Table Of ContentO P T I CS
LECTURES ON THEORETICAL PHYSICS, VOL. IV
BY ARNOLD SOMMERFELD
UNIVERSITY OF MUNICH
TRANSLATED BY
OTTO LAPORTE AND PETER A. MOLDAUER
UNIVERSITY OF MICHIGAN
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PREFACE
This volume is closely connected with "Electrodynamics," Vol. Ill of my
lectures. Not only the formalism of Maxwell's equations but also their intrinsic
character, the invariance with respect to the group of Lorentz transformations,
is adopted from Vol. Ill and is assumed to be known.
Chapter I is entitled "Reflection and Refraction of Light." Only the
(never realizable) ideal case of the monochromatic plane wave which is
necessarily completely (and, in general, elliptically) polarized is treated in
this chapter. Reflection and refraction are regarded throughout as boundary
value problems associated with a single boundary surface or (in the case of the
plate) with two boundary surfaces. It is surprising how much material falls in
this category : It extends from the classical Fresnel formulae to the very timely
problem of the tunnel effect and covers non-reflecting lenses, the Perot-Fabry
étalon, and the (no longer timely) problem of the "black submarine." The
fundamental question of the "coherence or non-coherence of light" is touched
upon briefly only in fig. 2 of this chapter. Not until the last chapter, Sec. 49,
will we return to the problem of characterizing white light.
Chapter II deals at once with the optics of moving media. Indeed, these
questions seem to me to be basically simpler and more fundamental than the
contents of the later chapters because one is dealing here with the universal
character of the velocity of light and with its physical and astronomical
consequences. The first doubts about the classical wave nature of light
appear at the end of this chapter in connection with the Doppler and photo-
electric effects, and the equivalent corpuscular nature of light makes its first
appearance.
Chapter III deals with the theory of dispersion from Drude's semi-
phenomenological point of view, which is based on the classically formulated
resonance oscillations of electrons bound to atoms. However, it seemed to me
unavoidable to add to this chapter a section in which the theory of dispersion
is treated wave-mechanically, that is, where the characteristic oscillations are
replaced by transitions between two different energy levels.
Chapter IV is dedicated to crystal optics, the favorite subject of physics
in the last century. Here again the treatment is phenomenological even in the
problem of the rotation of the plane of polarization in acentric crystals, which
turns out to be particularly simple thanks to our use of the complex notation.
v
VI PREFACE
Chapter V and most of Chapter VI are devoted to the problem of diffrac-
tion. Diffraction by gratings (including three-dimensional ones) is treated
first. Then follows Huygens' principle for scalar diffraction problems, which is
applied to the question of "light and shadow" with its manifold paradoxical
contradictions of geometrical optics. Chapter V closes with a presentation of
the rigorously solvable boundary value problem of the perfectly reflecting
straight edge.
Chapter VI begins with the problem of the narrow slit, which Lord
Rayleigh solved in the first approximation more than fifty years ago. The
problem leads to an integral equation from which higher approximations
can be derived if proper use is made of the insight gained in the problem of the
straight edge by regarding the behavior of the branched solutions at the edge
of the screen. In the succeeding paragraphs a more or less new comprehensive
point of view is applied to the question of the resolving powers of spectral
apparatus (including Michelson's mirrors for the measurement of the diameters
of fixed stars). Thomas Young's theory of diffraction in the formulation given
to it by Rubinowicz, and Debye's formulation of focal point diffraction are
presented next. Finally, the difference between the scalar and the vector
diffraction problems is emphasized and the vectorial generalization of
Huygens' principle is discussed. This latter discussion follows the most recent
and particularly lucid treatment of the problem by W. Franz.
The presentation of the Cerenkov electron in Sec. 47 reaches beyond the
limits of the conventional conception of optics and enters, so to speak, the
realm of velocities greater than that of light. Section 48 deals with the (so far
almost entirely neglected) geometrical optics. The introduction of the eikonal
(and the unit vector associated with it) enables us to give a very brief presenta-
tion of several of the fundamental problems of geometrical optics. The very
large field of physiological optics, on the other hand, could only be touched
on in the introduction even though it is of primary importance with regard
to our actual experience.
