Quantum Electronics. Maser Amplifiers and Oscillators

337 Pages·1969·8.11 MB·English

by  V. M. Fain

OTHER TITLES IN THE SERIES
IN NATURAL PHILOSOPHY
Vol. 1. DAVYDOV—Quantum Mechanics
Vol. 2. FoKKER—Time and Space, Weight and Inertia
Vol. 3. KAPLAN—Interstellar Gas Dynamics
Vol. 4. ABRIKOSOV, GOR'KOV and DZYALOSHINSKII—Quantum Field
Theoretical Methods in Statistical Physics
Vol. 5. OKUN'—Weak Interaction of Elementary Particles
Vol. 6. SHKLOVSKH—Physics of the Solar Corona
Vol. 7. AKHIEZER et al.—Collective Oscillations in a Plasma
Vol. 8. KiRZHNirs—Field Theoretical Methods in Many-body Systems
Vol. 9. KLIMONTOVICH—The Statistical Theory of Non-equilibrium
Processes in a Plasma
Vol. 10. KURTH—Introduction to Stellar Statistics
Vol. 11. CHALMERS—Atmospheric Electricity (2nd edition)
Vol. 12. RENNER—Current Algebras and their Applications
Vol. 13. FAIN and KHANIN—Quantum Electronics, Vol.1.—Basic Theory
Vol. 15. MARCH—Liquid Metals
Vol. 16. HoRi—Spectral Properties of Disordered Chains and Lattices
Vol. 17. SAINT JAMES, THOMAS and SARHA—Type II Superconductivity
Vol. 18. MARGENAU and KESTNER—Theory of Intermolecular Forces
Q U A N T UM
E L E C T R O N I CS
IN TWO VOLUMES
VOLUME 2: MASER AMPLIFIERS AND OSCILLATORS
V. M. FAIN AND YA. L KHANIN
TRANSLATED BY
H. S. H. MASSEY
EDITED BY
J.H.SANDERS
THC OUT IN« AWAMO
PERGAMON PRESS
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Copyright © 1969
Pergamon Press Ltd.
First English edition 1969
Distributed in the United States and Canada by M.I.T. Press, Cambridge, Massachusetts.
This book is a translation of Part II of KBanTOBaü ΡαΛΗθφΗ3Ηκα by V. Μ. Fain
and Ya. I. Khanin published in 1965 by Sovetskoye Radio, Moscow, and includes
corrections and revisions supplied by the authors.
Library of Congress Catalog Card No. 67-22832
PRINTED IN GERMANY
08 012238 8
Foreword
THE BIRTH of a new independent field of physics, now known by the name
of quantum electronics, was heralded about ten years ago by the creation of
the molecular oscillator. This field at once attracted the attention of a large
number of research workers, and rapid progress took place. Exten­
sive experimental and theoretical material has now been accumulated. The
present book attempts to give a resume of this material and, to a certain
extent, to generalize it.
We have tried to arrange the material so that, as far as is possible, the
reader need not continually refer elsewhere. The references to literature of a
theoretical nature make no pretence of completeness, but when citing experi­
mental work we have to give as full a list as possible since readers may be
interested in details for which there is no space in the book.
The theoretical sections of the books are by no means a survey of present
work. We have tried to highhght the basic principles and their results. It is
natural that slightly more attention has been paid to fields in which the
authors themselves have been involved. The experimental material is given
in the form of a survey, with only a brief description of the technical details
of devices.
The book as a whole is designed for the reader with a knowledge of
theoretical physics (quantum mechanics in particular) at university level. It
should be pointed out that the material in the various sections of the book is
of differing degrees of complexity. Readers will need less preparation for
Volume 2. The most difficult paragraphs are marked with an asterisk.
In conclusion we should mention that §§ 1-20, 22-40 and Appendix I were
written by V.M.Fain; §§ 41-49 and 51-58 by Ya.I.Khanin. At the author's
request §21 was written by V.N.Genkin, § 50 by E.G. Yashchin, Appen­
dix II by V.I.Talanov and Appendix III by Ye. L. Rozenberg.
We are grateful to Professor A. V.Gaponov and Professor V.L.Ginzburg
for reading the book in manuscript and making a number of useful sugges­
tions. We are also grateful to A.P.Aleksandrov, V.N.Genkin, G.M.Gen-
kin, N.G.Golubeva, G.L.Gurevich, G.K.Ivanova, Μ.I.Kheifets, Yu.G.
