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Diffraction Grating

Module by: Paul Padley. E-mail the author

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Summary: We derive the interference patter for a diffraction grating.

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Diffraction Grating

Consider the case of N slit diffraction, We have E 1 = ε L a R sin β β e i ( k R 1 − ω t ) E 1 = ε L a R sin β β e i ( k R 1 − ω t ) E 2 = ε L a R sin β β e i ( k R 2 − ω t ) E 2 = ε L a R sin β β e i ( k R 2 − ω t ) . . . . . . E N = ε L a R sin β β e i ( k R N − ω t ) E N = ε L a R sin β β e i ( k R N − ω t ) So we can just follow the steps of the two slit case and extend them and get (using R N = R − ( N − 1 ) d sin θ R N = R − ( N − 1 ) d sin θ ) E = ∑ n = 1 N E N = ∑ n = 1 N ε L a R sin β β e i ( k R − 2 ( n − 1 ) α − ω t ) = ε L a R sin β β ∑ n = 1 N e i ( k R − 2 ( n − 1 ) α − ω t ) = ε L a R sin β β e i ( k R − ω t ) ∑ n = 1 N e − i 2 ( n − 1 ) α = ε L a R sin β β e i ( k R − ω t ) ∑ j = 0 N − 1 e − i 2 j α E = ∑ n = 1 N E N = ∑ n = 1 N ε L a R sin β β e i ( k R − 2 ( n − 1 ) α − ω t ) = ε L a R sin β β ∑ n = 1 N e i ( k R − 2 ( n − 1 ) α − ω t ) = ε L a R sin β β e i ( k R − ω t ) ∑ n = 1 N e − i 2 ( n − 1 ) α = ε L a R sin β β e i ( k R − ω t ) ∑ j = 0 N − 1 e − i 2 j α This is the same geometric series we dealt with before ∑ n = 0 N − 1 x n = 1 − x N 1 − x ∑ n = 0 N − 1 x n = 1 − x N 1 − x so E = ε L a R sin β β e i ( k R − ω t ) ∑ j = 0 N − 1 e − i 2 j α = ε L a R sin β β e i ( k R − ω t ) 1 − e − i 2 N α 1 − e − i 2 α = ε L a R sin β β e i ( k R − ω t ) e − i N α e − i α e i N α e i α 1 − e − i 2 N α 1 − e − i 2 α = ε L a R sin β β e i ( k R − ω t ) e − i N α e − i α e i N α − e − i N α e i α − e − i α = ε L a R sin β β e i ( k R − ( N − 1 ) α − ω t ) sin N α sin α E = ε L a R sin β β e i ( k R − ω t ) ∑ j = 0 N − 1 e − i 2 j α = ε L a R sin β β e i ( k R − ω t ) 1 − e − i 2 N α 1 − e − i 2 α = ε L a R sin β β e i ( k R − ω t ) e − i N α e − i α e i N α e i α 1 − e − i 2 N α 1 − e − i 2 α = ε L a R sin β β e i ( k R − ω t ) e − i N α e − i α e i N α − e − i N α e i α − e − i α = ε L a R sin β β e i ( k R − ( N − 1 ) α − ω t ) sin N α sin α

Notice that this just ends up being multisource interference multiplied by single slit diffraction.

Squaring it we see that: I ( θ ) = I 0 sin 2 β β 2 sin 2 N α sin 2 α I ( θ ) = I 0 sin 2 β β 2 sin 2 N α sin 2 α

Figure 1
Figure 1 (IK1OTM04.png)
Interference with diffraction for 6 slits with d = 4 a d = 4 a

Figure 2
Figure 2 (IK1OTM05.png)
Interference with diffraction for 6 slits with d = 4 a d = 4 a

Figure 3
Figure 3 (IK1OTM06.png)
Interference with diffraction for10 slits with d = 4 a d = 4 a

Figure 4
Figure 4 (IK1OTM07.png)
Interference with diffraction for10 slits with d = 4 a d = 4 a

Principal maxima occur when sin N α sin α = N sin N α sin α = N or since α = k d ( sin θ ) / 2 α = k d ( sin θ ) / 2 k d sin θ = 2 n π    n = 0 , 1 , 2 , 3 k d sin θ = 2 n π    n = 0 , 1 , 2 , 3 or 2 π λ d sin θ = 2 n π 2 π λ d sin θ = 2 n π or sin θ = n λ d sin θ = n λ d

and just like in multisource interference minima occur at sin θ = n λ N d    n = 1 , 2 , 3 …   n N ≠ i n t e g e r sin θ = n λ N d    n = 1 , 2 , 3 …   n N ≠ i n t e g e r A diffraction grating is a repetitive array of diffracting elements such as slits or reflectors. Typically with N very large (100's). Notice how all but the first maximum depend on λ λ . So you can use a grating for spectroscopy.

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