Bi-Static Radar Cross Section for PEC Ellipsoid as a Deformed Sphere

Bi-Static Radar Cross Section for PEC Ellipsoid as a Deformed Sphere Baris Ates1, Alp Kustepeli2, Zebih Cetin3 1Department of Mathematics, Izmir Institute of Technology, Türkiye 2Department of Electrical & Electronics Eng., Izmir Institute of Technology, Türkiye 3Department of Physics, Izmir Institute of Technology, Türkiye Corresponding author: [email protected] ORCID IDs: First Author: 0000-0002-8138-7360 Second Author: 0000-0002-8995-6628 Third Author: 0000-0002-2858-7144 DOI : 10.5281/zenodo.18034082 Abstract In this study, the bi-static radar cross section (RCS) characteristics of electromagnetic plane wave scattering from perfectly conducting triaxial ellipsoids are investigated. The ellipsoids are treated as perturbed spheres, and the analytical framework previously developed for arbitrarily shaped scatterers is employed. Unlike the backward and forward scattering cases, the bi-static RCS patterns of ellipsoids exhibit significantly different behaviors compared to those of spheres, indicating that spherical results cannot be directly used as an approximation for ellipsoidal geometries. Comprehensive analyses are performed for various ellipsoid sizes, orientations, and aspect ratios, and the resulting RCS variations are examined with respect to both the polar angle θ and the azimuthal angle φ . The obtained results reveal that the scattering characteristics are highly dependent on these parameters, especially for nonsymmetric configurations. Numerical validations are carried out by comparing the analytical results with full-wave simulations and a good agreement is observed. These findings highlight the distinct scattering behavior of triaxial ellipsoids and demonstrate the accuracy and applicability of the analytical method in predicting bi-static RCS for non-spherical conducting bodies. 1 Introduction The analytical solution of Maxwell equations for a geometry requires the analytical solution of Helmholtz equation for the same geometry. It is known that the Helmhotz equation is separable in 11 coordinate systems [ 1 ]. Due to the complexity of the ellipsoidal wave equation (Lame Equation), which has three regular singularities, one irregular singularity at infinity, and five term recurrence relations and completeness problems [ 2 ], a closed-form solution does not exist for a perfect electric conductor ellipsoidal surface. To overcome this difficulty, one has to consult some approximate solutions or numerical methods [3]-[8]. However, in the present work, a method developed earlier for arbitrarily shaped deformed spheres is followed [ 9 ], [ 10 ]-[ 14 ]. Ellipsoids are considered as perturbed spheres. The advantage F1-1 KOSC-2025 Proceedings of the present method is that, it allows fully analytical calculations. In addition to this, all results are given in the simplicity of spherical Bessel (Hankel) functions and spherical harmonics. The considered ellipsoids are the most general ones, i.e., all of the axis lengths are different. All possible size orderings according to the lengths of x, y, z axis are considered to make comparison of radar cross-section results. The obtained results are compared with those of a full-wave solver, and it is observed that they are in good agreement. 