Wave Propagation in Linear Media | re-examined

ef,tria
trans,tria
evan,tria
p l = Ctria
+ Ctria
+ Ctria
p l = Ctria
+ Ctria
p l = Ctria
Z 1
1
Z 1
1
Z 1
,1
Z 1
1
Z 1
1
Z 1
,1
s +
tria ( ) cref( ) 2e j(,T 2 + k 1
p ,X , k)
,
8 Application of the quadrature routine
, e j(,T 2 +2 k 1
p ,X ,2 k) , e j(,T 2 +X ) d +
s , tria ( ) cref( ) 2e j(,T 2 , k 1
p +X , k) ,
, e j(,T 2 ,2 k 1
p +X ,2 k) , e j(,T 2 +X ) d +
s +
tria ( ) cref( ) 2e j(,T 2 + k 1
p ,X , k)
,
, e j(,T 2 +2 k 1
p ,X ,2 k) , e j(,T 2 ,X ) d
s +
tria ( ) ctun( ) 2e j(,T 2 + k 1
p
p +X 2,1, k)
,
, e j(,T 2 +2 k 1
p +X
s , tria ( ) ctun( ) 2e j(,T 2 , k 1
p ,X
p 2,1,2 k) , e j(,T 2 +X ) d +
p 2,1, k) ,
, e j(,T 2 ,2 k 1
p
p ,X 2,1,2 k) j(,T
, e 2 ,X )
d
s +
tria ( ) ctun( ) 2e j(,T 2 + k 1
p
p +X 2,1, k)
,
, e j(,T 2 +2 k 1
p +X
p 2,1,2 k) , e j(,T 2 +X ) d :
(8.20)
(8.21)
(8.22)
We distinguish four di erent parts of the wave function. Inside the tunnel we have the
evanescent and the transmitted portion, respectively. Outside the tunnel, we separate the
incident from the re ected part. For these two parts a division of the integration range into
three parts | by analogy with the tunnel | has no physical reason. Still we stick to this
structure because we then need only two modules for all parts of the wave function.
As the integrands consist of three distinct terms, the inde nite integrals are split in order
to be amenable to the quadrature function. Therefore, the computation of the transmitted
portion of the wave needs six individual quadrature calls. Obviously the oscillatory parts of
the integrands are all very similar. Inside the tunnel, they have the structure
otun( )=e j(a2 p
+b +c 2,1+d)
; (8.23)
on the outside they look like
oout( )=e j(a 2 +b +c +d) ; (8.24)
with only the signs of the parameters changing. This notice is important for the implementation
since it allows us to compute the coe cients inside the module for all function calls and
to change only their signs as they are actually passed to the quadrature function.
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