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
Properties of 1D FT
Properties of 1D FT
Some frequently used functions
Some frequently used functions
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)
Time duration and spectral width
Widths at 1/e, 3-dB, half-maximum
1
f(t)
t
= 2
2D Fourier transform
Superposition of plane waves
Properties of 2D FT
Properties of 2D FT
Properties of 2D FT
Fourier and Inverse Fourier Transform
( , )x yf f
Input placed
against lens
Input placed
in front of lens
Input placed
behind lens
back focal plane
Fourier Transform with Lenses
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.
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
Types of Lensesconvex:0f
concave:0f
22
2exp, yx
fkjyxtl
Collimating property of a convex lens
Fig. 1.21, Iizuka
zi
Plane wave!
How can a convex lens perform the FT
fo fo
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’
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 !
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
Invariance of the input location to FT
Imaging property of a convex lens
magnification
From an input point S to the output point P ;
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 ;
FT in cylindrical (polar) coordinates
In rectangular coordinate
In cylindrical coordinate
( , )( , )x yr
( , )
( , )x yf f
FT in cylindrical coordinates
FT in cylindrical coordinates
(Ex) Circular aperture : for the special case when
Special functions in Photonics
Special functions in Photonics
Special functions in Photonics
Appendix : Linear systems
Appendix : Shift-invariant systems
Appendix : Linear shift-invariant causal systems
p.180Example : The resonant dielectric medium
Susceptibility of a resonant medium :
Let,
Response to harmonic (monochromatic) fields :
Appendix : Transfer function
Homework
Show the FT properties