{'fraction_non_alphanumeric': 0.039034342060523634, 'fraction_numerical': 0.011696701802108127, 'mean_word_length': 5.091963545981773, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Local cross-sections and energy efficiencies in the multiple optical scattering by two conducting particles

In this work, exact mathematical expansions for the intrinsic electromagnetic (EM) or optical cross-sections (i.e., extinction, scattering and absorption) for a pair of perfectly conducting circular cylinders in a homogeneous medium are derived. The incident illuminating field is an axially-polarized electric field composed of plane travelling waves with an arbitrary angle of incidence in the polar plane. The formalism is based on the multipole modal expansion method in cylindrical coordinates and the translational addition theorem applicable to any range of frequencies. An effective EM field, incident on the probed cylinder, is defined first, which includes the initial and re-scattered field from the second cylinder. Subsequently, it is used jointly with the scattered field to derive the mathematical expressions for the intrinsic/local cross-sections based on integrating their corresponding time-averaged intensities over the surface of the probed object by applying the Poynting theorem. Numerical computations for the intrinsic extinction (or scattering) energy efficiencies for a pair of conducting circular cylinders with different radii in a homogeneous medium are considered. Emphasis is given on varying the interparticle distance, the angle of incidence, and the dimensionless sizes of the cylinders. The results computed a priori can be useful in the full characterization of a multiple scattering system of many particles, in conjunction with experimental data for the extrinsic cross-sections.


Introduction
The absorption, extinction and scattering cross-sections [1] (or energy efficiencies) are physical observables used for quantitative characterization in the interaction of electromagnetic (EM) [or light] waves with an object [2][3][4][5][6] or multiple objects. For example, they can be used in the accurate predictions of the interaction properties of aerosol particles in a cluster [7].
Acknowledging the advantages of developing rigorous analytical models to predict and compute such physical observables in multiple scattering systems by several objects [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21], various investigations [22][23][24][25][26] considered the modal expansion method with boundary matching in cylindrical coordinates for this purpose. In these works, analytical/mathematical equations for the extrinsic cross-sections have been developed, which provide quantifiable information on the global characteristics of the multiple interacting system of particles with the incident field. For example, the obtained extrinsic scattering cross-section accounts for the multiple/cross-interferences, with a clear dependence on the distance between two interacting objects. It can be correlated with experimental measures in the far-field. This has been acknowledged recently in the multiple acoustical scatterings by rigid and fluid cylinders [27,28]. This approach, however, is not adequate to quantify the inherent/intrinsic scattering, absorption and extinction properties, which exhibit local properties for each particle in the presence of other particles, quite distinguishable from the extrinsic ones. Thus, it is of some importance to develop a new analytical model for the intrinsic extinction, scattering and absorption cross-sections in order to predict the local properties of the multiple interacting system of particles with the incident EM/optical field.
The aim of this investigation is therefore directed toward this goal, where exact partial-wave series expansions for the intrinsic scattering, absorption and extinction cross-sections in the multiple scattering by a pair of perfectly conducting

