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.