Transfer function
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Transcript Transfer function
Appendix A : Fourier transform
Appendix : Linear systems
Appendix : Shift-invariant systems
Impulse-response function : h(t)
Transfer function : H(n)
Impulse-response function (the response of the system to an impulse, or a point, at the input)
Transfer function (the response to spatial harmonic functions)
Appendix : Transfer function
4. Fourier Optics
2D Fourier transform
Plane wave
In a plane (e.g. z=0)
(harmonic function)
One-to-one correspondence because
with
“+”, forward
“-” , backward
In general (superposition integral of harmonic functions)
The transmitted wave is a superposition of plane wave
Superposition of plane waves
Spatial harmonic function and plane wave
U x, y ,0 f x, y A exp j 2 n x x n y y
Spatial spectral analysis
Spatial frequency and propagation angle
Amplitude modulation
B. Transfer Function of Free Space
Impulse response function h
Transfer function H
Transfer Function of Free Space
: evanescent wave
We may therefore regard 1/l as the cutoff spatial frequency
(the spatial bandwidth) of the system
Fresnel approximation
Fresnel approximation for transfer function of free space
If a is the largest radial distance in the output plane, the largest angle qm a/d, and
we may write
Input - Output Relation
Impulse-Response Function of Free Space
Impulse-Response Function of Free Space
= Inverse Fourier transform of the transfer function
Paraboloidal wave
Free-Space Propagation as a Convolution
Huygens-Fresnel Principle and the impulse-response function
The Huygens-Fresnel principle states that each point on a wavefront generates a spherical wave.
The envelope of these secondary waves constitutes a new wavefront.
Their superposition constitutes the wave in another plane.
The system’s impulse-response function for propagation between the planes z = 0 and z = d is
In the paraxial approximation, the spherical wave is approximated by the paraboloidal wave.
Our derivation of the impulse response function is therefore consistent with the H.-F. principle.
In summary:
Within the Fresnel approximation, there are two approaches to
determining the complex amplitude g(x, y) in the output plane,
given the complex amplitude f(x, y) in the input plane:
Space-domain approach
in which the input wave is expanded in terms of paraboloidal elementary waves
Frequency-domain approach
in which the input wave is expanded as a sum of plane waves.
4.2 Optical Fourier transform
A. Fourier Transform in the Far Field
If the propagation distance d is sufficiently long,
the only plane wave that contributes to the complex amplitude at a point (x, y) in the output plane
is the wave with direction making angles
Proof!
Proof :
for Fraunhofer approximation
If f(x, y) is confined to a small area of radius b, and if the distance d is sufficiently large
so that the Fresnel number
is small,
Condition of Validity of Fraunhofer Approximation
when the Fresnel number
Three configurations
back focal
plane
Input placed
against lens
Input placed
in front of lens
Input placed
behind lens
Phase representation of a thin lens in paraxial approximation
f 0 : convex
k 2
2
tl x, y exp j
x y
2f
f 0 : concave
(a) The input placed directly against the lens
1
0
Pupil function ; P x , y
inside the lens aperture
otherwise
k
2
2
U l' x , y U l x , y P x , y exp j
x
y
2f
Ul
Ul’
From the Fresnel approximation when d = f ,
k
exp j
u2 2
k
2
2f
U f u,
U l' x , y exp j
x 2 y 2 exp j
xu y dxdy
jl f
2f
lf
k
exp j
u2 2
2
2f
U f u,
U l x , y P x , y exp j
xu y dxdy
jl f
lf
Quadratic phase factor
Fourier
transform
(b) The input placed in front of the lens
k
A exp j
2f
U f u,
d 2 2
1 f u
2
U l x, y exp j
xu y dxdy
jl f
lf
If d = f
A
U f u,
jl f
2
U
x
,
y
exp
j
xu
y
dxdy
l l f
Exact Fourier transform !
(c) The input placed behind the lens
Af f
f
U 0 ,
P ,
d d d
k 2
2
exp
j
t A ,
2d
k 2
A exp j
u 2
f
f
2
f t A , P , exp j
u dd
2d
U f u,
d
d
ld
jld
d
Scaleable Fourier transform !
As d reduces, the scale of the transform is made smaller.
In summary, convex lens can perform Fourier transformation
The intensity at the back focal plane of the lens is therefore
proportional to the squared absolute value of the Fourier
transform of the complex amplitude of the wave at the input plane,
regardless of the distance d.
4.3
Regimes of Diffraction
Diffraction of Light
A. Fraunhofer diffraction
Aperture function :
d
b
Note,
for focusing Gaussian beam
with an infinitely large lens,
Radius :
W0'
2
l
f
f
0.64l
D
D
B. Fresnel diffraction
Spatial filtering in 4-f system
Transfer Function of the 4-f Spatial Filter With Mask Transmittance p(x, y) :
The transfer function has the same shape as the pupil function.
Impulse-response function is
4.5 Holography
If the reference wave is a uniform plane wave,
Original object wave!!
Off-axis holography
qmin qs/2 )
2qs
Spreading-angle width : qs
Assume that the object wave has a complex amplitude
Ambiguity term
Fourier-transform holography
Holographic spatial filters
IFT
Called “ Vander Lugt filter” or “Vander Lugt correlator”
Volume holography
THICK
Recording
medium
Transmission hologram :
Reflection hologram :
Volume holographic grating
kg
Grating vector
kr
kg = k0 - kr
Grating period
L = 2/ |kg|
Proof !!
Volume holographic grating = Bragg grating
Bragg condition :
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100 Mbyte/s data-transfer rate
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C. Single-lens imaging system
Impulse response function
At the aperture plane :
Beyond the lens :
Assume d1 = f
Single-lens imaging system
Transfer function
Imaging property of a convex lens
From an input point S to the output point P ;
magnification
Fig. 1.22, Iizuka
Diffraction-limited imaging of a convex lens
From a finite-sized square aperture of dimension a x a
to near the output point P ;