp½i! d Xð!Þ ¼ 2 1 1 2€ ¼

1 2


 i exp  2 exp½i! d 2€ 1

Z

1 2

1

Z

1

  exp st02 exp½i! d

1





qffiffiffiffiffiffi 1 i=4 i € 2 1 1 1 2 € 2 exp !
pffiffiffi pffiffi exp  ! ¼ pffiffiffi e 2 4s 2  2  s sffiffiffiffiffi

 € e 1 i exp   € !2 ¼ 2s 4s 2 2 i=4

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ð57Þ

Intermittent Positioning in Micro-Optomechatronics

75

Hence we find 1 Eout ð!Þ ¼ pffiffiffiffiffiffiffiffiffi ei0 þi=4 Xð!  !0 Þ 2€ rffiffiffi

 ei0 1 1 i  € ð!  !0 Þ2 ¼ pffiffiffi exp  4s 2 4  s

ð58Þ

Hence, the field component of the transmitted pulses is given as the inverse Fourier transform of the above equation, Z1 Eout ðtÞ ¼ Eout ð!Þ expði!tÞ d! 1 rffiffiffi

 Z1 ei0 1 1 i € 02   ! expði!0 tÞ d!0 ¼ pffiffiffi exp  expði!0 tÞ 4s 2 4  s 1 " # ei0 a 1 1 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp  2 ¼ t 2a 1 þ 2€2 =ð4a4 Þ 2 a2  i€  i € 2
exp  t expði!0 tÞ 2 a4 þ €  a expði!0 t þ iÞ i€ t2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp  2ð
tÞ4 =ð4 ln 2Þ2 þ 2€ 2 ð
tÞ2 =ð4 ln 2Þ  i€  2 ln 2 t2 ð59Þ
exp  ð
tÞ2 1 þ €2 ð4 ln 2Þ2 =ð
tÞ4 This field component shows that the dispersive medium generates the phase modulation accompanied with linear chirping for the Gaussian pulses. It is also obvious that the width of the transmitted pulses always increases as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ €2 ð4 ln 2Þ2 ð60Þ
tout ¼
t ð
tÞ4 independently of the signature of the second-order phase dispersion. Figure 29 shows a typical calculation for temporal evolution of a Gaussian pulse propagating in a dispersive medium to certify the above intrinsic pulse width broadening phenomenon. Pulses with Phase Modulation (Chirped Pulses). The discussion can be readily extended to consideration on chirped Gaussian pulses represented as

 EðtÞ ¼ exp i!0 t  ðs þ ibÞt2 ð61Þ

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Chapter 3

Figure 29 Simulation of propagation of optical pulses with no chirp through a dispersive medium. (
t ¼ 5 ps).

The parameter b corresponds to a term of phase modulation. It is obvious that the modulation as expressed above gives a linear chirp, taking account of !

@’ ¼ !0  2bt @t

ð62Þ

The chirping behavior of the pulses is determined according to the signature of b: when b > 0 the frequency decreases with time (down chirping), and when b > 0 it increases (up chirping), as shown in Fig. 30. The field component of the transmitted pulses can be derived from a similar calculation as follows: rffiffiffiffiffiffiffiffiffiffiffiffi

 Z1 ei0 1 i 1 þ 2€ !02 Eout ðtÞ ¼ pffiffiffi expði!0 tÞ exp 4 b  is 4  s þ ib 1
expði!0 tÞ d!0

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expði!0 t þ i0 Þ a2 ð2a2 b þ iÞ pffiffiffi

  ¼ 4  ð1 þ 2ia2 bÞ € þ i 2a2 b€ þ a2  1 1 2
exp  2 t 2a 1 þ 4b€ þ 4b2 €2 þ €2 =a4 " #   2€b 2€b þ 1 þ €2 =a4 2 t
exp i 1 þ 4b€ þ 4b2 €2 þ €2 =a4

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ð63Þ

Intermittent Positioning in Micro-Optomechatronics

Figure 30

77

Electromagnetic fields of chirped pulses. (a) Down chirp. (b) Up chirp.

We estimate the envelope of the pulses as  1 1 2 fðtÞ ¼  2 t 2a 1 þ 4b€ þ 4b2 €2 þ €2 =a4  1 4 ln 2 1 2 t ¼  2 ð
tÞ2 1 þ 4b€ þ 4b2 €2 þ €2 ð4 ln 2Þ2 =ð
tÞ4

ð64Þ

Therefore we can derive the width of the transmitted pulses from the above equation as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 4b€ þ 4b2 €2 þ €2 ð4 ln 2Þ2 ð65Þ
tout ¼
t ð
tÞ4 To evaluate this pulse width change, we introduce the following parameter as a function of the second-order phase dispersion,  


tout 2 ð4 ln 2Þ2 ðÞ  ¼ 1 þ 4b þ ð66Þ 2
t ð
tÞ4 þ 4b2 This parameter indicates that when the second-order dispersion satisfies the following condition, jj <

4jbj ð4 ln 2Þ =ð
tÞ4 þ 4b2 2

ð67Þ

the width of the transmitted pulses becomes smaller than that of the initial pulses. The minimal width is readily calculated as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
tout 4b2 ¼ 1= 1 þ ð
tin Þ4 ð68Þ
tin ð4 ln 2Þ2

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78

Chapter 3

This minimization is achieved when the second-order dispersion has an optimal value opt given by opt ¼ 

2b ð4 ln 2Þ =ð
tÞ4 þ 4b2 2

ð69Þ

Scheme of Pulse Width Narrowing. The above consideration shows that pulse width narrowing is possible for chirped pulses by using dispersive media with an appropriate second-order phase dispersion. In this section, we consider the physical meaning of the pulse width narrowing phenomenon and then verify the narrowing scheme by using a simple numerical simulation. The physical picture of pulse width narrowing is clearly understood by considering the chromatic velocity variation dependent on the portion of the pulse that comes from the chirping. This technique uses a dispersive medium with an appropriate group delay to compensate the chirp of the pulses (Fig. 31). A filter changes the group delay maintaining a constant reflectivity. If the GVD is reverse to the chirp (see 31c), the filter can compress the chirped pulses (see 31d). The effect of pulse narrowing mentioned above will always occur if € satisfies the pulse compression condition, but the chirping parameter b must

Figure 31 Scheme of adiabatic pulse compression. (a) Intensity profile of initial pulse. (b) Frequency change by chirp. (c) Transmission through filter. (d) Pulse width narrowing.

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Figure 32 Pulse width change versus second-order phase dispersion of medium as a parameter of linear chirp coefficient b.

Figure 33 Simulation of propagation of chirped pulses through a dispersive medium. (
t ¼ 5 ps, b ¼ 1023 s2).

exceed a certain value to be practical. Figure 32 shows a pulse width ratio
tout/
tin as a function of €, assuming an initial pulse width of 5 ps. The narrowing effect is not clear for b < 2
1022(s2), but it is distinguishable above 1023(s2) at around €  3
1024 ðs2 Þ: the ratio becomes less than 0.5. Figure 33 shows a typical numerical simulation of chirped pulse propagation in a dispersive medium to verify the narrowing effect. Our interest is now focused on the phase modulation by the dispersive medium expressed as a function of the second-order dispersion  as   2b þ ð4 ln 2Þ2 =ð
tÞ4 þ 4b2 2 2 ð70Þ ðt, Þ ¼    t 1 þ 4b þ ð4 ln 2Þ2 =ð
tÞ4 þ 4b2 2

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80

Chapter 3

It is attractive that the phase modulation becomes zero for the above optimal second-order dispersion. 1.3.4

GTI as Tunable Phase Dispersion

As mentioned above, optimizing phase dispersion of a medium is essential for pulse width narrowing. Typical optical elements capable of controlling its dispersion include a Gires–Tournois interferometer (GTI) as well as filters and diffraction gratings. Theory of GTI. The GTI’s optical response offers changeable phase shift based on multiple reflection in the interferometer while maintaining the total reflection condition. Figure 34 illustrates a schematic diagram of a GTI consisting of an optical resonator with a half mirror and a total reflection mirror spaced by h to form a configuration similar to that of Fabry–Perot resonators. Let us numerically evaluate the phase dispersion of the GTI. We assume an incident planwave with an amplitude of unity for simplicity. Assuming that the half mirror has reflection and transmission coefficients r and t, respectively, the effective reflectance taking account of round-trips of light in the interferometer is expressed as pffiffiffiffi  pffiffiffiffi  2 reff ¼  R þ ð1  RÞei þ ð1  RÞei Rei þ ð1  RÞei rei þ   ¼ r þ 1  jrj2 ei

1 1  rei

with  ¼ !

Figure 34

Schematic diagram of GTI.

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ð71Þ

Intermittent Positioning in Micro-Optomechatronics

81

pffiffiffiffi where is the round-trip time of light in the interferometer. Using r ¼ R, the above equation is simplified to pffiffiffiffi ð1 þ RÞ cos  2 R  ið1  RÞ sin pffiffiffiffi ð72Þ reff ¼ 1 þ R  2 R cos Hence the phase shift given by the GTI is obtained as an argument of the above complex reflectance:  ð1  RÞ sinð! Þ 1 pffiffiffiffi ð!Þ ¼ Arg½reff  ¼ tan ð73Þ ð1 þ RÞ cosð! Þ  2 R Hence higher order phase dispersion terms are given as @ ð1  RÞ

pffiffiffiffi ð!Þ ¼  @! 1 þ R  2 R cosð! Þ pffiffiffiffi @2 2 2 ð1  RÞ R sinð! Þ  ð ! Þ ¼ pffiffiffiffi  2 @!2 1 þ R  2 R cosð! Þ

ð74Þ

ð75Þ

Design Criterion of GTI for Pulse Compression. Optical pulses can be compressed to the transform limit by a method based on dispersion compensation for chirped pulses. Since the spectral bandwidth is enlarged with reducing the pulse width, a wideband dispersion compensator is needed to generate ultrashort pulses. To obtain pulses with a width of a few picoseconds, the corresponding spectral width is a few nanometers. The compensator must satisfy the compression condition in this wavelength span. As Eq. (75) indicates, the GTI will be featured by its tunable dispersion from a negative to a positive value at a wavelength of interest, if its cavity length h is changeable. As shown in Fig. 35, the GTI surely has the second-order phase dispersion alternately changing from negative to positive at every half wave as a function of the cavity length. The amplitude of this dispersion undulation increases with cavity length. It also increases with increasing the reflectivity of the half mirror composed of the GTI. Therefore the GTI can be used for compensating both upchirped and downchirped pulses. The problem is that the band to maintain a large second-order phase dispersion is limited. This may be a drawback if the GTI is not optimized for the pulses to be compressed. Hence let us consider how a GTI is optimized for a subject that initial transform-limited pulses 5 ps in width should be compressed to a width of 2 ps. Assuming a chirp parameter of 1023 s2,

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Figure 35

Chapter 3

Tunability in the second-order phase dispersion of GTI.

Figure 36 Optimal design for GTIs. (a) Short cavity GTI with a cavity length of around 20 mm. (b) Long cavity GTI with a length of around 127 mm.

we find from Fig. 36 that the second-order phase dispersion must be around 1024 s2 in a wavelength region 1 nm in width for pulse compression. This requirement will be satisfied when a GTI has h ¼ 20 mm and R ¼ 0.85, or when it has h ¼ 127 mm and R ¼ 0.35. This means that both GTIs can be used for pulse compression. Hence we can select either of them, but we must note that a short-cavity GTI is easy to fabricate but needs extremely high positioning accuracy (see Fig. 36a), and that a long-cavity GTI is controllable by a conventional positioning mechanism but needs an optimal structure to compensate the diffraction loss due to its long cavity for multiple reflections in the cavity necessary for obtaining a high phase dispersion value.

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Compression Experiments Using Micromechanical GTI [23]. Let us look at certain optical pulse compression experiments using a short-cavitytype GTI fabricated on an Si substrate with a micro moving mechanism and an optical fiber. A short-cavity tunable GTI can be constructed by adjusting a cleaved facet of a single-mode optical fiber against a micro moving mirror with electrostatic comb actuators as shown in Fig. 37a. A high reflectivity of around 0.9 is achieved by using an Au coating on the mirror facet. The reflectivity of the fiber facet is also improved to 0.16 by using dielectric coatings. The adjustment of the fiber is carried out along a V-groove formed on the Si substrate to achieve a cavity length of 15 mm. Figure 37b shows a photograph of a fabricated GTI. Figure 38 shows a schematic system equipped with a tunable GTI as mentioned above for pulse compression experiments. A colliding-pulse mode-locked laser (CPM-LD) provides an initial pulse stream with a repetition rate of 192 GHz. The pulses are amplified by EDFA and then pass through a 20 km long single-mode dispersion-shifted fiber (DSF). Since the zero dispersion wavelength of the fiber (1565 nm) is almost equal to the center wavelength of the pulses, the transmitted pulses should be chirped by self-phase modulation in the fiber. These pulses are introduced to the GTI via a circulator, and the reflected pulses are analyzed to confirm the compression effect using a conventional optical spectrum analyzer (OSA) and autocorrelator to observe spectral and temporal waveforms of the pulses.

Figure 37 Micromechanically tunable GTI. (a) Schematic structure. (b) SEM image of fabricated GTI.

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Figure 38

Schematic setup for pulse compression experiments.

Figure 39

Pulse compression process.

Let us consider a more detailed pulse compression process. As can be seen in Fig. 39, the initial pulses exhibit
t

¼ 0.57, so they are not transform limited. After they are transmitted through the DSF, they are compressed to 0.62 ps in width. They also exhibit a broad spectral distribution 0.84 THz in width. The corresponding time-bandwidth product exhibit
t

is 0.49. This means that the pulses are still chirped. Then these transmitted pulses are introduced to the GTI. The GTI is optimized so that the pulses have the narrowest width. Consequently, we obtained the compressed pulses having a width of 0.43 ps. The corresponding timebandwidth product of 0.36 shows that the compressed pulses are nearly transform limited. This confirms that the GTI can compress optical pulses

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based on dispersion compensation of chirped pulses. This pulse compression concept is the same as that of general methods based on an adiabatic compression mechanism widely used for femtosecond optical pulses.

2 2.1

MICROMECHANICAL CONTROL OF CAVITIES BASED ON SLIDE TUNING MECHANISM AND ITS APPLICATIONS Practical Instrument Based on Slide Tuning [24,25]

We present a disk-shaped wavelength-tunable optical bandpass filter (the disk filter) to demonstrate the rotary slide tuning mechanism. Figure 40 shows a schematic configuration for implementation of the rotary slide tuning scheme. As a narrow collimated optical beam traces a folded path between optical fibers, it passes through the filter via two prism mirrors. The transmission center wavelength can be tuned by rotating the disk. The normal incidence condition is always maintained during the above tuning operation. This normal incidence enables polarization-independent wavelength tuning. The periphery of the disk with a diameter of 50 mm and a thickness of 1.2 mm has a slit pattern with a reference point providing the zero position (Fig. 41a). An optical sensor reads this pattern to generate encoded signals (Fig. 41b), which are analyzed by a processor to determine logically the passing position (Table 2). Arbitrary wavelength will be readily selected by positioning the disk by detecting the passing position, if all the wavelengths are calibrated as a function of the position.

Figure 40

Disk-shaped wavelength-tunable optical bandpass filter.

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Figure 41 Determination of absolute rotation angle by encoder scheme. (a) Schematic illustration of patterned marks on the disk and corresponding encoded signals. (b) Measured encoded signals.

Table 2

Logic for Determining the Absolute Rotation Angle

Timing B " or # A" A# B" B#

2.2

Logic (A ¼ L)^(Z ¼ L) B¼L B¼H B¼H B¼L A¼H A¼L A¼L A¼H

Function Reset(0) Up(þ1) Down(1) Up(þ1) Down(1) Up(þ1) Down(1) Up(þ1) Down(1)

Principle of Linearly Tunable Optical Filters

Let us explain the principle of the disk filter using a schematic filter structure as shown in Fig. 42. The filter is based on interference filtering using a structure that has a cavity layer with half the wavelength sandwiched by two high-reflection (HR) coatings. This structure is fabricated on a stable glass (SiO2) substrate. Such a disk filter is basically equivalent to a Fabry–Perot resonator and has the function of selecting the particular wavelength matched with the cavity layer. To achieve the wavelength tunability, the cavity layer thickness

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Figure 42 Schematic structure of disk filter with circular-wedge resonant cavity layer sandwiched by high-reflection coatings.

Figure 43 Schematic structure of an HR coating consisting of quarter-wave stacks with different refractive indices.

is changed. The disk filter has a circularly wedged cavity that has the cavity thickness linearly changing with the disk rotation angle. The wavelength selectivity is determined according to the performance of the HR coatings consisting of periodic stacks of two quarter-wave layers with different refractive indices (Fig. 43): as the band rejection performance is improved with increasing the number of stack layers of the HR coatings, the passband width is narrowed. Let us present a simple theoretical consideration on the transmission performance, taking account of the theory of Fabry–Perot etalons. According to the theory, we obtain the transmission center wavelength and filtering bandwith as functions of the resonant cavity thickness depending on the rotation angle h():

c ¼

2nh m


¼

ðm ¼ 1, 2, 3, . . .Þ

2nhð1  RÞ pffiffiffiffi  Rm2

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ð76Þ ð77Þ

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Figure 44 Typical transmission and reflection spectra of disk filter with 0.1 nm bandwidth at 3 dB reduction.

The transmission spectrum of the filter is also represented as Tð , c Þ ffi

1 1 þ 4½ð  c Þ=
2

ð78Þ

We can also exactly estimate the transmission performance using thin film optics based on the Maxwell equations as described in the following chapter. However, this simple estimation supports our understanding of how the disk filter works. Typical transmission characteristics of the disk filter are shown in Fig. 44. The presented transmission and reflection spectra with Lorentzian profiles trace the theory with good agreement, confirming that the filter theory can be applied to such wavelength-tunable optical bandpass filters with wedged cavities. Wavelength-tunable performance is also confirmed by measuring the transmission center wavelength versus the rotation angle as shown in Fig. 45. The figure also shows the uniformity of the filter: the bandwidth and the total transmission loss remain constant around 0.15 and 6 dB in the entire tuning range of 1515–1545 nm, respectively.

2.3

Positioning Using DC Servo Motors

The system to realize the intermittent positioning is based on a conventional closed-loop control technique using a voltage-controlled rotary actuator such as a DC servo motor widely used in hard disk drives. To evaluate the closed loop, the dynamic performance of the disk driven by the motor is

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Figure 45 Typical transmission characteristics of disk filter as a function of rotation angle.

analyzed by using an equation of motion, J

d 2 d  ðtÞ þ B ðtÞ ¼ KT IðtÞ 2 dt dt

ð79Þ

where (t) is the rotation angle as a function of time, J the total rotary inertia of the disk and motor, B the viscosity damping factor of the motor, KT the torque parameter, and I the driving current for the motor. The current is represented with other electric parameters according to Ohm’s law as L

d d IðtÞ þ RIðtÞ  KE ðtÞ ¼ VðtÞ dt dt

ð80Þ

Arranging the Laplace transforms of the above two equations, we readily obtain the transfer function of the motor, GðsÞ 

 ð sÞ 1 KT ¼ VðsÞ s ðJs þ BÞðLs þ RÞ  KE KT

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ð81Þ

90

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Figure 46 Intermittent positioning scheme. (a) Block diagram for position control. (b) Positioning performance.

An intermittent positioning to maintain a constant rotation angle is achieved in a closed loop. The loop is constructed with an appropriate loop filter for suppressing the oscillation of the loop, as schematically shown in Fig. 46a. Hence the transfer function of the closed loop is represented as HðsÞ ¼

FðsÞGðsÞ 1 þ FðsÞGðsÞ

KT FðsÞ  ¼  s ðJs þ BÞðLs þ RÞ  KE KT þ KT FðsÞ

ð82Þ

The loop filter is usually determined based on empirical considerations. Typical filters include the one expressed as 1 FðsÞ ¼ KP þ KI þ KD s s

ð83Þ

The gain of the transfer function defined as jH(i!)j becomes unity for an effective frequency range by optimizing the parameters (KP, KI and KD) of the loop filter. This means that the output rotation angle  is equal to the

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input angle 0, hence the positioning is achieved. Such a positioning scheme can be logically realized by programming to do so in an MPU. Experimental demonstrations were performed with a disk 40 mm in diameter and 1.2 mm in thickness. Figure 46b shows the result of a test where the intermittent positioning changed the absolute disk position from 1000 to 0 in a digitized scale providing 2048 counts per revolution. The positioning with a settling time of around 80 ms was achieved with no overshooting in an accuracy of 1 count. The positioning performance must be tested in more severe conditions for practical applications, but the positioning scheme was fundamentally verified. 2.4 2.4.1

Accurate Wavelength Control for WDM-Based Optical Networks Requirement of WDM-Based Optical Networks [26–29]

There is great interest in photonic networks for versatile internet services. For such a network as shown in Fig. 47, wavelength division multiplexing (WDM) is useful because it offers various functions necessary for operating the networks. These functions typically include optical add–drop multiplexing by handling the WDM channels, optical switching, or wavelength tuning. These functions can be realized using monolithically integrated photonic devices including passive or active devices fabricated on planar lightwave circuits, such as arrayed waveguide gratings (AWGs). Since the wavelengths of WDM channels are usually fixed, the wavelengths of modules including such as transmitters and receivers must be precisely adjusted to the fixed values to operate the photonic networks. Therefore the wavelength management in the networks is the most important task.

Figure 47 Typical topology for wide-area networks. (a) OADM ring and bus topology. (b) Logical structure with full-mesh links.

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Conventional centralized wavelength management systems are much too costly for practical network systems, so, simple and flexible decentralized management is desired. This management scheme needs a simple open-loop wavelength control scheme but supports a single network manager or local managers allocated at each network nodes. 2.4.2

Absolute Wavelength Control Scheme

This section presents a simple absolute wavelength control method using calibrated disk-shaped optical bandpass filters. There are some possible candidates for local wavelength standards. Optical spectrum analyzers using gratings are the most promising, but they are too large and costly. The compact, stable, and low-driving-power disk filter is the most promising candidate for a local wavelength standard. However, there remains a problem, which is the temperature dependence of the filter, which comes from the thermal expansion of the filter materials and the temperature dependence of the refractive index [30,31]. This drawback is readily eliminated by temperature compensation control of the filter. Figure 48 shows a principle of the compensation. A relationship between the digitized beam transmitting position given by the rotary encoder and the transmission center wavelength under a reference temperature of T ¼ T0 is assumed. Since the transmission center wavelength is almost linearly changed with temperature, the relationship is simply shifted by k(l)
T in the graph. Therefore a constant transmission center wavelength l0 is maintained by rotating the disk to the position where the central wavelength agrees with l0  k(l)
T. Consequently, accurate

Figure 48 Principle of temperature compensation to realize highly accurate wavelength control for disk-shaped tunable filter.

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Figure 49 Demonstration of highly accurate wavelength control of disk-shaped tunable filter at various temperatures.

wavelength control with small wavelength errors within 0.02 nm has been achieved for the entire tuning range at various temperatures (Fig. 49). 2.4.3

OADM Subsystem

Let us introduce the OADM subsystem (Fig. 50) to verify the presented autonomous network control scheme. The main switching element uses four OLIVE (oil latching interfacial tension variation effect) switches [32] which are connected to MUX/DEMUX utilizing temperature-controlled AWGs, and to doubled add and drop ports. The add ports were equipped with wavelength-tunable lasers, and the drop ports are equipped with optical amplifiers and corresponding wavelength-tunable optical bandpass filters to eliminate amplified spontaneous emission noises. These tunable devices can be suited to all the AWG channels by using their internal calibration tables. 2.4.4

Transmission Experiments

The disk filter has a wide variety of applications in OADM subsystems, including three typical examples: wavelength selectors, channel monitors, and wavelength-tunable lasers. This section focuses on the wavelengthtunable lasers.

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Figure 50 Schematic diagram of an OADM subsystem using wavelength-tunable modules for both transmitting and receiving.

Wavelength-tunable transmitters are a key for creating an arbitrary OADM channel at the add ports. Widely wavelength-tunable lasers can be constructed using the disk filter. They include a simple fiber ring configuration. The oscillation center wavelength can be determined by the filter based on the above wavelength-control scheme using calibration data. The intensity of the ring laser can be stabilized by installing an SOA in the laser cavity operated under the gain-saturated condition. Tests have been performed with an experimental setup (Fig. 51a) equipped with the above wavelength-tunable ring laser. The stable laser output is modulated at 10 Gbps using an electroabsorption modulator. Since the WDM channels given by the MUX/DEMUX are measured in advance, the wavelength of the laser can be controlled to any WDM channel in the open system using the calibration data of the filter. The tests have found that all the oscillation wavelengths agree with the WDM channels given by the MUX/DEMUX as shown in Fig. 51b. All corresponding temporal traces measured by an oscilloscope exhibit allowable eye patterns (Fig. 51c). This shows the feasibility of the presented autonomic network control scheme.

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Figure 51 Demonstration of autonomic channel control for an add port of the OADM subsystem using a wavelength-tunable ring laser using an optical disk filter. (a) Experimental setup. (b) Wavelength tuning of laser suitable to AWG’s grids. (c) Temporal waveforms of transmitted signals through add modules.

Figure 52 Performance of OLIVE switch. (a) Experimental setup. (b) Optical spectra observed at passing and drop ports. (c) Observation of switching operation.

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The tests have also clarified the switching performance using an experimental setup as shown in Fig. 52a. The optical switching performance of the OLIVE switches gives a sufficient extinction ratio of >45 dB, which is allowed in practical systems (Fig. 52b). Dynamic performance as shown in Fig. 52c has clarified that the net switching time of less than 10 ms is sufficiently fast. These experimental results have verified that optomechatronic devices can offer various high-performance functions.

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Bowers, J.E.; Morton, P.A.; Mar, A.; Corzine, S.W. Actively modelocked semiconductor lasers. IEEE J. Quantum Electron. 1989, 25, 1426– 1439. Chen, Y.-K.; Wu, M.C. Monolithic colliding-pulse mode-locked quantum-well laser. IEEE J. Quantum Electron. 1992, 28, 2176–2185. Katagiri, Y.; Takada, A. A harmonic colliding-pulse mode-locked semiconductor laser for stable sub-THz pulse generation. IEEE Photon. Technol. Lett. 1997, 9, 1442–1444. Helkey, R.J.; Derikson, D.J.; Mar, A.; Wasserbauer, J.G.; Bowers, J.E.; Thornton, R.L. Repetition frequency stabilization of passively mode-locked semiconductor lasers. Electron. Lett. 1992, 28, 1920–1922. Buckman, L.A.; Georges, J.B.; Park, J.; Vassilovski, D.; Kahn, J.M.; Lau, K.Y. Stabilization of millimater-wave frequencies from passively mode-locked semiconductor lasers using an optoelectronic phase-locked loop. IEEE Photon. Technol. Lett. 1993, 5, 1137–1140. Katagiri, Y.; Takada, A. Synchronised pulse-train generation from passively mode-locked semiconductor lasers by a phase-locked loop using optical sidebands. Electron. Lett. 1996, 32, 1892–1894. Hashimoto, E.; Takada, A.; Katagiri, Y. Synchronisation of subterahertz optical pulse train from PLL-controlled colliding pulse mode-locked semiconductor laser. Electron. Lett. 1998, 34, 580–582. Katagiri, Y.; Takada, A.; Nishi, S.; Abe, H.; Uenishi, Y.; Nagaoka, S. Repetition-rate tunable micromechanical passively mode-locked semiconductor laser. Electron. Lett. 1996, 32, 2354–2355. Katagiri, Y.; Takada, A.; Nishi, S.; Abe, H.; Uenishi, Y.; Nagaoka, S. Passively mode-locked micromechanically-tunable semiconductor lasers. IEICE Trans. Electron. 1998, E81-C, 151–159. Yamamoto, H.; Asada, M.; Suematsu, Y. Electric-field refractive index variation in quantum-well structure. Electron. Lett. 1985, 21, 579–580. Akimoto, K.; Katagiri, Y.; Takada, A.; Nagaoka, S. Micromechanically controlled GT interferometer for optical-pulse narrowing. Sensors and Actuators 2000, 82, 79–82. Mekada, N.; Al-Hamdan, A.; Chong, T.H.; Daut, D.G. Polarizaion independent, linear-tuned interference filter with constant transmission characteristics over 1530–1570-nm tuning range. IEEE Photon. Technol. Lett. 1997, 9 (6), 782–784. Katagiri, Y.; Tachikawa, Y.; Aida, K.; Nagaoka, S.; Ohira, F. Synchroscanned rotating tunable optical disk filter for wavelength discrimination. IEEE Photon. Technol. Lett. 1998, 10, 400–402. Vreeburg, C.; Uitterdijk, T.; Oei, Y.; Smit, M.; Groen, F.; Metaal, E.; Demeester, P.; Frankena, H. First InP-based reconfigurable integrated add– drop multiplexer. IEEE Photon. Technol. Lett. 1997, 9, 188. Teshima, M.; Koga, M. 100-GHz-spaced 8-channel frequency control of DBR lasers for virtual wavelength path cross-connect systems. IEEE Photon. Technol. Lett. 1996, 8, 1701.

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Yano, Y.; Ono, T. Absolute wavelength stabilization of LD lights for dense WDM systems using a multi-wavelength meter. OECC’97, 10B2-4, 1997, Seoul, Korea. Hashimoto, E.; Katagiri, Y. 10-GHz-spacing DWDM channel selector using disk filter cascade with distributed amplification. ECOC ‘2001, Th.F3 Tech. Digest 2001, 80–81. Drouard, E.; Chantome, P.-H.; Escoubas, L.; Flory, F. @n=@T measurements performed with guided waves and their application to the temperature sensitivity of wavlength-division multiplexing filters. Appl. Opt. 2002, 41, 3132–3136. Katagiri, Y.; Aida, K.; Tachikawa, Y.; Nagaoka, S.; Abe, H.; Ohira, F. Thermal stability in wavelength discrimination using synchro-scanned optical disk filter. Electron. Lett. 1998, 34, 1515–1516. Sakata, T.; Togo, H.; Makihara, M.; Shimokawa, F.; Kaneko, K. Improvement of switching time in a thermocapillarity optical switch. IEEE J. Lighwave Technol. 2001, 19, 1023–1027.

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4 Constant Velocity Positioning in Micro-Optomechatronics Constant velocity positioning, which maintains the constant velocity of a moving object, is regarded as a typical case of continuous positioning. The technique uses a closed loop in which an error signal detected by comparing the velocity of the object with a reference signal is fed back to an actuator that drives the object so that the error should become zero. Techniques to accomplish this control method include the phase-locked loop (PLL), which is widely used in various areas, particularly in wireless communication systems. The PLL consists of a velocity detector, a phase comparator to produce an error signal, and a feedback circuit, including a velocitycontrollable actuator. Such a scheme is featured by the capability of not only regulating a constant velocity but also synchronizing the motion with an electrical signal. This synchronization has contributed to the development of micro-optomechatronics. In this chapter, we will be dealing with a rotary system to understand how the PLL works. Of course the system can be readily extended to more general cases including rectilinear motion systems, but rotary motion is useful for many practical applications in that area of precision optical information instruments. Laser-beam printers and scanners, for example, use a rotating polygon mirror whose velocity is precisely controlled to deflect a laser beam. High-performance optical disk systems are another good example to show its usefulness; they control the angular velocity of a rotating disk to maintain a constant read-out bit rate, i.e., constant linear velocity, independently of the radial position of an optical head. Then, we will study an advanced technique for micro-optomechatronics based on the constant velocity positioning. This technique is linear wavelength scanning using a synchroscanned rotating disk-shaped wavelength-tunable optical filter. Linearity in the angular position, versus the transmission wavelength characteristics of the filter, is essential for realizing this technique. We will study how to obtain linearity. We also meet some 99

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useful applications of this technique implemented in optical measurement and telecommunication systems.

1 1.1

PHASE-LOCKED LOOP FOR CONSTANT VELOCITY POSITIONING Basic Formulation Based on Time–Domain Analysis

Let us consider a rotary system that can be controlled by constant velocity positioning. The system consists of a disk with slit marks on its periphery, an encoder sensor to read the marks, and a rotary actuator whose torque is controlled by a driving current (see Fig. 1a). The sensor generates digital encoded signals when the disk is rotated. When the rotation speed is ! and the number of slit marks is N, the encoded signals have a carrier frequency,  ¼ !N. As shown in Fig. 1b, the signals have a fundamental component at a frequency  and harmonic components at multiples of  in the frequency spectrum. The fundamental component can be extracted from the signals using a low-pass filter. The extracted component exhibits a sinusoidal

Figure 1 Schematic of angular velocity detection mechanism. (a) Rotary system. (b) Roll of low-pass filter.

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waveform,

 V0 ðtÞ ¼ a0 cos t þ ’0 ðtÞ

ð1Þ

where ’0(t) is a noise component and  the average. We call this signal a velocity signal. A transient angular frequency (t), corresponding to the angular velocity !(t), can be obtained by differentiating the phase of Eq. (1) as ðtÞ 

@ @  ¼  þ ’0 ðtÞ @t @t

ð2Þ

The second term corresponds to a velocity fluctuation. Now, how to detect the transient angular frequency is discussed as follows. We know the average frequency  in advance, and so we can provide a reference signal with the same average frequency as

 ð3Þ Vi ðtÞ ¼ ai sin t þ ’i ðtÞ This reference signal is mixed with the velocity signal using a phase comparator. The comparator generates the output voltage VC ðtÞ ¼ KC Vi ðtÞV0 ðtÞ



 ¼ KC ai sin t þ ’i ðtÞ a0 cos t þ ’0 ðtÞ    1 ¼ KC ai a0 sin 2t þ ’i ðtÞ þ ’0 ðtÞ þ sin½’i ðtÞ  ’0 ðtÞ 2

ð4Þ

where KC is the comparator’s gain. As the higher frequency component of the output is eliminated by a low-pass filter, we can obtain the second term [which is often called an intermediate frequency (IF) signal] as 1 VP ðtÞ ¼ KC ai a0 sin½’i ðtÞ  ’0 ðtÞ 2

ð5Þ

When the term ’i (t)  ’0(t) is very small, the output is approximated as   1 VP ðtÞ ¼ KC ai a0 ’i ðtÞ  ’0 ðtÞ 2

ð6Þ

This equation shows that the output is proportional to the phase error, so we can regard the output as a phase-error signal. Consider what happens when the phase-error signal is fed back to the velocity-controllable rotary actuator. For a gain of the actuator K0, the feedback signal generates a transient velocity change, that can be

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expressed as   d 1 ’0 ðtÞ ¼ K0 VP ðtÞ ¼ K0 KC ai a0 ’i ðtÞ  ’0 ðtÞ dt 2

ð7Þ

Defining a phase error function and loop gain respectively as ðtÞ  ’i ðtÞ  ’0 ðtÞ

ð8Þ

1 KL  K0 KC ai a0 2

ð9Þ

we can simplify Eq. (7) to d d ðtÞ ¼ ’i ðtÞ  KL ðtÞ dt dt

ð10Þ

We must consider that the actuator is not so quickly responsive to an electrical signal as electrical circuits. Hence, assuming F(t) as a response function of the actuator, we have to rewrite Eq. (7) in a convolution form as Zt d ’0 ðtÞ ¼ K0 Fðt  t0 ÞVP ðt0 Þ dt0 dt 0 Zt ffi KL Fðt  t0 Þðt0 Þ dt0 ð11Þ 0

This equation represents a more realistic mechanical behavior of the actuator. Using Eq. (8), we finally obtain Zt d d ðtÞ ¼ ’i ðtÞ  K0 Fðt  t0 ÞVP ðt0 Þdt0 dt dt 0 Zt d ¼ ’i ðtÞ  KL Fðt  t0 Þðt0 Þ dt0 ð12Þ dt 0 We can exactly simulate the motion of the actuator by solving this integral equation. 1.2

Frequency–Domain Analysis

In time-domain analysis, as mentioned above, it was necessary for solving a integral equation such as (12) to investigate the behavior of the rotary system. This integration is usually complicated. Hence we here present a more simple and useful method to deal with the issue. This method is based on frequency-domain analysis using Laplace transforms.

