Diffraction_Of_Light

Diffraction Of Light.—1. When light proceeding from a small source falls upon an opaque object, a shadow is cast upon a screen situated behind the obstacle, and this shadow is found to be bordered by alternations of brightness and darkness, known as “diffraction bands.” The phenomena thus presented were described by Grimaldi and by Newton. Subsequently T. Young showed that in their formation interference plays an important part, but the complete explanation was reserved for A. J. Fresnel. Later investigations by Fraunhofer, Airy and others have greatly widened the field, and under the head of “diffraction” are now usually treated all the effects dependent upon the limitation of a beam of light, as well as those which arise from irregularities of any kind at surfaces through which it is transmitted, or at which it is reflected.

2\. Shadows.—In the infancy of the undulatory theory the objection most frequently urged against it was the difficulty of explaining the very existence of shadows. Thanks to Fresnel and his followers, this department of optics is now precisely the one in which the theory has gained its greatest triumphs. The principle employed in these investigations is due to C. Huygens, and may be thus formulated. If round the origin of waves an ideal closed surface be drawn, the whole action of the waves in the region beyond may be regarded as due to the motion continually propagated across the various elements of this surface. The wave motion due to any element of the surface is called a secondary wave, and in estimating the total effect regard must be paid to the phases as well as the amplitudes of the components. It is usually convenient to choose as the surface of resolution a wave-front, i.e. a surface at which the primary vibrations are in one phase. Any obscurity that may hang over Huygens’s principle is due mainly to the indefiniteness of thought and expression which we must be content to put up with if we wish to avoid pledging ourselves as to the character of the vibrations. In the application to sound, where we know what we are dealing with, the matter is simple enough in principle, although mathematical difficulties would often stand in the way of the calculations we might wish to make. The ideal surface of resolution may be there regarded as a flexible lamina; and we know that, if by forces locally applied every element of the lamina be made to move normally to itself exactly as the air at that place does, the external aerial motion is fully determined. By the principle of superposition the whole effect may be found by integration of the partial effects due to each element of the surface, the other elements remaining at rest.  
  
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Fig. 1.  
  
We will now consider in detail the important case in which uniform plane waves are resolved at a surface coincident with a wave-front (OQ). We imagine a wave-front divided into elementary rings or zones—often named after Huygens, but better after Fresnel—by spheres described round P (the point at which the aggregate effect is to be estimated), the first sphere, touching the plane at O, with a radius equal to PO, and the succeeding spheres with radii increasing at each step by ½λ. There are thus marked out a series of circles, whose radii x are given by x² + r² = (r + ½nλ)², or x² = nλr nearly; so that the rings are at first of nearly equal area. Now the effect upon P of each element of the plane is proportional to its area; but it depends also upon the distance from P, and possibly upon the inclination of the secondary ray to the direction of vibration and to the wave-front.

The latter question can only be treated in connexion with the dynamical theory (see below, § 11); but under all ordinary circumstances the result is independent of the precise answer that may be given. All that it is necessary to assume is that the effects of the successive zones gradually diminish, whether from the increasing obliquity of the secondary ray or because (on account of the limitation of the region of integration) the zones become at last more and more incomplete. The component vibrations at P due to the successive zones are thus nearly equal in amplitude and opposite in phase (the phase of each corresponding to that of the infinitesimal circle midway between the boundaries), and the series which we have to sum is one in which the terms are alternately opposite in sign and, while at first nearly constant in numerical magnitude, gradually diminish to zero. In such a series each term may be regarded as very nearly indeed destroyed by the halves of its immediate neighbours, and thus the sum of the whole series is represented by half the first term, which stands over uncompensated. The question is thus reduced to that of finding the effect of the first zone, or central circle, of which the area is πλr.

We have seen that the problem before us is independent of the law of the secondary wave as regards obliquity; but the result of the integration necessarily involves the law of the intensity and phase of a secondary wave as a function of r, the distance from the origin. And we may in fact, as was done by A. Smith (Camb. Math. Journ., 1843, 3, p. 46), determine the law of the secondary wave, by comparing the result of the integration with that obtained by supposing the primary wave to pass on to P without resolution.

Now as to the phase of the secondary wave, it might appear natural to suppose that it starts from any point Q with the phase of the primary wave, so that on arrival at P, it is retarded by the amount corresponding to QP. But a little consideration will prove that in that case the series of secondary waves could not reconstitute the primary wave. For the aggregate effect of the secondary waves is the half of that of the first Fresnel zone, and it is the central element only of that zone for which the distance to be travelled is equal to r. Let us conceive the zone in question to be divided into infinitesimal rings of equal area. The effects due to each of these rings are equal in amplitude and of phase ranging uniformly over half a complete period. The phase of the resultant is midway between those of the extreme elements, that is to say, a quarter of a period behind that due to the element at the centre of the circle. It is accordingly necessary to suppose that the secondary waves start with a phase one-quarter of a period in advance of that of the primary wave at the surface of resolution.

Further, it is evident that account must be taken of the variation of phase in estimating the magnitude of the effect at P of the first zone. The middle element alone contributes without deduction; the effect of every other must be found by introduction of a resolving factor, equal to cos θ, if θ represent the difference of phase between this element and the resultant. Accordingly, the amplitude of the resultant will be less than if all its components had the same phase, in the ratio

∫ | +½π | cos θdθ : π,  
---|---|---  
-½π  
  
or 2 : π. Now 2 area /π = 2λr; so that, in order to reconcile the amplitude of the primary wave (taken as unity) with the half effect of the first zone, the amplitude, at distance r, of the secondary wave emitted from the element of area dS must be taken to be

dS/λr  (1).

By this expression, in conjunction with the quarter-period acceleration of phase, the law of the secondary wave is determined.

That the amplitude of the secondary wave should vary as r-1 was to be expected from considerations respecting energy; but the occurrence of the factor λ-1, and the acceleration of phase, have sometimes been regarded as mysterious. It may be well therefore to remember that precisely these laws apply to a secondary wave of sound, which can be investigated upon the strictest mechanical principles.

The recomposition of the secondary waves may also be treated analytically. If the primary wave at O be cos kat, the effect of the secondary wave proceeding from the element dS at Q is

dS | cos k(at − ρ \+ ¼λ) = − | dS | sin k(at − ρ).  
---|---|---|---  
λρ | λρ  
  
If dS = 2πxdx, we have for the whole effect

− | 2π | ∫ | ∞ | sin k(at − ρ)x dx | ,  
---|---|---|---|---|---  
λ | 0 | ρ  
  
or, since xdx = ρdρ, k = 2π/λ,

−k ∫ | ∞ |  sin k(at − ρ)dρ = [ −cos k(at − ρ) ] | ∞ |  | .  
---|---|---|---|---|---  
r | r  
  
In order to obtain the effect of the primary wave, as retarded by traversing the distance r, viz. cos k(at − r), it is necessary to suppose that the integrated term vanishes at the upper limit. And it is important to notice that without some further understanding the integral is really ambiguous. According to the assumed law of the secondary wave, the result must actually depend upon the precise radius of the outer boundary of the region of integration, supposed to be exactly circular. This case is, however, at most very special and exceptional. We may usually suppose that a large number of the outer rings are incomplete, so that the integrated term at the upper limit may properly be taken to vanish. If a formal proof be desired, it may be obtained by introducing into the integral a factor such as e-hρ, in which h is ultimately made to diminish without limit.

When the primary wave is plane, the area of the first Fresnel zone is πλr, and, since the secondary waves vary as r-1, the intensity is independent of r, as of course it should be. If, however, the primary wave be spherical, and of radius a at the wave-front of resolution, then we know that at a distance r further on the amplitude of the primary wave will be diminished in the ratio a : (r + a). This may be regarded as a consequence of the altered area of the first Fresnel zone. For, if x be its radius, we have

{(r + ½λ)² − x²} + √ {a² − x²} = r + a,

so that

x² = λar/(a + r) nearly.

Since the distance to be travelled by the secondary waves is still r, we see how the effect of the first zone, and therefore of the whole series is proportional to a/(a + r). In like manner may be treated other cases, such as that of a primary wave-front of unequal principal curvatures.

The general explanation of the formation of shadows may also be conveniently based upon Fresnel’s zones. If the point under consideration be so far away from the geometrical shadow that a large number of the earlier zones are complete, then the illumination, determined sensibly by the first zone, is the same as if there were no obstruction at all. If, on the other hand, the point be well immersed in the geometrical shadow, the earlier zones are altogether missing, and, instead of a series of terms beginning with finite numerical magnitude and gradually diminishing to zero, we have now to deal with one of which the terms diminish to zero at both ends. The sum of such a series is very approximately zero, each term being neutralized by the halves of its immediate neighbours, which are of the opposite sign. The question of light or darkness then depends upon whether the series begins or ends abruptly. With few exceptions, abruptness can occur only in the presence of the first term, viz. when the secondary wave of least retardation is unobstructed, or when a ray passes through the point under consideration. According to the undulatory theory the light cannot be regarded strictly as travelling along a ray; but the existence of an unobstructed ray implies that the system of Fresnel’s zones can be commenced, and, if a large number of these zones are fully developed and do not terminate abruptly, the illumination is unaffected by the neighbourhood of obstacles. Intermediate cases in which a few zones only are formed belong especially to the province of diffraction.

An interesting exception to the general rule that full brightness requires the existence of the first zone occurs when the obstacle assumes the form of a small circular disk parallel to the plane of the incident waves. In the earlier half of the 18th century R. Delisle found that the centre of the circular shadow was occupied by a bright point of light, but the observation passed into oblivion until S. D. Poisson brought forward as an objection to Fresnel’s theory that it required at the centre of a circular shadow a point as bright as if no obstacle were intervening. If we conceive the primary wave to be broken up at the plane of the disk, a system of Fresnel’s zones can be constructed which begin from the circumference; and the first zone external to the disk plays the part ordinarily taken by the centre of the entire system. The whole effect is the half of that of the first existing zone, and this is sensibly the same as if there were no obstruction.

When light passes through a small circular or annular aperture, the illumination at any point along the axis depends upon the precise relation between the aperture and the distance from it at which the point is taken. If, as in the last paragraph, we imagine a system of zones to be drawn commencing from the inner circular boundary of the aperture, the question turns upon the manner in which the series terminates at the outer boundary. If the aperture be such as to fit exactly an integral number of zones, the aggregate effect may be regarded as the half of those due to the first and last zones. If the number of zones be even, the action of the first and last zones are antagonistic, and there is complete darkness at the point. If on the other hand the number of zones be odd, the effects conspire; and the illumination (proportional to the square of the amplitude) is four times as great as if there were no obstruction at all.

The process of augmenting the resultant illumination at a particular point by stopping some of the secondary rays may be carried much further (Soret, Pogg. Ann., 1875, 156, p. 99). By the aid of photography it is easy to prepare a plate, transparent where the zones of odd order fall, and opaque where those of even order fall. Such a plate has the power of a condensing lens, and gives an illumination out of all proportion to what could be obtained without it. An even greater effect (fourfold) can be attained by providing that the stoppage of the light from the alternate zones is replaced by a phase-reversal without loss of amplitude. R. W. Wood (Phil. Mag., 1898, 45, p 513) has succeeded in constructing zone plates upon this principle.

In such experiments the narrowness of the zones renders necessary a pretty close approximation to the geometrical conditions. Thus in the case of the circular disk, equidistant (r) from the source of light and from the screen upon which the shadow is observed, the width of the first exterior zone is given by

dx = λ(2r)/4(2x),

2x being the diameter of the disk. If 2r = 1000 cm., 2x = 1 cm., λ = 6 × 10-5 cm., then dx = .0015 cm. Hence, in order that this zone may be perfectly formed, there should be no error in the circumference of the order of .001 cm. (It is easy to see that the radius of the bright spot is of the same order of magnitude.) The experiment succeeds in a dark room of the length above mentioned, with a threepenny bit (supported by three threads) as obstacle, the origin of light being a small needle hole in a plate of tin, through which the sun’s rays shine horizontally after reflection from an external mirror. In the absence of a heliostat it is more convenient to obtain a point of light with the aid of a lens of short focus.

The amplitude of the light at any point in the axis, when plane waves are incident perpendicularly upon an annular aperture, is, as above,

cos k(at − r1) − cos k(at − r2) = 2 sin kat sin k(r1 − r2),

r2, r1 being the distances of the outer and inner boundaries from the point in question. It is scarcely necessary to remark that in all such cases the calculation applies in the first instance to homogeneous light, and that, in accordance with Fourier’s theorem, each homogeneous component of a mixture may be treated separately. When the original light is white, the presence of some components and the absence of others will usually give rise to coloured effects, variable with the precise circumstances of the case.  
  
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Fig. 2.  
  
