Spatial frequency approaches

Angular spectra in beam propagation

 

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

   

 

   





 

      

    

      

Angular Spectrum

Directional cosines ()

Spatial spectrum

Introduction to Fourier Optics, J. Goodman, 3.10, p.55

Spatial frequency approaches

Spatial frequency approaches

A general solution is

For



In terms of angular spectrum,

 ²  ²

1

X Y

X Y

f f

f f

 

 

  





  

 

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

   

 

   





 

      

    

 

 

Angular Spectrum

Directional cosines ()

Propagation of the Angular Spectrum





, ; z , ; 0 exp j 2 1 z

    

   

    

     

         

     

 ²  ²

1

X Y

X Y

f f

f f

 

 

  





  

Propagation of the Angular Spectrum

2 1

2 : each plane-wave component propagates at a different angle

  

 ^{, ,}  ^{, ; 0 exp 2 1 2 2}

E x y z    j    z

  





   



 







 



 



 



 



circ ^{2 2} exp 2

2 1

2 : evanescent waves

   





, ; z , ; 0 exp j 2 1 z

    

   

    

        

   

     

For wave propagation

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   i   T  

 

     

      

     

     

• For the case of a unit amplitude plane wave incidence,

, ,

i

   

 

   

    

   

   

, , , ,

t     T   T  

 

       

          

       

       

By convolution theorem

Transfer function in free-space propagation - 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

    



 

    

 







 

  

 0

1 2

exp ,

2 2

Y X

y X

f z f

j f

f H



 

 

2 1

2  _Y 

X f

f

otherwise

Transfer function of wave propagation phenomenon in free space



 

 



  

, ;z , ; 0 H ,

     

 

     

      

     

     

Therefore,

(Demonstrate the spatial-frequency approach and the spatial-coordinate solution provide the same diffraction disturbance.)

In the case of the evanescent wave,

but the amplitude decays as z increases.

BPM (Beam Propagation Method)

 ^{, ,} ^]

[ )

(z average n x y z

n 







  



 



 



 



 , ;  , ; exp ( )^{2 2} (2 )² (2 )²



 



 

 





 









o

h z z A z j z n z k

A

  



 



 



 



 



 



 



 



 



  ^



 





 

 

 







z A  z z j x y d d

z y x

U_{h h} 2 ( )

exp

; , ,

,

on (x,y) plane







 



U , ,    _h , ,   exp   ( ) _o

Note that it is invalid for wide angle propagation -> WPM (wave propagation method)





, ;z , ; 0 exp j 2 1 z

    

   

    

       

   

      ^n(x,y,z) _z

z