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

				2.10				Particle on or near an Infinite Surface
$ M mn3 (ksr) %			= −	1				1	2π π/2−j∞ $ mπn|m|(β)
	3	(ksr)		2πjn+1						|m|
N							2n(n + 1)		0 0		(β)
	mn									τn
				$ τn|m|(β)				%	ejmαejk(β,α)·r sin βdβ dα ,
			+j					eα
				mπn|m|(β)

169

%

eβ

(2.204)

where (ks, β, α) are the spherical coordinates of the wave vector k, and (ek , eβ , eα) are the spherical unit vectors of k. Each reflected plane wave in (2.204) will contain a Fresnel reflection term and a phase term equivalent to exp(2jksz0 cos β). The reflected vector spherical wave functions can be expressed as

$

3,R

(ksr)

%

1

1

2π π/2

j

$

|m|

%

M mn

= −

−

∞

mπn

(β)

r (β)eβR

N 3,R(ksr)

2πjn+1

|m|

2n(n + 1)

0 0

mn

τn

(β)

$ τn|m|(β)

%

ejmαe2jksz0 cos β ejkR(βR,αR)·r

+ j

mπn|m|(β)

r (β)eαR

× sin β dβ dα ,

where βR = π − β, αR = α, (ks, βR, αR) are the spherical coordinates of the reflected wave vector kR, and (ekR, eβR, eαR) are the spherical unit vectors of kR. For r inside a sphere enclosed in the particle, we expand each plane wave in terms of regular vector spherical wave functions

$ %

eβR

eαR

ejkR·r =	∞ n1			n1		$		|m1|	%
				4j		jm1πn1			(π − β)
	− n1=1 m1=−n1						τn|m1 1|(π − β)
			2n1 (n1 + 1)
	1		jτn|m1 1|(π − β)					1		jm1α
	× M m1n1 (ksr) +			m1			N m1n1 (ksr) e−
				m1πn| 1	|(π − β)

and obtain the following expressions for the elements of the reflection matrix:

α

=

2jn1−n

π/2−j∞ m2π|m|(β)π|m|(π

−

β)r

(β)

mnn1

nn1(n + 1)(n1 + 1)

0

n

n1

+ τ |m|

(β)τ |m|(π

−

β)r

(β) e2jksz0 cos β sin β dβ ,

(2.205)

n

n1

β

=

2jn1−n

π/2−j∞ m

π|m|(β)τ |m|(π

−

β)r

(β)

mnn1

nn1(n + 1)(n1 + 1)

0

n

n1

+ τ |m|(β)π|m|(π

−

β)r

(β) e2jksz0 cos β sin β dβ ,

(2.206)

n

n1

170 2 Null-Field Method

γ

=

2jn1−n

π/2−j∞ m

τ |m|(β)π|m|(π

−

β)r

(β)

mnn1

nn1(n + 1)(n1 + 1)

0

n

n1

+ π|m|

(β)τ |m|(π

−

β)r

(β) e2jksz0 cos β sin β dβ ,

(2.207)

n

n1

δ

=

2jn1−n

π/2−j∞ τ |m|(β)τ |m|(π

−

β)r

(β)

mnn1

nn1(n + 1)(n1 + 1)

0

n

n1

+ m2πn|m|(β)πn|m1

|(π − β)r (β) e2jksz0 cos β sin β dβ ,

(2.208)

An approximate expression for the reflection matrix can be derived if we assume that the interacting radiation strikes the surface at normal incidence. Assuming r(0) = r (β) = −r (β), changing the variable from β to βR = π−β, and using the relations

πn|m|(π − βR) = (−1)n−|m|πn|m|(βR) , τn|m|(π − βR) = (−1)n−|m|+1τn|m|(βR) ,

yields the following simplified integral representations for the reflected vector spherical wave functions:

$%

M 3,R(ksr)

( 1)n−|m|r(0)

1

mn

=

−

−

N mn3,R(ksr)

2πjn+1

2n(n + 1)

2π π $

|m|

%

$

|m|

(βR)

%

−

mπn

(βR)

eβR + j

−

τn

×

eαR

0

π/2+j∞

τn|m|(βR)

mπn|m|(βR)

×ejmαR e−2jksz0 cos βR ejkR(βR,αR)·r sin βR dβR dαR .

