L1 Optics
1. Formation and propagation of light
You can sense with your eye the difference between bright and weak light. Moreover, in bright light, you can generate the one or other sensation by closing or opening your eyes; when your eyes are open, you say light enters. In other words, we speak of light as a cause, which can provoke inside us a light- sensation and recognize the eye as a mediator. However, we also sense occasionally brightness with closed eyes and also in darkness, for example, when your eye is struck or when an electric current flows in a definite direction through your head, in fact, in the case of any stimulation of the optical nerve which leads from the eye to the brain. Similarly to the sensing of sound, you can define the sensation of light as the characteristic mode of reaction of the eye to external stimulation. To all external stimulation! You call light the stimulus which is considered to be the normal agent of light sensation. This terminology, which has been derived from a particular sensation, is too narrow, because the same means of stimulation generate, when they act on your skin, a sensation of heat and they can also cause (photographic) processes which have nothing to do with sensation. However, in the sequel, we are merely concerned with something which can evoke the sensation of light.
Formation and propagation of light
The normal cause of light sensation is a source of light. It would appear that the simplest concept of how light spreads out from its source is that the source of light emits particles - corpuscles - which, as they encounter the eye, provoke the sensation of light. This corpuscular theory - emission theory of Newton 1672, according to which light is a substance - had only historical significance from about 1830 until the start of the Twentieth Century. Certain of its consequences contradicted experience; in particular, it was unable to explain - and this decided its fate - the interference phenomena of light.
What is to be understood by this? Imagine that the process is described by Fig. 351: Sound waves, emitted by the same source, reach simultaneously your ear by two different routes and under certain given conditions you cannot hear the sound. Now imagine that the ear is replaced by the eye, the source of sound by a source of light. You will now understand what is meant by interference phenomena of light: Phenomena during which light is added to light and under certain conditions the result may be darkness. We cannot imagine that, if light is a substance and creates brightness, it should be possible to create darkness by addition of the same substance to that substance*. However, we can explain satisfactorily interference phenomena by interpreting light as a wave motion like that of sound. The wave theory of light (Huygens. Fresnel, Young) assumes that certain particles in a shining body oscillate very fast, that these oscillations propagate from the body into space and provoke sensations of light as they encounter optical nerves. Certain phenomena force us to assume that the oscillations occur perpendicularly to their direction of propagation (Fig. 283) and not in that direction as in the case of sound (Fig. 284).
*However, as theory of light quanta, the emission theory has raised renewed interest; the latest discoveries (in 1935) have shown that the corpuscular theory and wave theory do not exclude each other, but fuse at a higher level. This new theory of quantum-mechanics and quantum-electrodynamics admits not only light, but matter itself as well as the concept of particles (atoms, electron-light-quanta) as well as that of waves as clear description of actual phenomena, which cannot be conceived otherwise; the applicability of the one or the other concept depends on the experimental conditions and it can be shown that contradictions do not arise. The wave nature of matter has been demonstrated irrevocably by interference experiments with electrons (cathode rays)
Hence there arises immediately the question: How do the oscillations of the source of light reach the eye? Sound is transmitted between the source and the ear by oscillations of matter - solid, fluid or gaseous; as a rule, it is air. But light passes through empty space. Hence it cannot be matter which propagates the oscillations of the source of light. The wave theory assumes that it is realized by the ether. To start with, it ascribes to it only the property that it exists everywhere, also at each point in empty space, and that it can oscillate and transmit energy. The transmission of light from its source to the eye has then to be envisaged as follows: To start with, the ether close to the source of light begins to oscillate, these oscillations communicate themselves to the ether between the source of light and the eye and eventually set in motion the ether next to the retina of the eye, the oscillations of which cause the sensation of light.
This theory of
light is a partner with that of sound. Since its general
acceptance (about 100 years ago) it has had to be changed
considerably, more accurately stated, extended. While the concept
of the propagation by waves - transverse waves - has remained, the model of light waves has changed.
