Spatial frequency approaches

Spatial Frequency Approaches Spatial Frequency Approaches

and and

( )

, ; 0 E x y_o, _o, 0 exp j2 x_o y_o dx dy_{o 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 y_o, _o, 0 exp j2 x_o y_o dx dy_{o 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

(

)

, ; 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

∞

−∞

⎛ ⎞ ⎛ ⎞

= ⎜ ⎟ ⎜ − − ⎟

⎝ ⎠ ⎝ ⎠

∫ ∫

(

)

⎦

⎢ ⎤

⎣

⎡ ⎟

⎠

⎜ ⎞

⎝

⎛ +

+

×circ ^{2 2} exp 2

2 1

2 : evanescent waves α + β >

(

)

, ; 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

(

) (

) (

)

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

U_{h h} 2 ( )

exp

; , ,

,

on (x,y) plane

(

)

(

) [

]

U , , +δ = _h , , + δ exp − δ ( ) _oδ

Invalid for wide angle propagation -> WPM (wave propagation method)