Mie scattering of Laguerre-Gaussian beams: photonic nanojets and near-field optical vortices

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by  Alexei D. Kiselev

Mie scattering of Laguerre-Gaussian beams: photonic nanojets
and near-field optical vortices
Alexei D. Kiselev1,∗ and Dmytro O. Plutenko1,†
1Institute of Physics of National Academy of Sciences of Ukraine,
prospekt Nauki 46, 03680 Ky¨ıv, Ukraine
(Dated: January 30, 2014)
We study Mie light scattering of Laguerre-Gaussian (LG) beams remodelled using
the method of far-field matching. The theoretical results are applied to examine the
optical field in the near-field region for purely azimuthal LG beams characterized
4 by the nonzero azimuthal mode number m . The mode number m is found to
LG LG
1
have a profound effect on the morphology of photonic nanojets and the near-field
0
2 structure of optical vortices associated with the components of the electric field.

n
a PACS numbers: 42.25.Fx, 42.68.Mj, 42.25.Bs
J
Keywords: light scattering; Laguerre-Gaussianbeams; photonic nanojets; optical vortices;
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2

]
s
c I. INTRODUCTION
i
t
p
o The problem of light scattering by particles of one medium embedded in another has a
.
s long history, dating back more than a century to the classical exact solution due to Mie [1].
c
The Mie solution applies to scattering by uniform spherical particles with isotropic dielectric
i
s
properties. The analysis of a Mie–type theory uses a systematic expansion of the electro-
y
h magnetic field over vector spherical harmonics [2–7]. The specific form of the expansions
p
is also known as the T–matrix ansatz that has been widely used in the related problem
[
of light scattering by nonspherical particles [5, 8, 9]. More recently this strategy has been

1
successfully applied to optically anisotropic particles [10–18].
v
5 In its original form the Mie theory assumes that the scatterer is illuminated with a plane
8
electromagnetic wave. For laser beams, it is generally necessary to go beyond the plane-wave
5
7 approximation that may severely break down when the beam width becomes of order of the
. scatterer size. The problem of light scattering from arbitrary shaped laser beams has now
1
0 a more than two decade long history [19–23] and has been the key subject of the Mie–type
4
theory — the so-called generalized Lorenz–Mie theory (GLMT) [7, 24] — extended to the
1
: case of arbitrary incident-beam scattering.
v
Mathematically, in such generalization of the Mie theory, the central and the most im-
i
X
portant task is to describe illuminating beams in terms of expansions over a set of basis
r
a wavefunctions (for the spherical coordinate system, it is the multipole expansion over the
basis of vector spherical wavefunctions). In GLMT, a variety of formally exact (the quadra-
ture and double quadrature formulas) and approximate (the finite series and localized ap-
proximations) methods [25] were developed to evaluate the expansion coefficients that are
referred to as the beam shape coefficients (for a recent review see Ref. [26] and references
therein).
∗ Email address: [email protected]
† Email address: [email protected]
2
The central problem with laser beams is due to the fact that in their standard mathemat-
ical form these beams are not radiation fields which are solutions to Maxwell’s equations.
Typically, the analytical treatment of laser beams is performed using the paraxial approxi-
mation [27] and the beams are described as pseudo-fields which are only approximate solu-
tions of the vector Helmholtz equation (higher order corrections can be used to improve the
accuracy of the paraxial approximation [27, 28]).
Unfortunately, multipole expansions do not exist for such approximate pseudo-fields.
Therefore, some remodelling procedure must be invoked to obtain a real radiation field
which can be regarded as an approximation to the original paraxial beam.
The basic concept that might be calledmatching the fields on a surface lies at the heart of
various traditional approaches to the laser beam remodelling and is based on the assumption
that there is a surface where the actual incident field is equal to the paraxial field. Examples
of physically reasonable and natural choice arescatterer-independent matching surfaces such
as a far-field sphere [29], the focal plane (for beams with well-defined focal planes) [29, 30],
andaGaussianreference sphere representing alens [31]. Giventheparaxialfielddistribution
onthe matching surface, the beam shape coefficients can beevaluated using either numerical
integration or the one-point matching method [29].
An alternative approach is to describe analytically propagation of a laser beam, which is
known in the paraxial limit, without recourse to the paraxial approximation. In Refs. [32–
36] this strategy has been applied to the important case of Laguerre–Gaussian (LG) beams
using different methods such as the vectorial Rayleigh–Sommerfeld formulas [33, 36], the
vector angular spectrum method [35], approximating LG beams by nonparaxial beams with
(near) cylindrical symmetry [32, 34].
