The absorption of electromagnetic (EM) waves by a spherical dilute suspension of nanoparticles

(1)

The absorption of electromagnetic (EM) waves by a spherical dilute suspension of nanoparticles

Mariana Dalarsson and Sven Nordebo

Department of Physics and Electrical Engineering Linnaeus University, SE-351 95, V¨axjo, Sweden

2017-05-23

In the present technical report, we review the problem of the electromagnetic (EM) wave propagation within a structure consisting of two lossy media, as shown in Fig. 1.

Medium 1 (E

1

, H

1

)

Incident EM wave (E

i

, H

i

)

Scattered EM wave (E

s

, H

s

)

Medium 2

z-axis Figure 1: The incident fields (E

_i

, H

_i

) give rise to the fields (E

₁

, H

₁

) inside the sphere (medium 1) and to the scattered fields (E

s

, H

s

) in the medium surrounding the sphere (medium 2).

From Fig. 1 we see that the sphere (medium 1) of radius a consists of a non-magnetic dilute suspension of nanoparticles. It is surrounded by a non-magnetic surrounding medium (medium 2) with no nanoparticles present. Since both media (medium 1 and medium 2) are assumed to be non-magnetic, their relative permeabilities are equal to unity

µ

p

= 1 (medium 1) , µ

m

= 1 (medium 2) (1) The complex relative permittivity of the lossy medium 1 inside the sphere is

ǫ

p

(ω) = ǫ

^′_p

(ω) − jǫ

^′′p

(ω) (2)

(2)

while the complex relative permittivity of medium 2 surrounding the particle is

ǫ

m

(ω) = ǫ

^′_m

(ω) − jǫ

^′′m

(ω) (3) In Fig. 1, the EM fields in medium 1 are denoted by (E

1

, H

1

), while the EM fields in medium 2 are denoted by (E

2

, H

2

), such that

E

₂

= E

i

+ E

s

, H

₂

= H

i

+ H

s

(4)

where (E

i

, H

i

) are the fields of the incident EM wave and (E

s

, H

s

) are the fields of the scattered EM wave in medium 2.

Let us now introduce the spherical coordinates (r, θ, ϕ) with the origin at the center of the sphere (medium 1). In such a case, the normal vector on the spherical boundary surface between the two media (r = a) is ˆ n = ˆ r . The EM boundary conditions at the boundary surface between the two media

ˆ

n × (E

¹

− E

²

) = 0 , n ˆ × (H

¹

− H

²

) = 0 (5) can then be rewritten as follows

(E

_i

+ E

_s

− E

1

) × ˆr = 0 , (H

_i

+ H

_s

− H

1

) × ˆr = 0 (6) Finally, we note that time dependence for complex field vectors is assumed to be of the form e

^jωt

, which is the usual convention in EM theory.

1 Incident fields

As indicated in Fig. 1, the incident EM wave is assumed to be a plane wave (E

_i

, H

_i

) propagating along the z-axis, with the electric field E

i

polarized along the x-axis. Thus we can write

E

_i

= E

₀

e

⁻^jkz

x ˆ (7)

H

_i

= 1 η

2

ˆ

z × E

i

=

r ǫ

0

ǫ

m

µ

0

µ

m

E

₀

e

⁻^jkz

ˆ z × ˆx = ω√ǫ

0

µ

0

ωµ

0

√ ǫ

_m

E

₀

e

⁻^jkz

y ˆ (8)

where η

2

= pµ

2

/ǫ

2

= pµ

0

µ

m

/ǫ

0

ǫ

m

is the wave impedance of medium 2. Using here

√ ǫ

0

µ

0

= 1/c and introducing the magnitude of the complex wave number

k = ω √ ǫ

₀

µ

₀

√ ǫ

_m

= ω c

√ ǫ

_m

= k

^′

− jk

^′′

(9)

we can rewrite (8) as follows

H

i

= k ωµ

0

E

0

e

⁻^jkz

y ˆ (10)

Let us now analyze the result (9), and write k = ω

c pǫ

^′_m

− jǫ

^′′m

= k

^′

− jk

^′′

= ω

c (u − jv) (11)

(3)

where

ǫ

^′_m

− jǫ

^′′m

= (u − jv)

²

= u

²

− v

²

− j2uv (12) such that

u

²

− v

²

= ǫ

^′_m

, 2uv = ǫ

^′′_m

⇒ u = ǫ

^′′_m

2v (13)

Thus we obtain

(ǫ

^′′_m

)

²

4v

²

− v

²

= ǫ

^′_m

⇒ v

⁴

+ ǫ

^′_m

v

²

− (ǫ

^′′_m

)

²

4 = 0 (14)

The solution to this equation is

v

²

= − ǫ

^′_m

2 +

r (ǫ

^′_m

)

²

4 + (ǫ

^′′_m

)

²

4 = p(ǫ

^′_m

)

²

+ (ǫ

^′′_m

)

²

− ǫ

^′m

2 (15)

or

v

²

= |ǫ

^m

| − ǫ

^′m

2 , |ǫ

^m

| = p(ǫ

^′_m

)