Khronopulo, Ye.E. Yakubovich and E.G. Yashchin for their great help in
reading the proofs.
Radiophysics Scientific Research Institute, V. M. FAIN
Gor'kii YA. I. KHANIN
xi
Preface to the English Edition
WE WERE very pleased to learn that our book was to be translated into
English and would thus become available to a wide circle of Enghsh-speaking
readers.
Research in the field of quantum electronics has continued since our
manuscript was handed over to the Soviet publishers, but there has been no
essential change in the basic theory or understanding of the physical pro­
cesses in quantum devices. Among the most interesting questions on which
work has been done of late, mention should be made of the development of
coherence theory and holography, which is based on it. Non-linear optics
and its application are developing rapidly. Although the book treats the
fundamentals of non-linear optics, one of the important non-linear optical
effects—the phenomenon of self-trapping—is not discussed in the book.
Among the other important problems recently worked on and not re­
flected in the book are the electrodynamics of gas lasers, the theory of the
natural width of lasers (gas lasers in particular) and the important work on
semiconductor lasers.
Despite the great importance of these questions we did not include them in
the present edition, not merely for lack of time but also because one is here
dealing with subjects under development, not all of whose facets are com­
pletely clear and whose discussion would be previous.
We are preparing material on the majority of these questions, and also a
detailed treatment of the theory of the non-linear properties of materials
(mainly solids) which is of great importance for non-linear optics and quan­
tum electronics; this will be included in the second Soviet edition, which is
planned for 1969.
In the present edition we have confined ourselves to correcting any errors
that have been found and making some slight additions.
February 1967
V. M. FAIN Institute of Solid-State Physics,
Moscow, Academy of Sciences U.S.S.R.
YA. I. KHANIN Radiophysics Scientific Research Institute, Gor'kii.
xii
Introduction
QUANTUM electronics as an independent field of physics came into prominence
in the middle fifties when the first quantum oscillators and amplifiers were
made. The immediate precursor of quantum electronics was radiofrequency
spectroscopy, which is now one of its branches. An enormous quantity of
experimental material concerned with the resonant properties of substances
had been accumulated by radiofrequency spectroscopy. Such research had
made it possible to establish the structure of levels, the frequencies and in­
tensities of transitions, and the relaxation characteristics of different sub­
stances. Investigations of paramagnetic resonance spectra in solids and the
inversion spectrum of ammonia have been of particular importance to quan­
tum electronics.
During radiofrequency investigations the state of a substance is not, as a
rule, subject to significant changes and remains close to thermodynamic equi­
librium. But besides the investigation of substances under undisturbed con­
ditions, other methods began to appear which were connected with the action
of strong resonant fields on a substance. These methods, which we can call
active, were first applied in nuclear magnetic resonance. They include nuclear
magnetic induction spin echo and the Overhauser effect. The main outcome
of these methods was the possibility of producing strongly non-equilibrium
states in quantum systems which could emit coherently. Therefore the actual
material was accumulated through radiofrequency spectroscopy, and resulted
in the birth of experimental ideas which were then used as the basis of quan­
tum oscillators and amplifiers.
The concept of stimulated emission, which is important for quantum
electronics, was first formulated by Einstein as early as 1917. Ginzburg (1947)
pointed out the importance of this phenomenon in radiofrequency spectro­
scopy.
The idea of amplifying electromagnetic waves by non-equilibrium quantum
systems was first mooted by Fabrikant, Vudynskii and Butaeva. The patent
(Fabrikant, Vudynskii and Butaeva, 1951) obtained by this team in 1951
contains a description of the principle of molecular amplification. Slightly
later, in 1953, Weber made a suggestion about a quantum amplifier. Baso ν
and Prokhorov (1954) discussed an actual design for a molecular oscillator
xiii
Introduction
and amplifier operating with a beam of active molecules and developed their
theory. Gordon, Zeiger and Townes independently had the same idea and,
in the same year, 1954, published a report on the construction of an oscillator
that operated with a beam of ammonia molecules. Gordon, Zeiger and Townes
introduced the now well-known term "maser".t
The successful operation of a beam molecular oscillator stimulated the
search for new methods and results were not long in coming. Basov and
Prokhorov (1954) suggested the principle of a three-level gas-beam oscillator.
In 1956 Bloembergen discussed the possibility of making a quantum amplifier
with a solid paramagnetic working medium. The estimates he made con
firmed that the idea was feasible and in 1957 such an instrument was made
by Scovil, Feher and Seidel. After this, reports appeared on the production
of a whole series of similar instruments based on different paramagnetic
crystals.
