MIT 2.71/2.710 Optics 10/27/04 wk8-b-1 The imaging problem object imaging optics (lenses, etc.) image.

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The imaging problem

objectimaging optics

(lenses, etc.) image

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The imaging problem

Illumination(coherent vsincoherent)

object imageimaging optics

(lenses, etc.)

free space free space

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The imaging problem

Illumination(coherent vsincoherent)

object image

(spatial) linear shift-invariant system

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The imaging problem

object image

(spatial) linear shift-invariant system

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Our approach

• Today: – linear shift invariant (LSI) systems in the space/spatial frequency domains – mathematical properties of Fourier transforms• Monday: – free space propagation: Fresnel and Fraunhofer diffraction • Wednesday: – examples of Fraunhofer diffraction: amplitude and phase diffraction gratings – wave description of light propagation through a lens – Fourier transformation and imaging using lenses

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Spatial filtering

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Spatial frequency representation

space domain3 sinusoids

Fourier domain(aka spatial frequency domai

n)

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Spatial frequency removal

Fourier domain(aka spatial frequency domai

n)

space domain2 sinusoids (1 removed)

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From space to spatial frequency: 2D2D Fourier analysis

Can I express an arbitrary g(x,y)

as a superposition of sinusoids?

... etc. ...

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Spatial frequency representation

space domaing(x,y)

Fourier domain(aka spatial frequency domai

n)

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Low-pass filtering

space domainFourier domain

(aka spatial frequency domain)

removed high-frequency content

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Band-pass filtering

removed high-and low-frequency content

space domainFourier domain

(aka spatial frequency domain)

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Example: optical lithography

Original nested Ls

original pattern(“nested

L’s”)

mildlow-pass filtering

Notice:(i) blurring at the edges(ii) ringing

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Example: optical lithography

Original nested Ls

original pattern(“nested

L’s”)

severelow-pass filtering

Notice:(i) blurring at the edges(ii) ringing

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The 2D2D Fourier integral

(aka inverse Fourier transform)

superposition sinusoids

complex weight,expresses relative amplitude

(magnitude & phase) of superposed sinusoids

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The 2D2D Fourier integral

The complex weight coefficients G(u,v),Aka Fourier transformFourier transform of g(x,y)

are calculated from the integral

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

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2D2D Fourier transform pairs

Image removed due to copyright concerns

(from Goodman, Introduction to Fourier Optics, page 14)

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Space and spatial frequency representations

SPACE DOMAIN

2D2D Fourier transform 2D2D Fourier integral

aka

inverse 2D2D Fourier transform SPATIAL FREQUENCY DOMAIN

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Fourier transform properties /1

•Fourier transforms and the delta function

•Linearity of Fourier transforms

if and

then

for any pair of complex numbers

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Fourier transform properties /2

Let

Shift theorem (space →frequency)

Shift theorem (frequency →space)

Scaling theorem

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Fourier transform properties /3

Let

Let

and

Convolution theorem (space →frequency)

Convolution theorem (frequency →space)

Let

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Fourier transform properties /4

Let

Let

and

Let

Correlation theorem (space →frequency)

Correlation theorem (frequency →space)

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2D2D linear shift invariant systems

input output

convolution with impulse responseimpulse response

multiplication with transfer functiontransfer function

Fo

uri

er

tran

sfo

rmIn

verse Fo

urier

transfo

rm

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2D2D linear shift invariant systems

SPACE DOMAIN

SPATIAL FREQUENCY DOMAIN

input output

Fo

uri

er

tran

sfo

rmIn

verse Fo

urier

transfo

rmconvolution with impulse responseimpulse response

multiplication with transfer functiontransfer function

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2D2D linear shift invariant systems

input output

Fo

uri

er

tran

sfo

rmIn

verse Fo

urier

transfo

rmconvolution with impulse responseimpulse response

multiplication with transfer functiontransfer function

are pair

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Sampling space and frequency

pixel size

field size

spacedomain

spatial frequency

domain

Nyquistrelationships:

frequencyresolution

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The Space–Bandwidth Product

Nyquist relationships:

from space → spatial frequency domain:

from spatial frequency → space domain:

: 1D Space–Bandwidth Product (SBP)

aka number of pixels in the space domain

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SBP: example

space domainFourier domain

(aka spatial frequency domain)