1911 Encyclopædia Britannica/Diffraction of Light/11

Diffraction of Light
§ 11. Dynamical Theory of Diffraction.

11. Dynamical Theory of Diffraction.—The explanation of diffraction phenomena given by Fresnel and his followers is independent of special views as to the nature of the aether, at least in its main features; for in the absence of a more complete foundation it is impossible to treat rigorously the mode of action of a solid obstacle such as a screen. But, without entering upon matters of this kind, we may inquire in what manner a primary wave may be resolved into elementary secondary waves, and in particular as to the law of intensity and polarization in a secondary wave as dependent upon its direction of propagation, and upon the character as regards polarization of the primary wave. This question was treated by Stokes in his “Dynamical Theory of Diffraction” (Camb. Phil. Trans., 1849) on the basis of the elastic solid theory.

Let x, y, z be the co-ordinates of any particle of the medium in its natural state, and χ, η, ζ the displacements of the same particle at the end of time t, measured in the directions of the three axes respectively. Then the first of the equations of motion may be put under the form

${\displaystyle {\frac {d^{2}\xi }{dt^{2}}}{=}b^{2}\left({\frac {d^{2}\xi }{dx^{2}}}+{\frac {d^{2}\xi }{dy^{2}}}+{\frac {d^{2}\xi }{dz^{2}}}\right)+(a^{2}-b^{2}){\frac {d}{dx}}\left({\frac {d\xi }{dx}}+{\frac {d\eta }{dy}}+{\frac {d\zeta }{dz}}\right),}$

where a2 and b2 denote the two arbitrary constants. Put for shortness

 ${\displaystyle {\frac {d\xi }{dx}}+{\frac {d\eta }{dy}}+{\frac {d\zeta }{dz}}=\delta }$ (1),

and represent by ∇2ξ the quantity multiplied by b2. According to this notation, the three equations of motion are

 ${\displaystyle {\frac {d^{2}\xi }{dt^{2}}}=b^{2}\nabla ^{2}\xi +(a^{2}-b^{2}){\frac {d\delta }{dx}}}$ ${\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}}$ (2). ${\displaystyle {\frac {d^{2}\eta }{dt^{2}}}=b^{2}\nabla ^{2}\eta +(a^{2}-b^{2}){\frac {d\delta }{dy}}}$ ${\displaystyle {\frac {d^{2}\zeta }{dt^{2}}}=b^{2}\nabla ^{2}\zeta +(a^{2}-b^{2}){\frac {d\delta }{dz}}}$

It is to be observed that S denotes the dilatation of volume of the element situated at (x, y, z). In the limiting case in which the medium is regarded as absolutely incompressible δ vanishes; but, in order that equations (2) may preserve their generality, we must suppose a at the same time to become infinite, and replace a2δ by a new function of the co-ordinates.

These equations simplify very much in their application to plane waves. If the ray be parallel to OX, and the direction of vibration parallel to OZ, we have ξ = 0, η = 0, while ζ is a function of x and t only. Equation (1) and the first pair of equations (2) are thus satisfied identically. The third equation gives

 ${\displaystyle {\frac {d^{2}\zeta }{dt^{2}}}{=}b^{2}{\frac {d^{2}\zeta }{dx^{2}}}}$ (3),

of which the solution is

 ζ = ƒ(bt − x) (4),

where ƒ is an arbitrary function.

The question as to the law of the secondary waves is thus answered by Stokes. “Let ξ = 0, η = 0, ζ = ƒ(btx) be the displacements corresponding to the incident light; let O1 be any point in the plane P (of the wave-front), dS an element of that plane adjacent to O1, and consider the disturbance due to that portion only of the incident disturbance which passes continually across dS. Let O be any point in the medium situated at a distance from the point O1 which is large in comparison with the length of a wave; let O1O = r, and let this line make an angle θ with the direction of propagation of the incident light, or the axis of x, and φ with the direction of vibration, or axis of z. Then the displacement at O will take place in a direction perpendicular to O1O, and lying in the plane ZO1O; and, if ζ′ be the displacement at O, reckoned positive in the direction nearest to that in which the incident vibrations are reckoned positive,

${\displaystyle \zeta ^{\prime }{=}{\frac {d\mathrm {S} }{4\pi r}}(1+\cos \theta )\sin \phi f^{\prime }(bt-r).}$

In particular, if

 ${\displaystyle f(bt-x){=}c\sin {\frac {2\pi }{\lambda }}(bt-x)}$ (5),

we shall have

 ${\displaystyle \zeta ^{\prime }{=}{\frac {cd\mathrm {S} }{2\lambda r}}(1+\cos \theta )\sin \phi \cos {\frac {2\pi }{\lambda }}(bt-r)}$ (6).”

