1911 Encyclopædia Britannica/Diffraction of Light/3

    Fig. 2.

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)

$-{\frac {d\mathrm {S} }{\lambda \rho }}\sin k(at-\rho ),$

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

$\rho =f\left\{1-{\frac {x\xi +y\eta }{f^{2}}}\right\}.$

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

$-{\frac {1}{\lambda f}}\iint \sin k\left\{at-f+{\frac {x\xi +y\eta }{f}}\right\}dxdy$

(1);

and for the intensity, represented by the square of the amplitude,

	${\big .}\mathrm {I} ^{2}={\frac {1}{\lambda ^{2}f^{2}}}\left[\iint \sin k{\frac {x\xi +y\eta }{f}}dxdy\right]^{2}$
	$+{\frac {1}{\lambda ^{2}f^{2}}}\left[\iint \cos k{\frac {x\xi +y\eta }{f}}dxdy\right]^{2}$	(2).

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 −1/2a and +1/2a, and for y to be −1/2b, +1/2b. We readily find (with substitution for k of 2π/λ)

$\mathrm {I} ^{2}={\frac {a^{2}b^{2}}{f^{2}\lambda ^{2}}}\cdot {\frac {\sin ^{2}{\frac {\pi a\xi }{f\lambda }}}{\frac {\pi ^{2}a^{2}\xi ^{2}}{f^{2}\lambda ^{2}}}}\cdot {\frac {\sin ^{2}{\frac {\pi b\eta }{f\lambda }}}{\frac {\pi ^{2}b^{2}\eta ^{2}}{f^{2}\lambda ^{2}}}}$

(3),

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 + 1/2)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

$y={\frac {1}{\mathrm {U} }}\left(1+{\frac {y}{\mathrm {U} }}+{\frac {y^{2}}{\mathrm {U} ^{2}}}+\ldots \right)-{\frac {y^{3}}{3}}-{\frac {2y^{5}}{15}}-{\frac {17y^{7}}{315}}.$

This equation is to be solved by successive approximation. It will readily be found that

$u=\mathrm {U} -y=\mathrm {U} -\mathrm {U} ^{-1}-{\frac {2}{3}}\mathrm {U} ^{-3}-{\frac {13}{15}}\mathrm {U} ^{-5}-{\frac {146}{105}}\mathrm {U} ^{-7}-\ldots$

(6).

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 + 1/2)π nearly, the successive values are not very different from

${\frac {4}{9\pi ^{2}}},{\frac {4}{25\pi ^{2}}},{\frac {4}{49\pi ^{2}}},$ &c.

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

$\int _{-\infty }^{+\infty }{\frac {\sin ^{2}u}{u^{2}}}du=\int _{-\infty }^{+\infty }{\frac {\sin u}{u}}du=\pi .$

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 1/4λ.

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

${\frac {1}{3\pi }}\int _{-{\frac {3}{2}}\pi }^{+{\frac {3}{2}}\pi }\cos \phi d\phi :1$

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

$\int _{-\infty }^{+\infty }\mathrm {I} ^{2}d\eta ={\frac {a^{2}b}{\lambda f}}{\frac {\sin ^{2}{\frac {\pi a\xi }{\lambda f}}}{\frac {\pi ^{2}a^{2}\xi ^{2}}{\lambda ^{2}f^{2}}}}$

(8),

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.

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—

$\iint _{-\infty }^{+\infty }\mathrm {I} ^{2}d\xi d\eta =\iint dxdy=\mathrm {A}$

(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

$\mathrm {A} ^{\prime }\mathrm {I} _{0}^{2}=\iint _{-\infty }^{+\infty }\mathrm {I} ^{2}d\xi d\eta ,$

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.