The convolution theorem

multiplication convolution

MIT 2.71/2.710 04/08/09 wk9-b-18

The convolution theorem

multiplication convolution

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Fraunhofer diffraction patterns

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z→∞

z→∞

z→∞

z→∞

Spatial filtering

space domain Fourier domain3 sinusoids

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Spatial filtering

space domain Fourier domain 2 sinusoids (one removed)

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Spatial filtering of a scene

Unfiltered: all spatial frequencies present

0

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Spatial filtering of a scene

Low-pass filtered: high spatial frequencies removed

0

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Spatial filtering of a scene

Band-pass filtered: high & low spatial frequencies removed

0

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The transfer function of Fresnel propagation

Fresnel (free space) propagation may be expressed as a

convolution integral

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Today

• Fourier transforming properties of lenses • Spherical - plane wave duality • The telescope (4F system) revisited

– imaging as a cascade of Fourier transforms – spatial filtering by a pupil plane transparency

Wednesday

• Spatial filtering in the 4F system • Point-Spread Function (PSF)

and Amplitude Transfer Function (ATF)

MIT 2.71/2.710 04/13/09 wk10-a- 1

Reminder: thin lens (geometrical optics)

f (focal length)

object at ∞

f (focal length)

image at ∞image at object at(plane wave) back focal plane front focal plane (plane wave)

amount of ray bending is proportional to the distance from the optical axis

point object point image

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at finite distance (spherical wave)

at finite distance (spherical wave)

so si

Thin lens complex transmittance

Δ(x,y)

Δ0

Δ(x,y): glass thickness at coordinate (x,y)

n: index of refraction R1, R2: radii of curvature

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Plane wave incident on a lens

back focal plane

f

spherical wave converging to u0λf

θ0 f =

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Spherical wave incident on a lens

spherical wave, diverging

front focal plane

plane wave plane wave converging to u0λfoff–axis

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Fourier transforming property of lenses

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2.71 / 2.710 Optics Spring 2009

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