Initially, it was assumed: The particles of the ether oscillate as a result of elastic forces, which act between them and therefore
form transverse waves. But transverse waves can only propagate
along a row of points (Fig. 283), if the elastic force acting in between them is shear (acoustic). However, only solids have shearing resistance;
hence the ether had to be given a characteristic property of a solid. However, the heavenly bodies, which
move all the time through the 
ether, are not impeded; hence one had to
ascribe to the fictitious solid body the property of a frictionless gas - an unacceptable contradiction! This contradiction is
avoided by the electro-magnetic theory of light of Maxwell
1864: Accordingly, the
waves are generated by electric and magnetic phenomena in the
ether, which repeat periodically at each particle of the ether
and are transmitted to their neighbours. Hence a source of light
is a spatially and timely periodic change of the electro-dynamic
field. A wave is this
periodic change in the ether in the same sense as the change
between compaction and rarefaction during the propagation of
sound in the atmosphere is a wave,
that is, in graphical terms: The curve which represents the change of state of the
electro-magnetic field has a wave form (Fig. 294).
Also this theory
had to be extended, because it could not explain the dispersion of the colour of 
light, while the electron theory of
dispersion based on the concepts of Lorentz agrees
well with reality. Moreover, the Theory of Relativity has invalidated the concept of a
material ether, while the electromagnetic field can be assumed to
be accepted empirically. - It is unimportant for the following
presentation whether the oscillations are imagined to have been
provoked by elastic or electro-magnetic forces, whence we will
throughout refer to elastic waves, which are more readily
interpreted.
Velocity of propagation of light
How long does it take light to travel from its source to our eye? Every day experience seems to deprive it of a practical sense, because we always have the impression that switching on of a light, however far away it is, is accompanied by an instantaneous sensation. However, this impression only arises, because the velocity of the waves of light is very large. They cover the distances between terrestrial sources of light and the eye in minute fractions of a second. Such a small time interval can only be detected with special equipment, which therefore is required to measure the velocity of light over terrestrial distances. Olaf Römer 1644-1710 1676 discovered in connection with the predictable eclipses of the moons of Jupiter annual deviations between the observed and theoretical values. When Earth was furthest away from Jupiter, they occurred 16 minutes 22 seconds later than when the distance between the two planets was shortest. If you set this distance equal to 3x108 km and the time used to 1000 seconds (instead of 982), you find for the velocity of light around 3x105 km/sec. James Bradley 1692-1762 1728 found a similar value by means of the aberration of fixed stars, discovered by him - a proof that the light of independent sources of light (fixed stars) propagates just as fast as that of dependent ones (planets). Today, astronomical methods for the measurement of the velocity of propagation of light are only of historical interest. They have been replaced by terrestrial methods, because the distances covered by the light can be determined more exactly. The oldest methods belong to Fizeau and Foucault 1854.
We will describe here the latest
method by which Michelson has measured the velocity of light (Fig. 610) between 1921 and 1926. The source of light is the slit
S, from which comes the light of an arc lamp. The light
encounters one of the two planes a of a vertical
octogonal prism, the planes of which are mirrors, and reaches via
the path abcDEfEDcb1a the eye O of the observer. Everything is
fixed, only the octogonal mirror prism rotates
about its vertical axis of symmetry. Hence the light, which
meets a, reaches the observer's eye only under one
condition: Since always after 1/8 turn of the prism, the path of
the light indicated in Fig.
610 is restored (by which alone
the light can go from S to O), the mirror
must make in the same time interval 1/8 turn, which the light needs
to go from bcDEfEDcb1 to a'. If this
condition is met, the observer sees in O all
the time the source of light. The distance, which the light had
thus to cover, was approximately 71 km, whence the time it
required (using 3·105 km) was approximately 0.00023
sec. This yielded the approximate rate of rotation which had to be given to the mirror in
order to meet the above condition. Hence the measurement was
reduced to an exact determination of the mirror's rotation rate
and an exact measurement of the path of the light. The rate of
rotation turned out to be 528, the path of the light, exact to
within 1 cm in 5 - 10 km, 2 x 35,373.21 m. After hundreds of
observations, the result for the velocity of light was 299,796 
4 km/sec.
