Spectra Diffraction Grating

In the equation (a + b) sin θ = λ, if (a + b) < λ, then sin θ > 1. But this is not possible. Hence the first order spectrum is absent. Similarly, the second, the third, etc. order spectra will be absent if (a + b) < 2λ, a + b) < 3λ etc. In general, if (a + b) < nλ, then the n th order spectrum will be absent.

The condition for absent spectra can be obtained from the following consideration. For the nth order principal maximum

(a + b) sin θn = nλ                    (i)

Further, if the value of a and θn are such that

a sin θn = λ                        (ii)

then, the effect of the wavefront from any particular slit will be zero. Considering each slit to be made up of two halves, the path difference between the secondary waves from the corresponding points will be λ/2 and they cancel one another’s effect. If the two conditions given by equations (i) and (ii) are simultaneously satisfied, then dividing (i) by (ii)

In equation (iii), the values of n = 1, 2, 3 etc. refer to the order of the principal maxima that are absent in the diffraction pattern.
    
(i) If  = 1; b = 0

In this case, the first order spectrum will be absent and the resultant diffraction pattern is similar to that due to a single slit,
    
(ii) If  = 2; a = b

i.e. the width of the slit is equal to the width of the opaque spacing between any two consecutive slits. In this case, the second order spectrum will be absent.

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