# 1911 Encyclopædia Britannica/Diffraction of Light/9

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​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. 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
$-{\frac {1}{\lambda f}}\int _{-h}^{0}\int _{-l}^{+l}\sin k\left\{at-f+{\frac {x\xi +y\eta }{f}}\right\}dxdy$
| $=-{\frac {2lh}{\lambda f}}\cdot {\frac {f}{k\eta l}}\sin {\frac {k\eta l}{f}}\cdot {\frac {2f}{k\xi h}}\sin {\frac {k\xi h}{2f}}\cdot \sin k\left\{at-f-{\frac {\xi h}{2f}}\right\}$ | (1), |
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
| $-{\frac {2lh}{\lambda f}}\cdot {\frac {f}{k\eta l}}\sin {\frac {k\eta l}{f}}\cdot {\frac {2f}{k\xi h}}\sin {\frac {k\xi h}{2f}}\cdot \sin k\left\{at-f-\mathrm {R} +{\frac {\xi h}{2f}}\right\}$ | (2). |
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＝u$^2$ + v$^2$ + 2uv cos α,
which becomes on putting for u, v, and α their values, and putting
| $\left\{{\frac {f}{k\eta l}}\sin {\frac {k\eta l}{f}}\right\}^{2}=\mathrm {Q}$ | (3), |
| $\mathrm {I} =\mathrm {Q} \cdot {\frac {4l^{2}}{\pi ^{2}\xi ^{2}}}\sin ^{2}{\frac {\pi \xi h}{\lambda f}}\left\{2+2\cos \left({\frac {2\pi \mathrm {R} }{\lambda }}-{\frac {2\pi \xi h}{\lambda f}}\right)\right\}$ | (4). |
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
| $\mathrm {I} ={\frac {2}{\xi ^{2}}}\sin ^{2}{\frac {\pi \xi h}{\lambda f}}\left\{1+\cos \left({\frac {2\pi \mathrm {R} }{\lambda }}-{\frac {2\pi \xi h}{\lambda f}}\right)\right\}$ | (5). |
If R = 1/2λ, 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/ξ$^2$ sin$^2$ 2πξh/λf, 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′ + ϖξ | (7), |
where ρ′ denotes the value of ρ at O, and ϖ is a constant, which is positive when the retarding plate is held at the side on which the blue of the spectrum is seen. The possibility of dark bands depends upon ϖ being positive. Only in this case can
cos {ρ′ + (ϖ − 2πh/λf) ξ}
retain the constant value -1 throughout the integration, and then only when
| ϖ＝2πh / λf | (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′ +(ϖ − 2πh/λf) ξ}＝cos {ρ′ − g$_1$ξ} ＝cos ρ′ cos g$_1$ξ + sin ρ′ sin g$_1$ξ,
we see that the second part vanishes when integrated, and that the remaining integral is of the form
$w=\int _{-\infty }^{+\infty }\sin ^{2}h_{1}\xi \cos g_{1}\xi {\frac {d\xi }{\xi ^{2}}},$
where
| h$_1$＝πh/λf,   g$_1$＝ω − 2πh/λf | (10). |
By differentiation with respect to g$_1$ it may be proved that
| w = 0 | from g$_1$ = −∞ | to g$_1$ = −2h$_1$, |
| w = 1/2π(2h$_1$ + g$_1$) | from g$_1$ = −2h$_1$ | to g$_1$ = 0, |
| w = 1/2π(2h$_1$ − g$_1$) | from g$_1$ = 0 | to g$_1$ = 2h$_1$, |
| w = 0 | from g$_1$ = 2h$_1$ | to g$_1$ = ∞. |
The integrated intensity, I′, or
2πh$_1$ + 2 cos ρw,
is thus
| I′＝2πh$_1$ | (11), |
when g$_1$ numerically exceeds 2h$_1$; and, when g$_1$ lies between ±2h$_1$,
| I＝π{2h$_1$ + (2h$_1$ − √ g$_1$$^2$) cos ρ′ | (12). |
It appears therefore that there are no bands at all unless ω lies between 0 and +4h$_1$, and that within these limits the best bands are formed at the middle of the range when ω = 2h$_1$. The formation of bands thus requires that the retarding plate be held upon the side already specified, so that ω be positive; and that the thickness of the plate (to which ω is proportional) do not exceed a certain limit, which we may call 2T$_0$. At the best thickness T$_0$ 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 /ϖ | (13). |
With the best thickness
| ϖ＝2πh/λf | (14), |
so that in this case
| e＝λf / 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.