To compute M 3mn,R and N 3mn,R we introduce the image coordinate system O x y z by shifting the original coordinate system a distance 2z0 along the positive z-axis. The geometry of the image coordinate system is shown in Fig. 2.16. Taking into account that kR · r = kR · r − 2ksz0 cos βR, where r = (x , y , z ), we identify in the resulting equation the integral representations for the radiating vector spherical wave functions in the half-space z < 0:

$ M 3,R(k r) %	= ( 1)n−|m|r(0)	$	M	3		(k	r ) %
mn s		−		mn		s	.
N mn3,R(ksr)	−	N mn3			(ksr )

In this case the interacting field is the image of the scattered field and the expansion (2.198) can be derived by using the addition theorem for vector spherical wave functions. The elements of the reflection matrix are the translation coe cients, and as a result, the amount of computer time required to solve the scattering problem is significantly reduced. In this regard it should

	2.10 Particle on or near an Infinite Surface		171
	z
	O	x
z	r
	0
	z	Σ
		P
z0
	r	x

O

Fig. 2.16. Image coordinate system

be mentioned that the formalism using the approximate expression for the reflection matrix has been employed by Videen [240–242].

In most practical situations we are interested in the analysis of the scattered field in the far-field region and below the plane surface, i.e., for θ > π/2. In this region we have two contributions to the scattered field: the direct electric far-field pattern Es∞(θ, ϕ),

	1		∞ n	n+1
Es∞(θ, ϕ) =		ks	n=1 m=−n (−j)		[fmnmmn(θ, ϕ) + jgmnnmn(θ, ϕ)] (2.209)

and the interacting electric far-field pattern ERs∞(θ, ϕ),

				1	∞ n		j)n+1 fmnmR
	ER	(θ, ϕ) =			(	−		(θ, ϕ) + jgmnnR	(θ, ϕ) ,
			ks
	s∞				n=1 m=−n		mn	mn
									(2.210)

where mmn and nmn are the vector spherical harmonics, and mRmn and nRmn are the reflected vector spherical harmonics,

mR

(θ, ϕ) =

1

e−2jksz0 cos θ

mn

2n(n + 1)

×

jmπ|m|(π

−

θ)r (π

−

θ)e

τ |m|(π

−

θ)r

(π

−

θ)e

ejmϕ ,

n

θ −

n

ϕ

nR

(θ, ϕ) =

1

e−2jksz0 cos θ

mn

2n(n + 1)

τ |m|(π

−

θ)r

(π

−

θ)e

θ

+ jmπ|m| (π

−

θ) r (π

−

θ)e

ejmϕ .

× n

n

ϕ

172 2 Null-Field Method

Thus, the solution of the scattering problem in the framework of the separation of variables method involves the following steps:

1.Calculation of the T matrix relating the expansion coe cients of the fields striking the particle to the scattered field coe cients.

2.Calculation of the reflection matrix A characterizing the reflection of vector spherical wave functions by the surface.

3.Computation of an approximate solution by solving the matrix equation (2.203).

4.Computation of the far-field pattern by using (2.209) and (2.210).

In practice, we must compute the integrals in (2.205)–(2.208), which are of the form

π/2−j∞

I = f (cos β) e2jq cos β sin βdβ .

0

Changing variables from β to x = −2jq(cos β − 1), we have

I =	e2jq ∞			x		x
		0	f 1 −			e−	dx
	2jq			2jq

and integrals of this type can be computed e ciently by using the Laguerre polynomials [15].

Scanning near-field optical microscopy [202,203] requires a rigorous analysis of the evanescent scattering by small particles near the surface of a dielectric prism [31, 142, 199, 252]. Scattering of evanescent waves can be analyzed by extending our formalism to the case of an incident plane wave propagating in the substrate (Fig. 2.17).