2 Maxwell Equation for Nonspherical Geometries In the present chapter, solution of the Maxwell equations will be briefly reviewed. Electric and magnetic fields both depend on time and spatial coordinates. If the time dependence is assumed to be e−iwt , Maxwell equations are given with the following relations in the absence of external sources [15] ∇ ×  E−iω  B= 0,∇ ×  H+iω  D= 0 (1) ∇ ·  D= 0,∇ ·  B= 0.(2) In the last two equations, the divergences of the fields are zero, thus electric and magnetic fields must be curl of some functions, which means that they must be in the form of  E = −1 ϵ∇ ×  F and  H = 1 µ∇×  A . Taking advantage of the first and second equations, the final form of the electric and magnetic fields is found to be  E = −1 ϵ∇×  F + 1 ωµϵ ∇×∇×  A,  H = 1 µ∇×  A + i ωµϵ ∇×∇×  F where vectors  A and  F are called Debye potentials. Choosing two special form of these fields, (one is  F ( r, θ, φ ) = 0 , A ( r, θ, φ ) = ˆrAr ( r, θ, φ )and another one is  F ( r, θ, φ ) = ˆrFr ( r, θ, φ ) , A ( r, θ, φ ) = 0) enable to find two independent fields [ 16 ]. Thanks to the linearity of the Maxwell equations, superposition principle allows to find general solution in the following form [16] Hr=X n,m ibnm ωµϵ (∂2 ∂r2+k2)( krzn(kr))Pm n(cos θ)eimφ (3) Hθ=X n,m imanm µr sin θkr zn(kr)Pm n(cos θ)eimφ +X n,m ibnm ωµϵr ∂ ∂r(kr zn(kr)) ∂ ∂θPm n(cos θ)eimφ (4) Hφ=−X n,m anm µr krzn(kr)∂ ∂θPm n(cos θ)eimφ −X n,m mbnm ωµϵr sin θ ∂ ∂r(kr zn(kr))Pm n(cos θ)eimφ (5) where r, θ, φ are standard spherical coordinates, anm , bnm are arbitrary constants, zn ( kr )stands for the spherical Hankel (Bessel) functions which is determined according to the considered domain, Pm n ( cos θ )are associated Legendre functions [ 17 ], ω, k are angular frequency, wave number, µ, ϵ are the permeability and permittivity of the considered region respectively. Since electric and magnetic fields are found, one can proceed to the scattering problem. 2.1 Solution of the Maxwell Equations for Deformed Conducting Spherical Boundaries Maxwell equations are separable in certain geometries [ 1 ]. Thus, for an arbitrarily shaped region, a coordinate system is not known whose constant values may describe the boundary of the region. F1-2 2nd Kocaeli Science Congress, November 19–21, 2025 2.1 Solution of the Maxwell Equations for Deformed Conducting Spherical Boundaries Because of this fact, for such kind of general boundaries one need to consult other advanced methods. In this work, because of this necessity, perturbation method is preferred because it enables fully analytical treatment and gives accurate results when higher order corrections are taken into account. For this purpose, second order perturbation method is used to get more accurate results. Since the present work deals with the triaxial ellipsoidal boundaries, in order to convert ellipsoidal boundaries to be eligible to the perturbation method, they are considered as perturbated spheres. The vector combining origin and points on deformed sphere is given by following relation [9] r(θ, φ)=R1+β f1(θ, φ) + β2f2(θ, φ)ˆr, (6) where β is smallness parameter, f1 ( θ, φ )and f2 ( θ, φ )are arbitrarily defined smooth deformation functions. Unit normal to the deformed sphere is ˆn=1−β21 2(∂f1 ∂θ )2−1 2 sin2θ(∂f1 ∂φ )2ˆr+−β∂f1 ∂θ +β2f1 ∂f1 ∂θ −β2∂f2 ∂θ ˆ θ +−β1 sin θ ∂f1 ∂φ +β2f1 sin θ ∂f1 ∂φ −β21 sin θ ∂f2 ∂φ ˆφ(7) In perturbation method, it is assumed that the fields are well behaved, that is, it is assumed that small changes on the boundary leads to the small changes on the fields −→ Btot ( r, θ, φ ) = −→ B0 ( r, θ, φ ) + β−→ B1 ( r, θ, φ ) + β2−→ B2 ( r, θ, φ ), where −→ Btot is total magnetic field, −→ B0 is the magnetic field for sphere, −→ B1 and −→ B2 are the first and second order correction fields. Boundary conditions for the total fields on deformed sphere is formulated as follows, ˆn× Etot ( ˜ R, θ, φ ) = 0,  Btot ( ˜ R, θ, φ ) = 0 Here, it should be noted that, the fields are evaluated on the the perturbed sphere, r = ˜ R . By employing Taylor expansion, these fields can be evaluated on the sphere [9] Fα(˜ R, θ, φ) = hFα(r, θ, φ) + βf1r∂ ∂rFα(r, θ, φ) +β2f2r∂ ∂rFα(r, θ, φ) + β2(f1)2r2 2 ∂2 ∂r2Fα(r, θ, φ)ir=R(8) where α is used to denote r, θ, φ components of the fields. Here vector field  F represent all electric and magnetic fields including fields for sphere and perturbative correction fields. Using these expansions boundary conditions on deformed sphere governing the relation between the fields are given by [9] 1 sin θ ∂f1 ∂φ E0 r+f1r∂ ∂rE0 φ+E1 φr=R= 0 (9) h1 2 sin θ ∂(f1)2 ∂φ E0 r−1 sin θ ∂(f1)2 ∂φ r∂ ∂rE0 r−1 sin θ ∂f1 ∂φ E1 r−f2 1r2 2 ∂2 ∂r2E0 φ−f1r∂ ∂rE1 φ−E2 φ −1 sin θ ∂f2 ∂φ E0 r−f2r∂ ∂rE0 φir=R= 0 (10) 2nd Kocaeli Science Congress, November 19–21, 2025 F1-3 KOSC-2025 Proceedings hf1r∂ ∂r B0 r(r, θ, φ) + B1 r(r, θ, φ)−∂f1 ∂θ B0 θ(r, θ, φ)−1 sin θ ∂f1 ∂φ B0 φ(r, θ, φ)ir=R= 0 (11) hf2 1r2 2 ∂2 ∂r2B0 r+f1r∂ ∂rB1 r+B2 r−1 2 ∂(f2 1) ∂θ r∂ ∂rB0 θ−∂f1 ∂θ B1 θ+1 2 ∂(f1)2 ∂θ B0 θ−1 2 sin θ ∂(f1)2 ∂φ r∂ ∂rB0 φ −1 sin θ ∂f1 ∂φ B1 φ+1 2 sin θ ∂(f1)2 ∂φ B0 φ+f2r∂ ∂rB0 r−∂f2 ∂θ B0 θ−1 sin θ ∂f2 ∂φ B0 φir=R= 0.(12) Solution of this system of equations enables to find perturbative correction fields. 2.2 Scattering of Electromagnetic Plane Waves from Deformed Spheres When perturbation method is used for the scattering problems, zeroth order field is consist of incident and scattered fields for the sphere. Hence total fields is of the following form, −→ Btot(r, θ, φ) = −→ Binc(r, θ, φ) + −→ Bsphr(r, θ, φ) + β−→ B1(r, θ, φ) + β2−→ B2(r, θ, φ)(13) Since zeroth order fields are known, starting from first order correction fields, all correction fields are found step by step by using boundary conditions given by (9)-(12). Determining fields means that determining the unknown coefficients anm and bnm ’s. Since radial component of the magnetic field contains only unkown coefficiens bnm , we start with the boundary condition (11) B1 r(R, θ, φ) = −f(θ, φ)r∂ ∂rB0 r(r, θ, φ) + Binc r(r, θ, φ) +fθ(θ, φ)B0 θ(r, θ, φ) + Binc θ(r, θ, φ)+fφ(θ, φ) sin(θ)B0 φ(r, θ, φ) + Binc φ(r, θ, φ)r=R .(14) If we multiply both side of the above equation with Pm′ l ( cos θ ) sin θe−im′φ and taking the angular integrals with the use of orthogonality type relations of the associated Legendre functions, one can find a single b1 nm coefficient for fixed values of n and m . In order to evaluate the right hand side of the expression exactly, one need to expand perturbation functions f1, ( f1 ) 2 and f2 in terms of spherical harmonics as f(θ, φ) = ∞ X j=0 j X s=−j fs jPs j(cos θ)eisφ (15) with the help of these expansions, the right hand side of the expression contains three associated Legendre functions, one coming from the definitions of fields in (3)-(5), and one spherical harmonics expansion of the perturbation functions in (15) and the final one comes from the multiplication factor. One of those integrals is given below Z2π 0Zπ 0 Pm n(cos θ) sin θ ∂Ps j(cos θ) ∂θ Pm′ l(cos θ)ei(m−m′)φsin θdθdφ (16) For the exact value of the above integral