²

+ (ǫ

^′′_m

)

²

(16) Substituting (16) into the first of the equations (13) gives

u

²

= v

²

+ ǫ

^′_m

= |ǫ

^m

| + ǫ

^′m

2 (17)

Using (11) with (16) and (17), we readily obtain k

^′

= ω

√ 2c p|ǫ

m

| + ǫ

^′m

, k

^′′

= ω

√ 2c p|ǫ

m

| − ǫ

^′m

(18)

From the results (18), we see that for a non-absorbing medium (ǫ

^′′_m

= 0), we have

|ǫ

m

| = ǫ

^′m

⇒ k

^′

= ω

c ǫ

^′_m

, k

^′′

= 0 ⇒ k = ω

c ǫ

^′_m

(19) Using now the result (11) in (7) and (10), we can write

E

_i

= E

0

e

⁻^jk^′^z

e

⁻^k^′′^z

x ˆ (20) H

^∗_i

= k

^′

+ jk

^′′

ωµ

0

E

₀

e

^+jk^′^z

e

⁻^k^′′^z

y ˆ (21) The Poynting vector associated with the incident EM wave is defined by

S

i

= 1

2 Re(E

i

× H

^∗i

) = E

₀²

2ωµ

0

(ˆ x × ˆy)e

⁻^2k^′′^z

Re(k

^′

+ jk

^′′

) (22) such that

S

_i

= E

₀²

k

^′

2ωµ

0

e

⁻^2k^′′^z

z ˆ , k

^′

= Re(k

^∗

) (23) Thus, the rate at which the energy of the incident EM wave is transferred across the closed spherical boundary of the medium 1 (r = a) is obtained from

W

i

= I

A

S

i

· ˆ n dA = − I

A

S

i

· ˆr a

²

sin θdθdϕ (24)

(4)

where the minus sign appears due to the inwards energy flow into the sphere. For a non-absorbing medium (k

^′′

= 0, ǫ

^′′_m

= 0), we have

W

i

= E

₀²

k

^′

a

²

2ωµ

0

Z

π 0

(ˆ z · ˆr) d(cos θ) Z

2π

0

dϕ (25)

On the other hand, we know that ˆ

r = sin θ cos ϕˆ x + sin θ sin ϕˆ y + cos θˆ z (26) such that ˆ z · ˆr = cos θ and

W

i

= E

₀²

k

^′

a

²

2ωµ

0

2π Z

π

0

cos θ d(cos θ) = πE

₀²

k

^′

a

²

ωµ

0

cos

²

θ 2

2

≡ 0 (27)

Thus in a non-absorbing medium (ǫ

^′′_m

= 0), the integral W

i

is identically equal to zero.

When the surrounding medium is absorbing, i.e. when k

^′′

6= 0, we have W

i

= E

₀²

k

^′

a

²

2ωµ

0

Z

π 0

e

⁻^2k^′′^{a cos θ}

cos θ d(cos θ) Z

2π

0

dϕ (28)

Introducing I

0

= (E

₀²

k

^′

)/(2ωµ

0

) and the variable substitution w = −2k

^′′

a cos θ, we can calculate the integrals in (28) as follows

W

i

= 2π E

₀²

k

^′

2ωµ

₀

1 4(k

^′′

)

²

Z

π 0

e

⁻^2k^′′^{a cos θ}

(−2k

^′′

a cos θ) d(−2k

^′′

a cos θ) =

= 2πI

0

4(k

^′′

)

²

Z

2k^′′a

−2k^′′a

we

^w

dw = 2πI

0

4(k

^′′

)

²

(we

^w

− e

^w

)

^2k₋_2k^′′^a′′a

=

= 2πI

0

4(k

^′′

)

²

[2k

^′′

a(e

^2k^′′^a

+ e

⁻^2k^′′^a

) − (e

^2k^′′^a

− e

⁻^2k^′′^a

)] (29) or finally

W

_i

= 2πI

₀

(k

^′′

)

²

k

^′′

a cosh(2k

^′′

a) − 1

2 sinh(2k

^′′

a)

(30) Thus in case of an absorbing medium (k

^′′

6= 0, ǫ

^′′m

6= 0), the integral W

i

does not vanish, and there is a non-zero rate at which the energy of the incident EM wave is transferred across the closed boundary of the sphere (medium 1). The phenomenon can be viewed as an attenuation of the incident plane wave in the medium, before it hits and scatters against the spherical boundary of medium 1. The ratio C

i

= W

i

/I

0

is referred to as the incident wave cross section.

