This expression has the period 2[pi]/n with respect to the time and
the perion 2[pi]/k with respect to x, so that the "time of vibration"
and the "wave-length" are given by T = 2[pi]/n, [lambda] = 2[pi]/k.
Further, it is easily seen that the phase belonging to certain values
of x and t is equal to that which corresponds to x + [Delta]x and t +
[Delta]t provided [Delta]x = (n/k)[Delta]t. Therefore the phase, or
the disturbance itself, may be said to be propagated in the direction
normal to the wave-fronts with a velocity (velocity of the waves) v =
n/k, which is connected with the time of vibration and the wave-length
by the relation
[lambda] = vT. (6)
In isotropic bodies the propagation can go on in all directions with
the same velocity. In anisotropic bodies (crystals), with which the
theory of light is largely concerned, the problem is more complicated.
As a general rule we can say that, for a given direction of the
wave-fronts, the vibrations must have a determinate direction, if the
propagation is to take place according to the simple formula given
above. It is to be understood that for a given direction of the waves
there may be two or even more directions of vibration of the kind, and
that in such a case there are as many different velocities, each
belonging to one particular direction of vibration.
7. _Wave-surface._--After having found the values of v for a particular
frequency and different directions of the wave-normal, a very
instructive graphical representation can be employed.
Let ON be a line in any direction, drawn from a fixed point O, OA a
length along this line equal to the velocity v of waves having ON for
their normal, or, more generally, OA, OA´, &c., lengths equal to the
velocities v, v´, &c., which such waves have according to their
direction of vibration, Q, Q´, &c., planes perpendicular to ON through
A, A^1, &c. Let this construction be repeated for all directions of
ON, and let W be the surface that is touched by all the planes Q, Q´,
&c. It is clear that if this surface, which is called the
"wave-surface," is known, the velocity of propagation of plane waves
of any chosen direction is given by the length of the perpendicular
from the centre O on a tangent plane in the given direction. It must
be kept in mind that, in general, each tangent plane corresponds to
one definite direction of vibration. If this direction is assigned in
each point of the wave-surface, the diagram contains all the
information which we can desire concerning the propagation of plane
waves of the frequency that has been chosen.
The plane Q employed in the above construction is the position after
unit of time of a wave-front perpendicular to ON and originally
passing through the point O. The surface W itself is often considered
as the locus of all points that are reached in unit of time by a
disturbance starting from O and spreading towards all sides. Admitting
the validity of this view, we can determine in a similar way the locus
of the points reached in some infinitely short time dt, the
wave-surface, as we may say, or the "elementary wave," corresponding
to this time. It is similar to W, all dimensions of the latter surface
being multiplied by dt. It may be noticed that in a heterogeneous
medium a wave of this kind has the same form as if the properties of
matter existing at its centre extended over a finite space.
8. _Theory of Huygens._--Huygens was the first to show that the
explanation of optical phenomena may be made to depend on the
wave-surface, not only in isotropic bodies, in which it has a spherical
form, but also in crystals, for one of which (Iceland spar) he deduced
the form of the surface from the observed double refraction. In his
argument Huygens availed himself of the following principle that is
justly named after him: Any point that is reached by a wave of light
becomes a new centre of radiation from which the disturbance is
propagated towards all sides. On this basis he determined the progress
of light-waves by a construction which, under a restriction to be
mentioned in § 13, applied to waves of any form and to all kinds of
transparent media. Let [sigma] be the surface (wave-front) to which a
definite phase of vibration has advanced at a certain time t, dt an
infinitely small increment of time, and let an elementary wave
corresponding to this interval be described around each point P of
[sigma]. Then the envelope [sigma]´ of all these elementary waves is the
surface reached by the phase in question at the time t + dt, and by
repeating the construction all successive positions of the wave-front
can be found.
