If R = ½λ, I vanishes at ξ= 0; but the whole illumination, represented by ∫+∞−∞I dξ, is independent of the value of R. If R = 0, I = (1/ξ²) sin² (2πξh/λƒ), in agreement with § 3, where a has the meaning here attached to 2h.
The expression (5) gives the illumination at ξ due to that part of the complete image whose geometrical focus is at ξ = 0, the retardation for this component being R. Since we have now to integrate for the whole illumination at a particular point O due to all the components which have their foci in its neighbourhood, we may conveniently regard O as origin. ξ is then the co-ordinate relatively to O of any focal point O′ for which the retardation is R; and the required result is obtained by simply integrating (5) with respect to ξ from −∞ to +∞. To each value of ξ corresponds a different value of λ, and (in consequence of the dispersing power of the plate) of R. The variation of λ may, however, be neglected in the integration, except in 2πR/λ, where a small variation of λ entails a comparatively large alteration of phase. If we write
ρ = 2πR/λ (6),
we must regard ρ as a function of ξ, and we may take with sufficient approximation under any ordinary circumstances
ρ = ρ′ +ωξ (7),
where ρ′ denotes the value of ρ at O, andωis a constant, which is positive when the retarding plate is held at the side on which the lue of the spectrumis seen. The possibility of dark bands depends uponωbeing positive. Only in this case can
cos {ρ′ + (ω− 2πh/λƒ) ξ}
retain the constant value -1 throughout the integration, and then only when
ω= 2πh / λƒ (8)
and
cos ρ′ = −1 (9).
The first of these equations is the condition for the formation of dark bands, and the second marks their situation, which is the same as that determined by the imperfect theory.
The integration can be effected without much difficulty. For the first term in (5) the evaluation is effected at once by a known formula. In the second term if we observe that
cos {ρ′ +(ω− 2πh/λƒ) ξ} = cos {ρ′ − g1ξ}= cos ρ′ cos g1ξ + sin ρ′ sin g1ξ,
we see that the second part vanishes when integrated, and that the remaining integral is of the form
where
h1= πh/λƒ, g1= ω − 2πh/λƒ (10).
By differentiation with respect to g1it may be proved that
The integrated intensity, I′, or
2πh1+ 2 cos ρw,
is thus
I′ = 2πh1(11),
when g1numerically exceeds 2h1; and, when g1lies between ±2h1,
I = π{2h1+ (2h1− √ g1²) cos ρ′} (12).
It appears therefore that there are no bands at all unless ω lies between 0 and +4h1, and that within these limits the best bands are formed at the middle of the range when ω = 2h1. The formation of bands thus requires that the retarding plate be held upon the side already specified, so that ω be positive; and that the thickness of the plate (to which ω is proportional) do not exceed a certain limit, which we may call 2T0. At the best thickness T0the bands are black, and not otherwise.
The linear width of the band (e) is the increment of ξ which alters ρ by 2π, so that
e = 2π /ω(13).
With the best thickness
ω= 2πh/λƒ (14),
so that in this case
e = λƒ / h (15).
The bands are thus of the same width as those due to two infinitely narrow apertures coincident with the central lines of the retarded and unretarded streams, the subject of examination being itself a fine luminous line.
If it be desired to see a given number of bands in the whole or in any part of the spectrum, the thickness of the retarding plate is thereby determined, independently of all other considerations. But in order that the bands may be really visible, and still more in order that they may be black, another condition must be satisfied. It is necessary that the aperture of the pupil be accommodated to the angular extent of the spectrum, or reciprocally. Black bands will be too fine to be well seen unless the aperture (2h) of the pupil be somewhat contracted. One-twentieth to one-fiftieth of an inch is suitable. The aperture and the number of bands being both fixed, the condition of blackness determines the angular magnitude of a band and of the spectrum. The use of a grating is very convenient, for not only are there several spectra in view at the same time, but the dispersion can be varied continuously by sloping the grating. The slits may be cut out of tin-plate, and half covered by mica or “microscopic glass,” held in position by a little cement.
If a telescope be employed there is a distinction to be observed, according as the half-covered aperture is between the eye and the ocular, or in front of the object-glass. In the former case the function of the telescope is simply to increase the dispersion, and the formation of the bands is of course independent of the particular manner in which the dispersion arises. If, however, the half-covered aperture be in front of the object-glass, the phenomenon is magnified as a whole, and the desirable relation between the (unmagnified) dispersion and the aperture is the same as without the telescope. There appears to be no further advantage in the use of a telescope than the increased facility of accommodation, and for this of course a very low power suffices.