The nonparaxial beams are solutions of Maxwell’s equations and the beam shape coeffi-
cients can be computed using the methods of GLMT. In recent studies of light scattering by
spherical and spheroidal particles illuminated with LG beams [37, 38], the analytical results
of Ref. [34] were used to calculate the beam shape coefficients.
It is now well known [39] that LG beams represent optical vortex beams that carry angu-
lar momentum of two kinds: spin angular momentum associated with the polarization state
of the beam and orbital angular momentum related to spatial variations of the field. These
variations derive from the helical structure of the wavefronts comprising the beam or, equiv-
alently, from a phase singularity at the beam axis. The topological charge characterizing the
phase singularity andassociated orbital angular momentum gives rise to distinctive phenom-
ena such as soliton generation [40], entanglement of photon quantum states, orbital angular
momentum exchange with atoms and molecules (in addition to the collection of papers [39],
see reviews in Ref. [41]), rotation and orbital motion of spherical particles illuminated with
LG beams [42, 43].
In this paper the problem of light scattering from LG beams that represent laser beams
exhibiting a helical phase front and carrying a phase singularity will be of our primary
interest. Inour calculationswe shall followRef.[15]anduse theT–matrixapproachinwhich
the far-field matching method is combined with the results for nonparaxial propagation of
LG beams [35, 36]. Our goal is to examine the near-field structure of electromagnetic field
depending on the parameters characterizing both the beam and the scatterer.
This structure has recently attracted considerable attention that was stimulated by an
upsurge of interest to the so-called photonic nanojets and their applications (for a review
see Ref. [44]). These nanojets were originally identified in finite-difference-time-domain
simulations [45, 46] as narrow, high-intensity electromagnetic beams that propagate into
3
background medium from the shadow-side surface of a plane-wave illuminated dielectric
microcylinder [45] or microsphere [46] of diameter greater than the illuminating wavelength.
In other words, a photonic nanojet can be regarded as a localized, subdiffractional, non-
evanescent light focus propagating along the line of incidence.
The bulk of theoretical studies devoted to nanojets [47–52] has been predominantly fo-
cused on the case of plane-wave illumination. In this paper we intend to fill the gap.
The layout of the paper is as follows. In Sec. II, we describe our theoretical approach
and then, in Sec. III, we obtain the analytical results for the beam shape coefficients of LG
beams. The numerical procedure and the results of numerical computations representing
the near-field intensity distributions and phase maps of electric field components for purely
azimuthal LG beams are presented in Sec. IV.
Finally, in Sec. V, we present our results and make some concluding remarks.
II. T–MATRIX FORMULATION OF LORENZ–MIE THEORY
WeconsiderscatteringbyasphericalparticleofradiusR embeddedinauniformisotropic
p
dielectric medium with dielectric constant ǫ and magnetic permeability µ . The dielec-
med med
tric constant and magnetic permittivity of the particle are ǫ and µ , respectively.
p p
In this subsection we remind the reader about the relationship between Maxwell’s equa-
tions in the region of a scatterer and the formulation of scattering properties in terms of the
T–matrix [2, 5]. Our formulation is slightly non-standard and closely follows to the line of
our presentation given in Ref. [15].
Weshall need to write theMaxwell equations for a harmonicelectromagnetic wave (time–
dependent factor is exp iωt ) in the form:
{− }
µ
ik−1∇ E = iH, (1a)
− i × n
i
n med, r > R
ik−1∇ H = iE, i = p (1b)
i × µ p, r < R
i ( p
where n = √ǫ µ is the refractive index outside the scatterer (in the ambient
med med med
medium), where r > R (i = med) and k = k = n k (k = ω/c = 2π/λ is
p i med med vac vac
the free–space wavenumber); n = √ǫ µ is the refractive index for the region inside the
p p p
spherical particle (scatterer), where r < R (i = p) and k = k = n k .