Huygens also considered the propagation of waves that are laterally
limited, by having passed, for example, through an opening in an
opaque screen. If, in the first wave-front [sigma], the disturbance
exists only in a certain part bounded by the contour s, we can confine
ourselves to the elementary waves around the points of that part, and
to a portion of the new wave-front [sigma]´ whose boundary passes
through the points where [sigma]´ touches the elementary waves having
their centres on s. Taking for granted Huygens's assumption that a
sensible disturbance is only found in those places where the
elementary waves are touched by the new wave-front, it may be inferred
that the lateral limits of the beam of light are determined by lines,
each element of which joins the centre P of an elementary wave with
its point of contact P´ with the next wave-front. To lines of this
kind, whose course can be made visible by using narrow pencils of
light, the name of "rays" is to be given in the wave-theory. The
disturbance may be conceived to travel along them with a velocity u =
PP´/dt, which is therefore called the "ray-velocity."
The construction shows that, corresponding to each direction of the
wave-front (with a determinate direction of vibration), there is a
definite direction and a definite velocity of the ray. Both are given
by a line drawn from the centre of the wave-surface to its point of
contact with a tangent plane of the given direction. It will be
convenient to say that this line and the plane are conjugate with each
other. The rays of light, curved in non-homogeneous bodies, are always
straight lines in homogeneous substances. In an isotropic medium,
whether homogeneous or otherwise, they are normal to the wave-fronts,
and their velocity is equal to that of the waves.
By applying his construction to the reflection and refraction of
light, Huygens accounted for these phenomena in isotropic bodies as
well as in Iceland spar. It was afterwards shown by Augustin Fresnel
that the double refraction in biaxal crystals can be explained in the
same way, provided the proper form be assigned to the wave-surface.
In any point of a bounding surface the normals to the reflected and
refracted waves, whatever be their number, always lie in the plane
passing through the normal to the incident waves and that to the
surface itself. Moreover, if [alpha]1 is the angle between these two
latter normals, and [alpha]2 the angle between the normal to the
boundary and that to any one of the reflected and refracted waves, and
v1, v2 the corresponding wave-velocities, the relation
sin [alpha]1/sin [alpha]2 = v1/v2 (7)
is found to hold in all cases. These important theorems may be proved
independently of Huygens's construction by simply observing that, at
each point of the surface of separation, there must be a certain
connexion between the disturbances existing in the incident, the
reflected, and the refracted waves, and that, therefore, the lines of
intersection of the surface with the positions of an incident
wave-front, succeeding each other at equal intervals of time dt, must
coincide with the lines in which the surface is intersected by a
similar series of reflected or refracted wave-fronts.
In the case of isotropic media, the ratio (7) is constant, so that we
are led to the law of Snellius, the index of refraction being given by
µ = v1/v2 (8)
(cf. equation 1).
9. _General Theorems on Rays, deduced from Huygens's
Construction._--(a) Let A and B be two points arbitrarily chosen in a
system of transparent bodies, ds an element of a line drawn from A to
B, u the velocity of a ray of light coinciding with ds. Then the
integral [int]u^(-1) ds, which represents the time required for a
motion along the line with the velocity u, is a minimum for the course
actually taken by a ray of light (unless A and B be too far apart).
This is the "principle of least time" first formulated by Pierre de
Fermat for the case of two isotropic substances. It shows that the
course of a ray of light can always be inverted.
(b) Rays of light starting in all directions from a point A and
travelling onward for a definite length of time, reach a surface
[sigma], whose tangent plane at a point B is conjugate, in the medium
surrounding B, with the last element of the ray AB.
(c) If all rays issuing from A are concentrated at a point B, the
integral [int]u^(-1) ds has the same value for each of them.
(d) In case (b) the variation of the integral caused by an infinitely
small displacement q of B, the point A remaining fixed, is given by
[delta][int]u^(-1) ds = q cos [theta]/v_B. Here [theta] is the angle
between the displacement q and the normal to the surface [sigma], in
the direction of propagation, v_B the velocity of a plane wave
tangent to this surface.