The original investigation of Stokes, here briefly sketched, extends also to the case where the streams are of unequal width h, k, and are separated by an interval 2g. In the case of unequal width the bands cannot be black; but if h = k, the finiteness of 2g does not preclude the formation of black bands.
The theory of Talbot’s bands with a half-coveredcircularaperture has been considered by H. Struve (St Peters. Trans., 1883, 31, No. 1).
The subject of “Talbot’s bands” has been treated in a very instructive manner by A. Schuster (Phil. Mag., 1904), whose point of view offers the great advantage of affording an instantaneous explanation of the peculiarity noticed by Brewster. A planepulse,i.e.a disturbance limited to an infinitely thin slice of the medium, is supposed to fall upon a parallel grating, which again maybe regarded as formed of infinitely thin wires, or infinitely narrow lines traced upon glass. The secondary pulses diverted by the ruling fall upon an object-glass as usual, and on arrival at the focus constitute a procession equally spaced in time, the interval between consecutive members depending upon the obliquity. If a retarding plate be now inserted so as to operate upon the pulses which come from one side of the grating, while leaving the remainder unaffected, we have to consider what happens at the focal point chosen. A full discussion would call for the formal application of Fourier’s theorem, but some conclusions of importance are almost obvious.
Previously to the introduction of the plate we have an effect corresponding to wave-lengths closely grouped around the principal wave-length, viz. σ sin φ, where σ is the grating-interval and φ the obliquity, the closeness of the grouping increasing with the number of intervals. In addition to these wave-lengths there are other groups centred round the wave-lengths which are submultiples of the principal one—the overlapping spectra of the second and higher orders. Suppose now that the plate is introduced so as to cover naif the aperture and that it retards those pulses which would otherwise arrive first. The consequences must depend upon the amount of the retardation. As this increases from zero, the two processions which correspond to the two halves of the aperture begin to overlap, and the overlapping gradually increases until there is almost complete superposition. The stage upon which we will fix our attention is that where the one procession bisects the intervals between the other, so that a new simple procession is constituted, containing the same number of members as before the insertion of the plate, but now spaced at intervals only half as great. It is evident that the effect at the focal point is the obliteration of the first and other spectra of odd order, so that as regards the spectrum of the first order we may consider that the two beamsinterfere. The formation of black bands is thus explained, and it requires that the plate be introduced upon one particular side, and that the amount of the retardation be adjusted to a particular value. If the retardation be too little, the overlapping of the processions is incomplete, so that besides the procession of half period there are residues of the original processions of full period. The same thing occurs if the retardation be too great. If it exceed the double of the value necessary for black bands, there is again no overlapping and consequently no interference. If the plate be introduced upon the other side, so as to retard the procession originally in arrear, there is no overlapping, whatever may be the amount of retardation. In this way the principal features of the phenomenon are accounted for, and Schuster has shown further how to extend the results to spectra having their origin in prisms instead of gratings.
10.Diffraction when the Source of Light is not seen in Focus.—The phenomena to be considered under this head are of less importance than those investigated by Fraunhofer, and will be treated in less detail; but in view of their historical interest and of the ease with which many of the experiments may be tried, some account of their theory cannot be omitted. One or two examples have already attracted our attention when considering Fresnel’s zones, viz. the shadow of a circular disk and of a screen circularly perforated.
Fresnel commenced his researches with an examination of the fringes, external and internal, which accompany the shadow of a narrow opaque strip, such as a wire. As a source of light he used sunshine passing through a very small hole perforated in a metal plate, or condensed by a lens of short focus. In the absence of a heliostat the latter was the more convenient. Following, unknown to himself, in the footsteps of Young, he deduced the principle of interference from the circumstance that the darkness of the interior bands requires the co-operation of light from both sides of the obstacle. At first, too, he followed Young in the view that the exterior bands are the result of interference between the direct light and that reflected from the edge of the obstacle, but he soon discovered that the character of the edge—e.g.whether it was the cutting edge or the back of a razor—made no material difference, and was thus led to the conclusion that the explanation of these phenomena requires nothing more than the application of Huygens’s principle to the unobstructed parts of the wave. In observing the bands he received them at first upon a screen of finely ground glass, upon which a magnifying lens was focused; but it soon appeared that the ground glass could be dispensed with, the diffraction pattern being viewed in the same way as the image formed by the object-glass of a telescope is viewed through the eye-piece. This simplification was attended by a great saving of light, allowing measures to be taken such as would otherwise have presented great difficulties.