p i p p vac
A. Vector spherical harmonics and Wigner D functions
The electromagnetic field can always be expanded using the vector spherical harmonic
basis, Y (φ,θ) Y (ˆr) (δ = 0, 1) [53], as follows:
j+δjm j+δjm
≡ ±
E = E = p(0)(r)Y(0)(ˆr)+p(e)(r)Y(e)(ˆr)+p(m)(r)Y(m)(ˆr) , (2a)
jm jm jm jm jm jm jm
Xjm Xjm h i
H = H = q(0)(r)Y(0)(ˆr)+q(e)(r)Y(e)(ˆr)+q(m)(r)Y(m)(ˆr) , (2b)
jm jm jm jm jm jm jm
Xjm Xjm h i
4
where Y(m) = Y and Y(e) = [j/(2j + 1)]1/2Y + [(j + 1)/(2j + 1)]1/2Y are
jm jjm jm j+1jm j−1jm
electric and magnetic harmonics respectively, and Y(0) = [j/(2j + 1)]1/2Y [(j +
jm j−1jm −
1)/(2j + 1)]1/2Y are longitudinal harmonics. In Ref. [15], it was shown that the
j+1jm
spherical harmonics can be conveniently expressed in terms of the Wigner D–functions [53,
54] as follows
Y(m)(ˆr) = N /√2 Dj∗ (ˆr)e (ˆr) Dj∗ (ˆr)e (ˆr) , (3a)
jm j m,−1 −1 − m,1 +1
Yj(em)(ˆr) = Nj/√2 (cid:8)Dmj∗,−1(ˆr)e−1(ˆr)+Dmj∗,1(ˆr)e+1(ˆr) (cid:9), (3b)
Yj(0m)(ˆr) = NjDmj∗,0(cid:8)(ˆr)e0(ˆr) = Yjm(ˆr)ˆr, Nj = [(2j +(cid:9)1)/4π]1/2, (3c)
where e (ˆr) = (e (ˆr) ie (ˆr))/√2; e (ˆr) ϑˆ = (cosθcosφ,cosθsinφ, sinθ), e (ˆr)
±1 x y x y
∓ ± ≡ − ≡
ϕˆ = ( sinφ,cosφ,0) are the unit vectors tangential to the sphere; φ (θ) is the azimuthal
−
(polar) angle of the unit vector ˆr = r/r = (sinθcosφ,sinθsinφ,cosθ) e (ˆr) e (ˆr).
0 z
≡ ≡
(Hats will denote unit vectors and an asterisk will indicate complex conjugation.)
Note that, for the irreducible representation of the rotation group with the angular num-
ber j, the D-functions, Dj (α,β,γ) = exp( imα)dj (β)exp( iµγ), give the elements of
mν − mµ −
the rotation matrix parametrized by the three Euler angles [53, 54]: α, β and γ. In formu-
las (3) and throughout this paper, we assume that γ = 0 and Dj (ˆr) Dj (φ,θ,0). These
mν ≡ mν
D-functions meet the following orthogonality relations [53, 54]
hDmj∗ν(ˆr)Dmj′′ν(ˆr)iˆr = 2j4+π 1 δjj′δmm′ , (4)
2π π
where f ˆr dφ sinθdθf(ˆr) and f(ˆr) f(φ,θ). The orthogonality condition (4)
h i ≡ ≡
Z0 Z0
and Eqs. (3) show that a set of vector spherical harmonics is orthonormal:
hYj(αm)∗(ˆr)·Yj(β′m)′(ˆr)iˆr = δαβ δjj′δmm′ . (5)
We can now use the relations [53]
N Dj∗(ˆr) = Y (ˆr), (6)
j m0 jm
i
N Dj∗ (ˆr) = n ∂ + ∂ Y (ˆr), n [j(j +1)]−1/2 (7)
j m±1 j ∓ θ sinθ φ jm j ≡
(cid:20) (cid:21)
where ∂ stands for a derivative with respect to x and Y (ˆr) is the normalized spherical
x jm
function
(2j +1)(j m)!
Y (φ,θ) = N exp(imφ)dj (θ) = ( 1)m − exp(imφ)P m(cosθ) (8)
jm j m,0 − s 4π(j +m)! j
expressed in terms of the associated Legendre polynomial of degree j and order m
( 1)m/(2jj!)(1 x2)m/2∂j+m(x2 1)j, m > 0
P m(x) = − − x − , (9)
j ((−1)|m|(j −|m|)!/(j +|m|)!Pj|m|(x), m < 0
5
and derive the following expressions for the magnetic and electric vector spherical functions
m
Y(m)(ˆr) = in [∂ Y ]ϕˆ i Y ϑˆ =
jm − j θ jm − sinθ jm
n LY = iˆr hY(e), i (10)
j jm − × jm
m
Y(e)(ˆr) = n [∂ Y ]ϑˆ +i Y ϕˆ =
jm j θ jm sinθ jm
n r∇Y = hiˆr Y(m), i (11)
j jm − × jm
where L is the operator of angular momentum given by
iL = r ∇ = ϕˆ ∂ ϑˆ [sinθ]−1∂ . (12)
θ φ
× −
Formulas (10) and (11) give the vector spherical harmonics (3) rewritten in the well-known
standard form [55].
