Arbitrary shaped beam scattering from a chiral-coated conducting object with arbitrary monochromatic illumination

An exact semi-analytical method of calculating the scattered fields from a chiral-coated conducting object under arbitrary shaped beam illumination is developed. The scattered fields and the fields within the chiral coating are expanded in terms of appropriate spherical vector wave functions. The unknown expansion coefficients are determined by solving an infinite system of linear equations derived using the method of moments technique and the boundary conditions. For incidence of a Gaussian beam, circularly polarized wave, zero-order Bessel beam and Hertzian electric dipole radiation on a chiral-coated conducting spheroid and a chiral-coated conducting circular cylinder of finite length, the normalized differential scattering cross sections are evaluated and discussed briefly.

where ω µ ε = k 0 0 0 , η µ ε = / 0 0 0 are respectively the wavenumber of the incident wave and the characteristic impedance of free space, and α mn , β mn are the unknown expansion coefficients to be determined.
A chiral medium can be characterized by the following constitutive relations 16 ε ε κ µ ε = +i D E H where κ, ε r and μ r denote, respectively, the chirality parameter, relative permittivity and permeability of the chiral medium.
The EM fields existing within the chiral coating (internal fields) can be represented by a combination of the spherical VWFs, in the following form 16 ∑ ∑ = + + + (1) ( 1) where η η µ ε = / r r 0 Eqs (1, 2, 5 and 6) are obtained in the MoM scheme, i.e., expanding the scattered and internal fields by using appropriate spherical VWFs as basis functions.
If the boundary conditions is writed, i.e., continuity of the tangential components of the EM fields at interface S between the chiral coating and free space  and vanishing of the tangential components of the electric field at S 1 (inner conducting object's surface) In Eqs (7-9), E i and H i denote the incident electric and magnetic fields, and, n and n 1 are respectively the outward unit normals to S and S 1 .
By virtue of Eqs (1, 2, 5 and 6), the boundary conditions in Eqs (7 and 8) are written as Eqs (10 and 11) are respectively multiplied (dot product) by the spherical VWFs 0 , and then integrated over S, the following equations are obtained (1 ) ( 1 ) (1 ) The explicit expressions of U m′n′mn , V m′n′mn , K m′n′mn and L m′n′mn are given by Eqs (12)(13)(14)(15) are interpreted as follows. The scattered and internal fields are excited due to the incident fields E i and H i . So, the incident EM beam can be considered as a "source", and the scattered and internal fields as the subsequent "responses". The spherical VWFs 0 are usually used to expand an incident EM beam, and then they are chosen as the weighting functions to derive Eqs (12-15) following the MoM procedure.
A substitution of Eqs (5 and 6) into Eq. (9) leads to The combinations of the spherical VWFs + in Eq. (20) describe two eigenwaves (right and left-handed Beltrami waves) within the chiral coating 8 , and they also respectively represent the Beltrami waves propagating towards or scattered from the inner conducting object when the superscript j = 1 or 3. Motivated by the derivation of Eqs (12-15), we have Eq. (20) multiplied (dot product) by the weighting functions + respectively, and then integrated over S 1 . After performing the above mathematical operations, we can readily obtain ∑ ∑   where (j = 1, 3)  Eqs (12-15 and 21 and 22) provide an infinite system of linear equations to determine the expansion coefficients α mn , β mn , c mn , ′ c mn , d mn and ′ d mn , as the explicit expressions of E i and H i are known. Since the beam description of EM field components is usually obtained in its own system O′x′y′z′, to evaluate numerically the surface integrals including E i and H i in Eqs (12-15) the following transformations ought to be carried out from O′x′y′z′ to the scatterer system Oxyz Generally, compared with the usual MoM solution such as in 11 , the advantages of the above MoM based semi-analytical theoretical procedure are obvious. Instead of using the triangular rooftop vector functions, the corresponding spherical VWFs are adopted as the basis and weighting functions, so most of the formulations are described by analytical expressions. As a result, the number of unknowns that have to be determined is greatly reduced, especially for an axisymmetric object, and then a significant saving of computer time and memory can be achieved to solve for them. Moreover, the current MoM scheme is directly applied to the boundary conditions rather than to the combined field integral equations based on the surface equivalence principle, which is simple in theory and also easy to manipulate mathematically.

