Wave Propagation in Linear Media | re-examined

8.1 Preparation of the wave integrals for quadrature
For rectangular initial waves, we nd from (8.4) and (8.1 { 8.3)
and
inc,rect
ref,rect
trans,rect
evan,rect
p l = Crect
+ Crect
+ Crect
p l = Crect
+ Crect
+ Crect
p l = Crect
+ Crect
p l = Crect
Z 1
1 Z 1
1 Z 1
,1
Z 1
1 Z 1
1 Z 1
,1
Z 1
1
Z 1
1
Z 1
,1
Crect =
j
p
2
(8.25)
s +
1
rect ( )=
1, 1 p
(8.26)
s , rect ( )=
1
1+ 1
p
s +
h j(,T
rect ( ) e 2 +2 k 1 p +X ,2 k) j(,T
, e 2 i
+X )
d +
s , rect ( )
s +
rect ( )
h j(,T
e 2 ,2 k 1
p ,X ,2 k) j(,T
, e 2 i
,X )
d +
h j(,T
e 2 +2 k 1
p +X ,2 k) j(,T
, e 2 i
+X )
d
s +
rect ( ) c h j(,T
ref( ) e 2 +2 k 1 p ,X ,2 k) j(,T
, e 2 i
,X )
d +
h j(,T
e 2 ,2 k 1 p +X ,2 k) j(,T
, e 2 i
+X )
d +
s , rect ( ) c ref( )
s +
rect ( ) c ref( )
h j(,T
e 2 +2 k 1
p ,X ,2 k) j(,T
, e 2 i
,X )
d
s +
rect ( ) ctun( ) e j(,T 2 +2 k 1
p
p +X 2,1,2 k)
, e j(,T 2 +X ) d +
s , rect ( ) ctun( ) e j(,T 2 ,2 k 1
p
p ,X 2,1,2 k)
, e j(,T 2 ,X ) d
s +
rect ( ) ctun( ) e j(,T 2 +2 k 1
p
p +X 2,1,2 k)
, e j(,T 2 +X ) d :
(8.27)
(8.28)
(8.29)
(8.30)
(8.31)
The basic structure of the integrals is the same as before, however in this case the computation
of the transmitted wave needs only four calls of the quadrature routine. Like with the
triangular wave, it is su cient towrite one driver module for all parts of the wave function
and alter only the parameters in the quadrature call.
195