mit 2.71/2.710 optics 10/25/04 wk8-a-1 the spatial frequency domain

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The spatial frequency domain

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Recall: plane wave propagation

path delay increaseslinearly with x

plane of observation

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Spatial frequency ⇔angle of propagation?

plane of observation

electric field

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Spatial frequency ⇔angle of propagation?

plane of observation

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The cross-section of the optical field with the optical axisis a sinusoid of the form

Spatial frequency ⇔angle of propagation?

i.e.it looks like

where

is called the spatial frequencyspatial frequency

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2D sinusoids

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2D sinusoids

min

max

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2D sinusoids

min

max

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2D sinusoids

min

max

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Periodic Grating /1: vertical

Spacedomain

Frequency(Fourier)domain

Periodic Grating /2: tilted

Spacedomain

Frequency(Fourier)domain

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Superposition: multiple gratings

Spacedomain

Frequency(Fourier)domain

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More superpositions

Spacedomain

Frequency(Fourier)domain

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Size of object vsfrequency content

Spacedomain

Frequency(Fourier)domain

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Superimposing shifted objects

Spacedomain

Frequency(Fourier)domain

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Linear shift invariant systemsin the time domain

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Linear shift invariant systems

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Linear shift invariant systems

impulse excitation impulse excitation

shift invarianceshift invariance

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Linear shift invariant systems

impulse excitation impulse excitation

linearitylinearity

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Linear shift invariant systems

arbitrary

sifting property of the δ-function convolution integral

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Linear shift invariant systems

sinusoid sinusoid

attenuationphase delay

transfer function transfer function

same frequency

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From time to frequency: Fourier analysis

Can I express an arbitrary g(t) as a superposition of sinusoids?

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The Fourier integral

(aka inverse Fourier transforminverse Fourier transform)

superposition sinusoids

complex weight,expresses relative amplitude

(magnitude & phase) of superposed sinusoids

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The Fourier transform

The complex weight coefficients G(ν),akaFourier transform Fourier transform o

f g(t)are calculated from the integral

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Fourier transform pairs

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2D2D Fourier transform pairs pairs

Image removed due to copyright concerns

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