For the incident vector plane wave given by (2.193), the transmitted (or the refracted) vector plane wave is

EeT(r) = Ee0T ,β eβT + Ee0T ,αeαT ejkeT ·r ,

where

Ee0T ,β = t (β0)ejksz0(cos β−mrs cos β0)Ee0,β ,

Ee0T ,α = t (β0)ejksz0(cos β−mrs cos β0)Ee0,α ,

β0 is the incident angle and (ekT, eβT, eαT) are the spherical unit vectors of the transmitted wave vector keT. The Fresnel transmission coe cients are given by

2mrs cos β0 t (β0) = cos β0 + mrs cos β ,

2mrs cos β0 t (β0) = mrs cos β0 + cos β ,

2.10 Particle on or near an Infinite Surface

173

z

	β 0
	k e	mrs
		Σ
z 0		k eT
		β
	O	x

mr

Fig. 2.17. Geometry of an axisymmetric particle situated near a plane surface. The external excitation is a vector plane wave propagating in the substrate

while the angle of refraction is computed by using Snell’s law:

sin β = mrs sin β0 ,

cos β = ± 1 − sin2 β .

Evanescent waves appear for real mrs and incident angles β0 > β0c, where β0c = arcsin(1/mrs). In this case, sin β > 1 and cosβ is purely imaginary. For negative values of z, we have

exp (jkeT · r) = exp (−jksz cos β + jksx sin β) = exp (jks |z| cos β + jksx sin β) ,

and we choose the sign of the square root such that Im{cos β} > 0. This choice guarantees that the amplitude of the refracted wave propagating in the negative direction of the z-axis decreases with increasing the distance |z|. The expansion coe cients of the transmitted wave are

aT

=

4jn1

jmπ|m|(π

β) ET

+ τ |m|(π

β) ET

,

mn1

− 2n1 (n1 + 1)

n1

−

e0,β

n1

−

e0,α

bT

=

4jn1+1

τ |m|(π

β) ET

jmπ|m|(π

β) ET

,

mn1

− 2n1 (n1 + 1)

n1

−

e0,β −

n1

−

e0,α

and we see that our previous analysis remains unchanged if we replace the total expansion coe cients amn1 and bmn1 , by the expansion coe cients of the transmitted wave aTmn1 and bTmn1 .

2.10.2 Particle on or near an Arbitrary Surface

In the precedent analysis, we considered scattering by a particle situated near a plane surface. The scattering problem of a particle situated in the

174 2 Null-Field Method

neighborhood of an arbitrary infinite surface can be solved by using the T -operator formalism. Specifically, we are interested to compute the vector spherical wave expansions of the reflected fields ERe , M 3mn,R and N 3mn,R in the case of an infinite surface. For this purpose, we consider the scattering problem of an infinite surface illuminated by an arbitrary incident field and follow the analysis of Kristensson [124].

We briefly recall the definitions and the basic properties of scalar and vector plane waves [14]. The scalar plane wave is defined by

χ(r, K±) = exp(jK± · r) = exp[j (Kxx + Ky y ± Kz z)] ,

where

K± = Kxex + Ky ey ± Kz ez .

Using the notation kT = Kx2 + Ky2, Kz can be expressed as Kz = k2 − kT2 , where the square root is always chosen to have a positive imaginary part. For

real k, Kz is given by Kz = k2 − kT2 , if kT ≤ k, and by Kz =j kT2 − k2 if kT > k. The case kT ≤ k corresponds to harmonic propagating waves, while

the case kT > k corresponds to evanescent waves. The vector plane waves M (r, K±) and N (r, K±) are defined in terms of scalar plane wave as

M (r, K±) =	1		× [ez χ (r, K±)] = j			KT		χ (r, K±) ,
	kT					kT
N (r, K±) =	1	× M (r, K±) = −		K±					KT
								×		χ (r, K±) ,
	k					k			kT

where KT is the transverse component of the wave vector K± and is given by

KT = K± × ez = Ky ex − Kxey .