B. Wave functions and T–matrix
(λ)
The electric field (2a) is completely described by the coefficients p (r) and similarly
{ jm }
(λ)
the magnetic field (2b) is described by q (r) with λ = 0,e,m . In order to find the
{ jm } { }
coefficient functions we can use separation of variables. This implies that the expansions (2)
must be inserted into Maxwell’s equations (1). The coefficient functions then can be derived
by solving the resulting system of equations. In the simplest case of an isotropic medium
the coefficient functions can be expressed in terms of spherical Bessel functions, j (x) =
j
[π/(2x)]1/2J (x), and spherical Hankel functions [56], h(1,2)(x) = [π/(2x)]1/2H(1,2) (x),
j+1/2 j j+1/2
and their derivatives.
Alternatively, it is well-known (a discussion of the procedure can be found, e.g., in
Ref. [57]) that solutions of the scalar Helmholtz equation, (∇2 + k2)ψ(r) = 0, taken in
the form
ψ(α) = n z(α)(ρ)Y(ˆr), n [j(j +1)]−1/2, (13)
jm j j j ≡
(α)
whereρ = kr andz (ρ)iseitherasphericalBessel orHankelfunction, canbeusedtoobtain
j
the following solenoidal solutions of the vector Helmholtz equation, ∇ [∇ Ψ] = k2Ψ:
× ×
M(α)(ρ,ˆr) = Lψ(α) = z(α)(ρ)Y(m)(ˆr), (14)
jm jm j jm
j(j +1)
N(α)(ρ,ˆr) = ik−1∇ M(α) = z(α)(ρ)Y(0)(ˆr)+Dz(α)(ρ)Y(e)(ˆr), (15)
jm − × jm ρ j jm j jm
p
whereDf(x) x−1∂ (xf(x)). Thevectorwave functions, M(α) andN(α), arelinkedthrough
≡ x jm jm
the identities
i∇ M(α) = kN(α), i∇ N(α) = kM(α) (16)
− × jm jm × jm jm
and their linear combination represents the expansions (2) over the vector spherical harmon-
ics.
6
There are three cases of these expansions that are of particular interest. They corre-
spond to the incident wave, E ,H , the outgoing scattered wave, E ,H and the
inc inc sca sca
{ } { }
electromagnetic field inside the scatterer, E ,H :
p p
{ }
E = α(α)M(α)(ρ ,ˆr)+β(α)N(α)(ρ ,ˆr) , α inc,sca,p (17a)
α jm jm i jm jm i ∈ { }
jm
X(cid:2) (cid:3)
H = n /µ α(α)N(α)(ρ ,ˆr) β(α)M(α)(ρ ,ˆr) , (17b)
α i i jm jm i − jm jm i
jm
X(cid:2) (cid:3)
j (ρ), α = inc
j
med, α inc,sca
i = ∈ { } , z(α)(ρ ) = h(1)(ρ), α = sca , (17c)
p, α = p j i  j
(
j (ρ ), α = p
j p
where ρ = k r ρ, ρ = k r nρ, and n = n /n is the ratio of refractive indexes
med med p p p med
≡ ≡
also known as the optical contrast.
Thus outside the scatterer the electromagnetic field is a sum of the incident wave field
(inc) (sca) (1)
with z (ρ) = j (ρ) and the scattered waves with z (ρ) = h (ρ) as required by the
j j j j
Sommerfeld radiation condition.
