Page:EB1911 - Volume 08.djvu/264

If B_m denote the brightness of the m^th lateral image, and B₀ that of the central image, we have

$\mathrm {B} _{m}:\mathrm {B} _{0}=\left[\int _{-{\frac {am\pi }{a+d}}}^{+{\frac {am\pi }{a+d}}}\cos xdx\div {\frac {2am\pi }{a+d}}\right]^{2}=\left({\frac {a+d}{am\pi }}\right)^{2}\sin ^{2}{\frac {am\pi }{a+d}}$

(1).

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

$\mathrm {B} _{m}:\mathrm {B} ={\frac {1}{m^{2}\pi ^{2}}}\sin ^{2}{\frac {am\pi }{a+d}}$

(2).

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 m^th 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

$\mathrm {B} _{m}:\mathrm {B} ={\frac {\sin ^{2}{\tfrac {1}{2}}m\pi }{m^{2}\pi ^{2}}}$

(3).

showing that, when m is even, B_m vanishes, and that, when m is odd,

B_m : 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

$1={\frac {1}{2}}+{\frac {1}{4}}+{\frac {2}{\pi ^{2}}}\left(1+{\frac {1}{9}}+{\frac {1}{25}}+\ldots \right),$

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

${\frac {a}{a+d}}=\left({\frac {a}{a+d}}\right)^{2}+{\frac {2}{\pi ^{2}}}\sum _{m=1}^{m=\infty }{\frac {1}{m^{2}}}\sin ^{2}\left({\frac {m\pi a}{a+d}}\right),$

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 B_m : 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 1/2λ, 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.^[1]

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 1/2(θ + φ) cos 1/2 (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 1/2λ, as appears from (5), when we suppose θ = 1/2π, φ = 1/2π.

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 m^th 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,^[2] 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.^[3]

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1. 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.]
2. Compare also F. F. Lippich, Pogg. Ann. cxxxix. p. 465, 1870; Rayleigh, Nature (October 2, 1873).
3. 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.