The orthogonality relations

M (r, K

±

)

·

M

r, K dx dy = 4π2e±j(Kz −Kz )z δ Kx

−

K , Ky

−

K

Σ

−

±

x

y

N (r, K

±

)

·

N

r, K dx dy = 4π2

K± · K±

e±j(Kz −Kz )z

Σ

−

±

kk

×δ Kx − Kx, Ky − Ky ,

M (r, K

±

)

·

N

r, K dx dy =

N (r, K

±

)

·

M

−

r, K dx dy = 0,

Σ

−

±

Σ

±

hold on a plane z = const, where

M (−r, K±) =

1

KT

× [ez χ (−r, K±)] = −j

χ (−r, K±) ,

kT

kT

1

K±

KT

N (−r, K±) =

× M (−r, K±) = −

×

χ (−r, K±) .

k

k

kT

2.10 Particle on or near an Infinite Surface

175

		z
Di		mrs
S		z = z>
		z = z<
D		k e
	s	β0

O	x

Fig. 2.18. Geometry of an infinite surface

Alternative expressions for the vector plane waves are
M (r, K±) = −jeαχ (r, K±) ,
N (r, K±) = −eβ χ (r, K±) ,	(2.211)
and
M (−r, K±) = jeαχ (−r, K±) ,
N (−r, K±) = −eβ χ (−r, K±) ,	(2.212)

where (ek , eβ , eα) are the spherical unit vectors of the wave vector K±.

We consider now the scattering geometry depicted in Fig. 2.18. The incident field Ee is a vector plane wave or any radiating vector spherical wave functions. To solve the scattering problem in the framework of the T -operator method we apply the Huygens principle to a surface consisting of a finite part of S and a lower half-sphere. Letting the radius of the sphere tend to infinity and assuming that the integrals over S exist and the integrals over the lower half-sphere vanish (radiation conditions), we obtain

E(r)	"
0	= Ee(r)+	S	[n (r ) × Ei (r )] · × G (ks, r , r)							(2.213)
										#
	+ [n (r ) × ( × Ei (r ))] ·					(ks, r , r) dS (r ) ,				r Ds
					G						,
										r Di
where	E(r) = Ee(r) + Es(r) is			the total field in					the domain Ds		and

G(ks, r , r) is the free space dyadic Green function of wave number ks. The dyadic Green function can be expanded in terms of vector plane waves, [14, 124]

176 2 Null-Field Method

jks

s

s

G(ks, r , r) =

R2 M −r , K+

M r, K+

8π2

+ N

r , Ks

N r, Ks

dKxs dKys

ksKzs

−

+

+

for z > z , and

(ks, r , r) =

jks

M

r , Ks

M

r, Ks

G

8π2

R2

−

−

−

+ N

r , Ks

N r, Ks

dKxs dKys

ksKzs

−

−

−

for z < z , where the superscript s indicates that the vector plane waves correspond to the domain Ds.

The incident field is a prescribed field, whose sources are assumed to be situated in Ds. This means that in any case, the sources are below the fictitious plane z = z> and we can represent the external excitation as an integral over up-going vector plane waves

Ee(r) =		A(K+s ) M (r, K+s ) + B(K+s ) N (r, K+s )	dKxs dKys	.
	2
R			k Ks
			s z

For r Di and z > z>, we use the plane wave expansion of the Green dyad and the orthogonality properties of the vector plane waves to obtain

A Ks+

B Ks+

= −

jks2

!

s

[n(r ) × Ei(r )] · N −r , K+

8π2

S

+

1

[n (r )

×

(

×

Ei (r ))]

·

M

r , Ks

ks

−

+

= −

jks2

!

s

[n(r ) × Ei(r )] · M −r , K+

8π2

S

+

1

[n (r )

(

×

Ei (r ))]

·

N

r , Ks

ks

×

−

+

(2.214)

"

dS (r ) ,

(2.215)

"

dS (r ) .

Taking into account that the tangential vector plane waves n ×M (·, Ki+) and n × N (·, Ki+), form a complete system of vector function on S, we represent the surface fields n × Ei and n × ( × Ei) as integrals over up-going vector plane waves, i.e.,

n (r ) × Ei (r ) = C Ki+ c Ki+ n (r ) × M r , Ki+ (2.216)