Although the matter can be fully treated only upon the basis of a dynamical theory, it is proper to point out at once that there is an element of assumption in the application of Huygens’s principle to the calculation of the effects produced by opaque screens of limited extent. Properly applied, the principle could not fail; but, as may readily be proved in the case of sonorous waves, it is not in strictness sufficient to assume the expression for a secondary wave suitable when the primary wave is undisturbed, with mere limitation of the integration to the transparent parts of the screen. But, except perhaps in the case of very fine gratings, it is probable that the error thus caused is insignificant; for the incorrect estimation of the secondary waves will be limited to distances of a few wave-lengths only from the boundary of opaque and transparent parts.

3\. Fraunhofer’s Diffraction Phenomena.—A very general problem in diffraction is the investigation of the distribution of light over a screen upon which impinge divergent or convergent spherical waves after passage through various diffracting apertures. When the waves are convergent and the recipient screen is placed so as to contain the centre of convergency—the image of the original radiant point, the calculation assumes a less complicated form. This class of phenomena was investigated by J. von Fraunhofer (upon principles laid down by Fresnel), and are sometimes called after his name. We may conveniently commence with them on account of their simplicity and great importance in respect to the theory of optical instruments.

If ƒ be the radius of the spherical wave at the place of resolution, where the vibration is represented by cos kat, then at any point M (fig. 2) in the recipient screen the vibration due to an element dS of the wave-front is (§ 2)

− | dS | sin k(at − ρ),  
---|---|---  
λρ  
  
ρ being the distance between M and the element dS.

Taking co-ordinates in the plane of the screen with the centre of the wave as origin, let us represent M by ξ, η, and P (where dS is situated) by x, y, z.

Then

ρ² = (x − ξ)² + (y − η)² + z², ƒ² = x² + y² + z²;

so that

ρ² = ƒ² − 2xξ − 2yη \+ ξ² + η².

In the applications with which we are concerned, ξ, η are very small quantities; and we may take

ρ = ƒ{ 1 − | xξ \+ yη | }.  
---|---|---  
ƒ²  
  
At the same time dS may be identified with dxdy, and in the denominator ρ may be treated as constant and equal to ƒ. Thus the expression for the vibration at M becomes

− | 1 | ∫∫sin k { at − ƒ + | xξ \+ yη | } dxdy  (1);  
---|---|---|---|---  
λƒ | ƒ  
  
and for the intensity, represented by the square of the amplitude,

I² = | 1 | [ ∫∫ sin k | xξ \+ yη | dxdy ] | ² |    
---|---|---|---|---|---|---  
λ²ƒ² | ƒ |    
+ | 1 | [ ∫∫ cos k | xξ \+ yη | dxdy ] | ² |  (2).  
λ²ƒ² | ƒ |    
  
This expression for the intensity becomes rigorously applicable when ƒ is indefinitely great, so that ordinary optical aberration disappears. The incident waves are thus plane, and are limited to a plane aperture coincident with a wave-front. The integrals are then properly functions of the direction in which the light is to be estimated.

In experiment under ordinary circumstances it makes no difference whether the collecting lens is in front of or behind the diffracting aperture. It is usually most convenient to employ a telescope focused upon the radiant point, and to place the diffracting apertures immediately in front of the object-glass. What is seen through the eye-piece in any case is the same as would be depicted upon a screen in the focal plane.

Before proceeding to special cases it may be well to call attention to some general properties of the solution expressed by (2) (see Bridge, Phil. Mag., 1858).

If when the aperture is given, the wave-length (proportional to k-1) varies, the composition of the integrals is unaltered, provided ξ and η are taken universely proportional to λ. A diminution of λ thus leads to a simple proportional shrinkage of the diffraction pattern, attended by an augmentation of brilliancy in proportion to λ-2.

If the wave-length remains unchanged, similar effects are produced by an increase in the scale of the aperture. The linear dimension of the diffraction pattern is inversely as that of the aperture, and the brightness at corresponding points is as the square of the area of aperture.

If the aperture and wave-length increase in the same proportion, the size and shape of the diffraction pattern undergo no change.

We will now apply the integrals (2) to the case of a rectangular aperture of width a parallel to x and of width b parallel to y. The limits of integration for x may thus be taken to be −½a and +½a, and for y to be −½b, +½b. We readily find (with substitution for k of 2π/λ)

  | · | sin² | πaξ | · | sin² | πbη |  (3),  
---|---|---|---|---|---|---|---  
I² = | a²b² | ƒλ | ƒλ  
ƒ²λ² | π²a²ξ² | π²b²η²  
  | ƒ²λ² | ƒ²λ²  
  
as representing the distribution of light in the image of a mathematical point when the aperture is rectangular, as is often the case in spectroscopes.

The second and third factors of (3) being each of the form sin²u/u², we have to examine the character of this function. It vanishes when u = mπ, m being any whole number other than zero. When u = 0, it takes the value unity. The maxima occur when

u = tan u, (4),

and then

sin²u / u² = cos²u  (5).

To calculate the roots of (5) we may assume

u = (m + ½)π − y = U − y,

where y is a positive quantity which is small when u is large. Substituting this, we find cot y = U − y, whence

y = | 1 | (1 + | y | + | y- | \+ ...) − | y³ | − | 2y5 | − | 17y7 | .  
---|---|---|---|---|---|---|---|---|---|---|---|---  
U | U | U² | 3 | 15 | 315  
  
This equation is to be solved by successive approximation. It will readily be found that

u = U − y = U − U−1 − | 2 | U−3 − | 13 | U−5 − | 146 | U−7 − ...  (6).  
---|---|---|---|---|---|---  
3 | 15 | 105  
  
In the first quadrant there is no root after zero, since tan u > u, and in the second quadrant there is none because the signs of u and tan u are opposite. The first root after zero is thus in the third quadrant, corresponding to m = 1. Even in this case the series converges sufficiently to give the value of the root with considerable accuracy, while for higher values of m it is all that could be desired. The actual values of u/π (calculated in another manner by F. M. Schwerd) are 1.4303, 2.4590, 3.4709, 4.4747, 5.4818, 6.4844, &c.

Since the maxima occur when u = (m + ½)π nearly, the successive values are not very different from

4 | , | 4 | , | 4 | , &c.  
---|---|---|---|---|---  
9π² | 25π | 49π²  
  
The application of these results to (3) shows that the field is brightest at the centre ξ = 0, η = 0, viz. at the geometrical image of the radiant point. It is traversed by dark lines whose equations are

ξ = mfλ / a, η = mfλ / b.

Within the rectangle formed by pairs of consecutive dark lines, and not far from its centre, the brightness rises to a maximum; but these subsequent maxima are in all cases much inferior to the brightness at the centre of the entire pattern (ξ = 0, η = 0).

By the principle of energy the illumination over the entire focal plane must be equal to that over the diffracting area; and thus, in accordance with the suppositions by which (3) was obtained, its value when integrated from ξ = ∞ to ξ = +∞, and from η = −∞ to η = +∞ should be equal to ab. This integration, employed originally by P. Kelland (Edin. Trans. 15, p. 315) to determine the absolute intensity of a secondary wave, may be at once effected by means of the known formula

∫ | +∞ | sin²u | du = ∫ | +∞ | sin u | du = π.  
---|---|---|---|---|---|---  
−∞ | u² | −∞ | u  
  
It will be observed that, while the total intensity is proportional to ab, the intensity at the focal point is proportional to a²b². If the aperture be increased, not only is the total brightness over the focal plane increased with it, but there is also a concentration of the diffraction pattern. The form of (3) shows immediately that, if a and b be altered, the co-ordinates of any characteristic point in the pattern vary as a−1 and b−1.

The contraction of the diffraction pattern with increase of aperture is of fundamental importance in connexion with the resolving power of optical instruments. According to common optics, where images are absolute, the diffraction pattern is supposed to be infinitely small, and two radiant points, however near together, form separated images. This is tantamount to an assumption that λ is infinitely small. The actual finiteness of λ imposes a limit upon the separating or resolving power of an optical instrument.

This indefiniteness of images is sometimes said to be due to diffraction by the edge of the aperture, and proposals have even been made for curing it by causing the transition between the interrupted and transmitted parts of the primary wave to be less abrupt. Such a view of the matter is altogether misleading. What requires explanation is not the imperfection of actual images so much as the possibility of their being as good as we find them.

At the focal point (ξ = 0, η = 0) all the secondary waves agree in phase, and the intensity is easily expressed, whatever be the form of the aperture. From the general formula (2), if A be the area of aperture,

I0² = A² / λ²ƒ²  (7).

The formation of a sharp image of the radiant point requires that the illumination become insignificant when ξ, η attain small values, and this insignificance can only arise as a consequence of discrepancies of phase among the secondary waves from various parts of the aperture. So long as there is no sensible discrepancy of phase there can be no sensible diminution of brightness as compared with that to be found at the focal point itself. We may go further, and lay it down that there can be no considerable loss of brightness until the difference of phase of the waves proceeding from the nearest and farthest parts of the aperture amounts to ¼λ.

When the difference of phase amounts to λ, we may expect the resultant illumination to be very much reduced. In the particular case of a rectangular aperture the course of things can be readily followed, especially if we conceive ƒ to be infinite. In the direction (suppose horizontal) for which η = 0, ξ/ƒ = sin θ, the phases of the secondary waves range over a complete period when sin θ = λ/a, and, since all parts of the horizontal aperture are equally effective, there is in this direction a complete compensation and consequent absence of illumination. When sin θ = 3⁄2λ/a, the phases range one and a half periods, and there is revival of illumination. We may compare the brightness with that in the direction θ = 0. The phase of the resultant amplitude is the same as that due to the central secondary wave, and the discrepancies of phase among the components reduce the amplitude in the proportion

1 | ∫ | +3⁄2π | cos φ dφ: 1  
---|---|---|---  
3π | −3⁄2π  
  
or -2⁄3π : 1; so that the brightness in this direction is 4⁄9π² of the maximum at θ = 0. In like manner we may find the illumination in any other direction, and it is obvious that it vanishes when sin θ is any multiple of λ/a.

The reason of the augmentation of resolving power with aperture will now be evident. The larger the aperture the smaller are the angles through which it is necessary to deviate from the principal direction in order to bring in specified discrepancies of phase—the more concentrated is the image.

In many cases the subject of examination is a luminous line of uniform intensity, the various points of which are to be treated as independent sources of light. If the image of the line be ξ = 0, the intensity at any point ξ, η of the diffraction pattern may be represented by

  | sin² | πaξ |  (8),  
---|---|---|---  
∫ | +∞ | I²dη = | a²b | λf  
−∞ | λƒ | π²a²ξ²  
  | λ²f²  
  
the same law as obtains for a luminous point when horizontal directions are alone considered. The definition of a fine vertical line, and consequently the resolving power for contiguous vertical lines, is thus independent of the vertical aperture of the instrument, a law of great importance in the theory of the spectroscope.

The distribution of illumination in the image of a luminous line is shown by the curve ABC (fig. 3), representing the value of the function sin²u/u² from u = 0 to u = 2π. The part corresponding to negative values of u is similar, OA being a line of symmetry.  
  
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Fig. 3.  
  
Let us now consider the distribution of brightness in the image of a double line whose components are of equal strength, and at such an angular interval that the central line in the image of one coincides with the first zero of brightness in the image of the other. In fig. 3 the curve of brightness for one component is ABC, and for the other OA′C′; and the curve representing half the combined brightnesses is E′BE. The brightness (corresponding to B) midway between the two central points AA’ is .8106 of the brightness at the central points themselves. We may consider this to be about the limit of closeness at which there could be any decided appearance of resolution, though doubtless an observer accustomed to his instrument would recognize the duplicity with certainty. The obliquity, corresponding to u = π, is such that the phases of the secondary waves range over a complete period, i.e. such that the projection of the horizontal aperture upon this direction is one wave-length. We conclude that a double line cannot be fairly resolved unless its components subtend an angle exceeding that subtended by the wave-length of light at a distance equal to the horizontal aperture. This rule is convenient on account of its simplicity; and it is sufficiently accurate in view of the necessary uncertainty as to what exactly is meant by resolution.

If the angular interval between the components of a double line be half as great again as that supposed in the figure, the brightness midway between is .1802 as against 1.0450 at the central lines of each image. Such a falling off in the middle must be more than sufficient for resolution. If the angle subtended by the components of a double line be twice that subtended by the wave-length at a distance equal to the horizontal aperture, the central bands are just clear of one another, and there is a line of absolute blackness in the middle of the combined images.

The resolving power of a telescope with circular or rectangular aperture is easily investigated experimentally. The best object for examination is a grating of fine wires, about fifty to the inch, backed by a sodium flame. The object-glass is provided with diaphragms pierced with round holes or slits. One of these, of width equal, say, to one-tenth of an inch, is inserted in front of the object-glass, and the telescope, carefully focused all the while, is drawn gradually back from the grating until the lines are no longer seen. From a measurement of the maximum distance the least angle between consecutive lines consistent with resolution may be deduced, and a comparison made with the rule stated above.

Merely to show the dependence of resolving power on aperture it is not necessary to use a telescope at all. It is sufficient to look at wire gauze backed by the sky or by a flame, through a piece of blackened cardboard, pierced by a needle and held close to the eye. By varying the distance the point is easily found at which resolution ceases; and the observation is as sharp as with a telescope. The function of the telescope is in fact to allow the use of a wider, and therefore more easily measurable, aperture. An interesting modification of the experiment may be made by using light of various wave-lengths.