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Let us start making the Laplace transform of Eq. (11) as 1 ’0 ðsÞ ¼ KL FðsÞ½’i ðsÞ  ’0 ðsÞ s

ð13Þ

We can extract many physical meanings from this equation; it nominally offers a closed loop, as shown in Fig. 2a, where ’i(s) is an input signal, ’0(s) the corresponding output, and KLF(s)/s the open-loop transfer function. This function can be decomposed into three independent elements, KL, F(s), and 1/s (see Fig. 2b). KL is a frequency-independent gain that translates the position error ’i  ’0 to a corresponding input voltage variation V(s) applied to the control element of the rotary actuator. F(s) is the frequency response of the actuator, and 1/s means a temporal integration of the

Figure 2 Block diagrams of PLL to achieve constant velocity positioning. (a) Fundamental block diagram. (b) Decomposed diagram to clarify physical meaning. (c) Diagram of PLL with additional feedback for higher stability.

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actuator’s output, which indicates the angular position ’0. The combination of these elements characterizes the behavior of the closed loop. We define a transfer function of the loop as ’0 ðsÞ  HðsÞ’i ðsÞ

ð14Þ

This transfer function can be readily obtained by eliminating the phase components as HðsÞ ¼

KL ðFðsÞ=sÞ 1 þ KL ðFðsÞ=sÞ

ð15Þ

This form is also given by considering the loop configuration as shown in Fig. 2a. This transfer function gives the important fact that a sufficiently large loop gain KL offers H(s) ¼ 1, which means ’0(s) ¼ ’i(s) This obviously means that the rotary actuator is synchronized with the reference signal. So we call the above loop a PLL. For higher stability, a DC component of the IF signal from the phase comparator is negatively fed back to the input signal through an appropriate low-pass filter and loop delay as shown in Fig. 2c. This feedback loop produces an additional damping force and thus contributes to improving the stability. Such control schemes achieve complete agreement in position and velocity between an object and an electrical signal. This is intrinsically equivalent to the control scheme as mentioned in Fig. 14 of Chap. 2. 1.3

Frequency Response Characteristics

We numerically evaluate the transfer function H(s) of the rotary system. The transfer function of the rotary actuator is given by FðsÞ ¼

 S2 þ 2!0 s þ !20

ð16aÞ

Then we obtain an expression for the transfer function, KL  HðsÞ ¼  2 s s þ 2!0 s þ !20 þ KL

ð16bÞ

This equation shows that the loop is stable independently of the loop gain KL. Here we must remember that the stability comes from mechanical damping of the rotary actuator, so that appropriate loop filters will be needed to suppress loop ocillation if the actuator is small enough to provide a negligible inertia. Substituting i! for s, we can discuss the dynamic response. We usually investigate this function by independently calculating the absolute

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Figure 3 Calculated frequency characteristics of transfer function of PLL.

amplitude term jH(i!)j and the corresponding phase term Arg[H(i!)]. Figure 3 shows an example of calculations. These calculations exhibit typical mechanical resonant oscillations of rotary actuators with their resonant frequencies at around !0. The PLL stably works at frequencies less than each resonant frequency. 1.4

Practical PLL System

A schematic diagram of a practical rotary system equipped with a PLL circuit is shown in Fig. 4. A mixer and a low-pass filter form a phase comparator to produce an IF signal. This signal is a beat signal between the reference and velocity signals, and it exhibits a sinusoidal undulation according to Eq. (5). However, it acts as a phase-error signal linearly changing with the phase difference according to V ¼ K
’ [see Eq. (6)], when the frequency of the beat signal is lower than a critical value. This value determines the capture range of the PLL. Such an IF signal is biased and negatively fed back to a voltage control element of the servomotor. This DC biasing is usually adjusted to obtain an angular velocity signal with a frequency close to that of the reference signal. The motor driver produces an alternating control signal for the motor. The velocity of the motor is controlled by this signal, and phase-locking is accomplished.

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Figure 4 PLL.

Schematic diagram of constant angular velocity positioning system using

Figure 5 Phase-locking process.

The locking process is illustrated in Fig. 5. The IF signal provides a beat signal with a decreasing frequency in the capture process of the PLL, and then it is reduced to zero in the phase-locking process. The IF signal as a phase-error signal is not perfectly zero in the phase-locked condition, but the residual slight phase error can be eliminated by optimizing the loop. 1.5

Numerical Evaluation of Stability [1,2]

We will estimate the stability of disk rotation controlled by a PLL. The estimation is carried out by analyzing a frequency spectrum of the angular velocity signal.

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Figure 6 Frequency domain analysis of the angular-velocity signal for numerical evaluation of rotation stability. (a) Power spectrum of the velocity signal. (b) Singlesideband phase noise derived from power spectrum.

As shown in Fig. 6a, the velocity signal exhibits a spectral waveform having a sharp spike corresponding to the carrier frequency accompanied with broad phase-noise components on both sides of the carrier. Hence we can numerically evaluate the stability by comparing an effective noise power derived from the phase-error components with the power of the carrier signal. Since the phase-noise components are symmetric about the carrier frequency, there are only single-sideband (SSB) phase-noise components. It is allowed that the velocity signal has no power fluctuation, so the phasenoise components L’( f ) can be directly obtained from the measured noised components S’( f ) as L’ ð f Þ ¼

S’ ð f Þ
f

ð17Þ

where
f is the frequency resolution for spectral measurement. The rootmean-squared (RMS) rotation jitter can be estimated using the singlesideband (SSB) phase noise L’( f ) of the position signal in a frequency range ( fL, fH) as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Z fH 1 L ’ ð! Þ df ð18Þ j ¼ 2
f 2fN fL where f is the carrier frequency. Let us calculate the RMS rotation jitter according to the above formulation. Figure. 6b shows the SSB phase-noise spectrum derived from

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the measured position signal at 160 kHz, which was produced by a disk with 1024 slits rotating at 156 revolutions per second. Carrying out the integration of Eq. (18) in the range of 100–1000 Hz, we obtain an RMS jitter of 1.5 nm.

2

LINEAR WAVELENGTH SCANNING

Applications of the disk filter operated under the intermittent positioning were presented in Chap. 3. The filter acted as a wavelength selector. In this chapter the filter operated in a sychroscan rotating mode acts as a wavelength sweeper. Such a usage is of great value for a wide variety of applications in optical measurements. Wavelength sweepers are usually required to scan the wavelength linearly. However, linear wavelength scanning cannot be achieved by merely using wavelength-tunable filters. Hence we explain how to achieve linear wavelength scanning. 2.1

Tunable Optical Filters for Linear Wavelength Scanning

There are many applications based on constant velocity positioning. However, it is often more important for changing physical parameters at a constant speed. This task is not realized only by constant velocity positioning. The applications based on this concept include linear wavelength scanning using a disk-shaped wavelength tunable optical filter. High linearity between angular position and transmission wavelength characteristics is as important as stable synchronous disk rotation, to achieve almost perfect linear scanning, as shown in Fig. 7. 2.2

Filter Design

We argued in Chap. 3 that the disk-shaped wavelength-tunable optical filter offers linearity in rotation angle versus transmission wavelength owing to its circularly wedge-shaped cavity. This argument is inadequate for discussing linearity. In this section we will be dealing with this issue using a theory of optical thin films based on Maxwell’s equations. Details of the theory are described in Chap. 6. The transmission characteristics of optical bandpass filters are given by 4n0 Re½Ns  T¼  pn0 þ q2

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ð19Þ

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Figure 7 Wavelength and optical frequency deviations from each linear approximation as functions of normalized resonant cavity length.

with



p 1  ðUH UL ÞM1 UH UC UH ðUL UH ÞM1 q y NS 0 Uj  @

cos ’

1 sin ’ y Nj A cos ’

i

iy Nj sin ’

’¼

 0 2

ð20Þ

ð21Þ

Here l0 is the nominally designed wavelength, Nj the complex refractive pffiffiffiffiffiffiffiffiffiffiffiffi index, y the admittance of vacuum "0 =0 , and n0 the complex refractive index of the incidence medium. UC is a matrix corresponding to the resonant cavity with a lower refractive index, which can be given by replacing ’ with

 0 1
D 1þ ’¼ 2 2 D0

ð22Þ

where D0 is the original length of the phase-shift section at l ¼ l0. Equation (19) cannot directly exhibit the linearity, the relationship between cavity length and transmission wavelength calculated according to this equation shows sufficient linearity at around l0 corresponding to the Bragg wavelength of the periodic structure of high-reflection coating regions

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as shown in Fig. 7. We surely find a perfect linear portion in the normalized cavity length range of
D/D0 ¼ 0.96–1.04, which corresponds to a wavelength range of 1530–1560 nm for l0 ¼ 1550 nm. We here must note that there is no linear portion in optical frequency versus transmission wavelength characteristics also as shown in the figure.

2.3

Fabrication of Filters [3,4]

Let us consider how to fabricate a wedge-shaped cavity. Dielectric multilayer filters are generally fabricated by techniques in which the cavity is formed by depositing molecules of the cavity materials on a substrate using evaporation or sputtering. Suppose that the deposition rate can be maintained constant. When a plate mask is displaced along the substrate surface at a constant speed, a wedge-shaped structure will be formed on the substrate along the moving direction as shown in Fig. 8a. This concept can be readily extended to a circularly wedge-shaped structure by using a fan-shaped mask rotating above a disk substrate as shown in Fig. 8b. Now, we will investigate transmission characteristics of a fabricated filter to verify the above method. Figure 9a shows the transmission center wavelength versus the rotation angle of a filter fabricated for the test. We can readily find the linear portion in the range of the rotation angle of 0–26,000 in the entire count of 60,000 per round. The linearity can be investigated by calculating wavelength deviation from a linear approximation of measured characteristics. The deviations are suppressed to less than 0.02 nm in the linear portion as shown in Fig. 9b.

Figure 8 How to fabricate a circular wedge cavity. (a) Method of fabricating wedge-shaped thin film by moving a mask. (b) Method of fabricating circularly wedge-shaped film using a fan-shaped mask.

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Figure 9 Linearity in transmission wavelength versus rotation angle. (a) Linear portion of actual filter. (b) Deviation from perfect linearity.

3

PRACTICAL EXAMPLES OF LINEAR WAVELENGTH SCANNING

3.1 3.1.1

Time–Domain Optical Spectrum Analyzer Outline

A most simple application of the linear wavelength scanning of wavelengthtunable filters is optical spectrum measurement. This section describes a time–domain optical spectrum analyzer using a disk filter rotated synchronously with an electrical signal. The measurement can be simply carried out by detecting the transmitted power. Figure 10 shows temporal waveforms for various laser wavelengths in the entire tuning range of 1530–1608 nm. The time of the peaks gives the corresponding wavelength. The time axis is absolutely calibrated by the laser locked to the HCN-gas absorption line at 1552.46 nm [5]. This calibration enables the displacement of the time axis by the wavelength axis. However, in practical measurement, we must take account of temperature dependence on transmission wavelength of the filter as well as the rotation angle versus the transmission wavelength characteristics. 3.1.2

High-speed Operation [6]

It is important to clarify the high speed limitation imposed on the measurement system. The mechanical limitation is in a higher range of around 10,000 rpm for existing DC servomotors widely used in HDD and optical disk systems. However, it may be decreased by mechanical

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Figure 10

Optical spectrum measurement for various laser wavelengths.

Figure 11

High-speed rotation stability.

fluctuations of not only the mechanism of the motor but also that of the disk. Such fluctuations are undesirable for such information instruments because of their limited tracking servo performance. They are also not allowed in the measurement instrument using the disk-shaped filter because they may produce incident-angle variation, thereby generating transmitted power fluctuations. We simply investigate the high-speed performance by measuring temporal waveforms of the transmitted power for various rotation speeds as shown in Fig. 11, and we find a maximal available speed of 8,200 rpm for a disk 50 mm in diameter and 2 mm in thickness. The maximal speed will be improved if the rotary mechanism is optimized.

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113

Resolution in Spectrum Measurement

It is necessary to define resolution in spectrum measurement using optical filters with a Lorentzian spectral response much different from the ideal sink function as shown in Fig. 12a. Such a Lorentzian spectral response is featured by its long tailing edge, which may degrade the power accuracy owing to the accumulation of wideband spectrum components of the light to be measured. Hence the wavelength resolution should be much larger than the transmission bandwidth of the filter usually defined as an FWHM. Let us consider the resolution issue further. A spectral component that can be measured by a filter is theoretically given by a convolution form: Z1 ð Þ ¼ ð 0  ÞFð 0 Þ d 0 ð22Þ 0

where F(l) is a spectral profile of the light to be measured, (l) the spectral response of the filter, and (l) the observable spectral profile for the light. This means that the actual spectral component at l0 ¼ l is diluted by other components. Therefore, we introduce a parameter of power accumulation that maximizes the influence as Z 0 Pð 0 Þ  ð 00 Þ d 00
100 ð%Þ ð23Þ 0

Figure 12b shows the calculated power accumulation as a function of normalized wavelength deviation   l/a, where a is a characteristic parameter of optical bandpass filters such as 1 j j  a 0, j j a Ideal ð24Þ ð Þ ¼ 2a ð Þ ¼

1 1 þ ð =aÞ2

Lorentzian

ð25Þ

Figure 12 Concept of spectrum analysis using optical bandpass filter. (a) Calculated spectral responses of optical filters. (b) Power assumptions.

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The power accumulation linearly increases with increasing , and it reaches 100% at  ¼ 1 for the ideal filter. However, the Lorentzian filter provides an accumulation slightly increasing with wavelength deviation. This means that the Lorentzian filter can be used for sharp spectral analysis when a lower power accumulation is required, but that it will be inadequate for highperformance measurement in optical networks when a higher power accumulation is required. 3.1.4

Double-Path Configuration

A simple method of eliminating the power accumulation issue of the Lorentzian filter is to use multiple transmission paths for modifying the spectral response. Although the number of the paths is limited, only the doubled Lorentizan exhibits a sharp rise in power accumulation equivalent to that of the ideal filter, as can be seen in Fig. 12b. This method is achieved using a cascade configuration with two spatial beams that trace each folded path (see Fig. 13). These two spatial beams have each intersection precisely adjusted along the radial direction of the disk to obtain the same rotation angle. This adjustment enables simultaneous wavelength tuning for the two beam paths. The problem is lack of uniformity in the transmission characteristics along the radial direction, which may readily disable the multiple path method. Taking account of further investigations on transmission characteristics of the filters, however, it is possible to fabricate filters with enough uniformity to perform the method. A transmission test is carried out using the disk filter with an FWHM of 0.8 nm. The input laser wavelengths are in the range of 1530–1580 nm,

Figure 13 Schematic of a multiple(double)-path method using fiber collimator couplers in a cascade configuration.

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Figure 14 Effective multiple path configuration for spectrum analysis. (a) Temporal traces of transmitted power through cascaded fiber collimators at various wavelengths. (b) Comparison between single and double path configurations.

which corresponds to the entire tuning range of the filter. Figure 14a shows the measured temporal traces of the transmitted power for these wavelengths. The trace surely reaches each maximum when the transmission wavelength of the filter agrees with each laser wavelength. This confirms the feasibility of the multiple path method. The method is then applied to discriminating two laser wavelengths spaced by 2.6 nm. The temporal trace measured by a conventional single path method is not sufficient for distinguishing these wavelengths, but that measured by the multiple path method is sufficient for doing so (see Fig. 14b). 3.2 3.2.1

Highly Sensitive Laser Wavelength Detection [7] Principle

Continuous wavelength locking of lasers is important for WDM-based optical networks. The simple synchroscanned disk filter easily enables discriminating the laser wavelength, which is a key to this laser wavelength locking. This section describes how much sensitive the laser wavelength detection is. Since the line width of the laser is much narrower than the transmission bandwidth of the filter, the temporal waveform of the transmitted power exhibits a typical Lorentizan profile as shown in Fig. 15. The waveform is almost symmetric about the transmission center wavelength. The time i corresponding to the peak of the waveform is related to the laser

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Figure 15

Schematic for wavelength detection.

wavelength li. It is difficult to find i because the gradient of the transmitted power becomes zero at the peak. However, a differential method can readily detect the time i and therefore enables accurate discrimination for laser wavelength. The differential method is a way to find a peak in a single-lobe curve. To carry out this method, a differential signal S is defined as the power difference measured at ¼ 0  . When the laser wavelength is equal to l0, the transmitted power becomes maximal. This means S ¼ 0. However, when the laser wavelength is slightly displaced from l0, the differential signal is not zero: it indicates the wavelength discrepancy and is regarded as a push– pull error signal. Hence using calibration data obtained in advance, it is possible to determine the laser wavelength from the time when the differential signal becomes zero. A more quantitative discussion is presented as follows. Suppose that the Lorentzian spectral response of the filter is given by Tð , 0 Þ ¼

1 1 þ 4ðð  0 Þ=
Þ2

ð26Þ

where l0 is the transmission center wavelength. Rotated synchronously with a clock, the filter can relate the wavelength to the relative time as

c ¼ s þ k!

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ð27Þ

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where ls is the wavelength at ¼ 0, k is the angular gradient of the peak wavelength change, and ! is the angular velocity of the rotation. The laser wavelength to be detected is li. The rotating filter gives the temporal transmission power trace for the laser wavelength that exhibits a delta function, so the temporal waveform is given by Z Pð , i Þ ¼ Tð , c ð ÞÞ ð  i Þ d (



i  c ¼ 1þ4


2 )1

(



k! ¼ 1þ4


2

)1 2

ð i  Þ

ð28Þ

where i is the time when the waveform reaches a peak. Suppose that the target wavelength l0 is given at 0. The differential signal is expressed using the two powers monitored at ¼ 0   as Sð i , 0 Þ  Pð 0 þ , i Þ  Pð 0  , i Þ

ð29Þ

The right-hand components are approximated by using Taylor’s expansion: 2 k! Pð 0 þ , i Þ ffi 1  4 ð 0  i þ Þ2 ð30Þ
Pð 0  , i Þ ffi 1  4

k! 2 ð 0  i  Þ2


ð31Þ

Therefore we find

4k! Sð i , 0 Þ ¼ ð 0  i Þ


2 ð32Þ

Introducing a parameter  ¼
l/2k! for convenience, we obtain Sð i , 0 Þ ¼

8k! ð i  0 Þ


ð33Þ

This differential signal linearly indicates the wavelength discrepancy between the laser and the target, and it becomes zero when i ¼ 0. 3.2.2

Experimental Demonstration

Now let us examine the laser wavelength detection method as mentioned above. Figure 16a shows a typical S-shaped curve according to Eq. (33). The curve can be measured by slightly changing the oscillation wavelength of the laser while maintaining 0   constant. Detailed figures of this curve can be investigated by finely changing the laser wavelength at around li ¼ l0, which

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Figure 16 Resolution of laser wavelength detection. (a) Measures S-shaped curve. (b) Detailed investigation.

must be detected using a heterodyne technique. In this region, the S-shaped curve becomes almost linear as shown in Fig. 16b. This curve gives a minimal detectable wavelength variation. This value is determined by the transmission bandwidth of the filter, but it is much smaller than the bandwidth. In the case presented here, the detectable wavelength variation is 100 MHz (0.8 pm). Such extremely high laser wavelength detection performance will be used in advanced light wave applications such as coherent systems. 3.3

Wavelength Scanning Laser

An optical spectral analysis scheme using optical filters has an intrinsic problem arising from the statistical behaviors of photons, as often discussed in quantum optics. The point of the problem is the increase in intensity uncertainty introduced by the spectral slicing of the filters. An effective way to avoid this problem is to use a wavelength scanning laser, whose statistical Poisson distribution never changes, although the power is attenuated. This means that the power of the filtered light is exactly measured in the narrow time window necessary for high-speed measurement. This method also makes possible ultra-high-resolution measurements supported by the linewidth of the laser, which is much narrower than the bandwidth of practical filters. 3.3.1

Principle and Fundamental Behaviors [8,9]

The disk filter and optical gain media in a ring configuration are used to construct a wavelength scanning laser. Figure 17a is a schematic of the laser

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Figure 17 spectrum.

119

Intensity stable fiber ring laser. (a) Configuration. (b) Oscillation

system. The disk filter is installed between two optical gain media The ring also has an isolator for unidirectional optical circulation, a polarization controller for suppressing polarization diversity, and a directional coupler for extracting an optical output from the ring. One of the optical gain media must be a semiconductor optical amplifier with strong gain saturation, while the other may be an optical fiber amplifier. The ring laser provides a lasing spectrum exhibiting multiple modes; it is confirmed, however, that the intensity stability is equal to that of a conventional single-mode laser. The spectrum spans over approximately 1 GHz at the transmission-peak wavelength of the filter, as shown in Fig. 17b. The center oscillation wavelength can be changed by controlling the disk filter. Such tunability is demonstrated in the range from 1530 to 1590 nm at every 5 nm. Output powers above 3 dBm and optical signal-tonoise ratios of around 45 dB are also obtained at every oscillation wavelength (see Fig. 18). 3.3.2

Demonstrations of Spectral Measurement [10–12]

The laser is even stable against external mechanical fluctuations. This performance is also maintained when the wavelength of the filter is changed. Consequently, we can continuously change the laser wavelength by tuning the disk filter. To confirm the continuous tunability, step-by-step spectral measurements are performed for acetylene 12C2H2 absorption lines (Fig. 19a) [13]. The absorption lines were clearly detected at the wavelengths corresponding to the authorized reference. This confirms continuous tunability without any oscillation instability. Then, faster laser wavelength scanning, at 670 nm/s, is also examined to obtain the corresponding transmission spectrum. Figure 19b shows the averaged temporal trace. Absorption lines

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Figure 18

Chapter 4

Wavelength tunable performance of ring laser.

Figure 19 Measurement on acetylene (12C2H2) absorption lines using wavelength tunable fiber laser. (a) Step-by-step measurement. (b) Measurement in synchroscan mode.

derived from this trace agree with those obtained by the step-by-step scheme, so successful discrimination is confirmed for the absorption lines from the measured trace. The scanning speed is much smaller than that limited mechanically, but it is sufficient for practical applications.

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Figure 20 Application to analyzing fine spectral response for various devices under scanning mode. (a) Reflection of fiber Bragg grating. (b) Transmission through etalon.

The laser also offers completely continuous wavelength tunability. This is useful for measuring spectral responses of optical devices that will be used in DWDM systems. Figure 20 shows spectral responses for typical devices including a fiber Bragg grating (FBG) with a bandwidth of 0.1 nm and an etalon with a free spectral range (FSR) of 50 GHz, which are measured in a sychroscan mode. These results not only confirm the continuous tunability but also show great potential for general measurement applications. 3.4 3.4.1

Pulsed Optical Frequency Sweeper [14–16] Concept and Principle

An optical frequency chain is a pulse train in which the optical frequency changes pulse by pulse by the same frequency interval. These chains provide a set of wavelength references in a way similar to conventional optical frequency combs. The merit of using the chains is the capability of determining arbitrary optical frequency only by extracting a corresponding optical pulse from the train on the time axis. The optical frequency chains are generated using an optical ring. The ring contains an optical frequency shifter and a directional coupler (Fig. 21a). Once an optical pulse is inserted into the ring,

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Figure 21 Wideband optical frequency chain generation. (a) Fundamental concept of optical frequency chain generation. (b) Method for extending chain.

it circulates in the ring, launching its replica at every round trip. The replicas form a pulse train on the time axis. As the optical frequency of the pulse is shifted by the optical frequency shifter at each passing, the pulse train exhibits an optical frequency chain as
0,
0 þ
f,
0 þ 2
f,
0 þ 3
f, and so on. We have considered that such chains have a wide variety of practical application areas, including optical measurements. Extending the chain length, however, is still a problem. The power profile of the pulse train exhibits exponential attenuation owing to losses in the ring, which includes the output coupler with 3 dB attenuation. Hence the length of the pulse train is effectively limited. To overcome this limitation, we have to compensate for the attenuation by optical amplification in the ring. Wideband optical fiber amplifiers are useful for this purpose. Moreover, they simultaneously cause undesirable accumulation of amplified spontaneous emission (ASE) noise. If a bandpass filter is scanned in the ring synchronously with the pulse circulation, ASE accumulation can be suppressed (Fig. 21b). We consider that the disk filter can be used as the scanning filter.

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3.4.2

123

Experimental Demonstration

Figure 22 shows an experimental setup. An acousto-optical frequency shifter (AOFS) operating at 1 GHz is used for the experiments. As mentioned in the previous section, the filter does not exhibit linear tuning performance for the optical frequency. Taking into account the discrepancy from linearity, a disk filter with a rather wide bandwith of 1.2 nm is adopted. An AO modulator launched the pulse into the cavity via a directional coupler. This coupler produced a replica of the circulating pulse. Two optical fiber amplifiers are used for attenuation compensation. The disk filter has two beam paths adjusted in the same condition as in the multiple-path method. One is used for the ring and the other for adjustment of the timing of the pulse circulation. The rotation speed can be determined according to the round trip time 0 and the slope in the wavelength versus rotation angle of the filter; however, the timing for continuous wavelength scanning cannot be determined. Hence to determine the timing of the initial pulse launch from the master laser with
0 oscillation, the transmitted power of a reference laser with an optical frequency of
t is monitored. Since the difference
t 
0 and the slope can be translated into the time delay D, the timing is exactly determined using an electrical delay generator. The optical frequency chain generation test using the ring is performed with a master laser oscillating at 194,675.1 GHz. The generated pulse train is

Figure 22

Setup for generating optical frequency chains.

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Figure 23 Performance of pulsed optical frequency chain. (a) Front portion of chain. (b) Entire chain.

observed with an oscilloscope. Figure 23a shows a front portion of the train, and Fig.23b shows the overall train. The ASE accumulation observed between the upper and lower envelopes is suppressed over a wide range of time, so a large circulation number is expected. Taking into account the round trip time of 3 ms and the observed pulse train length of 6,300 ms, we estimated the number as 2,100. As the frequency shift is 1 GHz per round, the sweep span of the generated optical frequency chain is estimated as 2.1 THz. To confirm this optical frequency sweep span, particular pulses are individually extracted and their wavelengths are measured using a conventional optical spectrum analyzer. Figure 24a shows the superposition of the measured spectra of 0th and 2000th pulses. The wavelength of the 0th pulse is adjusted as 1540.000 nm. The wavelength corresponding to the optical frequency displaced by 2000 GHz is 1556.996 nm. The 2000th pulse is successfully observed at this wavelength. The optical signal quality of the pulse train is also investigated. The continuous wave light from the master laser is modulated at 4 GHz to generate modulation sidebands and mixed with 0th and 2000th pulses individually to generate beat signals. Figure 24b shows the spectrum of a

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Figure 24 Optical quality of generated pulses. (a) Optical spectra of the first and final (2000th) pulse. (b) Optical frequency spectrum of the first pulse. (c) Optical frequency spectrum of the final pulse.

beat signal at around 4 GHz measured for the 0th pulse. It exhibits sufficient spectral purity. Figure 24c shows that for the 2000th pulse. The signal component at 4 GHz maintained spectral purity exhibiting high SNR, around 10 dB. This value is enough to discriminate the signal component from the background noises.

REFERENCES 1.

Suetake, H.; Tsuchinaga, H.; Tanaka, S.; Niihara, T.; Nakamura, S.; Mita, S. High-speed/high-density magneto-optic recording. Optical Data Storage ’92, SPIE Proceedings, San Jose, CA, 1992; Vol. 1663, 2–6. 2. Rodwell, M.J.W.; Bloom, D.M.; Weigarten, K.J. Subpicosecond laser timing stabilization. IEEE J. Quantum Electron. 1990, 26, 231–241. 3. Thelen, A. Circularly wedged optical coatings. I. Theory. Appl. Opt. 1965, 4, 977–981.

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4.

Apfel, J.H. Circular wedged optical coatings. II. Experimental Appl. Opt. 1965, 4, 983–985. Tachikawa, Y.; Suzuki, Y.; Arihara, M.; Ishikawa, J. Long-term stability in the frequency characteristics of a highly-frequency-stabilized light source in the 1.55-mm band. OFC’95 WE6 1995. Katagiri, Y.; Aida, K.; Tachikawa, Y.; Nagaoka, S.; Abe, H.; Ohira, F. Highspeed demonstration of wideband synchro-scanned optical disk filter for absolute laser wavelength discrimination. Electron. Lett. 1998, 34, 1310–1312. Katagiri, Y.; Aida, K.; Tachikawa, Y.; Nagaoka, S.; Abe, H.; Ohira, F. Highaccuracy laser-wavelength detection using a synchro-scanned optical disk filter. IEEE Photon. Technol. Lett. 1999, 11, 102–104. Barnett, S.M.; Radmore, R.M. Methods in Theoretical Quantum Optics; Clarendon Press: Oxford, 1997. Katagiri, Y.; Suzuki, K.-I.; Aida, K. Intensity stabilisation of spectrum-sliced Gaussian radiation based on amplitude squeezing using semicoductor optical amplifiers with gain saturation. Electron. Lett. 1999, 35, 1362–1364. Lammel, G.; Schwizer, S.; Schiesser, S.; Renaud, P. Tunable optical filter of porous silicon as key component for a MEMS spectrometer. J. Microelectromechanical Systems 2002, 11, 815–827. Strassner, M.; Luber, C.; Tarraf, A.; Chitica, N. Widely tunable constant bandwidth monolithic Fabry–Perot filter with a stable cavity design for WDM systems. IEEE Photon. Technol. Lett. 2002, 14, 1548–1550. Irmer, S.; Daleiden, J.; Rangelov, V.; Prott, C.; Romer, F.; Strassner, M.; Tarraf, A.; Hillmer. H. Ultralow biased widely continuously tunable Fabry– Perot filter. IEEE Photon. Technol. Lett. 2003, 15, 434–436. Gilbert, S.L.; Swann, V. Acetylene 12C2H2 Absorption Reference for 1550–1540 nm Wavelength Calibration. SRM 2517, NIST Special Publication 1998, 260-133. Hodgkinson, H.G.; Coppin, P. Pulsed operation of an optical feedback frequency synthesizer. Electron. Lett. 1990, 26, 1155–1157. Aida, K.; Nakagawa, K. Pulsed lightwave frequency synthesizer system using an EDFA and AOD in a fiber loop. IEICE Trans. Commun. 1995, E78-B, 664–673. Takesue, H.; Yamamoto, F.; Shimizu, K.; Horiguchi, T. 1 THz lightwave synthesised frequency sweeper with synchronously tuned bandpass filter. Electron. Lett. 1998, 34, 1507–1508.

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5 Follow-Up Positioning in Micro-Optomechatronics In this section, examples of follow-up positioning in micro-optomechatronics are explained. As shown in Fig. 1, there are active and passive forms in follow-up positioning. In active positioning, a feedback loop composed of a sensor and an actuator is used. There are two types of signal detection timings by the sensor: discrete and continuous. There are two principles of detecting light: use of optical intensity and phase. There are two directions of actuator movement: axial (out-of-plane), and vertical (in-plane). Passive positioning does not use sensors and actuators. In this section, micromechatronics devices are classified by the positioning methods, and typical examples are introduced. As examples of active control, in which we detect positional information as a continuous intensity signal, there are a focusing servo and a tracking servo in an optical disk. As an example of the use of a discrete intensity signal, there is a sampled servo. These are explained in Sec. 1. A flying optical head, an example of passive positioning, is explained in Sec. 2. A laser displacement sensor, an example of the use of a continuous phase signal, is explained in Sec. 3.

1 1.1

FOLLOW-UP POSITIONING IN CONVENTIONAL OPTICAL DISK [1,5] Outline of Principles of Optical Disk Recording, Reproducing, and Positioning

An outline of optical disk technology is repeated here. A schematic Structure of the CD-ROM drive is shown in Fig. 2. While a disk rotates, an optical head irradiates it by laser light beam from the lower side; thereby the readout from the disk is carried out. To read as small data marks as possible, the light beam is focused on the disk surface by a lens. 127

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Figure 1 Classification of tracking and positioning methods in this chapter.

Figure 2 CD-ROM drive. (From Ref. 1.)

There are several methods of performing data recording on a disk, as shown in Fig. 3. The first method uses relieflike uneven pits and offers readout operation only. Details are shown in Fig. 4. A pit occupies a half area of a spot of the laser beam, and upper parts are l/4n higher than lower parts. When a beam illuminates a pit, a reflection from an edge of the pitch has a different optical phase from that of the one from another edge, and their superposition changes its power by interference. The existence of pits is decided from the change in light strength. Pits are made on a plastic substrate by injection molding, and there is a thin aluminum film on its

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Figure 3 Recording methods of optical disks.

