# Generalized optical theorem for propagation invariant beams

Irving Rondón and F. Soto-Eguibar,
Instituto Nacional de Astrofísica Óptica y Electrónica,
Puebla, C.P. 72840 México

###### Abstract

Many practical applications require the analysis of electromagnetic scattering
properties of local structures using different sources of illumination. The
Optical Theorem (OT) is a useful result in scattering theory, relating the
extinction of a structure to the scattering amplitude in the forward
direction. The most common derivation of the OT is given for plane waves but
advances in optical engineering now allow laser beam shaping, which might
require an extended theorem where the impinging source is a structured field.
In this work, we derive an expression for the optical theorem based on
classical electromagnetic theory, for probe sources given in terms of
propagation invariant beams. We obtain a general expression for the
differential scattering cross section using the integral scattering amplitude
approximation in the far field. We also analyze the scattering problem of a
zero order Bessel beam by a dielectric sphere, under the Rayleigh
approximation by varying the angle of incidence.

##  1 Introduction

This work is focused on a fundamental relation in scattering theory, the so-
called optical cross-section theorem or simply the Optical Theorem (OT), which
describes the rate at which energy is distributed from a probing incident wave
field by a scattering object, due to re-radiation and absorption by the
scatterer [1]. The optical theorem has a long and interesting history; it
appeared in electromagnetic theory more than one hundred years ago and similar
theorems can be found in acoustical scattering and quantum mechanics [2]. This
classical problem is covered in advanced electromagnetic books where it is
always presented for plane waves [3]. Essentially, the optical theorem states
that the rate at which energy is extincted due to scattering and absorption at
the scatterer is proportional to the imaginary part of the forward scattering
amplitude, corresponding to the direction of propagation of the incident field
[4, 5, 6, 7]. The OT provides an analytic tool to determine some of the
objects physical properties (i.e., shape, size, concentration, density,
absorption, conductivity, etc.), from the scattered field.
Previous works have attempted to unify the OT. An example of those is the
generalized optical theorem for scalar fields [8], which is found in the near-
field optics [9]. In this context, the problem is solved for a non homogeneous
Helmholtz equation, $\nabla^{2}\psi+k^{2}\psi=-4\pi k^{2}\eta\psi$ being
$k=\omega/c$, $\eta$ the dielectric susceptibility, and $\psi$ the scattering
potential solution of the partial differential equation. A version of the
generalized optical theorem has been discussed in the scalar and vector
electromagnetic approaches in a wide context of applications [10]. An extended
formalism has been presented in the field of acoustics [11]. The solution of
the scattering problem in quantum mechanics for non-plane waves[12], for
anisotropic media [13], and for arbitrary scalar and vectorial fields in
cylindrical coordinates [14, 15] have been also analyzed. Recently, Marengo
and Tu provided an OT that describes the energy budget of wave scattering
phenomena in time domain [16] and its applications in transmission lines [17].
However, it is not the aim of this manuscript to list the full range of
applications and references where the OT has been used. Instead, we aim to
present an explicit derivation of a complete general expression of the OT
based on Maxwell equations that can be applied to any invariant beam. To the
best of our knowledge, this has not been presented before. An invariant beam,
also known as a “non-diffracting beam”, propagates indefinitely without
changes in its transverse intensity distribution [18, 19]. These beams can
also be represented as an infinite superposition of plane waves [20]. These
optical fields are solution of the transverse Helmholtz wave equation
$\nabla^{2}_{T}\varphi+k_{T}^{2}\varphi=0$ in Cartesian, circular cylindrical,
parabolic cylindrical, and elliptical cylindrical coordinates, where $k_{T}$
is the transversal wave vector; the solutions of this equation are the well
known plane, Bessel [21], Mathieu [22] and Weber beams [23], respectively. The
use of such fields in the experimental and theoretical works has attracted
considerable attention [24], from quantum mechanics [25, 26, 27, 28] to space
communications [29]. New scenarios have been opened for scattering problems
[30, 31, 32, 33]; however, none of them explores the possibility of using
these fields on any of the new technological challenges that range from nano
and micro-photonics to science and engineering of antennas, metamaterials and
electromagnetic devices, among others.
The main goal of this research is to present a derivation of the general OT
based on Maxwell equations and the standard literature for propagation
invariant beams, where the plane wave case is a particular case, using the
well-known far field approximation [34, 35]. In order to illustrate our
results we revisit the classical and standard scattering elastic problem of a
dielectric sphere in the Rayleigh regime, for which the incident field can be
any propagation invariant beam. The problem becomes clearer studying the case
when the electric incident field is a a zero order Bessel beam, due to its
wide range of applications [36, 37]. Finally, we present some conclusions and
possible applications.