In the case of isotropic bodies, for which the relation (8) holds, we
recover the theorems concerning the integral [int]µds which we have
deduced from the emission theory (§ 5).
10. _Further General Theorems._--(a) Let V1 and V2 be two planes in a
system of isotropic bodies, let rectangular axes of coordinates be
chosen in each of these planes, and let x1, y1 be the coordinates of a
point A in V1, and x2, y2 those of a point B in V2. The integral
[int]µds, taken for the ray between A and B, is a function of x1, y1,
x2, y2 and, if [xi]1 denotes either x1 or y1, and [xi]2 either x2 or
y2, we shall have
_ _
[dP]² / [dP]² /
------------------- | µ ds = ------------------- | µ ds.
[dP][xi]1 [dP][xi]2 _/ [dP][xi]2 [dP][xi]1 _/
On both sides of this equation the first differentiation may be
performed by means of the formula (3). The second differentiation
admits of a geometrical interpretation, and the formula may finally be
employed for proving the following theorem:
Let [omega]1 be the solid angle of an infinitely thin pencil of rays
issuing from A and intersecting the plane V2 in an element [sigma]2 at
the point B. Similarly, let [omega]2 be the solid angle of a pencil
starting from B and falling on the element [sigma]1 of the plane V1 at
the point A. Then, denoting by µ1 and µ2 the indices of refraction of
the matter at the points A and B, by [theta]1 and [theta]2 the sharp
angles which the ray AB at its extremities makes with the normals to
V1 and V2, we have
(µ1)² [sigma]1 [omega]1 cos [theta]1 =
(µ2)² [sigma]2 [omega]2 cos [theta]2.
(b) There is a second theorem that is expressed by exactly the same
formula, if we understand by [sigma]1 and [sigma]2 elements of surface
that are related to each other as an object and its optical image--by
[omega]1, [omega]2 the infinitely small openings, at the beginning and
the end of its course, of a pencil of rays issuing from a point A of
[sigma]1 and coming together at the corresponding point B of [sigma]2,
and by [theta]1, [theta]2 the sharp angles which one of the rays makes
with the normals to [sigma]1 and [sigma]2. The proof may be based upon
the first theorem. It suffices to consider the section [sigma] of the
pencil by some intermediate plane, and a bundle of rays starting from
the points of [sigma]1 and reaching those of [sigma]2 after having all
passed through a point of that section [sigma].
(c) If in the last theorem the system of bodies is symmetrical around
the straight line AB, we can take for [sigma]1 and [sigma]2 circular
planes having AB as axis. Let h1 and h2 be the radii of these circles,
i.e. the linear dimensions of an object and its image, [epsilon]1 and
[epsilon]2 the infinitely small angles which a ray R going from A to B
makes with the axis at these points. Then the above formula gives
µ1h1[epsilon]1 = µ2h2[epsilon]2, a relation that was proved, for the
particular case µ1 = µ2 by Huygens and Lagrange. It is still more
valuable if one distinguishes by the algebraic sign of h2 whether the
image is direct or inverted, and by that of [epsilon]2 whether the ray
R on leaving A and on reaching B lies on opposite sides of the axis or
on the same side.
The above theorems are of much service in the theory of optical
instruments and in the general theory of radiation.
11. _Phenomena of Interference and Diffraction._--The impulses or
motions which a luminous body sends forth through the universal medium
or aether, were considered by Huygens as being without any regular
succession; he neither speaks of vibrations, nor of the physical cause
of the colours. The idea that monochromatic light consists of a
succession of simple harmonic vibrations like those represented by the
equation (5), and that the sensation of colour depends on the frequency,
is due to Thomas Young[11] and Fresnel,[12] who explained the phenomena
of interference on this assumption combined with the principle of
super-position. In doing so they were also enabled to determine the
wave-length, ranging from 0.000076 cm. at the red end of the spectrum to
0.000039 cm. for the extreme violet and, by means of the formula (6),
the number of vibrations per second. Later investigations have shown
that the infra-red rays as well as the ultra-violet ones are of the same
physical nature as the luminous rays, differing from these only by the
greater or smaller length of their waves. The wave-length amounts to
0.006 cm. for the least refrangible infra-red, and is as small as
0.00001 cm. for the extreme ultra-violet.