It is here of interest to investigate the limit of the result (30) when k

^′′

a ≪ 1, in which case we can use

cosh(2k

^′′

a) ≈ 1 + (2k

^′′

a)

²

2! + · · · (31)

sinh(2k

^′′

a) ≈ 2k

^′′

a + (2k

^′′

a)

³

3! + · · · (32)

(5)

Substituting the approximations (31) and (32) into the result (30), we obtain W

i

≈ 2πI

0

(k

^′′

)

²

k

^′′

a + 4(k

^′′

a)

³

2 − k

^′′

a − 8(k

^′′

a)

³

12 = 2πI

0

(k

^′′

)

²

2(k

^′′

a)

³

− 2 3 (k

^′′

a)

³

= 2πI

0

a

³

k

^′′

2 − 2

3 = 8π

3 I

0

a

³

k

^′′

(33)

Using k

^′′

= ω/c Im{ √ ǫ

m

}, we finally obtain W

i

= I

0

8π 3 a

³

· ω

c Im{ √

ǫ

m

} (34)

2 Indirect calculation of absorption cross-section

From the results (4), we see that the fields in the medium surrounding the sphere (E

2

, H

2

) are superpositions of the incident fields (E

i

, H

i

) and the scattered fields (E

s

, H

s

). These fields must satisfy the Maxwell equations

∇ · E = 0 , ∇ · H = 0 (35)

∇ × E = −jωµH, ∇ × H = jωǫE (36)

Using Maxwell equations (35) and (36), we can calculate

∇ × (∇ × E) = jωµ (∇ × H) = jωµ (−jωǫE) = ω

²

µǫE (37)

∇ × (∇ × H) = −jωǫ (∇ × E) = −jωǫ (jωµH) = ω

²

µǫH (38) By means of the general vector identity

∇ × (∇ × A) = ∇(∇ · A) − ∇

²

A (39)

and using (35), we readily obtain

∇

²

E + k

²

E = 0 , ∇

²

H + k

²

H = 0 (40) where k

²

= ω

²

ǫµ. It should be noted here that it is not generally true that the individual components of E = (E

1

, E

2

, E

3

) and H = (H

1

, H

2

, H

3

) separately satisfy the scalar wave equation

∇

²

Ψ + k

²

Ψ = 0 (41)

It is only true in rectangular Cartesian coordinates.

2.1 Boundary conditions and energy conservation across the boundary

At the spherical boundary between the two media (r = a), the boundary conditions (5)

or (6) are valid. The rate, at which the the EM energy is transferred across a closed

(6)

spherical surface arbitrarily near the boundary surface inside medium 1 (r = a

⁻

), is given

by I

A

S

1

· ˆ ndA = I

A

ˆ

n · (E

¹

× H

¹

) dA (42)

Similarly, the rate at which EM energy is transferred across a closed spherical surface, arbitrarily near the boundary surface inside medium 2 (r = a

+

), is

I

A

S

2

· ˆ ndA = I

A

ˆ

n · (E

²

× H

²

) dA (43)

Using now the permutation rules for the triple scalar product

A · (B × C) = B · (C × A) = C · (A × B) (44) we can rewrite (42) and (43) as follows

I

A

S

₁

· ˆ ndA = I

A

H

₁

· (ˆ n × E

¹

) dA (45)

I

A

S

2

· ˆ ndA = I

A

E

2

· (H

²

× ˆ n) dA (46)

By means of the boundary conditions (5), i.e. ˆ n × E

¹

= ˆ n × E

²

and H

1

× ˆ n = H

2

× ˆ n, we can rewrite (45) and (46) as follows

I

A

S

1

· ˆ ndA = I

A

H

1

· (ˆ n × E

²

) dA (47)

I

A

S

2

· ˆ n dA = I

A

E

2

· (H

¹

× ˆ n) dA = I

A

H

1

· (ˆ n × E

²

) dA (48) From the results (47) and (48) we see that the boundary conditions (5) readily imply the energy conservation across that boundary

I

A

S

1

· ˆ ndA = I

A

S

2

· ˆ n dA (49)

where S

₁

and S

₂

are both calculated at the spherical boundary (r = a).

2.2 Energy considerations and extinction

If we now use (4), we can calculate the Poynting vector in medium 2, surrounding the sphere as follows

S

2

= 1

2 Re(E

2

× H

^∗2

) = 1

2 Re{(E

ⁱ

+ E

s

) × (H

^∗i

+ H

^∗_s

)} =

= 1

2 Re{E

ⁱ

× H

^∗i

} + 1

2 Re{E

^s

× H

^∗s

} + 1

2 Re{E

ⁱ

× H

^∗s

+ E

s

× H

^∗i

} =

= S

i

+ S

s

+ S

ext

(50)

(7)

where S

i

is associated with the incident EM wave and given by (23), while S

s

is associated with the scattered EM wave and S

ext

is the extinction interference term.

S

i

= 1

2 Re{E

ⁱ

× H

^∗i

} , S

s

= 1

2 Re{E

^s

× H

^∗s

} (51) S

_ext

= 1

2 Re{E

i

× H

^∗s

+ E

_s

× H

^∗i

} (52) The energy absorbed by the sphere per unit time is then equal to

W

abs

= − I

A

S

1

· ˆr dA = − I

A

S

2

· ˆr dA (53)

where S

1

and S

2

are both calculated at the spherical boundary (r = a), and we use the energy conservation formula (49). Since the radial unit vector ˆ r is chosen outward, the negative sign in (53) ensures that W

abs

is positive so that energy is indeed absorbed by the sphere. Substituting (50) into (53) gives