Like in the case of sound, the velocity of light does not depend on its strength. This differs from its dependence on the wave length. We explain first of all: Just as in the case of sound, there occur shorter and longer waves. Just as there correspond to waves of sound with different lengths tones of different height, different lengths of waves of light correspond to lights of different colour: Red corresponds to the longest waves, green in between, violet to the shortest waves. As we have asked earlier: Do tones of different height propagate equally fast? we ask now: Do kinds of light of different colour propagate equally fast? If high and low tones were to propagate at different speeds, then differently high tones, which are emitted simultaneously far away from you, would reach you one after another - a chord would reach you broken into individual sounds. The light, which reaches you from a star - say that during one of Jupiter's so frequently observed lunar eclipses - is composed of all possible colours, that is, it contains light of all possible wave lengths. If these were to propagate at different speeds, you would have to see one of these moons as it reappears in all possible colours and only in the end you would see it in its natural light. This contradicts observations: In empty space, short and long waves of light propagate equally fast.
It is different in weighable substances. We will encounter there the decomposition of light into a spectrum as a result of the different velocities of propagation of lights of different wave lengths. Of course, in most gases, for example, air, the differences for differently coloured lights are so small that light is treated by air like by empty space.
Astronomical methods measure the velocity of light in empty space, terrestrial methods in air. Hence you can find out by means of the latter whether light has in other substances a different velocity: For example, light passes through air 1.33 times faster than through water. The number which states the ratio of the velocities of light in other substances to that in air is called the refractive index n.
However, the
refractive index n of a substance, that is also the ratio of the just discussed velocities 
of propagation
of light, can be obtained much more simply by refraction. It also yields for water (for an
intermediate wave length) almost 1.33, but for carbon disulphide
1.64, while the measured velocities of propagation yield 1.75. Rayleigh 1877 discovered the reason: A source of light
emits light with different velocities of propagation. If there is
even only a very slight difference as in Fig. 353 between the waves a and b,
they become a wave pattern c, a wave group and - this is the important
aspect - the velocity of the group is not the same as that of the individual waves (phase velocity) except in a substance without or with negligible
dispersion (vacuum,
water, air). You understand by group velocity the velocity at
which the maximum (or also the minimum) of the amplitude
propagates. When you measure the velocity of propagation of
light, we measure its group velocity. If you measure the refractive
index refractometrically, you work only with (homogeneous) light
of one wave length, whence such a
refractive index is the quotient of the phase velocity of light in vacuum to that in
the substance under consideration. In substances with no or negligible dispersion, the group velocity
and phase velocity are numerically the same. However, in strongly
dispersing substances, the wave group propagates
more slowly (w) than an individual
wave, the velocity (v) of which you can compute
from the corresponding refractometrically determined n.
The reason is: While you have c/v = n,
you have c/w = n minus a quantity,
which consists of the wave length times the change of the wave length with the
refractive index. For vacuum, air, etc., the subtracted term is
negligible, whence v computed from n coincides
with the directly measured value of w, while for
strongly dispersing substances like carbon disulphide the
difference is significant.
The velocity of light does not depend on the motion of the source of light (Willem de Sitter 1872-1934 1913)
Does the motion of a source of light influence
the velocity of propagation of the 
light it emits? Does
its motion transfer to that of the emitted light? Observations of spectroscopic double
stars gave the answer: No! Double stars are
fixed star systems - about 20000 of them were known in 1935 -
which consist of two stars which orbit around each other; spectroscopic double stars for the visual
separation of which telescopes are insufficient, but the
spectrocsope allows to detect by the periodic shifts of the lines
in their spectrum (Doppler principle) that they orbit around each other;
occasionally even by the separation
of the lines of the
two components. The argument is as follows: A and B
in Fig.
611 are the components
of the double star. The motion of A, compared with that
of B, is so small, that A can be considered to
be at rest; B orbits it counter-clockwise. Earth lies at
the distance d in the orbital plane of the double 
star. Denote by c the velocity
of light, u the orbital velocity, 2T the
orbital time of B. We will focus out attention only at
those instants, at which B moves at its total orbital velocity either away
from or towards Earth, that is, when it passes through a or b.
They are the instants t = 0, 2T,
4T, ··· and t = 0, T,
3T, ···. According to the theory of Ritz, the velocity
of light c reduces (grows) by the orbital velocity u
at a or (b), whence: At t = 0, B sends a signal
with the velocity c - u, which reaches Earth
after d/(c - u) sec, that is, at t=d/(c-u).