In the far field region (ρ 1), the asymptotic behaviour of the spherical Bessel and
≫
Hankel functions is known [56]:
ij+1h(1)(ρ),ijDh(1)(ρ) exp(iρ)/ρ, (18)
j j ∼
( i)j+1h(2)(ρ),( i)jDh(2)(ρ) exp( iρ)/ρ, (19)
− j − j ∼ −
ij+1j (ρ),ij+1Dj (ρ) exp(iρ) ( 1)jexp( iρ) /(2ρ). (20)
j j+1
∼ − − −
(cid:2) (1) (cid:3)
So, the spherical Hankel functions of the first kind, h (ρ), describe the outgoing waves,
j
(2)
whereas those of the second kind, h (ρ), represent the incoming waves.
j
The incident field is the field that would exist without a scatterer and therefore includes
both incoming and outgoing parts (see Eq. (20)) because, when no scattering, what comes
in must go outwards again. As opposed to the spherical Hankel functions that are singular
at the origin, the incident wave field should be finite everywhere and thus is described by
the regular Bessel functions j (ρ).
j
(inc) (inc)
Now the incident wave is characterized by amplitudes α , β and the scattered out-
jm jm
(sca) (sca)
going waves are similarly characterized by amplitudes α , β . So long as the scattering
jm jm
(sca) (sca)
problem is linear, the coefficients α and β can be written as linear combinations of
jm jm
(inc) (inc)
α and β :
jm jm
α(sca) = T11 α(inc) +T12 β(inc) ,
jm jm,j′m′ j′m′ jm,j′m′ j′m′
jX′,m′h i
β(sca) = T21 α(inc) +T 22 β(inc) . (21)
jm jm,j′m′ j′m′ jm,j′m′ j′m′
jX′,m′h i
These formulae define the elements of the T–matrix in the most general case.
In general, the outgoing wave with angular momentum index j arises from ingoing waves
of all other indices j′. In such cases we say that the scattering process mixes angular
7
momenta [8]. The light scattering from uniformly anisotropic scatterers [15, 58] provides
an example of such a scattering process. In simpler scattering processes, by contrast, such
angular momentum mixing does not take place. Many quantum scattering processes and
classical Mie scattering belong to this category. For example, radial anisotropy keeps intact
spherical symmetry of the scatterer [10, 15, 18]. The T–matrix of a spherically symmetric
scatterer is diagonal over the angular momenta and the azimuthal numbers: Tnn′ =
jj′,mm′
δjj′δmm′Tjnn′.
In order to calculate the elements of T-matrix and the coefficients α(p) and β(p), we
jm jm
need to use continuity of the tangential components of the electric and magnetic fields as
boundary conditions at r = R (ρ = k R x).
p med p
≡
(p) (p)
So, the coefficients of the expansion for the wave field inside the scatterer, α and α ,
jm jm
are expressed in terms of the coefficients describing the incident light as follows
(inc)
α
(p) jm
iα = , µ = µ /µ , (22)
jm µ−1v (x)u′(nx) n−1v′(x)u (nx) p med
j j − j j
(inc)
β
(p) jm
iβ = , n = n /n , (23)
jm n−1v (x)u′(nx) µ−1v′(x)u (nx) p med
j j − j j
(1)
where x = k R , u (x) = xj (x) and v (x) = xh (x). The similar result relating the
med p j j j j
scattered wave and the incident wave
n−1u′(x)u (nx) µ−1u (x)u′(nx)
α(sca) = T11α(inc) = j j − j j α(inc), (24)
jm j jm µ−1v (x)u′(nx) n−1v′(x)u (nx) jm
j j − j j
µ−1u (x)u′(nx) n−1u′(x)u (nx)
β(sca) = T22β(inc) = j j − j j β(inc), (25)
jm j jm n−1v (x)u′(nx) µ−1v′(x)u (nx) jm
j j − j j
defines the T-matrix for the simplest case of a spherically symmetric scatterer. In addition,
since the parity of electric and magnetic harmonics with respect to the spatial inversion
ˆr ˆr ( φ,θ φ+π,π θ ) is different
→ − { } → { − }
Y(m)( ˆr) = ( 1)jY(m)(ˆr), Y(e)( ˆr) = ( 1)j+1Y(e)(ˆr), (26)
jm − − jm jm − − jm
where f(ˆr) f(φ,θ) and f( ˆr) f(φ + π,π θ), they do not mix provided the mirror
≡ − ≡ −
symmetry has not been broken. In this case the T-matrix is diagonal and T12 = T21 = 0.
j j
The diagonal elements T11 and T22 are also called the Mie coefficients.
j j
III. INCIDENT WAVE BEAMS
The formulas (22)- (25) are useful only if the expansion for the incident light beam is
known. First we briefly review the most studied and fundamentally important case where
the incident light is represented by a plane wave.
