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Spatial frequency approaches

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(1)

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

(2)

Spatial frequency approaches

Spatial frequency approaches

(3)

Spatial frequency approaches Spatial frequency approaches

A general solution is

For

Æ

(4)

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 )

(5)

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

=

=

=

(6)

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

(7)

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

(8)

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

= ⎞ ⎛

⎟ ⎜

⎠ ⎝

(9)

Therefore,

(10)
(11)

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)

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