R2

+ D Ki+ d Ki+ n (r ) × N r , Ki+ dKxi dKyi

2.10 Particle on or near an Infinite Surface 177

and

n (r ) × [ × Ei (r )] = ki C Ki+ c Ki+ n (r ) × N r , Ki+

R2

+ D Ki+ d Ki+ n (r ) × M r , Ki+ dKxi dKyi ,

(2.217)

respectively. The plane wave transmission coe cients in (2.216) and (2.217) are given by

i	=		2Kzs				i	s
c K+						e−j(Kz −Kz )z ,
		Ks		+ Ki
			z		z
i	=		2mrsKzs					j(Ki	Ks )z
d K+							e−	z −	z ,	(2.218)
		m2		Ks	+ Ki
			rs	z		z

where mrs = ki/ks is the relative refractive index of the domain Di with respect to the ambient medium, and the wave vectors Ks+ and Ki+ (in (2.218)) are related to each other by Snell’s law

kT = kTs = kTi = (Kxs )2 + (Kys )2 = (Kxi )2 + (Kyi )2 .

Inserting (2.216) and (2.217) into (2.214) and (2.215) yields

A K+s			Q	31	s i	C K+i	i i
B K+s		=			K+, K+	D K+i	dKx dKy , (2.219)
		R2
where
31	s	i	Q31 K+s , K+i 11 Q31 K+s , K+i 12
Q	K+, K+ =			Q31 K+s , K+i 21 Q31 K+s , K+i 22

and

Q31 Ks+, Ki+ 11

Q31 Ks+, Ki+ 12

Q31 Ks+, Ki+ 21

jks2

i

i

s

= −

S c(K+) n (r ) × M

r , K+

· N (−r , K+)

8π2

+ mrs

n (r )

×

N

r , Ki

M ( r , Ks ) dS (r ) ,

+

·

−

+

=

−

jks2

d(Ki ) n (r )

×

N

r , Ki

N ( r , Ks

)

8π2

S

+

+

·

−

+

+ mrs

n (r )

×

M

r , Ki

M (

−

r , Ks ) dS (r ) ,

+

·

+

=

−

jks2

c(Ki ) n (r )

×

M r , Ki

M ( r , Ks

)

8π2

S

+

+

·

−

+

+ mrs

n (r )

×

N

r , Ki

N ( r , Ks ) dS (r ) ,

+

·

−

+

178

2

Null-Field Method

31

s

i

22

jks2

i

i

s

Q

K+, K+

= −

S d(K+)

n (r ) × N

r , K+

· M (−r , K+)

8π2

+ mrs n (r )

×

M

r , Ki

N ( r , Ks

) dS (r ) .

+

·

−

+

We note that (Q31)αβ , α, β = 1, 2, are not functions in the usual sense and relations involving Q31 should be understood in a distributional sense. For a plane surface we have

Q31 K+s , K+i	11	= Q31 K+s , K+i	22
	plane		plane
		= ksKzs δ Kxi − Kxs , Kyi − Kys ,
Q31 K+s , K+i	11	= Q31 K+s , K+i	22	= 0
	plane		plane

and the system of integral equations (2.219) simplifies to a system of algebraic equations

A Ks+ = ksKzs C Ki+ ,

B Ks+ = ksKzs D Ki+ .

For solving the general case of an arbitrary surface, one has to invert a system of two-dimensional integral transforms and this is a formidable analytic and numerical problem. We can formally assume that the inverse transform exists and represent the solution as

C K+i			31	s i	1 A K+s	s s	. (2.220)
D K+i		=	Q	K+, K+ −	B K+s	dKx dKy
			R2

To compute the scattered field we consider (2.213) for r Ds, expand the free space dyadic Green function in terms of vector plane waves by assuming z < z<, and obtain

Es(r) =		F (Ks	)M (r, Ks		) + G(Ks		)N (r, Ks	)	dKxs dKys	, (2.221)
									k Ks
R	2	−		−		−	−
									s z
where the amplitudes F (Ks				) and G(Ks		) are given by
			−		−

F Ks−

G Ks−

=jks2

8π2

+1 ks

=jks2

8π2

+1 ks

S ! [n(r ) × Ei(r )] · N −r , K−s
[n (r )	(	×	Ei (r ))]	·	M			r , Ks "dS (r ) ,
	×					−			−
S ! [n(r ) × Ei(r )] · M −r , K−s
[n (r )	(	×	Ei (r ))]	·	N		r , Ks "dS (r ) .
	×					−			−