Numerical Results
In this section, we will focus on the far-zone scattered field which is often of practical significance. By using the asymptotic form of E s as k 0 r → ∞, the differential scattering cross section (DSCS) is defined in 8,9 σ θ φ π λ π θ φ θ φ In the following calculations, we are restricted to the Gaussian beam, CPW, ZOBB and HED radiated field illuminating from a chiral-coated conducting spheroid and finite-length circular cylinder, i.e., a conducting spheroid coated with a chiral spheroid layer (semimajor and semiminor axes denoted by a and b for the spheroid coating, and by a 1 and b 1 for the inner conducting spheroid) and a conducting cylinder coated with a chiral cylinder layer (length and cross section radius denoted by 2l 0 and r 0 for the cylinder coating, and by 2l 1 and r 1 for the inner conducting cylinder).
Scientific REPORTs | (2018) 8:12350 | DOI:10.1038/s41598-018-30596-2 Figure 2 is shown that the normalized DSCS πσ(θ, 0)/λ 2 for a conducting spheroid either with a chiral or dielectric spheroid coating, illuminated by a Gaussian beam (TE mode) following the Davis first-order expression 18 . In Fig. 2, the numerical results calculated by the present solution are also compared with those by the generalized Lorenz-Mie theory (GLMT) that gives an exact analytical procedure for a coated conducting spheroid in 19 and 20 . As expected, excellent agreements are observed in Fig. 2, which to a certain extent validates the proposed method.
The normalized DSCSs πσ(θ, π)/λ 2 are shown in Fig. 3 for a conducting circular cylinder coated either with a chiral or dielectric cylinder layer, under Gaussian beam illumination as in plotting Fig. 2.
From Figs 2 and 3 we can see that the conducting spheroid and finite-length circular cylinder, whether with a chiral or dielectric coating, have the maximum DSCS around θ = β, i.e., the maximum forward scattering. In addition, compared with the case of a dielectric coating, the difference in the normalized DSCS is obvious for a chiral-coated layer.
It is well-known that the left-and right-hand CPWs (electric and magnetic fields described as are different in their action on chiral media 21 . The difference in the normalized DSCS πσ(θ, π)/λ 2 is shown in Fig. 4 for a chiral-coated conducting spheroid and finite-length circular cylinder as in Figs 2 and 3. As a diffraction free beam, the ZOBB has gained growing attention in various fields [22][23][24] . A detailed description of the ZOBB propagating along the positive z′ axis in O′x′y′z′ has been given 23,24 . Figure 5 is shown the normalized DSCS πσ(θ, π)/λ 2 of a chiral-coated conducting spheroid and finite-length circular cylinder as in Figs 2 and 3 but under the illumination of a ZOBB. From Fig. 5 we can find that, as opposed to the case of a Gaussian beam, the maximum forward scattering dose not appear in the numerical results. In addition, the maximum DSCS for a Gaussian beam is usually larger than that for a ZOBB. Normalized DSCS πσ(θ, 0)/λ 2 for a chiral-coated conducting spheroid (k 0 a 1 = 6, a 1 /b 1 = 2, k 0 a = 9.14, a/b = 2, ε r = 4, μ r = 1, κ = 0.5) and that for a dielectric-coated conducting spheroid (similarly as the former but κ = 0), both illuminated by a Gaussian beam (TE mode, w 0 = 2λ, α = β = 0, x 0 = y 0 = z 0 = 0).

Conclusion
Based on a combination of the EM field expansions in infinite series of the spherical VWFs with the MoM scheme, a semi-analytical solution of the EM beam scattering from a chiral-coated conducting object is proposed. By taking as examples an incident Gaussian beam, VPW, ZOBB and HED radiation striking a chiral-coated conducting spheroid and finite-length circular cylinder, the normalized DSCS is calculated. The correctness of the present theory to a certain extent is validated by comparing the normalized DSCS for a conducting spheroid, either with a chiral or dielectric coating, illuminated by a Gaussian beam with those obtained by the GLMT that gives an exact analytical solution. Theoretically, the present MoM based scheme can be used to treat arbitrary EM beam scattering given their explicit expressions, even extended to an infinite cylinder when appropriate cylindrical VWFs are chosen as the basis and weighting functions. Figure 6. Normalized DSCS πσ(θ, 0)/λ 2 for a chiral-coated conducting spheroid and that for a chiral-coated conducting circular cylinder respectively as in Figs 2 and 3, under the illumination of the HED radiation (x 0 = 2λ, y 0 = z 0 = 0, α = π/6, β = π/4).