Micro-Optomechatronics (Optical Science and Engineering)

DK9834_half-series-title.qxd 11/2/04 9:08 AM Page 7 Micro-Optomechatronics Hiroshi Hosaka Graduate School of Fronti...

Author: Hiroshi Hosaka | Yoshitada Katagiri | Terunao Hirota | Kiyoshi Itao

61 downloads 901 Views 19MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

DK9834_half-series-title.qxd

11/2/04

9:08 AM

Page 7

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

9 8 7 6 5

4 3 2 1

PRINTED IN THE UNITED STATES OF AMERICA


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

iii


iv

Preface

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.


Preface

v

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


Contents

Preface

iii

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

43 45

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

99

Chapter 3

Chapter 4

15 17 20 30 41

85 96

100 108 111 125

vii


viii

Chapter 5

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


138 146 158

208 223

1 The World of Micro-Optomechatronics

1

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


2

Chapter 1

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].


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

3


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

4

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



5

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.

2

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


6

Chapter 1

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


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


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

7


8

Chapter 1

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.



3

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–


10

Chapter 1

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.

4

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



11

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).


12

Chapter 1

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.



13

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.


2 Technological Outline of Micro-Optomechatronics

1

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


16

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.


Technological Outline of Micro-Optomechatronics

2

17

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)


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


Positioning system Intermittent Positioning accuracy

Usage of optical power (>1 mm)

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


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.


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


ð57Þ


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 2ðtÞ4 =ð4 ln 2Þ2 þ 2€ 2 ðtÞ2 =ð4 ln 2Þ i€ 2 ln 2 t2 ð59Þ exp ðtÞ2 1 þ €2 ð4 ln 2Þ2 =ðtÞ4 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 ðtÞ4 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


76

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


ð63Þ


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 ðtÞ2 1 þ 4b€ þ 4b2 €2 þ €2 ð4 ln 2Þ2 =ðtÞ4

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 ðtÞ4 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 ðtÞ4 þ 4b2 This parameter indicates that when the second-order dispersion satisfies the following condition, jj
> 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.


ð26Þ

168

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


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


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



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.


172

Chapter 6

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


ð55Þ


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 ÞGðr, r0 Þ Gðr, r0 ÞXðr0 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


174

Chapter 6

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



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


176

Chapter 6

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



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


178

Chapter 6

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



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 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.



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.


184

Chapter 6

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


ð93Þ


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


ð98Þ

186

Chapter 6

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


ð104Þ


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


188

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



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.


190

Chapter 6

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.



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.


192

Chapter 6

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.



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.


194

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



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.


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.



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


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.



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 sE

ð135Þ

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

N sE 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


200

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



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 sH

ð149Þ

and

@ i!N2 E rot H ¼ þ " E¼ @t c2


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.



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.


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



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


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


ð189Þ


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.


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,



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


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


ð201Þ


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


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



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


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



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.


216

Chapter 6

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



217

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


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



219

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


220

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



221

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


222

Chapter 6

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.



223

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.


224

Chapter 6

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.


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


226

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


Fundamental Dynamics of Micro-Optomechatronics

227

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).


228

Chapter 7

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



229

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.


230

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



231

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


ð9Þ

ð10Þ

232

Chapter 7

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.


@2 w ¼0 @x2

@w ¼0 @x

@ @2 w EI 2 ¼ 0 @x @x


233

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


ð18Þ

234

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 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



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.


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



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


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


ð95Þ


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


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.



253

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.


ð108Þ

254

Chapter 7

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.



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


256

Chapter 7

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.



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


258

Chapter 7

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



259

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


260

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



261

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


262

Chapter 7

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.


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.


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


266

Chapter 8

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


Novel Technological Stream Toward Nano-Optomechatronics

267

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.)


268

Chapter 8

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.



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.


270

Chapter 8

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.



271

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


272

Chapter 8

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



273

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.


274

Chapter 8

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.



275

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


276

Chapter 8

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



277

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.


278

Chapter 8

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.



279

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).


280

Chapter 8

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.



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


282

Chapter 8

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.



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


284

Chapter 8

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



285

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


286

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.



287

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


288

Chapter 8

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.



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.

REFERENCES 1.

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.


290

Chapter 8

6.

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.

10.

11.


Micro-Optomechatronics (Optical Science and Engineering)

Electro-optical Displays (Optical Science and Engineering)

Physics of Optoelectronics (Optical Science and Engineering)

Laser Remote Sensing (Optical Science and Engineering)

Acousto-optics (Optical Science and Engineering)

Smart Biosensor Technology (Optical Science and Engineering)

Modern Optical Engineering: The Design of Optical Systems (Optical and Electro-Optical Engineering Series)

Terahertz Spectroscopy: Principles and Applications (Optical Science and Engineering Series)

Polymers for Lightwave and Integrated Optics (Optical Science and Engineering)

Smart CMOS Image Sensors and Applications (Optical Science and Engineering)

Biophotonics: Optical Science and Engineering for the 21st Century

Modern optical engineering

Modern Optical Engineering

Modern optical engineering

Handbook of Optical Engineering

Modern optical engineering

Computational science and engineering

Microwave and RF Engineering (Microwave and Optical Engineering)

Computational Science and Engineering

Computational Methods for Electromagnetic and Optical Systems (Optical Engineering)

Materials science and engineering

Optical Methods of Measurement: Wholefield Techniques, Second Edition (Optical Science and Engineering)

Flotation Science and Engineering

Optomechatronics: Fusion of Optical and Mechatronic Engineering

Mathematical Methods for Optical Physics and Engineering

Optical Methods of Engineering Analysis

Engineering Science,

The art and science of optical design

Optical Near-Field Recording: Science and Technology

Optical Near-Field Recording: Science and Technology

Mathematical modeling in optical science

Micro-Optomechatronics (Optical Science and Engineering)

Electro-optical Displays (Optical Science and Engineering)

Physics of Optoelectronics (Optical Science and Engineering)

Laser Remote Sensing (Optical Science and Engineering)

Acousto-optics (Optical Science and Engineering)

Smart Biosensor Technology (Optical Science and Engineering)

Modern Optical Engineering: The Design of Optical Systems (Optical and Electro-Optical Engineering Series)

Terahertz Spectroscopy: Principles and Applications (Optical Science and Engineering Series)

Polymers for Lightwave and Integrated Optics (Optical Science and Engineering)

Smart CMOS Image Sensors and Applications (Optical Science and Engineering)

Biophotonics: Optical Science and Engineering for the 21st Century

Modern optical engineering

Modern Optical Engineering

Modern optical engineering

Handbook of Optical Engineering

Modern optical engineering

Computational science and engineering

Microwave and RF Engineering (Microwave and Optical Engineering)

Computational Science and Engineering

Computational Methods for Electromagnetic and Optical Systems (Optical Engineering)

Materials science and engineering

Optical Methods of Measurement: Wholefield Techniques, Second Edition (Optical Science and Engineering)

Flotation Science and Engineering

Optomechatronics: Fusion of Optical and Mechatronic Engineering

Mathematical Methods for Optical Physics and Engineering

Optical Methods of Engineering Analysis

Engineering Science,

The art and science of optical design

Optical Near-Field Recording: Science and Technology

Optical Near-Field Recording: Science and Technology

Mathematical modeling in optical science

Recommend Documents