Fourier transformFourier transform - Hanyangoptics.hanyang.ac.kr/~shsong/3-Fourier...
Transcript of Fourier transformFourier transform - Hanyangoptics.hanyang.ac.kr/~shsong/3-Fourier...
Fourier transformFourier transform
{ }{ }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
π+∞−∞
−
= +∫ ∫
= F
{ }{ }
( , ) ( , )exp 2 ( )
( , )x y x yg f f f x y j f x f y dxdy
f x y
π+∞−∞= − +
=
∫ ∫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
Time duration and spectral widthTime 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)
Remind ! Spatial frequency and propagation angleRemind ! Spatial frequency and propagation angle
z
directional cosine : xα λν=
1
xνΛ =
FT in cylindrical (polar) coordinatesFT in cylindrical (polar) coordinates
In rectangular coordinate
In cylindrical coordinate
( , )( , )x yr θ
( , )
( , )x yf f
ρ φ
FT in cylindrical coordinatesFT in cylindrical coordinates
(Ex) Circular aperture : for the special case when
Input placed
against lens
Input placed
in front of lens
Input placed
behind lens
back focal plane
Fourier Transform with LensesFourier Transform with Lenses
R1>0 (concave)R2<0 (convex)
( ) ( ) ( )[ ]yxkyxknyx ,,, 0 Δ−Δ+Δ=φ
( ) [ ] ( ) ( )[ ]yxnjkjkyxtl ,1expexp, 0 Δ−Δ=
( ) ( ) ( )yxUyxtyxU lll ,,,' =
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡ +−−+
⎥⎥⎦
⎤
⎢⎢⎣
⎡ +−−−Δ=Δ 2
2
22
221
22
10 1111,R
yxRR
yxRyx
A thin lens as a phase transformationA thin lens as a phase transformation
( )' ,lU x y( ),lU x y
Intro. to Fourier Optics, Chapter 5, Goodman.
The Paraxial Approximation
( ) [ ] ( ) ⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−
+−−Δ=
21
22
011
21expexp,
RRyxnjkjknyxtl
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛−−≡
21
1111RR
nf
concave:0<fconvex:0>f
( ) ( )⎥⎦
⎤⎢⎣
⎡+−= 22
2exp, yx
fkjyxtl
Phase representation of a thin lens (paraxial approximation)
focal length
Collimating property of a convex lensCollimating property of a convex lens
Fig. 1.21, Iizuka
zi
Plane wave!
Fourier transforming property of a convex lensFourier 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
υπυ υ
λ λ
∞
−∞
⎡ ⎤+⎢ ⎥ ⎡ ⎤ ⎡ ⎤⎣ ⎦= + − +⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦∫ ∫
( )( )
( ) ( ) ( )2 2exp
2 2, , , expf l
kj uf
U u U x y P x y j xu y dxdyj f f
υπυ υ
λ λ
∞
−∞
⎡ ⎤+⎢ ⎥ ⎡ ⎤⎣ ⎦= − +⎢ ⎥
⎣ ⎦∫ ∫
Quadratic phase factor
From the Fresnel diffraction formula ( z = f ):
Fourier transform
Fourier transforming property of a convex lensFourier 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
υπυ υ
λ λ
∞
−∞
⎡ ⎤⎛ ⎞− +⎢ ⎥⎜ ⎟ ⎡ ⎤⎝ ⎠⎣ ⎦= − +⎢ ⎥
⎣ ⎦∫ ∫
If d = f
( ) ( ) ( )2, , expf lAU u U x y j xu y dxdy
j f fπυ υ
λ λ
∞
−∞
⎡ ⎤= − +⎢ ⎥
⎣ ⎦∫ ∫
Exact Fourier transform !
( )( )
df
dj
udkjA
uU f λ
υυ
⎥⎦⎤
⎢⎣⎡ +
=
22
2exp
, ( ) ( ) ηξυηξλπηξηξ ddud
jdf
dfPtA ⎥⎦
⎤⎢⎣⎡ +−⎟
⎠⎞
⎜⎝⎛× ∫ ∫
∞
∞−
2exp,,
Fourier transforming property of a convex lensFourier 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.
( ) ( ) ( )ηξηξηξηξ , 2
exp,, 220 Atd
kjdf
dfP
dAfU
⎭⎬⎫
⎩⎨⎧
⎥⎦⎤
⎢⎣⎡ +−⎟
⎠⎞
⎜⎝⎛=
Imaging property of a convex lensImaging 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 lensDiffraction-limited imaging of a convex lens
From a finite-sized square aperture of dimension a x a to near the output point P ;
p.180Example : The resonant dielectric medium
Susceptibility of a resonant medium :
Let,
Response to harmonic (monochromatic) fields :