##  2 Optical theorem

Let us consider a scatterer particle of arbitrary form and size, volume $V$,
and a complex permittivity

$\varepsilon_{p}\left(\vec{r}\right)=\varepsilon_{0}\varepsilon_{r}\left(\vec{r}\right)=\varepsilon_{0}\left[\varepsilon_{r}^{\prime}(\vec{r})+i\varepsilon_{r}^{\prime\prime}(\vec{r})\right],$
(1)

where $\varepsilon_{0}$ is the vacuum permittivity. The medium surrounding the
scatterer particle is lossless and its permittivity $\varepsilon$ is real. We
assume also that the scatterer particle and the surrounding medium are non
magnetic, so that they have magnetic permeability $\mu=1$, see Figure 1. The
fields $\vec{E}_{i}(\vec{r}),\,\vec{H}_{i}(\vec{r})$ correspond to an
invariant beam that strikes the scatterer particle; the scattered
electromagnetic field is represented by
$\vec{E}_{s}(\vec{r}),\,\vec{H}_{s}(\vec{r})$ and
$\vec{E}_{p}(\vec{r}),\,\vec{H}_{p}(\vec{r})$ describe the electromagnetic
field inside the scatterer particle, as shown in Figure 1.

Figure 1: Geometrical configuration for the generalization of the optical
theorem.

The time-averaged power absorbed by the scatterer object is given by [5, 4]

$P_{a}=\frac{\varepsilon_{0}}{2}\int_{V}\omega\,\varepsilon^{\prime\prime}_{r}\left(\vec{r}\right)\,\left|\vec{E}_{p}\left(\vec{r}\right)\right|^{2}dV.$
(2)

If the Poynting vector of the scattered field is given by the following
expression

$\vec{S}_{s}=\frac{1}{2}\mathbf{Re}\left(\vec{E}_{s}\times\vec{H}_{s}^{*}\right),$
(3)

then the time-average scattered power is [4]

$P_{s}=\int_{S_{\infty}}\vec{S}_{s}\cdot\hat{n}\,da=\int_{\mathrm{sup}}\vec{S}_{s}\cdot\hat{n}\,da,$
(4)

where $\mathrm{sup}$ is the surface around the volume $V$ of the scatterer
particle and $S_{\infty}$ is the surface at infinity. Assuming nothing else
that the correctness of the Maxwell equations, it is possible to show that [4]

$P_{a}+P_{s}=-\int_{\mathrm{sup}}\vec{S}^{\prime}\cdot\hat{n}da,$ (5)

with

$\vec{S}^{\prime}=\frac{1}{2}\mathbf{Re}\left(\vec{E}_{i}\times\vec{H}_{s}^{*}+\vec{E}_{s}\times\vec{H}_{i}^{*}\right).$
(6)

Bajer and Horak [18] have shown that for an invariant beam
$\nabla\cdot\left(\vec{S}_{i}+\vec{S}_{s}\right)=0$. Thus, using the
divergence theorem, it is easy to prove that [18, 19]

$\int_{\mathrm{sup}}\mathbf{Re}\left(\vec{E}_{i}\times\vec{H}_{s}^{*}\right)\cdot\hat{n}\,da=\int_{\mathrm{sup}}\mathbf{Re}\left(\vec{E}_{i}^{*}\times\vec{H}\right)\cdot\hat{n}\,da,$
(7)

and that

$\int_{\mathrm{sup}}\mathbf{Re}\left(\vec{E}_{s}\times\vec{H}_{i}^{*}\right)\cdot\hat{n}\,da=\int_{\mathrm{sup}}\mathbf{Re}\left(\vec{E}\times\vec{H}_{i}^{*}\right)\cdot\hat{n}\,da,$
(8)

where $\vec{E}$ y $\vec{H}$ denote the total fields; i.e.,
$\vec{E}=\vec{E}_{i}+\vec{E}_{s}$ and $\vec{H}=\vec{H}_{i}+\vec{H}_{s}$.
Substituting these last expressions in (5), using the divergence theorem and
expanding the divergence of the cross product obtained, we arrive to the
following expression for the total power absorbed and scattered,

