Sinusoidal amplitude grating

MIT 2.71/2.710 04/06/09 wk9-a- 5

θ

+1st order

-1st order

–θ0th order (or DC term)

incident plane wave

……

Λ

diffraction angle

spatial frequency

diffraction efficiencies

Example: binary phase grating

s q=+5

q=+4

q=+3 Λ

q=+2

q=+1

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

−30 −20 −10 0 10 20 300

0.25

0.5

0.75

1

x [!]

|gt| [

a.u.

]

−30 −20 −10 0 10 20 30 −pi

−pi/2

0

pi/2

pi

x [!]

phas

e(g t) [

rad]

Duty cycle = 0.5

Grating dispersion

…

Λ

…

white

Grating: blue light is diffracted at smaller angle than red:

anomalous dispersion

glassair

Prism: blue light is refracted at larger angle than red:

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 diffractionFresnel (free space) propagation

may be expressed as aconvolution integral

MIT 2.71/2.710 04/08/09 wk9-b- 2

x

Example: rectangular aperture

y z sinc pattern

x0

free space propagation by

l→∞

MIT 2.71/2.710 input field far field04/08/09 wk9-b- 3

Example: circular aperture

x

y z Airy pattern

2r0

free space propagation by

l→∞

MIT 2.71/2.710 input field far field04/08/09 wk9-b- 4

How far along z does the Fraunhofer pattern appear?Fresnel (free space) propagation

may be expressed as aconvolution integral

cos(πα2)

α

For example, if (x2+y2)max=(4λ)2, then z>>16λ to enter the Fraunhofer regime;if (x2+y2)max=(1000λ)2, then z>>106λ; in practice, the Fraunhofer intensity pattern is recognizable at smaller z than long shortthese 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

MIT 2.71/2.710 04/08/09 wk9-b- 6

x

g(x) [real] Re[e-i2πux]

Re[G(u)]= dx

(1D so we can draw it easily ... )

Frequency representation

x

g(x)=cos[2πu0x] Re[e-i2πux]

Re[G(u)]=

x Re[G(u)]=

dx =0, if u0≠u

dx =∞, if u0=u

G(u) G(u)=½ δ(u+u0)+½ δ(u−u0)

The negative frequency is physically meaningless,but necessary for mathematical rigor; it is the price to pay for the convenience of using

−u0 +u0 complex exponentials in the phasor representation

½ δ(u−u0)δ(u+u0)

½ u

MIT 2.71/2.710 04/08/09 wk9-b- 7

Commonly used functions in wave Optics

MIT 2.71/2.71004/08/09 wk9-b- 8 Goodman, Introduction to Fourier Optics (3rd ed.) pp. 12-14

Images from Wikimedia Commons, http://commons.wikimedia.org

Text removed due to copyright restrictions. Please see p. 12 inGoodman, Joseph W. Introduction to Fourier Optics.Englewood, CO: Roberts & Co., 2004. ISBN: 9780974707723.

Fourier transform pairsFunctions with radial symmetry

jinc(ρ)≡

MIT 2.71/2.71004/08/09 wk9-b- 9 Goodman, Introduction to Fourier Optics (3rd ed.) p. 14

Images from Wikimedia Commons, http://commons.wikimedia.org

Table removed due to copyright restrictions. Please see Table 2.1 inGoodman, Joseph W. Introduction to Fourier Optics. Englewood, CO: Roberts & Co., 2004.ISBN: 9780974707723.

Fourier transform properties

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

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 herehttp://en.wikipedia.org/wiki/Fourier_transform#Properties_of_the_Fourier_transform.

The spatial frequency domain: vertical grating

MIT 2.71/2.710 04/08/09 wk9-b-11

Space domain

Frequency (Fourier) domain

x

y

x

y

u

v

u

v

The spatial frequency domain: tilted grating

x

y

u

v

y v

x u

FrequencySpace (Fourier)domain domain MIT 2.71/2.710 04/08/09 wk9-b-12

Superposition: two gratings

+ +

MIT 2.71/2.710 04/08/09 wk9-b-13

Space domain

Frequency (Fourier) domain

Superposition: multiple gratings

discrete(Fourier series)

continuous(Fourier integral)

Frequency (Fourier) domain

Spacedomain

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

FrequencySpace (Fourier)domain domain MIT 2.71/2.710 04/08/09 wk9-b-16

The shift theorem

Space Frequency (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

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