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J. Electromagn. Eng. Sci > Volume 24(1); 2024 > Article
Bozorgi: Computation of TM Scattered Fields Induced by a Truncated Semi-Elliptic Cavity in a PEC

Abstract

This study investigated the scattered transverse-magnetic (TM) electromagnetic field from a truncated semi-elliptic embedded in a perfectly electric conducting plane. Boundary value analysis and the region-point-matching technique are employed to address the issues considered. The analyzed region is divided into three sub-regions, after which the tangential fields are expanded in terms of Mathieu functions into two different coordinate systems. By applying boundary conditions at two auxiliary boundaries and using the region-point-matching technique, several independent equations are constructed and subsequently simplified to generate a system of linear equations. For the validation of the obtained results, the suggested procedure is compared to the method of moment. Finally, the effects of cavity depth and incident angle on the scattering signature are inspected.

I. INTRODUCTION

Investigations into scattered waves arising from different objects and cavities offer useful insights for the design of electromagnetic structures used in practical applications, including optics, non-destruction testing, remote sensing, and radar cross section (RCS) reduction. Multiple methods have been proposed for analyzing the scattering of electromagnetic (EM) waves induced by various types of opened cavities, such as rectangular grooves [15], circular-arc channels [68], and triangular grooves [911].
EM scattering by elliptical cavities have also been conducted [1214]. A simple closed-form solution for the scattering generated from a semi-elliptic groove in the low-frequency limit was derived in [12], which was found to be valid for arbitrary polarization, arbitrary incidence and scattering directions, as well as arbitrary eccentricity. In [13], an analytic series solution based on the mode-matching technique was proposed for a semi-elliptic groove. Furthermore, [14] derived an analytic solution to the problem of scattering using a dielectric-coated conducting elliptic cylinder loading an elliptic groove in a perfect electric conductor (PEC). However, these solutions are only usable in the case of simple elliptic cavities. To calculate electromagnetic waves in a complex structure, such as elliptical boundaries, it seems reasonable to employ full numerical methods, such as the finite element method and the method of moment (MoM) meshing technique. However, when the size of the structure increases, the refinement of its grids produces enormous systems, solving which is computationally expensive.
This paper develops an efficient semi-analytic method to solve the issue of EM scattering originating from a truncated semi-elliptic cavity embedded in a PEC ground plane. The solution is based on boundary value analysis [13] and the region-point-matching method [15], which is a numerical matching procedure. As a result, the proposed method simultaneously benefits from the advantages of both the numerical and analytical methods. Furthermore, this methodology is easily applicable to complex problems related to elliptic boundaries, for which the Mathieu function addition theorem, characterized by difficult series and equations, has mostly been used. The Mathieu functions represent the solution to the wave equation with elliptical boundary conditions [16]. In 1868, Emile Mathieu presented a new differential equation, called the Mathieu equation, whose eigenvalues and corresponding periodic solutions led to the definition of a new class of functions, known as the Mathieu functions. Later, Whittaker and others developed novel theories and methods to compute the Mathieu functions [1619].
The steps involved in addressing the problem being dealt with in this research are as follows: first, two semi-elliptical auxiliary boundaries are introduced, following which the analyzed region is divided into three sub-regions. The tangential fields in the three sub-regions are expanded according to the Mathieu functions. By imposing matching boundary conditions at the first auxiliary interface (between sub-regions 1 and 2) and applying the orthogonal properties of the angular Mathieu functions, two sets of linear equations are analytically obtained. In the next step, the boundary conditions at the second semi-elliptical auxiliary border (between sub-regions 2 and 3) are satisfied by uniformly collocating the points in two different coordinate systems, leading to the construction of two other sets of linear equations. In this context, it is noteworthy that the collocation method is used to match the tangential fields for the purpose of avoiding the complex application of the addition theorem. Finally, the solution is summarized as a set of infinite algebraic equations that can be solved numerically by truncating infinite equations. Following this, the proposed method is compared to the MoM method used in the FEKO software to validate the obtained results. Finally, the effects of the cavity depth and incidence angle on the EM scattering signature are investigated.