Figure 4 Data reproducing principle of CD. (From Ref. 13.)

surface. This structure is suitable for mass production and is used in CDs, and in CD-ROMs. The second method uses hole punching, and data recording is performed only one time. The readout method is the same as that for uneven pits; the only different point is that pits are made by an optical head. In the writing operation, the laser increases its light power and partially melts the illuminated area of the recording layer, and a pit is made. In the reading operation, the laser power is decreased so that the layer is not

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melted. Since such a data recording scheme, by using melting, is so simple, this method is studied for next-generation probe memory. The third method uses the magneto-optic (MO) effect, which ideally shows a sufficient retention performance for an infinite read/write cycle number. There is a thin magnetic film on the surface of the disk, with data corresponding to the direction of magnetization; which are read by an optical head. When linearly polarized light is directed to a magnetic material, the direction of polarization of the reflected light changes according to the direction of magnetization (Kerr effect). Therefore, by observing the polarization direction, we can read the data marks. In writing, a magnetic field is applied to the disk, and a pit is formed on the disk by the laser beam and heated up by the laser. Consequently, the local magnetization at a pit is reversibly changed according to the direction of the external magnetic field. Ferromagnetic materials are characterized by a critical temperature (Curie temperature); they are ferromagnetic below the temperature but paramagnetic above it. The magnetization in a paramagnetic state is dominated by the external magnetic field; it vanishes when the magnetic field is removed, but it remains when the disk is rapidly quenched. The laser is used to control partially the temporal temperature of the disk; it raises only the temperature of a pit on the disk above the Curie temperature, so writing is achieved. The fourth method is based on the reversible phase change (PC) performance of chalcogenide materials such as Ge2Sb2Te5 (GST). This method also offers a retention performance with a large number of read/ write cycles. The material is heated by the laser and then quenched just by decreasing the laser power. Whether in crystalline or amorphous state, the material is controllable according to the temporal variation of the laser power. These two states offer different reflectivity, so we can carry out the readout operation by detecting the power of the reflected light. The system necessary for performing this operation is quite simple in comparison with that of the MO. Recent DVD-RAM adopts this mechanism. This system will be a prime recording method for future nanomemory. Conventional optical disk systems as mentioned above perform read/ write operations using a beam-converging optical system by lens for highdensity data recording. An optical disk is not perfectly flat, and the tracking center is not necessarily consistent with the rotation axis of the motor. Therefore the focal point of the laser must follow not only the surface of the rotating disk but also the tracking guide; in short, it needs three-dimensional follow-up control. The admissible positioning error for carrying out the read/write operations is 1 mm for focusing in the out-of-plane direction and 0.1 mm for tracking in the in-plane direction. But since there is a position detecting error caused by thermal expansion or machining error of

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leading grooves (mentioned below), the servo error must be about half of the total error, about 0.5 mm and 0.05 mm. The laser beam position is controlled by the rotation control of a motor in the circumference direction, by a tracking actuator in the radial direction, and by a focusing actuator in out-of-plane direction. In the control of circumference direction, the rotation of a motor is controlled so that pulse intervals of reproduced signal become constant. This controlling is classified as the constant velocity positioning discussed in Chap. 4, although the linear velocity (not the angular velocity) is constant in this case. Radial and axial (out-of-plane) positioning are due to miniature actuators in the head (Fig. 5). Positioning in axial and radial directions are called focusing and tracking controls, respectively. An objective lens is moved so that focus and tracking errors detected by reflected laser light become zero. A control block diagram was shown in Chap. 2. The head needs two actuators for two directions, horizontally, and vertically. The structure in Fig. 5 meets these demands. An optical detecting system consists of a quadruplet photodiode (PD). We can estimate both focus and track error signals by detecting the output signal difference between photodiode pairs formed by combining two elements neighboring horizontally and diagonally. The existence of a pit is detected from the sum of the outputs of all detectors. In the case of tracking control, not only following a specified track but also selecting any track on the disk, which is called seeking, is necessary. For this purpose, the laser beam must be moved over a wide range covering the

Figure 5 Lens actuator of the optical disk. (From Ref. 4.)

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Figure 6 Diagram of two-stage servo system.

entire disk. Therefore the whole optical head is moved by a coarse actuator. The coarse actuator covers a wide range but is too heavy to follow an oscillation with a high frequency. On the other hand, the fine actuator, which moves only the objective lens in the head, has a contrary property. Therefore in the actual tracking, an oscillation with high amplitude and low frequency, like an offset of the disk center, is followed by a coarse actuator. A slight oscillation at a high frequency, like an oscillation of bearings, is followed by a fine actuator. As shown in Figs. 6 and 7, the cooperation of two actuators enables a precise follow-up for an oscillation in a wide frequency range. Since optical disks are exchangeable the disk center can have a large offset by chucking. Since pits are circlular, the track width is small compared with the magnetic disk, which has a wide and short data area. Thus, an optical disk needs higher tracking performance than a magnetic disk. The optical disk solves this problem by two-stage actuators. In the following sections, methods for detecting a position in focusing and tracking controls are explained. 1.2

Position Detection in Focus Servo System

In an optical disk system, a laser beam must be focused on the surface of a rotating disk accompanied with an undesirable vibration with an amplitude of 0.5 mm. The focusing must be performed, suppressing a focus error of less than approximately 1 mm. To satisfy this requirement, an objective lens is activated so that the detected focus error of the reflected light of the read/ write laser beam becomes zero. There are several methods for focus error detection, and its representative is an astigmatic method. Figure 8 shows the principle. The system consists of a quadruplet photodetector (QPD), a cylindrical lens, a focusing lens, an objective lens, and a disk medium.

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Figure 7 Movement of actuator in two-stage servo mechanism. (From Ref. 5.)

Figure 8 Principle of focus error detection using astigmatic method. (From Ref. 4.)

A cylindrical lens works as a convex lens in the curvature direction (in-plane direction of the paper) and does not work in the axial direction of the cylinder perpendicular to the curvature direction (perpendicular direction to the paper). The laser beam from a head is aligned to form a collimated light beam and then focused on the disk surface by the objective lens. The

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reflected light from the disk is focused by the focusing lens. The optical system is set up so that the reflected light is focused at the back of the QPD in the axial direction of the cylinder, and in the curvature direction the light is focused in front of the QPD further by the cylindrical lens. The front and back focal points have the same distance from the QPD, forming a circular spot on the QPD. In this case, detected signals of the four PDs on the QPD become equal. Next, consider the case that the light is focused back of the disk owing to the disk vibration (nearer case). In this case, the reflected light from the cylindrical lens is defocused backward in both axial and curvature directions, resulting in that the focus point in the curvature direction moves nearer to the QPD than that in the axial direction. Then the beam profile on the QPD becomes elliptical with the minor axis in the curvature direction and the major axis in the axial direction. Consequently, we can estimate the position of the focus from the output signals of the QPD; the output of the PDs at up and down position is larger than that of the PDs at left and right position, and the difference between them is positive. The absolute difference becomes larger with increasing focus error. Next consider the case that the light is defocused in front of the disk (farther case). The light from the cylindrical lens is defocused in front of the QPD surface, and the beam shape becomes wide in the curvature direction, making the difference between PD outputs negative. Based on the above discussion, the focus error can be derived from the PD output difference. An optical head has an electromagenetic actuator shown in Fig. 5 or Fig. 17 of Chap. 2, which can control the objective lens up and down. The beam is always focused on the disk by applying a current proportional to the difference of the outputs, to a coil so as to compensate for the focus error. The dynamic model of the positioning system is the same as that described in Chap. 2. 1.3

Position Detection in Continuous Tracking Servo System

In an optical disk system, a laser beam must follow a pit row slightly rolling over by 70 mm in an accuracy of approximately 0.1 mm along a tracking guide. To satisfy this requirement, an objective lens is activated so that the detected position error by the reflected light of the laser beam becomes zero. There are several methods for position error detection, and its representative is a push–pull method. As shown in Fig. 9, the disk surface has a groove with a depth of l/8n (n is the refractive index) at both sides of the data recording area (called land area). When the center of the laser beam is shifted from the land area, the light will be diffracted by either groove, so its spatial intensity profile will become asymmetrical. Thus the tracking error can be detected. The laser light passes through a 1.2 mm thick substrate and

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Figure 9 Principle of tracking error using push–pull method. (From Ref. 3.)

Figure 10

Change of wavefront by tracking groove.

reflects at a groove area and a land area. The optical phase is different between the two refection beams; the phase of the reflected light from the groove area is advanced compared with that from the land area. Therefore the wavefront inclines to the left or right as shown in Fig. 10. Consequently, the wavefront of the reflected light also inclines to either side so that a sensor can detect the error. The tracking error detection system uses common optical components, i.e., a QPD with a focus error detection system. The output difference between the two left photodiodes and the two right photodiodes indicates a tracking error. An optical head has an electromagenetic actuator as shown

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in Fig. 5, which can move the objective lens from left to right. The beam can be always focused on the center of the land area by applying a current to the coil in proportion to the difference of the output. In tracking the optical disk, only the objective lens control mechanism is activated; a laser and a detector are fixed to a mount for reducing the inertia of the moving part. Owing to this individual arrangement, the optical axis of the laser usually disagrees with that of the lens, so the tracking error signal does not become zero even though the laser spot is positioned at the center of the track. This discrepancy is called offset error. We also must take into account the asymmetrical radiation profile of lasers, which may cause another error in the position detection operation.

1.4

Position Detection in Sampled Servo System [6]

Tracking methods in optical disk systems include a sampled servo control method in which the radial position of a head is detected using wobble marks accompanied with data mark rows given at every sector of the disk. Figure 11 shows the principle of track detection with the sampled servo system. The two pits of the wobble mark are shifted right and left by 1/4

Figure 11

Track detection method in sampled servo system.

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Figure 12

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Position detection method in sampled servo system.

track from the center of a track, and about one thousand of them are placed per track. A beam splitter extracts a reflected laser beam from the disk surface, and leads it to a photo detector. The difference of the reflected light from the two wobble pits determines the tracking signal. The relation between the laser beam position and the readout signal waveform is shown in Fig. 12. As mentioned above, wobble pits are slightly shifted from the center line in the radial direction. When the laser beam traces the center track line, the beam equivalently illuminates both wobble pits, and the decreases in the reflected light from the wobble pits are the same. Thus the difference between the readout signals is zero (the center area in Fig. 12). On the other hand, when the trace is displaced from the line to either direction, the beam does not equally illuminate the two pits. This causes a level difference between the reflected lights from the wobble pits. Then the intensity of the reflected light is a different, and the level difference is proportional to the amount of the off-track amount from the center line. This difference is used as an error signal for a feedback system, which controls an objective lens to accomplish the tracking operation. A slight position error much affects the intensity of the reflected light because the laser spot illuminates the edge of the wobble pit under the condition of on-tracking. When data pits are detected, the reflected light is not so greatly changed by a position error. This is because the laser spot traces the center of the pit row arranged along the center line of a track. Although that the wobble pit and the data pit have almost identical shapes, it is possible to detect the position error of the data pit with high sensitivity by use of the wobble pits.

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Table 1

Comparison Between Continuous and Sampled Servo Systems

Item

Continuous servo system

Guide method Track detection

Continuous groove Wobble pit Intensity distribution of Intensity difference of reflected the deflected light light from wobble pits Large Small

Cross talk with data signal Offset Large Optical system alignment Complicate Circuit Small

Sampled servo system

Small Simple Large

In the sampled servo system, the wobble marks are recorded at a leading part of each data block. The track error signal does not interfere with the data and focus error signals. Hence a high-sensitivity readout system can be achieved based on the precise lens control enabled by the sampled servo control. The optical system is also much simplified by this control compared with a continuous servo system (push–pull method). This system will also be a promising tracking method in future near-field optical disk systems. An experiment using a near field optical microscope (SNOM) simulates a future near field optical recording using a flying slider and clarifies that a tracking error can be detected with an accuracy of 10 nm for pit rows 200 nm in diameter. The sampled servo system also has the advantage of a negligible DC offset of the track error signal, which may be easily caused by objective lens movement and disk tilt in the case of the push–pull method. Sampled servo control has the shortcoming over continuous servo control that the controllable bandwidth is limited. This may degrade the tracking performance. This shortcoming can be suppressed by increasing the density of the embedded wobble marks, but this increase simultaneously decreases the recording capacity. Hence it is important to consider the bargaining point of these antithetical ideas according to the requirements of applications. Table 1 shows a comparison between the continuous servo system and the sampled servo system.

2 2.1

FOLLOW-UP POSITIONING OF OPTICAL DISK HEAD MOUNTED ON FLYING HEAD Outline of Equipment

Conventionally, a laser light beam is focused on a disk using an extremely high NA lens to obtain an ultimately minimal spot size. However, a large

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Figure 13

139

Construction of optical head using flying slider.

number of waves is required for this purpose (refer to Chap. 6), the necessary optical system becomes large owing to the many optical components. If a laser diode is arranged in the proximity of the disk surface, the emission from the laser will have a spatial profile as small as that of the emission area on the laser facet. This scheme offers an extremely simple optical system eliminating a conventional bulky optical system. This scheme will be realized if a flying slider used for magnetic disk drives is used. The flying slider maintains a constant spacing based on the hydrodynamics. The spacing is determined automatically by the balance between the hydrodynamic force generated in a thin airspace and the force imposed by a suspension, so it can be controlled by adjusting an optimal load and disk rotation velocity. By attaching a laser diode to such a slider, it becomes possible to make the laser diode set in proximity to a medium [7]. Figure 13 shows a schematic of the head. Since the flying slider has a size of a few millimeters, it is difficult to mount a conventional packaged device consisting of an individual laser diode and a photodiode. Hence a monolithic device that integrates these optical parts has been developed. The device has a taper-ridge laser, which has a spatially limited small illumination area using etched grooves (Fig. 14) [8]. It emits an almost circular beam about 0.8 mm in diameter. Such monolithically integrated devices can be fabricated by conventional LSI-compatible processing technology. The fabricated device is so small that it can be readily mounted to a flying slider [8] (Fig. 15). This scheme is quite different from the conventional optical disk systems; it eliminates passive optical components such as lenses for manipulating the laser beam, and a focus servomechanism is neglected. Consequently, the head is much simplified, and its resulting weight is dramatically decreased. This enables high-speed tracking as achieved in hard-disk systems.

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Figure 14

LD–PD integrated taper ridge laser. (From Ref. 8.)

Figure 15

Construction of slider-mounted optical head. (From Ref. 7.)

2.2

Readout Mechanism

The flying head scheme offers an extremely small head medium spacing of less than a micrometer, as hard-disk systems do. This configuration makes it harder for the head to use a conventional readout optical system to detect data signals from the reflected light without any undesirable scattering of incident illumination. An approach to an efficient readout mechanism uses

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light feedback effects, which are dominant in such a configuration. The effects include direct optical coupling of a semiconductor laser with a recording medium acting as an external mirror without any lens [9]. Let us consider the effect of the direct optical coupling. We assume an optical model suitable for the above configuration that consists of a Fabry– Perot laser diode with an external mirror in the proximity of a laser facet (Fig. 16). The laser facet facing the mirror has an antireflection (AR) coating so that light circulates in the external cavity constructed by the other laser facet and mirror while passing through the AR-coated facet. The external mirror consists of two regions with different reflectivity, corresponding to a practical recording medium with higher and lower reflectivity areas according to the recorded bit marks. The emission performance of this external cavity laser is characterized by mirror reflectivity: the higher reflectivity enables laser oscillation owing to the strong light feedback from the mirror (on-state), and the lower reflectivity keeps the laser off and emitting only weak spontaneously emitted light (off-state) (Figs. 17 and 18). Based on the performance of the external cavity laser, as mentioned above, an interesting effect, laser switching controlled by external mirror reflectivity, can be realized. The mirror reflectivity directly dominates the optical efficiency of the laser resonator, so its variation changes the threshold current of the laser. The external cavity laser provides two different current versus light output characteristics owing to the variety of mirror reflectivity. It is switched between stimulated and spontaneous emission states responding to the regions of the external mirror under the bias condition between the two thresholds (Fig. 19). The external cavity laser performing such switching is called an optically switched laser (OSL). OSL performance can be applied to the readout mechanism of the above flying head. When the head is scanned on the recording medium using

Figure 16

Schematic of laser with light feedback from external mirror.

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Figure 17

Laser emission control by external mirror.

Figure 18

Example of laser on–off experiment caused by optical feedback.

a flying slider to form the external cavity laser, it produces a stimulated or spontaneous emission according to the bit marks on the medium. (Fig. 20). The heads detect data signals by monitoring the light output. The laser oscillation for the on state is extremely stable because of the extremely short head medium spacing, which prohibits any unstable mode hopping. The fluctuation of the spacing is suppressed by the current flying slider technique. By using the large power difference between the on and off states, a high signal-to-noise ratio can be expected in such readout mechanism.

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Figure 19

Relationships among light output, injection current, and reflectivity.

Figure 20

Data signal detection by laser on–off control.

2.3

Passive Positioning by Flying Head

In the flying head, two types of follow-up positioning control are performed: flying space control equivalent to focusing control, and tracking control. At tracking control, error signal detection by beam splitting of reflected light like the push–pull method cannot be carried out. So sampled servo (refer to Sec. 1.4), which acquires a track error signal directly from a reproduced signal, is used. This section explains about flying space control peculiar to a flying head.

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Figure 21

Slider flying above disk. (From Refs. 1, 2.)

A flying slider has a simpler structure and small inertia compared with an active system, so it is suitable for high-speed accessing. The flying mechanism is shown in Fig. 21. At first the slider is pressed against the disk face by the support spring. When a disk rotates, surrounding air also moves and is stuffed into the tapered part of the front of the slider. Then the air pressure rises, a support spring bends, and a slider goes up from the disk surface. The air pressure at the sliding surface of the slider becomes higher as the gap becomes narrower. Thus the air pressure acts as a spring. The slider is located at the balancing point between this air spring and the support spring. The flying height of the slider and the follow-up performance to disk vibration are determined by the balance of air pressure, the rigidity of the support spring, and the inertia force of the slider. The air pressure is calculated by the Reynolds equation, which is obtained by simplifying the Navier–Stokes equation, which is a basic formula of hydrodynamics, on the condition that the air-film thickness is very small and is connected with the continuation equation. The Reynolds equation is   @h @h r ph3 rp ¼ 6U þ 12 @x @t

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ð1Þ

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where p is the air pressure, h the gap height, U the speed of the disk,  the viscosity coefficient of air, x the position in the longitudinal direction of the slider, and t the time. The left side of the equation shows the airflow generated by pressure gradient, the first term of the right side shows the airflow generated by in-plane motion of the disk, and the second term of the right side shows the airflow generated by out-of-plane motion of disk and slider. The whole equation shows that the sum of them is constant (continuity of airflow). Although a more strict equation that considers the molecular characteristics of air is actually used, the outline of the analysis is the same. The flying characteristic is calculated as follows: first, the Reynolds equation and the deformation equation of a support spring are solved for the steady-state condition. Then the average flying height and a pressure distribution are obtained. An example of the numerical calculation result of a pressure distribution is shown in Fig. 22. Next the Reynolds equation is linearized around the average flying height, on condition that the gap and pressure fluctuation from the average value is small. Then the air film becomes equivalent to the springs and dampers in translational and rotational directions. When these springs and dampers are combined with the support spring, the slider system comes down to a three-degreeof-freedom dynamic system. The motion directions are a rolling, a pitching, and a vertical translational direction. If vibrational movement or wavy deflection of a disk is given as an input, gap fluctuation can be

Figure 22

Pressure distribution on slider surface.

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calculated. Although the amount of calculation increases, there is also a method of solving directly the nonlinear equation of Eq. (1) in the time domain. Qualitatively, the follow-up performance can be evaluated from the characteristic of a one-degree-of-freedom system described in Sec. 4 of Chap. 2. The relative amplitude between disk and slider decreases, as the mass becomes smaller, as the air spring and the damper become larger, and as the support spring becomes smaller. Therefore a design that makes the slider as small as possible, the air film rigidity and damping as large as possible, and the rigidity of the support spring as small as possible is necessary for the formation of stable flying height. The flying height of sliders (for magnetic disks) is reducing year by year; it was 0.5 mm around 1980, became 0.2 mm around 1990, and is now reduced to less than 20 nm. 2.4

Fabrication Technology

A micro optical disk can be constructed by locating a laser diode in the proximity of a disk surface using a flying slider so as to form a short external cavity laser. However, practical disk systems need various peripheral components. For example, reflectance of laser facets facing the recording medium must be reduced to less than 0.01%. Such low-reflectivity laser facets are achieved by using a high-precision antireflection (AR) coating technology. This coating can reduce the influence of minute fluctuations of fling height and thereby contribute to high-quality signal detection. Also, in order to suppress temperature increase by the continuous-wave operation of a laser mounted to a slider with no heat sink, a material with high thermal conductivity has been developed for fabricating the flying slider. These component technologies developed for proximity flying type optical disk head are shown in Table 2.

3

DISPLACEMENT SENSORS BASED ON COUPLED CAVITY LASERS

3.1 3.1.1

Principle of Detection and Outline of Follow-Up Control Outline of Sensor

In this section, the usage of follow-up positioning in a displacement sensor is described. The sensor uses a coupled cavity laser (CCL). A CCL is constructed with a Fabry–Perot laser and an external mirror facing a laser

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Follow-Up Positioning in Micro-Optomechatronics Table 2

Component Technologies of Optical Head Mounted on Flying Slider

Subject of investigation Method of solving problem Micro aperture laser diode

High precision bonding

Laser diode cooling

Figure 23

147

Realization of the taper ridge waveguide laser by semicondcutor dry process technology Ion beam etching Development of precision bonder by laser length measuring technology Metalized ceramics Development of high thermal conductive slider (AIN) Development of diamond membrane.

Principle achievement InGaAsP/InP (1.3 mm)

FWHM < 1 mm Positioning accuracy < 1 mm

Pmax ¼ 25 mW (Room temperature, CW)

Configuration of a CCL displacement sensor.

facet to form an external cavity.[10–12] The laser therefore consists of two resonators: the internal laser diode cavity and the external cavity. Displacement sensors having the CCL configuration use the light output characteristics of the laser, which are sensitively changed according to the displacement of the external mirror. Figure 23 schematically shows the structure of such a CCL displacement sensor. The laser diode is monolithically integrated with a photodiode, as shown in Fig. 24. This integrated photonic device is mounted on a block with a TaF3 ball lens

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Figure 24

Chapter 5

Schematic structure of monolithic device for CCL sensor.

500 mm in diameter. Constructing such a compact and small-inertia optical component is necessary for improving dynamic performance for tracking. The core of the CCL sensor is constructed by combining this photonic component with a layered PZT actuator. The edge of the block mounting the laser can be quickly driven along the optical axis of the laser. The laser light emitted from the laser is collimated by the ball lens to illuminate an object (external mirror) to be measured. Adjusting the optical axis of the external mirror to be matched to the laser, the reflection from the mirror returns to the laser cavity. This optical feedback is essential for constructing a CCL with an external mirror. The external mirror is displaced along the optical axis, and the output of the CCL changes according to the displacement. This output change is readily monitored by the monolithically integrated photodiode, which produces a photocurrent while detecting the light power. This photocurrent can be turned into an electrical signal by using a conventional transimpedance transformation circuit. This electrical signal is compared with a reference voltage, and the consequent voltage difference is transiently obtained as an error signal. This error signal is negatively returned to the control element of the PZT actuator. This negative feedback displaces the edge of the CCL sensor block in the direction opposite to the mirror displacement and so compensates the mirror displacement. When the loop gain is optimized, the error signal is maintained at zero. This means that the edge position is controlled according to the mirror displacement, and the error signal corresponding

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to the transient displacement is accumulated using a computing unit to evaluate the mirror position numerically. 3.1.2

Follow-Up Mechanism

The CCL sensor uses an interference undulation for follow-up control. This undulation occurs at every half wavelength, displacing the external mirror as shown in Fig. 25, when the external cavity is around multiples of the effective internal laser cavity length (a product of the physical cavity length and the refractive index) [13,14]. This undulation is characterized by the linear portion in the light output versus the external mirror displacement. In this region, the light output directly indicates the amplitude and the direction of the mirror displacement. The variance of the light output is used for producing an error signal for the tracking control of the block to follow the external mirror. This is because the light output characteristics are different according not only to the individual photonic devices but also to the driving condition of the laser. Now let us explain the tracking control procedure. Consider the state of the laser on the sawtooth undulation curve with linear portions in every period of half the wavelength. The initial state is defined at the halfway point on the curve, where the light output corresponds to P0 used as a reference. A small temporal displacement
h of the external mirror is detected by a photodiode as a differential signal
P ¼ P  P0. This signal is filtered, amplified, and added to the control signal of the actuator to cancel the differential signal through an integrator. Consequently, a negative feedback loop is formed to maintain a constant external cavity length. The external cavity laser controlled in this loop is thereby stabilized. We can know both the transient and the total displacement from the initial position

Figure 25

Interference undulation by the change of external cavity length.

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Figure 26 Block diagram of control system of CCL sensor for maintaining a constant external cavity length.

by evaluating the differential signal and the integral of the signals, respectively. The absolute displacement performance of the employed actuator is readily calibrated and maintained over a long period of time. This promises accurate displacement measurement, independent of the oscillation performance of the employed laser diode and its driving condition. Figure 26 shows a block diagram of the loop. Such sensing mechanism has various merits as follows. Measurement of a small displacement based on the coherence of a single-mode laser light has both high resolution and high sensitivity, so it has been widely performed in various fields. However, conventional measurement schemes need stable, narrow-linewidth light sources consisting of relatively large and expensive optical components, so they are inadequate for general use. Coupled cavity lasers in a simple configuration exhibiting the interference performance are much in use for such displacement measurements. The problem is how to remove the oscillation instability simultaneously induced with displacing the external cavity for the measurement. This paradoxical problem is eliminated by using a coupled cavity laser stabilized with a mechanical negative feedback loop circuit. In the loop the position of the laser diode is controlled along the axial direction by using a high-resolution actuator to cancel the transient displacement of the external mirror while the light output is monitored. 3.1.3

Accuracy

This section focuses on the accuracy for the displacement sensing. A conventional method for estimating the sensitivity of displacement sensors analyzes the spectral response of the sensor to the sinusoidal input. Figure 27 shows a typical response spectrum of the CCL sensor. The external mirror is sinusoidally oscillating. This signal is readily filtered out from the noise by use of a conventional lock-in technique. The peak power corresponds

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Figure 27 Frequency spectrum of a signal detected for a single-frequency oscillation of the external mirror at 200 Hz.

Figure 28 Relationship between the square root of the signal power and the amplitude of the driving signal for mirror oscillation.

to the amplitude of the mirror oscillation: the squared peak power is linearly related to the amplitude. Maximum sensitivity for a particular oscillation frequency is obtained from power equal to the noise floor. This power is usually too small to detect, but it can be estimated numerically by extrapolation of the relationship between root peak power and oscillation pffiffiffiffiffiffiffi amplitude, as shown in Fig. 28. A maximum sensitivity of 0:018 nm= Hz is obtained at the frequency. This sensitivity is equivalent to that of conventional laser interferometers. While such lock-in measurement is useful in particular fields where the cantilever is oscillated, dynamic measurement is necessary for most applications. Hence an effective sensitivity is defined as the amplitude that corresponds to the power equal to the rms noise power. The rms noise amplitude is given by integration of the noise spectrum in the particular frequency range of interest. This effective sensitivity is 0.8 nm in the range of 0–500 Hz for a typical CCL sensor. The CCL sensor produces a sinusoidal

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signal at the same frequency, while following the external mirror. This CCL signal exhibits a sharp peak with broad noise components. 3.1.4

Dynamic Performance

Dynamic performance means the tracking performance of the CLL for the moving external mirror. This performance can be evaluated by measuring the CCL sensor response to a sinusoidal external mirror oscillation with a particular frequency. Figure 29 shows a typical dynamic response of the sensor to a sinusoidal input function. The relation between the input and the corresponding response is characterized by parameters of gain and phase: the gain indicates the amplitude response, and the phase indicates the delay. Such response is measured as a function of frequency, so the dynamic performance of the CCL sensor is clarified. Figure 30 shows measured dynamic characteristics. As the frequency of the external mirror oscillation decreases, the amplitude ratio of the output to the input decreases monotonically while the phase maintains a small variation in the range of zero to a few kilohertz. A transient displacement of the mirror can be measured with certainty in this frequency range. 3.1.5

Stabilization of CCL by a Negative Feedback Loop

The coupled cavity laser (CCL) can be stabilized by using a negative feedback loop. The laser position is controlled along the optical axis to cancel the transient displacement of the external mirror, while the light output variation is monitored. Figure 31 shows a block diagram of the loop. It is assumed that , 0, HA, X, HL are the Laplace transforms of the light output, reference light output, transfer function of the actuator, transient displacement of the external mirror, and transfer function of the laser diode,

Figure 29

Temporal response of CCL sensor to external mirror displacement.

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Figure 30

Dynamic response characteristics of the sensor in the closed loop.

Figure 31

Schematic diagram of displacement sensing.

153

respectively. We find the following relationship according to a conventional loop analysis:   0 ¼

X =HA 1 þ 1=ðHA HL Þ

ð2Þ

This means that the light output difference of the laser and reference will become zero if the gain of the actuator HA is sufficiently increased while the loop oscillation is suppressed. 3.2 3.2.1

Examples of Practical Instruments Stand-Alone Scanning Probe Microscope [15]

Scanning probe microscopes (SPMs), including atomic force microscopes using a cantilever having a sharp tip, are powerful tools for imaging surface

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Figure 32

Chapter 5

Tracking of a cantilever tracing surface profiles.

topography (Fig. 32). Since the surfaces of interest have nanometer-scale structures, measurement of the resulting extremely small distortion of the cantilever is required. Conventional methods using high-sensitivity laser interferometers and deflection angle detectors have been used for detecting such small distortion, but the large-scale setups they require are impediments to becoming widespread. The CCL has shown the potential in such measurements, because they are small and highly sensitive to detecting an extremely small displacement. Figure 33 shows a schematic diagram of an SPM system equipped with a detecting head having a displacement sensor. As the head approaches a sample on a mechanical stage, the tip contacts the sample surface. Further displacement of the head in the same direction generates distortion of the cantilever. Since the cantilever works together with an external mirror of the displacement sensor, the distortion is translated to the photodiode output variance. The absolute distortion of the cantilever, which is derived from the detected signal, corresponds to the load of the tip to the surface. The numerical evaluation of the load is based on the calculation using Fook’s law with values of stiffness of the cantilever. Hence an optimum load is adjustable according to the hardness of the sample. Once an optimum load is determined, the controller of the displacement sensor works so as to cancel a temporal distortion of the cantilever while the external cavity length maintains a constant value by working a tube actuator along the z axis. While the head is raster-scanned in the x–y plane by the tube actuator, we obtain images of the sample surface using the control signal in the z axis direction. A typical example is shown in Fig. 34 for an optical disk surface. The measured image faithfully reflects the surface with shallow tracking grooves 0.1 mm in depth and 1.6 mm in spacing.

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Figure 33 sensor.

Configuration of scanning probe microscope with CCL displacement

Figure 34

Application of a CCL sensor to scanning probe microscope imaging.

3.2.2

Application to Dynamic Detection of Small Forces [16]

Dynamic measurement of small forces is of great importance for mechanical systems. In the field of high-precision information instruments, including

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hard-disk and optical disk systems as required to be miniaturized, the forces of interest become smaller as the system dimension is reduced. For optimum design of the systems, such forces must be temporally measured under operating conditions. Such small forces can be measured from the distortion of a cantilever. Stiff cantilevers with a high resonance frequency are needed for measuring such forces dynamically over a wide frequency range. The distortion is obviously small, so highly sensitive distortion measurement is essential. A small sensor based on lasing characteristics of CCL sensors has been achieved for dynamically detecting small forces. Taking account of the linear relationship between the distortion and the force based on Hook’s law, the force can be derived from the distortion if the stiffness of the cantilever is measured in advance. Figure 35 shows a schematic diagram of such a force sensor: the CCL is formed with a mirror attached to the cantilever, and a laser diode integrated with a photodiode and a small ball lens facing the mirror; it detects a small distortion of the cantilever in the direction normal to the mirror surface. The small force is imposed on the cantilever along this direction. The stiffness of the cantilever is given as 500 mN/mm, and the corresponding resonant oscillation frequency is 3 kHz. The mechanical oscillation is however a shortcoming for such dynamic detection, it is suppressed via mechanical dampers attached to the cantilever. The combination of this cantilever with the CCL sensor has achieved a minimal detectable force of 0.4 mN in the frequency range of DC 2 kHz. To investigate the performance of the above force sensor, a temporal trace of the friction force, which occurs when a small object consisting of stainless steel, with surface roughness Ra ¼ 0.14 mm, is dragged over a rough stainless steel surface with Ra ¼ 0.10 mm, is measured as shown in Fig. 36. The trace conclusively reveals the transition process through the maximum friction force state to the kinetic force state: the friction force shows an

Figure 35 Schematic diagram of a small force detection sensor based on coupled cavity laser performance.

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Figure 36

Friction forces imposed on a small object dragged over a rough surface.

Figure 37

Experimental setup for measuring forces imposed on a flying head.

extreme rise that exponentially decays to the kinetic force level after reaching the maximum. The CCL force sensor is also applied to the investigation of dynamic properties of information instruments including hard-disk systems. In conventional hard-disk systems, a small read–write head mounted on a flying slider is used. During operation, random access is required for on-line use in systems, so the contact start–stop performance of the head is crucial. Hence it is important to monitor the temporal forces imposed on the head. The investigation is also used to clarify the abrasion of the head, which is

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Figure 38 Friction force imposed on head and corresponding relative displacement of a disk in a hard-disk system.

placed very close to the recording medium surface by a spacing of submicrometers. The force measurement is performed with a mechanical system consisting of the head and the disk as shown in Fig. 37. The head is attached to a suspended beam, with which the force sensor is assembled. Since the beam has an extremely high lateral stiffness, the force imposed by a rotating disk can be directly transmitted to the cantilever and measured. Figure 38 shows a typical temporal trace of the force and the corresponding disk displacement. The trace indicates that a static friction force of around 8 mN is imposed on the head in the direction opposite to the rotation under static friction conditions and that the head reaches the steady flying state after some instantaneous contacts with the disk. This is a typical stick–slip phenomenon that occurs at the contact start of flying sliders. REFERENCES 1. 2. 3. 4.

Ono, K.; Tagawa, N.; Nakayama, M.; Ichihara J.; Yoshimura, S. Memories and Printers; Ohm-sha, 1995; in Japanese. Japan Society Of Mechanical Engineers, Ed. Dynamics and Control of Information Equipment; Yohkendou, 1996; in Japanese. Itao, K. Precision Machinery Components. (2) Mechanism of Mechatronics; Corona-Sha, 1987; in Japanese. Itao, K., Ed. Mechatronics for Electrical Communications and Information Processing; Institute of Electronic, Information and Communication Engineers, 1992; in Japanese.

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Follow-Up Positioning in Micro-Optomechatronics 5. 6. 7. 8. 9.

10.

11. 12.

13. 14.

15.

16.