and for the explicit expressions of other integrals one may consult the work [ 9 ]. With the help of these integrals, one can obtain the unknown coefficients of the first order correction fields b1 lm′ . In order to find other coefficients a1 lm′ one can use the boundary condition (9), for the coefficients b2 lm′ one can use (12) and finally for the coefficients a2 lm′ one can use (10). This means, with this operations all perturbed fields have F1-4 2nd Kocaeli Science Congress, November 19–21, 2025 been evaluated explicitly [9]. 3 Bi-Static Radar Cross Section Results The radar cross section is defined by the following formula [18], σ= lim r→∞ 4πr2| Es|2 | Ei|2!.(17) Due to the perturbation of the boundary, similar to the fields, the radar cross section is also perturbed and thus it consists of correction terms, σ = σsph + βσ1 + β2σ2 + O ( β3 )[ 14 ]. A triaxial ellipsoid is given by the following equation x2 a2+y2 a2(1+β)2+z2 a2(1+βξ)2= 1.(18) This equation can be written in spherical coordinates with the help of the functions, f1 ( θ, φ ) = sin2θsin2φ + ξcos2θ , f2 ( θ, φ ) = 3 2, f2 1 ( θ, φ ) −f1 ( θ, φ ) + ( ξ−ξ2 ) ,cos2θ,  [ 12 ]. Choosing different values of β and ξ allows one to consider different ellipsoids in size. In Fig. 1, bi-static radar cross sections are given for constant φ = π/ 2. In Fig. 1(a), bi-static RCS results for ellipsoids are shown with the parameters ξ = 0 . 2, β = (0 . 2) 2, (0 . 3) 2, (0 . 4) 2 . The axis lengths of these three ellipsoids are x = 10 . 0 , y = 10 . 4 , z = 10 . 08, x = 10 . 0 , y = 10 . 9 , z = 10 . 18 and x = 10 . 0 , y = 11 . 6 , z = 10 . 32 respectively. The lengths are in order as y > z > x . It can be seen from the figure that the differences between the result of the sphere and the ellipsoidal ones are much more distinct in the range 0 ◦≤θ≤ 25 ◦ . In this range, all ellipsoidal results are greater than the spherical ones and when β increases, the RCS results also increase. In the range 25 ◦≤θ≤ 80 ◦ , the ordering is the reverse of the one in the range 0 ◦≤θ≤ 25 ◦ . In Fig. 2(a), bi-static RCS results for the constant angle θ = π/ 2are shown. In almost all the considered range of φ , the ellipsoidal RCS results are bigger than the spherical one. The difference between the ellipsoidal ones and the spherical one is much larger around the angles 0 ◦, 90 ◦, 270 ◦ and 360 ◦ . Comparisons of the result with CST shows a good agreement for both θ constant and ϕ constant cases. In Fig.1(b), RCS results for ellipsoids with the parameters ξ = 0 . 2, β = − (0 . 2) 2,− (0 . 3) 2,− (0 . 4) 2 are shown. The axis lengths of these three ellipsoids are x = 10 . 0 , y = 9 . 6 , z = 9 . 92, x = 10 . 0 , y = 9 . 1 , z = 9 . 82 and x = 10 . 0 , y = 8 . 4 , z = 9 . 68 respectively. The lengths are in order as x>z>y . It can be seen from the figure that, in the range 0 ≤θ≤ 30 ◦ , all ellipsoidal results are lower than the spherical ones and when the absolute value of β increases, the RCS results decrease. In the range 30 ≤θ≤ 80 ◦ the ordering is the reverse of that one in the range 0 ≤θ≤ 30 ◦ . For the rest of the range, there is no significant difference compared with the spherical result. For this choice of parameters, RCS results are shown in Fig. 2(b) from which one can conclude that, when compared with the spherical one, there is no significant difference except for the angles around 0◦,180◦and 360◦. In Fig.1(c), RCS results for ellipsoids obtained by the parameters ξ = − 0 . 