It is then verified that, after integration with respect to dS, (6) gives the same disturbance as if the primary wave had been supposed to pass on unbroken.

The occurrence of sin φ as a factor in (6) shows that the relative intensities of the primary light and of that diffracted in the direction θ depend upon the condition of the former as regards polarization. If the direction of primary vibration be perpendicular to the plane of diffraction (containing both primary and secondary rays), sin φ = 1; but, if the primary vibration be in the plane of diffraction, sin φ = cos θ. This result was employed by Stokes as a criterion of the direction of vibration; and his experiments, conducted with gratings, led him to the conclusion that the vibrations of polarized light are executed in a direction perpendicular to the plane of polarization.

The factor (1 + cos θ) shows in what manner the secondary disturbance depends upon the direction in which it is propagated with respect to the front of the primary wave.

If, as suffices for all practical purposes, we limit the application of the formulae to points in advance of the plane at which the wave is supposed to be broken up, we may use simpler methods of resolution than that above considered. It appears indeed that the purely mathematical question has no definite answer. In illustration of this the analogous problem for sound may be referred to. Imagine a flexible lamina to be introduced so as to coincide with the plane at which resolution is to be effected. The introduction of the lamina (supposed to be devoid of inertia) will make no difference to the propagation of plane parallel sonorous waves through the position which it occupies. At every point the motion of the lamina will be the same as would have occurred in its absence, the pressure of the waves impinging from behind being just what is required to generate the waves in front. Now it is evident that the aerial motion in front of the lamina is determined by what happens at the lamina without regard to the cause of the motion there existing. Whether the necessary forces are due to aerial pressures acting on the rear, or to forces directly impressed from without, is a matter of indifference. The conception of the lamina leads immediately to two schemes, according to which a primary wave may be supposed to be broken up. In the first of these the element dS, the effect of which is to be estimated, is supposed to execute its actual motion, while every other element of the plane lamina is maintained at rest. The resulting aerial motion in front is readily calculated (see Rayleigh, Theory of Sound, § 278); it is symmetrical with respect to the origin, i.e. independent of θ. When the secondary disturbance thus obtained is integrated with respect to dS over the entire plane of the lamina, the result is necessarily the same as would have been obtained had the primary wave been supposed to pass on without resolution, for this is precisely the motion generated when every element of the lamina vibrates with a common motion, equal to that attributed to dS. The only assumption here involved is the evidently legitimate one that, when two systems of variously distributed motion at the lamina are superposed, the corresponding motions in front are superposed also.

The method of resolution just described is the simplest, but it is only one of an indefinite number that might be proposed, and which are all equally legitimate, so long as the question is regarded as a merely mathematical one, without reference to the physical properties of actual screens. If, instead of supposing the motion at dS to be that of the primary wave, and to be zero elsewhere, we suppose the force operative over the element dS of the lamina to be that corresponding to the primary wave, and to vanish elsewhere, we obtain a secondary wave following quite a different law. In this case the motion in different directions varies as cosθ, vanishing at right angles to the direction of propagation of the primary wave. Here again, on integration over the entire lamina, the aggregate effect of the secondary waves is necessarily the same as that of the primary.

In order to apply these ideas to the investigation of the secondary wave of light, we require the solution of a problem, first treated by Stokes, viz. the determination of the motion in an infinitely extended elastic solid due to a locally applied periodic force. If we suppose that the force impressed upon the element of mass D dx dy dz is

DZ dx dy dz,

being everywhere parallel to the axis of Z, the only change required in our equations (1), (2) is the addition of the term Z to the second member of the third equation (2). In the forced vibration, now under consideration, Z, and the quantities ξ, η, ζ, δ expressing the resulting motion, are to be supposed proportional to eint, where i = √(-1), and n = 2π/τ, τ being the periodic time. Under these circumstances the double differentiation with respect to t of any quantity is equivalent to multiplication by the factor -n2, and thus our equations take the form

 ${\displaystyle (b^{2}\nabla ^{2}+n^{2})\xi +(a^{2}-b^{2}){\frac {d\delta }{dx}}=0}$ ${\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}}$ (7). ${\displaystyle (b^{2}\nabla ^{2}+n^{2})\eta +(a^{2}-b^{2}){\frac {d\delta }{dy}}=0}$ ${\displaystyle (b^{2}\nabla ^{2}+n^{2})\zeta +(a^{2}-b^{2}){\frac {d\delta }{dz}}=-\mathrm {Z} }$

