1911 Encyclopædia Britannica/Diffraction of Light/10

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. ^[1]

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

$\cos 2\pi \left({\frac {t}{\tau }}-{\frac {\delta }{\lambda }}\right)\cdot ds,$

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

${\frac {2\pi \delta }{\lambda }}{=}{\frac {\pi (a+b)s^{2}}{ab\lambda }}{=}{\frac {\pi }{2}}v^{2}$

(2),

the effect at B is

$\left\{{\frac {ab\lambda }{2(a+b)}}\right\}^{\frac {1}{2}}\left\{\cos {\frac {2\pi t}{\tau }}\int \cos {\tfrac {1}{2}}\pi v^{2}.dv+\sin {\frac {2\pi t}{\tau }}\int \sin {\tfrac {1}{2}}\pi v^{2}\cdot dv\right\}$

(3)

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 1/2πv2·dv}2 + { ∫ sin 1/2πv2·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

$\mathrm {C} =\int _{0}^{v}\cos {\frac {1}{2}}\pi v^{2}\cdot dv,\qquad \qquad \mathrm {S} =\int _{0}^{v}\sin {\frac {1}{2}}\pi v^{2}\cdot dv$

(5).

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 1/2, 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

$\mathrm {C} +i\mathrm {S} =\int _{0}^{v}e^{i.{\frac {1}{2}}\pi v^{2}}dv=e^{i.{\frac {1}{2}}\pi v^{2}}.v-{\tfrac {1}{3}}i\pi \int _{0}^{v}e^{i.{\frac {1}{2}}\pi v^{2}}dv^{3};$

and, by continuing this process,

$\mathrm {C} +i\mathrm {S} =e^{i.{\frac {1}{2}}\pi v^{2}}\left\{v-{\frac {i\pi }{3}}v^{3}+{\frac {i\pi }{3}}{\frac {i\pi }{5}}v^{5}-{\frac {i\pi }{3}}{\frac {i\pi }{5}}{\frac {i\pi }{7}}v^{7}+\ldots \right\}.$

By separation of real and imaginary parts,

	$\mathrm {C} =\mathrm {M} \cos {\tfrac {1}{2}}\pi v^{2}+\mathrm {N} \sin {\tfrac {1}{2}}\pi v^{2}$		$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$	(6),
	$\mathrm {S} =\mathrm {M} \sin {\tfrac {1}{2}}\pi v^{2}-\mathrm {N} \cos {\tfrac {1}{2}}\pi v^{2}$

where

$\mathrm {M} ={\frac {v}{1}}-{\frac {\pi ^{2}v^{5}}{3.5}}+{\frac {\pi ^{4}v^{9}}{3.5.7.9}}-\ldots$

(7),

$\mathrm {N} ={\frac {\pi v^{3}}{1.3}}-{\frac {\pi ^{3}v^{7}}{1.3.5.7}}+{\frac {\pi ^{5}v^{11}}{1.3.5.7.9.11}}\ldots$

(8).

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

1/2πv2＝u

(9),

we may write

$\mathrm {C} +i\mathrm {S} ={\frac {1}{{\sqrt {}}(2\pi )}}\int _{0}^{u}{\frac {e^{iu}du}{{\sqrt {}}u}}$

(10).

Again, by a known formula,

${\frac {1}{{\sqrt {}}u}}={\frac {1}{{\sqrt {}}\pi }}\int _{0}^{\infty }{\frac {e^{-ux}dx}{{\sqrt {}}x}}$

(11).

Substituting this in (10), and inverting the order of integration, we get

$\mathrm {C} +i\mathrm {S} ={\frac {1}{\pi {\sqrt {}}2}}\int _{0}^{\infty }{\frac {dx}{{\sqrt {}}x}}\int _{0}^{u}e^{u(i-x)}dx={\frac {1}{\pi {\sqrt {}}2}}\int _{0}^{\infty }{\frac {dx}{{\sqrt {}}x}}{\frac {e^{u(i-x)}-1}{i-x}}$

(12).