2, β = (0 . 2) 2, (0 . 3) 2 , (0 . 4) 2 are shown. The axis lengths of these three ellipsoids are x = 10 . 0 , y = 10 . 4 , z = 9 . 92, x = 10 . 0 , y = 10 . 9 , z = 9 . 82 and x = 10 . 0 , y = 11 . 6 , z = 9 . 68 respectively. The lengths are in 2nd Kocaeli Science Congress, November 19–21, 2025 F1-5 KOSC-2025 Proceedings 0 500 1000 1500 2000 2500 3000 3500 4000 0 20 40 60 80 100 120 140 160 180 PM Sphere kR=3 PM ξ=0.2 β=(0.2)2 kR=3 CST ξ=0.2 β=(0.2)2 kR=3 PM ξ=0.2 β=(0.3)3 kR=3 CST ξ=0.2 β=(0.3)2 kR=3 PM ξ=0.2 β=(0.4)2 kR=3 CST ξ=0.2 β=(0.4)2 kR=3 σ(mm2) θ(°) (a) 0 500 1000 1500 2000 2500 3000 3500 0 20 40 60 80 100 120 140 160 180 PM Sphere kR=3 PM ξ=0.2 β=-(0.2)2 kR=3 PM ξ=0.2 β=-(0.3)3 kR=3 PM ξ=0.2 β=-(0.4)2 kR=3 σ(mm2) θ(°) (b) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 0 20 40 60 80 100 120 140 160 180 PM Sphere kR=3 PM ξ=-0.2 β=(0.2)2 kR=3 PM ξ=-0.2 β=(0.3)3 kR=3 PM ξ=-0.2 β=(0.4)2 kR=3 σ(mm2) θ(°) (c) 0 500 1000 1500 2000 2500 3000 3500 0 20 40 60 80 100 120 140 160 180 PM Sphere kR=3 PM ξ=-0.2 β=-(0.2)2 kR=3 PM ξ=-0.2 β=-(0.3)3 kR=3 PM ξ=-0.2 β=-(0.4)2 kR=3 σ(mm2) θ(°) (d) 0 500 1000 1500 2000 2500 3000 3500 4000 0 20 40 60 80 100 120 140 160 180 PM Sphere kR=3 PM ξ=1.1 β=(0.20)2 kR=3 PM ξ=1.1 β=(0.25)2 kR=3 PM ξ=1.1 β=(0.30)2 kR=3 σ(mm2) θ(°) (e) 0 500 1000 1500 2000 2500 3000 3500 0 20 40 60 80 100 120 140 160 180 PM Sphere kR=3 PM ξ=1.1 β=-(0.20)2 kR=3 PM ξ=1.1 β=-(0.25)2 kR=3 PM ξ=1.1 β=-(0.30)2 kR=3 σ(mm2) θ(°) (f) Figure 1: Bi-static radar cross section for different ellipsoids when φ=π/2. order as y > x > z . It can be seen from the figure that the results are similar to the ones in Fig.1(a). The RCS results for the constant θ obtained with the same parameters are given in Fig. 2(c). It can be observed from the figure that the differences between the spherical result are much larger around the angles 0 ◦, 180 ◦ and 360 ◦ . In Fig.1(d), RCS results for ellipsoids obtained by the parameters ξ = − 0 . 2, β = − (0 . 2) 2 , ,− (0 . 3) 2,− (0 . 4) 2 are shown. The axis lengths of these three ellipsoids are x = 10 . 0 , y = 9 . 6 , z = 10 . 08, x = 10 . 0 , y = 9 . 1 , z = 10 . 18 and x = 10 . 0 , y = 8 . 4 , z = 10 . 32 respectively. The lengths are in order as z > x > y . The results are similar to the ones in Fig. 1(b) and as it can be seen, the ordering with respect to the amount of RCS results changes about the angles 27 . 5 ◦ and 86 ◦ . For the constant angle θ , the RCS results are smaller than the spherical one, and when the absolute value of β increases, the deviation from the spherical results also increases as shown in Fig. 2(d). In Fig.1(e), RCS for the ellipsoids obtained by the parameters ξ = 1 . 1, β = (0 . 20) 2, (0 . 25) 2 ,(0 . 30) 2 are shown. The axis lengths of these three ellipsoids are x = 10 . 0 , y = 10 . 4 , z = 10 . 44, x = 10 . 0 , y = 10 . 625 , z = 10 . 6875, and x = 10 . 0 , y = 10 . 9 , z = 10 . 99 respectively. The lengths are in order as z > y > x . The results are similar to the ones given in 1(c), in the range 20 ◦≤θ≤ 140 ◦ , the spherical results can be used as an approximation to those of the ellipsoidal case. The constant angle θ results are greater than the spherical ones in almost the entire region, as shown in Fig. 2(e). In Fig.1(f), RCS for the ellipsoids obtained by the parameters ξ = 1 . 1, β = − (0 . 2) 2 , − (0 . 25) 2,− (0 . 30) 2 are used to obtain the ellipsoids with axes lengths x = 10 . 0 , y = 9 . 