$\begin{split}P_{a}+P_{s}&=-\frac{1}{2}\mathbf{Re}\int_{V}\left[\left(\nabla\times\vec{E}_{i}^{*}\right)\cdot\vec{H}-\vec{E}_{i}^{*}\cdot\left(\nabla\times\vec{H}\right)\right.\\\
&\left.\qquad\qquad\qquad+\left(\nabla\times\vec{E}\right)\cdot\vec{H}_{i}^{*}-\vec{E}\cdot\left(\nabla\times\vec{H}_{i}^{*}\right)\right]dV.\end{split}$
(9)

If we substitute the Maxwell equations,

$\displaystyle\nabla\times\vec{E}$ $\displaystyle=i\omega\mu\vec{H},$ (10)
$\displaystyle\nabla\times\vec{H}$
$\displaystyle=-i\omega\varepsilon_{0}\vec{E},$ (11)

in (9), we can recover,

$\begin{split}P_{a}+P_{s}&=-\frac{1}{2}\mathbf{Re}\int_{V}\left[\left(-i\omega\mu\vec{H}_{i}^{*}\right)\cdot\vec{H}-\vec{E}_{i}^{*}\cdot\left(-i\omega\varepsilon_{p}\vec{E}\right)\right.\\\
&\left.\qquad\qquad\qquad+\left(i\omega\mu\vec{H}\right)\cdot\vec{H}_{i}^{*}-\vec{E}\cdot\left(i\omega\varepsilon_{0}\vec{E}_{i}^{*}\right)\right]dV\\\
&=\frac{1}{2}\,\mathbf{Im}\int_{V}\omega\left(\varepsilon_{p}-\varepsilon_{0}\right)\vec{E}_{i}^{*}\cdot\vec{E}\,dV,\end{split}$
(12)

where, for simplicity, we have avoided writing the $\vec{r}$-dependence of the
permittivity inside the particle. Equation (11) can be written as

$\nabla\times\vec{H}=-i\varepsilon_{0}\omega\vec{E}+\vec{J}_{\mathrm{eq}},$
(13)

where we have defined the current density

$\displaystyle\vec{J}_{\mathrm{eq}}(\vec{r})=\begin{cases}-i\omega\varepsilon_{0}\left[\varepsilon_{r}(\vec{r})-1\right]\vec{E},&\text{inside}\,\,V,\\\
0,&\text{outside}\,\,V,\end{cases}$ (14)

which can be thought as the producer of the scattered field. We introduce now
the Hertz potential

$\vec{\Pi}_{s}(\vec{r})=\frac{1}{-i\omega\varepsilon_{0}}\int_{V}G_{0}(\vec{r},\vec{r}^{\prime})J_{\mathrm{eq}}(\vec{r}^{\prime})dV^{\prime},$
(15)

where the Green function for the Hemholtz equation is

$G_{0}(\vec{r},\vec{r}^{\prime})=\frac{e^{ik|\vec{r}-\vec{r^{\prime}}|}}{4\pi|\vec{r}-\vec{r^{\prime}}|},$
(16)

which represents the scattered field. If we substitute (14) in (15), we obtain

$\displaystyle\vec{\Pi}_{s}(\vec{r})=\int_{V}\left[\varepsilon_{r}(\vec{r^{\prime}})-1\right]\vec{E}(\vec{r^{\prime}})G_{0}(\vec{r},\vec{r^{\prime}})dV^{\prime}.$
(17)

In terms of the Hertz potential the scattered electric and magnetic fields can
be written as [5]

$\displaystyle\vec{E}_{s}(\vec{r})$
$\displaystyle=\nabla\times\nabla\times\vec{\Pi}_{s}(\vec{r}),$ (18)
$\displaystyle\vec{H}_{s}(\vec{r})$
$\displaystyle=-i\omega\varepsilon\nabla\times\vec{\Pi}_{s}(\vec{r}).$ (19)

In the far field approximation, the quantity
$\frac{1}{|\vec{r}-\vec{r^{\prime}}|}$ can be approximated by $\frac{1}{R}$,
where $R=\left|\vec{r}\right|$. However, in the case of
$ik|\vec{r}-\vec{r^{\prime}}|$, the correct far field approximation is [34,
35]

