more and more, and special devices have to be adopted to separate the spectra, when observations are made in the higher orders. When spectra are to be recorded by photography, there is a similar overlapping but its range is different.

61. Dispersion of gratings. The maxima of two wave-lengths A, and A being in such positions that

e sin 01 = nλ1,

e sin 02 = nλ2,

the ratio (01-02)/(^1 — λ) may be taken to measure the angular dispersion of the grating. The ratio increases with increasing values of nλ and hence the dispersion increases with the order of the spectrum.

If the incident beam is oblique

e (sin - sin ) = nλ,

which, by differentiation, gives with a constant value of

e cos 0 d0 = ndλ,

so that in this case the angular dispersion is

do

αλ

= n/e cos 0.

When the diffracted beam leaves the grating nearly normally, cos varies much less rapidly than sin 0. In that case the dispersion is proportional to the order of the spectrum and independent of the wave-length, i.e. equal angular separation means equal differences of wave-length. We then say that the spectrum formed is "normal.”

62. Resolving power of gratings. The use of a grating as an analyser of light depends on its power to form a pure spectrum. To obtain a measure of the purity of a spectrum, we may imagine it to be projected on a screen, which has a narrow opening parallel to the original slit intended to transmit only that wave-length which has a maximum coinciding in position with the opening. It is then found

that the waves passing through even an indefinitely narrow aperture are not absolutely homogeneous. In Fig. 79 the curve a represents the

distribution of light on the screen for a given wave-length. OK indicates the position of a narrow opening placed so as to transmit the maximum amount of light having a given wave-length λ, the amount so transmitted being proportional to the intensity OK and to the width of the opening. If λ, be a wave-length near A, it will have its maximum a little to one side. Its intensity curve is represented by the second curve and an amount of its light proportional to OH passes through the opening. The curves of intensity having no definite limit, there is some light of every wave-length passing through the slit, but the intensity quickly diminishes and we need only consider those wave-lengths which are not very different from λ. If we wish to compare different spectrum-forming instruments with each other, it will be sufficient to limit the investigation to that light which lies between the two minima on either side of the maximum.

(1

1

(1-). The

It follows from Art. 59 that a wave-length λ, has its first minimum when there is maximum for A if nN (A, -A)/λ=+1. Hence we may say that the range of wave-lengths passing through the opening extend from a wave-length λ ( 1 + to a wave-length A Nn quantity Nn has been called the resolving power of the grating. Denoting it by R, we may say that very little light passes through the slit which differs in wave-length from λ by more than X/R. Resolving power will be further considered in Chapter VII.

63. Wire gratings. In certain cases, the intensity of the spectra of different orders may be calculated. If the grating is

formed by a number of equidistant thin wires of

equal thickness (Fig. 80), the periodicity of the grating is such that one portion does not obstruct gooooooo the passage of the light whilst the other is opaque. Take the incident light to be normal to the grating, and let the widths of each transparent and opaque portion be a and b respectively; the amplitude of the light diffracted at an angle to the normal is then (Art. 53) (A sin a)/a where a = πɑ sin 0/λ.

Fig. 80.

The maximum of the nth order is determined by

(a + b) sin 0 = nλ; so that a = Tan/(a + b).

The amplitudes at the maxima are therefore

For the central image, in which there is no dispersion, a=0 and the amplitude is A. The law of falling off in the intensities of the different images to the sides of the maxima is the same near all maxima,

so that for the ratio of the intensities of the images, we may substitute the ratio of the squares of the amplitudes at the maxima. For the calculation of the amplitude at the central maximum, it is sufficient to point out that the interposition of the grating reduces the amplitude in the ratio of its transparent portion to its total surface, i.e. in the ratio a/(a+b), and hence the intensity of the central image is {a/(a+b)}, if the intensity of the incident light is unity. This deter mines the value of A.

We now obtain for the intensities of the other images,

If ab, the sine factor is zero for all even values of n, so that the spectra of even order disappear, and the intensities of the spectra of odd orders are, in terms of the incident light, 32π2

1

1

1 n22·

The fraction 1/2 represents the maximum intensity which the spectrum of the first order can possibly have in this class of gratings, and shows what a considerable amount of light is lost when a grating is used as an analyser of light. If we desire to make the second order spectrum as intense as possible, we must make a/b equal to 1/3 or 3, but even in this case, we should only secure little more than two per cent. of the light.

