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النشر الإلكتروني

CHAPTER VI.

DIFFRACTION GRATINGS.

59. General theory of a grating. A grating is a surface having a periodical structure which impresses a periodical alteration of phase or intensity on a transmitted or reflected wave of light. The most common method of manufacturing a grating is to rule equidistant lines with a diamond point on a surface of glass or metal.

The diamond introduces a periodical structure, each portion of which is probably very irregular, but which is repeated at perfectly regular intervals, Fig. 75. If the grating is ruled on a plane surface, that surface is called the plane of the grating. Any plane passing through corresponding points of the grooves such as A1, A2, A3, is parallel to the plane of the grating. We distinguish between "reflexion gratings" and "transmission gratings" according as they are ruled on an opaque surface, the reflected or scattered light being used, or a transparent plate, through which the light is transmitted.

H

C1

C3

Fig. 75.

Let a plane wave-front be incident parallel to the grating. Waves spread out from the different portions of the grooves which may be considered as centres of secondary disturbances. If the light be received on a distant screen, the resultant of all vibrations at each point may be determined. Consider that point of the screen which lies in a direction A2 C22 from the grating, and draw a plane HK at right angles to that direction. As the optical distance from any point on HK to the corresponding point of the distant screen is the same, we may take the phases of the vibrations which are to be combined, to be the same as the phases at HK. We combine in the first place, those vibrations which are due to the secondary waves coming from one of the grooves. Selecting any point on the groove Ag, we may always express the phase of the resultant vibration due to the whole groove as that corresponding to an optical distance A,C,- €,

where is some length which depends on the shape of the groove and on the direction of A2 C2. The resultant amplitude similarly may be written ka, where a is the amplitude of the incident light and k a factor depending also on the shape of the grooves and the direction. The different distances from the points of the groove to the plane HK do not affect the amplitude because that plane is only an auxiliary surface, the amplitude really being required at the screen which is so far away that the small differences in distance from different points of the grating are negligible. Taking the resultant of the other grooves, we should find similarly that the resultant phases at HK may be derived from the optical distance A,C,-, A3C3-e, etc., A1, A2, A3, being corresponding points on the grating. The theory of the grating depends on the fact that the values of and k are the same for each groove. This involves the similarity of all the grooves, and if that similarity holds, the difference in phases between the resultant vibrations of two successive grooves is (A, C2- €) - (A, C1- €) and is therefore independent of ε. We may now draw a plane through any set of corresponding points of the groove and call it the plane

N

H

CC2 C3

-K

of the grating (Fig. 76), and in calculating the resultant phases at HK we need only consider the difference in the optical distance AC1, A2C2, A3C3. If that difference is a multiple of a wave-length, the phases at HK are identical and we must then obviously have a maximum of light, wherever those identical phases are brought together. This may either be the principal focus of a lens placed with its axis at The direction in which these maxima appear is be the angle between the normal to the grating and the direction A,C, and A,N be drawn at right angles to A,C1:

Fig. 76.

distant screen or the right angles to HK. easily obtained. If

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where e is the distance 4,4, between the grooves ruled on the grating, A the wave-length and n an integer number. The number of maxima is finite because sin cannot be greater than one, and the highest value which we can take is therefore that integer which is nearest but smaller than e/. If e were smaller than λ there could be no maximum except that for which n= 0. The amplitudes in the direction of the maxima are Nka, where N is the total number of grooves and k the constant already introduced, which may and does very seriously affect the amplitude. It is theoretically possible that k is zero for one of the directions defined by (1) and in that case that maximum would of

course be absent. It is also possible that k is unity, and in that case the whole of the light would be concentrated at or near that maximum.

The complete investigation of the grating includes the determination of the amplitudes of light in directions not necessarily confined to those at which the maxima appear. We proceed, therefore, to find the distribution of light in the neighbourhood of the maxima. The wavelength of a homogeneous beam incident on the grating being λ and having, as has been shown, a maximum in such directions that (Fig. 76) A2N=nλ, let the whole system of rays A1C1, A, C, etc. and with it the normal plane HK be turned round slightly so that AN now becomes n', where X' is a length differing little from A. The difference in phase between the vibrations at C2 and C, for the wave-length A becomes 2nλ'/λ or 2πn (λ' -A)/A, as we may add or subtract any multiple of four right angles to a phase difference. This is also the phase difference between the vibrations at C, and C2, etc. To obtain the complete resultant, we can therefore apply the proposition of Art. 5, which gives for the amplitude of N vibrations of equal amplitude ka, and constant phase difference 2a/N, a resultant amplitude

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In the present case, a = πnÑ (X' — λ)/λ.

The distribution of intensity corresponding to this amplitude has been discussed in Art. 53. Fig. 70 shows for different values of a, the amplitude (sin a)/a (dotted curve) and the intensity (sin2 a)/a2 (full curve). The intensity has secondary maxima which are not, however, important compared with the principal one, at which a = = 0.