Since the limitation of the width of the central band in the image of a luminous line depends upon discrepancies of phase among the secondary waves, and since the discrepancy is greatest for the waves which come from the edges of the aperture, the question arises how far the operation of the central parts of the aperture is advantageous. If we imagine the aperture reduced to two equal narrow slits bordering its edges, compensation will evidently be complete when the projection on an oblique direction is equal to ½λ, instead of λ as for the complete aperture. By this procedure the width of the central band in the diffraction pattern is halved, and so far an advantage is attained. But, as will be evident, the bright bands bordering the central band are now not inferior to it in brightness; in fact, a band similar to the central band is reproduced an indefinite number of times, so long as there is no sensible discrepancy of phase in the secondary waves proceeding from the various parts of the same slit. Under these circumstances the narrowing of the band is paid for at a ruinous price, and the arrangement must be condemned altogether.

A more moderate suppression of the central parts is, however, sometimes advantageous. Theory and experiment alike prove that a double line, of which the components are equally strong, is better resolved when, for example, one-sixth of the horizontal aperture is blocked off by a central screen; or the rays quite at the centre may be allowed to pass, while others a little farther removed are blocked off. Stops, each occupying one-eighth of the width, and with centres situated at the points of trisection, answer well the required purpose.

It has already been suggested that the principle of energy requires that the general expression for I² in (2) when integrated over the whole of the plane ξ, η should be equal to A, where A is the area of the aperture. A general analytical verification has been given by Sir G. G. Stokes (Edin. Trans., 1853, 20, p. 317). Analytically expressed—

∫∫ | +∞ | I² dξdη = ∫∫ dxdy = A  (9).  
---|---|---  
−∞  
  
We have seen that I0² (the intensity at the focal point) was equal to A²/λ²f². If A′ be the area over which the intensity must be I0² in order to give the actual total intensity in accordance with

A′ I0² = ∫∫ | +∞ | I² dξdη,  
---|---|---  
−∞  
  
the relation between A and A′ is AA′ = λ²f². Since A′ is in some sense the area of the diffraction pattern, it may be considered to be a rough criterion of the definition, and we infer that the definition of a point depends principally upon the area of the aperture, and only in a very secondary degree upon the shape when the area is maintained constant.

4\. Theory of Circular Aperture.—We will now consider the important case where the form of the aperture is circular.

Writing for brevity

kξ/f = p, kη/f = q,  (1),

we have for the general expression (§ 11) of the intensity

λ²f²I² = S² + C²  (2),

where

S = ∫∫ sin(px + qy)dx dy,  (3),

C = ∫∫ cos(px + qy)dx dy,  (4).

When, as in the application to rectangular or circular apertures, the form is symmetrical with respect to the axes both of x and y, S = 0, and C reduces to

C = ∫∫ cos px cos qy dx dy,  (5).

In the case of the circular aperture the distribution of light is of course symmetrical with respect to the focal point p = 0, q = 0; and C is a function of p and q only through √(p² + q²). It is thus sufficient to determine the intensity along the axis of p. Putting q = 0, we get

C = ∫∫ cos px dx dy = 2 ∫ | +R | cos px √(R² − x²) dx,  
---|---|---  
−R  
  
R being the radius of the aperture. This integral is the Bessel’s function of order unity, defined by

J1(z) = | z | ∫ | π | cos (z cos φ) sin² φ dφ (6).  
---|---|---|---|---  
π | 0  
  
Thus, if x = R cos φ,

C = π²R | 2J1(pR) |  (7);  
---|---|---  
pR  
  
and the illumination at distance r from the focal point is

  | · | 4J1² ( | 2πRr | ) |  (8).  
---|---|---|---|---|---  
I² = | π²R4 | ƒλ  
λ²f² |  ( | 2πRr | ) ²  
  | ƒλ  
  
The ascending series for J1(z), used by Sir G. B. Airy (Camb. Trans., 1834) in his original investigation of the diffraction of a circular object-glass, and readily obtained from (6), is

J1(z) = | z | − | z³ | + | z5 | − | z7 | \+ ...  (9).  
---|---|---|---|---|---|---|---|---  
2 | 2²·4 | 2²·4²·6 | 2²·4²·6²·8  
  
When z is great, we may employ the semi-convergent series

J1(z) = √( | 2 | ) sin (z − ¼π) { 1 + | 3·5·1 | ( | 1 | ) | ² | − | 3·5·7·9·1·3·5 | ( | 1 | ) | 4 | \+ ... }  
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---  
πz | 8·16 | z |   | 8·16·24·32 | z |    
\+ √( | 2 | ) cos (z − ¼π) {3/8 · 1/z − | 3·5·7·1·3 | ( | 1 | ) | ³ | + | 3·5·7·9·11·1·3·5·7 | ( | 1 | ) | 5 | − ... } (10).  
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---  
πz | 8·16·24 | z |   | 8·16·24·32·40 | z |    
  
A table of the values of 2z-1J1(z) has been given by E. C. J. Lommel (Schlömilch, 1870, 15, p. 166), to whom is due the first systematic application of Bessel’s functions to the diffraction integrals.

The illumination vanishes in correspondence with the roots of the equation J1(z) = 0. If these be called z1 z2, z3, ... the radii of the dark rings in the diffraction pattern are

ƒλz1 |  | ƒλz2 | , ...  
---|---|---|---  
2πR | 2πR  
  
being thus inversely proportional to R.

The integrations may also be effected by means of polar co-ordinates, taking first the integration with respect to φ so as to obtain the result for an infinitely thin annular aperture. Thus, if

x = ρ cos φ, y = ρ sin φ,

C = ∫∫ cos px dx dy = ∫ | R | ∫ | 2π | cos (pρ cos θ) ρdρ dθ.  
---|---|---|---|---  
0 | 0  
  
Now by definition

J0(z) = | 2 | ∫ | ½π | cos (z cos θ) dθ = 1 − | z² | + | z4 | − | z6 | \+ ...  (11).  
---|---|---|---|---|---|---|---|---|---|---  
π | 0 | 2² | 2²·4² | 2²·4²·6²  
  
The value of C for an annular aperture of radius r and width dr is thus

dC = 2 π J0 (pρ) ρ dρ,  (12).

For the complete circle,

C = | 2π | ∫ | pr | J0(z) zdz = | 2π | { | p²R² | − | p4R4 | + | p6R6 | − ...}= πR² · | 2J1(pR) | as before.  
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---  
p² | 0 | p² | 2 | 2²·4² | 2²·4²·6² | pR  
  
In these expressions we are to replace p by kξ/ƒ, or rather, since the diffraction pattern is symmetrical, by kr/ƒ, where r is the distance of any point in the focal plane from the centre of the system.

The roots of J0(z) after the first may be found from

z | = i − .25 + | .050561 | − | .053041 | + | .262051 |  (13),  
---|---|---|---|---|---|---|---  
π | 4i − 1 | (4i − 1)³ | (4i − 1)5  
  
and those of J1(z) from

z | = i + .25 − | .151982 | + | .015399 | − | .245835 |  (14),  
---|---|---|---|---|---|---|---  
π | 4i + 1 | (4i + 1)³ | (4i + 1)5  
  
formulae derived by Stokes (Camb. Trans., 1850, vol. ix.) from the descending series.1 The following table gives the actual values:—

i | z/π for J0(z) = 0 | z/π for J1(z) = 0  
---|---|---  
1 | 7655 | 1 2197  
2 | 1 7571 | 2 2330  
3 | 2 7546 | 3 2383  
4 | 3 7534 | 4 2411  
5 | 4 7527 | 5 2428  
6 | 5 7522 | 6 2439  
7 | 6 7519 | 7 2448  
8 | 7 7516 | 8 2454  
9 | 8 7514 | 9 2459  
10 | 9 7513 | 10 2463  
  
In both cases the image of a mathematical point is thus a symmetrical ring system. The greatest brightness is at the centre, where

dC = 2πρ dρ, C = π R².

For a certain distance outwards this remains sensibly unimpaired and then gradually diminishes to zero, as the secondary waves become discrepant in phase. The subsequent revivals of brightness forming the bright rings are necessarily of inferior brilliancy as compared with the central disk.

The first dark ring in the diffraction pattern of the complete circular aperture occurs when

r/ƒ = 1.2197 × λ/2R  (15).

We may compare this with the corresponding result for a rectangular aperture of width a,

ξ/ƒ =λ/a;

and it appears that in consequence of the preponderance of the central parts, the compensation in the case of the circle does not set in at so small an obliquity as when the circle is replaced by a rectangular aperture, whose side is equal to the diameter of the circle.

Again, if we compare the complete circle with a narrow annular aperture of the same radius, we see that in the latter case the first dark ring occurs at a much smaller obliquity, viz.

r/ƒ = .7655 × λ/2R.

It has been found by Sir William Herschel and others that the definition of a telescope is often improved by stopping off a part of the central area of the object-glass; but the advantage to be obtained in this way is in no case great, and anything like a reduction of the aperture to a narrow annulus is attended by a development of the external luminous rings sufficient to outweigh any improvement due to the diminished diameter of the central area.2

The maximum brightnesses and the places at which they occur are easily determined with the aid of certain properties of the Bessel’s functions. It is known (see Spherical Harmonics) that

J0′(z) = −J1(z),  (16);

J2(z) = (1/z) J1(z) − J1′(z)  (17);

J0(z) + J2(z) = (2/z) J1(z)  (18).

The maxima of C occur when

d | ( | J1(z) | ) = | J1′(z) | − | J1(z) |  = 0;  
---|---|---|---|---|---|---|---  
dz | z | z | z²  
  
or by 17 when J2(z) = 0. When z has one of the values thus determined,

2| J1(z) = J0(z).  
---|---  
z  
  
The accompanying table is given by Lommel, in which the first column gives the roots of J2(z) = 0, and the second and third columns the corresponding values of the functions specified. If appears that the maximum brightness in the first ring is only about 1⁄57 of the brightness at the centre.

z | 2z−1J1(z) | 4z−2J1²(z)  
---|---|---  
.000000 | +1.000000 | 1.000000  
5.135630 | − .132279 | .017498  
8.417236 | \+ .064482 | .004158  
11.619857 | − .040008 | .001601  
14.795938 | \+ .027919 | .000779  
17.959820 | − .020905 | .000437  
  
We will now investigate the total illumination distributed over the area of the circle of radius r. We have

I² = | π²R4 | · | 4J1²(z) |  (19),  
---|---|---|---|---  
λ²ƒ² | z²  
  
where

z = 2πRr/λf  (20).

Thus

2π ∫ I²rdr = | λ²f² | ∫ I²zdz = πR²· 2 ∫ z−1J1²(z)dz.  
---|---|---  
2πR²  
  
Now by (17), (18)

z-1J1(z) = J0(z) − J1′(z);

so that

z-1J1²(z) = − ½ | d | J0² − ½ | d | J1²(z),  
---|---|---|---|---  
dz | dz  
  
and

2 ∫ | z | z-1J1²(z)dz = 1 − J0²(z) − J1²(z)  (21).  
---|---|---  
0  
  
If r, or z, be infinite, J0(z), J1(z) vanish, and the whole illumination is expressed by πR², in accordance with the general principle. In any case the proportion of the whole illumination to be found outside the circle of radius r is given by

J0²(z) + J1²(z).

For the dark rings J1(z) = 0; so that the fraction of illumination outside any dark ring is simply J0²(z). Thus for the first, second, third and fourth dark rings we get respectively .161, .090, .062, .047, showing that more than 9⁄10ths of the whole light is concentrated within the area of the second dark ring (Phil. Mag., 1881).

When z is great, the descending series (10) gives

2J1(z) | = | 2 | √( | 2 | ) sin(z − ¼π)  (22);  
---|---|---|---|---|---  
z | z | πz  
  
so that the places of maxima and minima occur at equal intervals.

The mean brightness varies as z-3 (or as r-3), and the integral found by multiplying it by zdz and integrating between 0 and ∞ converges.

It may be instructive to contrast this with the case of an infinitely narrow annular aperture, where the brightness is proportional to J0²(z). When z is great,

J0(z) = √( | 2 | ) cos(z-1/4π).  
---|---|---  
πz  
  
The mean brightness varies as z-1; and the integral ∫∞0 J0²(z)z dz is not convergent.

5\. Resolving Power of Telescopes.—The efficiency of a telescope is of course intimately connected with the size of the disk by which it represents a mathematical point. In estimating theoretically the resolving power on a double star we have to consider the illumination of the field due to the superposition of the two independent images. If the angular interval between the components of a double star were equal to twice that expressed in equation (15) above, the central disks of the diffraction patterns would be just in contact. Under these conditions there is no doubt that the star would appear to be fairly resolved, since the brightness of its external ring system is too small to produce any material confusion, unless indeed the components are of very unequal magnitude. The diminution of the star disks with increasing aperture was observed by Sir William Herschel, and in 1823 Fraunhofer formulated the law of inverse proportionality. In investigations extending over a long series of years, the advantage of a large aperture in separating the components of close double stars was fully examined by W. R. Dawes.

