My PhD Thesis, PDF 3MB - Stanford University

the potentials in jth ( j ) ( j )
radial layer by a set of eigen-functions, { f n ( z ) , g n ( z ) ;
n 1, 2 , . .. , N }, as
where the eigenfunctions f n
( j)
n
and n
( j)
? ( j)
(r , z )
( j )
? ( r , z )
d 2
( j )
( j )
f ( z ) n
dz 2
d 2 ( j )
g n
dz 2
( z )
N
? ( j ) ( j )
( r ) f ( z ) , (3.2a)
n
n
n 1
N
( j ) ( j )
? (r ) g ( z ) . (3.2b)
n
n
n 1
( z ) and g n
( j )
( j )
[ k ( z ) ]
2
( z ) satisfy
( j )
( j )
f ( z ) [ ] n
n
2
- 59 -
( j )
f ( z ) , ( 3. 3a )
n
( j )
[ k ( z ) ]
2 ( j )
( j )
g ( z ) [ ] n
n
2 ( j )
g ( z ) , ( 3. 3 b )
n
are the corresponding eigenvalues to be determined. Solving equation (3.3)
is an eigenvalue problem. The detailed procedures of solving equation (3.3) will be
described in Section 3.6. I here give its solutions as follows:
( j )
f n
( j)
g n
2 N 1
( j )
( l, n ) exp [ik z ] , l (3.4a)
( z ) a p
l 1
2 N 1
( j )
(l , n ) exp [ ik z ] , l (3.4b)
(z ) a s
l 1
where, k l 2 ( l N 1) L is l th vertical wavenumber ( l 1, 2 , .. ., 2 N 1 ); L is the
periodic length (see Section 3.6); { a p
the following constrains:
( j )
( j ) ( j )
f ( z ) f n
m
L / 2
( z ) dz
L / 2
( l , n ) } and { a s
( j )
( l , n ) } are normalized by imposing
nm , (3.5a)
( j ) ( j )
g ( z ) g n
m
L / 2
( z ) dz
L / 2
nm . (3.5b)