159

Hatamura, Y., Ed. Technology for Information Equipment; University of Tokyo, 1993; in Japanese. Tamaru, N. Study on Positioning Control of Flexible Media for Information Storage, Ph. D. thesis; Waseda University, 1992; in Japanese. Ukita, H.; Katagiri, Y.; Fujimori, S. Supersmall flying optical head for phase change recording media. Appl. Opt. 1989, 28, 4360–4365. Uenishi, Y.; Isomura, Y.; Sawada, R.; Ukita, H.; Toshima, T. Beam converging laser diode by taper ridged waveguide. Electron. Lett. 1988, 24 (10), 623–624. Katagiri, Y.; Ukita, H. Improvement in signal-to-noise ratio of a longitudinally coupled cavity laser by internal faced reflectivity reduction. Japanese J. Appl. Phys. 1989, 28 (Suppl. 28-3), 177–182. Voumard, C.; Salathe, R.; Weber, H. Resonance amplifier model describing diode lasers coupled to short external resonators. Appl. Phys. 1977, 12, 369–378. Lang, R.; Kobayashi, K. External optical feedback effects on semiconductor injection laser properties. IEEE J. Quantum Electron. 1980, QE-16, 347–355. Acket, G.; Lenstra, D.; Boef, A.; Verbeek, B. The influence of feedback intensity on longitudinal mode properties and optical noise in index-guided semiconductor lasers. IEEE J. Quantum Electron. 1984, QE-20, 1163–1169. Katagiri, Y.; Hara, S. Increased spatial frequency in interferential undulations of coupled-cavity lasers. Appl. Opt. 1994, 33, 5564–5570. Olesen, H.; Henrik, J.; Tromborg, B. Nonlinear dynamics and spectral behavior for an external cavity laser. IEEE J. Quantum Electron. 1986, QE-22, 762–773. Katagiri, Y.; Hara, S. Scanning-probe microscope using an ultra-small coupled-cavity laser distortion sensor based on mechanical negative-feedback stabilization. Meas. Sci. Technol. 1998, 9, 1441–1445. Katagiri, Y.; Itao, K. Dynamic microforce measurement by distortion detection with a coupled-cavity laser displacement sensor stabilized in a mechanical negative-feedback loop. Appl. Optics 1998, 37, 7193–7199.

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6 Fundamental Optics of Micro-Optomechatronics This chapter presents the fundamentals of optics for technologies in microoptomechatronics. For a long time, many scientists argued about whether light is a wave or particle. This argument simply arises from the different phenomenological aspects of light. Light acts as particles and as waves. Light is an electromagnetic wave and is classified according to its wavelength. Techniques characteristic of optical micromechatronics are carried out in different ways based on these aspects for a wide variety of applications. Such versatile aspects of light are used in micro-optomechatronics as follows. Light as a wave offers characteristic behaviors typically including diffraction or interference characterized by phenomena of different colored lights of the rainbow making up the spectrum of reflection of white light. Such light waves are classified according to wavelength. In the order of shorter to longer wavelengths, we have gamma-rays, x-rays, ultraviolet, visible and infrared light, millimeter waves, microwaves, and finally radio waves (see Fig. 1). The wavelengths of most interest are usually in the range of 0.4–2 mm, so a small displacement has a strong influence on the behavior of light at these wavelengths. The positioning techniques in microoptomechatronics are based on this feature. These techniques are used for many practical applications such as tunable lasers, filters (Chaps. 3, 4), and high-precision information instruments including optical disk systems (Chap. 5). Light as particles interacts with objects based on mechanical dynamics that offer momentum or energy transfer. The momentum transfer by light is usually too small to detect, but it can have an influence on small particles. Typical applications include optical tweezers that manipulate small particles such as biological cells (Chap. 2). The energy transfer of light is characterized by absorption and emission of light. These optical processes are truly clarified by introducing the concept of photons, although they can 161

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Chapter 6

Figure 1 Various aspects of electromagnetic waves.

be phenomenologically explained by using wave equations. We have a wide variety of applications based on the energy transfer of light, typically including optical recording systems and photothermal oscillators. The optical processes also provide chemical effects on matter. A typical example is the generation of photoelectrons from metal surfaces. The essence of this phenomenon is applied to detecting photons by using photodiodes. As these optical processes are characterized by chemical potential, it is necessary to optimize the wavelength of light corresponding to the photon energy. Considerations on properties of light described in this chapter is based on the Maxwell equations. Section 1 presents fundamentals of the equations and various optical properties using equations of light wave propagation, polarization, interference, diffraction, localization including evanescent fields, and energy transfer. Section 2 describes optical resonators and their applications, typically including semiconductor lasers, based on wave theory. Section 3 particularly focuses on the characterization of multiple thin optical films based on the Maxwell equations, which are fundamentals of wavelength-tunable optical filters. Finally, Sec. 4 introduces novel electromagnetic waves existing at metal–dielectric interfaces, whose theoretical explanations are also based on the Maxwell equations. Such novel waves will be promising for realizing nano-optomechatronics.

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163

FUNDAMENTAL OPTICS [1] Maxwell Equations

In this section we describe the principles of light based on the Maxwell equations. We first derive a wave equation from these equations and present some typical solutions including a Gaussian wave, which is of great use for various optical systems. We also characterize light waves to understand physical phenomena such as energy transfer, signal transmission, and refraction. The Maxwell equations have been empirically given by characteristic electromagnetic phenomena independently related to electric or magnetic fields. We have four fundamental equations to describe all of the classical phenomena: div E ¼ 

ð1Þ

div H ¼ 0

ð2Þ

rot H ¼ "

@ E @t

rot E ¼ 

@ H @t

ð3Þ ð4Þ

These equations were empirically obtained, and it is easy to relate each equation to a corresponding physical picture as shown in Fig. 2. Equation (1) is related to the electric field generated by an electrical charge. The electric field is quantitatively evaluated using the concept of flux. A charge Q produces an electrical flux having a number of Q/". As shown in Fig. 2a, the electrical flux is not created or annihilated in a space with no electrical charge, hence in a closed space the influx is balanced with the efflux. A similar quantitative evaluation can be performed for the magnetic field by defining a magnetic flux as shown in Fig. 2b. However, we must note that there is no evidence for the existence of monopoles. So this gives Eq. (2). Equation 3 shows that a magnetic field is generated around a charge flow (current), as shown in Fig. 2c. The differential term is a nominal current in a vacuum coming from electromagnetic induction. Equation (4) shows that a loop current generates a magnetic field, as shown in Fig. 2d. This is equivalent to another picture in which an electric field is generated along a loop path by changing the magnetic flux running through the loop. To derive wave equations from these Maxwell equations, we introduce the vector potential A, defined as rot A  B ¼ H

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ð5Þ

164

Chapter 6

Figure 2 Schematic illustration for explaining Maxwell’s equations.

Substituting this in Eq. (4) we obtain

@ rot E þ A ¼ 0 @t

ð6Þ

This differential equation has the general solution Eþ

@ A ¼  grad  @t

ð7Þ

Selecting Coulomb gauge div A ¼ 0, we can determine a unique solution E¼

@ A @t

Substituting Eqs. (5) and (8) into Eq. (3), we obtain



1 @ @ rot rot A ¼ " A  @t @t

ð8Þ

ð9Þ

Taking account of the relation rot(rot A) ¼ grad(div A) 
A, we can simplify the above equation to
A  "

@2 A¼0 dt2

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ð10Þ

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which exhibits the form of a wave equation. Assuming a form A ¼ A(r) exp(i!t), the wave equation is represented by
AðrÞ þ "!2 AðrÞ ¼ 0 We find a general solution o Xn Aðr, tÞ ¼ Ak ðrÞ expðikrÞ þ A k ðrÞ expðikrÞ expði!tÞ

ð11Þ

ð12Þ

k

where (¼1 or 2) represents polarization states of light, and k is a wave number vector satisfying jkj2 ¼ k2 ¼ "!2. Considering the definition k  2/ , we find the light speed c from the relation kc ¼ ! as 1 c ¼ pffiffiffiffiffiffi "

ð13Þ

Using the Coulomb gauge featured by the condition grad  ¼ 0, we find divAðrÞ ¼ iAðrÞ k ¼ 0

ð14Þ

This means that the wave vector is normal to the vector potential (see Fig. 3). This indicates that the electromagnetic waves are transverse waves and offer polarization according to the vector potential. Considering general solutions of the wave vector, the electric and magnetic field components of the electromagnetic radiation can be written as X E¼ Ek ðrÞ expðikr  i!tÞ ð15Þ k

H¼

X

Hk ðrÞ expðikr  i!tÞ

ð16Þ

k

Figure 3 Relation of vector potential, wave vector, and electric magnetic fields.

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1.2 1.2.1

Chapter 6

Traveling Waves as Solutions of Wave Equations Plane Waves

Assuming that Ek(r) and Hk(r) remain constant independently of an arbitrary position given by r, we obtain a most simple solution of the wave equation called the plane wave, written as Ek ðrtÞ ¼ Eo expðikr  i!tÞ

ð17Þ

Hk ðrtÞ ¼ Ho expðikr  i!tÞ

ð18Þ

Consider a simple case which gives an electric field, assuming that the light wave propagates along the z axis: Ek ðrtÞ ¼ Ex expðikz  i!tÞex

ð19Þ

Substituting this equation and Eq. (18) into Eq. (4), we obtain rotðEx expðikz z  i!tÞex Þ ¼ i!Hk ðrtÞ

ð20Þ

which is readily simplified to Hk ðrtÞ ¼

k Ex expðikz  i!tÞey !

ð21Þ

We readily evaluate the amplitude ratio between the electric and the magnetic field components as   rffiffiffiffi Ek ðrtÞ  ¼ Z   ð22Þ " Hk ðrtÞ which is called the wave impedance. An aspect of such plane waves is schematically shown in Fig. 4.

Figure 4 Schematic of plane wave.

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167

Spherical Waves

Point sources play an important role in characterizing the propagation of light in free space. Radiation patterns can be calculated in the far field for arbitrary light sources using the concept of point sources, because they are regarded as an ensemble of point sources. Such point sources are described by spherical waves represented by 9 1 > > exp½ikðr  r ðr, r Þ ¼ Þ  i!t Eþ ð23Þ 0 0 = k jr  r0 j 1 > ð24Þ ; exp½ikðr  r0 Þ  i!t > E k ðr, r0 Þ ¼ jr  r0 j where þ means divergence from a point r0 and  means convergence to the point. The spatial aspect of spherical waves is shown in Fig. 5. 1.2.3

Gaussian Waves

This section describes Gaussian waves often used for describing light wave propagation for micro-optical devices. We introduce a scalar wave notation. We assume a wave propagating along the z-axis with a field component represented as ðrtÞ ¼ ðrÞ expðikz  i!tÞ

ð25Þ

Substituting this equation into the wave equation similar to Eq. (11), we obtain the wave equation
ðrtÞ þ k2 ðrtÞ ¼ 0

Figure 5 Schematic of spherical wave.

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ð26Þ

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Chapter 6

For a light wave localized near the z-axis, the second-order differential coefficient @2/@Z2 is negligible, so we find   @ 2 ð27Þ
ðrtÞ ¼ expðikz  i!tÞ
2 ðrÞ þ 2ik ðrÞ  k ðrÞ @z where
2 is the two-dimensional Laplacian (@2/@x2, @2/@y2). Therefore we obtain the wave equation
2 ðrÞ þ 2ik

@  ð rÞ ¼ 0 @z

ð28Þ

We can obtain a solution of this differential equation by assuming a solution form    k  2 x þ y2 ðrÞ ¼ exp i PðzÞ þ ð29Þ 2QðzÞ Substituting this equation into Eq. (28), we find  2 

 k @Q k2  2 i @P 2   þ y x  ð r Þ þ 2k  ð rÞ ¼ 0 Q @z Q2 @z Q2

ð30Þ

We must accept the conditions @Q/@Z ¼ 1 and @P/@Z ¼ i/Q to yield the above equation independently of x and y. We readily obtain QðzÞ ¼ z þ Qo

ð31Þ s ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

iz z2 z PðzÞ ¼ i log 1 þ ð32Þ ¼ i log 1  2 þ i tan1 Qo Qo Qo pffiffiffiffiffiffiffiffiffiffiffiffiffi where we use the relation logð1 þ ixÞ ¼ log 1 þ x2 þ i tan1 ðxÞ. We here define parameters R(z) and W(z) by 1 1

 þi QðzÞ RðzÞ WðzÞ2

ð33Þ

Considering R(1) ¼ 1 and W(0)¼ W0, we obtain Q0 ¼ iW2o = . Hence the above equation becomes 1 1

þi ¼ z þ iW2o = RðzÞ WðzÞ2 Noting the real and imaginary part of this equation, we obtain (

2 ) W2o RðzÞ ¼ z 1 þ

z

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ð34Þ

ð35Þ

Fundamental Optics of Micro-Optomechatronics

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffi

z 2 Wð zÞ ¼ Wo 1 þ W2o

169

ð36Þ

Hence we obtain

      2

 exp  x2 þ y2 =ðWðzÞ2 Þ x þ y2 k

z 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  tan ðrÞ ¼ exp i  2 W2o 2RðzÞ 1 þ z=ðW2o Þ ð37Þ This equation indicates that W(z) and R(z) correspond to the beam diameter and the wavefront of the Gaussian wave, respectively. 1.3

Polarization

It is obvious that light is a transverse wave, taking account of the solutions of wave equations represented by electromagnetic field components. A most simple plane wave maintains the direction of these field components while propagating. So the state of light is defined as linear polarization. Light emitted from lasers usually shows this linear polarization. Spontaneous emission such as Gaussian light from lamps, on the other hand, exhibits random polarization, because they are superposition of many linearly polarized light waves. Quantitative characterization on polarization is performed with a representation of a light wave propagating along the z-axis: E ¼ E0 expðikz  i!tÞ

ð38Þ

For convenience, we only take the real part of the field component as Ex ¼ ax cosðkz  !t þ ’x Þ   Ey ¼ ay sin kz  !t þ ’y

ð39Þ ð40Þ

where ’x and ’y are the polarization parameters. The point of the electric-field vector (Ex, Ey) shows a variety of traces on the xy-axis according to the polarization parameters as shown in Fig. 6. The linear polarization is given by ’x ¼ ’y ¼ ’

ð41Þ

However, a slight discrepancy between the parameters gives an elliptical trace. Assuming the discrepancy to be ’x  ’y ¼ 

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ð42Þ

170

Chapter 6

Figure 6 Schematic illustration of polarized state. (a) Linear polarization. (b) Circular polarization.

the point of the field vector is represented in a form that gives a elliptical trace: 2 2 Ey Ex Ex Ey 2 cos  þ ¼ sin2  ð43Þ ax ax ay ay For a particular case with polarization parameters, ax ¼ ay ¼ a ¼

 2

ð44Þ ð45Þ

the trace is given by ðEx Þ2 þ ðEy Þ2 ¼ a2

ð46Þ

This means that the point traces a circle. (The signature of delta indicates the direction of the trace: the plus corresponds to a clockwise rotation of the field vector, and the minus sign corresponds to a counterclockwise rotation.) Such circularly polarized light waves can be used to represent a wave in an arbitrary polarized state. The linearly polarized wave is, for example, expressed as a superposition of the circularly polarized waves as ! ! ! Ex a cosðkz  !t þ ’Þ a cosðkz  !t þ ’Þ ¼ þ Ey a sinðkz  !t þ ’ þ =2Þ a sinðkz  !t þ ’  =2Þ ! 2a cosðkz  !t þ ’Þ ¼ ð47Þ 0

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1.4

171

Interference

Periodic phenomena obtained by superposition of light waves include interference. A simple consideration is presented using two waves with the same linear polarization: E1 ¼ a expði!1 tÞ

ð48Þ

E2 ¼ a expði!2 t  iÞ

ð49Þ

The superposed light wave has the intensity

  I  jE1 þ E2 j2 ¼ 2jaj2 1 þ cos ð!2  !1 Þt þ 

ð50Þ

For !1 6¼ !2, the superposed wave produces a beat wave with a frequency equal to the optical frequency difference (see Fig. 7). The beat can be directly detected by using a photodiode. Assuming a negligible phase noise for these light waves, the phase difference  can be estimated from the beat signal. The phase term of the beat signal is differentiated in the time domain as @ @ ¼ !1  !2 þ @t @t

ð51Þ

This means that the phase difference corresponding to the transient change of the traveling path of light can be observed in the frequency domain. When an object linearly moves at a constant speed, for example, the differential term of @/@t becomes constant. This component is not observed in the frequency spectrum of the beat signal. On the condition that the optical path sinusoidally changes, the beat signal is phase modulated. This modulation produces a series of modulation sidebands around the carrier

Figure 7 Beat generation with two light waves.

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frequency. The change in the optical path can be numerically estimated from these sidebands. In the particular case of no frequency difference between these two light waves, the intensity of the superposition is independent of the time as I  jE1 þ E2 j2 ¼ 2jaj2 ½1 þ cos 

ð52Þ

Hence the accurate measurement of the intensity can precisely determine the optical path related to the . 1.5

Diffraction Theory and Spatial Control of Light Waves

We have fouud in the previous sections that light waves derived from the Maxwell equations are transverse waves. The propagation characteristics of such light waves, however, can be quantitatively evaluated by using the scalar wave picture, which uses a common field component replacing the electric or magnetic field component. 1.5.1

Basic Theory

According to the Huygens theorem, an arbitrary light wave can be represented as a linear combination of fundamental functions such as plane waves. This concept is extended to the diffraction theory. This section explains a theoretical method for describing propagating light waves based on this theory. Solutions of the wave equation can be obtained for particular boundary conditions by using appropriate Green’s functions. To discuss light wave propagation in free space, we introduce a Green function for diverging spherical light waves as Gðr, r0 Þ ¼ 



 1 exp ikr  r0  4jr  r0 j

ð53Þ

where r and r0 are arbitrary points in a three-dimensional space (see Fig. 8). Using this function we readily solve the wave equation. Assuming a general solution given by ðrtÞ ¼ XðrÞ exp½i!t

ð54Þ

the wave equation is rewritten in a form only with spatial terms as
XðrÞ þ k2 XðrÞ ¼ 0

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ð55Þ

Fundamental Optics of Micro-Optomechatronics

Figure 8

173

Schematic diagram for solving wave equation by using Green’s function.

where k is the wave number. The Green function is a solution of the wave equation and thus satisfies
Gðr, r0 Þ þ k2 Gðr, r0 Þ ¼ ðr  r0 Þ

ð56Þ

where (r) is the three-dimensional Dirac delta function defined as ðrÞ  ðxÞðyÞðzÞ Using Eqs. (55) and (56), we find Z  0  XðrÞ ¼ Xðr Þ
Gðr, r0 Þ  Gðr, r0 Þ
Xðr0 Þ dr0

ð57Þ

ð58Þ

V

R where V is a closed area and the formula A(r0 )(r  r0 ) dr0 ¼ A(r) is used. According to the Gauss theorem, the above integration is replaced by the integration on the surface of the closed area,  Z  @ @ Xðr0 Þ Gðr, r0 Þ  Gðr, r0 Þ Xðr0 Þ dr0 ð59Þ XðrÞ ¼ @n @n s where @/@n indicates differentiation along the direction normal to the surface. Substituting Eq. (53) into this equation, we find Z 

 ik 1 r  r0  Xðr0 Þ dr0 XðrÞ ¼  exp ik ðcos  þ cos Þ ð60Þ j r  r0 j 4 s where  is the angle between r  r0 and the z-axis and is also the angle between the wave vector and the z-axis (see Fig. 9). This formula means that we can evaluate the field X(r) at an arbitrary point located in the closed area if we completely know the field on the surface of the closed area. This equation corresponds to Kirchhoff‘s diffraction formula. Since  ¼ ¼ 0 can

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Figure 9 Schematic illustration for explaining inclination factor.

be assumed when light waves propagate around the z-axis, the above formula is simplified to Z 

 ik 1 exp ikr  r0  Xðr0 Þ dr0 ð61Þ XðrÞ ¼  2 s jr  r0 j We also simplify this formula in Fresnel regions where plane-wave expansions are available. This simplification offers a diffraction formula,

Z k expðikLÞ x2 þ y2 exp ik Xðx0 , y0 Þ Xðx, yÞ ¼ i 2L 2L s !

x 0 2 þ y0 2 xx0 þ yy0 ð62Þ
exp ik exp ik dx0 dy0 2L L where (x, y) is a point on the plane z ¼ L, and X(x0 , y0 ) is the field component at on (x0 , y0 ) the plane z ¼ 0. 1.5.2

Focusing of a Light Beam by a Lens

We can characterize the propagation of light waves in three-dimensional space from diffraction theory. We present a typical application of this theory, which is related to beam convergence by lenses. We demonstrate that a minimal size of a spot produced by convergence is limited by the diffraction of light waves. Assume that the optical axis of a lens with a focal length f to be considered agrees with the z-axis (see Fig. 10). We introduce a phase shift function of lenses expressed by

x2 þ y2 hðx, y, dÞ  exp ik ð63Þ 2d

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Figure 10

175

Converging light waves via lens.

where d denotes the propagation distance in the direction of the z-axis. This lens offers a field taking account of the aperture of the lens as o ðx, yÞ ¼ Ahðx, y,  fÞ ¼0

ra

r a

Hence we know the field at the focal plane of the lens z ¼ L ¼ f as

Z expðikfÞ x2 þ y2 ðx, yÞ ¼ i exp ik o ðx0 , y0 Þ

f 2f s !

x 0 2 þ y0 2 xx0 þ yy0
exp ik exp ik dx0 dy0 2f f

expðikfÞ x2 þ y2 exp ik ¼ i

f 2f

Z xx0 þ yy0
A exp ik dx0 dy0 f s

ð64Þ

ð65Þ

Since this integration is readily performed with the cylindrical coordinates using x0 ¼ r cos , y0 ¼ r sin , x ¼ R cos , and y ¼ R sin , we find Z a Z 2 i expðð2i= ÞfÞ A exp½ið2i= fÞRr cosð  Þr dr d XðRÞ ¼ 

f 0 0

i expðð2i= ÞfÞ R2 2a2 2 A exp i Ra ¼ J 1 ð Þ ¼

f

f

f ð66Þ

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Figure 11

Intensity distribution of beam spot formed by lens.

where Jn(z) is the Bessel function. Hence the intensity profile at the focal plane becomes



2 2   J 1 ð Þ 2 XðRÞ2 ¼ A2 2a ð67Þ

f This profile, as illustrated in Fig. 11, exhibits fringes. These fringes provide a minimal disk having a diameter that can be determined from the Bessel first zero point as a¼

0:61 f R

ð68Þ

This gives a minimal spot size under the diffraction limit. R is the minimal spot size and is related to the wavelength and numerical aperture f/a. 1.5.3

Solid Immersion Lens [2]

Light shrinks in a medium with a higher refractive index: the wavelength is inversely proportional to the index. This indicates that light can be confined in a small space in such a medium. This effect has been applied to an optical microscope with a high spatial resolution in which the space formed between the objective lens and the samples is filled with oil with a higher index than that of air. This immersion-lens technique has been used for applications even using a solid medium with a higher index. The technique offers a solid immersion lens (SIL). Figure 12 schematically explains the principle of this technique. Light is focused on an interface between the air and the medium with a higher index than that of air. Light is reflected at the interface owing

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Figure 12

177

Light wave behavior at dielectric surface.

Figure 13 Principle of super SIL. (a) Schematic structure. (b) Effect of super SIL on light beam propagation.

to the refractive index difference of the medium. A slight portion of the reflection leaks out into the air. This portion has a shorter wavelength as the light shrinks in the higher-index medium. This leakage does not produce any radiative modes in air, but high-resolution performance based on the effectively reduced wavelength can be used in the space near the interface. The performance of such an SIL is determined by the refractive index of the medium. Since usual optical media have an index in the 2–3 range, the SIL is not so effective as expected. Of course media with higher indices such as semiconductors can be used, but their transparency condition limits the available wavelength range in the infrared region. This shortcoming is, however, eliminated by improving the structure of the SIL. Figure 13a shows a schematic illustration for enhancing the

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wavelength-reducing effect. This effect can be theoretically explained via a simple geometrical consideration. As shown in Fig. 13b, a sphere R in radius is placed at the origin O of the orthogonal coordinate axes. A light beam is assumed to converge to the point on the x-axis apart from the point O by nR (n: refractive index of the sphere). A component of the beam with an angle  to the x-axis has an incident angle of  on the sphere surface. This simply provides a relation tan  ¼

R sinð þ Þ nR þ R cosð þ Þ

ð69Þ

Then we obtain sin  ¼ n sin 

ð70Þ

According to Snell’s law, this equation means that the refraction angle is  corresponding to the incident angle . Hence the light beam intersects the x-axis at the point apart from O by X. Then we obtain   X þ R cosð þ Þ tan   ¼ ð71Þ 2 R sinð þ Þ This equation gives a simple relation using Eq. (70): nX ¼ R

ð72Þ

This means that a focus is formed at X ¼ R/n on the x-axis in the sphere. The nominal numerical aperture is improved by a factor of n. Hence a minimal spot size determined according to the Airy disk is reduced with a squared refractive index to 2R ¼ 1:22

NA n2

ð73Þ

In case of NA ¼ 0.5, n ¼ 2, ¼ 0.7 mm, the spot size is given by 2R ¼ 0.43 mm. The SIL can provide a minimal spot size much smaller than that obtained in the air, and so is promisingly applied to next-generation optical disk systems. 1.6

Evanescent Field

Consider the light beam behavior at the interface between media that have refractive indices of n1(¼ n > 1) [medium 1] and n2(¼1) [medium 2]. It is assumed that a light beam with an incident angle  to the z-axis normal to the interface produces a corresponding refracted beam with an angle 

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Figure 14

179

Generation of evanescent field by total reflection.

(see Fig. 14). According to Snell’s law, the relation between these light beams is given by n sin  ¼ sin 

ð74Þ

Taking account of jsin j  1, a critical angle is defined as 1 sin c  ð<1Þ n

ð75Þ

When the condition   c is satisfied, the light beam is refracted at the interface according to Eq. (74). However, in case of  c, the refraction of light in the direction of medium1 ! medium2 is mathematically prohibited. This situation is explained by a real physical picture introducing the concept of total reflection. Let us consider the latter case further. Snell’s law can be satisfied if a complex notation is allowed for an angle. So it becomes important to consider the physical meaning of the complex angle. The wavenumber vector of the incident beam is assumed as 0 1 0 1 nk0 sin  kx @ ky A ¼ @ A ð76Þ 0 kz kz with  2 k0 2 ¼ jkx j2 þ ky  þ jkz j2

ð77Þ

This equation is combined with Eq. (73) to represent a relation ðkz Þ2 ¼ k0 2  ðk0 n sin Þ2

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ð78Þ

180

Chapter 6

which gives qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kz ¼ ik0 ðn sin Þ2  1

ð79Þ

The minus sign has no physical meaning, but the plus sign gives real existence that shows an exponentially damping wave from the interface along the z-axis as  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EðzÞ ¼ E exp½izkz  ¼ E exp zk0 ðn sin Þ2  1 ð80Þ This damping field localized near the interface is well known as the evanescent field. The reflection coefficient along the z-axis is nominally represented by r¼

sinð  Þ   expðiÞ sinð þ Þ

ð81Þ

with

tan  ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn sin Þ2  1 n cos 

ð82Þ

Note that a real energy flow exists along the x-axis in this evanescent field. This energy flow is essential for practical applications. 1.7 1.7.1

Energy Flow and Signal Transmission Evaluation of Energy Flow Rate

This section explains how the light wave transports energy. We define a Poynting vector, which evaluates an energy flow rate per unit cross section: S¼E
H

ð83Þ

The absolute value is related to the energy density of the electromagnetic field as jSj ¼ c wðrtÞ

ð84Þ

where the energy density w(rt) is given by 2 1  2 1  wðrtÞ ¼ "EðrtÞ þ HðrtÞ 2 2

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ð85Þ

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181

We evaluate the energy flow of a plane wave for example. Assuming that the wave propagating along the z-axis has field components E ¼ (Ex, 0, 0) and H ¼ (0, Hy, 0), the Poynting vector becomes S ¼ ð0, 0, Ex Hy Þ

ð86Þ

Noting the relation between the field components according to Maxwell’s pffiffiffiffiffiffiffiffi equations Hy ¼ "=Ex , we obtain the energy flow rffiffiffiffi " 2 1 jSj ¼ E ¼ " pffiffiffiffiffiffiffi E2x ¼ " cE2x ð87Þ  x "$ RT Using an averaging factor  ¼ ð1=TÞ 0 cos2 ð!tÞ dt ¼ 1=2, the average energy flow is evaluated as D  2 2 E 1 hjSji ¼ " c EðrtÞ ¼ " cEðrÞ 2

ð88Þ

This average value is useful for describing experiments. 1.7.2

Group Velocity and Signal Transmission

The energy flow of light waves is equivalent to the transmission of wave packets, which are often called optical signals (see Fig. 15). The velocity of this energy transmission is, however, not always equal to the speed of light but corresponds to the phase velocity, which is derived from the well-known relation 2c ¼ !. The velocity of the wave packets is some what different from the phase velocity and is represented by g

Figure 15

d! dk

Propagation of wave packet in refractive medium.

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ð89Þ

182

Chapter 6

This velocity is the group velocity. We rewrite the form of the group velocity using and obtain g¼cþk

dc dc ¼c dk d

ð90Þ

where c is the speed of light taking into account the refractive index of the medium. This group velocity agrees with the speed of light in a medium offering a constant light speed independently of wavelength, but it is changed when the medium has a wavelength dispersion that offers a wavelength dependence of a refractive index to change the light speed. Noting dc/d > 0 for normal dispersion, we have a group velocity always smaller than the phase velocity.

2 2.1

OPTICAL RESONATORS AND THEIR APPLICATIONS Principle and Variations of Resonators

Resonance is a universal wave phenomenon: a sinusoidal input produces an output with an amplitude diverging to infinity. It is observed in a wide variety of waves including mechanical, electrical, acoustic, and optical oscillations. Resonance is characterized by the amplitude of the output wave, which is larger than that of the input wave. A fundamental principle of resonance is based on harmonic superposition of waves: as shown in Fig. 16, the superposition of two sinusoidal waves with the same spatial frequency increases the wave amplitude under the same phase condition. Optical resonators enhance this amplitude increase by using multiple reflections or circulations in optical cavities. Figure 17 schematically illustrates various optical resonators including a

Figure 16

Superposition of light waves.

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Figure 17 resonator.

183

Variations of optical resonators. (a) Fabry–Perot resonator. (b) Ring

Fabry–Perot resonator with two facing mirrors (a) and a ring resonator (b). The Fabry–Perot resonator is characterized by a standing wave formed in the cavity as  ¼ A exp½iðk  !tÞ þ A exp½iðk  !tÞ ¼ 2A expði!tÞ cosðkÞ

ð91Þ

where  is a parameter of position. This standing wave obviously provides a periodic intensity distribution in the resonator along the optical axis. On the other hand, the ring resonator provides an enhanced wave with negligible time-delay terms:  ¼ MA exp½iðk  !tÞ

ð92Þ

where M is an integer to indicate the enhancement effect. There exist two possible circulating waves in the ring resonator, but an optical isolator can select one of the waves. This unidirectional traveling wave provides a uniform intensity distribution. 2.2

Resonant Wavelength

The harmonic superposition in a resonator to enhance the wave amplitude is enabled at particular wavelengths determined according to the cavity length (see Fig. 18). The amplitude falls as the wavelength deviates from these resonant wavelengths. The superposed waves with such resonant wavelengths are therefore regarded as the eigenmodes of the resonator.

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Figure 18

Wavelength selection by optical resonator.

Let us consider the transmission of light waves through a resonator as a function of wavelength. The transmittance reaches a maximum at a very resonant wavelength. The rest of the transmission is coupled to the backward waves as the reflection, and so goes back to the source. We can use this transmission behavior for wavelength selection. The optical resonators are also used with gain media for laser oscillation. The enhancement effect of the resonators is efficiently used for stimulated emission, and consequently laser oscillation occurs at a resonant wavelength. We have practically obtained laser light sources since 1960 when the laser oscillation was demonstrated using a ruby crystal in a Fabry– Perot resonator for the first time. This innovation has stimulated many people to create various developments in lasers using various media. These works include semiconductor lasers, which have been much developed since laser oscillation was achieved at room temperature by doubleheterojunction structures. These semiconductor lasers are very small and operate under low power so thus show the great potential for a wide variety of applications in various fields including information, measurement, communications, manufacturing, and medical appliances. Now let us consider the wavelength-selection function more quantitatively. It is assumed that a Fabry–Perot resonator L in length is constructed by two mirrors with reflectance R in a homogeneous medium with a refractive index n, as shown in Fig. 19. When the incident, transmitted, and reflected light waves have amplitudes E0, Et, and Er, respectively, we easily obtain the field components of the transmitted and reflected waves:   1R E0 Et ¼ E0 ð1  RÞ 1 þ Rei þ R2 e2i þ ¼ 1  Rei

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ð93Þ

Fundamental Optics of Micro-Optomechatronics

Figure 19

185

Light wave behavior concerned with Fabry–Perot resonator.

npffiffiffiffi pffiffiffiffi  o Er ¼ E0 R þ ð1  RÞ Rei 1 þ Rei þ R2 e2i þ   ð1  RÞei pffiffiffiffi ¼ 1þ RE0 1  Rei

ð94Þ

with ¼

4nL

ð95Þ

The power transmission and reflection coefficients are therefore estimated respectively as T

jEt j2 ð1  RÞ2 ¼ jE0 j2 ð1  RÞ2 þ 4R sin2 ð=2Þ

ð96Þ

R

jEr j2 4R sin2 ð=2Þ ¼ jE0 j2 ð1  RÞ2 þ 4R sin2 ð=2Þ

ð97Þ

We readily find T þ R ¼ 1, which confirms the energy conservation law. Figure 20 shows calculated transmittance as a function of wavelength. The peaks at every constant frequency spacing correspond to the resonant optical frequencies of the resonator. The phase-matching condition, where the phase shift produced at every round trip is a multiple of 2, is given by 2nL ¼ m ¼ m

c


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ð98Þ

186

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Figure 20

Eigenmodes of Fabry–Perot resonator.

where
is the optical frequency. Hence each resonant mode is specified for the integer m (mode number) as c m 2nL


m ¼

ð99Þ

The mode spacing, the free spectral range (FSR), is obtained using this equation as


mþ1 
m ¼

c 2nL

ð100Þ

A transmission bandwith is numerically evaluated by using a full width at half maximum (FWHM) f!. We have a relation using this equation,
 ¼

4nL

c

ð101Þ

Consider the condition ð1  RÞ2 1 ¼ 2 2 ð1  RÞ þ 4R sin ð! =2Þ 2

ð102Þ

This relation gives an equation 1R ! ! pffiffiffiffi ¼ sin ffi 2 2 2 R

ð103Þ

It is obvious that the parameter ! gives the FWHM as ! ¼

2nL f! c

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ð104Þ

Fundamental Optics of Micro-Optomechatronics

187

Figure 21 Wave length selectivity of Fabry–Perot resonator. (a) Spectral responses for various mirror reflectivities. (b) Transmission bandwidth versus mirror reflectivity.

