Файл: Doicu A., Wriedt T., Eremin Y.A. Light scattering by systems of particles (OS 124, Springer, 2006.pdf

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102 2 Null-Field Method

where

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

= −

mrAn (mrx) + n jn (x)

−

jn

−

1 (x)

(2.46)

 

 

 

 

 

 

 

x

 

 

 

Tn

 

 

 

 

 

mrAn (mrx) + nx hn(1) (x) − hn(1)−1 (x)

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

An (mrx)

n

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

+ x

jn (x) − jn−1 (x)

 

 

2

= −

 

 

mr

 

(2.47)

Tn

 

 

 

 

 

 

 

 

.

An (mrx)

 

n

(1)

(1)

 

 

 

 

 

 

 

 

 

+ x

hn (x) − hn−1 (x)

 

 

 

 

 

 

 

mr

 

Equations (2.46) and (2.47) relating the transition matrix to the size parameter and relative refractive index are identical to the expressions of the Lorenz–Mie coe cients given by Bohren and Hu man [17].

2.2 Homogeneous and Chiral Particles

The problem of scattering by isotropic, chiral spheres has been treated by Bohren [16], and Bohren and Hu man [17] using rigorous electromagnetic field-theoretical calculations, while the analysis of nonspherical, isotropic, chiral particles has been rendered by Lakhtakia et al. [135]. To account for chirality, the surface fields have been approximated by leftand right-circularly polarized fields and the same technique is employed in our analysis. The transmission boundary-value problem for a homogeneous and isotropic, chiral particle has the following formulation.

Given Ee, He as an entire solution to the Maxwell equations representing the external excitation, find the vector fields Es, Hs and Ei, Hi satisfying the Maxwell equations

× Es = jk0µsHs ,

 

× Hs = −jk0εsEs

(2.48)

in Ds, and

 

 

 

 

 

 

 

 

 

Ei

 

 

Ei

 

 

× Hi

 

 

= K

Hi

 

(2.49)

in Di, where

 

 

 

 

 

 

 

 

K =

1

 

 

 

βki2

jk0µi

(2.50)

 

 

 

−jk0εi βki2 .

1 − β2ki2

 

In addition, the vector fields must satisfy the transmission conditions (2.2) and the Silver–M¨uller radiation condition (2.3).

Applications of the extinction theorem and Huygens principle yield the null-field equations (2.6) and the integral representations for the scattered field coe cients (2.16). Taking into account that the electromagnetic fields propagating in an isotropic, chiral medium can be expressed as a superposition of vector spherical wave functions of leftand right-handed type (cf. Sect. 1.3), we represent the approximate surface fields as

2.2 Homogeneous and Chiral Particles

103

$ eiN (r ) %

N N

 

 

n (r ) × Lµ (kLir )

 

 

= cµ

 

 

 

 

 

 

 

 

hN (r )

 

j

 

εi n (r )

×

Lµ (kLir )

i

µ=1

−

 

µi

 

 

 

 

 

N

n (r ) × Rµ (kRir )

 

 

+dµ j

 

 

 

 

 

 

(k r ) ,

 

 

εi n (r )

×

R

µ

 

 

 

µi

 

 

 

 

 

Ri

 

where Lµ and Rµ are given by (1.58) and (1.59), respectively, and

kLi =

ki

, kRi =

ki

1 − βki

1 + βki

are the wave numbers of the leftand right-handed type waves. The transition matrix of a homogeneous, chiral particle then becomes

 

 

T =

−

Q11

(ks, ki) Q31

(ks, ki) −1 ,

 

 

 

 

chiral

chiral

 

 

 

where, for µi = µs, the elements of the Qchiral31

matrix are given by

31

11

 

jks2

 

3

 

Qchiral

νµ

=

 

 

S [n (r ) × Lµ (kLir )] · N

 

(ksr )

π

 

ν

Q31chiral 12νµ =

Q31chiral 21νµ =

and

Q31chiral 22νµ =

 

 

εi

 

 

 

 

 

 

 

 

3

 

 

 

 

 

 

 

"

 

+

 

 

 

 

 

 

 

 

 

[n (r )

× Lµ (kLir )] · M

 

 

 

 

 

 

(ksr )

dS (r ) , (2.51)

 

εs

ν

jks2

 

 

 

 

 

 

 

 

 

 

3

 

 

 

 

 

 

 

 

 

 

 

 

 

S [n (r ) × Rµ (kRir )] · N

 

 

 

 

 

 

(ksr )

 

 

