Overview from last week• Optical systems act as linear shift-invariant (LSI) filters (we have not yet seen why)• Analysis tool for LSI filters: Fourier transform

– decompose arbitrary 2D functions into superpositions of 2D sinusoids (Fourier transform)

– use the transfer function to determine what happens to each 2D sinusoid as it is transmitted through the system (filtering)

– recompose the filtered 2D sinusoids to determine the output 2D function (Fourier integral, aka inverse Fourier transform)

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Today

• Wave description of optical systems

• Diffraction– very short distances: near field, we skip

– intermediate distances: Fresnel diffraction expressed as a convolution

– long distances (∞): Fraunhofer diffraction expressed as a Fourier transform

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Space and spatial frequency representations

SPACE DOMAIN

2D Fourier transform 2D Fourier integralAka

inverse 2D Fourier transformSPATIAL FREQUENCY

DOMAIN

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2D linear shift invariant systems

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input output

convolution with impulse response

multiplication with transfer function

Fou

rier

tran

sfor

m

transform

inverse Fourier

Wave description of optical imaging systems

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Thin transparencies

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coherentillumination:

planewave

Transmission function:

Field before transparence:

Field after transparence:

assumptions: transparence at z=0transparency thickness can be ignored

Diffraction: Huygens principle

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incidentplane Wave

Field at distance d:contains contributions

from all spherical wavesemitted at the transparency,

summed coherently

Field after transparence:

Huygens principle: one point source

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incomingplane wave

opaquescreen

sphericalwave

Simple interference: two point sources

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incomingplane wave

opaquescreen

intensity

Two point sources interfering: math…

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intensity

Amplitude:(paraxial approximation)

Diffraction: many point sources

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incomingplane wave

opaquescreen

many spherical waves tightly packed

Diffraction: many point sources,attenuated & phase-delayed

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incomingplane wave

thintransparency

Diffraction: many point sources attenuated & phase-delayed, math

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incomingplane wave

field

Transmission functionThin transparency

continuouslimit

wave

Fresnel diffraction

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The diffracted field is the convolution of the transparency with a spherical wave

amplitude distributionat output plane

transparencytransmissionfunction (complex teiφ)

spherical wave@z=l

(aka Green’s function)

FUNCTION OF LATERAL COORDINATES:Interesting!!!

CONSTANT:NOT interesting

Example: circular aperture

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input field

Example: circular aperture

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input field

Example: circular aperture

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input field

Example: circular aperture

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(from Hecht, Optics, 4thedition, page 494)

input field

Image removed due to copyright concerns

Fraunhofer diffraction

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propagation distance lis “very large”

approximaton valid if

Fraunhofer diffraction

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The“far-field” (i.e. the diffraction pattern at a largelongitudinal distance l equals the Fourier transform

of the original transparencycalculated at spatial frequencies

Fraunhofer diffraction

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spherical waveoriginating at x

plane wave propagatingat angle –x/l⇔ spatial frequency –x/(λl)

Fraunhofer diffraction

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superposition ofspherical wavesoriginating atvarious pointsalong

superposition ofplane waves propagatingat corresponding angles –x/l⇔spatial frequencies –x/(λl)

Example: rectangular apertureinput

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input field far field

free spacepropagation by

Example: circular aperture

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also known asAiry pattern, or

free spacepropagation by

input field far field