6 , z = 9 . 56 and x = 10 . 0 , y = 9 . 375 , z = 9 . 3125, x = 10 . 0 , y = 9 . 1 , z = 9 . 01. The lengths are in order as x>y>z . The results are similar to the ones in Fig. 1(b), for the range 25 ◦≤θ≤ 120 ◦ the RCS results are quite close to the spherical F1-6 2nd Kocaeli Science Congress, November 19–21, 2025 0 100 200 300 400 500 600 700 800 0 50 100 150 200 250 300 350 PM Sphere kR=3 PM ξ=0.2 β=(0.2)2 kR=3 CST ξ=0.2 β=(0.2)2 kR=3 PM ξ=0.2 β=(0.3)3 kR=3 CST ξ=0.2 β=(0.3)2 kR=3 PM ξ=0.2 β=(0.4)2 kR=3 CST ξ=0.2 β=(0.4)2 kR=3 σ(mm2) ϕ(°) (a) 0 100 200 300 400 500 600 0 50 100 150 200 250 300 350 PM Sphere kR=3 PM ξ=0.2 β=-(0.2)2 kR=3 PM ξ=0.2 β=-(0.3)3 kR=3 PM ξ=0.2 β=-(0.4)2 kR=3 σ(mm2) ϕ(°) (b) 0 100 200 300 400 500 600 0 50 100 150 200 250 300 350 PM Sphere kR=3 PM ξ=-0.2 β=(0.2)2 kR=3 PM ξ=-0.2 β=(0.3)3 kR=3 PM ξ=-0.2 β=(0.4)2 kR=3 σ(mm2) ϕ(°) (c) 0 100 200 300 400 500 600 0 50 100 150 200 250 300 350 PM Sphere kR=3 PM ξ=-0.2 β=-(0.2)2 kR=3 PM ξ=-0.2 β=-(0.3)3 kR=3 PM ξ=-0.2 β=-(0.4)2 kR=3 σ(mm2) ϕ(°) (d) 0 100 200 300 400 500 600 0 50 100 150 200 250 300 350 PM Sphere kR=3 PM ξ=1.1 β=(0.20)2 kR=3 PM ξ=1.1 β=(0.25)3 kR=3 PM ξ=1.1 β=(0.30)2 kR=3 σ(mm2) ϕ(°) (e) 0 100 200 300 400 500 600 0 50 100 150 200 250 300 350 PM Sphere kR=3 PM ξ=1.1 β=-(0.20)2 kR=3 PM ξ=1.1 β=-(0.25)3 kR=3 PM ξ=1.1 β=-(0.30)2 kR=3 σ(mm2) ϕ(°) (f) Figure 2: Bi-static radar cross section for different ellipsoids when θ=π/2. one. When the constant angle θ results shown in Fig. 2(f) are considered, it can be concluded that deviation from the spherical result is not as much as the previous cases. 4 Conclusion In the present work, scattering of electromagnetic plane waves from perfect electric conductor ellipsoids are considered by following the results of [ 9 ]. Considering Ellipsoids as deformed spheres enabled to carry out all calculations in the simplicity of spherical functions whose analytical properties are much easier than that of the ellipsoidal ones. Radar cross section results for 18 different ellipsoids are given. Validation of the results are achieved by comparing obtained results with that of a full wave solver and it is observed that they are in a good agreement. In addition to the this, axis lengths of the ellipsoids are chosen differently, hence all 6different possible length ordering is considered to get an insight on the importance of the axis lengths on radar cross sections. Since all possible ordered cases are studied and presented for bi-static radar cross section, present work may enables to researchers take a forward step towards to understand the light scattering from ellipsoidal particles. References [1] Morse P. M. and Feshbach H., (1953), Methods of Theoretical Physics, McGraw - Hill Science. 2nd Kocaeli Science Congress, November 19–21, 2025 F1-7 KOSC-2025 Proceedings [2] Arscott, F. M., Taylor, P. J.,Zahar, R. V. M. (1983), On the Numerical Construction of Ellipsoidal Wave Functions, Mathematics of Computation, 40 (161) 367–380. [3] Huang, Y. and Kok, J. F. and Saito, M. and Muñoz, O., (2023), Single-scattering properties of ellipsoidal dust aerosols constrained by measured dust shape distributions, Atmospheric Chemistry and Physics, 23 (4) 2557-2577. [4] A. Charalambopoulos and G. Dassios, (1999), Scattering of a spherical wave by a small ellipsoid, IMA JAM, 62 117-136. [5] G. Perrusson, M. Lambert, D. Lesselier, A. Charalambopoulos, and G. 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