The resolving power of telescopes was investigated also by J. B. L. Foucault, who employed a scale of equal bright and dark alternate parts; it was found to be proportional to the aperture and independent of the focal length. In telescopes of the best construction and of moderate aperture the performance is not sensibly prejudiced by outstanding aberration, and the limit imposed by the finiteness of the waves of light is practically reached. M. E. Verdet has compared Foucault’s results with theory, and has drawn the conclusion that the radius of the visible part of the image of a luminous point was equal to half the radius of the first dark ring.

The application, unaccountably long delayed, of this principle to the microscope by H. L. F. Helmholtz in 1871 is the foundation of the important doctrine of the microscopic limit. It is true that in 1823 Fraunhofer, inspired by his observations upon gratings, had very nearly hit the mark.3 And a little before Helmholtz, E. Abbe published a somewhat more complete investigation, also founded upon the phenomena presented by gratings. But although the argument from gratings is instructive and convenient in some respects, its use has tended to obscure the essential unity of the principle of the limit of resolution whether applied to telescopes or microscopes.  
  
---  
Fig. 4.  
  
In fig. 4, AB represents the axis of an optical instrument (telescope or microscope), A being a point of the object and B a point of the image. By the operation of the object-glass LL′ all the rays issuing from A arrive in the same phase at B. Thus if A be self-luminous, the illumination is a maximum at B, where all the secondary waves agree in phase. B is in fact the centre of the diffraction disk which constitutes the image of A. At neighbouring points the illumination is less, in consequence of the discrepancies of phase which there enter. In like manner if we take a neighbouring point P, also self-luminous, in the plane of the object, the waves which issue from it will arrive at B with phases no longer absolutely concordant, and the discrepancy of phase will increase as the interval AP increases. When the interval is very small the discrepancy, though mathematically existent, produces no practical effect; and the illumination at B due to P is as important as that due to A, the intensities of the two luminous sources being supposed equal. Under these conditions it is clear that A and P are not separated in the image. The question is to what amount must the distance AP be increased in order that the difference of situation may make itself felt in the image. This is necessarily a question of degree; but it does not require detailed calculations in order to show that the discrepancy first becomes conspicuous when the phases corresponding to the various secondary waves which travel from P to B range over a complete period. The illumination at B due to P then becomes comparatively small, indeed for some forms of aperture evanescent. The extreme discrepancy is that between the waves which travel through the outermost parts of the object-glass at L and L′; so that if we adopt the above standard of resolution, the question is where must P be situated in order that the relative retardation of the rays PL and PL’ may on their arrival at B amount to a wave-length (λ). In virtue of the general law that the reduced optical path is stationary in value, this retardation may be calculated without allowance for the different paths pursued on the farther side of L, L′, so that the value is simply PL − PL′. Now since AP is very small, AL′ − PL′ = AP sin α, where α is the angular semi-aperture L′AB. In like manner PL − AL has the same value, so that

PL − PL′ = 2AP sin α.

According to the standard adopted, the condition of resolution is therefore that AP, or ε, should exceed ½λ/sin α. If ε be less than this, the images overlap too much; while if ε greatly exceed the above value the images become unnecessarily separated.

In the above argument the whole space between the object and the lens is supposed to be occupied by matter of one refractive index, and λ represents the wave-length in this medium of the kind of light employed. If the restriction as to uniformity be violated, what we have ultimately to deal with is the wave-length in the medium immediately surrounding the object.

Calling the refractive index μ, we have as the critical value of ε,

ε = ½λ0 /μ sin α,  (1),

λ0 being the wave-length in vacuo. The denominator μ sin α is the quantity well known (after Abbe) as the “numerical aperture.”

The extreme value possible for α is a right angle, so that for the microscopic limit we have

ε = ½λ0/μ (2).

The limit can be depressed only by a diminution in λ0, such as photography makes possible, or by an increase in μ, the refractive index of the medium in which the object is situated.

The statement of the law of resolving power has been made in a form appropriate to the microscope, but it admits also of immediate application to the telescope. If 2R be the diameter of the object-glass and D the distance of the object, the angle subtended by AP is ε/D, and the angular resolving power is given by

λ/2D sin α = λ/2R  (3).

This method of derivation (substantially due to Helmholtz) makes it obvious that there is no essential difference of principle between the two cases, although the results are conveniently stated in different forms. In the case of the telescope we have to deal with a linear measure of aperture and an angular limit of resolution, whereas in the case of the microscope the limit of resolution is linear, and it is expressed in terms of angular aperture.

It must be understood that the above argument distinctly assumes that the different parts of the object are self-luminous, or at least that the light proceeding from the various points is without phase relations. As has been emphasized by G. J. Stoney, the restriction is often, perhaps usually, violated in the microscope. A different treatment is then necessary, and for some of the problems which arise under this head the method of Abbe is convenient.

The importance of the general conclusions above formulated, as imposing a limit upon our powers of direct observation, can hardly be overestimated; but there has been in some quarters a tendency to ascribe to it a more precise character than it can bear, or even to mistake its meaning altogether. A few words of further explanation may therefore be desirable. The first point to be emphasized is that nothing whatever is said as to the smallness of a single object that may be made visible. The eye, unaided or armed with a telescope, is able to see, as points of light, stars subtending no sensible angle. The visibility of a star is a question of brightness simply, and has nothing to do with resolving power. The latter element enters only when it is a question of recognizing the duplicity of a double star, or of distinguishing detail upon the surface of a planet. So in the microscope there is nothing except lack of light to hinder the visibility of an object however small. But if its dimensions be much less than the half wave-length, it can only be seen as a whole, and its parts cannot be distinctly separated, although in cases near the border line some inference may be possible, founded upon experience of what appearances are presented in various cases. Interesting observations upon particles, ultra-microscopic in the above sense, have been recorded by H. F. W. Siedentopf and R. A. Zsigmondy (Drude’s Ann., 1903, 10, p. 1).

In a somewhat similar way a dark linear interruption in a bright ground may be visible, although its actual width is much inferior to the half wave-length. In illustration of this fact a simple experiment may be mentioned. In front of the naked eye was held a piece of copper foil perforated by a fine needle hole. Observed through this the structure of some wire gauze just disappeared at a distance from the eye equal to 17 in., the gauze containing 46 meshes to the inch. On the other hand, a single wire 0.034 in. in diameter remained fairly visible up to a distance of 20 ft. The ratio between the limiting angles subtended by the periodic structure of the gauze and the diameter of the wire was (.022/.034) × (240/17) = 9.1. For further information upon this subject reference may be made to Phil. Mag., 1896, 42, p. 167; Journ. R. Micr. Soc., 1903, p. 447.

6\. Coronas or Glories.—The results of the theory of the diffraction patterns due to circular apertures admit of an interesting application to coronas, such as are often seen encircling the sun and moon. They are due to the interposition of small spherules of water, which act the part of diffracting obstacles. In order to the formation of a well-defined corona it is essential that the particles be exclusively, or preponderatingly, of one size.

If the origin of light be treated as infinitely small, and be seen in focus, whether with the naked eye or with the aid of a telescope, the whole of the light in the absence of obstacles would be concentrated in the immediate neighbourhood of the focus. At other parts of the field the effect is the same, in accordance with the principle known as Babinet’s, whether the imaginary screen in front of the object-glass is generally transparent but studded with a number of opaque circular disks, or is generally opaque but perforated with corresponding apertures. Since at these points the resultant due to the whole aperture is zero, any two portions into which the whole may be divided must give equal and opposite resultants. Consider now the light diffracted in a direction many times more oblique than any with which we should be concerned, were the whole aperture uninterrupted, and take first the effect of a single small aperture. The light in the proposed direction is that determined by the size of the small aperture in accordance with the laws already investigated, and its phase depends upon the position of the aperture. If we take a direction such that the light (of given wave-length) from a single aperture vanishes, the evanescence continues even when the whole series of apertures is brought into contemplation. Hence, whatever else may happen, there must be a system of dark rings formed, the same as from a single small aperture. In directions other than these it is a more delicate question how the partial effects should be compounded. If we make the extreme suppositions of an infinitely small source and absolutely homogeneous light, there is no escape from the conclusion that the light in a definite direction is arbitrary, that is, dependent upon the chance distribution of apertures. If, however, as in practice, the light be heterogeneous, the source of finite area, the obstacles in motion, and the discrimination of different directions imperfect, we are concerned merely with the mean brightness found by varying the arbitrary phase-relations, and this is obtained by simply multiplying the brightness due to a single aperture by the number of apertures (n) (see Interference of Light, § 4). The diffraction pattern is therefore that due to a single aperture, merely brightened n times.

In his experiments upon this subject Fraunhofer employed plates of glass dusted over with lycopodium, or studded with small metallic disks of uniform size; and he found that the diameters of the rings were proportional to the length of the waves and inversely as the diameter of the disks.

In another respect the observations of Fraunhofer appear at first sight to be in disaccord with theory; for his measures of the diameters of the red rings, visible when white light was employed, correspond with the law applicable to dark rings, and not to the different law applicable to the luminous maxima. Verdet has, however, pointed out that the observation in this form is essentially different from that in which homogeneous red light is employed, and that the position of the red rings would correspond to the absence of blue-green light rather than to the greatest abundance of red light. Verdet’s own observations, conducted with great care, fully confirm this view, and exhibit a complete agreement with theory.

By measurements of coronas it is possible to infer the size of the particles to which they are due, an application of considerable interest in the case of natural coronas—the general rule being the larger the corona the smaller the water spherules. Young employed this method not only to determine the diameters of cloud particles (e.g. 1⁄1000 in.), but also those of fibrous material, for which the theory is analogous. His instrument was called the eriometer (see “Chromatics,” vol. iii. of supp. to Ency. Brit., 1817).

7\. Influence of Aberration. Optical Power of Instruments.—Our investigations and estimates of resolving power have thus far proceeded upon the supposition that there are no optical imperfections, whether of the nature of a regular aberration or dependent upon irregularities of material and workmanship. In practice there will always be a certain aberration or error of phase, which we may also regard as the deviation of the actual wave-surface from its intended position. In general, we may say that aberration is unimportant when it nowhere (or at any rate over a relatively small area only) exceeds a small fraction of the wave-length (λ). Thus in estimating the intensity at a focal point, where, in the absence of aberration, all the secondary waves would have exactly the same phase, we see that an aberration nowhere exceeding ¼λ can have but little effect.

The only case in which the influence of small aberration upon the entire image has been calculated (Phil. Mag., 1879) is that of a rectangular aperture, traversed by a cylindrical wave with aberration equal to cx³. The aberration is here unsymmetrical, the wave being in advance of its proper place in one half of the aperture, but behind in the other half. No terms in x or x² need be considered. The first would correspond to a general turning of the beam; and the second would imply imperfect focusing of the central parts. The effect of aberration may be considered in two ways. We may suppose the aperture (a) constant, and inquire into the operation of an increasing aberration; or we may take a given value of c (i.e. a given wave-surface) and examine the effect of a varying aperture. The results in the second case show that an increase of aperture up to that corresponding to an extreme aberration of half a period has no ill effect upon the central band (§ 3), but it increases unduly the intensity of one of the neighbouring lateral bands; and the practical conclusion is that the best results will be obtained from an aperture giving an extreme aberration of from a quarter to half a period, and that with an increased aperture aberration is not so much a direct cause of deterioration as an obstacle to the attainment of that improved definition which should accompany the increase of aperture.

If, on the other hand, we suppose the aperture given, we find that aberration begins to be distinctly mischievous when it amounts to about a quarter period, i.e. when the wave-surface deviates at each end by a quarter wave-length from the true plane.

As an application of this result, let us investigate what amount of temperature disturbance in the tube of a telescope may be expected to impair definition. According to J. B. Biot and F. J. D. Arago, the index μ for air at t° C. and at atmospheric pressure is given by

μ − 1 = | .00029 | .  
---|---|---  
1 + .0037 t  
  
If we take 0° C. as standard temperature,

δμ = -1.1 × 10-6.

Thus, on the supposition that the irregularity of temperature t extends through a length l, and produces an acceleration of a quarter of a wave-length,

¼λ = 1.1 lt × 10-6;

or, if we take λ = 5.3 × 10-5,

lt = 12,

the unit of length being the centimetre.

We may infer that, in the case of a telescope tube 12 cm. long, a stratum of air heated 1° C. lying along the top of the tube, and occupying a moderate fraction of the whole volume, would produce a not insensible effect. If the change of temperature progressed uniformly from one side to the other, the result would be a lateral displacement of the image without loss of definition; but in general both effects would be observable. In longer tubes a similar disturbance would be caused by a proportionally less difference of temperature. S. P. Langley has proposed to obviate such ill-effects by stirring the air included within a telescope tube. It has long been known that the definition of a carbon bisulphide prism may be much improved by a vigorous shaking.