Instruments based on quantum principles have a number of exceptional
properties when compared with ordinary amplifiers and oscillators. The
molecular beam maser oscillator is not particularly powerful but its stability
is far better than the stability of the best quartz oscillators. This has brought
about the use of the maser as a frequency standard. The paramagnetic maser
amplifier has an extremely low noise level and satisfactory gain and band
width characteristics.
The next stage in the development of quantum electronics was the extension
of its methods into the optical range. In 1958 Schawlow and Townes dis
cussed the question theoretically and came to the conclusion that it was per
fectly possible to make an optical maser oscillator. They suggested gases
and metal vapour as the working substances. The question of possible work
ing substances and the methods of producing the necessary non-equilibrium
states in them was also discussed in a survey by Basov, Krokhin and Popov
(1960). These authors discussed paramagnetic crystals and semiconductors
as well as gases.
In 1960 Maiman made the first pulsed ruby quantum optical generator
which is called a "laser".* For the first time science and technology had
available a coherent source of light waves. The future prospects of devices of
this kind were obvious and in a very short time a large number of teams had
come onto the scene of laser research. The list of crystals suitable for use in
lasers quickly grew. Then certain luminescent glasses and liquids were used
for the same purpose. In 1961 Javan, Bennett and Herriott made the first
continuous laser operating with a mixture of the inert gases neon and helium.
t Maser is an acronym formed from Microwave amplification by Stimulated ¿"mission of
i?adiation.
* The term laser is an acronym from light amplification by Stimulated Emission of
i?adiation. It must be pointed out that there is not yet any firmly established terminology in
quantum electronics. Besides "laser" the name "optical maser" is frequently used.
xiv
Introduction
Quantum electronics is very young; its basic trends are still far from clear.
A whole series of problems is still unsolved. Under these conditions the writ­
ing of a monograph discussing the theoretical and experimental basis of
quantum electronics is a rather complex affair. It must be understood that
the book reflects to only a limited extent the position as it is today.
Quantum electronics as a theoretical science possesses a number of char­
acteristic features which separate it both from quantum physics and from
electronics. Unlike ordinary "classical" electronics, quantum electronics is
characterized by the extensive application of the methods of quantum theory.
However, the application of quantum held theory to quantum electronics
has a specific feature which distinguishes it from ordinary quantum electro­
dynamics (see, e.g., Heitler, 1954*; Akhiezer and Berestetskii, 1959*).
Quantum electronics makes wide use of the resonant properties of matter
both for the study of matter itself (radiofrequency spectroscopy, paramagnetic
resonance), and for its use in quantum amplifiers and oscillators. It is obvious
that resonances with a high ö-factor in a substance are essential for both
purposes. To obtain sharp resonances discrete energy levels must exist in
the substance. The presence of discrete electron levels means that these
electrons cannot be free but must be in bound states in the atoms, molecules
or solid. We notice that the characteristic feature of ordinary "classical"
electrodynamics is the interaction of the radiation field with free electrons.
It is true that quasiclassical systems (harmonic oscillator, electron in a mag­
netic field, etc.) may also have a discrete spectrum but the essential feature
of classical and quasi-classical systems is that the energy levels of such sys­
tems are quasi-equidistant. For example, the harmonic oscillator has equi­
distant levels with no upper limit. The energy spectrum of quantum systems
is much more diverse than the spectrum of quasi-classical systems. In par­
ticular the energy levels may be so arranged that there are two levels whose
spacing is not the same as the spacing of any other levels in the same system.
Under certain conditions no attention need be paid (during an interaction
with radiation of the corresponding frequency) to any other levels of the
system and we can use the idealization of a two-level system. The ideahzations
of a three-level system, etc., are introduced likewise.
As we have already pointed out, wide use is made of the resonant properties
of matter in quantum electronics. As may be easily understood, during the
resonant interaction of matter with a field it is particularly important to
allow for different kinds of dissipative relaxation processes. UnHke ordinary
quantum electrodynamics in which, as a rule, we are not interested in relaxa­
tion processes in matter, in quantum electronics the concept, and thus the
description, of the different relaxation processes plays a major part.
The concept of stimulated emission plays an important and even pre­
dominant part in quantum electronics. All the active quantum-electronic
instruments—maser amplifiers and oscillators—use the phenomenon of sti-
XV
Introduction
mulated emission. The phenomenon of stimulated emission is closely linked
(as will become clear from the appropriate sections of the book) with the
non-linear properties of quantum systems used in quantum electronics. The
non-linear properties, in their turn, are caused by the non-equidistant nature
of the energy levels.