II. THEORETICAL FORMULATION

1. Description of Cavity

This study considers a semi-elliptical open cavity embedded in a PEC that is horizontally truncated at the bottom, as shown in Fig. 1. Two Cartesian coordinate systems and its two corresponding elliptic cylinder coordinate systems are also presented in Fig. 1. The origin of the global coordinate systems (x, y) or (ξ, η) is fixed at the center of the elliptic cavity, while the origin of the local coordinate system (x1, y2) or (ξ1, η1) is placed at the midpoint of the cavity bottom. The half lengths of the major and minor axes and the focal distance of the elliptic cavity are a, b, and c, respectively. Furthermore, the truncation depth of the semi-elliptical cavity is denoted as d, while the bottom width is referred to as 2W. The two lower corners of the cavity are located at η = −ηb and η = π + ηb with respect to the global elliptic coordinate. In the case of the two semi-elliptical auxiliary borders, the semi-elliptical boundary S1 is defined as {(ξ, η)|ξ = ξc0}, based on which the second half of the curve forms the cavity wall, and the semi-elliptical boundary S2 is considered {(ξ1, η1)|ξ1 = ξc1}, where the half lengths of the major and minor axes and focal distance of S2 are W, b1, and c1, respectively. As shown in Fig. 1, by introducing the auxiliary boundaries S1 and S2, the analyzed region can be divided into three sub-regions—an open region (Region 1) and two enclosed regions (Regions 2 and 3). In this context, it should be noted that the interface S2 should satisfy two important conditions:
  • 1) It should not cross the interface S1.

  • 2) The origin of o should be placed below the border S2 inside Region 2.

To satisfy these conditions, d < b1bd is considered. Notably, the shape of the boundary S2 can be altered as desired by changing its axial ratio.

2. Electric Field Expansion in Region 1

A transverse-magnetic (TM) polarized plane wave can be expressed as follows:
(1)
EZi=z^ejk0(xcos ϕi+ysin ϕi)         ϕi(0,π),
where the suppressed time dependence of exp (jωt) is incident on a truncated semi-elliptic cavity with the incidence angle ϕi, as shown in Fig. 1, where k0 is the free-space wave number. In the open Region 1, the sum of the incident and reflected fields in the global coordinate system (ξ, η) can be expanded using the Mathieu functions as follows:
(2)
Ezi+Ezr=4Σn=1inMsn(1)(ξ,q)sen(η,q)sen(ϕi,q),
where q is (k0a e)2/4. Furthermore, Msn(1)(.) and sen(.) are the odd radial and angular Mathieu functions of the first kind for order n [20, 21]. Meanwhile, the scattered field can be expressed as follows:
(3)
Ezs(ξ,η)=Σn=1AnMsn(4)(ξ,q)sen(η,q),
where Msn(4)(.) is the radial Mathieu function of the fourth kind [20, 21].
Subsequently, the total electric field EzI(ξ,η) in Region 1 can be expressed as follows:
(4)
EzI=Ezi+Ezr+Ezs=4Σn=1inMsn(1)(ξ,q)sen(η,q)sen(ϕi,q)+Σn=1AnMsn(4)(ξ,q)sen(η,q),
where An refers to the unknown coefficients that need to be determined.

3. Electric Field Expansion in Region 2

As mentioned previously, the center of the semi-elliptical boundary S1 is located outside Region 2. Therefore, there are no singular points in this region, which indicates that the z-components of the electric field in this region can be expressed using a proper wave function, as follows:
(5)
EzII=Σn=1[BnMsn(1)(ξ,q)+CnMsn(4)(ξ,q)]sen(η,q)+[DnMcn(1)(ξ,q)+EnMcn(4)(ξ,q)]cen(η,q),
where Mcn(1)(.),Mcn(4)(.) and cen(.) are the even radial Mathieu functions of the first and fourth kinds and the angular Mathieu function of the first kind for order n, respectively [20, 21]. In Eq. (5), the expansion coefficients Bn, Cn, Dn and En are determined by applying the boundary conditions.

4. Electric Field Expansion in Region 3

Region 3 comprises a semi-elliptical auxiliary boundary S2 and a flat PEC boundary at the bottom of the cavity (Γb). According to the coordinate system (ξ1, η1), the electric field Ez(III)(ξ1,η1) in the closed Region 3 should be zero at η1 = 0 and η1 = π. Therefore, it can be denoted in the following form:
(6)
Ez(III)(ξ1,η1)=n=1FnMsn(1)(ξ1,q1)sen(η,q1)
where q1 is (k0W e1)2/4. Notably, the complex expansion coefficients Fn are unknown.

