Spatial Frequency Approaches Spatial Frequency Approaches
and and
Angular SpectrumAngular Spectrum( )
, ; 0 E x yo, o, 0 exp j2 xo yo dx dyo o
α β α β
ε π
λ λ λ λ
∞
−∞
⎡ ⎤
⎛ ⎞ = − ⎛ + ⎞
⎜ ⎟ ⎢ ⎜ ⎟⎥
⎝ ⎠ ∫ ∫ ⎣ ⎝ ⎠⎦
Angular Spectrum
Directional cosines (α, β, γ )
Spatial spectrum
Spatial frequency approaches
Spatial frequency approaches
Spatial frequency approaches Spatial frequency approaches
A general solution is
For
Æ
Angular Spectrum Angular Spectrum
( )2 ( )2
1
X Y
X Y
f f
f f
α λ
β λ
γ λ λ
=
=
= − −
( )
, ; 0 E x yo, o, 0 exp j2 xo yo dx dyo o
α β α β
ε π
λ λ λ λ
∞
−∞
⎡ ⎤
⎛ ⎞ = − ⎛ + ⎞
⎜ ⎟ ⎢ ⎜ ⎟⎥
⎝ ⎠
∫ ∫
⎣ ⎝ ⎠⎦Angular Spectrum
Directional cosines (α, β, γ )
Propagation of the Angular Spectrum Propagation of the Angular Spectrum
(
2 2)
, ; z , ; 0 exp j 2 1 z
α β α β π
ε ε α β
λ λ λ λ λ
⎛ ⎞ ⎛ ⎞ ⎧ ⎫
∴ ⎜ ⎟ = ⎜ ⎟ ⎨ − − ⎬
⎝ ⎠ ⎝ ⎠ ⎩ ⎭
( )2 ( )2
1
X Y
X Y
f f
f f
α λ
β λ
γ λ λ
=
=
= − −
Propagation of the Angular Spectrum Propagation of the Angular Spectrum
2 1
2 : each plane-wave component propagates at a different angle α +β <
( , , ) , ; 0 exp 2 1 2 2
E x y z ε α β j π α β z
λ λ λ
∞
−∞
⎛ ⎞ ⎛ ⎞
= ⎜ ⎟ ⎜ − − ⎟
⎝ ⎠ ⎝ ⎠
∫ ∫
(
α β)
j π αλ x λβ y ⎥d αλ d λβ⎦
⎢ ⎤
⎣
⎡ ⎟
⎠
⎜ ⎞
⎝
⎛ +
+
×circ 2 2 exp 2
2 1
2 : evanescent waves α + β >
(
2 2)
, ; z , ; 0 exp j 2 1 z
α β α β π
ε ε α β
λ λ λ λ λ
⎛ ⎞ = ⎛ ⎞ ⎧⎨ − − ⎫⎬
⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎩ ⎭
For wave propagation
Effect of Aperture in Angular spectra Effect of Aperture in Angular spectra
( ) ( )
( )
, ; 0
, , ; 0
t A
i
Amplitude function of an aperture E x y
t x y
E x y
=
, , ,
t α β i α β T α β
ε ε
λ λ λ λ λ λ
⎛ ⎞ = ⎛ ⎞ ⊗ ⎛ ⎞
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
• For the case of a unit amplitude plane wave incidence,
, ,
i
α β α β
ε δ
λ λ λ λ
⎛ ⎞ = ⎛ ⎞
⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
, , , ,
t α β α β T α β T α β
ε δ
λ λ λ λ λ λ λ λ
⎛ ⎞ = ⎛ ⎞⊗ ⎛ ⎞ = ⎛ ⎞
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
By convolution theorem
Transfer function in free
Transfer function in free--space propagationspace propagation -- a linear spatial filter a linear spatial filter --
( X, Y; ) ( X, Y;0 exp) 2 1 ( X ) ( )2 Y 2
f f z f f j z f f
ε ε π λ λ
λ
⎡ ⎤
= ⎢⎣ − − ⎥⎦
( )
( ) ( )⎪⎪
⎩
⎪⎪⎨
⎧ ⎥⎦⎤
⎢⎣⎡ − −
= 0
1 2
exp ,
2 2
Y X
y X
f z f
j f
f H
λ λ λ
π λ
2 1
2 + Y 〈
X f
f
otherwise
Transfer function of wave propagation phenomenon in free space
(
fx, f zy;) (
fx, fy; 0) (
H fx, fy)
ε = ε
, ;z , ; 0 H ,
α β α β α β
ε ε
λ λ λ λ λ λ
⎛ ⎞ = ⎛ ⎞ ⎛ ⎞
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
Therefore,
BPM (Beam Propagation Method) BPM (Beam Propagation Method)
( , , )] [
)
(z average n x y z
n =
⎟⎟⎠
⎞
⎜⎜⎝
⎛ − −
⎟⎠
⎜ ⎞
⎝
= ⎛
⎟⎠
⎜ ⎞
⎝
⎛ , ; + , ; exp ( )2 2 (2 )2 (2 )2
λ π β λ
π α λ δ
β λ δ α
λ β λ α
o
h z z A z j z n z k
A
( ) ⎟
⎠
⎜ ⎞
⎝
⎟ ⎛
⎠
⎜ ⎞
⎝
⎟ ⎛
⎠
⎜ ⎞
⎝
⎛ +
⎟⎠
⎜ ⎞
⎝
⎛ +
= + ∫ ∫∞
∞
− λ
β λ
β α λ α
δ π λ
β λ
δz A α z z j x y d d
z y x
Uh h 2 ( )
exp
; , ,
,
on (x,y) plane
(
x y z z)
U(
x y z z) [
j n z k z]
U , , +δ = h , , + δ exp − δ ( ) oδ
Invalid for wide angle propagation -> WPM (wave propagation method)