# An analytical approximation to measure the extinction cross-section using:
Localized Waves

Irving Rondón  School of Computational Sciences, Korea Institute for Advanced
Study,
Seoul 0245, Republic of Korea<EMAIL_ADDRESS>
###### Abstract

We present a general expression for the optical theorem in terms of Localized
Waves. This representation is well-known and commonly used to generate Frozen
waves, Xwaves, and other propagation invariant beams. We analyze several
examples using different input beam sources on a circular detector to measure
the extinction cross-section.

##  1 Introduction

Recently, considerable interest has increased to understanding the transversal
and longitudinal energy distribution in optics and acoustic waves. In this
direction, many applications in several scenarios such as microscopy, material
manipulation, spectroscopy, telecommunication, holography, quantum mechanics,
Etc. Interesting reviews have been reported in order to show the recent
development in engineering applications, and promising opportunities using
multimode light shaping [1], and structured beams [2, 3].

Due to this demand of different experiments in optics and acoustic [1, 2, 3],
it is essential from the physical point of view to have analytical expressions
able to measure the energy distribution (intensity) in a particular
experiment. Generally speaking, this measure can be done using the optical
theorem (OT) [4] . Essentially, the OT states that the rate at which energy is
extinguished due to scattering and absorption at the scattered object [5]. It
can help to understand the mechanics and how the absorption and the scattering
effects wave propagation throughout a medium and to obtain meaningful features
of the object such as the characterization and its physical properties[6].

In the last few years, many interesting problems using beam shape methods
related to optical manipulation and applications of the Lorenz-Mie theory [7,
8], and several interesting papers have been published to generalize the
optical theorem in electromagnetic theory and acoustic [9, 10, 11, 12, 13, 14,
15, 16, 17, 18, 19, 20, 21], in quantum mechanics [22], seismologic waves
[23]. Also, recently the vector extinction effect in the OT using radially
polarized cylindrical beams [24].

Technically the OT for arbitrary can be reviewed for plane waves [25, 26], or
using a general representation for structured beams such as the Whitaker
integral [27]. Using this representation directly to obtain the OT makes a
challenging task to solve the extinction integrals analytically and
numerically, even for isotropic scattering, and only numerical results can be
applied, which usually brings some inconveniences for further analysis, such
as uncertainties concerning the fast oscillating field components [28].

However, so far, to our knowledge, there is a gap to describe the OT using
Localized Waves (LW) [29], this is the principal motivation and contribution
of this work. In this letter, we present for the first time a general
derivation for the optical theorem (OT) for arbitrary beams using LW. To avoid
redundancy about these propagating waves, we refer the reader for a complete
list of works and references in these books [30].

Using the LW representation is possible to generate several structured beams
[30], just by designing a specific longitudinal spectrum $S(k_{z})$ under the
proper physical conditions, which can allow producing the transversal optical
(or acoustical) intensity shape [31].
Considerable importance in the last few years using LW have been reported in
different scenarios such as imaging, microscopy, remote sensing, and optical
manipulation [RABSuarez, 34]. One good example of LW is the so-called ”Frozen
Waves” (FW) [33]. This beam has unique characteristics and is suited to many
applications, such as optical tweezers, remote sensing, atom guiding [30, 31,
33]. For example, recently in [32] the authors have demonstrated the
propagation arrays of ”Frozen Waves experimentally” in free space, using
synthetic optical holography on a spatial light modulator.

Our analysis considers the standard derivation approach for the OT for plane
waves in the far-field approximation but using LW. We analyze numerically our
results reported by [35] using plane waves to, alternatively, measure the
power extinguished by a single particle in terms of physical energy fluxes for
collimated beam [25, 26, 28], here for structured beams. We validate our
analysis using numerical simulations for two different illumination structured
beam probes upon the particle and explore the scattering measured on a
circular detector.