In theoretical investigations these problems are usually treated as of two dimensions only, everything being referred to the plane passing through the luminous point and perpendicular to the diffracting edges, supposed to be straight and parallel. In strictness this idea is appropriate only when the source is a luminous line, emitting cylindrical waves, such as might be obtained from a luminous point with the aid of a cylindrical lens. When, in order to apply Huygens’s principle, the wave is supposed to be broken up, the phase is the same at every element of the surface of resolution which lies upon a line perpendicular to the plane of reference, and thus the effect of the whole line, or rather infinitesimal strip, is related in a constant manner to that of the element which lies in the plane of reference, and may be considered to be represented thereby. The same method of representation is applicable to spherical waves, issuing from apoint, if the radius of curvature be large; for, although there is variation of phase along the length of the infinitesimal strip, the whole effect depends practically upon that of the central parts where the phase is sensibly constant.10
In fig. 17 APQ is the arc of the circle representative of the wave-front of resolution, the centre being at O, and the radius QA being equal to a. B is the point at which the effect is required, distant a + b from O, so that AB = b, AP = s, PQ = ds.
Taking as the standard phase that of the secondary wave from A, we may represent the effect of PQ by
where δ = BP − AP is the retardation at B of the wave from P relatively to that from A.
Now
δ = (a + b) s²/2ab (1),
so that, if we write
the effect at B is
the limits of integration depending upon the disposition of the diffracting edges. When a, b, λ are regarded as constant, the first factor may be omitted,—as indeed should be done for consistency’s sake, inasmuch as other factors of the same nature have been omitted already.
The intensity I², the quantity with which we are principally concerned, may thus be expressed
I²={ ∫cos ½πv²·dv}² +{ ∫sin ½πv²·dv}² (4).
These integrals, taken from v = 0, are known as Fresnel’s integrals; we will denote them by C and S, so that
When the upper limit is infinity, so that the limits correspond to the inclusion of half the primary wave, C and S are both equal to ½, by a known formula; and on account of the rapid fluctuation of sign the parts of the range beyond very moderate values of v contribute but little to the result.
Ascending series for C and S were given by K. W. Knockenhauer, and are readily investigated. Integrating by parts, we find
and, by continuing this process,
By separation of real and imaginary parts,
where
These series are convergent for all values of v, but are practically useful only when v is small.
Expressions suitable for discussion when v is large were obtainedby L. P. Gilbert (Mem. cour. de l’Acad. de Bruxelles, 31, p. 1). Taking
½πv² = u (9),
we may write
Again, by a known formula,
Substituting this in (10), and inverting the order of integration, we get
Thus, if we take
C = ½ − G cos u + H sin u, S = ½ − G sin u − H cos u (14).
The constant parts in (14), viz. ½, may be determined by direct integration of (12), or from the observation that by their constitution G and H vanish when u = ∞, coupled with the fact that C and S then assume the value ½.
Comparing the expressions for C, S in terms of M, N, and in terms of G, H, we find that
G = ½ (cos u + sin u) − M, H = ½ (cos u − sin u) + N (15),
formulae which may be utilized for the calculation of G, H when u (or v) is small. For example, when u = 0, M = 0, N = 0, and consequently G = H = ½.
Descending series of the semi-convergent class, available for numerical calculation when u is moderately large, can be obtained from (12) by writing x = uy, and expanding the denominator in powers of y. The integration of the several terms may then be effected by the formula
and we get in terms of v
The corresponding values of C and S were originally derived by A. L. Cauchy, without the use of Gilbert’s integrals, by direct integration by parts.
From the series for G and H just obtained it is easy to verify that
We now proceed to consider more particularly the distribution of light upon a screen PBQ near the shadow of a straight edge A. At a point P within the geometrical shadow of the obstacle, the half of the wave to the right of C (fig. 18), the nearest point on the wave-front, is wholly intercepted, and on the left the integration is to be taken from s = CA to s = ∞. If V be the value of v corresponding to CA, viz.
we may write
or, according to our previous notation,
I²=(½ − Cv)² + (½ − Sv)² = G² + H² (21).