Method
The procedure of determining the expressions for the intrinsic scattering, absorption and extinction cross-sections consists first of obtaining the expressions for the effective incident and scattered fields for the probed cylinder [29]. Therefore, an analysis for the multiple scattering of the incident field by the pair of conducting cylinder of different radii (equivalent to its acoustic counterpart [30]) is required.
Consider an electric field with axial polarization (i.e. along the z-direction -see Fig. 1) composed of plane progressive waves. The field is incident upon a pair of two perfectly conducting circular cylinders of different radii a and b, respectively. The interparticle distance d separates the center of mass of the cylinders, and the field has an angle of incidence  in the polar plane as shown in Fig. 1. The waves propagate in a non-conducting nonmagnetic homogenous medium without any dissipation (i.e. lossless medium). The axial component for the incident electric field is expressed as where E0 is the field amplitude,  is the angular frequency, r is the vector position, and k is the wave-vector.
In the polar system (r,) centered on the first circular cylinder, Eq.(1) is expanded as a partial-wave series as  where ,1 n C and ,2 n C are the modal coefficients, which depend on the material properties of the cylinders, and are coupled with the incident field.     1 n H  is the cylindrical Hankel function of the first kind of order n.
Eq. (5) shows that the scattered axial electric field component from the second cylinder is written in terms of the primed system of coordinates. To determine the coupled expansion coefficients, suitable boundary matching conditions should be satisfied, all written in the same system of coordinates. Therefore, the translational addition theorem is utilized for this purpose. Accordingly, the following equalities hold [34], It follows that the total axial electric field component in the host medium of wave motion close to the first (second, respectively) cylinder such that r < d (r' < d, respectively) is rewritten in either coordinate system as, Fig. 2. Panels (a) and (b) display the plots for the intrinsic extinction energy efficiencies for two perfectly conducting circular cylinders having ka = 0.1 and kb = 1 versus the dimensionless interparticle distance kd and incidence angle .   Since the incident electric field is axially polarized (perpendicularly to the polar plane -known also as TM polarization), the boundary conditions that must be satisfied on the surface of each perfectly conducting cylinder require that [30]   tot , , 0, where a and b are the radii of cylinders 1 and 2, respectively (Fig. 1).
Using the orthogonality condition, the coupled linear system of equations is obtained from Eqs. (10) and (11) as follows, This system of equations is solved by matrix inversion procedures such that the modal coefficients ,1 n C and ,2 n C can be determined numerically by imposing a truncation limit in the series such that Nmax = [max(max(ka, kb), kd)] + 30. This limit ensures suitable convergence of the series and negligible truncation numerical error [30].
As discussed previously, the modal coefficients depend on both objects as shown from Eqs. (12) and (13). Column vectors are of dimension 1  (2Nmax + 1) elements, while the square matrices are of dimension (2Nmax + 1)  (2Nmax + 1) elements. 7 The procedure of determining the expressions for the intrinsic extinction, scattering and absorption cross-sections consists of integrating the associated time-averaged intensities over the surface of each cylinder. When multiple scattering effects arise between the particle pair, the probed cylinder will be subjected to an incident effective field composed of the primary incident EM field in addition to the multiple re-scattering field due to the presence of the other cylinder. Therefore, to obtain the mathematical expressions for the cross-sections related to a particular cylinder, an effective incident field on a particular scatterer should be defined and determined initially. This method provides the adequate means to obtain the intrinsic cross-sections on the probed cylinder as the problem in hand is reduced to that of the single object case, but with an incident effective field.
The determination of the incident effective electric (and magnetic) fields on each cylinder in each system of coordinates is accomplished based on Eqs. (8) and (9) by recognizing that the scattered (outgoing) waves from the probed particle cannot be simultaneously directed toward its own surface. Therefore, the effective electric field component due to the primary incident waves in the vicinity of the first cylinder (i.e. r < d) in the presence of the second cylinder is obtained from Eq.(8) as, In the primed system of coordinates, the effective electric field component incident on the second cylinder is deduced from Eq.(9) such that, Fig. 4. The same as in Fig. 2 Subsequently, the formulation of the scattering in the far-field [35] can be used, since the problem in hand is now reduced to the single-object case. This procedure leads to exact mathematical expressions without any approximations.
In the far-field limit, the cylindrical Bessel and Hankel functions of the first kind take the following forms, respectively,     The expression for the intrinsic absorption cross-section is also determined from [35] by noticing that the polarization of the incident plane waves is TM (i.e., axial polarization of the incident field). Therefore, based on Eqs.(9)-(11) in [35], the intrinsic absorption cross-section is obtained as is also obtained based on Faraday's law.
Finally, the expression for the extinction cross-section is obtained from Eqs. (16) and (17) as Subsequently, the corresponding dimensionless energy efficiency factors are determined from Eqs.(19)- (24), such that,