Hence we finally obtain f! ¼

ð1  RÞc ð1  RÞFSR pffiffiffiffi ¼ pffiffiffiffi 2nL R  R

ð105Þ

Figure 21a shows transmission spectra as parameters of mirror reflectivity. The transmission bandwith is narrower, as the reflectivity increases. Figure 21b shows the bandwidth as a function of the mirror reflectivity. The above spectral response is useful for extracting a particular wavelength. The extraction performance is numerically evaluated using a finesse defined as pffiffiffiffi FSR  R F ¼ ð106Þ f! 1R

2.3

Optics of Semiconductor Lasers

Semiconductor lasers are coherent light sources operated under the lowpower condition that electronic energy is efficiently transformed into light energy. The features of semiconductor lasers are, typically, the capability of direct modulation at higher frequencies, compactness, low driving power, and applicability to monolithic integration. Such sources play an essential part in micro-optomechatronics. 2.3.1

Basic Properties of Semiconductor Lasers [3–5]

Light Emission Processes in General Materials. We start with the emission mechanism of light in general materials, which of course include

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Chapter 6

Figure 22 processes.

Level diagrams of atomic systems for exhibiting three basic optical

semiconductors. The emission mechanism is explained using a level diagram having two level systems (see Fig. 22). There are three major optical processes, absorption, spontaneous emission, and stimulated emission. Absorption means that the atom in the ground state (state 1) makes a transition to the excited state (state 2) by absorbing energy from a photon. The photon is required to have a larger energy than the gap between these states for the absorption process. Spontaneous emission means that the excited state makes a transition to the ground state according to a natural probability independently of external triggers with simultaneous emission of a photon whose energy corresponds to the energy gap. Stimulated emission means that the transition with such a photon emission occurs triggered by the emission field according to the probability proportional to the emission field density. The emitted photon is cooperative to the existing field. Hence this stimulated emission enhances the emission field. The natural probabilities for these three emission processes are numerically evaluated using a rate equation given by d d N1 ¼  N2 ¼ N2 A21 þ ð!ÞðN1 B12 þ N2 B21 Þ dt dt

ð107Þ

with the following values: (!): the energy density of the radiation field N1: the population of state 1 N2: the population of state 2 Here A21, Einstein’s A-coefficient, is a transition probability from State 2 to State 1. The parameter B12, Einstein’s B-coefficient, means a transition probability from State 2 to State 1 under the emission field. Consider an electromagnetic field in thermal states. Since the timedependent terms are negligible and the population ratio of the two states

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189

follows the Boltzmann distribution given by N2 ¼ expð  h! Þ N1

¼

1 KB T

ð108Þ

the energy density of the emission field becomes ð!Þ ¼

A21 expð  h! Þ B12  B21 expð  h! Þ

ð109Þ

with the value KB the Boltzmann constant. This equation gives the relation between the two Einstein coefficients: A21 h!3  ¼ 3 2 B12 c 

ð110Þ

Light amplification is theoretically possible by using the stimulated emission, but it is inhibited by the absorption dominance in the medium because N2 < N1 for every temperature. In order to obtain a net gain in light amplification, N2 > N1 is required. This means that the reverse population nominally exhibits a negative temperature. It is of great importance for all lasers to create the mechanisms for yielding such a reverse population. Mechanism of Laser Oscillation in Semiconductors. Semiconductors have a characteristic electronic structure as shown in Fig. 23; they have a band structure consisting of valance and conduction bands. The emission processes of light in semiconductors are similar to the three-level model as described above. Electrons in a valence band are thermally excited to a

Figure 23

Emission of photons in semiconductors.

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conduction band while they leave positive holes in the valence band. The excited electrons can be recombined with the holes. Relaxation processes of this recombination include photon emissions whose energy h! corresponds to the band-gap energy
E. This band-gap energy is strongly dependent on the materials. Semiconductor lasers can be obtained in various emission wavelength bands from visible to infrared ranges, if appropriate materials are selected for constructing devices of interest. In order to achieve continuous photon emission by current injection, we have used a PN-junction structure as shown in Fig. 24. This structure consists of two types of semiconductors involving impurities. One is an N-type semiconductor whose impurities provide electrons for the conduction band. The other is a P-type semiconductor whose impurities accept excited electrons in the valence band to create positive holes. When these two types of semiconductors are joined together, the interface region can include both electrons and holes; hence when we make a closed loop circuit by applying a positive voltage to the P-type semiconductor while connecting the N-type semiconductor to the ground level, we obtain a stationary charge flow according to the amount of recombination in the junction. Therefore we obtain stationary photon emissions. We present a double heterojunction structure to achieve inverse populations in semiconductors as shown in Fig. 25. Heterojunction means connecting two kinds of semiconductors with different chemical elements. Such a junction offers high potential barriers for carriers. We have achieved strong carrier confinement by a sandwich structure using two

Figure 24

Emissions of photons at PN junctions.

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Figure 25

191

Double heterostructure for lasers.

heterojunctions. This structure, called a double heterojunction (DH) structure, is widely used for achieving inverse populations in semiconductor devices. In order to do laser oscillation, however, we further need an optical confinement structure to increase the laser field intensity. Thus optical waveguides in which the core region shows a higher refractive index while the outside cladding layers show a lower indices are used to keep the laser light in the core. Lasers with DH structures have the shortcoming that the core region shows a rather large propagation loss because the gain medium simultaneously acts as an absorption medium. Separated confinement heterostructure (SCH) lasers are used to eliminate this problem (see Fig. 26). SCH lasers have an electron confinement region usually consisting of very thin (several nanometers) layers, to construct quantum wells (QWs) and barriers, which are included in the core region. Since the core region is designed to have a wider band-gap compared with the energy of the emitted photons in the QWs, light waves guided in the core show low-loss propagation performance. Practical lasers need three-dimensional confinement structures for both carriers and light waves. These structures are designed and realized based on the relationship between the chemical composition of the materials and the corresponding refractive index. We have developed a stripe-geometry laser structure as shown in Fig. 27, which basically consists of a waveguide with a layered vertical confinement structure and a buried lateral guide structure. This laser structure also has the function of optical resonance by using two facet mirrors to enhance the field intensity.

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Figure 26 Schematic of separated confinement heterostructure (SCH) lasers with quantum wells (QWs) for electron confinement sandwiched by semiconductor cladding regions having a wide band-gap to offer transparency for emitted photons at the wells.

Figure 27 Schematic cross section of buried heterostructure (BH) lasers with double heterostructure active region.

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193

Threshold Condition and Oscillation Mode. As an injection current to a semiconductor laser increases, the population of electrons in the conduction band increases. In the lower current injection region, the light amplification in the semiconductor medium is so insufficient that the light attenuates while circulating in the laser cavity. However, the light is dramatically amplified above an injection current level that gives a roundtrip with no attenuation. This condition is called the threshold condition for laser oscillation. A characteristic injection current versus light output curve is given in Fig. 28. This feature is explained by a microscopic consideration: that electrons injected into semiconductor lasers are used for increasing the optical gain below the threshold, while injected electrons are used to produce light by stimulated emission with high efficiency. We evaluate the threshold using a simple stripe-geometry Fabry–Perot laser. It is assumed that the laser consists of two facet mirrors and a waveguide with structural parameters including a cavity length of L and facet reflectivities of R1 and R2. Using gain and loss coefficients g and , an amplitude round-trip gain a is defined as pffiffiffiffiffiffiffiffiffiffiffi a ¼ R1 R2 exp½ðg  ÞL þ 2ikL ð111Þ Noting a ¼ 1 under oscillation, we obtain the oscillation conditions 9  m m ¼ 1, 2, 3, . . . > km ¼ = nL 1 1 > ; gth ¼ þ log 2L R1 R2

Figure 28

ð112Þ

Light output vs. injection current characteristics of semiconductor laser.

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Chapter 6

The first condition corresponds to the resonant modes of the Fabry–Perot cavity, and the second condition gives the threshold condition necessary for laser oscillation. Consider the second condition to evaluate the threshold current condition. The gain coefficient g is given as a function of injection current density J;

J g ¼   J0 ð113Þ d where J0 is the nominal current, d the thickness of the active region, and  a constant. Substituting this equation into Eq. (112), we find the threshold current density as

d 1 1 þ log Jth ¼ ð114Þ þ J0 d  2L R1 R2 We also discuss the laser oscillation mode using the first oscillation condition. The laser eigenmodes determined by Eq. (112) include many candidates for laser oscillation. However, a unique mode with a maximum gain coefficient remains, while the others fade out during circulation in the cavity. For more high-performance applications, semiconductor lasers are desired to show single-mode oscillation performance. This oscillation condition is readily obtained by using a grating as a mode selector to construct a laser resonator (see Fig. 29).

Figure 29 a cavity.

Laser oscillation growing up from amplified spontaneous emission in

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2.3.2

195

Coupled-Cavity Lasers [6–14]

Coupled-cavity lasers with a simple configuration consisting of a Fabry– Perot laser diode and an external mirror (see Fig. 30) have been the subject of intense investigation because of their attractive oscillation performance readily controlled by external mirrors [11–15]. Substitution of the external cavity with an effective mirror having a complex reflectivity is also available although the external cavity is relatively long compared with the optically switched lasers. The reflectivity is represented considering multiple reflections in the external cavity as pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi 2h R2 R3 expði Þ reff ¼ R2  ð1  R2 Þ pffiffiffiffiffiffiffiffiffiffiffi ð115Þ

¼ c 1  R2 R3 expði Þ with the following values: : oscillation angular frequency c: light speed in a vacuum h: external cavity length R2: reflectivity of laser facet facing external cavity R3: reflectivity of external mirror The oscillation frequency is pulled into one of the eigenmodes of the laser diode owing to the strong optical gain, hence it is represented as c ð116Þ  ¼ !0 þ m nLo with the following values represented: m: longitudinal-mode number; L0: length of laser diode;

Figure 30 Variations of coupled-cavity lasers (CCLs) realized by light feedback from external mirror. (a) CCL with an extremely small external cavity. (b) CCL with a collimated lens in an external cavity.

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196

Chapter 6

N: refractive index of laser diode. !0: optical angular frequency of an eigenmode We can readily derive the interference undulation with a period of half the wavelength from the representation of the effective reflectivity, assuming a single-mode oscillation with no mode-hopping. However, the undulation actually exhibits various increased spatial frequencies such as /4, /6, /8, etc., according to the external cavity length (see Fig. 31). Such coupledcavity lasers are explained by a mode selection rule: one of the eigenmodes with maximum reflectivity, corresponding to the minimum threshold, is selected as the oscillation mode. The substitution of the external cavity with the effective reflectivity is insufficient for discussing asymmetrical sawtooth undulations. We must take account of the time-dependent field component of the light in the external cavity laser. Assuming the stationary condition, this consideration results in oscillation frequency change dependent on the external cavity length as given by rffiffiffiffiffiffi R3 c sinð Þ ð117Þ  ¼ !m þ ð1  R2 Þ R2 2nLo When the equation gives a unique solution for , the frequency change   !m corresponds to the phase change in the interference undulations at a period of half the wavelength. This phase change generates the sawtooth undulation curves.

Figure 31 Interference undulations by light feedback from an external mirror located away from the laser facet by several millimeters.

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3

197

OPTICS OF DIELECTRIC THIN FILMS [15]

In this section the fundamental optics of dielectric optical thin films are described based on Maxwell’s equations for designing optical bandpass filters.

3.1 3.1.1

Macroscopic Picture of Lightwave Propagation in Dielectric Media Complex Refractive Index and Admittance

We all know that light waves are slowed down and attenuated when they propagate in a dielectric medium. This phenomenon is macroscopically described by introducing the concept of complex refractive index. Taking account of the charge transfer in the dielectric medium, we must rewrite Maxwell’s equation (3) as rot H ¼ J þ "

@ E @t

ð118Þ

where J is the current density. The following representation is easily obtained:

@ @ @ rot rot E ¼ 
E þ div E ¼ 
E ¼  rot H ¼  Jþ" E @t @t @t ð119Þ Using Ohm’s law with electronic conductivity , J ¼ E

ð120Þ

we obtain
E  

@ @2 E  " 2 E ¼ 0 @t @t

ð121Þ

Similarly, we can obtain a wave equation for the magnetic field
H  

@ @2 H  " 2 H ¼ 0 @t @t

ð122Þ

Consider a plane wave with an optical frequency !, E ¼ E0 exp½iðk r  !tÞ

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ð123Þ

198

Chapter 6

where k ¼ ks is the wavenumber vector specified by the magnitude k ¼ 2/ and the unit vector along the propagation directions. Then we obtain 2

  @ @2 @2
E ¼ E0 þ þ exp i kx x þ ky y þ kz z expði!tÞ 2 2 2 @x @y @z ¼ k2 E

ð124Þ

Using the relations @ E ¼ i!E @t

ð125Þ

@2 E ¼ !2 E @t2

ð126Þ

we obtain a dispersion relation as k2 ¼ i!  þ "!2

ð127Þ

As the ratio of wave number to the optical frequency is given by k 1 1 ¼ ¼ !
v

ð128Þ

we can rewrite the above relation using the light speed in vacuum pffiffiffiffiffiffiffiffiffiffiffi c ¼ 1= "0 0 as c2  r ¼ "r r þi ð129Þ v ! "0 with "r  "/"0 and r  /0. Since the refractive index is nominally defined as a ratio of a light speed in a medium to that in vacuum, the above equation gives the index as N2 ¼ "r r þi

 r ! "0

ð130Þ

Since this means that the refractive index of the medium is a complex generally expressed by N ¼ n þ i, the parameters n, and  are related to each other by n 2  2 ¼ " r  r 2n ¼

 r ! "0

ð131Þ ð132Þ

As we can obtain the complex refractive index, let us obtain a complex admittance that gives the ratio of the field components of light waves.

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The unit vector s is given by   s ¼ ð, , Þ ¼ cos x , cos y , cos z

199

ð133Þ

The electric field is expressed as E ¼ E0 ðtÞ exp½ik r ¼ E0 ðtÞ exp½ikðx þ y þ zÞ  2N ðx þ y þ zÞ ¼ E0 ðtÞ exp i

ð134Þ

We can readily derive the following relation using the above equation: rot E ¼

  2iN 2iN expðik rÞ ex þ ey þ ez
E0 ðtÞ ¼ s
E

ð135Þ

Substituting this equation into Maxwell’s equation Eq. (4), we obtain H¼

N s
E c

ð136Þ

Hence the complex admittance is given as y

H N ¼ ¼ Ny E c

y ¼

1 ¼ c 0

ð137Þ

with

3.1.2

rffiffiffiffiffiffi "0 : 0

Light Wave Behavior at Dielectric Surface

Dielectric surfaces are regarded as interfaces between two optical media with different refractive indices. A part of a light wave is transmitted while the other is reflected at the interface. This section characterizes the transmission and reflection of light at such dielectric surfaces. Consider a P polarized incident light wave going into an interface of two media having the admittance of y0 and y1 as shown in Fig. 32a. As the field component along the surface must satisfy the continuity, we obtain the relations Ei cos 0 þEr cos 0 ¼ Et cos 1

ð138Þ

Hi  Hr ¼ Ht

ð139Þ

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Chapter 6

Figure 32 Transmission and reflection at dielectric surfaces. (a) P-polarized light wave incidence. (b) S-polarized incidence.

Eq. (139) can be rewritten using the admittance as y0 ðEi  Er Þ ¼ y1 Ht

ð140Þ

Hence we can obtain reflectance and transmittance as  p

Er y0 cos 1  y1 cos 0 y0 =cos 0  y1 =cos 1 0  1 ¼ ¼ ¼ Ei y0 cos 1 þ y1 cos 0 y0 =cos 0 þ y1 =cos 1 0 þ 1

ð141Þ

2y0 cos 1 20 ¼ y0 cos 1 þy1 cos 0 0 þ 1

ð142Þ

p ¼

Here h is an effective admittance defined for P polarization as ¼

y cos 

ð143Þ

A similar analysis can be applied to S polarized light waves as shown in Fig. 32b to obtain the reflectance and transmittance based on the condition of continuity: Hi cos 0 Hr cos 0 ¼ Ht cos 1

ð144Þ

Ei þ Er ¼ Et

ð145Þ

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201

The nominal expressions for reflectance and transmittance are the same as those for the P polarized light waves, but the effective admittance must be replaced by  ¼ y cos 

ð146Þ

Our purpose is to obtain the power reflectance and transmittance, but they are not directly derived from the above coefficients. This is because the plane waves are quite ideal and unreal: existing light waves have a finite cross-sectional area, and the change of the area must be taken into account when the light waves are refracted. As this effect is negligible for reflection, the power reflectance is simply estimated as   0  1 2 Ir 2   ð147Þ R ¼ ¼ jj ¼  0 þ 1  Ii The power transmittance, however, must be calculated taking account of the effect. It is given by 2  It n1 2 n1  20  T ¼ ¼ j j ¼  ð148Þ n0  0 þ  1  Ii n 0 with 1 Ii ¼ n0 y jEi j2 2

3.1.3

1 It ¼ n1 y j j2 jEi j2 2

Derivation of Effective Admittance

The effective admittance was used for expressing the coefficients of reflection and transmission, but they should be derived from Maxwell’s equations. Note the expressions for the magnetic field using Eq. (136), rot H ¼

2iN s
H

ð149Þ

and

@ i!N2 E rot H ¼  þ " E¼ @t c2

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ð150Þ

202

Chapter 6

Combining the above equations, we obtain yE ¼  s
H

ð151Þ

The field components are expressed for P polarized light waves as E ¼ ðEx, 0, Ez Þ

ð152Þ

  H ¼ 0,Hy, 0

ð153Þ

Assuming an incidence direction as s ¼ (sin 0, 0, cos 0), we can estimate Eq. (151) as   yE ¼  s
H ¼ Hy cos 0 , 0, Hy sin 0 ð154Þ Since the field components of interest are Ex and Hy, we obtain the admittance as Hy y ¼  Ex cos 0

ð155Þ

Similar analyses are available for the S polarized light waves:   E ¼ 0, Ey , 0

ð156Þ

H ¼ ðHx , 0, Hz Þ

ð157Þ

We can readily derive a simple relation   H ¼ ys
E ¼ yEy cos 0 , 0, yEy sin 0

ð158Þ

Hence we obtain Hx ¼ y cos 0   Ey

3.2

ð159Þ

Propagation of Light Waves in Multilayer Structures

Fundamental light wave behavior at dielectric surfaces was characterized in the previous section. We can estimate the propagation of light waves in dielectric multilayer structures such as optical filters based on the characterization.

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3.2.1

203

Transfer Matrix

In this section, we derive a transfer matrix that relates the field components between neighboring layers. We can calculate transmission characteristics of any multilayer structure by using this matrix. Consider a multilayer structure consisting of M dielecric layers. We now focus on the layers m  1, m, and m þ 1 and investigate the propagation of light waves as shown in Fig. 33a. The field components in the direction þ along the þ z-axis are Eþ m,m1 and Hm,m1 , and those along the  z-axis are   Em,m1 and Hm,m1 at the interface of the (m  1)th and mth layers. þ   Similarly, we have the field components Eþ m,m , Hm,m , Em,m , and Hm,m at the interface of mth and (m þ 1)th layers. We introduce the phase parameter m ¼

2Nm dm cos m

ð160Þ

where Nm and dm are the complex refractive index and the thickness of mth layer, respectively. The phase parameter relates each field component as þ Eþ m,m ¼ Em,m1 expðim Þ

ð161Þ

þ Hþ m,m ¼ Hm,m1 expðim Þ

ð162Þ

Figure 33 Schematic illustration of light wave propagation through thin films. (a) Neighboring layers. (b) Entire multilayered structure.

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204

Chapter 6  E m,m ¼ Em,m1 expðim Þ

ð163Þ

 H m,m ¼ Hm,m1 expðim Þ

ð164Þ

Similarly we obtain  Em1 ¼ Eþ m,m1 þ Em,m1

ð165Þ

 Hm1 ¼ Hþ m,m1 þ Hm,m1

ð166Þ

 Em ¼ Eþ m,m þ Em,m

ð167Þ

 Hm ¼ Hþ m,m þ Hm,m

ð168Þ

We can obtain the relationship between the neighboring layers by carrying out calculations with these field components. The detailed calculations are carried out as follows. The field components at the mth layer are represented by





þ

 Eþ Em,m Em 1 1 m,m þ Em,m ¼ ¼ ð169Þ þ  m Em,m  m Em,m E Hm m m m,m Hence we obtain þ Em,m 1 ¼ E m m,m

1 m

1

Em Hm

¼

1 2m



m m

1 1



Em Hm

ð170Þ

On the other hand, the field components at the interfaces can be estimated using the phase parameter by

þ

þ

Em,m Em,m1 0 expðim Þ ¼ ð171Þ E E 0 exp i ð Þ m m,m m,m1 Hence we obtain

Eþ m,m1 E m,m1

¼

1 2 expðim Þ 1 2 expðim Þ

1 2m

expðim Þ 1  2m expðim Þ

!

Replacing m with m  1 in Eq. (169), we obtain



 Eþ Em1 1 m,m1 þ Em,m1 ¼ ¼  Hm1  m Eþ   E m m m,m1 m,m1

Em Hm



1 m

ð172Þ



Eþ m,m1 E m,m1

ð173Þ

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Combining Eqs. (170) and (173), we obtain





iðsin m =m Þ Em1 cos m Em ¼ im sin m Hm1 cos m Hm This equation defines the transfer matrix as

iðsin m =m Þ cos m Um  im sin m cos m

3.2.2

205

ð174Þ

ð175Þ

Evaluation of Transmission Characteristics Using Transfer Matrix

We can now evaluate the field components for any neighboring pairs of layers in the multilayer structure as



Em Em1 ¼ Um ð176Þ Hm1 Hm This relationship can be readily extended to the entire structure as







E0 E1 E2 E3 ¼ U1 ¼ U1 U2 ¼ U1 U2 U3 ¼ H0 H1 H2 H3



M EM EM ¼ U1 U2 U3     UM ¼  Um ð177Þ i¼1 HM HM We are only interested in the light wave transmitted through such a structure as that shown in Fig. 33b. Consider the field just inside the first layer specified as E0, H0. We can define an admittance for these fields as Y¼

H0 E0

ð178Þ

Similarly, we define an admittance on the transmitted side as s ¼

HM EM

Hence Eq. (177) is simply written as



M 1 1 E0 ¼ EM  Um Y s i¼1

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ð179Þ

ð180Þ

206

Chapter 6

This equation gives the structural parameters as 

M p 1   Um q s i¼1

ð181Þ

The matrix elements can be calculated using Snell’s law, N0 sin 0 ¼ N1 sin 1 ¼ N2 sin 2 ¼ ¼ NM sin M

ð182Þ

The reflection coefficient was already derived from Eq. (141). This expression can be extended to the multilayer structure. We obtain ¼

0  Y p0 q ¼ 0 þ Y p0 þq

Hence the power reflectance is given by



p0  q p0  q R¼ p0 þ q p0 þ q

ð183Þ

ð184Þ

where * denotes taking the complex conjugate of the operand. The power transmittance can be found by taking account of energy flow. The light power flowing into the structure is represented as  1 1 I0 ¼ Re E0 H 0 ¼ Re½ pq EM E M 2 2

ð185Þ

This energy flow is equal to the power deduced from the incidence power by the reflection power as I0 ¼ Iin ð1  RÞ

ð186Þ

Hence we estimate the incidence power as Iin ¼

1 I0 Re½ pq EM E M ¼ 1  R 2ð1  RÞ

ð187Þ

We can also estimate the transmitted power as  1 1 IM ¼ Re EM H M ¼ Re½s EM E M 2 2

ð188Þ

Therefore we find the power transmittance as T¼

IM 40 Rebs c ¼  Iin  p0 þ q2

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ð189Þ

Fundamental Optics of Micro-Optomechatronics

3.3

207

Transmission Characteristics of Optical Bandpass Filter

The filter is modeled on a stack of dielectric layers, which forms a periodic structure consisting of a half-wave cavity sandwiched by quarter-wave mirrors (Fig. 34). Each layer alternatively has a higher or lower refractive index. Specifying these layers as H (higher index) and L (lower index), the filter is expressed by



 ðHLÞM ðLHÞM ¼ ðHLÞM1 H ðLLÞ H ðLHÞM1 ð190Þ where M is an integer ( 2). This periodic structure follows that of the conventional optical resonance with a /2 phase shift section corresponding to the central portion specified by the half-wave cavity LL. This resonance is explained as follows. The quarter-wave regions provide eigenmodes A(z), B(z), C(z), D(z). These modes are optimally coupled, so the phase shift section forms a resonant cavity with a total phase shift of 2. We can obtain the structural parameters for the above structure using Eq. (181) as



1 p  ðUH UL ÞM1 UH UC UH ðUL UH ÞM1 ð191Þ y NS q where NS is the complex refractive index of the substrate. Here the transfer matrix for quarter-wave layers is represented as

 0 cos ’ iðsin ’=y Nj Þ Uj  ð j ¼ H, LÞ ’¼ iy Nj sin ’ cos ’ 2 ð192Þ

Figure 34 Schematic model of a quarter-wave optical bandpass filter having the structure of an optical resonant tunnel.

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208

Chapter 6

Figure 35 lengths.

Power transmission spectra of optical bandpass filters for various cavity

Uc is the matrix for the half-wave cavity, and it can be calculated by using the above expression with replacement of ’:

 0 1
D ’¼ 1þ ð193Þ 2 2 D0 where D0 is the original length of the phase shift section at ¼ 0. Let us carry out numerical evaluations for existing wavelength-tunable filters consisting of two materials, Ta2O5 and SiO2. The calculations are carried out at d ¼ 0.7 /2n, 0.8 /2n, 0.9 /2n, /2n, 1.1 /2n, 1.2 /2n, 1.3 /2n, 1.4 /2n for the bandpass filter with refractive indices of H ¼ Ta2O5 (nH ¼ 2.16) and L ¼ SiO2 (nL ¼ 1.46) [16] and M ¼ 11. High-reflection quarter-wave sections are not changed. Assuming 0 ¼ 1.55 mm, the calculated spectral responses are shown in Fig. 35 in the range of 1450 to 1700 nm for various cavity lengths. When the cavity length is changed in the range by 70–140%, the transmission center wavelength is changed while maintaining a constant stopband of 1400–1800 nm. 4

EXTRAORDINARY ELECTROMAGNETIC WAVES IN CONDENSED MATTER WITH FREE ELECTRONS

The technology stream is now going from micro- to nano-optomechatronics. How to remove the diffraction limit, from which many applications,

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209

Figure 36 Schematic setup for light wave transmission experiment through nanoholes with a diameter much smaller than the wavelength.

particularly including high-density optical recording, suffer, is a key to the nanoworld. How narrow a hole can light waves pass through? Conventional theories based on the Maxwell equations give the answer that the minimal diameter of the hole that light can pass through is half the wavelength. This limit corresponds to the cutoff frequency of holes. However, a very recent paper [17] has reported the extraordinary transmission of light through a hole whose diameter is much smaller than the wavelength. This extraordinary transmission, occuring in thin metal films with such nanoholes, is related to electromagnetic waves being bound at metal–dielectric interfaces, i.e., surface plasmon polaritons (SPPs) [18–24] (see Fig. 36). Through detailed investigation into SPPs at single metal–dielectric interfaces, it is found that free-space photons can launch the SPPs inside a metal nanogap, a sub-wavelength-thick dielectric spacing between two semi-infinite metals [25]. Hence such SPPs are expected to become a breakthrough in photonics and related technology in wide application areas including optical integration, memories, and processing. In this section, fundamentals of SPPs are described based on electromagnetic dynamics to understand why and how SPPs are so extraordinary compared with conventional light waves. 4.1 4.1.1

Dynamics of Free Electrons Dynamic Response by an Alternating Electric Field

Suppose that a single electron stands still at the origin of the coordinate axes. When an alternating electric field is applied to the electron, the

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210

Chapter 6

equation of motion is written as m€s þ _s ¼ eE0 ei!t

ð194Þ

where s is the coordinate parameter to indicate the position of the electron, m the mass of the electron, e the fundamental charge of the electron, the nominal damping factor, and ! the angular optical frequency of the electric field of the light; the dots indicate the temporal differential operators. For convenience, the Fourier transform of the equation is used for the following considerations. Taking account of the relation of Fourier transforms for each coordinate parameter, Z1 sð!Þ expði!tÞ d! ð195Þ sðtÞ ¼ 1

Z

1

s_ ðtÞ ¼ i

! s ð!Þ expði!tÞ d!

ð196Þ

!2 sð!Þ expði!tÞ d!

ð197Þ

1

Z

1

s€ ðtÞ ¼  1

we obtain the equation of motion in the frequency domain as  2  m! þ i ! sð!Þ ¼ eEð!Þ

ð198Þ

Hence the response of the electron against the alternating electric field is represented in the frequency domain as sð!Þ ¼

4.1.2

e Eð!Þ m!2 þ i !

ð199Þ

Dielectric Function of Free Electrons

The dynamics for a single electron as described above must be extended to a system containing many electrons to characterize the behavior of electrons using macroscopic parameters. Assuming a number density of electrons N, we can define a current density Jð!Þ ¼ eN_sð!Þ

ð200Þ

Taking account of the Ohm law J ¼ E, we can rewrite the above equation as s_ ð!Þ ¼ 

 Eð!Þ Ne

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ð201Þ

Fundamental Optics of Micro-Optomechatronics

211

In the stationary condition (! ! 0), the term s€ should be zero, and we have an equation for an equivalent electron flow e s_ ð!Þ ¼  Eð!Þ

ð202Þ

Combining Eqs. (201) and (202), we obtain the damping factor, ¼

Ne2 

ð203Þ

Hence we finally obtain the dynamic response of electrons as sð!Þ ¼

1 e  2  Eð!Þ 2 ! m þ i Ne =!

ð204Þ

Here we introduce an electric polarization, which can be defined using a displacement of electrons as Pð!Þ ¼ Ne sð!Þ

ð205Þ

Substituting the electrical response for the coordinate parameter in the above equation, we find the dielectric function of free electrons to be "ð!Þ ¼ 1 

 !  i ! "0 1 þ ð !Þ2

¼

m Ne2

ð206Þ

where is the relaxation time of an electron in the metal. When the dielectric function is described as " ¼ "R þ i"I, the real and imaginary components are expressed as "R ¼ 1 

"I ¼



1 !p 2 ¼1 2 2 "0 1 þ ð !Þ ! þ 1= 2

 1

!p 2 ¼ 2 ! "0 1 þ ð !Þ ! 1 þ ð! Þ2

ð207Þ

ð208Þ

where !p is a frequency defined as the plasma oscillation frequency of the free-electron system and defined as !p 2 



"0

ð209Þ

Figure 37 illustrates the real and imaginary parts as a function of angular frequency normalized by the plasma oscillation frequency. Note that the real part of the dielectric function can be negative when the frequency ! is sufficiently small (! < !p). Figure 38 shows the measured real part of

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212

Chapter 6

Figure 37

Real and imaginary part of dielectric function of free-electron gas.

Figure 38

Wavelength dependence of permittivity for various metals.

dielectric functions for various metals. They surely exhibit negative permittivity as described above. 4.2

Plasma Oscillation in Free-Electron Gas

Consider a free-electron gas in a condensed matter such as a metal. It is assumed that the electron–electron interaction due to the Coulomb force is

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213

ignored and the total charge is neutralized by positive metal ions arranged to form a crystal lattice. Electrons in such matter can be described by the above gas model. The electrons offer wavelike behaviors; both longitudinal and transverse waves can exist in such an electron gas. These behaviors come from the local displacement of electrons and so are described by the change of electric polarization P. Hence we can characterize these waves using electric polarization. 4.2.1

Longitudinal Plasma Oscillation

For the longitudinal wave, we have the condition div P 6¼ 0 and rot P ¼ 0. This means that the electron density may be locally changed but a DC current does not exist. Suppose that a wave propagates along the x-axis; the polarization wave is represented as Pðx, tÞ ¼ P expðiqx  i!tÞ

ð210Þ

where q is the wave number. The corresponding electric field is represented as Eðx, tÞ ¼

Pðx, tÞ P expðiqx  i!tÞ  ¼   "0 "ð! Þ  1 "0 "ð! Þ  1

ð211Þ

This equation readily relates the electric field to the electric polarization in the frequency domain as Eð!Þ ¼

Pð!Þ   "0 "ð!Þ  1

ð212Þ

This equation is readily rewritten as Dð!Þ ¼ "0 "ð!ÞEð!Þ ¼

"ð!ÞPð!Þ "ð! Þ  1

ð213Þ

Hence we obtain div Dð!Þ ¼

"ð!Þ div Pð!Þ "ð!Þ  1

ð214Þ

Since Maxwell’s equation offers div D ¼ 0, "(!)¼ 0 is essential for the existence of the longitudinal polarization wave. Consider the particular case of  ! 1. This extreme situation is inadequate in the lower frequency range but allowable in much higher optical frequencies, above several terahertz. In this case, the dielectric

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214

Chapter 6

function is given by "ð!Þ ¼ 1 

!p 2 !2

ð215Þ

Hence the longitudinal wave is given at the plasma oscillation frequency. 4.2.2

Transverse Plasma Oscillation

For transverse plasma oscillation, we have the condition div P ¼ 0 and rot P 6¼ 0. This means that the electric polarization is not created or annihilated in the electron gas. Such a transverse wave can interact with the electromagnetic field. Taking account of the wave equation represented using Fourier transform
E ¼ "E€ , we can readily have the relation @2 @x2

Z

1

Eð!Þ expðiqx  i!tÞ d! Z @2 1 ¼  "0 2 "ð!ÞEð!Þ expðiqx  i!tÞ d! @t 1 1

ð216Þ

This equation is modified to q2 Eð!Þ ¼ !2 Dð!Þ ¼ !2  "0 "ð!ÞEð!Þ

ð217Þ

This gives the dispersion relation q2 c2 ¼ "ð!Þ!2

ð218Þ

Using Eq. (215) as a representation of the dielectric function of the electron gas, we obtain qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ! ¼ q2 c2 þ !p

ð219Þ

where c is the light speed in vacuum. Figure 39 illustrates this dispersion relation. The linear relation indicates the dispersion of vacuum, ! ¼ qc. For ! !p, the wave number q is a real number and the group velocity is always smaller than the light speed. This means that a light wave that satisfies the above frequency condition can excite plasma oscillation, and that the plasma oscillation is nonradiative. On the other hand, for ! < !p, the wave number is an imaginary number, so a light with such an optical frequency is damped while creating a backward light, when it

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Figure 39

215

Dispersion relation of transverse plasma oscillation.

comes into metals. This shows the mechanism of total reflection at metal surfaces. 4.3 4.3.1

Surface Plasmon Polariton Waves Concept of Surface Plasmon Polariton [26]

Transverse plasma oscillation can be coupled with an electromagnetic field. The coupled waves are radiative, and their phase velocity exceeds the light speed in vacuum. However, when the electron gas has a boundary condition, the situation is quite different from that mentioned above. Consider a micrometallic sphere as shown in Fig. 40a. The electrons are displaced by the electric field, but surface charges appear on the upper and lower side of the sphere owing to the boundary that strictly confines the electrons. These charges mean that electric polarization is generated. This electric polarization also generates a corresponding antielectric field that acts as a damping force. When this force is synchronized with the initial electric field with a 180 degree phase shift, plasma oscillation occurs. This concept of plasma oscillation in small particles is extended to the wave at the metal–dielectric surface, as shown in Fig. 40b. This means that such surface plasma oscillation can be coupled with the transverse electromagnetic waves. It is noteworthy that the surface oscillation is nonradiative, like the longitudinal plasma oscillation. Therefore it is bounded at the surface. This is a novel light confinement concept, and details are described in the following sections.