It will now be convenient to introduce the quantities.ϖ1, ϖ2, ϖ3 which express the rotations of the elements of the medium round axes parallel to those of co-ordinates, in accordance with the equations

 ${\displaystyle \varpi _{3}={\frac {d\xi }{dy}}-{\frac {d\eta }{dx^{\prime }}},\qquad \varpi _{1}={\frac {d\eta }{dz}}-{\frac {d\zeta }{dy^{\prime }}},\qquad \varpi _{2}{\frac {d\zeta }{dx}}-{\frac {d\xi }{dz^{\prime }}}}$ (8).

In terms of these we obtain from (7), by differentiation and subtraction,

 (b2Δ2 + n2) ϖ3 = 0 ${\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}}$ (9). (b2Δ2 + n2) ϖ1 = dZ/dy (b2Δ2 + n2) ϖ2 = −dZ/dx

The first of equations (9) gives

 ϖ3＝0 (10).

For ϖ1, we have

 ${\displaystyle \varpi _{1}={\frac {1}{4\pi b^{2}}}\iiint {\frac {d\mathrm {Z} }{dy}}{\frac {e^{-ikr}}{r}}dxdydz}$ (11),

where r is the distance between the element dx dy dz and the point where ϖ1 is estimated, and

 k＝n/b＝2π/λ (12),

λ being the wave-length.

(This solution may be verified in the same manner as Poisson’s theorem, in which k = 0.)

We will now introduce the supposition that the force Z acts only within a small space of volume T, situated at (x, y, z), and for simplicity suppose that it is at the origin of co-ordinates that the rotations are to be estimated. Integrating by parts in (11), we get

${\displaystyle \int {\frac {e^{-ikr}}{r}}{\frac {d\mathrm {Z} }{dy}}dy=\left[\mathrm {Z} {\frac {e^{-ikr}}{r}}\right]-\int \mathrm {Z} {\frac {d}{dy}}\left({\frac {e^{-ikr}}{r}}\right)dy,}$

in which the integrated terms at the limits vanish, Z being finite only within the region T. Thus

${\displaystyle \varpi _{1}={\frac {1}{4\pi b^{2}}}\iiint \mathrm {Z} {\frac {d}{dy}}\left({\frac {e^{-ikr}}{r}}\right)dxdydz.}$

Since the dimensions of T are supposed to be very small in comparison with λ, the factor ${\displaystyle {\frac {d}{dy}}\left({\frac {e^{-ikr}}{r}}\right)}$ is sensibly constant; so that, if Z stand for the mean value of Z over the volume T, we may write

 ${\displaystyle \varpi _{1}={\frac {\mathrm {TZ} }{4\pi b^{2}}}\cdot {\frac {y}{r}}\cdot {\frac {d}{dr}}\left({\frac {e^{-ikr}}{r}}\right)}$ (13).

In like manner we find

 ${\displaystyle \varpi _{2}=-{\frac {\mathrm {TZ} }{4\pi b^{2}}}\cdot {\frac {x}{r}}\cdot {\frac {d}{dr}}\left({\frac {e^{-ikr}}{r}}\right)}$ (14).

From (10), (13), (14) we see that, as might have been expected, the rotation at any point is about an axis perpendicular both to the direction of the force and to the line joining the point to the source of disturbance. If the resultant rotation be ω, we have

${\displaystyle \varpi ={\frac {\mathrm {TZ} }{4\pi b^{2}}}\cdot {\frac {{\sqrt {}}(x^{2}+y^{2})}{r}}\cdot {\frac {d}{dr}}\left({\frac {e^{-ikr}}{r}}\right)={\frac {\mathrm {TZ} \sin \phi }{4\pi b^{2}}}{\frac {d}{dr}}\left({\frac {e^{-ikr}}{r}}\right),}$

φ denoting the angle between r and z. In differentiating eikr/r with respect to r, we may neglect the term divided by r2 as altogether insensible, kr being an exceedingly great quantity at any moderate distance from the origin of disturbance. Thus

 ${\displaystyle \varpi ={\frac {-ik\cdot \mathrm {TZ} \sin \phi }{4\pi b^{2}}}\cdot {\frac {e^{-ikr}}{r}}}$ (15),

which completely determines the rotation at any point. For a disturbing force of given integral magnitude it is seen to be everywhere about an axis perpendicular to r and the direction of the force, and in magnitude dependent only upon the angle (φ) between these two directions and upon the distance (r).