 

π

 

ν

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

"

 

 

 

εi

 

 

 

 

 

 

 

 

 

 

 

3

 

 

 

 

 

 

−

 

 

 

 

[n (r )

× Rµ (kRir )] · M

 

 

 

 

 

(ksr )

dS (r ) , (2.52)

εs

ν

 

 

 

jks2

 

 

 

 

 

 

 

 

 

 

3

 

 

 

 

 

 

 

 

 

 

S [n (r ) × Lµ (kLir )] · M

 

 

 

 

(ksr )

 

 

 

π

 

ν

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

r )" dS (r ) , (2.53)

 

+

εi

 

[n (r )

×

L (k r )]

·

N

3

 

 

(k

 

 

 

 

 

 

 

 

εs

 

 

µ

Li

 

ν

 

 

 

 

s

 

 

 

jks2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3

 

 

 

 

 

 

 

 

S [n (r ) × Rµ (kRir )] · M

 

 

(ksr )

 

 

 

π

 

 

ν

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(k r )" dS (r ) . (2.54)

 

 

 

εi

[n (r )

×

R

 

(k r )]

·

N

3

 

 

−

 

 

 

 

 

 

 

 

 

 

 

 

 

εs

 

 

 

µ

Ri

 

 

ν

 

 

 

 

s

 

The expressions of the elements of the Q11chiral matrix are similar but with M 1ν and N 1ν in place of M 3ν and N 3ν , respectively. It must be noted that in case the

104 2 Null-Field Method

particle is spherical, the transition matrix becomes diagonal, and the resulting solution tallies exactly with that given by Bohren [16] for chiral spheres. In addition, if β = 0, i.e., the particle becomes a nonchiral sphere, the Lorenz– Mie series solution is obtained. Finally, simply by setting kLi = kRi = ki and β = 0, the solution for a nonspherical, nonchiral particles is recovered. In our analysis, the ambient medium is nonchiral and a generalization of the present approach to the scattering by a chiral particle in a chiral host medium has been addressed by Lakhtakia [128].

2.3 Homogeneous and Anisotropic Particles

The scattering by anisotropic particles is mostly restricted to simple shapes such as cylinders [232] or spheres [265]. Liu et al. [143] solved the electromagnetic fields in a rotationally uniaxial medium by using the method of separation of variables, while Piller and Martin [193] analyzed three-dimensional anisotropic particles by using the generalized multipole technique. In our analysis we follow the treatment of Kiselev et al. [119] which solved the scattering problem of radially and uniformly anisotropic spheres by using the vector quasi-spherical wave functions for internal field representation. The transmission boundary-value problem for a homogeneous and uniaxial anisotropic particle has the following formulation.

Given Ee, He as an entire solution to the Maxwell equations representing the external excitation, find the vector fields Es, Hs and Ei, Hi satisfying the Maxwell equations

× Es = jk0µsHs , × Hs = −jk0εsEs

(2.55)

in Ds, and

 

 

 

 

 

× Ei = jk0Bi , × Hi = −jk0Di ,

 

· Bi = 0 ,

 

· Di

= 0

(2.56)

in Di, where

 

 

 

 

 

Di =

 

iEi , Bi = µiHi ,

(2.57)

ε

and

 

 

 

 

 

 

 

 

 

εi 0

0

 

 

 

 

i

= 0 εi

0 .

(2.58)

ε

 

 

 

 

0 0 εiz

 

 

In addition, the vector fields must satisfy the transmission conditions (2.2)

and the Silver–M¨uller radiation condition (2.3).

Considering the null-field equations (2.6) and the integral representations for the scattered field coe cients (2.16), and taking into account the results

e,h mn
e,h mn

2.4 Inhomogeneous Particles

105

established in Sect. 1.3 regarding the series representations of the electromagnetic fields propagating in anisotropic media, we see that the scattering problem can be solved if we approximate the surface fields by finite expansions of vector quasi-spherical wave functions X and Y (cf. (1.48)–(1.53)),

$

eNi (r ) hNi (r )

% N

N

 

 

n (r ) × Xµe (r )

 

= cµ

 

 

 

 

 

 

 

 

 

−

j

 

 

εi n (r )

×

Xh

(r )

µ=1

 

 

 

 

µi

 

µ

 

 

N

n (r ) × Y µe (r )

 

 

+dµ

 

 

 

 

 

 

Y h (r ) .

