# wk9b - Today Diffraction from periodic transparencies...

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-1 Today • Diffraction from periodic transparencies: gratings • Grating dispersion • Wave optics description of a lens: quadratic phase delay • Lens as Fourier transform engine

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-2 Diffraction from periodic array of holes Λ incident plane wave Period : Λ Spatial frequency : 1/ Λ A spherical wave is generated at each hole; we need to figure out how the periodically-spaced spherical waves interfere
MIT 2.71/2.710 Optics 11/03/04 wk9-b-3 Diffraction from periodic array of holes Λ incident plane wave Period : Λ Spatial frequency : 1/ Λ Interference is constructive in the direction pointed by the parallel rays if the optical path difference between successive rays equals an integral multiple of λ (equivalently, the phase delay equals an integral multiple of 2 π ) Optical path differences

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-4 Diffraction from periodic array of holes Λ d θ d = Λ sin θ Period : Λ Spatial frequency : 1/ Λ From the geometry we find Therefore, interference is constructive iff Λ = = Λ λ θ m m sin sin
MIT 2.71/2.710 Optics 11/03/04 wk9-b-5 Diffraction from periodic array of holes m =1 Λ incident plane wave m =3 m =2 m =–1 m =–2 m =–3 m =0 “straight-through” order (aka DC term) Grating spatial frequency : 1/ Λ Angular separation between diffracted orders : ∆θ ≈ λ / Λ several diffracted plane waves “diffraction orders” 1 st diffracted order 2 nd diffracted order –1 st diffracted order

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MIT 2.71/2.710 Optics 11/03/04 wk9-b-6 Fraunhofer diffraction from periodic array of holes Λ z →∞ Λ λ θ m sin z m x Λ
MIT 2.71/2.710 Optics 11/03/04 wk9-b-7 Sinusoidal amplitude grating () ( ) ( ) + + + + × + = 0 0 0 0 2 2 2 out 4 1 4 1 2 1 exp e , zv y zu x zv y zu x y x z y x z i y x g z i λ δ π x y u x v y Λ ψ [] ) ( 2 cos 1 2 1 , 0 0 in y v x u y x g + + = 0 0 2 0 2 0 tan , 1 v u v u = + = Λ ψ

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