We will now consider the application of the principle to the formation of images, unassisted by reflection or refraction (Phil. Mag., 1881). The function of a lens in forming an image is to compensate by its variable thickness the differences of phase which would otherwise exist between secondary waves arriving at the focal point from various parts of the aperture. If we suppose the diameter of the lens to be given (2R), and its focal length ƒ gradually to increase, the original differences of phase at the image of an infinitely distant luminous point diminish without limit. When ƒ attains a certain value, say ƒ1, the extreme error of phase to be compensated falls to ¼λ. But, as we have seen, such an error of phase causes no sensible deterioration in the definition; so that from this point onwards the lens is useless, as only improving an image already sensibly as perfect as the aperture admits of. Throughout the operation of increasing the focal length, the resolving power of the instrument, which depends only upon the aperture, remains unchanged; and we thus arrive at the rather startling conclusion that a telescope of any degree of resolving power might be constructed without an object-glass, if only there were no limit to the admissible focal length. This last proviso, however, as we shall see, takes away almost all practical importance from the proposition.

To get an idea of the magnitudes of the quantities involved, let us take the case of an aperture of 1⁄5 in., about that of the pupil of the eye. The distance ƒ1, which the actual focal length must exceed, is given by

√ (ƒ1² + R²) − ƒ1 = ¼λ;

so that

ƒ1 = 2R²/λ (1).

Thus, if λ = 1⁄40000, R = 1⁄10, we find

ƒ1 = 800 inches.

The image of the sun thrown upon a screen at a distance exceeding 66 ft., through a hole 1⁄5 in. in diameter, is therefore at least as well defined as that seen direct.

As the minimum focal length increases with the square of the aperture, a quite impracticable distance would be required to rival the resolving power of a modern telescope. Even for an aperture of 4 in., ƒ1 would have to be 5 miles.

A similar argument may be applied to find at what point an achromatic lens becomes sensibly superior to a single one. The question is whether, when the adjustment of focus is correct for the central rays of the spectrum, the error of phase for the most extreme rays (which it is necessary to consider) amounts to a quarter of a wave-length. If not, the substitution of an achromatic lens will be of no advantage. Calculation shows that, if the aperture be 1⁄5 in., an achromatic lens has no sensible advantage if the focal length be greater than about 11 in. If we suppose the focal length to be 66 ft., a single lens is practically perfect up to an aperture of 1.7 in.

Another obvious inference from the necessary imperfection of optical images is the uselessness of attempting anything like an absolute destruction of spherical aberration. An admissible error of phase of ¼λ will correspond to an error of 1⁄8λ in a reflecting and ½λ in a (glass) refracting surface, the incidence in both cases being perpendicular. If we inquire what is the greatest admissible longitudinal aberration (δƒ) in an object-glass according to the above rule, we find

δƒ = λα-2 (2),

α being the angular semi-aperture.

In the case of a single lens of glass with the most favourable curvatures, δƒ is about equal to α²ƒ, so that α4 must not exceed λ/ƒ. For a lens of 3 ft. focus this condition is satisfied if the aperture does not exceed 2 in.

When parallel rays fall directly upon a spherical mirror the longitudinal aberration is only about one-eighth as great as for the most favourably shaped single lens of equal focal length and aperture. Hence a spherical mirror of 3 ft. focus might have an aperture of 2½ in., and the image would not suffer materially from aberration.

On the same principle we may estimate the least visible displacement of the eye-piece of a telescope focused upon a distant object, a question of interest in connexion with range-finders. It appears (Phil. Mag., 1885, 20, p. 354) that a displacement δf from the true focus will not sensibly impair definition, provided

δƒ < ƒ²λ/R²  (3),

2R being the diameter of aperture. The linear accuracy required is thus a function of the ratio of aperture to focal length. The formula agrees well with experiment.

The principle gives an instantaneous solution of the question of the ultimate optical efficiency in the method of “mirror-reading,” as commonly practised in various physical observations. A rotation by which one edge of the mirror advances ¼λ (while the other edge retreats to a like amount) introduces a phase-discrepancy of a whole period where before the rotation there was complete agreement. A rotation of this amount should therefore be easily visible, but the limits of resolving power are being approached; and the conclusion is independent of the focal length of the mirror, and of the employment of a telescope, provided of course that the reflected image is seen in focus, and that the full width of the mirror is utilized.  
  
---  
Fig. 5.  
  
A comparison with the method of a material pointer, attached to the parts whose rotation is under observation, and viewed through a microscope, is of interest. The limiting efficiency of the microscope is attained when the angular aperture amounts to 180°; and it is evident that a lateral displacement of the point under observation through ½λ entails (at the old image) a phase-discrepancy of a whole period, one extreme ray being accelerated and the other retarded by half that amount. We may infer that the limits of efficiency in the two methods are the same when the length of the pointer is equal to the width of the mirror.

We have seen that in perpendicular reflection a surface error not exceeding 1⁄8λ may be admissible. In the case of oblique reflection at an angle φ, the error of retardation due to an elevation BD (fig. 5) is

QQ′ − QS = BD sec φ(1 − cos SQQ′) = BD sec φ (1 + cos 2φ) = 2BD cos φ;

from which it follows that an error of given magnitude in the figure of a surface is less important in oblique than in perpendicular reflection. It must, however, be borne in mind that errors can sometimes be compensated by altering adjustments. If a surface intended to be flat is affected with a slight general curvature, a remedy may be found in an alteration of focus, and the remedy is the less complete as the reflection is more oblique.

The formula expressing the optical power of prismatic spectroscopes may readily be investigated upon the principles of the wave theory. Let A0B0 be a plane wave-surface of the light before it falls upon the prisms, AB the corresponding wave-surface for a particular part of the spectrum after the light has passed the prisms, or after it has passed the eye-piece of the observing telescope. The path of a ray from the wave-surface A0B0 to A or B is determined by the condition that the optical distance, ∫ μ ds, is a minimum; and, as AB is by supposition a wave-surface, this optical distance is the same for both points. Thus

∫ μ ds (for A) = ∫ μ ds (for B)  (4).

We have now to consider the behaviour of light belonging to a neighbouring part of the spectrum. The path of a ray from the wave-surface A0B0 to the point A is changed; but in virtue of the minimum property the change may be neglected in calculating the optical distance, as it influences the result by quantities of the second order only in the changes of refrangibility. Accordingly, the optical distance from A0B0 to A is represented by ∫(μ \+ δμ)ds, the integration being along the original path A0 ... A; and similarly the optical distance between A0B0 and B is represented by ∫ (μ \+ δμ)ds, the integration being along B0 ... B. In virtue of (4) the difference of the optical distances to A and B is

∫ δμ ds (along B0 ... B) − ∫ δμ ds (along A0 ... A)  (5).

The new wave-surface is formed in such a position that the optical distance is constant; and therefore the dispersion, or the angle through which the wave-surface is turned by the change of refrangibility, is found simply by dividing (5) by the distance AB. If, as in common flint-glass spectroscopes, there is only one dispersing substance, ∫ δμ ds = δμ·s, where s is simply the thickness traversed by the ray. If t2 and t1 be the thicknesses traversed by the extreme rays, and a denote the width of the emergent beam, the dispersion θ is given by

θ = δμ(t2 − t1)/a,

or, if t1 be negligible,

θ = δμt/a  (6).

The condition of resolution of a double line whose components subtend an angle θ is that θ must exceed λ/a. Hence, in order that a double line may be resolved whose components have indices μ and μ \+ δμ, it is necessary that t should exceed the value given by the following equation:—

t = λ/δμ (7).

8\. Diffraction Gratings.—Under the heading “Colours of Striated Surfaces,” Thomas Young (Phil. Trans., 1802) in his usual summary fashion gave a general explanation of these colours, including the law of sines, the striations being supposed to be straight, parallel and equidistant. Later, in his article “Chromatics” in the supplement to the 5th edition of this encyclopaedia, he shows that the colours “lose the mixed character of periodical colours, and resemble much more the ordinary prismatic spectrum, with intervals completely dark interposed,” and explains it by the consideration that any phase-difference which may arise at neighbouring striae is multiplied in proportion to the total number of striae.

The theory was further developed by A. J. Fresnel (1815), who gave a formula equivalent to (5) below. But it is to J. von Fraunhofer that we owe most of our knowledge upon this subject. His recent discovery of the “fixed lines” allowed a precision of observation previously impossible. He constructed gratings up to 340 periods to the inch by straining fine wire over screws. Subsequently he ruled gratings on a layer of gold-leaf attached to glass, or on a layer of grease similarly supported, and again by attacking the glass itself with a diamond point. The best gratings were obtained by the last method, but a suitable diamond point was hard to find, and to preserve. Observing through a telescope with light perpendicularly incident, he showed that the position of any ray was dependent only upon the grating interval, viz. the distance from the centre of one wire or line to the centre of the next, and not otherwise upon the thickness of the wire and the magnitude of the interspace. In different gratings the lengths of the spectra and their distances from the axis were inversely proportional to the grating interval, while with a given grating the distances of the various spectra from the axis were as 1, 2, 3, &c. To Fraunhofer we owe the first accurate measurements of wave-lengths, and the method of separating the overlapping spectra by a prism dispersing in the perpendicular direction. He described also the complicated patterns seen when a point of light is viewed through two superposed gratings, whose lines cross one another perpendicularly or obliquely. The above observations relate to transmitted light, but Fraunhofer extended his inquiry to the light reflected. To eliminate the light returned from the hinder surface of an engraved grating, he covered it with a black varnish. It then appeared that under certain angles of incidence parts of the resulting spectra were completely polarized. These remarkable researches of Fraunhofer, carried out in the years 1817-1823, are republished in his Collected Writings (Munich, 1888).

The principle underlying the action of gratings is identical with that discussed in § 2, and exemplified in J. L. Soret’s “zone plates.” The alternate Fresnel’s zones are blocked out or otherwise modified; in this way the original compensation is upset and a revival of light occurs in unusual directions. If the source be a point or a line, and a collimating lens be used, the incident waves may be regarded as plane. If, further, on leaving the grating the light be received by a focusing lens, e.g. the object-glass of a telescope, the Fresnel’s zones are reduced to parallel and equidistant straight strips, which at certain angles coincide with the ruling. The directions of the lateral spectra are such that the passage from one element of the grating to the corresponding point of the next implies a retardation of an integral number of wave-lengths. If the grating be composed of alternate transparent and opaque parts, the question may be treated by means of the general integrals (§ 3) by merely limiting the integration to the transparent parts of the aperture. For an investigation upon these lines the reader is referred to Airy’s Tracts, to Verdet’s Leçons, or to R. W. Wood’s Physical Optics. If, however, we assume the theory of a simple rectangular aperture (§ 3); the results of the ruling can be inferred by elementary methods, which are perhaps more instructive.

Apart from the ruling, we know that the image of a mathematical line will be a series of narrow bands, of which the central one is by far the brightest. At the middle of this band there is complete agreement of phase among the secondary waves. The dark lines which separate the bands are the places at which the phases of the secondary wave range over an integral number of periods. If now we suppose the aperture AB to be covered by a great number of opaque strips or bars of width d, separated by transparent intervals of width a, the condition of things in the directions just spoken of is not materially changed. At the central point there is still complete agreement of phase; but the amplitude is diminished in the ratio of a : a + d. In another direction, making a small angle with the last, such that the projection of AB upon it amounts to a few wave-lengths, it is easy to see that the mode of interference is the same as if there were no ruling. For example, when the direction is such that the projection of AB upon it amounts to one wave-length, the elementary components neutralize one another, because their phases are distributed symmetrically, though discontinuously, round the entire period. The only effect of the ruling is to diminish the amplitude in the ratio a : a + d; and, except for the difference in illumination, the appearance of a line of light is the same as if the aperture were perfectly free.

The lateral (spectral) images occur in such directions that the projection of the element (a + d) of the grating upon them is an exact multiple of λ. The effect of each of the n elements of the grating is then the same; and, unless this vanishes on account of a particular adjustment of the ratio a : d, the resultant amplitude becomes comparatively very great. These directions, in which the retardation between A and B is exactly mnλ, may be called the principal directions. On either side of any one of them the illumination is distributed according to the same law as for the central image (m = 0), vanishing, for example, when the retardation amounts to (mn ± 1)λ. In considering the relative brightnesses of the different spectra, it is therefore sufficient to attend merely to the principal directions, provided that the whole deviation be not so great that its cosine differs considerably from unity.