The last section is concerned with the nature of white light which
possesses not a trace of periodicity and attains its wave character only upon
passing through a spectral apparatus. The wave representation which appears
here only as a secondary attribute of light is missing entirely in geometrical
optics and is replaced by a corpuscular conception in Fermat's principle.
The corpuscular concept points the-way to the modern theory of photons and
the complementarity of wave and corpuscle which was already stated at the
end of the second chapter. Finally, it is impressed upon the reader that
our presentation, which is essentially based upon the classical wave concept,
forms only a part of the entire field of optics; in particular it does not en-
PREFACE vil
compass the primary processes in the retina because these are photoelectric
in nature and therefore their discussion must be based on the theory of photons
and not on the wave theory.
The text of this volume is based upon a careful record of my lectures on
optics made by L. Waldmann in 1934. However, the last few subjects
discussed go considerably beyond the contents of my lectures at that time.
As in the case of Vol. Ill, I enjoyed the invaluable cooperation of
Mr. J. Jaumann in the preparation of this volume. In our many discussions
he not only communicated to me his rich experience in experimental optics,
but in many instances he prepared the first drafts for the manuscript. I men-
tion, in particular, Sections 3 C, 6 C, 7 C, 30 C, 41, and 42. His part in the
writing of this book should not be underestimated. My colleague, Dr. O. Buhl,
subjected the entire manuscript to his critical inspection and has helped me
with many useful remarks. Dr. P. Mann has kindly checked the exercises.
Munich, end of 1949.
Arnold Sommerfeld
TRANSLATORS' NOTE
The translators of this volume have endeavored to adhere to the spirit
of the original as much as possible and to keep changes to a minimum. In
addition to certain changes in notation, some modifications of the text have
proved to be inevitable. These are especially contained in Sections 27 and 28
which were kindly contributed by Professor P. P. Ewald. Furthermore,
Sec. 47 should be read in the light of a recent paper by H. Motz and L. I. Schiff,
Am. J. Phys. 21, 258, 1953. A completely new author and subject index was
prepared.
O. L.
P. A. M.
PREFACE vil
compass the primary processes in the retina because these are photoelectric
in nature and therefore their discussion must be based on the theory of photons
and not on the wave theory.
The text of this volume is based upon a careful record of my lectures on
optics made by L. Waldmann in 1934. However, the last few subjects
discussed go considerably beyond the contents of my lectures at that time.
As in the case of Vol. Ill, I enjoyed the invaluable cooperation of
Mr. J. Jaumann in the preparation of this volume. In our many discussions
he not only communicated to me his rich experience in experimental optics,
but in many instances he prepared the first drafts for the manuscript. I men-
tion, in particular, Sections 3 C, 6 C, 7 C, 30 C, 41, and 42. His part in the
writing of this book should not be underestimated. My colleague, Dr. O. Buhl,
subjected the entire manuscript to his critical inspection and has helped me
with many useful remarks. Dr. P. Mann has kindly checked the exercises.
Munich, end of 1949.
Arnold Sommerfeld
TRANSLATORS' NOTE
The translators of this volume have endeavored to adhere to the spirit
of the original as much as possible and to keep changes to a minimum. In
addition to certain changes in notation, some modifications of the text have
proved to be inevitable. These are especially contained in Sections 27 and 28
which were kindly contributed by Professor P. P. Ewald. Furthermore,
Sec. 47 should be read in the light of a recent paper by H. Motz and L. I. Schiff,
Am. J. Phys. 21, 258, 1953. A completely new author and subject index was
prepared.
O. L.
P. A. M.
INTRODUCTION
1. Geometrical, Physical, and Physiological Optics. Historical Chart
The eye is our noblest sense organ. It is therefore not surprising that
even the natural philosophers of antiquity were concerned with the science
of light. Leonardo da Vinci called optics "the paradise of mathematicians".