5. Boundary Conditions Applying

Before applying the boundary conditions, it is necessary to obtain the tangential component of the magnetic field. The tangential magnetic field Hη in each region is derived using Maxwell’s equations, as follows:
(7)
Hη=1jωμ0ccosh2(ξ)-cos2(η)Ezξ.
The tangential fields in Regions 1, 2, and 3 should satisfy the following boundary conditions:
(8)
On   S1{EzI(ξ,η)=EzII(ξ,η)at ξ=ξc0,   0ηπEηI(ξ,η)=EηII(ξ,η)at ξ=ξc0,   0ηπ,
(9)
On   S2{EzII(ξ,η)=EzIII(ξ,η)at ξ1=ξc1,   0η1πEηII(ξ,η)=HηIII(ξ1,η1)at ξ1=ξc1,   0η1π
(10)
On   ΓlEzII(ξ,η)=0at ξ=ξc0,   πηπ+ηb,
(11)
On   ΓrEzII(ξ,η)=0at ξ=ξc0,-ηbη2π.
By applying the six boundary conditions, the unknown coefficients An, Bn, Cn, Dn, En and Fn can be obtained. The above expressions show that Eqs. (8), (10), and (11) are expressed in the elliptical coordinate system (ξ, η), while the equations pertaining to (9) are in two different elliptical coordinate systems. In such a case, the Mathieu function addition theorem is the first solution that is usually considered for transferring the Mathieu functions located in multiple elliptical coordinate systems. However, this theorem is extremely challenging to employ. This study proposes a solution to avoid this difficulty by using the region-point matching technique at border S2.

6. Equations Solving

To solve Eqs. (8)(11) for the unknown expansion coefficients, both sides of Eqs. (8), (10), and (11) are first multiplied by sem(η, q) and then integrated with the corresponding boundary to obtain four independent linear equations, as follows:
(12)
Σn=1[AnMsn(4)(ξc0,q)-BnMsn(1)(ξc0,q)-CnMsn(4)(ξc0,q)]π2δnm-[DnMcn(1)(ξc0,q)+EnMcn(4)(ξc0,q)]Znm=-4Σn=1inMsn(1)(ξc0,q)π2δnm,
(13)
Σn=1[AnMsn(4)(ξc0,q)-BnMsn(1)(ξc0,q)-CnMsn(4)(ξc0,q)]π2δnm-[DnMcn(1)(ξc0,q)+EnMcn(4)(ξc0,q)]Znm=-4Σn=1inMcn(4)(ξc0,q)π2δnm,
(14)
Σn=1[BnMsn(1)(ξc0,q)+CnMsn(4)(ξc0,q)]Ynm+[DnMcn(1)(ξ,q)+EnMcn(4)(ξ,q)]Knm=0,
(15)
Σn=1[BnMsn(1)(ξc0,q)+CnMsn(4)(ξc0,q)]Lnm+[DnMcn(1)(ξ,q)+EnMcn(4)(ξ,q)]Inm=0,
where δnm is the Kronecker delta, while functions Znm, Ynm, Knm, Lnm, and Inm have been defined in Appendix. Furthermore, the prime (′) notation is used to signify differentiation with respect to the corresponding function. Subsequently, to convert Eq. (9) into linear equations, point collocation—a mesh-free technique where the electric and magnetic fields are matched at discrete places at S2, whose residuals are zero—is used. For this purpose, a sequence of points along S2 is uniformly considered. The coordinate of the mth collocation point on S2 in coordinate systems (ξ1, η1) and (ξ, η) are represented as (ξc1, η1m) and (ξm, ηm), respectively. To convert the coordinate (ξc1, η1m) into coordinate (ξm, ηm), coordinate (ξc1, η1m) was first transformed from elliptic to Cartesian coordinates using the following expressions:
(16)
{x1m=c1cosh(ξc1)cos (η1m)y1m=c1sinh(ξc1)sin (η1m).
Assuming d is the distance between coordinate systems (x, y) and (x1m, y1m), xm = x1m and ym = y1m + d. The formulas to transform from Cartesian to elliptic coordi-nates, as noted in [25], are as follows:
(17)
{ηm=arcsin(pm)ξm=12ln(1-2rm+2rm2-rm.
Notably, parameters pm and rm in (17) can be defined as follows:
(18)
pm=-Bm+Bm2+4c2ym22c2,rm-Bm±Bm2+4c2ym22c2,
where Bm=xm2+ym2-c2. Since the number of sampling points on S2 is M, the angular coordinate of the mth collocation point is η1m = /M. Considering Eq. (9) and truncating the summation indices n according to the M terms, two other independent equations are derived, as follows:
(19)
n=1N[BnMsn(1)(ξm,q)+CnMsn(4)(ξm,q)]sen(ηm,q)+[DnMcn(1)(ξm,q)+EnMcn(4)(ξm,q)]cen(ηm,q)-FnMsn(1)(ξc1,q1)sen(η1m,q1)=0,         m=1,2,Mm
(20)
n=1N[BnMsn(1)(ξm,q)+CnMsn(4)(ξm,q)]sen(ηm,q)+[DnMcn(1)(ξm,q)+EnMcn(4)(ξm,q)]cen(ηm,q)-FnMsn(1)(ξc1,q1)sen(η1m,q1)=0,         m=1,2,M.
To compute the unknown coefficients numerically, it is essential to truncate the infinite series in (12)–(15) into finite numbers. Therefore, in Eqs. (12)(15), the summation index n is truncated into M terms. The truncated term number M depends on the accuracy requirement. Subsequently, Eqs. (12)(15) and (19)(20) constructed a set of linear systems with regard to the unknown expansion coefficients (An, Bn, Cn, Dn, En and Fn) and could be solved by matrix methods. Finally, the scattered wave in Region 3 was directly calculated using (3). Meanwhile, in the far zone (ξ → ∞), Ezs can be simplified and expressed as a function of η, as follows [13]:
(21)
Ezs(ρ,η)=2πk0ρe-j(k0ρ-π4)Σn=1N(j)nAnsen(η,q),
where ρcsinhξ(ξ) Finally, the backscattering echowidth for TM polarization ( σTM2D) was defined as:
(22)
σTM2D=limρ2πρEzs2Ezi2.

