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Transcript 
The spatial frequency domain
MIT 2.71/2.710 Optics
10/25/04 wk8-a-1
Recall: plane wave propagation
path delay increases
linearly with x
plane of observation
MIT 2.71/2.710 Optics
10/25/04 wk8-a-2
Spatial frequency ⇔angle of propagation?
electric field
plane of observation
MIT 2.71/2.710 Optics
10/25/04 wk8-a-3
Spatial frequency ⇔angle of propagation?
plane of observation
MIT 2.71/2.710 Optics
10/25/04 wk8-a-4
Spatial frequency ⇔angle of propagation?
The cross-section of the optical field with the optical axis
is a sinusoid of the form
i.e.it looks like
where
is called the spatial frequency
MIT 2.71/2.710 Optics
10/25/04 wk8-a-5
2D sinusoids
MIT 2.71/2.710 Optics
10/25/04 wk8-a-6
2D sinusoids
min
max
MIT 2.71/2.710 Optics
10/25/04 wk8-a-7
2D sinusoids
min
max
MIT 2.71/2.710 Optics
10/25/04 wk8-a-8
2D sinusoids
min
max
MIT 2.71/2.710 Optics
10/25/04 wk8-a-9
Periodic Grating /1: vertical
Space
domain
MIT 2.71/2.710 Optics
10/25/04 wk8-a-10
Frequency
(Fourier)
domain
Periodic Grating /2: tilted
Space
domain
MIT 2.71/2.710 Optics
10/25/04 wk8-a-11
Frequency
(Fourier)
domain
Superposition: multiple gratings
Space
domain
MIT 2.71/2.710 Optics
10/25/04 wk8-a-12
Frequency
(Fourier)
domain
More superpositions
Space
domain
MIT 2.71/2.710 Optics
10/25/04 wk8-a-13
Frequency
(Fourier)
domain
Size of object vsfrequency content
Space
domain
MIT 2.71/2.710 Optics
10/25/04 wk8-a-14
Frequency
(Fourier)
domain
Superimposing shifted objects
Space
domain
MIT 2.71/2.710 Optics
10/25/04 wk8-a-15
Frequency
(Fourier)
domain
Linear shift invariant systems
in the time domain
MIT 2.71/2.710 Optics
10/25/04 wk8-a-16
Linear shift invariant systems
MIT 2.71/2.710 Optics
10/25/04 wk8-a-17
Linear shift invariant systems
impulse excitation
shift invariance
MIT 2.71/2.710 Optics
10/25/04 wk8-a-18
impulse excitation
Linear shift invariant systems
impulse excitation
linearity
MIT 2.71/2.710 Optics
10/25/04 wk8-a-19
impulse excitation
Linear shift invariant systems
arbitrary
sifting property of the δ-function
MIT 2.71/2.710 Optics
10/25/04 wk8-a-20
convolution integral
Linear shift invariant systems
sinusoid
sinusoid
attenuation
phase delay
same frequency
transfer function
MIT 2.71/2.710 Optics
10/25/04 wk8-a-21
From time to frequency: Fourier analysis
Can I express an arbitrary g(t)
as a superposition of sinusoids?
MIT 2.71/2.710 Optics
10/25/04 wk8-a-22
The Fourier integral
(aka inverse Fourier transform)
superposition
sinusoids
complex weight,
expresses relative amplitude
(magnitude & phase)
of superposed sinusoids
MIT 2.71/2.710 Optics
10/25/04 wk8-a-23
The Fourier transform
The complex weight coefficients G(ν),
akaFourier transform Fourier transform of
g(t)
are calculated from the integral
MIT 2.71/2.710 Optics
10/25/04 wk8-a-24
Fourier transform pairs
MIT 2.71/2.710 Optics
10/25/04 wk8-a-25
2D Fourier transform pairs
Image removed due to copyright concerns
(from Goodman,
Introduction to
Fourier Optics,
page 14)
MIT 2.71/2.710 Optics
10/25/04 wk8-a-26
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