W

abs

= − I

A

S

_i

· ˆr dA − I

A

S

_s

· ˆr dA − I

A

S

_ext

· ˆr dA = W

ⁱ

− W

^s

+ W

ext

(54) where

W

_i

= − I

A

S

_i

· ˆr dA , W

_s

= + I

A

S

_s

· ˆr dA (55)

W

ext

= − I

A

S

_ext

· ˆr dA (56)

The minus sign before W

s

in (54) and consequently the positive sign in the definition of W

s

in (55) indicates that the scattered Poynting vector points in the same direction as the outwardly oriented radial unit vector ˆ r. We know from (27) that W

i

≡ 0 for a non-absorbing medium (k

^′′

= 0, ǫ

^′′_m

= 0), in which case

W

ext

= W

abs

+ W

s

(57)

and the extinction power W

ext

is just a sum of the energy absorption rate W

abs

and the energy scattering rate W

s

.

2.3 Solutions to the vector wave equations

As shown in (40), any physically realizable time-harmonic EM fields (E, H) in a linear, isotropic, homogeneous medium must satisfy the wave equations

∇

²

E + k

²

E = 0 , ∇

²

H + k

²

H = 0 (58) where k

²

= ω

²

ǫµ, and (E, H) must have zero divergence

∇ · E = 0, ∇ · H = 0 (59)

In addition, E and H are mutually inter-dependent by means of Maxwell equations

∇ × E = −jωµH , ∇ × H = jωǫE (60)

(8)

In order to construct suitable vector fields that satisfy all of the above equations, we can use the solution Ψ(r) of the scalar wave equation (41), i.e.

∇

²

Ψ + k

²

Ψ = 0 (61)

Given the scalar function Ψ(r) and some yet unspecified vector c, we can try to create a vector field M as a curl of the vector cΨ(r) which, by definition, has zero divergence. If we, to begin with, assume that c is a constant vector, then

M = ∇ × (cΨ) = −c × ∇Ψ = ∇Ψ × c (62)

where we used the vector formula that can easily be verified in Cartesian coordinates

∇ × (cΨ) =

ˆ

x y ˆ z ˆ

∂

∂x

∂

∂y

∂

∂z

c

x

Ψ c

y

Ψ c

z

Ψ

=

ˆ

x y ˆ z ˆ

∂Ψ

∂x

∂Ψ

∂y

∂Ψ

∂z

c

x

c

y

c

z

= ∇Ψ × c (63)

From (62), we readily see that M is perpendicular to c and that

∇ · M = ∇ · [∇ × (cΨ)] ≡ 0 (64)

Furthermore, we have

∇

²

M = ∇

²

[∇ × (cΨ)] (65)

If we now recall the general result that the Laplacian of the curl is equal to the curl of the Laplacian, i.e.

∇

²

(∇ × A) = ∇ × (∇

²

A) (66)

we can rewrite (65) as follows

∇

²

M = ∇ × [∇

²

(cΨ)] (67)

Using next the general vector formula

∇

²

(cΨ) = c∇

²

Ψ + 2(∇Ψ · ∇)c + Ψ∇

²

c (68) which for a constant vector c reads

∇

²

(cΨ) = c∇

²

Ψ (69)

the result (67) becomes

∇

²

M = ∇ × [c∇

²

Ψ] (70)

Thus, we can write

∇

²

M + k

²

M = ∇ × [c∇

²

Ψ] + k

²

[∇ × (cΨ)] = ∇ × [c(∇

²

Ψ + k

²

Ψ)] = 0 (71) since Ψ is a solution of the scalar wave equation (61). Thus the vector

M = ∇ × (cΨ) , c = constant (72)

satisfies the equations (58) and (59), i.e.

∇

²

M + k

²

M = 0 , ∇ · M = 0 (73)

(9)

If we assume that M is for example an “electric field”, then in order to satisfy Eq. (60), we need to introduce yet another vector field to play the role of the “magnetic field”. A suitable such field is obviously

N = ∇ × M

k , k = scalar constant (74)

since it then immediately satisfies the Maxwell equation (60)

∇ × M = kN (75)

From (74) we readily see that

∇ · N = 1

k ∇ · (∇ × M) ≡ 0 (76)

Furthermore we can calculate

∇

²

N = 1

k ∇

²

(∇ × M) = 1

k ∇ × (∇

²

M ) (77)

such that

∇

²

N + k

²

N = 1

k ∇ × (∇

²

M ) + 1

k ∇ × (k

²

M ) = 1

k ∇ × (∇

²

M + k

²

M ) ≡ 0 (78) Thus the N -field also satisfies the equations (58) and (59), i.e.

∇

²

N + k

²

N = 0 , ∇ · N = 0 (79)

Finally, let us calculate

∇ × N = 1

k ∇ × (∇ × M ) = 1

k ∇(∇ · M) − ∇

²

M

(80) Using here (73), that is ∇

²

M = −k

²

M and ∇ · M = 0, we obtain

∇ × N = kM (81)

and we see that the N -field also satisfies the Maxwell equation (60), that is, we see that the curl of the N -field is also proportional to the M -field. Now, it remains to determine a suitable vector c in order to specify the vector fields M and N . The choice of the vector c is dependent on the choice of coordinates used to describe the geometry at hand.

