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, A1, &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 σ 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 σ. Then the envelope σ′ 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 σ, 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 σ′ whose boundary passes through the points where σ′ 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 α1 is the angle between these two latter normals, and α2 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 α1/sin α2 = 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 ∫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 σ, 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 ∫u−1ds 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 δ∫u−1ds = q cos θ/vB. Here θ is the angle between the displacement q and the normal to the surface σ, in the direction of propagation, vB 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 ∫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 ∫μds, taken for the ray between A and B, is a function of x1, y1, x2, y2 and, if ξ1 denotes either x1 or y1, and ξ2 either x2 or y2, we shall have
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 ω1 be the solid angle of an infinitely thin pencil of rays issuing from A and intersecting the plane V2 in an element σ2 at the point B. Similarly, let ω2 be the solid angle of a pencil starting from B and falling on the element σ1 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 θ1 and θ2 the sharp angles which the ray AB at its extremities makes with the normals to V1 and V2, we have
(b) There is a second theorem that is expressed by exactly the same formula, if we understand by σ1 and σ2 elements of surface that are related to each other as an object and its optical image—by ω1, ω2 the infinitely small openings, at the beginning and the end of its course, of a pencil of rays issuing from a point A of σ1 and coming together at the corresponding point B of σ2, and by θ1, θ2 the sharp angles which one of the rays makes with the normals to σ1 and σ2. The proof may be based upon the first theorem. It suffices to