Of course, by optics he meant only geometrical or ray optics, the theory of
perspective and the distribution of light and shadow. How much more justified
would his assertion have been had he known the wave optics with its marvelous
color phenomena arising from diffracted light or the polarized light of crystals.
It is in particular these latter phenomena which one has in mind when one
speaks of physical optics. Physical optics is related to ray optics in the same
way in which wave mechanics is related to classical mechanics. This fact was
recognized by Schrodinger on the basis of the profound work of Hamilton.
There is, however, still a third branch of optics which is called physiological
, optics after the title of Helmholtz's principal work. Also in this field funda-
mental laws hold which, however, are based on the operation of the sense
organs and the mind. But these laws are not encompassed by our physical
theory. It was the tragedy in the life of Goethe that he would not recognize
the distinction between physical and physiological optics; this was the
reason for his fruitless fight against Newton. Today we understand without
difficulty that the sensation yellow which is caused by the ZMines of sodium
is a phenomenon which is entirely different from the wavelengths λ = 5890 Â
and λ = 5896 Â by which we must describe these lines physically. For, we
know that the psychological response to an event is something entirely
different from the physical event itself; the two are different in nature and
incommensurable.
In this volume we shall be able to deal only briefly with ray optics and
unfortunately not at all with physiological optics. Wave optics, which we
shall develop directly from the results of Vol. Ill and which, through spectro-
scopy, opens the way to modern atomic physics, will give us enough to do.
We shall not, for instance, enter upon the interesting field of the theory of
color which was formulated in a classical manner by Thomas Young and
Helmholtz, was further developed particularly by Grassmann, Maxwell and
1
2 INTRODUCTION 1
Schrödinger, and is even today not a closed subject. We shall here only
demonstrate very briefly that, quite aside from the quality of colors and their
contrast effects, there exists a profound difference between subjective percepr
tion and objective fact even in regard to the quantitative determination of
intensity. The phenomenon in question is that of the so-called "half-shadow".
This phenomenon played a role in the earliest attempts to determine the
wavelength of X-rays. On X-ray plates there appear half-shadow regions
between the complete shadow and the
region of full illumination. These are
due to secondary X-rays which origi-
nate, for instance, at the edges of a
slit. To the eye these half-shadow
regions appear as bright and dark
fringes which were at first interpreted
as interference lines. However, Haga
and Wind were able to show that these
fringes were subjective in origin and
they called attention to a phenomenon
which had been investigated by E.
Mach1 and had also been recognized
by H. Seeliger in his studies of eclipses
FFiig. 1.
Rotating disc for the demonstration of °f the m00n· We sha11 deSCribe {t here
a physiological optical illusion. as our sole example of physiological-
optical phenomena.
Consider a white circular cardboard disc which is partially blackened as
shown in fig. 1. The boundary between the black and white fields consists
of two spirals of Archimedes and portions of a radius of the disc. Let us
consider the average brightness (or blackness) along each circle concentric
with the edge of the disc; this quantity determines, in accordance with a
law due to Talbot, the perception of brightness when the disc is rotated
sufficiently fast. The center of the disc is then perfectly black and so is its
edge. Between the center and the edge there is a zone of maximum brightness.
The transition between darkness and brightness consists of two half-shadow
regions. Since the radius vectors of the spirals of Archimedes increase (or
decrease) linearly with the central angle, the intensity in the half-shadow
region also increases (or decreases; linearly with the distance from the center
of the disc. If the disc is set into rapid rotation on the axis of a motor, then
1See, for instance, his book Prinzipien der Physikalischen Optik, p. 158, J. A. Barth,
publ. 1921.
