Table Of ContentANALOGIES IN OPTICS AND MICRO ELECTRONICS
Analogies in Optics
and
Micro Electronics
Selected Contributions on Recent Developments
Edited by
Willem van Haeringen
Eindhoven University of Technology, Eindhoven, The Netherlands
and
Daan Lenstra
Eindhoven University of Technology and University of Leiden,
The Netherlands
KLUWER ACADEMIC PUBLISHERS
DORDRECHT / BOSTON / LONDON
Library of Congress Cataloging in Publication Data
Analogies in optics and micro electronIcs: selected contributions on
recent developments I edited by Willem van Haeringen and Daan
Lenstra.
p. cm.
1. Optics. 2. Microelectronics. I. Haeringen, Willem van, 1933-
II. Lenstra, Daan, 1947-
TA1675.A49 1990
535--dc20 90-4175
ISBN-13: 978-94-010-7400-1 e-ISBN-13: 978-94-009-2009-5
DOl: 10.1007/978-94-009-2009-5
Published by Kluwer Academic Publishers,
P.O. Box 17, 3300 AA Dordrecht, The Netherlands.
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the publishing programmes of
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© 1990 by Kluwer Academic Publishers
Softcover reprint of the hardcover 1st edition 1990
No part of the material protected by this copyright notice may be reproduced or
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CONTENTS
Preface vii
P ART I: PRINCIPLES AND BASIC CONCEPTS 1
1. Playing with electrons and photons in rings 3
D. Lenstra and W. van Haeringen
2. What to expect from similarities between the Schrodinger and
Maxwell equations 21
M. Kaveh
3. Synergetics as a theory of analogous behavior of systems 35
H. Haken
4. Density of states of electrons and electromagnetic waves in
one-dimensional random media 49
P. ErdOs and Z. Domanski
5. Classical light waves and spinors 69
R. Bhandari
PART II: COHERENT OPTICS 83
6. Towards observation of Anderson localization of light 85
M.P. van Albada, A. Lagendijk and M.B. van der Mark
7. Photon localization: the inhibition of electromagnetism in
certain dielectrics 105
S. John
8. Photonic band structure 117
E. Yablonovitch
9. Optical level crossings 135
J.P. Woerdman and R.J.C. Spreeuw
10. Berry's phases in optics 151
R. Y. Chiao
vi CONTENTS
PART III: COHERENT ELECTRONICS 163
11. Aharonov-Bohm oscillations and non-local electronic
conduction 165
C. Van Haesendonck
12. Phase-sensitive voltage measurements 185
M. Biittiker
13. Quantum point contacts and coherent electron focusing 203
H. van Houten and C. w.J. Beenakker
14. Charge build-up and intrinsic bistability in resonant
tunneling 227
L. Eaves
15. Electrons as guided waves in laboratory structures:
strengths and problems 243
R. Landauer
Index 259
PREFACE
This book gives an account of a number of recent developments in two different
subfields of research, optics and micro--electronics. The leading principle in
presenting them together in one book is the striking similarity between a variety of
notions in these two research areas. We mention in this respect tunneling,
quantum interference and localization, which are important concepts in
quantummechanics and more specifically in condensed matter physics.
Miniaturization in solid state engineering has led to new phenomena in which these
concepts play their significant roles. As it is the wave character of electrons which
is strongly emphasized in these phenomena one's attention is quite naturally
directed to the field of optics in which the above quantum-mechanical notions all
seem to have their direct classical wavemechanical counterparts.
Both micro--electronics and optics have been and still are in a mode of intensifying
activity. The possibilities to technically "translate" devices developed within one
research field to similar devices in the other field are strongly increasing. This
opens, among other things, a door leading to "quantummechanics" on a
macroscopic scale with visible light under relatively easily accessible experimental
conditions, or to "wave optics in the domain of solid state physics. Thinking in
II
terms of analogies is important anyhow, but it is especially the cross-fertilization
between optics and micro--electronics which according to the editors will lead to
deepened insights and a new type of technology.
We have chosen for a presentation in which a restricted number of selected
researchers in both fields sketch developments to which they have significantly
contributed themselves. This will make the book of utmost importance for workers
in the respective fields. Furthermore, the didactical skills of the contributing
authors guarantee accessibility of the book to a much broader group of readers, e.g.
graduate students, physicists and electrical engineers who are not particularly
specialized in this area, but wish to obtain an overview. The book contains fifteen
presentations which we have classified into three groups: I. principles and basic
concepts, II. coherent optics and III. coherent electronics. Apart from giving an
important collection of new articles on recent developments in optics and micro
electronics, the objective of the editors has been to demonstrate the apparent
potentialities of thinking in terms of analogies.
Many interesting other analogies such as, e.g., between superconducting and
optical devices are not dealt with in the present volume. Also several important
earlier developments in the field of electron beam "optics" and X-ray diffraction in
crystals have not been covered. For a book of this set up and size it is impossible
to give a complete account. So, the outline of this first edited work on analogies in
optics and micro--electronics reflects very much the opinions and preferences of the
editors.
We would like to express our sincere gratitude to Ria Coopmans-van Basten for
vii
viii PREFACE
the secretarial and technical support she gave with great commitment to the book.
Special thanks also to Ria Groenendijk and Brigitte Senden for their assistance at
critical moments during the final stage of the project.
May this book serve to intensify the "analogy" kind of approach, thus leading to
even more cross-fertilization and if possible new fascinating results in the
borderline between two important technologies in the fields of optics and micro
electronics.
