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