1 GEOMETRICAL, PHYSICAL, AND PHYSIOLOGICAL OPTICS 3
the intensity distribution presented to the eye is that represented by the dotted
line in fig. la. But what does the eye see? Instead of the linearly varying
half-shadows the eye perceives a uniform average brightness; where the half-
shadow borders on the completely black regions it perceives dark fringes
which are considerably blacker than the regions of complete blackness; at
the limits of full brightness it sees bright fringes which appear much brighter
than the region of full brightness. The eye (or the mind?) is, as it were,
startled by the transition from the half-shadow to full illumination; it
exaggerates the contrast. The
same exaggeration takes place I j
at the transition from the
half-shadow to complete black-
ness. The eye (or the mind)
judges only contrasts and not
objective intensity values; it
is affected more by the deriv-
atives of the intensity curve
than by the absolute values of
its ordinate. The bright and
dark fringes (which on the Center
rotating disc are, of course,
Fig. la.
circles about the center) are
The subjective intensity distribution perceived by
so definitely pronounced that
the eye (full line) and the objective distribution of
a naive observer would swear intensity (dotted line) when the disc is rotated.
to their genuineness.
Similar fringes are seen wherever extended light sources produce half-
shadows according to geometrical optics, as for instance, behind a pencil
which is illuminated by a Welsbach mantle. Also the bright border which
one sees about ones own shadow on the road when the sun is behind ones
back and which has the effect of a sort of halo about the head and limbs is
at least partly due to this optical illusion. Such fringes also played a part in
certain occasional arguments between the author and a group of Munich
painters which revolved around the old controversy "Goethe vs. Newton".
The opponents in these discussions understandably enough considered these
subjective phenomena as objective and offered them as proof of the falseness
of the physical theories.
It might be thought that this illusion could not be photographed and would
thereby betray its subjective character. This is not so. Even though the
number of blackened grains on the photographic plate corresponds to the
correct intensity, the eye interprets the photographic image in the same way
4 INTRODUCTION 1
as the original object and is deceived by its subjective contrast perceptions.
This is illustrated by the following experiment1: a micrometer slit illuminated
from the rear with parallel light is photographed. At the beginning of the
exposure the slit may have a width 2b. It is then slowly and uniformly opened
to a width 2a, whereupon the exposure is terminated. Thus the center por-
tion 2b of the photographic plate is continuously illuminated during the
exposure; the adjoining portions a-b are illuminated a shorter time which
decreases linearly to zero. On the photograph one sees again bright and dark
fringes at the limits of the half-shadows (if the slit is opened non-uniformly,
there appear also secondary fringes inside the half-shadow regions a-b which
correspond to discontinuities in the derivative of the curve depicting
illumination vs. time).
So much (or rather so little) for physiological optics. In order to provide
a general summary of the wealth of material to be covered in this volume we
continue with a historical list of the most important optical discoveries.
Snell's law of refraction (which became known only through Huygens) and
Descartes, Dioptrices, 1637. The first theory of the rainbow is also due to
Descartes.
Grimaldi, Physico-mathesis de lumine, coloribus et iride, Bononiae (Bologna)
1655; first textbook on optics; deviations from rectilinear ray paths;
diffraction.
Olaf Roemer, 1675; determination of the velocity of light from the eclipses
of the satellites of Jupiter.
Christian Huygens, Traité de la Lumière, Leiden 1690; wave theory without
closer investigation of the nature of the oscillations (whether longitudinal
or transverse). Huygens* principle; wave surfaces. Double refraction
in calcite.
Newton, Opticks 1706, English 1675. Colors of thin plates. Spectral colors
and their composition into white light. Theory of emission with lateral
"fits".
Bradley, 1728; aberration of light.
Thomas Young, Lectures on Natural Philosophy, 1807. Interference of light;
diffraction; theory of color; the color triangle; Young also deciphered
hieroglyphics.
Malus, Sur une propriété des forces répulsives qui agissent sur la lumière, 1809.
Polarization by reflection.
Biot, Brewster, Arago, crystal physics, Arago, 1811, rotatory power of quartz.
1J. Drecker, Physikal. ZS. 2, 145, 1900.