$|\vec{r}-\vec{r^{\prime}}|=\sqrt{R^{2}-2R\vec{r^{\prime}}\cdot\hat{o}+\vec{r}^{2}}\approx
R-\vec{r^{\prime}}\cdot\hat{o},$ (20)

where $\hat{o}=\frac{\vec{r}}{R}$ is the unit vector directed to the
observation point. As shown in Figure 2, the Green function (16) takes the
form,

$G_{0}(\vec{r},\vec{r^{\prime}})=\frac{e^{ikR-
ik\hat{r^{\prime}}\cdot\hat{o}}}{4\pi R}.$ (21)

Figure 2: Diagram of an incident field upon an object in the far field
approximation.

It is straightforward to show that

$\nabla\left(\frac{e^{ikR}}{R}\right)\approx ik\hat{o}\frac{e^{ikR}}{4\pi R}.$
(22)

Thus, the scattered field, Equation (18), can be written as

$\vec{E}_{s}(\vec{r})=\frac{e^{ikR}}{4\pi
R}k^{2}\int_{V}\Big{[}\vec{E}-\hat{o}\left(\hat{o}\cdot\vec{E}\right)\Big{]}\left[\varepsilon_{r}(\vec{r^{\prime}})-1\right]e^{-ik\vec{r^{\prime}}\cdot\hat{o}}dV^{\prime}.$
(23)

The previous equation can be written as

$\vec{E}_{s}(\vec{r})=\frac{e^{ikR}}{R}\vec{F}(\hat{i},\hat{o}),$ (24)

where we have defined the scattering amplitude

$\vec{F}(\hat{i},\hat{o})=\frac{k^{2}}{4\pi}\int_{V}\left[\vec{E}-\hat{o}(\hat{o}\cdot\vec{E})\right]\left[\varepsilon_{r}(\vec{r}^{\prime})-1\right]e^{-ik\vec{r^{\prime}}\cdot\hat{o}}dV^{\prime},$
(25)

and where the unit vector $\hat{i}$ is the direction of the impinging electric
field; physically, equations (24) and (25) represent how a spherical wave is
modulated. The scattering amplitude expresses how the particle responds to the
incident field and where that field is observed after the interaction with the
particle.
We make now $\hat{o}=\hat{i}$ and we get the forward scattering amplitude

$\vec{F}(\hat{i},\hat{i})=\frac{k^{2}}{4\pi}\int_{V}\vec{E}(\vec{r}^{\prime})\left[\varepsilon_{r}(\vec{r}^{\prime})-1\right]e^{-ik\vec{r}^{\prime}\cdot\hat{i}}dV^{\prime}.$
(26)

We write the propagation invariant incident beam in its plane-wave
representation [38, 22, 23, 20]

$\displaystyle\vec{E}_{i}(\vec{r})=\int_{-\pi}^{\pi}\hat{e}_{i}A(\phi)e^{-ik\hat{i}\cdot\vec{r}}d\phi,$
(27)

where $\hat{e}_{i}$ is the polarization of the incident field, $k$ is the wave
vector that can be related to the cone angle defined for invariant beams [21];
the function $A(\phi)$ represents the relative amplitude of the plane wave
superposition [20]. Multiplying Eq. (26) by $A\left(\phi\right)$ and
integrating it with respect to $\phi$ from $-\pi$ to $+\pi$, we get

$\displaystyle\int_{-\pi}^{\pi}\vec{F}(\hat{i},\hat{i})\cdot\hat{e}_{i}A(\phi)d\phi$
$\displaystyle=\frac{k^{2}}{4\pi}\int_{-\pi}^{\pi}d\phi\int_{V}\vec{E}(\vec{r}^{\prime})\left[\varepsilon_{r}(\vec{r}^{\prime})-1\right]\cdot\hat{e}_{i}A(\phi)e^{-ik\vec{r}^{\prime}\cdot\hat{i}}dV^{\prime}$
$\displaystyle=\frac{k^{2}}{4\pi}\int_{V}\vec{E}(\vec{r}^{\prime})\left[\varepsilon_{r}(\vec{r}^{\prime})-1\right]dV^{\prime}\cdot\int_{-\pi}^{\pi}\,\hat{e}_{i}A(\phi)e^{-ik\vec{r}^{\prime}\cdot\hat{i}}d\phi;$
(28)

thus, using the representation (27) for an invariant incident field,

$\displaystyle\vec{F}(\hat{i},\hat{i})\cdot\hat{e}_{i}\int_{-\pi}^{\pi}A(\phi)d\phi=\frac{k^{2}}{4\pi}\int_{V}\left[\varepsilon_{r}(\vec{r}^{\prime})-1\right]\vec{E}(\vec{r}^{\prime})\cdot\vec{E}_{i}(\vec{r}^{\prime})dV^{\prime}.$
(29)