The intensity of light is, however, more usually expressed in terms of the actual displacement in the plane of the wave. This displacement, which we may denote by ζ′, is in the plane containing z and r, and perpendicular to the latter. Its connexion with ϖ is expressed by ϖ = dζ′/dr; so that

 ${\displaystyle \zeta ^{\prime }={\frac {\mathrm {TZ} \sin \phi }{4\pi b^{2}}}\cdot {\frac {e^{\prime (at-kr)}}{r}}}$ (16),

where the factor eint is restored.

Retaining only the real part of (16), we find, as the result of a local application of force equal to

 DTZ cos nt (17),

the disturbance expressed by

 ${\displaystyle \zeta ^{\prime }={\frac {\mathrm {TZ} \sin \phi }{4\pi b^{2}}}\cdot {\frac {\cos(nt-kr)}{r}}}$ (18).

The occurrence of sin φ shows that there is no disturbance radiated in the direction of the force, a feature which might have been anticipated from considerations of symmetry.

We will now apply (18) to the investigation of a law of secondary disturbance, when a primary wave

 ζ＝sin(nt − kx) (19)

is supposed to be broken up in passing the plane x = 0. The first step is to calculate the force which represents the reaction between the parts of the medium separated by x = 0. The force operative upon the positive half is parallel to OZ, and of amount per unit of area equal to

b2D dζ/dxb2kD cos nt;

and to this force acting over the whole of the plane the actual motion on the positive side may be conceived to be due. The secondary disturbance corresponding to the element dS of the plane may be supposed to be that caused by a force of the above magnitude acting over dS and vanishing elsewhere; and it only remains to examine what the result of such a force would be.

Now it is evident that the force in question, supposed to act upon the positive half only of the medium, produces just double of the effect that would be caused by the same force if the medium were undivided, and on the latter supposition (being also localized at a point) it comes under the head already considered. According to (18), the effect of the force acting at dS parallel to OZ, and of amount equal to

2b2kD dS cos nt,

will be a disturbance

 ${\displaystyle \zeta ^{\prime }{=}{\frac {d\mathrm {S} \sin \phi }{\lambda r}}\cos(nt-kr)}$ (20),

regard being had to (12). This therefore expresses the secondary disturbance at a distance r and in a direction making an angle φ with OZ (the direction of primary vibration) due to the element dS of the wave-front.

The proportionality of the secondary disturbance to sin φ is common to the present law and to that given by Stokes, but here there is no dependence upon the angle θ between the primary and secondary rays. The occurrence of the factor λr−1, and the necessity of supposing the phase of the secondary wave accelerated by a quarter of an undulation, were first established by Archibald Smith, as the result of a comparison between the primary wave, supposed to pass on without resolution, and the integrated effect of all the secondary waves (§ 2). The occurrence of factors such as sin φ, or 12(1 + cos θ), in the expression of the secondary wave has no influence upon the result of the integration, the effects of all the elements for which the factors differ appreciably from unity being destroyed by mutual interference.

The choice between various methods of resolution, all mathematically admissible, would be guided by physical considerations respecting the mode of action of obstacles. Thus, to refer again to the acoustical analogue in which plane waves are incident upon a perforated rigid screen, the circumstances of the case are best represented by the first method of resolution, leading to symmetrical secondary waves, in which the normal motion is supposed to be zero over the unperforated parts. Indeed, if the aperture is very small, this method gives the correct result, save as to a constant factor. In like manner our present law (20) would apply to the kind of obstruction that would be caused by an actual physical division of the elastic medium, extending over the whole of the area supposed to be occupied by the intercepting screen, but of course not extending to the parts supposed to be perforated.

On the electromagnetic theory, the problem of diffraction becomes definite when the properties of the obstacle are laid down. The simplest supposition is that the material composing the obstacle is perfectly conducting, i.e. perfectly reflecting. On this basis A. J. W. Sommerfeld (Math. Ann., 1895, 47, p. 317), with great mathematical skill, has solved the problem of the shadow thrown by a semi-infinite plane screen. A simplified exposition has been given by Horace Lamb (Proc. Lond. Math. Soc., 1906, 4, p. 190). It appears that Fresnel’s results, although based on an imperfect theory, require only insignificant corrections. Problems not limited to two dimensions, such for example as the shadow of a circular disk, present great difficulties, and have not hitherto been treated by a rigorous method; but there is no reason to suppose that Fresnel’s results would be departed from materially.