Here, we are interested in spherical geometry, and it turns out that a suitable choice of the vector c is the radius vector r, i.e.

c = r = (x, y, z) (82)

such that

M = ∇ × (rΨ) = ∇Ψ × r (83)

A potential problem here is that the radius vector r is not a constant vector. However,

it turns out that all the results derived in the present section remain valid. One critical

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result here is (83), or (62) with (82), that needs to be verified. The general vector formula reads

∇ × (rΨ) = ∇Ψ × r + Ψ∇ × r = ∇Ψ × r (84)

since ∇×r ≡ 0. Another critical result, that needs to be verified, is the result (70). Using

∇

²

r ≡ 0 (85)

and

2(∇Ψ · ∇)r = 2 ∂Ψ

∂x x ˆ + ∂Ψ

∂y y ˆ + ∂Ψ

∂z ˆ z

· ˆ x ∂

∂x + ˆ y ∂

∂y + ˆ z ∂

∂z

r =

= 2 ∂Ψ

∂x

∂r

∂x + ∂Ψ

∂y

∂r

∂y + ∂Ψ

∂z

∂r

∂z

= 2 ∂Ψ

∂x x ˆ + ∂Ψ

∂y y ˆ + ∂Ψ

∂z z ˆ

= 2∇Ψ (86) the general result (68), with c = r, gives

∇

²

(rΨ) = r∇

²

Ψ + 2∇Ψ (87)

and we readily see that the result (69) is not valid and must be replaced by (87). However the result (70) remains valid since ∇ × (∇Ψ) ≡ 0, such that

∇

²

M = ∇ × ∇

²

(rΨ) = ∇ × (r∇

²

Ψ + 2∇Ψ) = ∇ × (r∇

²

Ψ) (88) and we see that (88) is identical to (70) with c = r. The results (62) and (70) are the only results that used the assumption that c is a constant vector, and since they remain valid for c = r, all the properties of the vector fields M and N discussed above remain valid, despite the fact that c = r is not a constant vector. Thus we can choose

M = ∇ × (rΨ) = ∇Ψ × r (89)

N = 1

k ∇ × M = 1

k ∇ × (∇Ψ × r) (90)

The next step is to determine the solutions Ψ(r) of the scalar wave equation (61) in spherical coordinates

∇

²

Ψ + k

²

Ψ = 0 (91)

Using the definition of the Laplacian ∇

²

in spherical coordinates, we can rewrite (91) as follows

1 r

²

∂

∂r

r

²

∂Ψ

∂r

+ 1

r

²

sin θ

∂

∂θ

sin θ ∂Ψ

∂θ

+ 1

r

²

sin θ

∂

²

Ψ

∂ϕ

²

+ k

²

Ψ = 0 (92) By separating the variables using

Ψ(r, θ, ϕ) = R(r)Θ(θ)Φ(ϕ) (93)

we obtain three one-dimensional differential equations d

²

Φ

dϕ

²

+ m

²

Φ = 0 (94)

(11)

1 sin θ

d dθ

sin θ dΘ dθ

+

n(n + 1) − m

²

sin

²

θ

Θ = 0 (95)

d dr

r

²

dR

dr

+ [k

²

r

²

− n(n + 1)]R = 0 (96) The even (e) and odd (o) solutions of equation (94) are

Φ

e

= cos(mϕ), Φ

o

= sin(mϕ) (97)

Since we require that Φ(ϕ) is a single valued function, after an increment of 2π, we must have

Φ(ϕ + 2π) = Φ(ϕ) (98)

which requires that m is an integer or zero. For our purposes, using the solutions (97), it is sufficient to use positive integers for m to generate all linearly independent solutions of (94).

The solutions of the equation (95), that are finite at the singular points (θ = 0 and θ = π, where sin θ = 0), are associated Legendre functions of the first kind P

_n^m

(cos θ) of degree n and order m, where n is an integer

n = 0, 1, 2, . . . (99)

and m is limited to n (m ≤ n). The Legendre functions are normalized as follows Z

+1

−1

P

_n^m

(µ)P

_q^m

(µ)dµ = Z

π

0

P

_n^m

(cos θ)P

_q^m

(cos θ) sin θdθ = δ

_nq

2 2n + 1 · (n + m)!

(n − m)! (100) The Legendre functions P

_n^m

(µ) can be calculated using the Rodrigues formula, i.e.