##  2 The optical theorem for arbitrary beam in the far field approximation

The optical theorem can be derived when a plane wave along a positive $z$ axis
is incident on an object [4], using the approximation that the wave amplitude
is larger away from the scattered object. The localized particle is in the
origin of the coordinates system and embedded in an infinite homogeneous, and
nonabsorbing medium [5, 6].
For simplicity, the time-harmonic factor dependence $e^{-i\omega t}$ is
omitted.
In general, the total scalar field is given by

$\psi(r)\approx\varphi(r)+f\frac{e^{ikr}}{r},$ (1)

where the first term represents the incident field and the second is related
to the scattered field. $k$ is the wave vector, and $f$ is the scattering
amplitude function (angular dependence). In this study $f$ is considered
constant such as in [25, 26, 28]. In the far-field, $r$ can be approximated as
$r\approx z+(x^{2}+y^{2})/2R$, where $R$ is the distance from the observation
point to the scattered object. Note that for this condition, when $\varphi(r)$
is a plane wave, this expression must satisfy the plane radiation condition in
the far-field [4, 5, 6].
Using this equation the total intensity is
$I\approx\psi(r)\psi^{*}(r)\approx|\psi(r)|^{2}$.

$I\approx\int_{S}\varphi^{*}\varphi\,\,da+\frac{2}{R}\mathbf{Im}\left(\int_{S}f^{*}\varphi\,e^{ik(x^{2}+y^{2})/2R}\right)da.$
(2)

In this expression, the first integral refers to the quantity of energy that
the particle receives. The second one, in what follows, is related to the
optical theorem or the extinction process. This integration is in the $XY$
plane using the far-field approximation.
If $\varphi(r)=e^{ikz}$ is a plane wave, the intensity is

$I\approx A-\frac{4\pi}{k}\,\mathbf{Im}[f],$ (3)

where $A$ is the transversal area, the second term is the extinction as a
function of the scattering amplitude.
An interesting question is what happens in this approximation the input scalar
field (incident wave) is a structured beam. In general, the most general
representation to describe any arbitrary incident beam is in terms of
Whittaker’s integral [27].

$\varphi(r)=e^{ik_{z}z}\int_{-\pi}^{\pi}A(\phi)e^{ik_{t}(x\cos\phi+y\sin\phi)}d\phi,$
(4)

where $A(\theta)$ is the angular spectrum of the field, the transverse and
longitudinal wave-vector components satisfy the relation
$k^{2}=k_{t}^{2}+k_{z}^{2}$ where $k$ is the wave vector. Note that the
Whittaker’s integral can be analytically expressed only for particular
functions $A(\phi)$. For example, higher-order Bessel beams are defined by
$A(\phi)=e^{im\phi}$, where $m$ is the azimuthal order of the beam. Mathieu
beams are defined by $A(\phi)=C(m,q,\phi)+iS(m,q,\phi)$ are the Mathieu cosine
and sine functions. Weber beams are defined by the angular spectrum given by
$A(a,\phi)=\frac{1}{2(\pi|\sin\phi|^{1/2})}e^{ia\ln|\tan\phi/2|}$, a
pedagogical review for invariant beams see [37] and references therein.

Substituting these different input invariant beams, represented by Eq. 4 after
substitution into Eq. 2, we found that it is not easy to analytically solve
(the second term), even for isotropic scattering ($f$ constant). Only
numerical results can be applied. Solving Eq. 2 is one of the primary and
essential contributions reported in this work. In order to compare and
validate our results, we adapted the approximation using plane waves reported
in Ref. [28].

For this purpose, we explore other general mathematical representations of
generating structured beams, and we found that the Localized Waves is the best
one to fulfill the physical solution.
In this problem, an arbitrary incident LW with cylindrical symmetry impinging
on a circular detector. Following the same physical consideration as in Refs.
[35, 25, 26, 28], where the surface is perfectly flat and coincides with the
$z=R$ plane, centered in the direction of the incidence wave, with the
physical condition $D/2<<R$, where $D$ is the detector diameter, and $R$ is
the distant point from the scattered object to the detector surface. A
straightforward theoretical setup for this configuration is in [28].

##  3 Basic theoretical background

In order to solve the problem given by Eq. 2, using Localized Waves (LW),
solutions to the scalar wave equation with cylindrical symmetry, as a physical
superposition of zero-order Bessel beams [29, 30].