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Figure 40 Plasma oscillation coupled with electromagnetic field. (a) Plasma oscillation in a small metal particle. (b) Field configurations of electromagnetic waves at metal surface.

4.3.2

Analysis of SPP Waves Based on Electromagnetic Dynamics

To describe plasma oscillation directly coupled to an electromagnetic field, a vector potential is defined as A ¼ A0 fðyÞ expðiqz  i!tÞ

ð220Þ

with fðyÞ ¼ expð1 yÞ

y 0

fðyÞ ¼ expð2 yÞ

y0

ð221Þ

assuming that the wave propagates in the direction of the z-axis along the dielectric ( y > 0) and metal ( y < 0) interface. Taking account of the definition of the vector potential B  rot A, we rewrite the wave equation for

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convenience as
A þ !2 "A ¼ 0

ð222Þ

Combining the above equations, we obtain pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ q2 !2  "0 "1 2 ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q2 !2  "0 "ð!Þ

ð223Þ ð224Þ

where "1 is the dielectric constant of the dielectric medium. The continuity condition must be satisfied for the components along the x-axis at the interface. The components include the x-component of the vector potential: Ax ¼ 

 @ Hx "0 " @x

ð225Þ

which can be easily given by the Maxwell equations in the form using the Fourier transform: rot Eð!Þ ¼ i!Hð!Þ

ð226Þ

rot Hð!Þ ¼ i!"Eð!Þ

ð227Þ

Hence we obtain the relation 2 "1 ¼ 1 "ð!Þ

ð228Þ

We can therefore obtain the dispersion relation, eliminating the unknown parameters 1 and 2, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! "1 "ð!Þ ð229Þ q¼ c "1 þ "ð!Þ We also obtain the field components as q Eðy1Þ ¼  Hx ðy > 0Þ ! "1 Eðy2Þ ¼  Eðz1Þ ¼ i

q Hx !"ð!Þ

ðy < 0Þ

1 2 Hx ¼ i Hx ¼ Eðz2Þ ! "1 ! "2

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ð230Þ ð231Þ ð232Þ

218

4.3.3

Chapter 6

Dispersion of SPP Waves

We can numerically characterize the SPP using this equation. Figure 41 shows dispersion relations of the SPP at Au–vacuum and Au–SiO2 interface, respectively, calculated using measured permittivity. The !–q dispersion curve of the SPP has no intersection with the dispersion of the corresponding dielectric medium. This means that the light propagating in the dielectric medium is not directly coupled to the SPP wave. However, the light with a much smaller propagation speed can be directly coupled to the SPP wave. Such direct coupling between the light and the SPP is enabled by using an evanescent field. We can also estimate the propagation loss coefficient using the complex representation for the permittivity of metals. Figure 42 shows loss coefficients calculated for various combinations of metals and dielectric media. This numerical evaluation clarifies that the loss is sufficiently small at infrared frequencies as used for fiber-optic telecommunications. This lowloss transmission performance is not enough for practical applications, but surely remains a possibility for future photonic integration. 4.3.4

Confinement of an Electromagnetic Field in a Small Space

The SPP wave provides a spatial profile normal to the interface that the intensity is dramatically damped from the interface particularly in the metal, as shown in Fig. 43. Using this performance, light can be confined into an ultrasmall space beyond the diffraction limit.

Figure 41 interfaces.

Dispersion relation of SPP waves at Au–vacuum and Au–SiO2

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Figure 42 Propagation loss coefficients as functions of wavelength for SPP waves using measured permittivity.

Figure 43 Field intensity profile of SPP wave at metal–insulator interface at l ¼ 1.55 mm.

An approach for achieving this concept uses a nanotunnel blocked out by metal cladding, as shown in Fig. 44a. In such a tunnel, the SPP can have two possible modes: symmetrical and asymmetrical (see Fig. 44b). It has been known that the asymmetrical mode has no cutoff frequency. This mode

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Chapter 6

Figure 44 SPP waveguide with dielectric core embedded by semi-infinite metals. (a) Cross-sectional structure. (b) Field configurations.

has field components given by 

h fðyÞ ¼ cosh 1 y  2

0yh

 h ¼ cosh 1 exp½2 ðy  hÞ 2  h ¼ cosh 1 exp½2 y 2

y0

ð233Þ

y h

ð234Þ

ð235Þ

where h is the thickness of the dielectric tunnel. The condition for continuity gives the relation "1 1 þ "ð!Þ2 tanhð2 Þ ¼ 0

ð236Þ

Hence we can numerically evaluate the dispersion relation and loss coefficient as functions of wavelength. Figure 45 shows dispersion curves for various thicknesses of the tunnel and corresponding loss coefficients as a function of wavelength. Independently of the thickness, there exists an SPP mode in the tunnel. Such a loss is also efficiently small at larger wavelengths above around 1 mm. Figure 46 shows the typical intensity profile of an SPP mode 1.55 mm in wavelength existing in a tunnel 100 nm in thickness. Of course, the SPP mode exists in a tunnel with a much smaller thickness. The two metals sandwiching the dielectric tunnel show enough blocking-out

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Figure 45 Propagation properties of SPP along metal waveguides with dielectric cores consisting of Au/SiO2/Au layer structure for asymmetric modes. (a) Dispersion relation. (b) Wavelength-dependent loss cofficient.

Figure 46 Nano tunneling using SPP mode. Field intensity profiles of SPP waves of l ¼ 1.55 mm for tunnel-type waveguides with dielectric core (h ¼ 100 nm) sandwiched by metal cladding.

effect for the electromagnetic field, so as to provide the strong opticalconfinement performance beyond the diffraction limit. 4.3.5

Experimental Demonstrations [27–31]

Long-range propagation of such symmetrical–SPP waves is demonstrated at infrared frequencies. Only short-range propagation over a few tens of micrometers has been demonstrated at visible frequencies owing to the

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extremely large propagation loss, but the use of practical laser sources with longer wavelengths that have been developed for optical communications has enabled efficient SPP transmission experiments. Figure 47a shows a cross-sectional structure of the SPP waveguides prepared for the experiment. The waveguides have metal patterns on a dielectric substrate without any lateral confinement structure as required for conventional dielectric waveguides. The metal patterns are fabricated by using a conventional photolithographic technique from a sputtered Au film with a thickness of 0.25 mm on an InP substrate covered with a 0.2 mm thick SiO2 layer. An adhesion layer consisting of a 5 nm thick Cr film formed between the Au and SiO2 layers is considered to have negligible influence on the SPP propagation. Devices prepared for experiment have a single stripe 0.5 mm in length and 10 and 20 mm in width (see Fig. 47b).

Figure 47

Structure of SPP waveguide. (a) Schematic structure. (b) SEM image.

Figure 48 SPP transmission experiment: illumination images for simple stripegeometry waveguides. (a) W ¼ 10 mm. (b) W ¼ 20 mm.

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The SPPs can be excited by illuminating the facet with a TM-polarized single-mode laser beam guided by a vertically adjusted tapered optical fiber. Clear spot images are observed for both samples, while optimizing the position of the fiber tip at each center of the stripe (see Fig. 48). On the other hand, the spot images are faded out as the illumination laser beam becomes TE polarized. Such polarization dependence confirms that the observed spot comes from the SPP propagation, eliminating the possibility of guiding TE-polarized light waves along the metal stripe.

REFERENCES 1. 2. 3. 4. 5. 6. 7.

8. 9. 10.

11. 12. 13.

14.

15.

Born, M.; Wolf, E. Principles of Optics, 6th Ed; Pergamon Press: Oxford, UK, 1980. Mansfield, S.M.; Kino, G.S. Solid immersion microscope. Appl. Opt. 1990, 57, 2615–2616. Meystre, P.; Sargent, M., III. Elements of Quantum Optics; Springer-Verlag: New York, 1991. Verdeyen, J.T. Laser Electronics, 3rd Ed.; Prentice Hall. Casey, H.C., Jr.; Panish, M.B. Heterostructure Lasers; Academic Press, 1978. Morikawa, T.; Mitsuhashi, Y.; Shimada, J. Return-beam-induced oscillations in self-coupled semiconductor lasers. Electron. Lett. 1971, 12, 435–436. Voumard, C.; Salathe, R.; Weber, H. Resonance amplifier model describing diode lasers coupled to short external resonators. Appl. Phys. 1977, 12, 369–378. Lang, R.; Kobayashi, K. External optical feedback effects on semiconductor injection laser properties. IEEE J. Quantum Electron. 1980, QE-16, 347–355. Fleming, M.; Mooradian, A. Spectral characteristics of external-cavity controlled semiconductor lasers. IEEE J. Quantum Electron. 1981, QE-17, 44–59. Acket, G.; Lenstra, D.; Boef, A.; Verbeek, B. The influence of feedback intensity on longitudinal mode properties and optical noise in index-guided semiconductor lasers. IEEE J. Quantum Electron. 1984, QE-20, 1163–1169. Agrawal, G. Line narrowing in a single-mode injection lasers due to external optical feedback. IEEE J. Quantum Electron. 1984, QE-20, 468–471. Katagiri, Y.; Hara, S. Increased spatial frequency in interferential undulations of coupled-cavity lasers. Appl. Opt. 1994, 33, 5564–5570. Spano, P.; Piazzolla, S.; Tamburrini, M. Theory of noise in semiconductor lasers in the presence of optical feedback. IEEE J. Quantum Electron. 1984, QE-20, 350–357. Olesen, H.; Henrik, J.; Tromborg, B. Nonlinear dynamics and spectral behavior for an external cavity laser. IEEE J. Quantum Electron. 1986, QE-22, 762–773. Macleod, H.A. Thin-film Optical Filters; Adam Hilger: Bristol, UK, 1986.

Copyright © 2005 Marcel Dekker, Inc.

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16.

Smith, D.; Baumeister, P. Refractive index of some oxide and fluoride coating materials. Appl. Opt. 1979, 18, 111–115. Ebbesen, T.W.; Lezec, H.J.; Ghaemi, H.F.; Thio, T.; Wolff, P.A. Extraordinary optical transmission through sub-wavelength hole arrays. Nature 1998, 391, 667. Chang, R.K.; Campillo, A.J. Optical Processes in Microcavities; World Scientific, 1996. Raether, H. Surface Plasmons on Smooth and Rough Surfaces and on Gratings; Springer-Verlag: Berlin, Heidelberg, 1988. Ritchie, R.H. Surface plasmons in solids. Surface Science 1973, 34, 1–19. Ruppin, R. Surface effects on optical phonons and on phonon-plasmon modes. Surface Science 1973, 34, 20–32. Economou, E.N. Surface plasmons in thin films. Phys. Rev. 1969, 182, 539–554. Ngai, K.L. Interaction of ac Josephson currents with surface plasmons in thin superconducting films. Phys. Rev. 1969, 182, 555–568. Burke, J.J.; Stegeman, G.I.; Tamir, T. Surface-polariton-like waves guided by thin, lossy metal films. Phys. Rev. B 1986, 33, 5186–5201. Evans, D.J.; Ushioda, S.; McMullen, J.D. Raman scattering from surface polaritons in a GaAs film. Phys. Rev. Lett. 1973, 31, 369–372. Takahara, J.; Yamagishi, S.; Taki, H.; Morimoto, A.; Kobayashi, T. Guiding of a one-dimensional optical beam with nanometer diameter. Opt. Lett. 1997, 22, 475–477. Berini, P. Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures. Phys. Rev. B 1999, 61, 10484–10503. Lamprecht, B.; Krenn, J.R.; Schider, G.; Ditlbacher, H.; Salerno, M.; Felidj, N.; Leitner, A.; Aussenegg, F.R. Surface plasmon propagation in microscale metal stripes. Appl. Phys. Lett. 2001, 79, 51–53. Charbonneau, R.; Berini, P.; Berolo, E.; Lisicka-Shrek, E. Experimental observation of plasmon-polariton waves supported by a thin metal film of finit width. Opt. Lett. 2000, 25, 844–846. Weeber, J.-C.; Krenn, J.R.; Dereux, A.; Lamprecht, B.; Lacroute, Y.; Goudonnet, J.P. Near-field observation of surface plasmon polariton propagation on thin metal stripes. Phys. Rev. B 2001, 64, 045411. Ferguson, R.E.; Wallis, F.R.; Chauvet, G. Surface plasma waves in the noble metals. Surface Science 1979, 82, 255–269.

17.

18. 19. 20. 21. 22. 23. 24. 25. 26.

27.

28.

29.

30.

31.

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7 Fundamental Dynamics of Micro-Optomechatronics

1

DYNAMICS OF MICROSIZED OBJECTS

In optical micromechatronics, the S/N ratio is raised and the system configuration is simplified by the introduction of space movement. However, since processing speed is decided by mechanical positioning time, speed is slow compared with a solid-state element, and it tends to become a bottleneck on the speed of the whole information system. Therefore improvement of the processing speed is strongly required for optical micromechatronics apparatus, and it is necessary to lighten the weight of a movable part and to raise its natural frequency. Both are achieved by the miniaturization of mechanisms. For this reason, it is necessary to understand the dynamics of microsized objects to design optical micromechatronics apparatus, of which the fundamental theory is described in this chapter. Generally, when a moving object is small, surface force dominates volume force. As shown in the example of a rolling ball, Fig. 1, there are air flow resistance, solid friction, and surface tension in the surface force. In micromechatronics, there are cases where they are used positively, or they become performance prevention factors. Examples of both are shown in Table 1. An example of using air flow force positively is a magnetic disk slider. The flying slider is geometrically similar to a jumbo jet flying several mm above the ground. Such critical movement becomes possible since the slider is smaller than the jumbo jet by about 100,000 times in length, so the viscous force (surface force) of air becomes large compared with weight (volume force). That is, although they are similar geometrically, they are not similar dynamically. As examples of performance prevention by air flow force, there is the damping of the tapping mode of a probe sensor (e.g., a 225

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Chapter 7

Figure 1 Suface forces working on a microsized object.

SNOAM: scanning near field optical and atomic force microscope) [1] and the oscillation in crystal oscillators. If damping increases, the Q factor decreases, and measurement resolution of the resonance frequency decreases. As an example of the usage of solid friction, there is the oscillating motor for the calendar display of a wristwatch [2], and as an example of prevention, there is the stick–slip in a micropart assembly. As an example of the usage of surface tension, there is the optical switch that moves refractive-index watching oil using heat capillarity [3], and as an example of prevention, there is the adsorption of the SNOAM probe to the measured surface. In the following sections, the influence of air flow resistance and friction to a microsized object is explained for a cantilever, which is the simplest movement mechanism. The mechanics of materials for beams, the hydrodynamics of the surrounding air, the air resistance that works on the oscillating beam combining both, and the movement of the cantilever under friction force are described.

2 2.1

EQUATION OF MOTION OF THE BEAM Dynamic Models of the Beam

Let us consider a cantilever beam made from homogeneous material and of which the section is rectangular. When a force is applied to the beam, as shown in Fig. 2a, elastic deformation of bending (Fig. 2b) and shearing (Fig. 2c) appear. Bending is a deformation in which a cross section vertical to the centerline of the beam keeps the right angle. The reaction force against bending is caused by elastic compression zd along the beam axis. Shearing is a deformation caused by change of angle between a cross section

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Table 1 Examples of Useful Effect and Performance Prevention of Surface Force in Micromechatronics Example of useful effect

Example of performance prevention

Airflow resistance

Follow-up positioning by the flying slider

Q-value decrease of oscillator sensor

Drive of oscillating motor

Decrease of positioning accuracy in assembly of miniature components

Optical-path change in optical switch

Adsorption of SPM head

Electrostatic actuator

Adsorption of dust

Friction

Surface tension

Electrostatic force

and the centerline of the beam. The reaction force is caused by shearing force. Movement of the beam results in translational movement (Fig. 2d) and rotation (Fig. 2e), and they cause inertial force and moment, which are proportional to mass and moment of inertia, respectively. When the beam is sufficiently long and narrow, error is negligible even if we ignore the shearing force (Fig. 2c) and the rotational inertia (Fig. 2e).

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Figure 2 Deflection and movement of beam. (a) Beam deflection. (b) Bending. (c) Shearing. (d) Translational motion. (e) Rotation.

The dynamic model of such a beam is called the Euler–Bernoulli beam. On the other hand, the dynamic model that considers all factors (Fig. 2b–e) is called the Timoshenko beam. To understand bending and shearing forces and translational and rotational inertia intuitively, let us express these beams as many-degree-offreedom systems. Figure 3 shows the Euler–Bernoulli beam expressed as such a system. In this model, T-shaped rigid members are joined to the neighboring ones via pivots, and there are springs at both ends of the member. When the beam is bent, restitutive force is caused by the elasticity of the springs. The center axis of the beam keeps a right angle to sections of the beam and shearing deformation does not appear. The mass of a member concentrates at the center of the member and does not cause rotational inertia. Figure 4 shows the Timoshenko beam expressed as a many-degree-offreedom system [4]. The T-shaped member is divided into two members, and

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Figure 3 Discrete model of Euler–Bernoulli beam.

Figure 4 Discrete model of Timoshenko beam. (From Ref. 4.)

they are joined so that rotation is free. There are springs kb, which produce reaction force proportional to the gap between the vertical members and the springs ks, which produce reaction force proportional to the rotation between the vertical members and the horizontal members. The spring kb expresses the bending rigidity, and ks expresses shearing rigidity. The mass of the beam is distributed over vertical members and causes translational and rotational inertia. In this chapter, we will analyze the motion of the beam using the Euler–Bernoulli model. The equation of motion of the beam can be derived using various theories of dynamics. Then we explain the following three methods: the balance of forces, the energy principle, and the limit of the many-degree-of-freedom system.

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2.2

Chapter 7

Derivation of the Equation of Motion of Beam Using Force Balance

First of all, let us obtain the reaction force generated by the strain of the beam shown in Fig. 2a. From the longitudinal balance of forces, the centerline of the beam becomes a neutral axis that never expands or contracts. If the neutral line deforms in the arc of curvature of radius r as shown in Fig. 2b, the following strain " is caused at upper and lower parts of the neutral line: "¼

zd’ z ¼ dx r

ð1Þ

This strain generates a reaction force caused by compression at the upper part of the beam and the stretch at the lower part of the beam. They cause a moment that puts the beam back. The magnitude of the moment M around the neutral axis is obtained by integrating the product of the distance from the neutral axis z by the stress (the product of the strain " and Young’s modulus E) with respect to the cross section of the beam: Z EI E"z dA ¼ M¼ r A ð2Þ Z z2 dA

I¼ A

A represents the cross section of the beam. I is called the geometrical moment of inertia and is a constant determined by the shape of the cross section of the beam. For a beam with a rectangular cross section, I is given by I ¼ bh3/12. On the other hand, the displacement w and the curvature radius r have the relation 1 @2 w ¼ r @x2

ð3Þ

From Eqs. (2) and (3) we derive the relation between the magnitude of the moment M and the displacement w: M ¼ EI

@2 w @x2

ð4Þ

Now let us find the moment caused by the external force on the beam member. Figure 5 shows a beam on which the distributed loading p is acting. In this case, the moment M at x is given by the product of the loading acting

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Figure 5 Distributed loading and moment acting on a beam.

on the right side of x and its distance from x: Zl MðxÞ ¼ pðÞð  xÞ d

ð5Þ

x

where  is the position at which the loading is acting. Differentiating Eq. (5) by x gives the formula Zl @M ¼ pðÞ d ¼ QðxÞ ð6Þ @x x Since the integral in the middle of Eq. (6) shows the total of the loading acting on the right side of x, Q(x) expresses the shearing force acting on the cross section at x of the beam. Differentiating Eq. (6) by x gives @2 M ¼ pðxÞ ð7Þ @x2 If the beam remains stationary, the moment by the strain and the moment by the external force are balanced and M in Eq. (4) is equal to M in Eq. (5). Therefore, by substituting Eq. (4) into Eq. (7), the relation between the pressure acting on the beam and the displacement of the beam is derived: @4 w ¼p ð8Þ @x4 If the beam moves, the inertial force is added to the distributed loading. When only the vertical translational movement is considered, the force of inertia per unit length is EI

@2 w @t2 Thus the equation of motion of the beam is p ¼ A

A

@2 w @4 w þ EI 4 ¼ p 2 @t @x

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ð9Þ

ð10Þ

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where p on the right side shows the external force except for the inertial force. Equation (10) can be solved by using proper boundary conditions and initial conditions. There are several boundary conditions such as fixed, free, simply supported, translational movement free/angle restricted, and so on. Table 2 shows their mathematical expressions. In the case of the cantilever beam, the displacement and inclination are zero at the fixed end (x ¼ 0), and the moment M and shearing force Q are zero at the free end (x ¼ l ). Its boundary conditions are given as wð0Þ ¼ 0

ð11Þ

@wð0Þ ¼0 @x

ð12Þ

@2 wðl Þ ¼0 @x2

ð13Þ

@3 wðl Þ ¼0 @x3

ð14Þ

Table 2

Boundary Conditions of Beam

Physical image

Equation

Fixed

w¼0 @w ¼0 @x

Free

@2 w ¼0 @x2

@ @2 w EI 2 ¼ 0 @x @x

EI

Simply supported

wð0, tÞ ¼ 0 EI

Displacement free angle fixed

Source: Ref. 5.

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@2 w ¼0 @x2

@w ¼0 @x

@ @2 w EI 2 ¼ 0 @x @x

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Since Eq. (10) is a fourth-order differential equation, these four equations are the complete boundary conditions. The initial conditions are the displacement w and the velocity @w/@t at t ¼ 0. 2.3

Derivation of the Equation of Motion Using the Energy Principle [5]

The equation of motion of the beam is also found by using the energy principle. In the case of a simple beam such as the Euler–Bernoulli beam, use of the force balance is easier for analysis. In the case of more complicated systems, in which rotation and translational movement are mixed, or electrostatic and electromagnetic force exist, use of the energy principle facilitates analysis of the system. There are various expressions for the energy principle, and Hamilton’s principle is suited for the motion of the beam. Hamilton’s principle expresses that the integration of W (increase of work done by external force)  T (increase of the kinetic energy) þ U (increase of the potential energy) with time is zero for an arbitrary minute displacement w around the actual displacement w, provided that w is chosen as w ¼ 0 or @(w)/@x ¼ 0 at the point on which the displacement or rotation is restricted by boundary conditions. In the case of the Euler–Bernoulli beam, the kinetic energy T is 2 Zl 1 @w A dx ð15Þ T¼ 2 @t 0 The potential energy U is given by integrating E"2/2 throughout the beam. Equations (1), (2), and (3) give

Z l Z h=2 1 2 E" b dz dx U¼ 0 h=2 2 2 2 Zl 1 @ w EI ¼ dx ð16Þ @x2 0 2 W, the work done by the external force p, determined by the minute displacement w, is Zl W ¼ p w dx ð17Þ 0

From Hamilton’s principle, Z t2 ðT  U þ WÞ dt ¼ 0 t1

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ð18Þ

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Chapter 7

The symbol  can be treated as an operator and has a characteristic similar to differentiation. For example, we have the relation w2 ¼ 2w w. It also can be exchangeable with differentiation. For example, we have (@w/@x) ¼ (@/@x)(w). If we use these properties, and repeat the integration by parts of Eq. (18), and use boundary conditions such as w(0) ¼ w(0) ¼ 0 and @w(0)/@x ¼ @(w(0))/@x ¼ 0, we obtain

Z t2  Z l @4 w @4 w @3 w A 4  EI 4 þ p w dx þ EI 3 ðlÞ wðlÞ @x @x @x 0 t1

 2 @ w @w ðlÞ dt ¼ 0 ð19Þ EI 2 ðlÞ @x @x Since w, w(l ), and (dw(l )/dx) are arbitrary, we find A

@2 w @4 w þ EI ¼p @x2 @x4

0<x
ð20Þ

@2 w ¼0 @x2

x¼l

ð21Þ

@3 w ¼0 @x3

x¼l

ð22Þ

Equation (20) expresses the equation of motion of the beam. Equations (21) and (22) express boundary conditions at the free end of the beam. If we add the boundary condition w(0) ¼ @w(0)/@x ¼ 0 at the fixed end of the beam, Eq. (20) will equal the equation of motion obtained in the previous section. Equations (21) and (22) are derived from the energy principle automatically and are called natural boundary conditions. 2.4

Derivation of the Equation of Motion Using a Many-Degree-of-Freedom System

The equation of motion of the beam is also derived by making the equation of motion for the many-degree-of-freedom system shown in Fig. 3 and by letting the number of the members be infinite. Let number of the members be n. The number is 1 at the fixed end of the beam and n at the free end of the beam. First of all, let us determine the spring constant k, the mass m, and the external force Fi of the system. The mass and the external force are set equal to those of the original beam in a unit length. Fi ¼ pi
x

ð23Þ

m ¼ A
x

ð24Þ

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where pi is the value of p at the center of the member i. We determine the spring constant so that the strain energy of the many-degree-of-freedom system is equal to that of the original beam for the length
x ¼ l/n. We consider that the strain along the original beam is uniform at the interval
x where the bending shown in Fig. 2b is acting. The strain is derived by letting the deformation at the upper part of the beam ui be as shown in Fig. 3. 2zui ð25Þ "¼ h
x The strain energy Ub of the beam at the interval
x is Z h=2 1 2 2EI 2 E" b dz ¼ u ð26Þ Ub ¼
x
x i h=2 2 On the other hand, because the expansion of a spring at the member i is ui as shown in Fig. 3, the strain energy of a spring in the many-degree-of-freedom system is 1 Ui ¼ ku2i ð27Þ 2 There are two springs in a member. Letting Ub ¼ 2Ui gives 2EI ð28Þ k¼ 2 h
x The spring constant can also be derived by noting that the moment acting on a member i of the system equals the moment acting on the original beam. Let us find the equation of motion of the system. We call the displacement of the center of a member wi, the angle of the member i, and the expansion of the upper spring ui. First of all, the strain energy of the spring i is Ui. Here Un ¼ 0, because the spring at the free end of the beam does not expand. Moreover, because rotational inertia is neglected, the member at the free end does not generate a moment, and the spring at member n-1 does not expand (Un1 ¼ 0). From geometrical relations, h ð29Þ ui ¼ ðiþ1  i Þ 2 wiþ1  wi i ¼ ð30Þ
x are obtained. From the above, the total strain energy of the system becomes

n2 n2 X kh2 X wiþ2  2wiþ1 þ wi 2 U¼2 Ui ¼ ð31Þ 4 i¼1
x i¼1 Next let us find the kinetic energy. We assume that i is minute and that the rotational motive energy of each member can be neglected (hypothesis of the

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Euler–Bernoulli beam). Then the kinetic energy of the member i is 1 ð32Þ Ti ¼ mw_ 2i 2 where w1 ¼ w2 ¼ 0 because the member at the fixed end of the beam does not move or rotate, and therefore T1 ¼ T2 ¼ 0. The total kinetic energy of the system becomes n X 1 mw_ 2i ð33Þ T¼ 2 i¼3 By using the Lagrangian method, the equation of motion of the system can be found from U and T. The Lagrange function L is defined as L¼TU

ð34Þ

and the equation of motion is

d dL @L ¼ Fj j ¼ 3 to n  dt dw_ j @wj

ð35Þ

where j ¼ 3 to n because w1 ¼ w2 ¼ 0 and the degree of freedom of the system is n to 2. By using Eqs. (31) and (32) on T, U, of Eq. (35), the following equation is obtained from j ¼ 3 to n  2:  kh2  mw€ j þ wjþ2  4wjþ1 þ 6wj  4wj1 þ wj2 ¼ Fj j ¼ 3 to n  2 2 4
x ð36Þ For j ¼ n  1 and n, the following equations stand. mw€ n þ

kh2 ðwn  2wn1 þ wn2 Þ ¼ Fn 2
x2

mw€ n1 þ

j¼n

ð37Þ

kh2 ð2wn þ 5wn1  4wn2 þ wn3 Þ ¼ Fn1 2
x2

j¼n1 ð38Þ

The number of unknown quantities is n  2 (w3 to wn), and the number of equations is n  2 [Eqs. (36), (37), and (38)]. These are the equations of motion of the many-degree-of-freedom system. Next let us find the equation of motion of the beam by using the limit
x ! 0. Applying Eqs. (23), (24), and (25) to Fj, k, m in Eq. (26) gives Aw€ j þ EI

wjþ2  4wjþ1 þ 6wj  4wj1 þ wj2 ¼ pj
x4

j ¼ 3 to n  2 ð39Þ

The fraction at the second term of the left side of Eq. (39) is transformed to (see page 237)

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ð40Þ

Fundamental Dynamics of Micro-Optomechatronics

wjþ2  4wjþ1 þ 6wj  4wj1 þ wj2
x4 wjþ2  wjþ1 wjþ1  wj wjþ1  wj wj  wj1 wjþ1  wj wj  wj1 wj  wj1 wj1  wj2    
x
x
x
x
x
x
x
x  
x
x
x
x 
x
x ¼
x

237

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238

Chapter 7

The term (wiþ1  wi)/
x is the finite differential expression of @w/@x. Equation (40) is the result of repeating that four times and is the expression of @4w/@x4. Thus the limit
x ! 0 gives the differential equation A

@2 w @4 w þ EI ¼p @t2 @x4

ð41Þ

This agrees with the equation of motion found in Secs. 2.2 and 2.3. Now let us study the remaining equations. Applying Eqs. (23), (24), and (27) to Eq. (37) gives
x2 Aw€ n þ

EI wn  2wn1 þ wn2 ¼
x2 pn 2
x2

ð42Þ

In the limit
x ! 0, the first terms on the left and right sides disappear and the second term on the left side becomes the differential expression @2w/@x2. Thus we obtain @2 w ¼0 @x2

x¼l

ð43Þ

Applying Eqs. (23), (24), (27), and (41) to Eq. (38) gives
xAðw€ n1 þ w€ n Þ  EI

wn  3wn1 þ 3wn2  wn3 ¼
xðpn1 þ Pn Þ
x3 ð44Þ

In the limit
x ! 0, the first terms on the left and right sides disappear and the second term on the left side becomes the differential expression @3w/@x3. Thus we obtain @3 w ¼0 @x3

x¼l

ð45Þ

Equations (35) and (36) agree with the boundary conditions at the free end of the beam in Sec. 2.2, i.e., the natural boundary conditions. Four boundary conditions are determined by adding a boundary condition at fixed end of the beam (w1 ¼ 1 ¼ 0). 2.5

Solution for Free Vibration and Eigenmode

In this section we will obtain the free vibration solution where the external loading p is zero in the equation of motion (10) as the simplest example. We assume a solution w as wðx, tÞ ¼ ðxÞTðtÞ where  and T are unknown functions.

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ð46Þ

Fundamental Dynamics of Micro-Optomechatronics

239

Substituting Eq. (46) into Eq. (10), we obtain 1 d2 T d2  ¼ a2 4 ¼ !2 2 T dt dx

ð47Þ

where sffiffiffiffiffiffiffi EI a¼ A

ð48Þ

The value of Eq. (47) does not depend on both x and t, because the left side and the middle do not depend on x and t, respectively. Therefore we put this value as an unknown constant value !2. We attain the two following differential equations about t and x from Eq. (47). d2 T þ !2 T ¼ 0 ct2

ð49Þ

d4   k4  ¼ 0 dx4

ð50Þ

where k2 ¼

! a

ð51Þ

That is, by assuming the solution of Eq. (10) in the form of Eq. (46), one partial differential equation was resolved to two ordinary differential equations. The general solution of Eq. (49) is given by T ¼ A0 cos !t þ B0 sin !t

ð52Þ

Equation (50) is a fourth-order linear differential equation with a constant coefficient, and its solution is  ¼ esx

ð53Þ

After substituting Eq. (53) into Eq. (50), we obtain s4  k4 ¼ 0

ð54Þ

Because the solutions of Eq. (54) are s ¼ k, ik, the general solution of Eq. (50) becomes  ¼ C01 eikx þ C0x eikx þ C03 ekx þ C04 ekx

ð55Þ

Equation (55) is rewritten as Eq. (58) by using trigonometric and hyperbolic functions when we use the relation of Eqs. (56), (57) and rewrite arbitrary

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240

Chapter 7

constants C0i : ei ¼ cos  þ i sin 

ð56Þ

e ¼ cosh  þ sinh 

ð57Þ

 ¼ C1 cos kx þ C2 sin kx þ C3 cosh kx þ C4 sinh kx

ð58Þ

Therefore the general solution of Eq. (10) becomes w ¼ ðC1 cos kx þ C2 sin kx þ C3 cosh kx þ C4 sinh kxÞ
ðA cos !kx þ B sin !kxÞ

ð59Þ

Next, let us obtain the solution that corresponds to the boundary condition of the cantilever. Since Eqs. (11)–(14) must stand regardless of t, the equations should stand when w is replaced by . We obtain the following equations when substituting Eq. (58) into (11) and (12): C1 þ C3 ¼ 0

ð60Þ

C2 þ C4 ¼ 0

ð61Þ

We obtain the following equations when considering these and computing Eqs. (13) and (14): C1 ð cos  cosh Þ þ C2 ð sin  sinh Þ ¼ 0

ð62Þ

C1 ðsin  sinh Þ þ C2 ð cos  cosh Þ ¼ 0

ð63Þ

where ¼ kl. Since either C1 or C2 is not zero, the coefficient determinant of C1, and C2 for Eqs. (62), (63) must be zero. ðcos þ cosh Þ2 þ sin2  sinh2 ¼ 0

ð64Þ

Equation (64) is simplified to cos cosh þ 1 ¼ 0

ð65Þ

There exist many solutions for Eq. (65). If they are denoted as 1, 2, . . . in ascending order, they are given as

1 ¼ 1:875 . . .