It is instructive to note that the grating reduces the intensity of the total light transmitted in the ratio a/(a+b), which is also the ratio in which the amplitude of the central image is reduced. The difference between a/(a + b) and {a/(a + b)}2 gives the amount of light which goes to form the lateral spectra.

A2

2 mms!

64. Gratings with predominant spectra. Rulings of gratings may be devised which concentrate most of the light into one spectrum. Fig. 81 represents the section of such a grating ruled on a reflecting surface. If the oblique portions of the grating are at such an angle that light incident in the direction of the arrow would, by the laws of geometrical optics, be reflected in the direction A,C, then all the rays from each of the oblique portions would be in equal phase at a plane HK, drawn at right angles to A,C1. If, further, the differ

C2

Fig. 81.

ence in optical length at HK between A,C, and A,C, be a wave-length, there is coincidence of phase between the rays from successive rulings.

Hence the amplitude at a point which is at the same optical distance from HK (e.g. the focus of a lens adjusted for infinity) is the same as if the whole wave-front HK were reflected in the ordinary way. The resultant amplitude is therefore less than the resultant amplitude of the incident wave, only on account of the contraction in the width of the beam due to obliquity. If be the angle between A,C, and the incident beam, it would follow that the intensity of the first order spectrum is cos2 in terms of the resultants of the incident light. This loss of light is accounted for by the light reflected from the other set of inclined faces. If the ruling is such that the first order spectrum is at an angle of 30° from the normal, three-quarters of the whole light would go to form that spectrum. For normal incidence we have as before, sin λ/e, and the reflecting facets must be inclined at an angle 0/2. The condition for maximum light can only be fulfilled for one wave-length at a time, but a slight tilting of the grating supplies the means of adjustment for any desired wave-length. Transmission

H

=

A 2

A1

C2

C

K

gratings may be ruled on the same principle, the condition being that the angles of the inclined facets are such that the incident rays in each little prisın formed are refracted along paths at right angles to HK, and that there is a retardation of a wave-length between two corresponding rays AC and A,C1. Mr T. Thorp has been able to demonstrate the practical possibility of manufacturing gratings of the kind considered. Triangular grooves were cut in a metallic surface, and a layer of liquefied celluloid was allowed to float and solidify over this grooved surface. On removal, the celluloid film showed in transmitted light spectra which were all very weak except that of the first order on one side. HK (Fig. 82) gives the direction of the wave-front of the diffracted wave which carries the maximum intensity for the wavelength X.

W

Fig. 82.

A

K

A2

A3

C2

C3

C1

H

65. Echelon gratings. If a reflecting grating were constructed on a principle similar to that of the last article, but subject to the additional condition that rays which go to form a particular spectrum return along the path of the incident light, the spectrum formed by reflexion would contain the whole intensity of the incident light. This consideration leads to Michelson's echelon grating. In Fig. 83 let a number of plates, T1, T2, T3, etc. be placed so that the different portions of a wave-front WF are reflected back parallel

Fig. 83.

| ន

TT T

to themselves from each of the plates, then if the depths of the steps AC1, A2C2, AC, are all equal to nλ, a multiple of a wave-length, the reflected beam has intensity equal to the incident beam, neglecting the loss of light at reflexion. For that particular wave-length, there cannot therefore be light in any other direction. The reasoning holds for all those wave-lengths for which the step is an exact multiple of a wave-length, and we may, if n is great, have a great number of maxima of light all overlapping in the same direction.

At a surface HK inclined to WF at a small angle 0, the retardation of successive corresponding rays is e0, where e is the width of each step. Hence there is coincidence of phase for a wave-length X' at corresponding points of HK if

For the dispersion 6/(X-X') we thus obtain n/e. But only a very small part of each spectrum is visible because the intensity of light falls off very rapidly to both sides of the normal direction.

At a wave-front parallel to WF, the relative retardation of two waves A and A', for the light reflected by the last element, is Nn (λ-X') if there is coincidence of phase for light reflected at the first element. Hence equation (2) holds, and the resolving power is Nn, as with ordinary gratings.

A reflecting grating of the kind described would be difficult to construct, but excellent results have been obtained by Michelson with a transmission grating based on the same principle.

A number of equal plates of thickness t are arranged as in Fig. 84.

and if e be the distance between corresponding points A1, A2, the angle through which the front is turned is LK/e or: