Question

Derive the condition for absent spectra in grating.

Answer 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) sin 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.