$\displaystyle\varphi(\rho,z)=\varphi_{0}\int_{-\omega/c}^{\omega/c}J_{0}\left(\rho\sqrt{\omega^{2}/c^{2}-k_{z}^{2}}\right)S(k_{z})\exp{(ik_{z}z)}dk_{z},$
(5)

where $\varphi_{0}$ is a certain amplitude, $\rho=\sqrt{x^{2}+y^{2}}$ is
expressed in cylindrical coordinates. $J_{0}(\xi)$ is a zero-order Bessel
beam, $S(k_{z})$ is a weight function (the longitudinal spectrum) and it is
defined over $-\omega/c<k_{z}<\omega/c$, for non evanescent waves. After
selecting a longitudinal physical spectrum $S(k_{z})$, it is possible to
generate different invariant beams with axial symmetry from the physical point
of view. It is important to note that Eq. 5 satisfies the Maxwell’s equations
[29, 30]. The spectral function $S(k_{z})$ can be continuous or discrete [31].
Using LW solutions allows generating paraxial and nonparaxial beams via the
longitudinal wavenumber $k_{z}=\omega/c$. For paraxial beams, the bandwidth is
$\Delta k_{z}<<\omega/c$ while nonparaxial beams have a wide spectrum
bandwidth or the same order of $k_{z}$. In [36] the authors reported a good
description using Maxwell’s equations for nonparaxial beams with axial
symmetry. This work focuses on nonparaxial beams using a discrete case
representation for the longitudinal spectrum to measure the optical cross-
section.
In the following illustrative sections, we briefly describe how to use the LW,
given by Eq. 5. This work put forward this representation and our method to
measure the optical(acoustical) cross-section for any arbitrary structured
field.

###  3.1  Longitudinal spectrum representation for Localized wave solution

Different LW wave solutions for Eq. 5 can be obtained for a discrete and
continuous functions $S(k_{z})$. It is possible to obtain a generalized
solution for the longitudinal spectrum $S(k_{z})$ using the Fourier series
[38]. For additional technical details, see [36]. The discrete case is given
by

$S(k_{z})=\sum_{n=-\infty}^{\infty}a_{n}\exp{\left(i\frac{2\pi
n}{\Lambda}k_{z}\right)},$ (6)

where the coefficient $a_{n}$ can be obtained as follow

$a_{n}=\frac{1}{\Lambda}\int_{-\omega/c}^{\omega/c}S(k_{z})\exp{\left(-i\frac{2\pi
n}{\Lambda}k_{z}\right)}dk_{z},$ (7)

where $\Lambda=2\omega/c$. For a continuum case see [32, 31].

###  3.2 Example 1, LW: using a constant longitudinal spectrum

The localized wave solution for a constant spectrum is given by

$\varphi(\rho,z)={\rm
sinc}\left[\sqrt{\frac{\omega^{2}}{c^{2}}\rho^{2}+\frac{\omega^{2}}{c^{2}}z^{2}}\right],$
(8)

where $S(k_{z})=c/2\omega$ [36] was solved using [39, 40].

###  3.3 Example 2, LW: using a longitudinal Gaussian spectrum

For a longitudinal Gaussian spectrum taking
$S(k_{z})=\frac{b}{\sqrt{\pi}}\exp{\left[-b^{2}(k_{z}-\bar{k}_{z})^{2}\right]}$,
where $\bar{k}_{z}$ is related with the central position for the spectrum and
$b$ is the amplitude [36]. After substituting Eq. 7 into the Fourier
coefficients formula, we obtain that the coefficient are given by
$a_{n}\approx\exp{\left(-i\frac{2\pi}{\Lambda}n\bar{k}_{z}-\frac{\pi^{2}n^{2}}{\Lambda^{2}b^{2}}\right)}$
in the interval $0\leq k_{z}\leq\omega/c$. Under these considerations the
equation 5 is given by

$\varphi(\rho,z)\approx\sum_{n=0}^{\infty}\exp{\left(-i\frac{2\pi}{\Lambda}n\bar{k}_{z}-\frac{\pi^{2}n^{2}}{\Lambda^{2}b^{2}}\right)}\,\,{\rm
sinc}\left[\sqrt{\frac{\omega^{2}}{c^{2}}\rho^{2}+\left(\frac{\omega}{c}z+\pi
n\right)^{2}}\right],$ (9)

when $n=0$, we recover the former constant longitudinal spectrum solution.
Note that the equation Eq. 8 and Eq. 9 satisfies the Maxwell’s equations. For
instance, it is possible to study the numerical conditions, convergence, and
explanation used to validate this approximation following Refs. [30, 36].
We obtain different scalar beams shape modulated by $a_{n}$ and the band
limited $sinc$ function, called the ”sampling function” commonly used in
digital signal processing and information theory [38]. This function is
related to the spherical Bessel function $j_{n}(x)$ of the first kind.
It is worth mentioning that using the LW representation is straightforward to
generate different vector beams. It allows to study fundamentals properties
for electromagnetic beams [41], the transversal component of the Poynting
vector [42] and the orbital angular momentum for nonparaxial beams [43] among
others problems. The following section shows how an extended optical theorem
using these results.