When describing the processes of the interaction of matter with radiation
we must, strictly speaking, use quantum theory, i.e. quantum theory is used
to treat the matter and the field. For many problems, however, the classical
description of an electromagnetic field is a fully justified approximation. This
is because the fields discussed in quantum electronics are large, and because
the mean quantum values of the electric and magnetic fields are accurately
described by the classical Maxwell equations. It is essential to allow for the
quantum properties of the field when investigating the quantum fluctuations
of the field, in particular when studying the noise properties of amplifiers
and oscillators.
The arrangement and selection of the material in the present book have
been made with these features of quantum electronics in mind. The book is
composed of two parts: Volume 1 "Basic Theory" and Volume 2 "Maser
Amplifiers and Oscillators". A large amount of material has been kept for the
Appendixes.
In Volume 1 an attempt is made to give the basic theory of quantum elec­
tronics. In this part we have tried to show how the concepts and equations
used in quantum electronics follow from the basic principles of theoretical
physics. When doing this we make frequent use of very simple models so as
not to complicate the treatment. Such models are necessary for the under­
standing of a particular process, but the models can often not be used for
direct comparison with experiment.
The first chapter of the book deals with general questions of the interaction
of radiation with matter. The basic concepts of quantum theory are briefly
treated in this chapter. The reader's attention is particularly drawn to the
density matrix description of the quantum state. This is because in its various
applications quantum electronics deals with mixed states and not with pure
states. Quantum theory allows us, by the use of the density matrix, to give
a unified description of both pure and mixed states. In the first chapter we
discuss in sequence the quantum theory of fields in resonators, in waveguides
and in free space and also the concept of phase in quantum field theory, of
the indeterminacy relation between the phase and the number of particles,
the question of the transition to classical physics, etc. Section 4 discusses in
more detail than usual the question of the different forms of interaction
energy between a field and charged particles.
The second chapter deals with the general question of relaxation. When
there are relaxation processes present the behaviour of quantum systems is
governed by the cause of the dissipation—adissipative system which possesses
xvi
Introduction
a continuous spectrum and an infinite number of degrees of freedom. In
this case we must derive approximate equations which will take into account
the relaxation processes (the transport equations). Therefore Chapter II deals
essentially with the applicability of the different equations used in quantum
electronics. In particular, by proceeding from basic principles, we derive
the conditions for applicability of the frequently used population balance
equations. The same chapter discusses the questions of the irreversibility of
real systems and the principle of the increase of entropy. We also show how
the transport equations can be used to describe fluctuations. Some long cal­
culations are given in this chapter but they may be omitted on the first read­
ing without making it difficult to understand the other parts of the book.
The results which are necessary for reading subsequent chapters are given in
the introduction to this chapter.
In Chapter III we have gathered together the possible quantum effects
in ordinary electronics which may appear at very high frequencies and
at low temperatures. These effects, as a rule, are small. In Chapter III
an account is also given of the quantum theory of real resonators with
finite ρ.
In Chapters IV and V we discuss the behaviour of quantum systems in
fields, which are here described classically. Particular attention is paid to
the response of a system to such fields. This response, for example in the
form of the mean magnetization of the system, is described in terms of the
susceptibility. A number of general susceptibility properties are discussed,
particularly the dispersion relations, the fluctuation-dissipation theorem, and
the symmetry properties. In these same chapters we treat the idealizations
of two- and three-level systems and find the equations of motion for these
systems. In § 20 of Chapter IV we show how it is possible to give a rigorous
description of systems which are not subject to the equations derived in
Chapter II. The method of moments is used; the rigorous basis of this
method is given in §20. In §21 it is used to examine cross-relaxation
processes.
In Chapters VI, VII and VIII we deal with a number of questions concern­
ing the theory of spontaneous and stimulated emission. In particular we
discuss the connection with classical theory, the part played by non-linearity,
the phase relations, etc. Chapter VII treats the theory of coherent spontane­
ous emission in free space and the theory of the natural line width. In
Chapter VIII we deal with the physical nature of the processes of spontane­
ous and stimulated emission in a resonator.
Recently a new branch of quantum electronics—non-linear optics—has
appeared. The development of non-linear optics, connected with success in
the field of optical quantum light generators (lasers), is only just beginning.
However, in our view a number of the essential features of the interaction of
matter with optical waves can already be stated. The ninth and last chapter
xvii