P

_n^m

(µ) = 1

2

ⁿ

n! (1 − µ

²

)

^m/2

d

^(n+m)

dµ

^(n+m)

(µ

²

− 1)

ⁿ

(101)

Introducing a new variable ρ = kr and a new function Z(ρ) = √ρR(ρ) = √

krR(kr), the radial equation (96) becomes

ρ d dρ

ρ dZ

dρ

+

"

ρ

²

−

n + 1

2

#

Z = 0 (102)

which is a Bessel differential equation with Bessel functions of the first and second kind as two linearly independent solutions

Z(ρ) =

( J

_n+¹

2

(ρ)

Y

_n+¹

2

(ρ)

)

(103)

where the order n + 1/2 is half-integral. Using R(ρ) = Z(ρ)/√ρ, we then obtain the solutions of Eq. (96) as follows

R(ρ) = j

n

(ρ) y

n

(ρ)

=





 q

π

2ρ

J

_n+¹

2

(ρ)

q

_π

2ρ

Y

_n+¹

2

(ρ)







(104)

(12)

where j

n

(ρ) and y

n

(ρ) are spherical Bessel functions of order n. The first few spherical Bessel functions are simple elementary functions

j

0

(ρ) = sin ρ

ρ , j

1

(ρ) = sin ρ

ρ

²

− cos ρ

ρ (105)

y

₀

(ρ) = − cos ρ

ρ , y

₁

(ρ) = − cos ρ

ρ

²

− sin ρ

ρ (106)

and higher orders can be generated by recurrence. Any linear combination of j

n

(ρ) and y

n

(ρ) is also a solution to (102). Instead of spherical Bessel functions j

n

(ρ) and y

n

(ρ), the so-called spherical Hankel functions can also be used

h

⁽¹⁾_n

(ρ) = j

n

(ρ) + jy

n

(ρ) (107) h

⁽²⁾_n

(ρ) = j

_n

(ρ) − jy

n

(ρ) (108) Using (93), we can then write down the even (e) and odd (o) solutions of the scalar wave equation (92) as follows

Ψ

emn

= cos(mϕ)P

_n^m

(cos θ)z

n

(kr) (109) Ψ

omn

= sin(mϕ)P

_n^m

(cos θ)z

n

(kr) (110) where z

n

is any of the four spherical Bessel functions j

n

, y

n

, h

⁽¹⁾n

and h

⁽²⁾n

. Using the solutions (109) and (110) in the definitions (89) and (90), we obtain the following vector solutions

M

emn

= ∇ × (rΨ

^emn

) , M

omn

= ∇ × (rΨ

^omn

) (111) N

_emn

= 1

k ∇ × M

^emn

, N

_omn

= 1

k ∇ × M

^omn

(112)

Using the definition of the curl in spherical coordinates

∇ × A = 1 r sin θ

∂

∂θ (A

ϕ

sin θ) − ∂A

θ

∂ϕ

ˆ r + 1

r

1 sin θ

∂A

r

∂ϕ − ∂

∂r (rA

ϕ

)

θ ˆ

+ 1 r

∂

∂r (rA

θ

) − ∂A

r

∂θ

ˆ

ϕ (113)

and the definition of the radius vector in spherical coordinates

r = rˆ r (114)

we readily obtain

M = ∇ × (rΨ) = ∇ × (rΨˆr) = 1 r sin θ

∂

∂ϕ (rΨ)ˆ θ − 1 r

∂

∂θ (rΨ) ˆ ϕ

= 1

sin θ

∂Ψ

∂ϕ θ ˆ − ∂Ψ

∂θ ϕ ˆ (115)

Substituting here (109) and (110), we obtain respectively M

emn

= −m

sin θ sin(mϕ)P

_n^m

(cos θ)z

n

(ρ)ˆ θ − cos(mϕ) dP

_n^m

(cos θ)

dθ z

n

(ρ) ˆ ϕ (116)

(13)

M

_omn

= +m

sin θ cos(mϕ)P

_n^m

(cos θ)z

n

(ρ)ˆ θ − sin(mϕ) dP

_n^m

(cos θ)

dθ z

n

(ρ) ˆ ϕ (117) and we see that M is perpendicular to the vector c = r as required. Using again the definition of the curl in spherical coordinates (113), and the results (116) and (117), we also obtain

N

_emn

= 1

k ∇ × M

^emn

= 1 kr sin θ

"

∂

∂θ

M

^(ϕ)_emn

sin θ

− ∂M

^(θ)_emn

∂ϕ

# ˆ r −

− 1 kr

∂

∂(kr)

krM

^(ϕ)_emn

θ ˆ + 1 kr

∂

∂(kr)

krM

^(θ)_emn

ˆ ϕ =

= z

_n

(ρ)

ρ cos(mϕ)

− 1 sin θ

d dθ

sin θ dP

_n^m

dθ

+ m

²

sin

²

θ P

_n^m

ˆ r +

+ cos(mϕ) dP

_n^m

dθ

1 ρ

d

dρ (ρz

n

) ˆ θ − m sin(mϕ) P

_n^m

sin θ

1 ρ

d

dρ (ρz

n

) ˆ ϕ =

= z

_n

(ρ)

ρ cos(mϕ)n(n + 1)P

_n^m

(cos ϕ) ˆ r + + cos(mϕ) dP

_n^m

(cos θ)

dθ 1 ρ

d

dρ [ρz

_n

(ρ)]ˆ θ − m sin(mϕ) P

_n^m

(cos θ) sin θ

1 ρ

d

dρ [ρz

_n

(ρ)] ˆ ϕ (118) and

N

omn

= 1

k ∇ × M

^omn

= 1 kr sin θ

"