Figure 1:  Upper: the $sinc$ beam given by Eq. 12. Bottom: the Gaussian beam
Eq. 4.3 using $n=3$. The functions are labeled in terms of $kR$. In both
cases, the energy flux is integrated over a circular aperture of radius $D$.
This oscillatory behavior of the intensity between negative and positive
amplitudes has been previously reported. (See Ref. [28] and references
therein).

##  4 Results and Methods:

###  4.1 the optical cross section using a longitudinal wave spectrum

Using the LW wave spectrum represented by Eq. 5 and substituting into Eq 2, we
put forward a general optical theorem using LW for arbitrary beam using a
longitudinal spectrum as

$\displaystyle I\approx\int_{S}\varphi^{*}\varphi\,\,da$ (10)
$\displaystyle+\frac{2\pi}{R}\mathbf{Re}\int_{S}\left(f^{*}\int_{-\omega/c}^{\omega/c}J_{0}\left(\rho\sqrt{\omega^{2}/c^{2}-k_{z}^{2}}\right)S(k_{z})\exp{\left(ik_{z}z+ik\frac{\rho^{2}}{2R}\right)}dk_{z}\right)da,$

It is well know for the OT using a plane wave, the first integral is related
to the absorption [4, 5, 6], it can be denoted as
$A(s)=\int_{S}\varphi^{*}\varphi\,\,da$, the second expression is related with
extinction or the optical theorem [35, 25, 26, 28] . After an integration, due
to the cylindrical symmetry in the interval $0\leq\theta\leq 2\pi$, we obtain

$\displaystyle I\approx A(s)$ (11)
$\displaystyle+\frac{4\pi}{R}\mathbf{Re}\left(f^{*}\int_{0}^{D}\int_{-\omega/c}^{\omega/c}J_{0}\left(\rho\sqrt{\omega^{2}/c^{2}-k_{z}^{2}}\right)S(k_{z})\exp{\left(ik_{z}z+ik\frac{\rho^{2}}{2R}\right)}dk_{z}d\rho\right).$

After the evaluation, the first term after a well-defined region is a constant
value, and the second integral term is related to the extinction, precisely
solvable. It is worth here to remark that these results recover the OT plane
wave case: when $\rho\rightarrow 0$, $J_{0}(0)=1$ and $z=0$, $S(k_{z})$ takes
a constant value.
In the following section, we show analytical solutions for Eq. 11 using two
different probe beams. It means using two longitudinal spectrum $S(k_{z})$, a
cylindrical beam using Eq. 8 and a Gaussian beam using Eq. 10.

###  4.2 Example 1: Optical cross section for constant longitudinal spectrum

Using a cylindrical $sinc$ beam Eq. 8 into Eq. 11. It allows to obtain an
optical cross-section solution as

$\displaystyle I\approx$
$\displaystyle\,\,\mathbf{Re}\Biggl{\\{}\frac{\pi\sqrt{\pi}}{k}\frac{\gamma
f^{*}}{\sqrt{kR}}\exp{\left[-ik\left(R^{2}+z^{2}\right)/2R\right]}\times\Bigg{(}\mathbf{erfi}\left[\frac{\gamma}{\sqrt{kR}}(R-z)\right]$
(12)
$\displaystyle+\mathbf{erfi}\left[\frac{\gamma}{\sqrt{kR}}(R+z)\right]-\mathbf{erfi}\left[\frac{\gamma}{\sqrt{kR}}\left(R-\sqrt{D^{2}+z^{2}}\right)\right]$
$\displaystyle-\mathbf{erfi}\left[\frac{\gamma}{\sqrt{kR}}\left(R+\sqrt{D^{2}+z^{2}}\right)\right]\Bigg{)}\Biggr{\\}},$

where $\gamma=(1+i)/2$ and $k=\omega/c$ is the wave vector. The analytical
results are given in measurable parameters such as $D$ ( detector diameter’s),
the propagation distance $z$, and the scaled product of $kR$. (See Refs [25,
26, 28, 35] For the plane wave case). In Fig 1, we show the numerical behavior
when the size $D$ detector is fixed, and $z$ varies is shown. The extinction
amplitude decreases to zero as expected, while the intensity directly vanishes
when $D\rightarrow 0$ (point particle).

