Fourier relations in Optics

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Fourier relations in Optics
Near field Far field
Frequency Pulse duration
Frequency Coherence length
Beam waist Beam divergence
Spatial dimension Angular dimension
Focal plane of lens The other focal plane
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Huygens Principle
E(R)
E(r)
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Fourier theorem A complex function f(t) may be
decomposed as a superposition integral of
harmonic function of all frequencies and complex
amplitude
(inverse Fourier transform) The component
with frequency ? has a complex amplitude F(?),
given by
(Fourier transform)
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Useful Fourier relations in optics between t and
?, and between x and ?.


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Useful Fourier relations in optics between t and
?, and between x and ?.
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Position or time
Angle or frequency
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Angle or frequency
Position or time
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Single- slit diffraction
Application of Fourier relation
a
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The applications of the Fourier relation
-Spatial harmonics and angles of propagation
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?
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Frequency, time, or position
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N
w0
Time
Dw
Frequency
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N
w0
Time
Dw
Frequency
Mode-locking
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N
x0
Angle
Dx
Position
Diffraction grating, radio antenna array
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The applications of the Fourier relation

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Finite number of elements
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-Graded grating for focusing -Fresnel lens
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Fourier transform between two focal planes of a
lens
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The use of spatial harmonics for analyses of
arbitrary field pattern
Consider a two-dimensional complex electric field
at z0 given by
where the ?s are the spatial frequencies in the
x and y directions.
The spatial frequencies are the inverse of the
periods.
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Thus by decomposing a spatial distribution of
electric field into spatial harmonics, each
component can be treated separately.
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Define a transfer function (multiplication
factor) in free space for the spatial harmonics
of spatial frequency ?x and ?y to travel from z0
to zd as
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Define a transfer function (multiplication
factor) in free space for the spatial harmonics
of spatial frequency ?x and ?y to travel from z0
to zd as
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Source
E
E
z0
z0
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To generalize
Grating momentum
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Stationary gratings vs. Moving gratings
Deflection Frequency shift
Deflection
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The small angle approximation (1/? ltlt?) for the H
function
???
A correction factor for the transfer function for
the plane waves
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F(x)
H(?x)F(x)
D
z0
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Express F(x,z) in ?x/z
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Express F(x,z) in ?x/z
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The effect of lenses

A lens is to introduce a quadratic phase shift to
the wavefront given by .
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Fourier transform using a lens
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Huygens Principle
E(R)
E(r)
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Recording of full information of an optical
image, including the amplitude and phase.
Holography
Amplitude only
Amplitude and phase
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A simple example of recording and
reconstruction
?
k2
k1

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A simple example of recording and
reconstruction
?
k2
k1

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k2
k1
?/2

?
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Another example Volume hologram
?
k2
k1

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Volume grating

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k1

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k1
k2
k1

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D
C
A
?
d
B
Bragg condition
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D
C
A
?
d
B
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Another example Image reconstruction of a point
illuminated by a plane wave.
Writing
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Reading
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E(x,y)
Er
Recorded pattern
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Recorded pattern
Diffracted beam when illuminated by ER
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