The star travels from a around
the semi-circle I. At time t=T, it
sends a signal with the velocity c + u, which
reaches Earth after d/(c + u) sec,
that is, at time T + d/(c + u);
for the observer on Earth, B has used for the travel
around the semi-circle I the time interval T + d/(c + u) - d/(c
- u) = T - 2ud/(c² - u²)
= T - 2ud/c² sec. (You can neglect u²
compared with c² !) At t = 2T, B sends
again the signal with the velocity c-u, which
reaches Earth at t = 2T + d/(c - u).
Hence, for the observer on Earth, the star has travelled around
the semi-circle II during T - 2ud/c²
sec. Hence, if Ritz's assumption is correct, the observer
would get the impression that B travels around the star A
along I during another time than along II;
however, the value T - 2ud/c² could
vanish for a by all means admissible choice of T, U,
d. This would cause in the spectroscope a certain
anomaly of line shift (according to the Doppler principle). This anomaly has never been observed; it
could not have escaped the observer's attention, if it were
present. Hence, the signals of light, which the star B
sends to Earth must at the instants, at which it moves towards
Earth, have the same velocity as when it moves away from earth;
in other words: The velocity of light does not depend on the
motion of its source. This fact is called the principle of the independence of the
vacuum velocity of light from the motion of the light's source. It
forms one of the main supports of the Theory of Relativity.
Straight line spreading of light
As long as light and sound propagate unimpeded, there is no fundamental difference between their spreading in space. However, matters change as soon as they encounter a wall, which impedes their propagation, but contains an opening, through which they can reach beyond. The difference is indicated by the fact that you can hear but not look around a corner. We will start with an explanation of one of the fundamental laws of the theory of light.
Not being able to look around a corner means: You can only see a point - a point-like source of light - when a straight line - air line - uninterruptedly leads from the point to the eye, that is, it does not intersect an object of any kind (we also exclude transparent bodies and every kind of optical instrumentation). Sound is not subject to such a restrictive condition. In words: Light only propagates along straight lines. If you intercept light coming from a point-source L by a wall WW (Fig. 612) and only let it pass through a hole AB to a table W1W1, you see on the - otherwise dark - table a spot of light, the outlines of which are determined by straight lines drawn from I through the border of AB (assuming the table to be parallel to the opening). If L were a source of sound, an ear on the table would hear it whether the straight line through the opening passed through L or not.
"The law of spreading of light along straight rays like any other fundamental law of Physics has not been based on special observations, designed for this purpose. . . It takes its validity from agreement of its consequences with experience. Everywhere in daily life, and in full strength in Astronomy and Geodesy, one relies on its unconditional validity ; . . . and further conclusions drawn from it have always been in total agreement with the initial assumption. Those innumerable, partly critical confirmations of the law have yielded it a soundness and general acceptance unequalled by hardly any other law of Nature" (Siegfried Szapski).
If the law applies unconditionally: Firstly, the border between the lighted and dark sections of the table W1W1, should be sharp. However, it is fuzzy. At the border of the bundle of light, where it passes exactly the rim of AB, the light deviates from the straight line. (To be sure, this deviation is minute and the amount of the deviating light vanishes compared with the light propagated along straight lines,so that, as a rule, it need not be taken into consideration.) Secondly, the border of the spot of light would always have to be determined, however wide or narrow is AB, by the straight lines drawn from L to the border of AB and extended to W1W1. Also this is not true. If you reduce prgressively the size of the opening, the border of AB becomes more uncertain, loses eventually all similarity with the opening and the spot of light becomes larger than it would be with straight line propagation. When eventually the opening p becomes a point, the light spreads out over the table MN (Fig. 613); the opening p then behaves as if the light spread out from it. Through such a narrow opening, light spreads exactly as we know is the case for sound.
The fundamental difference between the spreading of sound and that of light is related to the size of the opening through which they pass. If we introduce now the later required concept of the length of a wave of light, we can state briefly the result of a relevant investigation.