As $\int_{-\pi}^{\pi}A(\phi)d\phi$ is a constant, equating integrals (12) and
(29) gives us

$P_{a}+P_{s}=\frac{2\pi}{k\eta}\mathbf{Im}\,\left[\vec{F}(\hat{i},\hat{i})\cdot\hat{e}_{i}\right],$
(30)

which is the well known optical theorem [1, 2, 3, 4, 5]. The results presented
so far are valid for any incident beam which can be written in the form given
in (27).
Using the definition of the extinction transversal section as
$\sigma_{ext}=\sigma_{a}+\sigma_{s}$, where $\sigma_{a}$ is the absorbed
transversal section, and $\sigma_{s}$ is the scattering transversal section
[3], we can write the optical theorem as

$\sigma_{ext}=\frac{4\pi}{k|\langle\vec{S}_{i}\rangle|}\mathbf{Im}\,\left[\vec{F}(\hat{i},\hat{i})\cdot\hat{e}_{i}\right],$
(31)

where $|\langle\vec{S}_{i}\rangle|$ is the averaged incident Poynting vector.

##  3 Example for Rayleigh scattering

In this section, the scattering of an incident propagation invariant beam by a
dielectric sphere with constant relative permittivity $\varepsilon_{r}$ and
radius $a$ is presented [5, 1, 3, 4]. It is well known that the electric field
$\vec{E}$ inside a dielectric sphere immersed in an electric field
$\vec{E}_{i}$ is given by [1, 5]

$\vec{E}=\frac{3}{2+\varepsilon_{r}}\vec{E}_{i}.$ (32)

As the impinging field is an invariant beam, we substitute its plane wave
representation, given by Eq. (27), into (32) and the result in Eq. (25). Then,
under the Rayleigh approximation $e^{-ik\vec{r}^{\prime}\cdot\hat{o}}\approx
1$ [39, 40], the integral over the volume of the sphere can be easily
calculated to yield

$F(\hat{i},\hat{o})=k^{2}a^{3}\frac{\varepsilon_{r}-1}{\varepsilon_{r}+2}\left[\hat{e}_{i}-\left(\hat{o}\cdot\hat{e}_{i}\right)\hat{o}\right]\int_{-\pi}^{\pi}d\phi
A(\phi)e^{-ik\hat{i}\cdot\vec{r}},$ (33)

where $k$ is the wave vector and $a$ is the sphere radius. Using (33) as the
differential scattering cross section
$\sigma_{\text{d}}\equiv|F(\hat{i},\hat{o})|^{2}$ and the appropriated
relative amplitude of the plane wave superposition $A\left(\phi\right)$, the
differential scattering cross section for any invariant beam can be obtained.
Integration of this equation over the solid angle $d\Omega$ gives the
scattering cross section [5]; the polarization as a function of the scattered
radiation can be derived as well [1]. It is important to note that when the
angular modulation function $A(\phi)$ is a delta function, and the incident
beam is a plane wave, the classical solution for Rayleigh scattering is
recovered [1].
Using these results, we are also able to study the case when the incident
field is a Bessel beam. These beams were introduced by Durnin [21], and have
attracted considerable attention due to their properties of transverse
propagation invariance and self-reconstruction, among others [24]. In
addition, it is well know that Bessel beams carry both linear and angular
momentum that can be transferred to atoms, molecules and particles [36, 37].
For this case $A(\phi)=e^{im\phi}$ [11], and after substitution into (33), we
obtain

