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A Plane Transmission Diffraction Grating (N-Slits Diffraction)

A plane diffraction grating is an arrangement consisting of a large number of close, parallel, straight, transparent and equidistant slits, each of equal width a, with neighboring slits being separated by an opaque region of width b. A grating is made by drawing a series of very fine, equidistant and parallel lines on an optically plane glass plate by means of a fine diamond pen. The light cannot pass through the lines drawn by diamond; while the spacing between the lines is transparent to the light. There can be 15,000 lines per inch or more is such a grating to produce a diffraction of visible light. The spacing (a + b) between adjacent slits is called the diffraction element or grating element. If the lines are drawn on a silvered surface of the mirror (plane or concave) then light is reflected from the positions of mirrors in between any two lines and it forms a plane concave reflection grating.

 

Since the original gratings are quite expensive for practical purposes their photographic reproductions are generally used.

 

The commercial gratings are produced by taking the cast of an actual grating on a transparent film such as cellulose acetate. A thin layer of collodin solution (celluloid dissolved in a volatile solvent) is poured on the surface of ruled grating and allowed to dry. Thin collodin film is stripped off from grating surface. This film, which retains the impressions of the original grating, is preserved by mounting the film between two glass sheets. Now-a-days holographic gratings are also produced. Holograpic gratings have a much large number of lines per cm than a ruled grating Theory of Grating: Suppose a plane diffraction grating, consisting of large number of N parallel slits each of width a and separation b, is illuminated normally by a plane wave front of monochromatic light of wavelength A. as shown in Figure 5.8. The light diffracted through N slits is focused by a convex lens on screen XY placed in the focal plane of the lens L. The diffraction pattern obtained on the screen with very large number of slit consists of extremely sharp principle interference maximum; while the intensity of secondary maxima becomes negligibly small so that these are not visible in the diffraction pattern.

Thus, if we increase the number of slits (N), the intensity of principal maxima increases.

The direction of principal maxima are given by

 

sin β = 0, i.e., β = ± nπ, where n = 0, 1, 2, 3, … ·

 

π/λ  (a + b) sin θ = ± nπ => (a + b) sin θ = ± n λ.                              … (3)

 

If we put n = 0 in equation (3), we get θ = 0 and equation (3) gives the direction of zero

order principal maximum. The first, second, third, … order principal maxima may be obtained

by putting n = 1, 2, 3, . .. in equation (3).

.

Minima: The intensity is minimum, when

sin Nl3 = 0; but sin 13 :# 0

Therefore

Nβ = ± mπ

 

N π/λ (a + b) sin θ = ± mπ

 

N (a + b) sin θ = ± mλ .. ,(4)

 

Here can have all integral values except 0, N, 2N, 3N, … because for these values of m, sin 13 = 0 which gives the positions of principal maxima. Positive and negative signs shows that the minima lie symmetrically on both sides of the central principal maximum.

 

It is clear from equation (4) that form= 0, we get zero order principal maximum, m = 1, 2, 3,4, = (N -1) gives minima governed by equation (4) and then at m = N, we get principal maxima of first order. This indicates that, there are (N -1) equispaced minima between zero and first orders maxima. Thus, there are (N – 1) minimum between two successive principal maxima.

 

Secondary Maxima: The above study reveals that there are (N- 1) minima between two successive principal maxima. Hence there are (N -2) other maxima coming alternatively with the minima between two successive principal maxima. These maxima are called secondary maxima. To find the positions of the secondary maxima, we first differentiate equation (1) with respect to 13 and equating to zero

 

DI/dβ = A 2 sin2 a /a2 . 2 [sin Nβ/sin β ] N cos Nβ sin β- sin N cos β/sin 2β =0

N cos Nβ sin β = sin Nβ cos β = 0

tan Nβ  N tan β

To find the intensity of secondary maximum, we make these of the triangle shown in Figure 5.9

 

We have sin Nβ = N tan β/√ (1+N 2 tan 2 β)

FIGURE 5.9

Therefore                    sin 2β /sin 2β =(n 2 tan 2β /sin 2β (1+n2 tan 2β )

sin 2Nβ /sin 2β =(n 2 tan 2β(1+N tan 2)  /sin 2β=  N 2(1+n2 sin 2β )

Putting  this  value of sin 2 Nβ/ sin 2 β in   equation (1), we get  mtens1ty o secondary maxima  as

 

IS =A2 sin 2α /α2 =N2/[1+(N2-1) SIN2 β

 

This indicates the intensity of secondary maxima is proportional to N 2 /[1+(N2-1) sin 2 β]

 

whereas the intensity of principal maxima is proportional to N2.

 

 

5.8.1 Absent Spectra with a Diffraction Grating

 

It may be possible that while the first order spectra is clearly visible, second order may be not be visible at all and the third order may again be visible. It happen when for again angle of diffraction 0, the path difference between the diffracted ray from the two extreme ends of one slit is equal to an integral multiple of A if the path difference between the secondary waves from the corresponding point in the two halves will be A/2 and they will can all one another effect resulting is zero intensity. Thus the mining of single slit pattern are obtained in the direction given by.

 

a sin θ= mλ                                                     …(1 )

 

where m = 1, 2, 3, …… excluding zero but the condition for nth order principles maximum in

the grating spectrum is

(a + b) sin θ = nλ                                            … (2)

 

If the two conditions given by equation (2) are simultaneously satisfied then the direction in which the grating spectrum should give us a maximum every slit by itself will produce darkness in that direction and hence the most favourable phase for reinforcement will not be able to produce an illumination i.e., the resultant intensity will be zero and hence the absent spectrum. Therefore dividing equation (2) by equation (1)

(a+ b) sine θ/a sin θ =n/ m

(a+ b) /a =n/m

 

This is the condition for the absent spectra in the diffraction pattern

 

If a= b i.e., the width of transparent portion is equal to the width of opaque portion then

from equation (3) n = 2m

 

i.e., 2nd, 4th, 6th etc., orders of the spectra will be absent corresponds to the minima due to

single slit given by m = 1, 2, 3 etc.

b = 2a

n=3m

 

i.e., 3rd, 6th, 9th etc., order of the spectra will be absent corresponding to a minima due to a

single slit given by m = 1, 2, 3 etc.

 

5.8.2 Number of Orders of Spectra with a Grating

 

The number of spectra that are visible in a given grating can be easily calculated with the help

of the equation.

(a + b) sin θ = n λ

n=(a+b) sin θ/λ

 

Here (a+ b) is the grating element and is equal to 1/N = 2.54 N cm, N being number of lines per inch in the grating. Maximum possible value of the angle of diffraction e is 90°,

Therefore sin θ = 1 and the maximum possible order of spectra.

 

N max=(a+b)/λ

 

If (a + b) is between  λ and 2 λ. i.e., grating element (a + b) < 2 λ then,

n max <2 λ / λ < 2

and hence only the first order of spectrum is seen if (a+ b) is between 2A and 311. first two order

will obtained and so on.