###  4.3 Example 2: Optical cross section for a Gaussian beam

Substituting Eq. 9, into Eq. 2 the optical cross section for a longitudinal
Gaussian beam is given by

$\displaystyle I\approx$
$\displaystyle\,\,\mathbf{Re}\sum_{n=0}^{\infty}\Biggl{\\{}-\frac{2\pi\sqrt{\pi}}{k}\frac{\gamma
f^{*}}{\sqrt{kR}}\exp{\left[-\frac{i\bar{k}_{z}n\pi}{2\Lambda}-\frac{\pi^{2}n^{2}}{b^{2}\Lambda^{2}}-\frac{i\left(k^{2}R^{2}+\left(n\pi+kz\right)^{2}\right)}{2kR}\right]}\times$
$\displaystyle\Bigg{(}\mathbf{erfi}\left[\frac{\gamma\left(kR-\sqrt{(kz+\pi
n)^{2}}\right)}{\sqrt{kR}}\right]-\mathbf{erfi}\left[\frac{\gamma\left(kR+\sqrt{(kz+\pi
n)^{2}}\right)}{\sqrt{kR}}\right]$
$\displaystyle+\mathbf{erfi}\left[\frac{\gamma\left(kR+\sqrt{D^{2}k^{2}+(kz+\pi
n)^{2}}\right)}{\sqrt{kR}}\right]+\mathbf{erfi}\left[\frac{\gamma\left(kR-\sqrt{D^{2}k^{2}+(kz+\pi
n)^{2}}\right)}{\sqrt{kR}}\right]\Bigg{)}\Biggr{\\}},$

Fig 1 shows the numerical behavior in the nonparaxial regime [36] using the
following values: $\bar{k}_{z}=0.4\omega/c$, $\Lambda=2\omega/c$ and
$b=4/\bar{k}_{z}$ where $\omega=c=1$. After several simulations varying $n$
for small values, the series converges, the cross-section has oscillatory
behavior, and the amplitude decreases. These results are in agreement with the
case of a circular detector reported, which showed fast and decaying
oscillations in time [35, 28]. In this work, only a circular sensor and fixed-
size detector were considered. Additional analysis such as the numerical
convergence would be interesting to include the effects of interference terms
[35]. It may be of interest to make further numerical simulations and
comparisons with several reported publications, also to explore using
different cylindrical beams, but it is outside the scope of this work, is open
for future research, and can be addressed elsewhere.

##  5 Summary

We presented a generalized optical theorem using Localized Waves. We show an
analytical solution for arbitrary LW beams for the scattering cross-section
problem using this representation. Our findings suggest that (Eq. 10) can be
applied in optics and acoustics beams scattering problems. We show general
solutions for the optical theorem for two cases, a cylindrical beam (Eq. 12)
and a longitudinal Gaussian beam (Eq. 4.3). This new heuristic method can work
under the appropriate $S(k_{z})$ under well-defined physical initial
conditions and experimental parameters. Another application is to measure the
cross-section in optical/acoustic problems, for example, to measure the
scattering experimentally using ”Frozen waves” [31, 32] or another LW. Since
the parameters presented can be connected. Also, considering the vector effect
[24] in the extinction. Several interesting acoustic scattering problems as
the spin propagation [44], orbital angular momentum [45], vortex modes in
synthetic fluids [46], evanescent Bessel waves [47] could be analyzed using
LW. This method can suit to study experimentally new structured beams, such as
carving beams recently reported [48].

##  6 Acknowledgments

The author acknowledged financial support by Basic Science Research Program
through the National Research Foundation of Korea (NRF) funded by the Ministry
of Science and ICT [NRF-2017R1E1A1A01077717].

##  7 References

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