$F(\hat{i},\hat{o})=k^{2}a^{3}\frac{\varepsilon_{r}-1}{\varepsilon_{r}+2}\left[\hat{e}_{i}-\left(\hat{o}\cdot\hat{e}_{i}\right)\hat{o}\right]J_{m}(k_{T}\rho)e^{im\phi},$
(34)

where $\rho=\sqrt{x^{2}+y^{2}}$ is the transverse radius, $m$ is an integer,
$k_{T}=k\sin\beta$ is the transversal wave vector, and $\beta$ is the value of
the half-cone angle [21]; the expression (34) is the scattering field
amplitude for any Bessel beam of $m$-th order. It is important to remark that
in [44, 45], the authors have used the partial wave series in the far field
approximation to calculate a form function equivalent to equation (34), for
the scattering of an acoustic helicoidal Bessel beam by a sphere centered on
the axis; they also calculated the acoustic radiation force exerted by the
beam on the sphere in an inviscid fluid. Remarkably, using the forward
scattering amplitude, as presented in [52, 51, 53] for the Mie regime, allows
this approach to be extended due to the fact that any invariant beam can be
constructed as a plane wave superposition. For our case, the differential
scattering cross section is given by

$\sigma_{\text{d}}(\theta)=k^{4}a^{6}\left(\frac{\varepsilon_{r}-1}{\varepsilon_{r}+2}\right)^{2}J_{m}^{2}(k_{T}\rho)\left[(1+\cos^{2}\theta)/2\right],$
(35)

where we have used the standard scattering geometry [52, 51, 53] to relate the
scattering angle $\theta=\arccos(\hat{i}\cdot\hat{o})$ to the unit vectors
$\hat{i}$ and $\hat{o}$ at a particular point P, where the scattered radiation
is observed. On the other hand, as in the case of a plane wave, if the
incident field is unpolarized the differential scattering function is the
average over parallel and perpendicular incidents fields. This gives
$\sigma_{\text{d}}(\theta)=\frac{1}{2}\left[\sigma^{\perp}_{\text{d}}(\theta)+\sigma^{\parallel}_{\text{d}}(\theta)\right]$
and $\hat{o}\cdot\hat{e}_{i}=0$, if $\hat{e}_{i}$ is perpendicular to the
scattering plane, and $\hat{o}\cdot\hat{e}_{1}=\sin\theta$, if $\hat{e}_{i}$
lies in the plane. Using this information we can calculate the polarization
scattering function [1]

$\Pi(\theta)=\frac{\sigma^{\perp}_{\text{d}}(\theta)-\sigma^{\parallel}_{\text{d}}(\theta)}{\sigma^{\perp}_{\text{d}}(\theta)+\sigma^{\parallel}_{\text{d}}(\theta)}=\frac{\sin^{2}\theta}{1+\cos^{2}\theta}J_{m}^{2}\left(k_{T}\rho\right).$
(36)

By integrating the equation (35) over $d\Omega$, it is straightforward to
obtain $\sigma_{\text{s}}$, the scattering transversal section, as

$\displaystyle\sigma_{\text{s}}(\theta)=\int_{0}^{\pi}\int_{0}^{2\pi}\sigma_{\text{d}}(\theta)d\Omega=\frac{8}{3}(\pi
a^{2})k^{4}a^{4}\left(\frac{\varepsilon-\varepsilon_{0}}{\varepsilon+2\varepsilon_{0}}\right)^{2}J_{m}^{2}\left(k_{T}\rho\right),$
(37)

while the absorbed transversal section $\sigma_{\text{a}}$ has to be obtained
from the Poynting vector [1, 3, 4, 5], since it would result in
$\sigma_{\text{ext}}=0$ otherwise. Considering a transparent sphere (non-
absorbing medium) for which
$\sigma_{\text{ext}}=\sigma_{\text{a}}+\sigma_{\text{s}}$, (meaning that
$\sigma_{a}\approx 0$ [3]), and using equation (37) the scattering extinction
efficiency yields

$\displaystyle
Q_{\text{ext}}=\frac{\sigma_{\text{s}}(\theta)}{A_{\text{t}}}=\frac{8}{3}(ka)^{4}\left(\frac{\varepsilon-\varepsilon_{0}}{\varepsilon+2\varepsilon_{0}}\right)^{2}J_{m}^{2}\left(k_{T}\rho\right),$
(38)

where $A_{t}$ is the transversal area. This equation can be conveniently
written as

$\displaystyle
Q_{\text{ext}}=\frac{8}{3}x^{4}\left(\frac{n^{2}-1}{n^{2}+2}\right)^{2}J_{m}^{2}\left(k_{T}\rho\right),$
(39)

where $x=ka$ is a scattering parameter factor related to the wavelength and
the sphere radius [3], and we have represented the index of refraction as
$n^{2}\equiv\varepsilon/\varepsilon_{0}$. Note that in the limiting case $m=0$
and $\rho\rightarrow 0$ , $J_{m}\left(k_{T}\rho\right)\rightarrow 1$; thus,
this expression physically behaves as a plane wave and the Rayleigh scattering
law is recovered.