Now in the integrals represented by G and H every element diminishes as V increases from zero. Hence, as CA increases, viz. as the point P is more and more deeply immersed in the shadow, the illuminationcontinuouslydecreases, and that without limit. It has long been known from observation that there are no bands on the interior side of the shadow of the edge.
The law of diminution when V is moderately large is easily expressed with the aid of the series (16), (17) for G, H. We have ultimately G = 0, H = (πV)−1, so that
I² = 1/π²V²,
or the illumination is inversely as the square of the distance from the shadow of the edge.
For a point Q outside the shadow the integration extends overmorethan half the primary wave. The intensity may be expressed by
I² = (½ + Cv)² + (½ + Sv)² (22);
and the maxima and minima occur when
whence
sin ½πV² + cos ½πV² = G (23).
When V = 0, viz. at the edge of the shadow, I² = ½; when V = ∞, I² = 2, on the scale adopted. The latter is the intensity due to the uninterrupted wave. The quadrupling of the intensity in passing outwards from the edge of the shadow is, however, accompanied by fluctuations giving rise to bright and dark bands. The position of these bands determined by (23) may be very simply expressed when V is large, for then sensibly G = 0, and
½πV² = ¾π + nπ (24),
n being an integer. In terms of δ, we have from (2)
δ = (3⁄8+ ½n)λ (25).
The first maximum in fact occurs when δ =3⁄8λ − .0046λ, and the first minimum when δ =7⁄8λ − .0016λ, the corrections being readily obtainable from a table of G by substitution of the approximate value of V.
The position of Q corresponding to a given value of V, that is, to a band of given order, is by (19)
By means of this expression we may trace the locus of a band of given order as b varies. With sufficient approximation we may regard BQ and b as rectangular co-ordinates of Q. Denoting them by x, y, so that AB is axis of y and a perpendicular through A the axis of x, and rationalizing (26), we have
2ax² − V²λy² − V²aλy = 0,
which represents a hyperbola with vertices at O and A.
From (24), (26) we see that the width of the bands is of the order √{bλ(a + b)/a}. From this we may infer the limitation upon the width of the source of light, in order that the bands may be properly formed. If ω be the apparent magnitude of the source seen from A, ωb should be much smaller than the above quantity, or
ω < √{λ(a + b)/ab} (27).
If a be very great in relation to b, the condition becomes
ω < √(λ / b) (28).
so that if b is to be moderately great (1 metre), the apparent magnitude of the sun must be greatly reduced before it can be used as a source. The values of V for the maxima and minima of intensity, and the magnitudes of the latter, were calculated by Fresnel. An extract from his results is given in the accompanying table.
A very thorough investigation of this and other related questions, accompanied by fully worked-out tables of the functions concerned, will be found in a paper by E. Lommel (Abh. bayer. Akad. d. Wiss.II. CI., 15, Bd., iii. Abth., 1886).
When the functions C and S have once been calculated, the discussion of various diffraction problems is much facilitated by the idea, due to M. A. Cornu (Journ. de Phys., 1874, 3, p. 1; a similar suggestion was made independently by G. F. Fitzgerald), of exhibiting as a curve the relationship between C and S, considered as the rectangular co-ordinates (x, y) of a point. Such a curve is shown in fig. 19, where, according to the definition (5) of C, S,
The origin of co-ordinates O corresponds to v = 0; and the asymptotic points J, J′, round which the curve revolves in an ever-closing spiral, correspond to v = ±∞.
The intrinsic equation, expressing the relation between the arc σ (measured from O) and the inclination φ of the tangent at any points to the axis of x, assumes a very simple form. For
dx = cos ½πv²·dv, dy = sin ½πv²·dv;
so that
σ =∫√(dx² + dy²) = v, (30),
φ = tan−1(dy/dx) = ½πv² (31).
Accordingly,
φ = ½πσ² (32);
and for the curvature,
dφ / dσ = πσ (33).
Cornu remarks that this equation suffices to determine the general character of the curve. For the osculating circle at any point includes the whole of the curve which lies beyond; and the successive convolutions envelop one another without intersection.