The amount of light is everywhere small when a is greater than 2π; hence if Nn is large, the light is concentrated nearly in those directions for which (λ' -λ)/A is very small. It is owing to the rapid falling off of the light at both sides of the principal maxima, that the grating can be made use of to separate the different components of non-homogeneous light, without any great overlapping of different wave-lengths.

The condition for the first minimum a = π, may be obtained in the most suitable form by considering that a series of waves with constant differences of path neutralize each other's effect, when the difference in optical length between the first and last is a wave-length. There being N lines, the total difference in optical length is Nn (X-A), and for the first minimum this must be equal to X'. The condition that the minimum of light for a wave-length A' is coincident with the maximum of a wave-length λ is therefore

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It will be shown in Chapter VII. that a spectroscope resolves a double line, the components of which have wave-lengths A and X', when the maximum of the diffraction image of one line coincides with the first minimum of the other. The greater the value of Nn, the smaller is the difference - λ' which may be resolved. We may therefore take nN to be a measure of the resolving power.

If we extend the above investigation to directions which are not near those of the maxima, the total light is found to be negligible; for a vibration diagram representing phases at C1, C2, by means of vectors OP1, OP2, and including all N vibrations, would have the points P1, P2,...P distributed nearly symmetrically, so that the distance of their centre of gravity from the centre of the circle must be small compared with the radius of the circle.

The incident wave-front has so far been taken as parallel to the plane of the grating. For oblique incidence, consider a grating formed by ruling lines on a glass surface, and let a plane wave be transmitted

Av

obliquely through it. Let A1, A2 (Fig. 77) be corresponding points on successive grooves, and LM the incident wave-front, inclined at an angle to the plane of the grating. Draw two rays LA1, MA2, and consider the light diffracted in the direction A,C1, inclined at an angle to the normal of the grating. Draw AN and AT at right angles to A,C, and A,L respectively. The difference in phase between C, and C2 is then

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Fig. 77.

e (sin sin ),

and there is a maximum when

e (sin - sin 0) = ± nλ...

..(3).

and are here taken as having the same sign when they are both on opposite sides of the normal.

Writing y for 4-0, the angle between the incident and diffracted

beams, the condition for a minimum or maximum of deviation is which leads to do=de. By differentiating (3) we obtain

=

cos pdp- cos 0d0 = 0.

dy

=0,

do

If do = do it follows that cos = cos 0, i.e. $ = ± 0. and 0 cannot be equal unless n 0, which case need not be considered. For the condition of maximum-minimum we have therefore =0, which shows that the incident and diffracted light form equal angles with the plane of the grating. Further consideration shows that it is a minimum and not a maximum deviation that is involved.

If = the deviation is 20. Equation (3) becomes in that case

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=

60. Overlapping of spectra. The maxima of light for normal incidence have been shown to take place when e sin nλ. For each value of n, the maxima of the different wave-lengths take place along different directions, and hence the grating "analyses" the light which falls on it and produces homogeneous light. It acts in this respect like a prism, but splits up the light into a number of spectra, each value of n giving a separate spectrum. For n= 0, there is a maximum, but there is no spectrum because the position of the maximum is independent of the wave-length. The direction of this maximum is the direction of the incident light in a transmission grating, or in a grating which acts by reflexion; it is the direction in which the incident beam would be reflected from a polished surface coincident with the grating. For n = 1, we have the so-called spectrum of the first order, which spreads over the quadrant between 0=0 for λ = 0 and 0 = 1⁄2π for λ=e. Similarly the spectrum of the second order, for which n = 2, spreads over the same quadrant, the limits of wave-length being λ= 0 for =0, and λ=e/2 for 0. For each value of we have therefore an infinite number of overlapping maxima corresponding to all wave-lengths which are submultiples of e sin 0. If we confine ourselves to eye-observations, we need only consider the wave-lengths lying between 4 x 10-5 and 8 × 10−5. The limits and 0 of the spectra of different orders are then

=

for n=1; 4 × 10−5 = e sin ' and 8 × 10-5 = e sin 0,
for n=2;
8 × 10-5 = e sin ' and 16 × 10−5:
3; 12 × 10-5 = e sin 0'

for n=
for n=4; 16 × 10-5 e sin 0′

=

-5

= e sin 0,

and 24 × 10- = e sin 0,

and 32 × 10-5 = e sin 0.

In Fig. 78 the extension of the different spectra is marked by

40 36 32 28 24 20 16 12 8 40

Fig. 78.

straight lines lying above each other to avoid actual overlapping. If the wave-lengths marked are those corresponding to the first order spectrum, we may obtain the wave-length of the spectrum of order n, by dividing these numbers by n.

The visible spectrum of the first order stands out clear of the rest,

but the second and third overlap to a great extent, the range between λ = 6 × 10-5 and λ=8×10−5 of the second order being coincident with the range of λ = 4 × 10−5 to λ = 5·3 × 10-5 of the third order. The spectra of higher orders spreading over greater ranges of 0 overlap

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