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 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 Grating dispersion … Λ air glass white Grating: Prism: … blue light is diffracted at blue light is refracted at smaller angle than red: larger angle than red: anomalous dispersion normal dispersion MIT 2.71/2.710 04/06/09 wk9-a-10 Today • Fraunhofer diffraction • Fourier transforms: maths • Fraunhofer patterns of typical apertures • Fresnel propagation: Fourier systems description – impulse response and transfer function – example: Talbot effect Next week • Fourier transforming properties of lenses • Spatial frequencies and their interpretation • Spatial filtering MIT 2.71/2.710 04/08/09 wk9-b- 1 Fraunhofer diffraction Fresnel (free space) propagation may be expressed as a convolution integral MIT 2.71/2.710 04/08/09 wk9-b- 2 Example: rectangular aperture x y z sinc pattern free space propagation by l→∞ x0 MIT 2.71/2.710 input field far field 04/08/09 wk9-b- 3 Example: circular aperture x y z Airy pattern free space 2r0 propagation by l→∞ MIT 2.71/2.710 input field far field 04/08/09 wk9-b- 4 How far along z does the Fraunhofer pattern appear? Fresnel (free space) propagation may be expressed as a convolution integral cos(πα2) α 2 2 2 For example, if (x +y )max=(4λ) , then z>>16λ to enter the Fraunhofer regime; 2 2 2 6 if (x +y )max=(1000λ) , then z>>10 λ; in practice, the Fraunhofer intensity pattern is recognizable at smaller z than long short these predictions (but the correct Fraunhofer phase takes longer to form) propagation distance z MIT 2.71/2.710 04/08/09 wk9-b- 5 Fourier transforms • One dimensional – Fourier transform – Fourier integral • Two dimensional – Fourier transform – Fourier integral (1D so we can draw it easily ... ) g(x) [real] Re[e-i2πux] Re[G(u)]= x dx MIT 2.71/2.710 04/08/09 wk9-b- 6 Frequency representation -i2πux g(x)=cos[2πu0x] Re[e ] x Re[G(u)]= dx =0, if u0≠u x Re[G(u)]= dx =∞, if u0=u G(u) δ(u+u0) δ(u−u0) G(u)=½ δ(u+u0)+½ δ(u−u0) ½ ½ The negative frequency is physically meaningless, u but necessary for mathematical rigor; it is the price to pay for the convenience of using −u0 +u0 complex exponentials in the phasor representation MIT 2.71/2.710 04/08/09 wk9-b- 7 Commonly used functions in wave Optics Text removed due to copyright restrictions. Please see p. 12 in Goodman, Joseph W. Introduction to Fourier Optics. Englewood, CO: Roberts & Co., 2004. ISBN: 9780974707723. Images from Wikimedia Commons, http://commons.wikimedia.org MIT 2.71/2.710 04/08/09 wk9-b- 8 Goodman, Introduction to Fourier Optics (3rd ed.) pp. 12-14 Fourier transform pairs Functions with radial symmetry Table removed due to copyright restrictions. Please see Table 2.1 in Goodman, Joseph W. Introduction to Fourier Optics. Englewood, CO: Roberts & Co., 2004. ISBN: 9780974707723. jinc(ρ)≡ Images from Wikimedia Commons, http://commons.wikimedia.org MIT 2.71/2.710 04/08/09 wk9-b- 9 Goodman, Introduction to Fourier Optics (3rd ed.) p. 14 Fourier transform properties Text removed due to copyright restrictions. Please see pp. 8-9 in Goodman, Joseph W. Introduction to Fourier Optics. Englewood, CO: Roberts & Co., 2004. ISBN: 9780974707723. A general discussion of the properties of Fourier transforms may also be found here http://en.wikipedia.org/wiki/Fourier_transform#Properties_of_the_Fourier_transform. IMPORTANT! A note on notation: Goodman uses (fX, fY) to denote spatial frequencies along the (x,y) dimensions, respectively. In these notes, we will sometimes use (u,v) instead. MIT 2.71/2.710 04/08/09 wk9-b-10 Goodman, Introduction to Fourier Optics (3rd ed.) pp. 8-9 The spatial frequency domain: vertical grating y v x u y v x u Frequency Space (Fourier) domain domain MIT 2.71/2.710 04/08/09 wk9-b-11 The spatial frequency domain: tilted grating y v x u y v x u Frequency Space (Fourier) domain domain MIT 2.71/2.710 04/08/09 wk9-b-12 Superposition: two gratings + + Frequency Space (Fourier) domain domain MIT 2.71/2.710 04/08/09 wk9-b-13 Superposition: multiple gratings discrete (Fourier series) continuous (Fourier integral) Frequency Space (Fourier) domain domain MIT 2.71/2.710 04/08/09 wk9-b-14 Spatial frequency representation of arbitrary scenes 0 MIT 2.71/2.710 04/08/09 wk9-b-15 The scaling (or similarity) theorem Frequency Space (Fourier) domain domain MIT 2.71/2.710 04/08/09 wk9-b-16 The shift theorem Frequency Space (Fourier) domain domain MIT 2.71/2.710 04/08/09 wk9-b-17 The convolution theorem multiplication convolution MIT 2.71/2.710 04/08/09 wk9-b-18 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. .