Project 14 2 1 839G1.22.pdf


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Limiting the discussion to fields of the mathematical form of:
U r, t   a r  exp  j r  exp  j 2 t  ,
the wave equation, then, degenerates into Helmholtz equation (Eq.6





1
2 E  k 2 E  0 (k is the wave number and it satisfies k   2  2 )6).
c
c







1
6 2 E  k 2 E  0 (k is the wave number and it satisfies k   2  2 )
c
c


Now, given some abstract distribution of an electro-magnetic field at plane
1
z=0, u  x, y;0 , it can be shown , using the Helmholtz equation (Eq.6) , that
the distribution at plane z=z‟, u  x, y; z ' must satisfy,



 z  



       

; x 



, y  ,



   
 2
2
2
 A   ,  ;0  exp  j  1    



7 u  x, y; z     


       

 
exp  j 2  x  y   d   d  
       



 






whereas,
 

8 u  x, y;0  

 





  A   ,  ;0  exp  j2   x   y  d    d   

 

where,   sin  x 

ky
kx
,   sin  y 
k
k

and, A(νx,νy; 0) is referred to as, the Angular Spectrum of u(x,y;0),
 

 





 

9 A  , ;0     u  x, y;0  exp  j 2  x 

     



y   dxdy


Alternatively:




   
2


10 u  x, y; z     A  ,   exp  j
1  2   2 z
 
 
  
 
 
Angular
Spectrum


  
H  , ; z  transfer function of free space
  
of u  x , y ;0

where  in Eq.10 is the 2 dimensional Fourier Transform such that for any





2-dimensional function, g, it‟s Fourier transform is given by G

1

  1 2   2