Fourier transform Fourier 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 y
g 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. Goodman Fundamentals of Photonics, B. Saleh &M. Teich
Properties of 1D FT Properties of 1D FT
Properties of 1D FT Properties of 1D FT
Some frequently used functions Some frequently used functions
Some frequently used functions Some frequently used functions
Time duration and spectral width Time duration and spectral width
The power rms width
(most of the measurement quantities)
The rms width
(Principles of optics 7thEd, 10.8.3, p615)
Time duration and spectral width Time duration and spectral width
Widths at 1/e, 3-dB, half-maximum Widths at 1/e, 3-dB, half-maximum
1 f(t)
t
= 2τ.
2D Fourier transform 2D Fourier transform
Superposition of plane waves
Properties of 2D FT Properties of 2D FT
Properties of 2D FT Properties of 2D FT
Properties of 2D FT Properties of 2D FT
Fourier and Inverse Fourier Transform
α α β β
( fx, fy)
Input placed against lens
Input placed in front of lens
Input placed behind lens
back focal plane
Fourier Transform with Lenses
Fourier Transform with Lenses
R1>0 (concave) R2<0 (convex)
( )x, y = knΔ( )x, y + k[Δ0 − Δ( )x, y ]
φ
( )x y [jk ] [jk(n ) ( )x y ]
tl , = exp Δ0 exp −1 Δ ,
( ) ( ) ( )x y t x y U x y
Ul' , = l , l ,
( ) ⎥⎥
⎦
⎤
⎢⎢
⎣
⎡ − − +
⎥ +
⎥⎦
⎤
⎢⎢
⎣
⎡ − − +
− Δ
=
Δ 2
2 2 2
2 2 1
2 2
1
0 1 1 1 1
, R
y R x
R y R x
y x
A thin lens as a phase transformation A thin lens as a phase transformation
( )
' ,
Ul x y
( , )
Ul x y
Intro. to Fourier Optics, Chapter 5, Goodman.
The Paraxial Approximation
( )
[ ]
( ) ⎥⎦
⎢ ⎤
⎣
⎡ ⎟⎟
⎠
⎜⎜ ⎞
⎝
⎛ −
− +
− Δ
=
2 1
2 2
0
1 1
1 2 exp
exp
, R R
y n x
jk jkn
y x tl
( ) ⎟⎟
⎠
⎜⎜ ⎞
⎝
⎛ −
−
≡
2 1
1 1 1
1
R n R
f
concave :
< 0 f
convex :
> 0 f
( ) ( )⎥
⎦
⎢ ⎤
⎣
⎡− +
= 2 2
exp 2
, x y
f j k
y x tl
Æ Phase representation of a thin lens (paraxial approximation) focal length
Types of Lenses
convex :
> 0 f
concave :
< 0 f
( )
( )
⎥⎦
⎢ ⎤
⎣
⎡− +
= 2 2
exp 2
, x y
f j k y
x tl
Collimating property of a convex lens Collimating property of a convex lens
Fig. 1.21, Iizuka
zi
Plane wave!
How can a convex lens perform the FT How can a convex lens perform the FT
fo fo
Fourier transforming property of a convex lens Fourier transforming property of a convex lens
The input placed directly against the lens
Pupil function ; ( ) 1 in sid e th e len s a p ertu re , 0 o th erw ise
P x y ⎧
= ⎨⎩
( ) ( ) ( ) ( )
' 2 2
, , , exp
l l 2
U x y U x y P x y j k x y f
⎡ ⎤
= ⎢− + ⎥
⎣ ⎦
( ) ( )
( ) ( ) ( )
2 2
' 2 2
exp 2 2
, , exp exp
f l 2
j k u
f k
U u U x y j x y j xu y dxdy
j f f f
υ π
υ υ
λ λ
∞
−∞
⎡ ⎤
⎢ + ⎥ ⎡ ⎤ ⎡ ⎤
⎣ ⎦
= ⎢ + ⎥ ⎢− + ⎥
⎣ ⎦ ⎣ ⎦
∫ ∫
( ) ( )
( ) ( ) ( )
2 2
exp 2 2
, , , exp
f l
j k u
U u f U x y P x y j xu y dxdy
j f f
υ π
υ υ
λ λ
∞
−∞
⎡ + ⎤
⎢ ⎥ ⎡ ⎤
⎣ ⎦
= ⎢− + ⎥
⎣ ⎦
∫ ∫
Quadratic phase factor From the Fresnel diffraction formula ( z = f ):
Fourier transform
Ul Ul’
Fourier transforming property of a convex lens Fourier transforming property of a convex lens
The input placed in front of the lens
( ) ( )
( ) ( )
2 2
exp 1
2 2
, , exp
f l
k d
A j u
f f
U u U x y j xu y dxdy
j f f
υ π
υ υ
λ λ
∞
−∞
⎡ ⎛ ⎞ ⎤
− +
⎢ ⎜⎝ ⎟⎠ ⎥ ⎡ ⎤
⎣ ⎦
= ⎢− + ⎥
⎣ ⎦
∫ ∫
If d = f
( ), ( ), exp 2 ( )
f l
U u A U x y j xu y dxdy
j f f
υ π υ
λ λ
∞
−∞
⎡ ⎤
= ⎢− + ⎥
⎣ ⎦
∫ ∫
Exact Fourier transform !
( ) ( )
d f d
j d u j k A
u U f
λ
υ
υ ⎥⎦
⎢⎣ ⎤
⎡ +
=
2 2
exp 2
, ( ) ( ξ υη) ξ η
λ η π
ξ η
ξ u d d
j d d
f d
P f tA
⎥⎦⎤
⎢⎣⎡− +
⎟⎠
⎜ ⎞
⎝
×∫ ∫∞ ⎛
∞
−
exp 2 ,
,
Fourier transforming property of a convex lens Fourier transforming property of a convex lens
The input placed behind the lens
Scaleable Fourier transform !
By decreasing d, the scale of the transform is made smaller.
( )ξ η ξ η (ξ η ) ( )ξ,η
exp 2 ,
, 2 2
0 tA
d j k d
f d
P f d U Af
⎭⎬
⎫
⎩⎨
⎧ ⎥⎦⎤
⎢⎣⎡− +
⎟⎠
⎜ ⎞
⎝
= ⎛
Invariance of the input location to FT Invariance of the input location to FT
Imaging property of a convex lens 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 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 FT in cylindrical (polar) coordinates
In rectangular coordinate
In cylindrical coordinate ( , )
( , ) x y r θ
( , ) ( , )
x y
f f ρ φ
FT in cylindrical coordinates FT in cylindrical coordinates
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 Special functions in Photonics
Special functions in Photonics Special functions in Photonics
Appendix : Linear systems Appendix : Linear systems
Appendix : Shift-invariant systems Appendix : Shift-invariant systems
Appendix : Linear shift-invariant causal systems Appendix : Linear shift-invariant causal systems
p.180
Example : The resonant dielectric medium
Susceptibility of a resonant medium :
Let, Response to harmonic (monochromatic) fields :
Appendix : Transfer function Appendix : Transfer function
