Question

Discuss the diffraction at a grating and obtain the condition for the mth maximum.

Answer 1

1. Grating has multiple slits with equal widths of size comparable to the wavelength of diffracting light.
2. Grating is a plane sheet of transparent material on which opaque rulings are made with a fine diamond pointer.
3. The rulings act as obstacles having a definite width b and the transparent space between the rulings act as slit of width a.
4. The combined width of a ruling and a slit is called grating element (e = a + b). Points on successive slits separated by a distance equal to the grating element are called corresponding points.Diffraction grating experiment
5. Let a plane wavefront of monochromatic light with wave length λ be incident normally on the grating.
6. As the slits size is comparable to that of wavelength, the incident light diffracts at the grating.
7. A diffraction pattern is obtained on the screen when the diffracted waves are focused on a screen using a convex lens.
8. The path difference δ between the diffracted waves from one pair of corresponding points is,
   δ = (a + b) sin θ
   This path difference is the same for any pair of corresponding points. The point P will be bright, when
   δ = mλ where m = 0,1,2,3
   Combining the above two equations, we get,
   (a + b) sin θ = mλ
   Here, m is called an order of diffraction.
   
   1. Condition for zero order maximum, m = 0
      For (a + b) sinθ = 0, the position, θ = 0. sin θ = 0 and m = 0. This is called zero order diffraction or central maximum.
   2. Condition for first-order maximum, m = 1
      If (a + b) sin θ₁ = λ, the diffracted light meet at an angle θ₁ to the incident direction and the first-order maximum is obtained.
   3. Condition for second-order maximum, m = 2
      Similarly, (a + b) sin θ₂ = 2λ forms the second-order maximum at the angular position θ₂.
      Condition for higher-order maximum
      On either side of central maxima, different higher orders of diffraction maxima are formed at different angular positions.
      If we take,
      N = `1/("a + b")`
      `1/"N"` sin θ = mλ (or) sin θ = Nmλ