The utility of the curve depends upon the fact that the elements of arc represent, in amplitude and phase, the component vibrations due to the corresponding portions of the primary wave-front. For by (30) dσ = dv, and by (2) dv is proportional to ds. Moreover by (2) and (31) the retardation of phase of the elementary vibration from PQ (fig. 17) is 2πδ/λ, or φ. Hence, in accordance with the rule for compounding vector quantities, the resultant vibration at B, due to any finite part of the primary wave, is represented in amplitude and phase by the chord joining the extremities of the corresponding arc (σ2− σ1).
In applying the curve in special cases of diffraction to exhibit the effect at any point P (fig. 18) the centre of the curve O is to be considered to correspond to that point C of the primary wave-front which lies nearest to P. The operative part, or parts, of the curve are of course those which represent the unobstructed portions of the primary wave.
Let us reconsider, following Cornu, the diffraction of a screen unlimited on one side, and on the other terminated by a straight edge. On the illuminated side, at a distance from the shadow, the vibration is represented by JJ′. The co-ordinates of J, J′ being (½, ½), (−½, −½), I² is 2; and the phase is1⁄8period in arrear of that of the element at O. As the point under contemplation is supposed to approach the shadow, the vibration is represented by the chord drawn from J to a point on the other half of the curve, which travels inwards from J′ towards O. The amplitude is thus subject to fluctuations, which increase as the shadow is approached. At the point O the intensity is one-quarter of that of the entire wave, and after this point is passed, that is, when we have entered the geometrical shadow, the intensity falls off gradually to zero,without fluctuations. The whole progress of the phenomenon is thus exhibited to the eye in a very instructive manner.
We will next suppose that the light is transmitted by a slit, and inquire what is the effect of varying the width of the slit upon the illumination at the projection of its centre. Under these circumstances the arc to be considered is bisected at O, and its length is proportional to the width of the slit. It is easy to see that the length of the chord (which passes in all cases through O) increases to a maximum near the place where the phase-retardation is3⁄8of a period, then diminishes to a minimum when the retardation is about7⁄8of a period, and so on.
If the slit is of constant width and we require the illumination at various points on the screen behind it, we must regard the arc of the curve as ofconstant length. The intensity is then, as always, represented by the square of the length of the chord. If the slit be narrow, so that the arc is short, the intensity is constant over a wide range, and does not fall off to an important extent until the discrepancy of the extreme phases reaches about a quarter of a period.
We have hitherto supposed that the shadow of a diffracting obstacle is received upon a diffusing screen, or, which comes to nearly the same thing, is observed with an eye-piece. If the eye, provided if necessary with a perforated plate in order to reduce the aperture, be situated inside the shadow at a place where the illumination is still sensible, and be focused upon the diffracting edge, the light which it receives will appear to come from the neighbourhood of the edge, and will present the effect of a silver lining. This is doubtless the explanation of a “pretty optical phenomenon, seen in Switzerland, when the sun rises from behind distant trees standing on the summit of a mountain.”11
II.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
where a² and b² denote the two arbitrary constants. Put for shortness
and represent by Δ²χ the quantity multiplied by b². According to this notation, the three equations of motion are
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
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(bt − x) be the displacements corresponding to the incident light; let O1be 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 O1which 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,
In particular, if
we shall have
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 vibrationsof polarized light are executed in a directionperpendicularto 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 themotionat dS to be that of the primary wave, and to be zero elsewhere, we suppose theforceoperative 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
It will now be convenient to introduce the quantities.ω1,ω2,ω3which express therotationsof the elements of the medium round axes parallel to those of co-ordinates, in accordance with the equations
In terms of these we obtain from (7), by differentiation and subtraction,
The first of equations (9) gives
ω3= 0 (10).
Forω1, we have
where r is the distance between the element dx dy dz and the point whereω1is 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
in which the integrated terms at the limits vanish, Z being finite only within the region T. Thus
Since the dimensions of T are supposed to be very small in comparison with λ, the factor d/dy (e−ikr/r) is sensibly constant; so that, if Z stand for the mean value of Z over the volume T, we may write
In like manner we find
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
φ denoting the angle between r and z. In differentiating e−ikr/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
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 withωis expressed byω= dζ′/dr; so that
where the factor eintis 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
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/dx = b²kD 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. Thesecondary 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
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 ½(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.