<< Chapter < Page
  Waves and optics   Page 1 / 1
Chapter >> Page >
We derive the interference patter for a diffraction grating.

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 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 ) 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 ) 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 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

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

Interference with diffractionfor 6 slits with d = 4 a

Interference with diffractionfor 6 slits with d = 4 a

Interference with diffractionfor10 slits with d = 4 a

Interference with diffractionfor10 slits with d = 4 a

Principal maxima occur when sin N α sin α = N or since α = k d ( sin θ ) / 2 k d sin θ = 2 n π    n = 0 , 1 , 2 , 3 or 2 π λ d sin θ = 2 n π or 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 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 butthe first maximum depend on λ . So you can use a grating for spectroscopy.

Get the best College algebra course in your pocket!





Source:  OpenStax, Waves and optics. OpenStax CNX. Nov 17, 2005 Download for free at http://cnx.org/content/col10279/1.33
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Waves and optics' conversation and receive update notifications?

Ask