Another important part of Fresnel's work is his treatment of diffraction
on the basis of Huygens's principle. If, for example, light falls on a
screen with a narrow slit, each point of the slit is regarded as a new
centre of vibration, and the intensity at any point behind the screen is
found by compounding with each other the disturbances coming from all
these points, due account being taken of the phases with which they come
together (see DIFFRACTION; INTERFERENCE).
12. _Results of Later Mathematical Theory._--Though the theory of
diffraction developed by Fresnel, and by other physicists who worked on
the same lines, shows a most beautiful agreement with observed facts,
yet its foundation, Huygens's principle, cannot, in its original
elementary form, be deemed quite satisfactory. The general validity of
the results has, however, been confirmed by the researches of those
mathematicians (Siméon Denis Poisson, Augustin Louis Cauchy, Sir G. G.
Stokes, Gustav Robert Kirchhoff) who investigated the propagation of
vibrations in a more rigorous manner. Kirchhoff[13] showed that the
disturbance at any point of the aether inside a closed surface which
contains no ponderable matter can be represented as made up of a large
number of parts, each of which depends upon the state of things at one
point of the surface. This result, the modern form of Huygens's
principle, can be extended to a system of bodies of any kind, the only
restriction being that the source of light be not surrounded by the
surface. Certain causes capable of producing vibrations can be imagined
to be distributed all over this latter, in such a way that the
disturbances to which they give rise in the enclosed space are exactly
those which are brought about by the real source of light.[14] Another
interesting result that has been verified by experiment is that,
whenever rays of light pass through a focus, the phase undergoes a
change of half a period. It must be added that the results alluded to in
the above, though generally presented in the terms of some particular
form of the wave theory, often apply to other forms as well.
13. _Rays of Light._--In working out the theory of diffraction it is
possible to state exactly in what sense light may be said to travel in
straight lines. Behind an opening _whose width is very large in
comparison with the wave-length_ the limits between the illuminated and
the dark parts of space are approximately determined by rays passing
along the borders.
This conclusion can also be arrived at by a mode of reasoning that is
independent of the theory of diffraction.[15] If linear differential
equations admit a solution of the form (5) with A constant, they can
also be satisfied by making A a function of the coordinates, such
that, in a wave-front, it changes very little over a distance equal to
the wave-length [lambda], and that it is constant along each line
conjugate with the wave-fronts. In cases of this kind the disturbance
may truly be said to travel along lines of the said direction, and an
observer who is unable to discern lengths of the order of [lambda],
and who uses an opening of much larger dimensions, may very well have
the impression of a cylindrical beam with a sharp boundary.
A similar result is found for curved waves. If the additional
restriction is made that their radii of curvature be very much larger
than the wave-length, Huygens's construction may confidently be
employed. The amplitudes all along a ray are determined by, and
proportional to, the amplitude at one of its points.
14. _Polarized Light._--As the theorems used in the explanation of
interference and diffraction are true for all kinds of vibratory
motions, these phenomena can give us no clue to the special kind of
vibrations in light-waves. Further information, however, may be drawn
from experiments on plane polarized light. The properties of a beam of
this kind are completely known when the position of a certain plane
passing through the direction of the rays, and _in_ which the beam is
said to be polarized, is given. "This plane of polarization," as it is
called, coincides with the plane of incidence in those cases where the
light has been polarized by reflection on a glass surface under an angle
of incidence whose tangent is equal to the index of refraction
(Brewster's law).