Figure 3: Polar plot of the angular extinction cross section (39) and the
polarization scattered function (36) for a zero order Bessel beam varying the
angle of incidence to $\beta=\\{25^{\circ},35^{\circ},55^{\circ}\\}$.

Some numerical simulations in which a sphere of radius $a$, centered at the
origin and surrounded by a non absorbing medium, scatters a Bessel beam can be
considered at this point; for the sake of simplicity, we will consider only a
zero order Bessel beam. Since the zero order Bessel beam has the typical
characteristics of a Bessel beam, it can be easily obtained in the laboratory
by different means such as cylindrical lenses (axicon) [41], a tunable
acoustic gradient index of refraction lenses [42], or using a spatial light
modulator [43]. To our knowledge, the Rayleigh scattering problem using this
approximation for the invariant beam has not been addressed before. However,
the scattering problem of a rigid sphere in acousto-optics using a zero order
Bessel beam was firstly reported in [44, 45]; also, the case of a dielectric
sphere has been studied in [46, 47, 48] for the Mie regime.
Here we look at the analytical scattering solution for a dielectric sphere in
the Rayleigh regime $ka<<1$, where the transversal vector wave is
$k_{T}=\frac{2\pi}{\lambda}\sin\theta\sin{\beta}$ [48], being $n$ the index of
refraction, $\lambda$ the wavelength and $\beta$ the cone angle respectively
[21]. The behavior of the scattering extinction efficiency, given by equation
(39), and the polarization scattering function, represented by equation (36),
are shown in Figure 3. We study particular cases for which the angle of
incidence takes the values $\beta=25^{\circ},35^{\circ}$ and $55^{\circ}$; in
addition, the parameter $m=0$, the field wavelength $\lambda=1$ m, the
refractive index $n=1.5$, the sphere radius $a=0.02$ mm and the adimensional
scattering parameter $k=0.188496$ have been fixed. For $\beta=25^{\circ}$, the
total scattering extinction efficiency has two lobes in the forward and
backward direction, while four lobes were found for the polarization scattered
function. When $\beta=35^{\circ}$, two additional lobes appear in the total
scattering extinction efficiency, in addition to a forward and backscattering
peaks. The former can be interpreted as the peaks of the backscattering and
forward scattering components of the plane waves constructing the Bessel beam
[44, 45, 46]; in this case the polarization scattered function is zero at
$\theta=0^{\circ}$ and $\theta=180^{\circ}$. Increasing the angle to
$\beta=55^{\circ}$ narrows the forward and backscattering peaks, while
polarization curves for the forward and backward scattering vanish at
$\theta=0^{\circ}$ and $\theta=180^{\circ}$. Similar radiation patterns were
found in [49, 50] for other members of the family of invariants beams, but it
has only been studied for the Mie regime $ka>>1$ and still remains unknown for
Rayleigh scattering.

##  4 Conclusions

We have obtained a general optical theorem for any propagation invariant beam,
where the plane wave case is a particular case. To demonstrate the link
between our generalized formulation and prior results in this area, we have
shown that the presented ordinary form of the optical theorem renders the
particular case for free space derived in previous works. Using the amplitude
scattering function in the far field approximation we have obtained a general
representation for the Rayleigh scattering regime that can be applied to any
invariant beam. We have studied the scattering of a Bessel beam, assuming that
the incident wave is linearly polarized and considering a zero Bessel beam as
a function of the incident impinging angle $\beta$. Our method can be extended
to other propagation invariant beam members, both to calculate the scattering
amplitude function and to evaluate the forward-scattering approach. This
method can be applied to arbitrary probing fields as any invariant beam can be
written as a plane wave superposition.

##  5 Acknowledgments

The authors are grateful to B.M. Rodríguez-Lara for his continuous support and
for fruitful discussions. I. Rondón-Ojeda acknowledges financial support
through CONACyT doctoral grant CVU $423320$.

## References

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