Eindhoven, Spring 1990
Willem van Haeringen
Daan Lenstra
PART I
PRINCIPLES AND BASIC CONCEPTS
PLAYING WITH ELECTRONS AND PHOTONS IN RINGS
Daan Lenstra and Willem van Haeringen
A "one-dimensional" ring configuration is a simple non-trivial system, particularly
suited for a theoretical study of wave propagation dynamics under the influence of
an external driving force. This ring system, although initially designed as an
academic play field with non-interacting model electrons, becomes most relevant
and experimentally accessible in the case of real-world optics.
1. INTRODUCTION
A ring is a very interesting system which combines two useful properties in an
intriguing if not controversial way: it is a finite system with regard to its
dimensions but endless for waves or particles traveling around and around. In
optics both the ring resonator and the ring laser have been thoroughly investigated
which has led to many applications related to their subtle and rich
phenomenology. We mention 'the laser gyroscope [1], bistability effects in ring
lasers [2] and, more recently, 1f"-phase jumps [3] and optical bandstructures in ring
resonators [4], the latter item being adressed to in the chapter by Woerdman and
Spreeuw (Chap.g).
In submicron electronics the ring system has been used for measuring
Aharonov-Bohm type magneto-resistance oscillations [5]. The chapter by Van
Haesendonck is devoted to this and other nonlocal effects (Chap.l1). Supercond
ucting rings with Josephson junctions have been thoroughly investigated [6,7]. For
the theoretical physicist the ring has become an important model system for
studying fundamental properties of electron dynamics and electrical conduction as
it avoids the necessity of introducing ad hoc boundary conditions which are to
reflect externally controlled fluxes of incoming or outgoing particles (or waves).
On the one hand, a ring measures precisely one primitive cell, but on the other
hand it is an infinite periodic system as well. Particles or waves which are
traveling around in the ring can be accelerated due to an external force.
Furthermore, they can be scattered (in fact, only back scattered in a ID ring) and
thus a nontrivial dynamics can be created. We will consider only elastic scattering,
i.e., described by a certain structure in the potential energy function (electrons) or
refractive index (photons). For these scattering processes the coherence of the
waves before, during and after scattering is fully conserved. This is an interesting
limiting situation in case of electrons because it resembles the situation at very low
temperatures, where the increasing amount of experimental evidence shows that
electrical conduction cannot be fully described anymore in terms of diffusion of
carriers, but should rather be treated in terms of coherent wave propagation [8].
The chapters by Van Haesendonck (Chap.l1), Biittiker (Chap.12) and van Houten
and Beenakker (Chap.13) deal with this regime, while also the concept of resonant
tunneling, studied by Eaves in Chap.14, implies coherent propagation of charge
3
W. van Haeringen and D. Lenstra (eds.), Analogies in Optics and Micro Electronics, 3-19.
@ 1990 Kluwer Academic Publishers.
4 D. LENSTRA AND W. VAN HAERINGEN
carriers.
One could argue that analogies between electrons and light can best be studied
on the level of the Dirae and Maxwell equations. However, it is well known that
the physics of electrons in modern semieonducting materials and structures can
succesfully be described within a SchrOdinger-equation framework. It would
therefore be extremely instructive and useful if existing analogies between eertain
aspects of optics and micro-electronics can be discussed within the framework of a
Schrodinger equation covering both cases. Therefore, in Sec.2 we take up that
ehallenge and arrive at the conclusion that photons can indeed adequately be dealt
with by means of a non-relativistic Schrodinger equation.
In Sec.3 we use the ring configuration in order to present a unified treatment of
wave dynamics in response to a slowly varying external perturbation. We find
general properties of the eigenstates and corresponding eigenvalues and we treat
the dynamical response in the adiabatic approximation. In Sec.4 we discuss the
experimental feasibility of Bloch oscillations in a ring. By estimating the various
time scales involved in such an experiment, we conclude that the best opportun
ities for observing Bloch oscillations are found in the optical ring system.
Non-adiabatic response, involving transitions between energy bands, is
discussed in Sec.5. In this regime we can study, at least theoretically, the
fundamental problem of coherent (i.e. temperature T = OK) wave dynamics due to
relatively large driving fields, without performing any of the usual approximations
such as linear-response theory or Boltzmann-equation approach. In the case of
electrons, the solution of the time-dependent Schrodinger equation turns out to be
chaotic, which is related to the fact that the time-dependent phases (modulo 27r)
in the one-electron wave function generally don't repeat themselves after each
roundtrip in a regular or commensurate way. This gives the system self
randomizing properties due to pseudo-stochastics, that is, on short time scales.
The intrinsic coherence of the mUltiply scattered waves becomes noticeable on
longer time scales as a weak form of localization.
2. A SCHRODINGER EQUATION FOR PHOTONS
Since one of our aims is to exploit the analogy between electron waves and optical
waves, it would be beneficial to deal with one and the same equation valid for
either case. Confining ourselves to the rich variety of electron propagation
phenomena which are succesfully described by a Schrodinger equation, for which
the solutions and their properties are well known, we want to investigate the
possibility of describing photons by means of a Schrodinger equation. In order to
be able to study dynamical properties, it has to be a time-dependent Schrodinger
equation. The derivation will quite naturally lead to the quantity which plays the
role of potential; this will make it possible and easy to "translate" many typical
quantummechanical results directly into the field of optics [9].
Our starting point is the wave equation for the electric-field component E in a
eonfiguration involving dielectric materials,
(1)
For simplicity we consider the scalar case only; it is possible, however, to include
the vector nature of the light and draw a parallel between the polarization of light
