Page 134 - Bio Engineering Approaches to Cancer Diagnosis and Treatment

132 CHAPTER 6 Laser-assisted cancer treatment

6.1.1 Mie theory
This theory was introduced by Gustav Mie in 1908 [1]. This model is the analytical
solution of Maxwell’s equations to calculate the scattering properties of spherical
and nonmagnetic particles. Maxwell equations are:
( ) t,
∂Er
( ) =t,
∇ × Hr ε () r , (6.1)
∇×Hr,t=εpr∂Er,t∂t, p ∂t
∂Hr
( ) t,
∇ × Er µ() r , (6.2)
( ) = −t,
∇×Er,t=−µr∂Hr,t∂t, ∂t
( ) =t,
∇⋅Er,t=0, ∇⋅Er 0, (6.3)
( ) =t,
∇⋅Hr,t=0, ∇⋅ Hr 0, (6.4)
εp µ where H, E, ε , and µ are the magnetic field, the electric field, the electric permittiv-
p
ity, and the magnetic permeability, respectively. By applying Fourier transform to the
Eq. (6.4), the time variable in electromagnetic field meets the Helmholtz equations
[1]:
2
2
2
∇ E+ k12E=0, ∇ E + k E = 0, (6.5)
1
2
2
∇ H+k12E=0, ∇ H + k E = 0, (6.6)
2
1
() =
2
ε
r
k12r=wrεprµrC w where wave vector is kr ω () () r µ() r while ω and c are the frequency and
2
1
p
C 2
speed of light, respectively. By assuming the spherical harmonics shapes, E and
H are replaced by vectors M and N . The subscript, n, represents different vec-
n
n
tor spherical harmonics to describe the polar contribution to the scattered fields
(n = 1 dipolar, n = 2 quadripolar). The scattered field is calculated using series
below [1]:
∞
E = ∑ ( − bM ), (6.7)
EiaN
Esca=∑n=1∞EnianNein−bnMoin, sca n =1 n n ein n oin
∞
H sca = ∑ (ib N oin − a M ein ), (6.8)
E
n
n
n
Hsca=∑n=1∞EnibnNoin−anMein, n =1
 2 n +1 
En=inE 2n+1nn+1 where E = iE and E is the incident filed. The subscript of o and e are odd
n
( +1
0 n 0   nn )  0
and even parameter of the azimuthal solution to the vector. The a and b represent
n
n
Mie coefficients. These values are calculated as fallow [1]:
ψ ( )
ψ () −
m ψ ( ) ′ x ψ () ′ mx
mx
x
a n = n n n n (6.9)
ψ ( )
x
ξ′ x
mx
an=mψnmxψ′nx−ψnxψ′nmxmψnmxξ′nx−ξnxψ′nmx m ψ ( ) () − ξ () ′ mx
n
n
n
n
ψ ( )
x
mx
ψ () − m
ψ ( ) ′ x ψ () ′ mx
b n = n n n n (6.10)
ψ ( )
ξ′ x
x
mx
bn=ψnmxψ′nx−mψnxψ′nmxψnmxξ′nx−ξnxψ′nmx ψ ( ) () − ξ () ′ mx
n
n
n
n