Thus, if we take

$\mathrm {G} ={\frac {1}{\pi {\sqrt {}}2}}\int _{0}^{\infty }{\frac {e^{-ux}{\sqrt {}}x{.}dx}{1+x^{2}}},\mathrm {H} ={\frac {1}{\pi {\sqrt {}}2}}\int _{0}^{\infty }{\frac {e^{-ux}dx}{{\sqrt {}}x{.}(1+x^{2})}}$

(13),

C＝1/2 − G cos u + H sin u,   S＝1/2 − G sin u − H cos u

(14).

The constant parts in (14), viz. 1/2, 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 1/2.

Comparing the expressions for C, S in terms of M, N, and in terms of G, H, we find that

G＝1/2 (cos u + sin u) − M,   H＝1/2 (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 = 1/2.

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

$\int _{0}^{\infty }e^{-y}y^{q-{\frac {1}{2}}}dy=\Gamma (q+{\tfrac {1}{2}})=(q-{\tfrac {1}{2}})(q-{\tfrac {3}{2}})\ldots {\tfrac {1}{2}}{\sqrt {}}\pi ;$

and we get in terms of v

$\mathrm {G} ={\frac {1}{\pi ^{2}v^{3}}}-{\frac {1{.}3{.}5}{\pi ^{4}v^{7}}}+{\frac {1{.}3{.}5{.}7{.}9}{\pi ^{6}v^{11}}}-\ldots$

(16),

$\mathrm {H} ={\frac {1}{\pi v}}-{\frac {1{.}3}{\pi ^{3}v^{5}}}+{\frac {1{.}3{.}5{.}7}{\pi ^{5}v^{9}}}-\ldots$

(17).

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

${\frac {d\mathrm {H} }{dv}}=-\pi v\mathrm {G} ,\qquad \qquad {\frac {d\mathrm {G} }{dv}}=\pi v\mathrm {H} -1$

(18).

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.

$\mathrm {V} ={\sqrt {\big .}}\left\{{\frac {2(a+b)}{ab\lambda }}\right\}{.}\mathrm {CA} ,$

(19).

we may write

$\mathrm {I} ^{2}=\left(\int _{v}^{\infty }\cos {\tfrac {1}{2}}\pi v^{2}{.}dv\right)^{2}+\left(\int _{v}^{\infty }\sin {\tfrac {1}{2}}\pi v^{2}{.}dv\right)^{2}$

(20),

or, according to our previous notation,

I2＝(1/2 − Cv)2 + (1/2 − 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＝(1/2 + Cv)2 + (1/2 + Sv)2

(22);

and the maxima and minima occur when

$({\tfrac {1}{2}}+\mathrm {C} _{v}){\frac {d\mathrm {C} }{d\mathrm {V} }}+({\tfrac {1}{2}}+\mathrm {S} _{v}){\frac {d\mathrm {S} }{d\mathrm {V} }}=0,$

whence

sin 1/2πV2 + cos 1/2πV2＝G

(23).

When V = 0, viz. at the edge of the shadow, I² = 1/2; 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

1/2πV2＝3/4π + nπ

(24),

n being an integer. In terms of δ, we have from (2)

δ＝(3/8 + 1/2n)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)

$\mathrm {BQ} ={\frac {a+b}{a}}\mathrm {AD} =\mathrm {V} {\sqrt {\big .}}\left\{{\frac {b\lambda (a+b)}{2a}}\right\}$

(26).

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,

$x=\int _{0}^{v}\cos {\tfrac {1}{2}}\pi v^{2}{.}dv,\qquad y=\int _{0}^{v}\sin {\tfrac {1}{2}}\pi v^{2}{.}dv$

(29).

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 1/2πv²·dv,   dy＝sin 1/2πv²·dv;

so that

σ＝∫ √(dx2 + dy2)＝v,

(30),

φ＝tan−1(dy/dx)＝1/2πv2

(31).

Accordingly,

φ = 1/2πσ2

(32);

and for the curvature,

dφ / dσ = πσ

(33).

Cornu remarks that this equation suffices to determine the general character of the curve. For the osculating circle at any point includes the whole of the curve which lies beyond; and the successive convolutions envelop one another without intersection.

Fig. 19.

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 (1/2, 1/2), (−1/2, −1/2), 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.”^[2]

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1. 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.
2. 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.