The manner in which a wave motion spreads through an opening is decided by the length of the wave in relation to the width of the opening. If the opening is so large or the length of the wave so small, that related to the width of the opening the wave length can be considered to be exceedingly small, the wave motion affects only those points beyond the hole which lie on a straight line between the point of the opening and the centre of excitation of the wave motion - this is the case for visible light, because its wave lengths are minute (0.0004 - 0.0008 mm). However, if the wave is so long or the opening so small that the opening is not for practical purposes infinitely large compared to the wave length - the common case for sound (the musically employable wave lengths lie between 7 cm and 8 m) and under the previously stated conditions for light - then the wave motion behaves exactly as if the centre of the waves were lying in the opening; it then propagates from the opening to all sides (diffraction of light).
As long as the opening AB has a certain width, we can by a reduction of its size separate smaller and smaller cones of light from the sphere of rays about the source of light without the spot of light - apart from an unimportant lack of sharpness of its boundary - deviating in form and size from what one expects. Hence we can say: A cone of light can be decomposed into smaller bundles of light, which are independent of each other. However, we can only do this as long as the opening AB has not less than a certain size. If it is narrowed down to a point (Fig. 613), it is not at all possible to obtain a spot of light of the size of a point; in other words, there do not exist single rays.
Hence we can only state with reservation that light spreads along straight rays. Nevertheless, we can retain this statement, because the strict theory of light, which explains completely the deviating phenomena, shows in agreement with experience that bundles of light of finite (that is, not point like) cross-section behave in many respects as if they are composed out of individual, independent of each other rays along which the light propagates. Only in the phenomena of interference and diffraction and then often only with special auxiliary means can we detect exceptions to this rule, although it never holds strictly. Also at the borders of bundles of finite cross-section, light's behaviour deviates from this rule, but the amount of deviating light vanishes compared with the regularly propagating amount. Hence, unless you are concerned with refraction and interference, you can hold on to the much simpler (and to everyday experiences accessible) assumptions, not admitted by the strict theory, and derive a large section of the theoretical results by much simpler means and yet with sufficient approximation to reality. Hence we will exclude in the immediate sequel, until further notice, those exceptional cases and base our study on rectilinear spreading of light and independence of parts of the bundles of light from each other.
Speaking fundamentally, the difference between recilinear spreading of light and the non-rectilinear spreading of sound depends on the length of the waves. The same difference explains, on the one hand, in the case of light the formation of shadows, on the other hand, the absence (more strictly: Their occurrence under special conditions) of an analogous phenomenon for sound. If you place between the source of light L and the wall WW a plate F (Fig. 614), which the light cannot penetrate, there arises on the otherwise illuminated wall ww a shadow, the border of which is determined in the same manner as that of the spot of light in Fig. 612, and the form of which, if F is parallel to WW, is similar to that of the shadow producing body. An eye within the shadow cannot see L, while an ear could hear the sound from a source of sound at L. Also here, the the border of the shadow is uncertain and becomes more so the smaller is the shadow producing F; it loses all similarity to F and becomes much smaller than it could be as a result of strictly linear propagation of the light. If the shadow producing object becomes still smaller, it eventually does not produce a shadow - the light waves behave towards it exactly as waves of sound behave in the presence of bodies on their way. They circumvent it.
Can you cause sound to propagate through an opening in a wall recitilinearly like light? You would then have to make the opening in the wall so large that the wave length of the sound is negligible. Then there would arise for sound the same conditions under which light, as a rule, passes through an opening. We would then have to operate with dimensions which would be two to three million times those arising with the phenomena of light. There would correspond to an opening 1 millimetre wide for light's passage one which is 1 kilometre wide for sound. Under every day conditions, under which we hear, openings through which sound passes are therefore not wide enough compared with its wave length, in order to cause it to spread rectilinearly. However, on exceptional occasions, you can observe sound shadows.
An explosion on the River Mersey (Liverpool lies at its mouth) was heard for many miles from its location E in the direction of the arrow ( Fig. 615), but not at X closeby, which was situated behind a hill. The sound happened to pass opposite to its source through a valley (in the hill), an opening which was immensely large compared with the wave of the explosion. In other words, it propagated rectilinearly from E through the opening, and the location X was in its sound shadow.