∂

∂θ

M

^(ϕ)_omn

sin θ

− ∂M

^(θ)_omn

∂ϕ

# ˆ r −

− 1 kr

∂

∂(kr)

krM

^(ϕ)_omn

θ ˆ + 1 kr

∂

∂(kr)

krM

^(θ)_omn

ˆ ϕ =

= z

n

(ρ)

ρ sin(mϕ)

− 1 sin θ

d dθ

sin θ dP

_n^m

dθ

+ m

²

sin

²

θ P

_n^m

ˆ r +

+ sin(mϕ) dP

_n^m

dθ

1 ρ

d

dρ (ρz

_n

) ˆ θ + m cos(mϕ) P

_n^m

sin θ

1 ρ

d

dρ (ρz

_n

) ˆ ϕ =

= z

n

(ρ)

ρ sin(mϕ)n(n + 1)P

_n^m

(cos ϕ) ˆ r + + sin(mϕ) dP

_n^m

(cos θ)

dθ 1 ρ

d

dρ [ρz

n

(ρ)] ˆ θ + m cos(mϕ) P

_n^m

(cos θ) sin θ

1 ρ

d

dρ [ρz

n

(ρ)] ˆ ϕ (119) where we have used the differential equation (95), in the form

− 1 sin θ

d dθ

sin θ dP

_n^m

dθ

+ m

²

sin

²

θ P

_n^m

= n(n + 1)P

_n^m

(120) to simplify the radial components of the N -fields in (118) and (119). The vector spherical harmonics (116)-(119) constitute a complete set of vector functions, such that any electric field E can be written as a series expansion in these functions

E =

∞

X

n=1 n

X

m=1

(B

emn

M

emn

+ B

omn

M

omn

+ A

emn

N

emn

+ A

omn

N

omn

) (121)

(14)

2.4 Expansion of the incident fields

The general expansion (121) implies that the incident electric field (7), i.e.

E

i

= E

0

e

⁻^jkz

x ˆ = E

0

e

⁻^{jkr cos θ}

(sin θ cos ϕ ˆ r + cos θ cos ϕ ˆ θ − sin ϕ ˆ ϕ) (122)

can also be expanded in terms of vector spherical harmonics (116)-(119) as follows

E

_i

=

∞

X

n=1 n

X

m=1

(B

emn

M

_emn

+ B

omn

M

_omn

+ A

emn

N

_emn

+ A

omn

N

_omn

) (123)

In order to determine the coefficients B

emn

for this particular expansion, we can construct the following integral

Z

π 0

sin θdθ Z

2π

0

dϕE

i

· M

^emn

=

∞

X

q=1 q

X

p=1

B

epq

Z

π 0

sin θdθ Z

2π

0

dϕM

emn

· M

^epq

+ B

opq

Z

π 0

sin θdθ Z

2π

0

dϕM

emn

· M

^opq

+A

epq

Z

π 0

sin θdθ Z

2π

0

dϕM

emn

· N

^epq

+ A

opq

Z

π 0

sin θdθ Z

2π

0

dϕM

emn

· N

^opq

(124) In order to simplify the right-hand side of the equation (124), we need to establish four orthogonality conditions between the vector spherical harmonics. Let us begin with the second integral on the right-hand side of (124), and utilize the explicit expressions (116) and (117) to obtain

Z

π 0

sin θdθ Z

2π

0

dϕM

emn

· M

^opq

= Z

π

0

sin θdθ h

− mp

sin

²

θ P

_n^m

P

_q^p

z

n

(ρ)z

q

(ρ)

· Z

2π

0

sin(mϕ) cos(pϕ)dϕ + dP

_n^m

dθ

dP

_q^p

dq z

n

(ρ)z

q

(ρ) · Z

2π

0

cos(mϕ) sin(pϕ)dϕ

≡ 0 (125)

where we used the orthogonality relation for the trigonometric functions Z

2π

0

cos(aϕ) sin(bϕ)dϕ ≡ 0 for a, b = integers (126) Next we consider the third integral on the right-hand side of (124), and utilize the explicit expressions (116) and (118) to obtain

Z

π 0

sin θdθ Z

2π

0

dϕM

emn

· N

^epq

= Z

π

0

sin θdθ

− m

sin θ P

_n^m

dP

_q^p

dθ z

n

(ρ) 1 ρ

d

dρ [ρz

q

(ρ)]

· Z

2π

0

sin(mϕ) cos(pϕ)dϕ + p

sin θ P

_q^p

dP

_n^m

dθ z

_n

(ρ) · 1 ρ

d

dρ [ρz

_q

(ρ)]

Z

2π 0

cos(mϕ) sin(pϕ)dϕ

≡ 0

(127)

(15)

where we again used the orthogonality relation for the trigonometric functions (126).