The researches of Fresnel and Arago left no doubt as to the direction of
the vibrations in polarized light with respect to that of the rays
themselves. In isotropic bodies at least, the vibrations are exactly
transverse, i.e. perpendicular to the rays, either in the plane of
polarization or at right angles to it. The first part of this statement
also applies to unpolarized light, as this can always be dissolved into
polarized components.
Much experimental work has been done on the production of polarized rays
by double refraction and on the reflection of polarized light, either by
isotropic or by anisotropic transparent bodies, the object of inquiry
being in the latter case to determine the position of the plane of
polarization of the reflected rays and their intensity.
In this way a large amount of evidence has been gathered by which it has
been possible to test different theories concerning the nature of light
and that of the medium through which it is propagated. A common feature
of nearly all these theories is that the aether is supposed to exist not
only in spaces void of matter, but also in the interior of ponderable
bodies.
15. _Fresnel's Theory._--Fresnel and his immediate successors
assimilated the aether to an elastic solid, so that the velocity of
propagation of transverse vibrations could be determined by the formula
v = [root](K/[rho]), where K denotes the modulus of rigidity and [rho]
the density. According to this equation the different properties of
various isotropic transparent bodies may arise from different values of
K, of [rho], or of both. It has, however, been found that if both K and
[rho] are supposed to change from one substance to another, it is
impossible to obtain the right reflection formulae. Assuming the
constancy of K Fresnel was led to equations which agreed with the
observed properties of the reflected light, if he made the further
assumption (to be mentioned in what follows as "Fresnel's assumption")
that the vibrations of plane polarized light are perpendicular to the
plane of polarization.
Let the indices p and n relate to the two principal cases in which the
incident (and, consequently, the reflected) light is polarized in the
plane of incidence, or normally to it, and let positive directions h
and h´ be chosen for the disturbance (at the surface itself) in the
incident and for that in the reflected beam, in such a manner that, by
a common rotation, h and the incident ray prolonged may be made to
coincide with h´ and the reflected ray. Then, if [alpha]1 and [alpha]2
are the angles of incidence and refraction, Fresnel shows that, in
order to get the reflected disturbance, the incident one must be
multiplied by
[alpha]_p = -sin ([alpha]1 - [alpha]2) / sin ([alpha]1 + [alpha]2) (9)
in the first, and by
[alpha]_n = tan ([alpha]1 - [alpha]2) / tan ([alpha]1 + [alpha]2) (10)
in the second principal case.
As to double refraction, Fresnel made it depend on the unequal
elasticity of the aether in different directions. He came to the
conclusion that, for a given direction of the waves, there are two
possible directions of vibration (§6), lying in the wave-front, at right
angles to each other, and he determined the form of the wave-surface,
both in uniaxal and in biaxal crystals.
Though objections may be urged against the dynamic part of Fresnel's
theory, he admirably succeeded in adapting it to the facts.
16. Electromagnetic Theory.--We here leave the historical order and pass
on to Maxwell's theory of light.
James Clerk Maxwell, who had set himself the task of mathematically
working out Michael Faraday's views, and who, both by doing so and by
introducing many new ideas of his own, became the founder of the
modern science of electricity,[16] recognized that, at every point of
an electromagnetic field, the state of things can be defined by two
vector quantities, the "electric force" E and the "magnetic force" H,
the former of which is the force acting on unit of electricity and the
latter that which acts on a magnetic pole of unit strength. In a
non-conductor (dielectric) the force E produces a state that may be
described as a displacement of electricity from its position of
equilibrium. This state is represented by a vector D ("dielectric
displacement") whose magnitude is measured by the quantity of
electricity reckoned per unit area which has traversed an element of
surface perpendicular to D itself. Similarly, there is a vector
quantity B (the "magnetic induction") intimately connected with the
magnetic force H. Changes of the dielectric displacement constitute an
electric current measured by the rate of change of D, and represented
in vector notation by