8
A. Plane waves
The electric field of a transverse plane wave propagating along the direction specified by
a unit vector kˆ is
inc
E = E(inc)exp(ik r), E(inc) = E(inc)e (kˆ ), k = kkˆ . (27)
inc inc · ν ν inc inc inc
ν=±1
X
where the basis vectors e (kˆ ) are perpendicular to kˆ . Then the vector version of the
±1 inc inc
well known Rayleigh expansion (see, for example, [2])
∞ l
exp(iρkˆ ˆr) = 4π ilj (ρ)Y (ˆr)Y∗ (kˆ), ρ kr (28)
· l lm lm ≡
l=0 m=−l
X X
which is given by
e (kˆ)exp[iρ kˆ ˆr ] = α Dj (kˆ) iνM (ρ,ˆr) N (ρ,ˆr) , ν = 1, (29)
ν · j mν jm − jm ±
(cid:0) (cid:1) Xjm n o
where α = ij+1[2π(2j + 1)]1/2, immediately gives the expansion coefficients for the plane
j
wave
α(inc) = iα Dj (kˆ )νE(inc), β(inc) = α Dj (kˆ )E(inc), (30)
jm j mν inc ν jm − j mν inc ν
ν=±1 ν=±1
X X
where Dj is the Wigner D-function.
mm′
In the far field region, the electric field of scattered wave is related to the polarization
vector of the plane wave through the scattering amplitude matrix as follows [2, 8, 59]
Eν(sca) ≡ (eν∗(kˆsca),Esca) = ρ−1exp(iρ) Aνν′(kˆsca,kˆinc)Eν(i′nc), ν = ±1 (31)
ν′=±1
X
where kˆ = ˆr. For a spherically symmetric scatterer, the expression for the scattering
sca
amplitude matrix in terms of T-matrix is given by
Aνν′(kˆsca,kˆinc) = Ajνν′(kˆsca,kˆinc) =
j
X
= i (j +1/2)D˜j (kˆ ,kˆ ) νν′T 11 iνT12 +iν′T21 +T22 , (32a)
− νν′ sca inc j − j j j
j
X (cid:2) (cid:3)
D˜j (kˆ ,kˆ ) = Dj∗(kˆ )Dj (kˆ ). (32b)
νν′ sca inc mν sca mν′ inc
m
X
Equation (32b) shows that the scattering amplitude matrix (32a) depends only on the angle
between kˆ and kˆ . All far-field scattering characteristics of the system can be computed
inc sca
from the scattering amplitude matrix.
9
B. Far-field matching
Now we consider a more general case where an incident electromagnetic wave is written
as a superposition of propagating plane waves:
E (r) E (ρ,ˆr) = exp(iρkˆ ˆr)E (kˆ) , E (kˆ) = E (kˆ)e (kˆ), (33a)
inc inc inc kˆ inc ν ν
≡ h · i
ν=±1
X
n
H (r) H (ρ,ˆr) = exp(iρkˆ ˆr) kˆ E (kˆ) , (33b)
inc inc inc kˆ
≡ µ h · × i
(cid:2) (cid:3)
2π π
where f dφ sinθ dθ f.
kˆ k k k
h i ≡
Z0 Z0
Our first step is to examine asymptotic behavior of the wave field (33) in the far-field
region, ρ 1. The results can be easily obtained by using the asymptotic formula for a
≫
plane wave (see, e.g., [5])
2πi
exp(iρkˆ ˆr) − exp(iρ)δ(kˆ ˆr) exp( iρ)δ(kˆ +ˆr) at ρ 1, (34)
· ∼ ρ − − − ≫
(cid:2) (cid:3)
where δ(kˆ ˆr) is the solid angle Dirac δ-function symbolically defined through the expansion
∓
∞ l
δ(kˆ ˆr) = Y ( ˆr)Y∗ (kˆ). (35)
∓ lm ± lm
l=0 m=−l
X X
Applying the relation (34) to the plane wave superposition (33a) gives the electric field of
the incident wave in the far-field region
1
E (ρ,ˆr) E(∞)(ρ,ˆr) = exp(iρ)E (ˆr)+exp( iρ)E (ˆr) , (36)
inc ∼ inc ρ out − in
E (ˆr) = E ( ˆr), (cid:2) (cid:3) (37)
in out
− −
where E (ˆr) is the far-field angular distribution for the outgoing part of the electric field
out
of the incident wave:
E (ˆr) = 2πiE (ˆr) = E(out)(ˆr)e (ˆr)+E(out)(ˆr)e (ˆr), (38)
out − inc θ θ φ φ
whereas the incoming part of the incident wave is described by the far-field angular distri-
bution E (ˆr).