We have now to consider the amplitude due to a single element, which we may conveniently regard as composed of a transparent part a bounded by two opaque parts of width ½d. The phase of the resultant effect is by symmetry that of the component which comes from the middle of a. The fact that the other components have phases differing from this by amounts ranging between ± amπ/(a + d) causes the resultant amplitude to be less than for the central image (where there is complete phase agreement). If Bm denote the brightness of the mth lateral image, and B0 that of the central image, we have

Bm : B 0 = [∫ | \+ amπ/(a+d) | cosx dx ÷ | 2amπ | ] | ² | = ( | a + d | ) | ² | sin² | amπ |  (1).  
---|---|---|---|---|---|---|---|---|---|---|---|---  
− amπ/(a+d) | a + d |   | amπ |   | a + d  
  
If B denotes the brightness of the central image when the whole of the space occupied by the grating is transparent, we have

B0 : B = a² : (a + d)²,

and thus

Bm : B = | 1 | sin² | amπ |  (2).  
---|---|---|---|---  
m²π² | a + d  
  
The sine of an angle can never be greater than unity; and consequently under the most favourable circumstances only 1/m²π² of the original light can be obtained in the mth spectrum. We conclude that, with a grating composed of transparent and opaque parts, the utmost light obtainable in any one spectrum is in the first, and there amounts to 1/π², or about 1⁄10, and that for this purpose a and d must be equal. When d = a the general formula becomes

Bm : B = | sin² ½mπ |  (3),  
---|---|---  
m²π²  
  
showing that, when m is even, Bm vanishes, and that, when m is odd,

Bm : B = 1/m²π².

The third spectrum has thus only 1⁄9 of the brilliancy of the first.

Another particular case of interest is obtained by supposing a small relatively to (a + d). Unless the spectrum be of very high order, we have simply

Bm : B = {a/(a + d)}²  (4);

so that the brightnesses of all the spectra are the same.

The light stopped by the opaque parts of the grating, together with that distributed in the central image and lateral spectra, ought to make up the brightness that would be found in the central image, were all the apertures transparent. Thus, if a = d, we should have

1 = ½ + ¼ + 2/π² (1 + 1⁄9 \+ 1⁄25 \+ ...),

which is true by a known theorem. In the general case

a | = ( | a | ) | ² | + | 2 | Σ | m=∞ | 1 | sin²( | mπa | ),  
---|---|---|---|---|---|---|---|---|---|---|---|---  
a + d | a + d |   | π² | m=1 | m² | a + d  
  
a formula which may be verified by Fourier’s theorem.

According to a general principle formulated by J. Babinet, the brightness of a lateral spectrum is not affected by an interchange of the transparent and opaque parts of the grating. The vibrations corresponding to the two parts are precisely antagonistic, since if both were operative the resultant would be zero. So far as the application to gratings is concerned, the same conclusion may be derived from (2).  
  
---  
Fig. 6.  
  
From the value of Bm : B0 we see that no lateral spectrum can surpass the central image in brightness; but this result depends upon the hypothesis that the ruling acts by opacity, which is generally very far from being the case in practice. In an engraved glass grating there is no opaque material present by which light could be absorbed, and the effect depends upon a difference of retardation in passing the alternate parts. It is possible to prepare gratings which give a lateral spectrum brighter than the central image, and the explanation is easy. For if the alternate parts were equal and alike transparent, but so constituted as to give a relative retardation of ½λ, it is evident that the central image would be entirely extinguished, while the first spectrum would be four times as bright as if the alternate parts were opaque. If it were possible to introduce at every part of the aperture of the grating an arbitrary retardation, all the light might be concentrated in any desired spectrum. By supposing the retardation to vary uniformly and continuously we fall upon the case of an ordinary prism: but there is then no diffraction spectrum in the usual sense. To obtain such it would be necessary that the retardation should gradually alter by a wave-length in passing over any element of the grating, and then fall back to its previous value, thus springing suddenly over a wave-length (Phil. Mag., 1874, 47, p. 193). It is not likely that such a result will ever be fully attained in practice; but the case is worth stating, in order to show that there is no theoretical limit to the concentration of light of assigned wave-length in one spectrum, and as illustrating the frequently observed unsymmetrical character of the spectra on the two sides of the central image.4

We have hitherto supposed that the light is incident perpendicularly upon the grating; but the theory is easily extended. If the incident rays make an angle θ with the normal (fig. 6), and the diffracted rays make an angle φ (upon the same side), the relative retardation from each element of width (a + d) to the next is (a + d) (sinθ \+ sinφ); and this is the quantity which is to be equated to mλ. Thus

sinθ \+ sinφ = 2sin ½(θ \+ φ) cos ½ (θ − φ) = mλ/(a + d)  (5).

The “deviation” is (θ \+ φ), and is therefore a minimum when θ = φ, i.e. when the grating is so situated that the angles of incidence and diffraction are equal.  
  
---  
Fig. 7.  
  
In the case of a reflection grating the same method applies. If θ and φ denote the angles with the normal made by the incident and diffracted rays, the formula (5) still holds, and, if the deviation be reckoned from the direction of the regularly reflected rays, it is expressed as before by (θ \+ φ), and is a minimum when θ = φ, that is, when the diffracted rays return upon the course of the incident rays.

In either case (as also with a prism) the position of minimum deviation leaves the width of the beam unaltered, i.e. neither magnifies nor diminishes the angular width of the object under view.

From (5) we see that, when the light falls perpendicularly upon a grating (θ = 0), there is no spectrum formed (the image corresponding to m = 0 not being counted as a spectrum), if the grating interval σ or (a + d) is less than λ. Under these circumstances, if the material of the grating be completely transparent, the whole of the light must appear in the direct image, and the ruling is not perceptible. From the absence of spectra Fraunhofer argued that there must be a microscopic limit represented by λ; and the inference is plausible, to say the least (Phil. Mag., 1886). Fraunhofer should, however, have fixed the microscopic limit at ½λ, as appears from (5), when we suppose θ = ½π, φ = ½π.  
  
---  
Fig. 8.  
  
We will now consider the important subject of the resolving power of gratings, as dependent upon the number of lines (n) and the order of the spectrum observed (m). Let BP (fig. 8) be the direction of the principal maximum (middle of central band) for the wave-length λ in the mth spectrum. Then the relative retardation of the extreme rays (corresponding to the edges A, B of the grating) is mnλ. If BQ be the direction for the first minimum (the darkness between the central and first lateral band), the relative retardation of the extreme rays is (mn + 1)λ. Suppose now that λ \+ δλ is the wave-length for which BQ gives the principal maximum, then

(mn + 1)λ = mn(λ \+ δλ);

whence

δλ/λ = 1/mn  (6).

According to our former standard, this gives the smallest difference of wave-lengths in a double line which can be just resolved; and we conclude that the resolving power of a grating depends only upon the total number of lines, and upon the order of the spectrum, without regard to any other considerations. It is here of course assumed that the n lines are really utilized.

In the case of the D lines the value of δλ/λ is about 1/1000; so that to resolve this double line in the first spectrum requires 1000 lines, in the second spectrum 500, and so on.

It is especially to be noticed that the resolving power does not depend directly upon the closeness of the ruling. Let us take the case of a grating 1 in. broad, and containing 1000 lines, and consider the effect of interpolating an additional 1000 lines, so as to bisect the former intervals. There will be destruction by interference of the first, third and odd spectra generally; while the advantage gained in the spectra of even order is not in dispersion, nor in resolving power, but simply in brilliancy, which is increased four times. If we now suppose half the grating cut away, so as to leave 1000 lines in half an inch, the dispersion will not be altered, while the brightness and resolving power are halved.

There is clearly no theoretical limit to the resolving power of gratings, even in spectra of given order. But it is possible that, as suggested by Rowland,5 the structure of natural spectra may be too coarse to give opportunity for resolving powers much higher than those now in use. However this may be, it would always be possible, with the aid of a grating of given resolving power, to construct artificially from white light mixtures of slightly different wave-length whose resolution or otherwise would discriminate between powers inferior and superior to the given one.6

If we define as the “dispersion” in a particular part of the spectrum the ratio of the angular interval dθ to the corresponding increment of wave-length dλ, we may express it by a very simple formula. For the alteration of wave-length entails, at the two limits of a diffracted wave-front, a relative retardation equal to mndλ. Hence, if a be the width of the diffracted beam, and dθ the angle through which the wave-front is turned,

adθ = mn dλ,

or

dispersion = mn/a  (7).

The resolving power and the width of the emergent beam fix the optical character of the instrument. The latter element must eventually be decreased until less than the diameter of the pupil of the eye. Hence a wide beam demands treatment with further apparatus (usually a telescope) of high magnifying power.

In the above discussion it has been supposed that the ruling is accurate, and we have seen that by increase of m a high resolving power is attainable with a moderate number of lines. But this procedure (apart from the question of illumination) is open to the objection that it makes excessive demands upon accuracy. According to the principle already laid down it can make but little difference in the principal direction corresponding to the first spectrum, provided each line lie within a quarter of an interval (a + d) from its theoretical position. But, to obtain an equally good result in the mth spectrum, the error must be less than 1/m of the above amount.7

There are certain errors of a systematic character which demand special consideration. The spacing is usually effected by means of a screw, to each revolution of which corresponds a large number (e.g. one hundred) of lines. In this way it may happen that although there is almost perfect periodicity with each revolution of the screw after (say) 100 lines, yet the 100 lines themselves are not equally spaced. The “ghosts” thus arising were first described by G. H. Quincke (Pogg. Ann., 1872, 146, p. 1), and have been elaborately investigated by C. S. Peirce (Ann. Journ. Math., 1879, 2, p. 330), both theoretically and experimentally. The general nature of the effects to be expected in such a case may be made clear by means of an illustration already employed for another purpose. Suppose two similar and accurately ruled transparent gratings to be superposed in such a manner that the lines are parallel. If the one set of lines exactly bisect the intervals between the others, the grating interval is practically halved, and the previously existing spectra of odd order vanish. But a very slight relative displacement will cause the apparition of the odd spectra. In this case there is approximate periodicity in the half interval, but complete periodicity only after the whole interval. The advantage of approximate bisection lies in the superior brilliancy of the surviving spectra; but in any case the compound grating may be considered to be perfect in the longer interval, and the definition is as good as if the bisection were accurate.  
  
---  
Fig. 9.—x². | Fig. 10.—y². | Fig. 11.—x³. | Fig. 12.—xy².  
---  
Fig. 13.—xy.  Fig. 14.—x²y.  Fig. 15.—y³.  
  
The effect of a gradual increase in the interval (fig. 9) as we pass across the grating has been investigated by M. A. Cornu (C.R., 1875, 80, p. 655), who thus explains an anomaly observed by E. E. N. Mascart. The latter found that certain gratings exercised a converging power upon the spectra formed upon one side, and a corresponding diverging power upon the spectra on the other side. Let us suppose that the light is incident perpendicularly, and that the grating interval increases from the centre towards that edge which lies nearest to the spectrum under observation, and decreases towards the hinder edge. It is evident that the waves from both halves of the grating are accelerated in an increasing degree, as we pass from the centre outwards, as compared with the phase they would possess were the central value of the grating interval maintained throughout. The irregularity of spacing has thus the effect of a convex lens, which accelerates the marginal relatively to the central rays. On the other side the effect is reversed. This kind of irregularity may clearly be present in a degree surpassing the usual limits, without loss of definition, when the telescope is focused so as to secure the best effect.

It may be worth while to examine further the other variations from correct ruling which correspond to the various terms expressing the deviation of the wave-surface from a perfect plane. If x and y be co-ordinates in the plane of the wave-surface, the axis of y being parallel to the lines of the grating, and the origin corresponding to the centre of the beam, we may take as an approximate equation to the wave-surface

z = | x² | \+ Bxy + | y² | \+ αx³ + βx²y + γxy² + δy³ + ...  (8);  
---|---|---|---|---  
2ρ | 2ρ′  
  
and, as we have just seen, the term in x² corresponds to a linear error in the spacing. In like manner, the term in y² corresponds to a general curvature of the lines (fig. 10), and does not influence the definition at the (primary) focus, although it may introduce astigmatism.8 If we suppose that everything is symmetrical on the two sides of the primary plane y = 0, the coefficients B, β, δ vanish. In spite of any inequality between ρ and ρ’, the definition will be good to this order of approximation, provided α and γ vanish. The former measures the thickness of the primary focal line, and the latter measures its curvature. The error of ruling giving rise to α is one in which the intervals increase or decrease in both directions from the centre outwards (fig. 11), and it may often be compensated by a slight rotation in azimuth of the object-glass of the observing telescope. The term in γ corresponds to a variation of curvature in crossing the grating (fig. 12).

When the plane zx is not a plane of symmetry, we have to consider the terms in xy, x²y, and y³. The first of these corresponds to a deviation from parallelism, causing the interval to alter gradually as we pass along the lines (fig. 13). The error thus arising may be compensated by a rotation of the object-glass about one of the diameters y = ± x. The term in x²y corresponds to a deviation from parallelism in the same direction on both sides of the central line (fig. 14); and that in y³ would be caused by a curvature such that there is a point of inflection at the middle of each line (fig. 15).

All the errors, except that depending on α, and especially those depending on γ and δ, can be diminished, without loss of resolving power, by contracting the vertical aperture. A linear error in the spacing, and a general curvature of the lines, are eliminated in the ordinary use of a grating.

