Fourier analysis in linear systems - Hanyangoptics.hanyang.ac.kr/~shsong/a2-Fourier analysis...
Transcript of Fourier analysis in linear systems - Hanyangoptics.hanyang.ac.kr/~shsong/a2-Fourier analysis...
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Fourier analysis in linear systems
1
( , ) ( , )exp 2 ( )
( , )
x y x y x y
x y
f x y g f f j xf yf df df
g f f
= F
( , ) ( , )exp 2 ( )
( , )x y x yg f f f x y j f x f y dxdy
f x y
F
( , ) ( , )
( , ) ( , )
FT IFT
x y
x y
f x y g f f
f x y g f f
Introduction to Fourier Optics, J. GoodmanFundamentals of Photonics, B. Saleh &M. Teich
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Properties of 1D FT
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Properties of 1D FT
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Some frequently used functions
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Some frequently used functions
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Time duration and spectral width
The power rms width(most of the measurement quantities)
The rms width
(Principles of optics 7th Ed, 10.8.3, p615)
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Time duration and spectral width
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Widths at 1/e, 3-dB, half-maximum
1
f(t)
t
= 2
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2D Fourier transform
Superposition of plane waves
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Properties of 2D FT
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Properties of 2D FT
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Properties of 2D FT
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Fourier and Inverse Fourier Transform
( , )x yf f
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Input placed
against lens
Input placed
in front of lens
Input placed
behind lens
back focal plane
Fourier Transform with Lenses
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R1>0 (concave)R2<0 (convex)
yxkyxknyx ,,, 0
yxnjkjkyxtl ,1expexp, 0
yxUyxtyxU lll ,,,'
2
2
22
221
22
10 1111,R
yxRR
yxRyx
A thin lens as a phase transformation
' ,lU x y ,lU x y
Intro. to Fourier Optics, Chapter 5, Goodman.
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The Paraxial Approximation
21
22
011
21expexp,
RRyxnjkjknyxtl
21
1111RR
nf
concave:0fconvex:0f
22
2exp, yx
fkjyxtl
Phase representation of a thin lens (paraxial approximation)
focal length
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Types of Lensesconvex:0f
concave:0f
22
2exp, yx
fkjyxtl
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Collimating property of a convex lens
Fig. 1.21, Iizuka
zi
Plane wave!
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How can a convex lens perform the FT
fo fo
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Fourier transforming property of a convex lensThe input placed directly against the lens
Pupil function ; 1 in side the lens aperture,
0 otherw iseP x y
' 2 2, , , exp2l lkU x y U x y P x y j x yf
2 2
' 2 2
exp2 2, , exp exp
2f l
kj uf kU u U x y j x y j xu y dxdyj f f f
2 2exp
2 2, , , expf l
kj uf
U u U x y P x y j xu y dxdyj f f
Quadratic phase factor
From the Fresnel diffraction formula ( z = f ):
Fourier transform
Ul Ul’
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Fourier transforming property of a convex lensThe input placed in front of the lens
2 2exp 1
2 2, , expf l
k dA j uf f
U u U x y j xu y dxdyj f f
If d = f
2, , expf lAU u U x y j xu y dxdy
j f f
Exact Fourier transform !
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df
dj
udkjA
uU f
22
2exp
, ddud
jdf
dfPtA
2exp,,
Fourier transforming property of a convex lensThe input placed behind the lens
Scaleable Fourier transform !
By decreasing d, the scale of the transform is made smaller.
, 2
exp,, 220 Atd
kjdf
dfP
dAfU
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Invariance of the input location to FT
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Imaging property of a convex lens
magnification
From an input point S to the output point P ;
Fig. 1.22, Iizuka
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Diffraction-limited imaging of a convex lens
From a finite-sized square aperture of dimension a x a to near the output point P ;
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FT in cylindrical (polar) coordinates
In rectangular coordinate
In cylindrical coordinate
( , )( , )x yr
( , )
( , )x yf f
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FT in cylindrical coordinates
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FT in cylindrical coordinates
(Ex) Circular aperture : for the special case when
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Special functions in Photonics
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Special functions in Photonics
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Special functions in Photonics
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Appendix : Linear systems
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Appendix : Shift-invariant systems
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Appendix : Linear shift-invariant causal systems
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p.180Example : The resonant dielectric medium
Susceptibility of a resonant medium :
Let,
Response to harmonic (monochromatic) fields :
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Appendix : Transfer function
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Homework
Show the FT properties