in
The result for the far-field distribution of the magnetic field (33b) can be written in the
similar form:
1
H (ρ,ˆr) H(∞)(ρ,ˆr) = exp(iρ)H (ˆr)+exp( iρ)H (ˆr) , (39)
inc ∼ inc ρ out − in
H (ˆr) = H ( ˆr), (cid:2) (cid:3) (40)
in out
− −
µ/nH (ˆr) = ˆr E (ˆr), µ/nH (ˆr) = ˆr E ( ˆr). (41)
out out in out
× × −
Formulas (36)-(41) explicitly show that, in the far-field region, the incident wave field is
defined by the angular distribution of the outgoing wave (38). In particular, from these
10
formulas, it is not difficult to obtain the far-field expression for the Poynting vector of the
incident wave S = c/(8π)Re(E H∗ )
inc inc × inc
S (ρ,ˆr) S(∞)(ρ,ˆr) = ρ−2 S (ˆr)+S (ˆr) , (42)
inc ∼ inc in out
S (ˆr) = S ( ˆr), µ/nS (ˆr) = c/(8π) E (ˆr) 2ˆr, (43)
in out (cid:8)out (cid:9)out
− − | |
where E (ˆr) 2 = (E (ˆr) E∗ (ˆr)). From this expression it immediately follows that the
| out | out · out
flux of Poynting vector of the outgoing wave, S (ˆr), through a sphere of sufficiently large
out
radius is exactly balanced by the flux of Poynting vector of the incoming wave, S (ˆr).
inc
Alternatively, the far-field distribution of an incident light beam, E (ˆr), can be found
out
from the expansion over the vector spherical harmonics (17a). The far-field asymptotics for
the vector wave functions that enter the expansion for the incident wave (17)
( i)j+1
M(inc)(ρ,ˆr) − exp(iρ)Y(m)(ˆr) exp( iρ)Y(m)( ˆr) , (44)
jm ∼ 2ρ jm − − jm −
( i)j (cid:2) (cid:3)
N(inc)(ρ,ˆr) − exp(iρ)Y(e)(ˆr) exp( iρ)Y(e)( ˆr) , (45)
jm ∼ 2ρ jm − − jm −
(cid:2) (cid:3)
can be derived from Eqs. (14)-(15) with the help of the far-field relation (20). Substituting
Eqs. (44) and (45) into the expansion (17a) gives the far-field distribution of the form (36)
with
E (ˆr) = 2−1 ( i)j+1α(inc)Y(m)(ˆr)+( i)jβ(inc)Y(e)(ˆr) . (46)
out − jm jm − jm jm
Xjmh i
The coefficients of theincident wave cannow beeasily found astheFourier coefficients of the
far-field angular distribution, E , expanded using the vector spherical harmonics basis (3).
out
The final result reads
αj(imnc) = 2ij+1hYj(mm)∗(ˆr)·Eout(ˆr)iˆr = iαj νhDmjν(kˆ)Eν(kˆ)ikˆ, (47a)
ν=±1
X
βj(imnc) = 2ijhYj(em)∗(ˆr)·Eout(ˆr)iˆr = −αj hDmjν(kˆ)Eν(kˆ)ikˆ. (47b)
ν=±1
X
A comparison between the expressions on the right hand side of Eq. (47) and those for
the plane wave (30) shows that, in agreement with the representation (33a), the result for
plane waves represents the limiting case where the angular distribution is singular: E (kˆ) =
ν
E(inc)δ(kˆ kˆ ).
ν inc
−
By using Eqs. (10) and (11) formulas (47) can be conveniently rewritten in the explicit
form
αj(imnc) = 2njij+1hYj∗m(ˆr)(L·Eout(ˆr))iˆr =
2π π
2n ij dφ dθY∗ (φ,θ) ∂ (sinθE(out)) ∂ E(out) , (48a)
j jm θ φ − φ θ
Z0 Z0 h i
βj(imnc) = −2nj ijhYj∗m(ˆr)(r∇·Eout(ˆr))iˆr =
2π π
2n ij dφ dθY∗ (φ,θ) ∂ (sinθE(out))+∂ E(out) , (48b)
− j jm θ θ φ φ
Z0 Z0 h i