The explanation of the difference of focus upon the two sides as due to unequal spacing was verified by Cornu upon gratings purposely constructed with an increasing interval. He has also shown how to rule a plane surface with lines so disposed that the grating shall of itself give well-focused spectra.  
  
---  
Fig. 16.  
  
A similar idea appears to have guided H. A. Rowland to his brilliant invention of concave gratings, by which spectra can be photographed without any further optical appliance. In these instruments the lines are ruled upon a spherical surface of speculum metal, and mark the intersections of the surface by a system of parallel and equidistant planes, of which the middle member passes through the centre of the sphere. If we consider for the present only the primary plane of symmetry, the figure is reduced to two dimensions. Let AP (fig. 16) represent the surface of the grating, O being the centre of the circle. Then, if Q be any radiant point and Q’ its image (primary focus) in the spherical mirror AP, we have

1 | + | 1 | = | 2cosφ | ,  
---|---|---|---|---|---  
v1 | u | a  
  
where v1 = AQ′, u = AQ, a = OA, φ = angle of incidence QAO, equal to the angle of reflection Q′AO. If Q be on the circle described upon OA as diameter, so that u = a cos φ, then Q′ lies also upon the same circle; and in this case it follows from the symmetry that the unsymmetrical aberration (depending upon a) vanishes.

This disposition is adopted in Rowland′s instrument; only, in addition to the central image formed at the angle φ′ = φ, there are a series of spectra with various values of φ’, but all disposed upon the same circle. Rowland’s investigation is contained in the paper already referred to; but the following account of the theory is in the form adopted by R. T. Glazebrook (Phil. Mag., 1883).

In order to find the difference of optical distances between the courses QAQ′, QPQ′, we have to express QP − QA, PQ′ − AQ′. To find the former, we have, if OAQ = φ, AOP = ω,

QP² = u² + 4a²sin²½ω − 4au sin ½ω sin (½ω − φ)  
= (u + a sin φ sin ω)² − a² sin²φ sin²ω \+ 4a sin² ½ω(a − u cosφ).

Now as far as ω4

4 sin² ½ω = sin²ω \+ ¼sin4ω,

and thus to the same order

QP² = (u + a sin φ sin ω)²  
− a cos φ(u − a cos φ) sin²ω \+ ¼ a(a − u cos φ) sin4 ω.

pose that Q lies on the circle u = a cos φ, the middle term vanishes, and we get, correct as far as ω4,

QP = (u + a sin φ sin ω) √ {1 + | a² sin² φ sin4ω | };  
---|---|---  
4u  
  
so that

QP − u = a sin φ sin ω \+ 1⁄8a sin φ tan φ sin4 ω (9),

in which it is to be noticed that the adjustment necessary to secure the disappearance of sin²ω is sufficient also to destroy the term in sin³ω.

A similar expression can be found for Q’P − Q′A; and thus, if Q′A = v, Q′AO = φ′, where v = a cos φ′, we get

QP + PQ′ − QA -AQ′ = a sin ω (sin φ − sin φ′)  
\+ 1⁄8a sin4 ω (sin φ tan φ \+ sin φ′ tan φ′)  (10).

If φ′ = φ, the term of the first order vanishes, and the reduction of the difference of path via P and via A to a term of the fourth order proves not only that Q and Q′ are conjugate foci, but also that the foci are exempt from the most important term in the aberration. In the present application φ′ is not necessarily equal to φ; but if P correspond to a line upon the grating, the difference of retardations for consecutive positions of P, so far as expressed by the term of the first order, will be equal to ± mλ (m integral), and therefore without influence, provided

σ (sin φ − sinφ′) = ± mλ (11),

where σ denotes the constant interval between the planes containing the lines. This is the ordinary formula for a reflecting plane grating, and it shows that the spectra are formed in the usual directions. They are here focused (so far as the rays in the primary plane are concerned) upon the circle OQ′A, and the outstanding aberration is of the fourth order.

In order that a large part of the field of view may be in focus at once, it is desirable that the locus of the focused spectrum should be nearly perpendicular to the line of vision. For this purpose Rowland places the eye-piece at O, so that φ = 0, and then by (11) the value of φ′ in the mth spectrum is

σ sin φ’ = ± mλ (12).

If ω now relate to the edge of the grating, on which there are altogether n lines,

nσ = 2a sin ω,

and the value of the last term in (10) becomes

1⁄16 nσsin³ ω sin φ′ tan φ′,

or

1⁄16 mnλ sin³ω tan φ′ (13).

This expresses the retardation of the extreme relatively to the central ray, and is to be reckoned positive, whatever may be the signs of ω, and φ′. If the semi-angular aperture (ω) be 1⁄100, and tan φ′ = 1, mn might be as great as four millions before the error of phase would reach ¼λ. If it were desired to use an angular aperture so large that the aberration according to (13) would be injurious, Rowland points out that on his machine there would be no difficulty in applying a remedy by making σ slightly variable towards the edges. Or, retaining σ constant, we might attain compensation by so polishing the surface as to bring the circumference slightly forward in comparison with the position it would occupy upon a true sphere.

It may be remarked that these calculations apply to the rays in the primary plane only. The image is greatly affected with astigmatism; but this is of little consequence, if γ in (8) be small enough. Curvature of the primary focal line having a very injurious effect upon definition, it may be inferred from the excellent performance of these gratings that γ is in fact small. Its value does not appear to have been calculated. The other coefficients in (8) vanish in virtue of the symmetry.

The mechanical arrangements for maintaining the focus are of great simplicity. The grating at A and the eye-piece at O are rigidly attached to a bar AO, whose ends rest on carriages, moving on rails OQ, AQ at right angles to each other. A tie between the middle point of the rod OA and Q can be used if thought desirable.

The absence of chromatic aberration gives a great advantage in the comparison of overlapping spectra, which Rowland has turned to excellent account in his determinations of the relative wave-lengths of lines in the solar spectrum (Phil. Mag., 1887).

For absolute determinations of wave-lengths plane gratings are used. It is found (Bell, Phil. Mag., 1887) that the angular measurements present less difficulty than the comparison of the grating interval with the standard metre. There is also some uncertainty as to the actual temperature of the grating when in use. In order to minimize the heating action of the light, it might be submitted to a preliminary prismatic analysis before it reaches the slit of the spectrometer, after the manner of Helmholtz.

In spite of the many improvements introduced by Rowland and of the care with which his observations were made, recent workers have come to the conclusion that errors of unexpected amount have crept into his measurements of wave-lengths, and there is even a disposition to discard the grating altogether for fundamental work in favour of the so-called “interference methods,” as developed by A. A. Michelson, and by C. Fabry and J. B. Pérot. The grating would in any case retain its utility for the reference of new lines to standards otherwise fixed. For such standards a relative accuracy of at least one part in a million seems now to be attainable.

Since the time of Fraunhofer many skilled mechanicians have given their attention to the ruling of gratings. Those of Nobert were employed by A. J. Ångström in his celebrated researches upon wave-lengths. L. M. Rutherfurd introduced into common use the reflection grating, finding that speculum metal was less trying than glass to the diamond point, upon the permanence of which so much depends. In Rowland’s dividing engine the screws were prepared by a special process devised by him, and the resulting gratings, plane and concave, have supplied the means for much of the best modern optical work. It would seem, however, that further improvements are not excluded.

There are various copying processes by which it is possible to reproduce an original ruling in more or less perfection. The earliest is that of Quincke, who coated a glass grating with a chemical silver deposit, subsequently thickened with copper in an electrolytic bath. The metallic plate thus produced formed, when stripped from its support, a reflection grating reproducing many of the characteristics of the original. It is best to commence the electrolytic thickening in a silver acetate bath. At the present time excellent reproductions of Rowland’s speculum gratings are on the market (Thorp, Ives, Wallace), prepared, after a suggestion of Sir David Brewster, by coating the original with a varnish, e.g. of celluloid. Much skill is required to secure that the film when stripped shall remain undeformed.

A much easier method, applicable to glass originals, is that of photographic reproduction by contact printing. In several papers dating from 1872, Lord Rayleigh (see Collected Papers, i. 157, 160, 199, 504; iv. 226) has shown that success may be attained by a variety of processes, including bichromated gelatin and the old bitumen process, and has investigated the effect of imperfect approximation during the exposure between the prepared plate and the original. For many purposes the copies, containing lines up to 10,000 to the inch, are not inferior. It is to be desired that transparent gratings should be obtained from first-class ruling machines. To save the diamond point it might be possible to use something softer than ordinary glass as the material of the plate.

9\. Talbot’s Bands.—These very remarkable bands are seen under certain conditions when a tolerably pure spectrum is regarded with the naked eye, or with a telescope, half the aperture being covered by a thin plate, e.g. of glass or mica. The view of the matter taken by the discoverer (Phil. Mag., 1837, 10, p. 364) was that any ray which suffered in traversing the plate a retardation of an odd number of half wave-lengths would be extinguished, and that thus the spectrum would be seen interrupted by a number of dark bars. But this explanation cannot be accepted as it stands, being open to the same objection as Arago’s theory of stellar scintillation.9 It is as far as possible from being true that a body emitting homogeneous light would disappear on merely covering half the aperture of vision with a half-wave plate. Such a conclusion would be in the face of the principle of energy, which teaches plainly that the retardation in question leaves the aggregate brightness unaltered. The actual formation of the bands comes about in a very curious way, as is shown by a circumstance first observed by Brewster. When the retarding plate is held on the side towards the red of the spectrum, the bands are not seen. Even in the contrary case, the thickness of the plate must not exceed a certain limit, dependent upon the purity of the spectrum. A satisfactory explanation of these bands was first given by Airy (Phil. Trans., 1840, 225; 1841, 1), but we shall here follow the investigation of Sir G. G. Stokes (Phil. Trans., 1848, 227), limiting ourselves, however, to the case where the retarded and unretarded beams are contiguous and of equal width.

The aperture of the unretarded beam may thus be taken to be limited by x = -h, x = 0, y = -l, y= +l; and that of the beam retarded by R to be given by x = 0, x = h, y= -l, y = +l. For the former (1) § 3 gives

− | 1 | ∫ | 0 | ∫ | +l | sin k{ at − ƒ + | xξ \+ yη | }dxdy  
---|---|---|---|---|---|---|---|---  
λƒ | −h | −l | ƒ  
= − | 2lh | · | ƒ | sin | kηl | · | 2ƒ | sin | kξh | · sin k{at − ƒ − | ξh | } (1),  
---|---|---|---|---|---|---|---|---|---|---|---|---  
λƒ | kηl | ƒ | kξh | 2ƒ | 2ƒ  
  
on integration and reduction.

For the retarded stream the only difference is that we must subtract R from at, and that the limits of x are 0 and +h. We thus get for the disturbance at ξ, η, due to this stream

− | 2lh | · | ƒ | sin | kηl | · | 2ƒ | sin | kξh | · sin k{at − ƒ − R + | ξh | } (2)  
---|---|---|---|---|---|---|---|---|---|---|---|---  
λƒ | kηl | ƒ | kξh | 2ƒ | 2ƒ  
  
If we put for shortness π for the quantity under the last circular function in (1), the expressions (1), (2) may be put under the forms u sin τ, v sin (τ − α) respectively; and, if I be the intensity, I will be measured by the sum of the squares of the coefficients of sin τ and cos τ in the expression

u sin τ \+ v sin (τ − α),

so that

I = u² + v² + 2uv cos α,

which becomes on putting for u, v, and α their values, and putting

{ | ƒ | sin | kηl | } | ² | = Q  (3),  
---|---|---|---|---|---|---  
kηl | ƒ |    
I = Q · | 4l² | sin² | πξh | {2 + 2 cos( | 2πR | − | 2πξh | )} (4).  
---|---|---|---|---|---|---|---|---  
π²ξ² | λƒ | λ | λƒ  
  
If the subject of examination be a luminous line parallel to η, we shall obtain what we require by integrating (4) with respect to η from −∞ to +∞. The constant multiplier is of no especial interest so that we may take as applicable to the image of a line

I = | 2 | sin² | πξh | {1 + cos( | 2πR | − | 2πξh | )} (5).  
---|---|---|---|---|---|---|---|---  
ξ² | λƒ | λ | λƒ  
  
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, and ω is a constant, which is positive when the retarding plate is held at the side on which the lue of the spectrum is seen. The possibility of dark bands depends upon ω being 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

w = ∫ | +∞ | sin² h1ξ cos g1ξ | dξ | ,  
---|---|---|---|---  
−∞ | ξ²  
  
where

h1 = πh/λƒ,  g1 = ω − 2πh/λƒ  (10).

By differentiation with respect to g1 it may be proved that

w = 0 | from g1 = −∞ | to g1 = −2h1,  
---|---|---  
w = ½π(2h1 \+ g1) | from g1 = −2h1 | to g1 = 0,  
w = ½π(2h1 − g1) | from g1 = 0 | to g1 = 2h1,  
w = 0 | from g1 = 2h1 | to g1 = ∞.  
  