 

j

 

εi n (r )

×

 

−

 

 

µi

 

 

 

µ

 

 

The transition matrix of an uniaxial anisotropic particle then becomes

T = −Q11anis (ks, ki, mrz ) Q31anis (ks, ki, mrz ) −1 ,

where mrz = εiz /εs, and for µi = µs, the elements of the Q31anis matrix are given by

31

11

=

jks2

 

e

 

 

3

 

+

 

 

εi

 

 

h

3

 

 

Qanis

νµ

 

 

 

n × Xµ

 

· N

 

 

 

 

 

 

 

 

 

n × Xµ · M

 

 

dS ,

(2.59)

 

π

S

ν

 

 

εs

ν

 

 

 

 

jks2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

31

12

=

 

 

e

 

 

3

 

+

 

εi

 

 

h

3

 

Qanis νµ

 

 

 

n × Y

µ

 

· N

 

 

 

 

 

 

 

n × Y µ · M

 

dS ,

(2.60)

 

π

S

ν

 

 

εs

ν

 

 

 

jks2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

31

21

=

 

e

 

 

 

3

+

 

εi

 

h

3

 

Qanis

νµ

 

 

 

n × Xµ

 

· M

 

 

 

 

 

 

 

n × Xµ · N

 

 

dS ,

(2.61)

 

π

S

ν

 

 

 

 

εs

ν

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

jks2

 

 

 

 

 

 

 

 

 

 

 

 

 

31

22

=

 

 

e

 

 

 

3

+

 

εi

 

h

3

 

Qanis νµ

 

 

 

n × Y

µ

 

· M

 

 

 

 

 

 

 

n × Y µ · N

 

dS .

(2.62)

 

π

S

ν

 

 

 

 

εs

 

ν

The expressions of the elements of the Q11anis matrix are similar but with M 1ν and N 1ν in place of M 3ν and N 3ν , respectively. Using the properties of the

vector quasi-spherical wave functions (cf. (1.47)) it is simple to show that for εiz = εi, the present approach leads to the T -matrix solution of an isotropic particle.

2.4 Inhomogeneous Particles

In this section, we consider electromagnetic scattering by an arbitrarily shaped, inhomogeneous particle with an irregular inclusion. Our treatment follows the analysis of Peterson and Str¨om [189] for multilayered particles and is similar to the approach used by Videen et al. [243] for a sphere with an irregular inclusion. Note that an alternative derivation using the Schelkuno ’s equivalence principle has been given by Wang and Barber [248].

106 2 Null-Field Method

2.4.1 Formulation with Addition Theorem

In the present analysis we will derive the expression of the transition matrix by using the translation properties of the vector spherical wave functions. The completeness property of the vector spherical wave functions on two enclosing surfaces, which is essential in our analysis, is established in Appendix D.

The scattering problem is depicted in Fig. 2.2. The surface S1 is defined with respect to a Cartesian coordinate system O1x1y1z1, while the surface S2 is defined with respect to a Cartesian coordinate system O2x2y2z2. By assumption, the coordinate system O2x2y2z2 is obtained by translating the coordinate system O1x1y1z1 through r12 and by rotating the translated coordinate system through the Euler angles α, β and γ. The boundary-value problem for the inhomogeneous particle depicted in Fig. 2.2 has the following formulation.

Given the external excitation Ee, He as an entire solution to the Maxwell equations, find the scattered field Es, Hs and the internal fields Ei,1, Hi,1and Ei,2, Hi,2 satisfying the Maxwell equations

× Es = jk0

µsHs ,

× Hs = −jk0εsEs in

Ds ,

(2.63)

× Ei,1 = jk0

µi,1Hi,1 ,

× Hi,1 = −jk0εi,1Ei,1

in Di,1 , (2.64)

and

 

 

 

 

 

× Ei,2 = jk0µi,2Hi,2 , × Hi,2 = −jk0εi,2Ei,2

in

Di,2 ,

(2.65)

the boundary conditions

 

 

 

 

 

n1 × Ei,1 − n1 × Es = n1 × Ee ,

 

 

 

n1 × Hi,1 − n1 × Hs = n1 × He ,

 

 

(2.66)

 

M1

 

 

 

n1

 

 

 

 

 

 

r29

 

 

M2

 

P

r2 r19

 

r199

 

 

 

r1

 

r299 n2

S1

 

 

O

 

 

 

 

2

 

 

O

1

r12

S2

 

 

 

 

 

 

 

 

 

 

 

Di,1

 

Di,2

 

 

 

 

 

 

Ds

Fig. 2.2. Geometry of an inhomogeneous particle

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