Numerical examples
In the following, numerical examples for the intrinsic energy efficiencies given by Eqs. (25) and (26) illustrate the analysis for a pair of conducting cylinders of radii a and b respectively, where the separation distance between their center of mass is denoted by d (Fig. 1). Since the cylinders are considered perfectly conducting (and the medium of wave propagation is vacuum, i.e.  = 1), the following properties hold, that is,   In panel (b), a different behavior is anticipated such that the emergence of amplitude variations occurs at  = 0 and weakens when  reaches 180. In both panels, the amplitude variations versus  and kd are related to the strength of the many interactions between the two conducting cylinders. The waves scattered from the first impenetrable perfectly conducting cylinder induce reflected waves (i.e., specular reflection) from the second cylinder along with circumferential circumnavigating waves (i.e., Franz waves [36]) propagating in the exterior homogeneous medium. This phenomenon occurs repetitively during the multiple scattering process. It enhances or reduces the intrinsic extinction (or scattering) energy efficiency factor. From geometry considerations as shown in Fig.  1, clearly the angle of incidence modifies the path length for the circumnavigating Franz waves and their interactions/interferences with the specularly reflected waves, as  varies.
The increase of the size parameter ka for the first particle is also considered such that ka = kb = 1. The related results are exhibited in panels (a) and (b) of Fig. 3. Moreover, panels (c) and (d) display the one-line profile plots versus kd for four different incidence angles,  = 45, 90, 135 and 180. The comparison between the plots in each panel demonstrates that for this particular case where ka = kb, the following property for the local extinction energy efficiency factors holds, such that, The non-dimensional size of the first cylinder is further increased to ka = 5, whereas kb = 1. The related computational plots for the local extinction (or scattering) energy efficiencies are displayed in panels (a) and (b) of Fig. 4, respectively, while panels (c) and (d) display the one-line profile plots versus kd for four different incidence angles,  = 45, 90, 135 and 180. In this case, the multiple scatterings exhibited by the number of the amplitude variations in the plots are changed as ka (or kb) varies compared to the previous plots in Figs. 2 and 3.
Consider now the case of a small Rayleigh conducting cylinder such that kb = 0.1 at a fixed dimensionless interparticle distance kd = 7 while varying the size parameter ka of the first cylinder as well as the angle of incidence. Panels  The effect of increasing kb (= 1) is shown in the plots of panels (a)-(d) of Fig. 6, which display similar behaviors to those shown previously in Fig. 5. Nonetheless, the amplitude variations in the plots are different.
The effect of increasing the dimensionless interparticle distance to kd = 20 is further analyzed for the second Rayleigh (small) cylinder (kb = 0.1, Fig. 7) and for a larger one (kb = 1, Fig. 8). As noticed from the plots in Figs. 7 and 8, the interparticle distance, the angle of incidence as well as the size parameters of both cylinders alter the behavior of the intrinsic extinction (or scattering) energy efficiency factors, where multiple amplitude variations arise. As kd increases, the variations in the amplitudes become more frequent. Although kb is fixed while computing

Conclusion and perspectives
In summary, exact mathematical series expansions are derived for the intrinsic/local extinction, absorption and scattering cross-sections and their related energy efficiency factors. The modal multipole expansion method in cylindrical coordinates is used in conjunction with the translational addition theorem. An effective incident field is determined first, then used to obtain the series expansions for the intrinsic/local cross-sections based on Poynting's theorem. Assuming an axially-polarized electric field component composed of plane travelling waves of arbitrary incidence angle in the polar plane, numerical computations for a pair of perfectly conducting cylinders in a homogeneous free-space are presented and discussed. Particular emphases are given on the interparticle distance separating both cylinders, the angle of incidence as well as their sizes. The present formalism sheds light on the intrinsic (i.e., local) extinction or scattering properties in a system containing a pair of interacting cylindrical particles in EM waves. characterization of the additional cross-scattering term by separating the measured extrinsic cross-section from the intrinsic scattering cross-sections (for each object) computed a priori. Thus, in conjunction with the experimental data, the intrinsic cross-sections computed numerically a priori can be useful in the full characterization of a multiple scattering system of many particles.