The integrated intensity, I′, or

2πh1 \+ 2 cos ρw,

is thus

I′ = 2πh1 (11),

when g1 numerically exceeds 2h1; and, when g1 lies 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 T0 the 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-covered circular aperture 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 plane pulse, i.e. a disturbance limited to an infinitely thin slice of the medium, is supposed to fall upon a parallel grating, which again may be 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 beams interfere. 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.  
  
---  
Fig. 17.  
  
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 a point, 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

cos 2π( | t | − | δ | )· ds,  
---|---|---|---|---  
r | λ  
  
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

2πδ| = | π(a + b)s² | = | π | v²  (2),  
---|---|---|---|---|---  
λ | abλ | 2  
  
the effect at B is

{ | abλ | } | ½ | { cos | 2πt | ∫ cos ½πv²·dv + sin | 2πt | ∫ sin ½πv²·dv } (3)  
---|---|---|---|---|---|---|---|---  
2(a + b) |   | τ | τ  
  
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

C = ∫ | v | cos ½πv²·dv,  S = ∫ | v | sin ½πv²·dv  (5).  
---|---|---|---|---  
0 | 0  
  
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

C + iS = ∫ | v | e | i·½πv² | dv = e | i·½πv² | · v − 1⁄3 iπ ∫ | v | e | i·½πv² | dv³;  
---|---|---|---|---|---|---|---|---|---|---  
0 |   |   | 0 |    
  
and, by continuing this process,

C + iS = e | i·½πv² | { v − | iπ | v³ + | iπ | iπ | v5 − | iπ | iπ | iπ | v7 \+ ... }.  
---|---|---|---|---|---|---|---|---|---|---|---  
  | 3 | 3 | 5 | 3 | 5 | 7  
  
By separation of real and imaginary parts,

C = M cos ½πv² − N sin ½πv² | } (6),  
---|---  
S = M sin ½πv² − N cos ½πv²  
  
where

M = | v | − | π²v5 | + | π4v9 | − ...  (7),  
---|---|---|---|---|---|---  
1 | 3·5 | 3·5·7·9  
N = | πv³ | − | π3v7 | + | π5v11 | ...  (8).  
---|---|---|---|---|---|---  
1·3 | 1·3·5·7 | 1·3·5·7·9·11  
  
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 obtained by L. P. Gilbert (Mem. cour. de l’Acad. de Bruxelles, 31, p. 1). Taking

½πv² = u  (9),

we may write

C + iS = | 1 | ∫ | u | eiudu |  (10).  
---|---|---|---|---|---  
√(2π) | 0 | √u  
  
Again, by a known formula,

1 | = | 1 | ∫ | ∞ | e−uxdx |  (11).  
---|---|---|---|---|---|---  
√ u | √π | 0 | √x  
  
Substituting this in (10), and inverting the order of integration, we get

C + iS = | 1 | ∫ | ∞ | dx | ∫ | u | e | u(i − x) | dx = | 1 | ∫ | ∞ | dx | eu(i − x) − 1 |  (12).  
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---  
π√2 | 0 | √x | 0 |   | π√2 |   | √x | i − x  
  
Thus, if we take

G = | 1 | ∫ | ∞ | e−ux √x · dx | , H = | 1 | ∫ | ∞ | e−ux dx |  (13),  
---|---|---|---|---|---|---|---|---|---|---  
π√2 | 0 | 1 + x² | π√2 | 0 | √x · (1 + x²)  
  
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

∫ | ∞ | e−y yq−½dy = Γ(q + ½) = (q − ½)(q − 3⁄2) ... ½ √π;  
---|---|---  
0  
  
and we get in terms of v

G = | 1 | − | 1·3·5 | + | 1·3·5·9 | − ...  (16),  
---|---|---|---|---|---|---  
π²v³ | π4v7 | π6v11  
H = | 1 | − | 1·3 | + | 1·3·5·7 | − ...  (17).  
---|---|---|---|---|---|---  
πv | π³v5 | π5v9  
  
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

dH | = − πvG,  | dG | = πvH − 1  (18).  
---|---|---|---  
dv | dv  
  
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.

V = √{ | 2(a + b) | } · CA,  (19),  
---|---|---  
abλ  
  
we may write

I² = ( ∫ | ∞ | cos ½πv² · dv ) | ² | \+ ( ∫ | ∞ | sin ½πv² · dv ) | ² |  (20),  
---|---|---|---|---|---|---|---|---  
v |   | v |    
  
or, according to our previous notation,

I²=(½ − Cv)² + (½ − Sv)² = G² + H²  (21).  
  
---  
Fig. 18.  
  
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 illumination continuously decreases, 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 over more than 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

(½ + Cv) | dC | \+ (½ + Sv) | dS | = 0,  
---|---|---|---|---  
dV | dV  
  
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)

BQ = | a + b | AD = V √ { | bλ(a + b) | } (26).  
---|---|---|---|---  
a | 2a  
  
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.

  | V | I²  
---|---|---  
First maximum | 1.2172 | 2.7413  
First minimum | 1.8726 | 1.5570  
Second maximum | 2.3449 | 2.3990  
Second minimum  | 2.7392 | 1.6867  
Third maximum. | 3.0820 | 2.3022  
Third minimum | 3.3913 | 1.7440  
  
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,

x = ∫ | v | cos ½πv²·dv,  y = ∫ | v | sin ½πv²·dv  (29).  
---|---|---|---|---  
0 | 0  
  
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).  
  
---  
Fig. 19.  
  
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 is 1⁄8 period 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 is 3⁄8 of a period, then diminishes to a minimum when the retardation is about 7⁄8 of 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 of constant 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

d²ξ | = b² ( | d²ξ | + | d²ξ | + | d²ξ | ) \+ (a² − b²) | d² | ( | d²ξ | + | d²η | + | d²ζ | ),  
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---  
dt² | dx² | dy² | dz² | dx | dx² | dy² | dz²  
  
where a² and b² denote the two arbitrary constants. Put for shortness

d²ξ | + | d²η | + | d²ζ | ≈ δ (1),  
---|---|---|---|---|---  
dx² | dy² | dz²  
  
and represent by Δ²χ the quantity multiplied by b². According to this notation, the three equations of motion are

d²ξ | b²Δ²ξ \+ (a² − b²) | dδ | } (2).  
---|---|---|---  
dt² | dx  
d²η | b²Δ²η \+ (a² − b²) | dδ  
dt² | dy  
d²ζ | b²Δ²ζ \+ (a² − b²) | dδ  
dt² | 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

d²ζ | = b² | d²ζ |  (3),  
---|---|---|---  
dt² | dx²  
  
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 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,

ζ′ = | dS | (1 + cos θ) sin φ ƒ′(bt − r).  
---|---|---  
4πr  
  
In particular, if

ƒ(bt − x) = c sin | 2π | (bt − x) (5),  
---|---|---  
λ  
  
we shall have

ζ′ = | cdS | (1 + cos θ) sin φcos | 2π | (bt − r) (6).”  
---|---|---|---|---  
2λr | λ  
  
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

(b²Δ² + n²)ξ \+ (a² − b²) | dδ | = 0 | } (7).  
---|---|---|---  
dx  
(b²Δ² + n²)η \+ (a² − b²) | dδ | = 0  
dy  
(b²Δ² + n²)ζ \+ (a² − b²) | dδ | = −Z  
dz  
  
It will now be convenient to introduce the quantities.ω1, ω2, ω3 which express the rotations of the elements of the medium round axes parallel to those of co-ordinates, in accordance with the equations

ω3 = | dξ | − | dη | , ω1 = | dη | − | dζ | , ω2 | dζ | − | dξ |  (8).  
---|---|---|---|---|---|---|---|---|---|---|---|---  
dy | dx′ | dz | dy′ | dx | dz′  
  
In terms of these we obtain from (7), by differentiation and subtraction,

(b²Δ² + n²) ω3 = 0 | } (9).  
---|---  
(b²Δ² + n²) ω1 = dZ/dy  
(b²Δ² + n²) ω2 = −dZ/dx  
  
The first of equations (9) gives

ω3 = 0 (10).

For ω1, we have

ω1 = | 1 | ∫∫∫ | dZ | e−ikr | dx dy dz (11),  
---|---|---|---|---|---  
4πb² | dy | r  
  
where r is the distance between the element dx dy dz and the point where ω1 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

∫ | e−ikr | dZ | dy = [ Z | e−ikr | ] − ∫ Z | d | ( | e−ikr | ) dy,  
---|---|---|---|---|---|---|---|---|---  
r | dy | r | dy | r  
  
in which the integrated terms at the limits vanish, Z being finite only within the region T. Thus

ω1 = | 1 | ∫∫∫ Z | d | ( | e−ikr | ) dx dy dz.  
---|---|---|---|---|---|---  
4πb² | dy | r  
  
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

ω1 = | TZ | · | y | · | d | ( | e−ikr | ) (13).  
---|---|---|---|---|---|---|---|---  
4πb² | r | dr | r  
  
In like manner we find

ω2 = − | TZ | · | x | · | d | ( | e−ikr | ) (14).  
---|---|---|---|---|---|---|---|---  
4πb² | r | dr | r  
  
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

ω = | TZ | · | √(x² + y²) | · | d | ( | e−ikr | ) = | TZ sin φ | d | ( | e−ikr | ),  
---|---|---|---|---|---|---|---|---|---|---|---|---|---  
4πb² | r | dr | r | 4πb² | dr | r  
  
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

ω = | −ik · TZ sin φ | · | e−ikr |  (15),  
---|---|---|---|---  
4πb² | r  
  
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 ωis expressed by ω = dζ′/dr; so that

ζ′ = | TZ sin φ | · | e′ (at−kr) |  (16),  
---|---|---|---|---  
4πb² | r  
  
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

ζ′ = | TZ sin φ | · | cos (nt − kr) |  (18).  
---|---|---|---|---  
4πb² | r  
  
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. 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

ζ′ = | dS sin φ | cos (nt − kr) (20),  
---|---|---  
λr  
  
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.

(R.)

* * *

1 The descending series for J0(z) appears to have been first given by Sir W. Hamilton in a memoir on “Fluctuating Functions,” Roy. Irish Trans., 1840.

2 Airy, loc. cit. “Thus the magnitude of the central spot is diminished, and the brightness of the rings increased, by covering the central parts of the object-glass.”

3 ”Man kann daraus schliessen, was moglicher Weise durch Mikroskope noch zu sehen ist. Ein mikroskopischer Gegenstand z. B, dessen Durchmesser = (λ) ist, und der aus zwei Theilen besteht, kann nicht mehr als aus zwei Theilen bestehend erkannt werden. Dieses zeigt uns eine Grenze des Sehvermogens durch Mikroskope” (Gilbert’s Ann. 74, 337). Lord Rayleigh has recorded that he was himself convinced by Fraunhofer’s reasoning at a date antecedent to the writings of Helmholtz and Abbe.

4 The last sentence is repeated from the writer’s article “Wave Theory” in the 9th edition of this work, but A. A. Michelson’s ingenious échelon grating constitutes a realization in an unexpected manner of what was thought to be impracticable.—[R.]

5 Compare also F. F. Lippich, Pogg. Ann. cxxxix. p. 465, 1870; Rayleigh, Nature (October 2, 1873).

6 The power of a grating to construct light of nearly definite wave-length is well illustrated by Young’s comparison with the production of a musical note by reflection of a sudden sound from a row of palings. The objection raised by Herschel (Light, § 703) to this comparison depends on a misconception.

7 It must not be supposed that errors of this order of magnitude are unobjectionable in all cases. The position of the middle of the bright band representative of a mathematical line can be fixed with a spider-line micrometer within a small fraction of the width of the band, just as the accuracy of astronomical observations far transcends the separating power of the instrument.

8 “In the same way we may conclude that in flat gratings any departure from a straight line has the effect of causing the dust in the slit and the spectrum to have different foci—a fact sometimes observed.” (Rowland, “On Concave Gratings for Optical Purposes,” Phil. Mag., September 1883).

9 On account of inequalities in the atmosphere giving a variable refraction, the light from a star would be irregularly distributed over a screen. The experiment is easily made on a laboratory scale, with a small source of light, the rays from which, in their course towards a rather distant screen, are disturbed by the neighbourhood of a heated body. At a moment when the eye, or object-glass of a telescope, occupies a dark position, the star vanishes. A fraction of a second later the aperture occupies a bright place, and the star reappears. According to this view the chromatic effects depend entirely upon atmospheric dispersion.

10 In experiment a line of light is sometimes substituted for a point in order to increase the illumination. The various parts of the line are here independent sources, and should be treated accordingly. To assume a cylindrical form of primary wave would be justifiable only when there is synchronism among the secondary waves issuing from the various centres.

11 H. Necker (Phil. Mag., November 1832); Fox Talbot (Phil. Mag., June 1833). “When the sun is about to emerge ... every branch and leaf is lighted up with a silvery lustre of indescribable beauty.... The birds, as Mr Necker very truly describes, appear like flying brilliant sparks.” Talbot ascribes the appearance to diffraction; and he recommends the use of a telescope.

* * *