Today • Gratings: – Sinusoidal amplitude grating – Sinusoidal phase grating – in general: spatially periodic thin transparency Wednesday • Fraunhofer diffraction • Fraunhofer patterns of typical apertures • Spatial frequencies and Fourier transforms MIT 2.71/2.710 04/06/09 wk9-a- 1 Gratings Amplitude grating Phase grating 1 1 0.75 0.75 | [a.u.] | [a.u.] t 0.5 t 0.5 |g |g 0.25 0.25 0 0 −30 −20 −10 0 10 20 30 −30 −20 −10 0 10 20 30 x [λ] x [λ] pi pi pi/2 pi/2 ) [rad] t ) [rad] t 0 0 −pi/2 phase(g −pi/2 phase(g −pi −pi −30 −20 −10 0 10 20 30 −30 −20 −10 0 10 20 30 x [λ] x [λ] Λ: period; m: contrast; φ: phase shift Λ: period; m: phase contrast; φ: phase shift MIT 2.71/2.710 04/06/09 wk9-a- 2 Gratings Amplitude grating Phase grating x x y z y z MIT 2.71/2.710 04/06/09 wk9-a- 3 Sinusoidal amplitude grating … OPLD = Λsin θ constructive interference if OPLD = qλ (q integer) st +1 order Only the n=0th and n=±1st diffraction orders are generated The n=0th diffraction order Λ θ is also known as the DC term incident –θ plane wave −1st order u : Grating 0 spatial … period frequency MIT 2.71/2.710 04/06/09 wk9-a- 4 Sinusoidal amplitude grating … +1st order Λ θ 0th order (or DC term) incident –θ plane diffraction angle wave -1st order spatial frequency … diffraction efficiencies MIT 2.71/2.710 04/06/09 wk9-a- 5 Sinusoidal phase grating s q=+5 “surface relief” grating q=+4 Transmission function q=+3 Λ q=+2 q=+1 (n 1)s m 2π − ≡ λ q=0 “phase contrast” incident q= –1 plane Useful math property: wave q= –2 + ∞ q= –3 exp iα sin (θ) = Jq (α) exp (iqθ) q= ! " #−∞ q= –4 st Jm: Bessel function of 1 kind, q= –5 m-th order glass refractive index n MIT 2.71/2.710 04/06/09 wk9-a- 6 Sinusoidal phase grating s q=+5 Field after grating is expressed as: q=+4 q=+3 Λ amplitude q=+2 q=+1 q=0 plane waves (diffraction orders), propagation angle incident q= –1 plane wave q= –2 q= –3 The q-th exponential physically is q= –4 a plane wave propagating at angle θq q= –5 i.e. the q-th diffraction order glass refractive index n MIT 2.71/2.710 04/06/09 wk9-a- 7 Fresnel diffraction from a grating as a Fourier series More generally, a periodic transparency’s complex amplitude transmission may be s q=+5 expressed as a Fourier series expansion: q=+4 q=+3 Λ q=+2 amplitudes q=+1 q=0 incident q= –1 plane plane waves (diffraction orders), wave q= –2 propagation angle q= –3 q= –4 absorptive q= –5 diffraction efficiencies bands glass refractive index n MIT 2.71/2.710 04/06/09 wk9-a- 8 Example: binary phase grating 1 0.75 s | [a.u.] q=+5 t 0.5 |g 0.25 q=+4 0 −30 −20 −10 0 10 20 30 x [!] q=+3 pi Λ pi/2 q=+2 ) [rad] t 0 −pi/2 q=+1 phase(g Duty cycle = 0.5 −pi −30 −20 −10 0 10 20 30 x [!] q=0 incident q= –1 plane wave q= –2 q= –3 q= –4 q= –5 glass refractive index n MIT 2.71/2.710 04/06/09 wk9-a- 9 MIT OpenCourseWare http://ocw.mit.edu 2.71 / 2.710 Optics Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. .