2 ¼ 4:694 . . .

n  ð2n1 Þ=2

n 3

An unknown constant ! is decided for each i. Constant !i corresponds to i,s, sffiffiffiffiffiffiffi

2i EI ð66Þ !i ¼ 2 A l

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241

The parameter !i is called the ith mode natural frequency or an eigen angular frequency. Because A and I are proportional to the square and the 4th power of the length, respectively, when a beam is proportionarily enlarged or miniaturized, !i is inverse by a quantity proportional to the length. That is, the smaller the beam is, the bigger the natural frequency becomes. We conclude that the positioning time is reduced as the beam becomes smaller. Next, we obtain . Equations (62) and (63) become subordinate for the

that was obtained above, and C2 is derived as follows: C2 ¼ i C1 cos i þ cosh i i ¼ sin i þ sinh i

ð67Þ

When C3 and C4 are obtained from Eqs. (60), (61) and are substituted into Eq. (58),  becomes i ¼ C1 ðcos ki x  cosh ki x  i ðsin ki x  sinh ki xÞÞ,

ð68Þ

where  corresponding to i is denoted as i. The function i is called the ith eigenmode or eigenfunction. The functions i and Ti obtained above are solutions of Eq. (10) for the arbitrary number i. Because Eq. (10) is linear, the superposition of Tii also becomes a solution. Thus the general solution of the free vibration of the cantilever beam becomes wðx, tÞ ¼

1 X

Ti ðtÞi ðxÞ ¼

i¼1

1 X

ðAi cos !i t þ Bi sin !i tÞ

i¼1


ðcos ki x  cosh ki x  i ðsin ki x  sinh ki xÞÞ

ð69Þ

where

i ki ¼ l

1 ¼ 1:875

2 !i ¼ 2i l

sffiffiffiffiffiffiffi EI A

2 ¼ 4:694

i ¼

cos i þ cosh i sin i þ sinh i

i ð i 3Þ 

ð2n1 Þ 2

Here Ai and Bi are Ai ¼ A0i Ci and Bi ¼ B0i Ci ; these are decided by the initial conditions. The number of Ai and Bi is infinite, whereas the number of initial conditions is only two: speed and displacement. It appears that the number of conditions is not enough to determine the unknown parameters. Actually, Ai and Bi can be uniquely determined, because the displacement and the speed are the functions of x and are given at an infinite number of points. For example, if the initial condition is

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242

Chapter 7

w(x, 0) ¼ cos k1x  cosh k1x  1(sin k1x  sinh k1x), the parameters are determined as A1 ¼ 1, B1 ¼ 0, A i ¼ Bi ¼ 0(i ¼ 2, 3, . . .). The eigenfunction i is orthogonal. Since i satisfies Eq. (50), the following equation stands for the two modes i and j: d4  i ¼ k4i i dx4

ð70Þ

d4  j ¼ k4j j ð71Þ dx4 Multiplying the first formula by j and the second by i, integrating them from 0 to l, and subtracting them from each other, we obtain

Z l 4  Z l d i d4  j 4 4 i j dx ¼ j 4  i 4 dx ki  kj ð72Þ dx dx 0 0 Integrating the right side of Eq. (72) by parts gives  3 l d i d3 j dj d2 i di d2 j þ j 3  i 3  dx dx dx dx2 dx dx2 0

ð73Þ

In the case of the cantilever beam, i and j satisfy the boundary conditions i ð0Þ ¼ j ð0Þ ¼ 0 di dj ð0Þ ¼ ð0Þ ¼ 0 dx dx d2 i d2  j ðlÞ ¼ ðlÞ ¼ 0 ð74Þ dx2 dx2 3 3 d i d j ðlÞ ¼ 3 ðlÞ ¼ 0 3 dx dx Therefore, Eq. (73) becomes zero. Since the condition ki 6¼ kj stands if i 6¼ j, the following equation is obtained from the left side of Eq. (72): Zl i j dx ¼ 0 i 6¼ j ð75Þ 0

By multiplying Eq. (70), by j, integrating from 0 to l, and using Eq. (75), we obtain Zl d4 i j 4 dx ¼ 0 i¼ 6 j ð76Þ dx 0 Equations (75) and (76) are called orthogonality of eigenmode functions. These relations stand not only for cantilever boundary conditions but also for any boundary conditions, such as simple support. We use orthogonality in Sec. 4.

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Fundamental Dynamics of Micro-Optomechatronics

3 3.1

243

FLUID DYNAMICS AROUND MICROSIZED OBJECTS [6,7] Basic Equation of Fluid Flow

Since air consists of gas molecules, its movement has to be analyzed, strictly speaking, based on gas molecular dynamics. However, when an airflow around a beam with a size of about 100 micrometers is analyzed, the error is small even if the particle nature and also compressibility are neglected. Molecular dynamics is required for strict analysis of more minute sizes. For example, Ref. 8 gives its description. The movement of air is determined by the equation of motion (77), called the Navier–Stokes equation, showing the relation between pressure and flow velocity, and the equation of continuity (78), showing air flowing without a break, if the air is regarded as a continuous fluid and is incompressible: rp ¼ 
v  a ðvErÞv þ a r Ev ¼ 0

@v @t

ð77Þ ð78Þ

In both equations, the flow velocity v and the pressure p are unknown functions. The boundary conditions are as follows: pressure and flow velocity are zero at infinity, and the flow velocity coincides with the velocity of the beam on the beam surface. We explain the meaning of the Navier– Stokes equation (77) briefly. The left side expresses the pressure gradient in the fluid, and the right side shows that it is generated by change of the flow velocity. The first term on the right side shows the pressure gradient caused by viscous force, the second term shows that by the stationary inertial force, and the third term shows that by the oscillating inertial force. The oscillating inertial force is generated by the oscillating fluid like a sound wave; it is proportional to acceleration and equivalent to the inertial force of the vibrating solid. The stationary inertial force corresponds to centrifugal force in the movement of a solid; it is proportional to the square of the velocity and the curvature of the streamline. Even if the flow does not change in time, the change of velocity (direction or magnitude) along the streamline generates acceleration in each fluid molecule, and inertial force appears. For example, the centrifugal force working to the outside shore of the winding river is this force. Viscous force is a force that a fluid receives from the adjoining streamline according to the speed difference between streamlines; it is equivalent to the shearing force in an elastic body. Equation (77) is difficult to solve because it is a nonlinear equation. The magnitude of each term is estimated on condition that the beam is small, and the formula is simplified. A coordinate system is taken as shown

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244

Chapter 7

Figure 6 Coordinate system of cantilever.

in Fig. 6. First, the maximum flow velocity is calculated. This is about the same amplitude as the oscillating speed of the free end of the beam from the boundary conditions on the surface of the beam. As a typical example, the beam is assumed to be a rectangular parallelepiped of length l, width b ¼ l/10, and thickness h ¼ l/100, and whose material is silicone (Young’s modulus 155 Gpa and density b ¼ 2300 kg/m3). Its oscillating amplitude at the end is a ¼ l/1000. On this condition, the first mode natural frequency is given by sffiffiffiffiffiffiffi 1:8752 EI 83:2 ffi rad=s ð79Þ !¼ A l l2 The unit of l is m here. Since the maximum speed at the end of the beam v0 is the product of that and the amplitude a, it is given by v0 ¼ a! ¼ 8:32
103

m=s

ð80Þ

Next, the differential of v is estimated. In the flow around a minute beam, the flow velocity change from the maximum to zero in the distance about the width of the beam in both the thickness direction and the width direction. Therefore the differentiation of v0 about y and z becomes of the same order as v0 multiplied by 1/b. Moreover, since the oscillation amplitude is small compared with the beam length, the flow velocity change in the x direction is neglected. That is, the following approximations can be used for the differential operation: @ @2 ffi 2ffi0 @x @x

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ð81Þ

Fundamental Dynamics of Micro-Optomechatronics

245

@ @ 1 ffi ffi @y @z b

ð82Þ

@2 @2 1 ffi ffi 2 2 2 b @y @z

ð83Þ

If v is replaced by v0 and the differential is replaced by the upper formula in Eq. (77), the amplitude of each term of the right side can be estimated. The ratio of the first term to the second term, Re, is given by   a ðvErÞEv a v0 b ffi ffi 600l ð84Þ Re ¼ k
vk  The notation k k expresses the order of the norm of a vector. The parameter Re is called the Reynolds number. It expresses the ratio of viscous force to stationary inertial force. The Reynolds number is an important numerical value for hydrodynamics, and we explain its meaning briefly. Inertial force will become dominant if the Reynolds number is large. On the other hand, viscous force will become dominant if the Reynolds number is small. It is proportional to the objective representative length b and the speed v0 and is inversely proportional to the kinetic viscosity /a of a fluid. When two steady-state fluid systems (oscillating inertial force is zero) are considered, if the object shapes are similar and the Reynolds numbers are equal, the Navier–Stokes equation, the equation of continuity, and the boundary conditions of each system can be made equal by transforming the coordinates. This is applicable to a model experiment. For example, considering the case where an enlarged (or miniaturized) experiment is performed using the same fluid, since the kinetic viscosities are equal, if objective speed is decreased (increased), the same streamline is obtained and a similar experiment can be made. As another example, in order to make an enlarged experiment without changing speed, what is necessary is just to use a fluid with a higher kinetic viscosity. Moreover, the driving mechanism of living things and machines underwater can be qualitatively explained from the consideration of Reynolds numbers. If Re  1, it is advantageous to use inertial force, and the typical example is reaction force when water is pushed out. The jets of cuttlefish, the fillets of fish, the oars of a boat, and the screws of a vessel use inertial force. If Re  1, it is advantageous to use viscous force, and the typical example is shearing friction (rubbing movement) with water. There are flagella and cilia of microbes, etc., as examples. The Reynolds number of artificial and living things is calculated from objective length and speed, and these are shown in Ref. 9.

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246

Chapter 7

Let us return to the movement of a beam. In Eq. (84), Re becomes less than 1 for l < 1.6 mm, and it turns out that in a cantilever with a length of about 100 mm the second term can be neglected. Next, Rv, the ratio of the first and the third terms, is given by   a @v=@t a !b2 ffi Rv ¼ ffi 60,000l ð85Þ k
vk  The parameter Rv is called the oscillating Reynolds number or Valensi number; it expresses the ratio of viscous force to oscillating inertial force. Since Rv becomes larger than 1 for l > 16 mm for the usual cantilever, it turns out that the first and third terms are almost the same magnitude. Thus the equation that is obtained by neglecting the second term of the right side of Eq. (77) becomes the basic formula of minute vibration. This formula is called the Stokes equation: rp ¼ 
v þ a

3.2

@v @t

ð86Þ

Approximate Analysis by the Bead Model

The Stokes equation is linear for the unknown functions p and v. It is simpler than the Navier–Stokes equation, but an analytical solution cannot be obtained for the boundary condition of the vibrating rectangular parallelepiped. So another model is proposed. In this model, an analytical solution can be obtained and the fluid force is almost the same as in the previous model. It is possible to get an analytical solution for a vibrating sphere for both the Stokes equation and the equation of continuity [10]. The vibrating speed of the sphere is denoted as U ¼ U0 ei!t. When the actual vibration is U0 cos !t or U0 sin !t, the real part or imaginary part should be used in the solution. The direction of the flow velocity at a point in a plane that includes the vibration center and perpendicular to the vibration direction is parallel to vibration direction. The magnitude of the flow velocity is given by [7]  



 1   2 3R ½ðrRÞ=ð1þiÞ  þ 2 þ e þ þ u¼ i U r 2r 2r2 2r3 2 r3 sffiffiffiffiffiffi 2 ð87Þ ¼ !



R3 3 R3 3 32 þ ¼ 1þ i 2R 2 2 2R 2R2

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Fundamental Dynamics of Micro-Optomechatronics

247

where r is the distance from the center of the sphere and R is sphere radius. The fluid force that works on the sphere has only the vibrating direction component. The proportional part of the speed is given by 



 R 2R R F ¼ 6U 1 þ i 1þ ð88Þ  9  When sphere speed U is eliminated from Eqs. (87) and (88), the relationship between flow velocity and the fluid force is determined. u ¼ dF

  1 1=r þ =r2 þ 2 =r3 er= þ 2 =r3   d¼ 6R ð1 þ R=Þ  ið1 þ ð2RÞ=ð9ÞÞðR=Þ 1þi  ¼ 2 3R R= 1 ¼ e 2

R3 3 3 2 þ 2 1þ 2 ¼  R 2 R

ð89Þ

Next we need an approximate solution of the fluid force working on the beam, by using the fluid force of the sphere. Consider a flow around a plate placed in a uniform flow. When the Reynolds number is large (when the flow is fast), the streamline has a complicated shape as in Fig. 7a. But when the Reynolds number is small (when the flow is slow), the viscous force exceeds it, and the streamline becomes smooth as in Fig. 7b. In this case, stagnant flow areas exist in the vicinity of the plate, and the flow behaves as if the plate had become fat and round. The streamline becomes close to that of a flow around a cylinder. When streamlines around two objects are equal, the right sides of Eq. (86) of the two flows become equal, and the pressures on the left side also become equal. Therefore when the Reynolds number is low (slow flow), fluid forces of the plate and the cylinder become close. Additionally, in

Figure 7 Relationship between Reynolds number and streamline. (a) Re  1. (b) Re  1. (c) Cylinder.

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248

Chapter 7

Figure 8 Bead model.

the chain of spheres (the beads) and the cylinder, fluid resistance is close, so the plate resistance is replaced by the resistance of the beads (Fig. 8). As will be shown below, the fluid forces of the plate and the bead are calculated from the fluid forces of the single sphere. It is also possible to obtain the fluid force of the plate from that of a cylinder, although the calculation becomes complicated. It is shown in Ref. 11 that the calculated forces obtained from the flows of the beads and the cylinder are very close. Next we express the flow around the beads by using the linear sum of the flow of the single sphere. The flow in Eq. (89) is superposed for all spheres. Because Eqs. (86) and (78) are linear, the superposed flow satisfies them. Since at an infinitely far point, both the velocity and the pressure of the superposed flow become zero, the boundary conditions at infinity are satisfied. Therefore if the superposition satisfies the boundary condition of the sphere surface (i.e., velocity of the fluid ¼ velocity of the sphere), the flow of the superposition equals the flow of the beads. But the superposition cannot satisfy boundary conditions all over the sphere surface. This is a contradiction caused by the defect of our method, which expresses the solution of the partial differential equation, which has infinite degrees of freedom, by a finite number (the number of spheres) of known functions (flow around a sphere). Therefore the boundary condition is approximately satisfied by satisfying it at only one point on each sphere’s surface. As the point at which it is satisfied, we adopt the contact point of the spheres, where calculation becomes easy. That is, the flow in Eq. (89) is weighted and added to by the number of the spheres, and the weight factor is decided so that the velocity of the flow at the contact point equals the velocity of the sphere. Equation (89) includes the unknown fluid forces F, so F is used as a weight factor in order to satisfy the boundary condition. Both the number of F and the number of points that satisfy the boundary condition are the number of the spheres, so all the Fs are decided uniquely. Then the simultaneous equation regarding the fluid forces Fj working to the sphere j is

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Fundamental Dynamics of Micro-Optomechatronics

249

obtained as ui ¼

N X

dij Fj

i ¼ 1N

ð90Þ

j¼1

where the dij is d in Eq. (89) where r is set to the distance between the sphere j and the point i (the contact point of spheres i and i þ 1), Ui is velocity of the point i, and N is the number of the spheres. The relationship between the velocity of the beads and the fluid forces is decided by Eq. (90). It generally expresses the fluid force that works on each sphere when many spheres vibrate. Using this formula, for example, we can also analyze vibration couplings of two beams by the airflow by replacing two beams with two sphere chains [7]. When the number of beams is one, Eq. (90) is further simplified. Under the conditions of the beam that is used in the evaluation of the Reynolds number in the previous section, if the length of the beam is more than 37 mm, the velocity of the flow in Eq. (87) decreases to less than 1/10 of the sphere velocity at the distance of sphere diameter away from the sphere surface. Therefore, in most microcantilevers, the series in Eq. (90) should consider only the spheres adjoining the contact point. Furthermore, when the deformation of the beads is smooth, i.e., the differences of the velocities between the adjoining spheres are small, Eq. (90) can be simplified to ui ¼ 2d11 Fi

ð91Þ

This shows that the fluid force Fi, which works on each sphere of the beads, becomes half of the force working on a single sphere [first formula in Eq. (89)]. If the beam is long and slim, and the number of spheres in the beads is very large, Fi can be considered as distributed continually, and the fluid force of Fi/b can be considered to work per unit length of the beam. By the above discussion, an approximate solution of the fluid forces was obtained.

4 4.1

MOVEMENT OF THE BEAM WITH AIR RESISTANCE Vibration for Sinusoidal Input

We derive a dynamic equation with fluid force, by combing the equation of motion of the beam and the equation of air resistance that we derived in previous sections. According to the bead model, the fluid force Fi/b ¼ w_ /2d11b (w_ is the velocity of the beam) works on the beam per unit length. When we include this force in the equation of motion of an

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250

Chapter 7

undamped beam, we can derive the dynamic equation of the beam with air resistance: 0 S

@2 w  @w @2 w þ EI þ ¼ fei!t @t2 2b @t @x4

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3  ¼ 3b þ b2 2 ! 4

ð92Þ

Here, w, I, x, and f stand for the displacement of the beam, the moment of inertia of the area, the position in the longitudinal direction, and the amplitude of the external force, respectively, and  stands for the real part of 1/d11. The reason we take only the real part of the air resistance is that the imaginary part works only as the force in proportion to the accelerated velocity (additional mass) and is small enough to be ignored compared with the inertial force of the beam. The first term on the left side stands for the inertial force, the second term for the fluid resistance, and the third term for the rigidity of beam; the right side stands for the external force. The external force is given as a complex number, so that the calculation becomes easy. We can take the real part of w when the external force is f cos !t, and we can take the imaginary part when the force is f sin !t. The partial differential equation (92) can be solved by mode expansion. First, we obtain solutions for the free vibration of the undamped beam. They can be given as !n (natural frequency) and n (eigenmode) in Sec. 2. The function n has orthogonality as shown here: Zl i j dx ¼ 0 i 6¼ j Z

0 l

d4  j i 4 dx ¼ 0 i 6¼ j dx 0 Z Z d4  i !2i 2i dx ¼ i 4 dx dx

ð93Þ

These were derived in Sec. 2.5. Mode expansion is a method of deriving the solution w of Eq. (92) as a linear combination of n. Because Eq. (92) is not that for free vibration, n is not a solution. But the superposition of n can become the solution. Let wn be an unknown time function, and let w be w¼

1 X

wn ðtÞn ðxÞ

ð94Þ

n¼1

We substitute Eq. (94) into Eq. (92), multiply both sides by n, and take Eq. (93) into consideration; thus we obtain mn

d2 wn dwn þ kn wn ¼ fnei!t þ cn 2 dt dt

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ð95Þ

Fundamental Dynamics of Micro-Optomechatronics

251

Here Z

l

mn ¼ b bh 0

2n dx

mn 2b bh2 kn ¼ mn !2n Zl fn ¼ fn dx cn ¼

ð96Þ

0

Equation (95) has the same form as the equation for a single-degree-offreedom system under forced vibration; the steady-state solution is wn ¼ Gn ð!Þfn ei!t Gn ð!Þ ¼

!2

ð97Þ

1 1 1 ¼ 2 þ cn i! þ kn kn 1  ð!=!n Þ þ 2in !=!n

mn cn n ¼ 2mn !n

ð98Þ

The parameter n is called the damping ratio of the nth mode. The parameter that represents the fluid force is , and this appears only in cn of Eq. (96). We can understand that the fluid force is affected as the damping ratio to the vibration of the beam. When we substitute Eqs. (97) and (98) into Eq. (94), we can derive the steady-state response with the sinusoidal input for the beam in fluid as w ¼ ei!t

1 X

fn Gn ð!Þn ðxÞ

ð99Þ

n¼1

4.2

Damping Ratio of Microsized Beam

As mentioned in the previous section, air resistance eventually results in a damping ratio. Thus we will explain the physical meaning of the damping ratio. We also explain other factors for vibrational damping and describe how an actual beam vibrates. First, we assume that the frequency of the external force ! is located near the natural frequency of the nth mode !n. Because n  1, as we will explain later, among the of Eq. (99), the nth term is overwhelmingly larger than the other terms. Therefore, when the external frequency ! is close to !n, we can approximate as follows: w ffi ei!t fn Gn ð!Þn ðxÞ

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ð100Þ

252

Chapter 7

That is, when ! ffi !n, the movement of the beam is the same as that of a single-degree-of-freedom system. Thus we describe the meaning of damping in a single-degree-of-freedom system below. We consider that the external force F(t) ¼ fn sin !nt acts on the mass as shown in Fig. 9. If we denote the displacement of the mass by x(t), x is given as the imaginary part of wn as follows: x¼

1 fn cosð!n tÞ 2n kn

ð101Þ

The maximum kinetic energy W of the mass in one cycle is given by

1 fn 2 f2 ¼ 2n ð102Þ W ¼ mn !n 2 2n kn 8n kn On the other hand, the work that the external force does during one cycle
W is the same as the energy consumed by cn, and is given by

2 Z 2=!n I I !n fn cn sinð!n tÞ dt
W ¼ cn x_ dx ¼ cn x_ 2 dt ¼ 2n kn 0 ¼

f2n cn ! n

ð103Þ

The ratio of the two energies is
W ¼ 4n W

ð104Þ

Thus the physical meaning of n is given by the relative energy consumption at the resonant frequency divided by 4p. When we want to maintain a

Figure 9 Vibration system with a single degree of freedom.

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certain vibrational amplitude with as little input energy as possible, like that of a quartz crystal, n should be smaller. Next we assume that the beam vibrates freely without external force. When we take the initial conditions appropriately, the movement of the system becomes as follows: x ¼ a0 e"t cos !d t qffiffiffiffiffiffiffiffiffiffiffiffiffi !d ¼ !n 1  n2

ð105Þ

" ¼ !n n If we take the ratio of neighboring peak heights as shown in Fig. 10, it becomes ai ¼ e"T ffi 1 þ 2n ð106Þ aiþ1 where we assumed n  1. Equation (106) means that n is given by the amplitude decaying ratio in free vibration divided by 2p. In positioning controlling, when we want to remove the residual vibration as fast as possible, we should set n as large as possible. Next we study the relationship between static deflection and resonance amplitude. Static deflection xst caused by an external force fn is given by xst ¼

fn kn

ð107Þ

The ratio of resonance amplitude xres and static deflection is xres ¼ 2n xst

Figure 10

Relationship between peak height and  in damped vibration.

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ð108Þ

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That is, n can be defined as the ratio of resonance amplitude and static reflection divided by 2. When we want to take as large an amplitude as possible, as in a vibrational gyroscope, n should be made as small as possible. Finally, we consider the meaning of n in the frequency domain (Fig. pffiffiffi 11). When we derive the frequency ! where the amplitude becomes 1= 2 of the resonance amplitude from Eqs. (97) and (98), they are given by ! ¼ !n ð1  n Þ

ð109Þ

The ratio of difference of these frequencies
! and resonance frequency !n is !n 1 ¼ ¼Q
! 2n

ð110Þ

This value shows the sharpness of the resonance peak and is called the Q-value. This formula is used for obtaining the damping ratio n experimentally, and the experimental method is called the half-bandwidth method. When the resonant frequency changes, the amplitude becomes larger as the Q-value is larger. Thus the resonant frequency change is measured more accurately as the Q-value is larger. In the sensors that use resonant frequency change such as gas sensors and SPMs (scanning probe microscopes), the sensitivity improves as n becomes small.

Figure 11

Q and  in frequency region.

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Figure 12

255

Vibration damping factors.

There are other damping factors for cantilevers, supporting-part loss and internal friction (Fig. 12). As these factors depend on the vibration type and supporting method, they are difficult to analyze theoretically as for air resistance. For both of them some approximation methods are known, and we will explain typical methods. As for internal friction, there is the structural damping theory. According to this theory, when the beam vibrates harmonically, internal friction is in proportion to the amplitude of strain of the beam, and its phase is 90 degrees delayed by distortion. In Sec. 2, I(@4w/@x4) was a term proportional to the deflection force obtained by integrating the strain (z@2w/@x2) in the cross-sectional area. When the beam vibrates harmonically as sin !t, the vibration that is delayed by 90 degrees is cos !t; it is given by (1/!)@(sin !t)/@t. Thus the equation of the motion of the beam is

4 @2 w  @ @ w þ1 ¼ fei!t ð111Þ b bh 2 þ EI @t ! @t @x4 The parameter  is called the structural damping coefficient and is around 105 for aluminum and 104 for glass; more detailed values are shown in Ref. 12. If we apply the mode expansion method to Eq. (111) as to air resistance, we can derive a damping ratio for each mode. As a result, the damping ratio becomes, independent of mode number and beam shape, ¼

 2

ð112Þ

For supporting-part loss, in such as bolted connections, where a tiny slide occurs, we must obtain the damping ratio by experiment for each case. However, the damping ratio can be theoretically derived when the beam and the base are integrated as one elastic object. For micro-oscillators, where the beam is formed by etching from bulk material, this condition is usually

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satisfied. In this case, the base deflects elastically, and the vibration damping occurs by the dissipation of kinetic energy through base vibration. When the width of the base is equal to that of the beam and the volume of the base is much larger than that of the beam, we can get the analytical solution of the energy loss by using the two-dimensional theory of elasticity. The amount of work the beam does to the base is given approximately as follows. The details of this derivation are shown in Ref. 13.
W 2:9h3 ¼ 3 W l

ð113Þ

By applying the relation between energy loss and damping ratio, we can derive the damping ratio by supporting-part loss, ¼

0:23h3 l3

ð114Þ

The damping ratio in an actual beam is the total of air resistance, internal friction, and supporting-part loss. Calculated results of each damping ratio in the first mode are shown in Fig. 13, where the beam is made of silicon and its shape is a proportional parallelepiped with length l, width l/10, and thickness l/100. Internal friction depends only on the material and is around 105. Supporting-part loss is so small, around 107, that it is not included in the figure. These damping ratios do not depend on the size. On the other hand, air resistance becomes larger as the beam becomes smaller; it is in

Figure 13

Length and damping ratio of silicon beam.

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Figure 14

257

Length and damping ratio of mild steel.

proportion to the 1 to 0.5 power of the length. This is because, in Eq. (98), the natural frequency is in inverse proportion to the length. In this figure, air resistance is always dominant. The damping ratio of the beam made of mild steel is shown in Fig. 14. Because mild steel has a large structural damping coefficient, when the beam length is more than several mm, the internal friction is dominant; otherwise the air resistance is dominant. When the beam length is more than several mm, the damping ratio is almost constant, and when the length is shorter than that, the damping ratio increases rapidly as the length of the beam decreases. It becomes advantageous for positioning control to make the structure smaller, because positioning accuracy increases with the natural frequency and the damping increase as mentioned in Sec. 4 of Chap. 2.

5

STICK–SLIP CAUSED BY FRICTION FORCE [14,15]

As a minute object slides on a solid surface, the influence of friction becomes important. Especially when the difference between the static friction and the kinetic friction is large, stick–slip vibration occurs and the error of positioning increases. In this chapter, as the simplest example of the

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Figure 15

Single-degree-of-freedom model of sliding mechanism.

assembly of optical parts, we study the case wherein the optical fiber of cantilever is positioned on a glass substrate, which corresponds to intercmittent positioning. The problem that positions an optic fiber on a glass plate (e.g., a waveguide substrate) is modeled by the single-degree-of-freedom system in Fig. 15 when the frequency of the driving force is much smaller than the second-resonance frequency of the fiber. The slider is connected to the point P with a spring and a dashpot, and P is driven at a constant velocity. The slider is pushed to the base by gravity, spring-back force, or electrostatic force. Here we regard the coordinate system fixed to P and study the model in which the base moves at a constant speed. In this model, the slider moves as shown in Fig. 16a. At first the slider is pulled by the static friction force from the base (stick: 0 < t < t1), and then it is pulled back by the spring-back force (slip: t1 < t < t2). As it becomes damped vibration, the friction force to the slider changes to kinetic friction force from static friction force. The equation of motion and the initial condition in this period are mx€ þ cx_ þ kc ¼ k p x0 ¼

s p  cv k

ð115Þ

x_ 0 ¼ v

where m is mass, k is the spring constant, c is the damping coefficient, v is the velocity of the base, p is the pressing force, k is the coefficient of static

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Figure 16 Dynamic characteristic of slider. (a) Movement of slider in stick–slip condition. (b) Movement of slider in smooth condition.

friction, s is the coefficient of kinetic friction, and x0 and v0 are the displacement and velocity of the slider at the moment of starting the relative motion. At the moment when the slider velocity changes to the base velocity, the friction force changes to a static friction force, and the slider is caught by the base. Then the slider begins to move at the same velocity as the base. These motions are called stick–slip. Therefore the condition of generating stick–slip is whether the relative velocity between the slider and the base becomes 0 or not. If it does not become 0, the slider shows the usual damped vibration and ends up in the equilibrium position of the spring-back force and the kinetic friction force (Fig. 16b). The stick–slip condition is obtained by calculating the slider velocity from Eq. (115) and judging whether the maximum velocity exceeds the base velocity. It is given as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð116Þ  2 þ v2cr > vcr ðs  k Þp pffiffiffiffiffiffiffi  2vcr  ¼ eð3=2þ’Þ mk rffiffiffiffi v  k c cr ’ ¼ arctan ¼ !¼ m 2m!  ¼

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Figure 17

Chapter 7

Generation of stick–slip in phase plane.

where vcr is the maximum base velocity that generates stick–slip and   1 is assumed. Let’s consider the condition of generating stick–slip in the phase plane whose horizontal axis represents the displacement of the slider and whose vertical axis represents the velocity of the slider (Fig. 17). At first, the slider stays at x ¼ 0 (A) and moves in the þx direction by the velocity v (A ! B) caught by the base. When it reaches the point B, which is the equilibrium position of the spring-back force, the damping force, and the static friction force, the slider takes off the base. It begins damped free vibration and moves in the phase plain in a spiral shape (solid line of B ! C). When the spiral reaches the point C, which represents x_ ¼ v, and the relative velocity changes to 0, the slider is caught by the base again. It thus repeats the motion C ! A ! B. If the damping of the spiral is large and the maximum velocity x_ (point D) does not reach v, the slider is not caught by the base and the spiral goes to the point E (dotted line B ! D ! E). Since the condition of generating stick–slip depends on whether the spiral crosses the line of x_ ¼ v or not, the larger the diameter of the spiral is, that is, the larger the distance between the point B and the point E is [i.e., (s  k)P larger, k smaller, c smaller], and the smaller the damping of the spiral is (i.e.,  smaller), and the nearer the line of x_ ¼ v is to the center of the spiral (i.e., v smaller), the easier the stick–slip occurs. Let us study the maximum velocity vcr of generating stick–slip when the system is miniaturized proportionally. In the same system, the smaller v

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is, the easier is the generation of stick–slip. So in the system that has a large vcr, it is easy to generate stick–slip. When Eq. (116) is calculated under the condition   1, vcr becomes as follows. When p is in proportion to L2 ( p is due to the spring-back force), vcr has no relation to L. When p is in proportion to L3 ( p is due to gravity), vcr increases with L. If p is in proportion to less than L1 ( p is due to electrostatic force), vcr decreases with L. Therefore, in the miniature object, as the electrostatic force is dominant, vcr increases with miniaturization of the system and it becomes easier to generate stick–slip. Next the comparison of positioning error between movements with and without stick–slip is explained. Here we assume that the coordinate system is fixed to the base; the base is fixed and the point P moves (Fig. 15). The point P is driven at a constant velocity and then stops suddenly. The difference between the displacement of the slider and that of P is defined as the error of positioning. First, if stick–slip is not generated, the constant

Figure 18 stick–slip.

Positioning error of sliding optical fiber. (a) Without stick–slip. (b) With

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kinetic friction force works to the slider. If m and c are small, the friction force equals the spring-back force, and the positioning error is pk/k. Second, if stick–slip is generated, the static friction and the kinetic friction appear in turns. So the error depends on the timing of stick–slip, and the maximum value of error is ps/k (maximum static friction force ¼ springback force). Generally, as s is larger than k, the maximum error of positioning is large when stick–slip is generated. Figure 18 shows examples of positioning error in the case that an optical fiber is slid on a glass rod. One end of the fiber is driven with constant velocity; the other end is pressed to the rod, and the displacement of the pressed point is measured. As the pressing force P is given by the spring-back force, p/k equals z (z is pressed height). The error of non-stick– slip is kz (a) and the error of stick–slip case is a little less than sz (b). Therefore it is verified that the error increases by stick–slip. REFERENCES 1. 2.

3.

4.

5. 6.

7. 8.

9. 10. 11.

Ootsu, M.; Kawata, S. Near-field Nanophotonics Handbook; Optronics, 1997; in Japanese. Iino, A.; Kotanagi, S.; Suzuki, M.; Kasuga, M. Development of ultrasonic micro-motor and application to vibration alarm analog quartz watch. Advances in Information Storege Systems 1999, 10, 263–273. Sato, M.; Shimokawa, F.; Inegaki, S.; Nishida, Y. Micromechanical intersecting waveguide optical switch based on thermo-capillary. NTT R&D 1999, 48 (1), 9–14. in Japanese. Crandall, S.H.; Karnopp, D.C.; Kurtz, E.F. Jr.; Pridmore-Brown, D.C. Dynamics of Mechanical and Electromechanical Systems; Robert E. Krieger: Malabar, Florida, 1982. Williams, J.H., Jr. Fundamentals of Applied Dynamics; John Wiley: New York, 1997. Hosaka, H.; Itao, K. Theoretical and experimental study on airflow damping of vibrational microcantilevers. Trans. ASME, J. Vibration and Acoustics 1999, 121, 64–69. Hosaka, H.; Itao, K. Coupled vibration of microcantilever array induced by airflow force. Trans. ASME, J. Vibration & Acoustics 2002, 124, 26–32. Fukui, S. Hardware technology for information equipment—molecular gas film lubrication for magnetic disk storage. J. Japan Soc. Precision Eng. 1996, 62 (9), 1242–1246. in Japanese. Motokawa, T. Times for Elephants and Mice. Chuko-Shinsho, 1992; in Japanese. Landau, L.D.; Lifshits, E.M. Fluid Mechanics; Pergamon Press: London, 1959; 95. Hosaka, H. Study on airflow damping of microoscillator. Micromechatronics. 1998, 42 (3), 38–45. in Japanese.

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Fundamental Dynamics of Micro-Optomechatronics 12.

