38
3 Study of Electrophysical Characteristics of Blood …
Expressions (3.43) and (3.46) establish relationships of the coefficients for the sphere
that is located at the center of the coordinate system and the sphere that is shifted
relative to the center.
The boundary conditions are written as
[E I 1 ( j) − E I 2 ( j)]| r 2 =b × e r = 0, [H I 1 ( j) − H I 2 ( j)]| r 2 =b × e r = 0,
(3.47)
[E i ( j) + E s ( j) − E I 1 ( j)]| r 1 =a × e r = [H i ( j) + H s ( j) − H I 1 ( j)] r 1 =a × e r (3.48)
E I 1θ ( j)| r 2 =b = E I 2 θ ( j)| r 2 =b , E I 1 φ ( j)| r 2 =b = E I 2 φ ( j)| r 2 =b ,
(3.49)
H I 1θ ( j)| r 2 =b = H I 2 θ ( j)| r 2 =b , H I 1φ ( j))| r 2 =b = H I 2 φ ( j)| r 2 =b ,
(3.50)
E iθ ( j)| r 1 =a + E sθ ( j)| r 1 =a = E I 1θ ( j)| r 1 =a ,
(3.51)
E iφ ( j)| r 1 =a + E sφ ( j)| r 1 =a = E I 1φ ( j)| r 1 =a ,
(3.52)
H iθ ( j)| r 1 =a + H sθ ( j)| r 1 =a = H I 1θ ( j)| r 1 =a ,
(3.53)
H iφ ( j)| r 1 =a + H sφ ( j)| r 1 =a = H I 1φ ( j)| r 1 =a .
(3.54)
Substituting the expressions for the fields that are expanded in terms of vector spherical harmonics with allowance for relationships (3.43)−(3.46) in boundary conditions
(3.49)−(3.54) and taking into account the orthogonality of the spherical harmonics,
we derive a system of equations for unknown coefficients. Note that the scattering
coefficients that are found using this system can be represented as
a
j
mn = a
j
n 1 p
p
j
mn + a
j
n 1q
q
j
mn , b
j
mn = b
j
n 1 p
p
j
mn + b
j
n 1q
q
j
mn ,
(3.55)
where
b
j
n 1 p
= −
b
j
n 11 p
b
j
n 21
, b
j
n 1q
= −
b
j
n 11q
b
j
n 21
,
b
j
n 11 p
= (A
m
n n )
(2)
ψ n ((ka)
j
)ξ
(2)
n ((ka)
j
)ξ
(1)
n ((k 1 a)
j
)F 21 ξ
(1)
n ((k 1 a)
j
)+
+(A
m
n n )
2
ψ n ((ka)
j
)F 12 ξ
(1)
n ((k 1 a)
j
)ξ
(1)
n ((ka)
j
)ξ
(2)
n ((k 1 a)
j
)+
+(A
m
n n )
2
ψ n ((ka)
j
)F 12 (ξ
(1)
n ((k 1 a)
j
))
2
ξ
(1)
n ((ka)
j
)F 21 −
−(A
m
n n )
2
ψ n ((ka)
j
)ξ
(2)
n ((k 1 a)
j
)ξ
(1)
n ((ka)
j
)F 21 ξ
(1)
n ((k 1 a)
j
)−
−(A
m
n n )
2
ψ n ((ka)
j
)F 12 ξ
(1)
n ((k 1 a)
j
)ξ
(1)
n ((ka)
j
)ξ
(2)
n ((k 1 a)
j
)−