Now we turn to the fourth integral on the right-hand side of (124), and utilize the explicit expressions (116) and (119) to obtain

Z

2π 0

dϕM

_emn

· N

opq

= − m

sin θ P

_n^m

dP

_q^p

dθ · z

n

(ρ) 1 ρ

d

dρ [ρz

_q

(ρ)]

Z

2π

0

sin(mϕ) cos(pϕ)dϕ−

−P

q^p

dP

_n^m

dθ

p

sin θ · z

ⁿ

(ρ) 1 ρ

d

dρ [ρz

q

(ρ)]

Z

2π 0

cos(mϕ) cos(pϕ)dϕ =

= −πδ

^mp

z

n

(ρ) ρ

d

dρ [ρz

q

(ρ)] m sin θ

P

_n^m

dP

_q^m

dθ + P

_q^m

dP

_n^m

dθ

=

= −mπδ

^mp

z

n

(ρ) ρ

d

dρ [ρz

q

(ρ)] 1 sin θ

d

dθ P

_n^m

P

_q^m

(128) where we used the orthogonality relations for the trigonometric functions

Z

2π 0

cos(mϕ) cos(pϕ)dϕ = Z

2π

0

sin(mϕ) sin(pϕ)dϕ = πδ

_mp

(129) with m and p being integers. Using (128) we can further calculate

Z

π 0

sin θdθ Z

2π

0

dϕM

emn

· N

^opq

= −mπδ

^mp

z

n

(ρ) ρ

d

dρ [ρz

q

(ρ)] · (P

n^m

P

_q^m

)

^π₀

≡ 0 (130) where we used the formula (101), which clearly indicates that for all m > 0, we have

P

_n^m

(±1) ≡ 0 (131)

or

P

_n^m

(cos π) = P

_n^m

(cos 0) = 0 (132) Thus we have shown that the second, third and fourth integrals on the right-hand side of (124) vanish. Let us now consider the first integral on the right-hand side of (124), and utilize the explicit expression (116) to obtain

Z

2π 0

dϕM

_emn

· M

epq

= mp

sin

²

θ P

_n^m

(cos θ)P

_q^p

(cos θ) · z

n

(ρ)z

_q

(ρ) Z

2π

0

sin(mϕ) sin(pϕ)dϕ

+ dP

_n^m

(cos θ) dθ

dP

_q^p

(cos θ)

dθ · z

ⁿ

(ρ)z

q

(ρ) Z

2π

0

cos(mϕ) cos(pϕ)dϕ

= πδ

mp

z

n

(ρ)z

q

(ρ) dP

_n^m

dθ

dP

_q^m

dθ + P

_n^m

P

_q^m

sin

²

θ

(133) where we used the orthogonality relations for the trigonometric functions (129). Using (133) we can write the complete integral

Z

π 0

sin θdθ Z

2π

0

dϕM

_epq

· M

emn

= πδ

_mp

z

_n

(ρ)z

_q

(ρ) Z

π

0

sin θ dP

_n^m

dθ

dP

_q^m

dθ + P

_n^m

P

_q^m

sin

²

θ

dθ

(134)

(16)

In order to calculate the integral on the right-hand side of (134), we refer to the differential equation (95), which implies

− 1 sin θ

d dθ

sin θ dP

_n^m

dθ

+ m

²

sin

²

θ P

_n^m

= n(n + 1)P

_n^m

(135)

− 1 sin θ

d dθ

sin θ dP

_q^m

dθ

+ m

²

sin

²

θ P

_q^m

= q(q + 1)P

_q^m

(136) and construct the following expressions

sin θ dP

_n^m

dθ

dP

_q^m

dθ = d

dθ

sin θP

_n^m

dP

_q^m

dθ

− P

n^m

d dθ

sin θ dP

_q^m

dθ

(137)

sin θ dP

_n^m

dθ

dP

_q^m

dθ = d

dθ

sin θP

_q^m

dP

_n^m

dθ

− P

q^m

d dθ

sin θ dP

_n^m

dθ

(138) Using (137)-(138), we can write

sin θ dP

_n^m

dθ

dP

_q^m

dθ = 1

2 sin θ dP

_n^m

dθ

dP

_q^m

dθ + sin θ dP

_q^m

dθ

dP

_n^m

dθ

=

= 1 2

d dθ

sin θP

_n^m

dP

_q^m

dθ + sin θP

_q^m

dP

_n^m

dθ

+

+ 1

2 sin θP

_n^m

−1 sin θ

d dθ

sin θ dP

_q^m

dθ

+ 1

2 sin θP

_q^m

−1 sin θ

d dθ

sin θ dP

_n^m

dθ

=

= 1 2

d dθ

sin θ

P

_n^m

dP

_q^m

dθ + P

_q^m

dP

_n^m

dθ

+

+ 1

2 sin θP

_n^m

− m

²

sin

²

θ P

_q^m

+ q(q + 1)P

_q^m

+ 1

2 sin θP

_q^m

− m

²

sin

²

θ P

_n^m

+ n(n + 1)P

_n^m

=

= 1 2

d dθ

sin θ

P

_n^m

dP

_q^m

dθ + P

_q^m

dP

_n^m

dθ

+

− sin θ m

²

sin

²

θ P

_q^m

P

_n^m

+ 1

2 [n(n + 1) + q(q + 1)] sin θ P

_n^m

P

_q^m

(139) Rearranging (139) we obtain the integrand of the integral on the right-hand side of (134) in the form