263

Lazan, B.J. Damping of Materials and Members in Structural Mechanics; Pergamon Press: New York, 1968. 13. Jimbo, Y.; Itao, K. Energy loss of a cantilever vibrator. J. Horological Inst. Japan. 1968, 47, 1–15. in Japanese. 14. Suzuki, T.; Itao, K. The micro-motion and micro-positioning mechanism design of micromechanical information devices. Advances in Information Storage Systems 1999, 10, 249–261. 15. Hosaka, H.; Nagaki, N.; Suzuki, T.; Itao, K. Vibrational positioning method for optical fibers sliding on a frictional surface. Microsystem Technologies 2002, 8, 244–249.

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8 Novel Technological Stream Toward Nano-Optomechatronics

1

THE COMING OF NANOTECHNOLOGY [1]

The twentieth century was the era when communications, information (processing and memory), and sensing technology made remarkable progress. The technology is characterized by microscience based on quantum mechanics for semiconductors. A sophisticated informationoriented society was established, supported by such microtechnology. By the end of the century, microtechnology had reached maturity. High-speed data transmission and processing was the first priority in technological development. Such development was carried out with various optical elements, typically including semiconductor lasers, planar light wave circuits, and logic elements including microprocessors and memories. Another technological trend of the century was large-capacity data storage realized by high-density optical disk systems even including a rewritable DVD. In the information-sensing field, various precision measurement technologies were developed. In the twenty-first century, we expect a new technical stream toward nanoscaled mechatronics and biotechnology. As shown in Table 3 of Chap. 1, we have various technological sprouts in the field of optical micromechatronics, which are expected to grow into new technological trends specified by physics measured on a nanometer scale. These new trends are represented by nanomechatronics, characterized by wide scope including nanomachines, nanocontrolling, and nanosensing, and we expect bionanomechatronics to develop. Figure 1 shows a road map of the new technological stream. In the field of data and telecommunications, improving the capacity of fiber-optic transmission systems is a continuous project to develop versatile internet 265

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Figure 1 Trend to optical nanotechnology.

services; we want to realize an ultrafast signal processing technology. In future, a processing speed of hundreds of gigahertz will be necessary, and conventional methods that use photodiodes for detecting light signals suffer the intrinsic speed limitation imposed by existing electronics, Hence it is necessary to make a breakthrough to overcome this limitation. A new concept of manipulating photons based on the interaction between an electromagnetic field and matter may be a promising candidate for ultrafast signal processing. This new concept includes the idea of controlling a single photon by using the interaction between an electromagnetic field and an electron bound to atomic states in a small cavity. The fundamental principle is based on electromagnetic dynamics in cavities. We all expect much improvement in signal processing speed based on such a novel principle dealing with the single photon and the electron. Although we are far from the goal, we have many useful hints for innovation including existing fine nanoscale structures and related devices such as quantum wires and dots. In the field of information processing, we also need a major breakthrough

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in transmission speed. Key processing devices for ultrafast communication systems described above are also useful in information systems. In the field of information memory, further improvement in memory density is necessary for large-capacity data storage. Various researches are being carried out for realizing a terabyte optical memory; they include a trial to obtain a minimal beam spot with a diameter even in the nanometer range. To accomplish this purpose, the evanescent field attracts great interest to eliminate the diffraction limit [3] that determines the minimal data mark size in conventional optical disk systems. In the sensing field, atoms and molecules are observed and manipulated. Optical trapping technology was developed to manipulate cells. The key device is an optical tweezers [4], and remote control systems have been realized and used for various biological examinations including cell fusion for genetic recombination. This technology has developed into a more sophisticated one, which can manipulate atoms and molecules. Figure 2 shows a typical example. A DNA is labeled with a minute polystyrene sphere using an enzymatic reaction and manipulated by laser trapping in a thin water butt 30–40 microns in thickness. The dynamic properties of DNA can be investigated using flowing liquid [5] to obtain an effective force applied to the molecules of DNA. This technology will be more sophisticated and will manipulate molecules of protein and contribute to the fields of bioscience and medical science characterized by molecular biology, biochemistry, immunology, and so on. In future, mechanisms of unknown viruses will be clarified to make antibodies to them. In the field of sensing and measurement, it is also important to observe objects whose sizes are much smaller than wavelength. Optical microscopes

Figure 2

Molecule manipulation technology in biotechnology field. (From Ref. 5.)

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are quite popular, but their spatial resolution is limited by diffraction. Scanning probe microscopes including scanning near field optical microscopes (SNOMs) eliminate this limitation, enabling observation of objects much smaller than wavelength scale. As can be seen in the above, all technological streams in various fields seem to be bound for nanotechnology. These trends are not independent but interactive. Manipulation technology characterized by optical tweezers for molecules and atoms is also a key to fabricating fine structures useful for ultrafast signal processing. Near field technology developed for observing micro- and nanostructures is now promising for next-generation ultra-high-density optical data storage. This technology leads to ultimate large-capacity information memories realized by extremely small memory cells consisting of atomic or molecular clusters. The interesting fact that nanotechnology is a key in each field mentioned above seems to have an important intrinsic meaning for technology at the turning point. We consider that all these trends are supported by optical nanomechatronics, which connects the nanotechnical world with actual human society.

2

NANO-OPTOMECHATRONICS FOR OPTICAL STORAGE

One of the important application fields of nano-optomechatronics technology is that of optical recording. In this section, we describe nano-optical memory based on near field optics as a recent research topic on ultrahigh-density optical storage. Over the last few decades, optical recording technology has made great progress. However, we need higher recording densities to meet a growing demand for large-capacity storage devices that can be used in multimedia applications. It will be difficult to improve recording densities. If the recording mark size is in the range of nanometers, conventional optical systems cannot be used. Hence near field recording as a novel scheme is most promising. This scheme has no limitation imposed by diffraction, but extremely precise positioning is required: in near field recording, the optical head must be set very close to the recording medium under a constant spacing of several tens of nanometers. Thus we must newly construct a control scheme for recording systems based on mechartonics. In this section, we describe, the optical first surface recording based on nano-optomechatronics and near field optics, which is expected to be a breakthrough in this field.

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2.1

269

Background of Near Field Optical Recording

The recording density of existing optical disk memories is now maximal, because practical low-cost lasers at wavelengths shorter than 400 nm are not absolutely promising, while LEDs at near ultraviolet wavelengths are commercially available. Hence the density is limited according to the wavelength of existing lasers. As described in Chap. 6, the minimum spot size is given by d ¼ 1:22

0 NA

ð1Þ

where d is the spot size, 0 the wavelength of laser light in vacuum, and NA is the numerical aperture of the focusing lens (Fig. 3). This numerical estimation gives the minimum spot size under ideal conditions with no aberration. In typical cases, the mark length is about one-third of the spot diameter in conventional optical recording systems. NA is defined as n sin , where n is the refractive index of the material in which the light propagates, and  a half-cone angle of the condensing light beam. So as long as the optics is placed in the air, the theoretical maximal NA is about 1.0. Hence the ideal minimal mark length is estimated as 0.4 0. There are only two ways to minimize the spot size. Since further improvement of the NAs of lenses is extremely difficult, we have no choice except by shortening the laser wavelength.

Figure 3 Optical diffraction limit in conventional optical recording.

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The next-generation DVD systems adopt a blue-violet laser of

0 ¼ 405 nm and an objective lens of NA ¼ 0.85, so a minimal recording mark size is around 0.19 mm. It is difficult to reduce this value, because existing recording systems have many practical impediments to this effort. There are few transparent materials in the ultraviolet wavelength region except for some particular materials such as sapphire glasses: plastics widely used for practical disk systems may be not transparent owing to the photochemical reactions. This means that such an ultraviolet wavelength is not allowed in conventional optical recording systems using plastic materials for the optical system and the recording media. Increasing NA is also difficult, because comma aberration caused by the inclination of a disk surface proportionally increases as t/NA3 (where t is the thickness of the cover layer of the medium). In the next-generation DVDs, the thickness of the cover layer is 0.1 mm to achieve a high NA of 0.85. Since the defocus effect is smaller for a thinner cover layer, in other words, the influence of the surface contamination become serious, the recording density is limited to around 20 to 30 Gbit/inch2, assuming a minimum recording mark length of 150 to 200 nm according to the above estimation procedure. We consider that this is the limit of recording density for conventional optical recording technology, although the physical limit determined by the thermodynamic stability of the recording media is much higher. As long as a conventional object lens is used, there is no way to overcome the limit of the focused spot size caused by the diffraction effect. Instead of using a lens, near field recording uses a small aperture to make a small illumination spot on the recording medium surface. The spot size is not dependent on the wavelength but on the size of the aperture. Hence this scheme is free from the diffraction limit. As shown in Fig. 4, when focused light is introduced into the aperture having an extremely small diameter, under the diffraction limit, i.e., several tens of nanometers, almost all of the incident light beam reflects back. Particularly, when the aperture size is less than half the wavelength, no transmission mode can exist in the aperture, so all the light reflects and transmission is completely inhibited. In this condition, in the aperture, there exists an evanescent field, which exponentially damps from the surface. When the thickness of the metal film on the aperture side is equal to or less than the decay depth, the evanescent field reaches the opposite side of the metal film. Transmission from the aperture is strictly prohibited owing to the small aperture having a diameter of less than half the wavelength. However, when a high-reflective-index dielectric or metal particle is placed in this evanescent field, electric dipoles are excited by the field, and these dipoles radiate light in the free space.

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Figure 4 Principle of near field optical recording.

Near field optics using submicron-size aperture has already been put to practical use in the scanning near field optical microscope (SNOM). Under laboratory conditions, it is reported that resolution as small as 12 nm (1/43 of wavelength) is possible [6], and detection of light from a single molecule is also reported. This value of resolution corresponds to several hundred Gbit to Tbit/inch2. It should be noted that we have to be careful when we discuss the aerial density; it cannot be predicted only from the resolution, because the readout speed is also a critical property in storage applications as will be described later. But at least near field optics has the potential to realize Tbit-scale nano-optical storage. 2.2

Nano-Optomechatronics in Near Field Optical Recording

As described in the previous section, the principle of near field recording is simple, but the mechanical requirements are quite different from those of

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conventional optical recording. In this section, we discuss the nanooptomechatronics required in near field recording. First we have to know what is the critical factor that will determine the recording density of near field recording, and what type of recording device is suitable to take advantage of the characteristics of near field recording. The major factors that determine the performance of aperture-type near field recording can be summarized as follows. 1.

The resolution does not depend on the wavelength of the light but instead is determined by the aperture size. But when the aperture becomes small, the detectable optical power drops drastically, so the minimum aperture size depends on the minimum power required in signal detection. The aerial density is restricted by the optical transmittance of the aperture rather than the resolution limit. The wavelength of the light is also important. As described below, the wavelength of the light determines the decay length of the field amplitude, as well as the optical transmittance. A shorter wavelength has higher optical transmittance. Following is the theoretical result for radiation through a circular aperture in an infinitely thin, perfect conducting plate.



d

4 when

d

ð2Þ

where d is the diameter of the aperture, is wavelength, and

represents the transition coefficient, which is the ratio between the incident power per the aperture area and the transmission power. Therefore when the incident power density per unit area is constant, transmission power is proportional to d 6. Assuming an infinitely thin perfect conducting plate is not realistic. When d is smaller than a half of the wavelength, no propagating optical mode can exist in the aperture, and increasing the thickness of the plate, diminishes the transmission exponentially. For a more complete discussion on the transmittance of a small aperture, see the literature [7,8]. Of course the recording material also limits the recording density. In magneto-optical media, there exists a minimum volume of the recording mark to keep the magnetizing stable. A similar limitation also exists in the phase-change medium. But at least concerning several hundred Gbit/inch2 of aerial density, these limitations are not yet serious. For example, recording marks on magneto-optical media, in this range of aerial density, is

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demonstrated, and the potential of the recording medium is already confirmed. 2. Aperture-to-medium spacing should be minimized. Interaction between the medium and the near field light generated at the aperture decrease exponentially when the spacing becomes larger. Approximately, the decay length (1/2k) of near field light is given by 1

1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2kz 4 ð j =2d Þ2  1

ð3Þ

where is the wavelength in the material where the field exists, d is the aperture size, and j is the mode number ( j¼1 is most significant). For example, when ¼ 532 nm, d ¼ 220 nm and, the decay length is about 62 nm. When d is small enough compared to , the decay length is approximately 0.16d. Thus in near field recording, control of the head-to-medium spacing under several tens of nanometers is required. This means it is quite difficult to assure the removability of the medium. 3. In conventional optical recording, the track following is done by moving an objective lens, but in near field recording, the aperture itself should follow the track on the recording medium. This requires that the head containing the near field optics should be light and compact. Especially, the mechanical requirement, described above, is quite important for achieving high-speed access, more specifically, a high-speed surface scan rate, and precise tracking control. These requirements are similar to those of hard disk recording, so it is natural that the basic mechanical structure of the near field recording device is adapted from that of the hard disk. A flying slider mechanism mounted on a swing arm tracking mechanism driven by a voice coil actuator is considered to be the most probable mechanical architecture. From this perspective, near field recording can be regarded as a fusion of optical and magnetic recording technologies, with high spatial resolution beyond the diffraction limit, which is derived from the SNOM, and a high scanning rate with nanometer level spacing control, which is derived from hard disk drive. On the other hand, to achieve this fusion, we needed many new developments in nanooptomechatronics, for example, the mounting technique of an optical system on a millimeter-sized flying slider.

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In Fig. 5, the relationship between the technological components and the performance for aperture-type near field recording is summarized. The characteristic value required achieving 100 Gbit/inch2 aerial density is also indicated for each component. The data transfer rate is determined by the total transmittance of the aperture. In near field recording, the transmittance is decreased when the aperture is minimized, so high-sensitivity detection of the light is required to improve the recording density. In such cases, a major component of noise is shot noise. Therefore the signal-to-noise ratio (SNR) is determined by how many photons can be used to read out one bit of the recorded data. For example, to achieve 16 to 20 dB of SNR, which is a minimum requirement for digital signal transferring or processing, about 400 to 1000 of photons per bit is required (on condition of 10% quantum efficiency). To estimate the total output power from the aperture, this value is multiplied by the transfer rate. Forty to 100 nW (at 532 nm, and noise factor ¼ 1.3) of light power is required for a 100 MHz (200 Mbps) data transfer rate. It should be

Figure 5 Relationship between the technological components and the performance for aperture type near field recording.

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noted that the parameters used above have been estimated based on our demonstration system; it is not a general result. For example, the quantum efficiency may be too small. To improve the linear recording density, we not only need a minimization of the aperture size but also a reduction of head-to-medium spacing. Although this requirement is the same as in magnetic recording, the aspect ratio of the recording mark, which is nearly equal to 1 in aperturetype near field recording, is smaller than that in magnetic recording, in which the ratio is greater than 5, so the requirement for the spacing control, to achieve the same linear recording density, is higher than that in magnetic recording. To achieve 100 Gbit/in.2, which corresponds to a 70 nm mark size, under 30 nm of flying height is required. To improve the track density, we need a minimization of the recording mark size, a high resonance frequency of the head assembly to improve the accuracy of the tracking control, and a servo technology. In general, to improve the accuracy of the tracking control, the control bandwidth should be improved proportional to the one-half power of the preciseness. The control bandwidth is approximately one-sixth to one-third of the resonance frequency of the head assembly. A typical value of the tracking control is about one-tenth of the track pitch. Because the aspect ratio of the data mark of near field recording is smaller than that of magnetic recording, the requiement for the track density is more severe than in magnetic recording. To achieve 100 Gbit/in.2, which corresponds to a 100 nm track pitch, over 20 kHz of resonance frequency of the head assembly is required. To reduce the flying height, tribology between head and media, in other words, surface smoothness, wear toughness, and lubrication are quite important. At the same time, the minimization of the flying head slider is also a critical factor. The following capability of the flying slider to the high special frequency waviness is mainly dependent on the slider size. The glide height, which is the minimum average flying height without contact between the slider and the medium, is determined by the surface roughness (height of microprotrusion) in the microscale region and by microwaviness in the slider sized macroscale region. For reference, in magnetic recording, the height of microprotrusions on the polished medium decreases with improvements of polishing technique, but improvement of the microwaviness in the slider size scale is relatively small. Therefore to achieve several tens of nanometers, which has already been achieved in recent magnetic recording technology, the slider size of near field recording should be at least the same as that of a magnetic recording head. In Table 1, we see the size and the important mechanical properties of a 30% typical flying slider for magnetic recording (IDEMA standard), and the required value to achieve 100 Gbit/in.2 in near

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Table 1 Comparison Between 30% Flying Slide for Magnetic Recording and Mechanism Required for Aperture Type Near-Field Recording

Slider size Slider mass Track pitch Tracking error Resonance frequency of head assembly Tracking control bandwidth Flying height

30% slider (Pico slider) (2000 summer)

Near-field recording (required value for 100 Gbit/inch)

1.25
1.0
0.3 mm About 1.5 mg 1000 nm 100 nm 4 5 kHz

Same Same 100 nm 10 nm 20 kHz

About 700 Hz 15 nm

4 kHz 30 nm

field recording (indicated performance of the 30% flying slider is a value of the year 2000.

2.3

Head-to-Medium Spacing Control of Near Field Recording [9]

In this section, we discuss the head-to-medium spacing control of near field recording. First of all, the technique, which is used to produce a monolithic type head slider for near field recording, is described. This monolithic production of the optical head slider is a core issue both on the head-tomedium spacing control and on tracking control, because the size and weight of the head assembly are the most important characteristics, as explained in the previous section. An optical head slider produced using this technique is called a flexible optical head slider. Then for the monolithic type (flexible) flying slider, we discuss the spacing control and the air bearing design, which is peculiar to near field optical recording. The results of experimental evaluations of the prototypes are also shown. Figure 6 is a conceptual illustration of the flexible optical head slider. An air-bearing pad pattern is formed on the apex of a cantileverlike polymeric waveguide; with the cantilever itself serving as a suspension of the slider, the functions of the flying slider, suspension, and waveguide are all incorporated into one body structure. The optical waveguide itself works as a flying slider. The cantilever’s flexibility works as the slider suspension. This structure (a flexible optical head slider) can be expected to offer great advantages in miniaturizing head assemblies and simplifying both the

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Figure 6 The concept of the flexible optical head slider.

assembly and the optical trimming processes. Furthermore, the lightweight head assembly allows a wider tracking bandwidth. Laser light is introduced into the waveguide at the supported end of the cantilever and is delivered directly to the cantilever tip, where the submicron aperture is located. The end of the waveguide has been ground to an angle, so it works as a total reflection mirror that reflects the light toward the aperture and the recording medium. The aperture is fabricated using a focused ion beam (FIB) process, penetrating the shading metal film deposited on the air-bearing surface of the slider. The head slider is scanned over the recording medium at a velocity of several to several tens of meters per second, keeping constant the head-tomedium spacing of several tens of nm. Transmitted or scattered light power modulated by the recorded data pattern is detected by the photo detector on the opposite side of the recording medium. Except for the grinding process that forms the total reflection mirror, the entire process of fabricating this flexible optical head slider is a lithographic technique. The waveguide is formed on a silicone substrate, and the excellent flatness of the substrate is transferred to the slider pad surface.

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Unlike the case with hard sliders used in magnetic recording, the optical head slider is made of flexible material. This difference results in different kinds of constraints on slider design. First, we cannot presume the flatness of the slider air-bearing surface. The geometrical attitude of the air-bearing surface is affected by, for example, a change in pressure distribution, or a motion following the out-ofplane movement of the medium. Therefore we cannot apply the air-bearing designs of conventional polished ceramic head sliders, since these require several tens of nm proximity for an entire slider pad of several mm length. We should also consider deformation or warpage caused by changes in temperature or humidity, or generated in production processes. Second, the functionality of the suspension is limited (see Fig. 7). In the flexible optical head slider, the slider cantilever also works as a suspension, and it is difficult to design the ideal suspension function. In the conventional head slider assembly, a load beam suspension structure is applied; the head slider is supported by point contact and can be rotated freely. In contrast, it is difficult to decrease the spring constant of rotation for the suspension of the flexible slider, and there is a cross-term between the motions of rotation and out-of-plane translation. A compliance center is

Figure 7 Comparison of suspension structure. (a) Conventional suspention. (b) Flexible slider.

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located on the middle of the cantilever, so the out-of-plane disturbance of the slider directly affects the head slider as a pitching moment. The stiffness of the air film of the slider against rotational disturbance is not enough to support such a large pitching moment. The air bearing must therefore be designed to allow for the change in slider pitching angle. Considering the limitations mentioned above, we propose two types of slider design. The first, shown in Fig. 8a, is characterized by the large pitting angle of the slider pad and small air-bearing pad area. The large pitching angle of the slider reduces the influence of fluctuations in the pitching angle that arise mainly from out-of-plane disturbances. A small air-bearing pad area reduces the influence of rotational disturbance. The second design, shown in Fig. 8b, is characterized by the multiple and separated air-bearing pads. In this design, the slider consists of three airbearing pads, each of which forms an independent microcantilever. Two airbearing pads, located on the outer side of the slider, absorb a major portion of the disturbance and keep the center air-bearing pad parallel to the medium. In other words, the slider works as a dual-stage slider. The flexible optical head slider must be designed to accommodate a high frequency of the cantilever resonance mode, a large allowable range of out-of-plane disturbances, and warpage of material. The resonance modes of a dual-stage type cantilever are calculated using the finite element method (FEM). The lowest mode, with a 4.3 kHz resonance frequency, is the bending mode, and the lowest sway mode that determines

Figure 8 Two design strategies of air-bearing pad pattern for the flexible slider. (a) High pitch angle þ small slider area. (b) Multi and separated trailing edge (dual-stage slider).

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a tracking control band is around 8.8 kHz. Even though this design has not yet been optimized, these values are almost comparable to the resonance frequencies of commercially available suspensions used for magnetic recording. The allowable range of out-of-plane disturbances in the type A and type B prototype designs is about þ/2.5 mm and þ/10 mm, respectively. To evaluate the flying characteristics of the flexible optical head slider, we observed a flying attitude of prototype samples flying over a transparent glass disk using a laser interferometer. A sample slider is shown in Fig. 9.

Figure 9 Photograph of the flexible optical head slider.

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Figure 10

281

The interference fringe pattern observed on the air-bearing pad.

Figure 10 shows the interference fringe pattern observed on the air-bearing pad of the Type B prototype. The vertical interval of each fringe was about 190 nm. In this figure, the spacing at the trailing edge of the center cantilever was about 100 nm, and it was possible to reduce it to about 70 nm. From the fringe pattern, we can see that a large part of the center air-bearing pad kept its proximity to the medium. From the clarity of the fringe pattern, we can qualitatively see that the spacing control is stable. As a result of experimental evaluation, stable flying was verified at a flying height of under 100 nm. The suspension successfully absorbed about 5 mn of out-ofplane disturbance. To evaluate the dynamic response of the flexible optical head slider, the supporting point of the slider cantilever was shaken in an out-of-plane direction, and the fluctuation of spacing was observed. As a result, the frequency range of 1 kHz to 10 kHz—even at 4.6 kHz, which is the frequency of the second bending resonance mode—the amplitude ratio of the slider was under30 dB.

2.4

Continuous Tracking Error Detection for Near Field Recording [10]

Tracking control of 10-nanometer-order accuracy constitutes another challenging subject. For example, assuming 100 Gb/in.2 density requires a 70 nm data mark size and a 100 nm track pitch, thus tracking accuracy should be reached to approximately 10 nm, which is about one-sixth of required accuracy for today’s conventional optical disk storage. As described in Chap. 5, in regard to tracking error detection methods for near field optical storage, it is said that the sampled servo method is the

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most feasible, since other methods for conventional optical storage are based on the principle of optical diffraction or require detectors that are not applicable in the near field optics. In this section, another type of tracking control method, which is unique to aperture type near field recording, is described. In the conventional sampled servo method, as shown in Fig. 11a, servo zones are arranged along each track at fixed intervals. Each servo zone consists of displaced pits which are sifted half of the track pitch in the radial direction. When the head passes over these servo zones, a position error signal (PES) is obtained from the readout intensity of the displaced pit. The bandwidth of PES depends on the interval of the servo zones. In light of these phenomena, we propose a continuous tracking method. As shown in Fig. 11b, instead of using displaced pits, a dedicated aperture for tracking is placed on the edge of the data pits. As a result of interference between the near field and the data pits, scattered light is generated. By detecting this scattered light, we can obtain data readout information as a higher frequency component and the position error signal as a lower frequency component. This method is based on the uniqueness of near field optics in that the shape of the sensing spot can be arbitrarily controlled, unlike in conventional optics. For tracking error detection, high space resolution of the scanning direction is not needed, so an aperture shape with narrow radial width is adequate. Although the PES obtained using the method described above is also affected by the change of flying height, the signal level of the data readout

Figure 11

Schematic diagram of near field head submicron tracking system.

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Figure 12

283

Supposed spectrum of noise component from various sources.

head can be used to estimate the flying height and to compensate for this effect. In light of certain assumptions, its performance can be evaluated experimentally by the analysis of the SNR of the SNOM signal. The noise is assumed to contain the following major components, as shown in Fig. 12: (a) noise from media inhomogeneity, (b) noise from a laser light source (power drift, undesired oscillation), (c) noise from PMT and a current-tovoltage conversion system, and (d) noise from the measurement system. The noise generated by the mechanical vibrations of components such as the actuator, the flying head slider on which the optical head is loaded, and its supporting mechanism was neglected. Thus, as shown in Fig. 13, the tracking error (in this case, precision or uncertainty of PES) is determined by the total noise level. Now we can estimate the tracking error due to the noise component of PES based on the signal-to-noise ratio (SNR) of the PES. In the previous section we discussed the SNR, with signal level in that discussion corresponding to the difference between the signal level under the ontrack condition and on the middle of two tracks. Tracking error DEt is

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Figure 13

Estimation of tracking error owing to noise of PES.

written as DEt ¼

Nt t : DS 2p

ð4Þ

where t is track pitch, Nt is total noise (which is defined in previous section), and DS is the difference of signal levels as mentioned above. The result of tracking error estimation based on the method described above is that when the flying height is less than 50 nm, the tracking error would be less than 10 nm. The fiber probe used for this estimation is not designed as a tracking error detector; thus the spatial resolution is much too high, and the output signal level is relatively small. So the result should be regarded as an underestimation. In addition, as mentioned above, it is essential to consider the disturbance of the vibration created by the actuator and by the mechanism of the flying head slider. In summary, we could estimate the tracking error due to the noise derived mainly from the optoelectonic conversion system. In this simulative experiment, the optical efficiency of the head was relatively small compared with that of the planer or tapered type aperture; thus the result should be regarded as an underestimation. The results of the estimate for the specific condition were as follows: under the assumption of the use of a tapered fiber probe with an

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aperture size of 70 nm, a control bandwidth of 5 kHz, and a track pitch of 200 nm, the required tracking accuracy will be achieved if the flying height of the fiber probe can be set at 50 nm or less. On this condition, the tracking error, which arises from the sensor reading noise, will be about 10 nm. 2.5

Signal Detection and Readout Demonstration [11]

In this section, we discuss the signal detection and the detector-related mechanism. A readout demonstration using a flexible optical head slider is also shown. In near field optical recording, the optical power detected as a signal is quite different from that in conventional optical recording. In conventional optical recording, the power of the readout signal is of mW order. In contrast, the power detected in near field recording is of nW order. This is a fundamental characteristic of near field recording, which is derived from its principle. Instead of the diffraction effect, the optical transmittance of the miniature aperture limits the aerial density of the near field recording, thus the optical power of the readout becomes as small as possible. Then what is the minimum optical power required in signal detection? In such cases, the major component of the noise is shot noise. Therefore the signal-to-noise ratio (SNR) of the detection is determined by how many photon can be used to read out one bit of the recoded data. In other words, the minimum optical power is decided by the required SNR and the data transfer rate. For example, to achieve 16 to 20 dB of SNR, which is a minimum requirement for the digital signal transferring or processing, about 400 to 1000 of photons per bit is required (in condition of 10% quantum efficiency). To estimate the total output power from the aperture, this value is multiplied by the transfer rate. 40 to 100 nW (at 532 nm, and noise factor ¼ 1.3) of light power is required for a 100 MHz (200Mbps) data transfer rate. Based on the characteristics discussed above, what is the most probable light detection system for near field recording? In the following, one example we considered is described. The experimental setup for readout signal evaluation is shown in Fig. 14. The test medium (metal-patterned disk) was prepared by the procedure described here: a 40 nm thick chromium metal film and a 10 nm carbon overcoat were deposited on the 2.5 inch glass disk substrate for a commercial HDD. Then a line-and-space (L&S) pattern of various line widths, from 8 mm down to 0.35 mm, was formed using optical lithography

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Figure 14

Chapter 8

Experimental apparatus.

to etch the metal layer. The size of the patterned region was about 1 mm square. As shown in Fig. 14, each line width of the pattern consists of an 8 mm wide index line, an isolated single line, and the continuous L&S pattern. Next the patterned disk was coated with a 1 nm thick lubricant layer and burnished. Additionally, on a portion between the continuous and the isolated pattern, a 0.3 mm, a 0.2 mm, and a 0.15 mm wide L&S pattern was formed using FIB etching. The light source used was a 532 nm wavelength SHG laser, and light was introduced into the waveguide using a fiber focuser to focus the light on the input end of the waveguide core. As a detector, a photomultiplier tube (PMT) with a 0.4 NA objective lens was used.

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Figure 15 Photograph of the fabricated aperture. (a) Light emission from the aperture. (b) SIM image of the aperture.

In Fig. 15, the envelope of the readout signal observed when the slider passed over the entire pattern is shown. In this figure, the optical power is positive downward. Uniformity of the signal envelope indicates that the flying of the slider is stable and that the head-to-medium spacing is maintained almost constant while the slider passes over the 1 mm long patterned zone. Figure 16 shows the readout signals obtained from the experiment. The noise component of the readout signal can be predominantly attributed to shot noise in photoelectric conversion. The theoretical value of the signalto-noise ratio is 14 dB when we assume that the detected light power is 12 nW, that the bandwidth is 50 MHz, and that quantum efficiency is 10%. The SNR is proportional to the square root of the light power, so it can be improved by increasing input light power or by improvement of optical transmittance. 40 to 100 nW of light power is required to obtain 16 to 20 dB SNR at a 100 MHz (200 Mbps) bandwidth. The limiting factor of maximizing input power is thermal destruction of the aperture. For our prototype, the maximum power is around several tens of mW, and the power used in the experiment is already near this limitation. One of the promising ways to improve the optical transmittance is by the reduction of the waveguide core size. When the core size is reduced to 3 mm, the transmittance is expected to be increased 10 times, and the illuminated spot size becomes comparable to that of 0.22 NA focusing optics. Another possibility is adding a focusing feature to the flexible optical head slider. The total reflection mirror at the apex of the head can be modified to the focusing mirror. Or a lens-shaped air gap can be inserted near the end of the core. Another desirable improvement is to fill the gap in the aperture with material having a high reflective index (e.g., UV epoxy or

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Figure 16 Readout signal from the L&S pattern. (a) 0.15 mm. (b) 0.2 mm. (c) 0.35 mm. (d) 0.5 mm.

TiO2). Avoiding an exponential loss in the aperture, the optical transmittance can be improved. In the present case involving a 0.22 mm sized aperture of the 100 nm thick Ti shading metal, an approximately five fold improvement in the transmittance can be expected. Optimization of the aperture structure is also important to improve optical transmittance or heat dissipation. Next the miniaturization of the system including its detection optics should be discussed. In the demonstration described above, we used a conventional objective lens system and a photomultiplier tube (PMT) for signal detection; thus the size of the detection optics was larger than the head slider. In the following, the feasibility of the lensless detection system in which an APD sensor is directly located in close proximity to the medium is studied. In Fig. 17 a direct detection system is shown. Using a thin smalldiameter disk (1 inch, 0.25 mm thick), and reducing the thickness of plastic mold (0.1 mm), we expect the equivalent NA to be improved to about 0.6 or higher.

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Figure 17

3

289

Lensless detection using the APD device.

SUMMARY

In this chapter, we describe nano-optical memory based near field optics as a recent research on ultra-high-density optical storage. Through the overview of recent up-to-date technology, we can see that there are many unprecedented technological aspects to optomechatronics, including nanometer precision tracking, nanometer precision spacing control, and nanometer scanning mechanisms. To achieve the required performance of these mechanism, we had to propose novel construction methods of optical systems, and as with the flying slider mechanism, we even had to import technologies from another field. From micro- to nanoscale there is much room for expansion of the optomechatronics field, and continuous research effort is desired.

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Ishijima, A.; Kojima, H.; Tanaka, H. Application of nano-meter scale optics to biology. OYO BUTURI 1999, 68 (5), 556–560. in Japanese. 2. Baba, T. Semiconductor minute resonator and natural discharge control. Kotai Butsuri 1997, 32 (11), 859–869. in Japanese. 3. Betzig, E.; Harootunian, A.; Lewis, A.; Isaacson, M. Near-field diffraction by a slit: implications for superresolution microscopy. Appl. Opt. 1986, 25 (12), 1890–1900. 4. Sato, S.; Inaba, F. Cell processing by laser light. Laser Eng. 1992, 20 (11), 835–844. in Japanese. 5. Perkins, T.T.; Quake, S.R.; Smith, D.E.; Chu S. Relaxation of a single DNA molecule observed by optical microscopy. Science 1994, 264, 822–825.

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Betzig, E.; Trautman, J.K.; Harris, T.D.; Weiner, J.S.; Kostelak, R.L. Breaking the diffraction barrier: optical microscopy on nanometric scale. Science 1991, 251, 1468–1470. Bouwkamp, C.J. Diffraction theory, rep. Progress Phys. 1954, 17, 35–100. Roberts, A. Electromagnetic theory of diffraction by a circular aperture in thick perfectly conducting Screen. J. Opt. Soc. A. A, 1987, 4 (10), 1970–1983. Hirota, T.; Ohkubo, T.; Itao, K.; Yoshikawa, H.; Ando, Y. Air bearing design and flying characteristic of flexible optical head slider combining with visible laser light guide. Microsystem Technologies 2002, 8 (2-3), 155–160. Hirota, T.; Takahashi, Y.; Ohkubo, T.; Hosaka, H.; Itao, K.; Osumi, H.; Mitsuoka, Y.; Nakajima, K. Simulative experiment on precise tracking for high-density optical storage using a scanning near-field optical microscopy tip human friendly mechatronics. Elsevier, Amsterdam, 2000, 173–178. Hirota, T.; Ohkubo, T.; Itao, K.; Yoshikawa, H.; Ando, Y. Readout characteristics of flexible monolithic optical head slider combining with visible laser light guide. Microsystem Technologies 2003, 9 (5), 346–351.

7. 8. 9.

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