III. RESULTS

This section presents some examples to prove the validity of the solution developed in this study for computing near and far fields. Furthermore, to examine the influence of the problem parameters, such as the incidence of the angle (ϕi), the observation angle (ϕd) and the cavity depth (d), on the scattered field, sample numerical results are presented in Figs. 27.
The results obtained using the proposed method were examined by comparing them to the data obtained from the FEKO software, which uses MoM. During the calculation, the infinite series had to be truncated appropriately. However, the size of the cavity is a primary factor that affects the truncation term number M. Taking this into account, this study investigated the validity of the suggested method. Fig. 2 illustrates variations in the amplitudes of the electromagnetic fields |Ez|,|Hx|, and |Hy| in terms of x position at y = 0 for the cavity presented in Fig. 1, with a/λ = 1.5, b/λ = 1, d/λ = 0.75 when ϕi = 90°. Following this, the results shown in Fig. 3 were considered for an oblique incident case, i.e., ϕi = 60°. As shown in Fig. 2, the distribution of fields on the cavity is symmetrical for the perpendicular incident case. The results depicted in Fig. 2 were obtained using the proposed method and MoM. A comparison of the results observed in these figures demonstrates the accuracy of the proposed method. Furthermore, consequent examinations indicate that M = 50 is adequate for the given example to yield reliable results.
In the previous example, the convergence behavior of the echowidth obtained by the proposed method was verified by calculating the normalized errors of the series coefficient Am in (21) for different values of M using the following equation [22]:
(23)
error(M)=Σn=0MAnM-AnM-12Σn=0MAnM2.
The results of this exercise are plotted against the number of samples of M in Fig. 2(d), depicting that when M increases, the error increases initially and then undergoes a decline.
Furthermore, to study the influence of incident angle on the backscattering echowidth σTM2D (where ϕd = ϕi), as well as to validate the results obtained for the various incidence angles, two different cavity sizes were considered. In the first case (small cavity), the scattering echowidth versus the incident angle ϕd was calculated for a/λ = 0.5, b/λ = 0.25, and d/λ = 0.2, as illustrated in Fig. 4(a).
In the second case (large cavity), as shown in Fig. 4(b), a/λ = 1.5, b/λ = 1, and d/λ = 0.75 were considered. Their echowidths were obtained, with the incident angle ϕi varying from 0° to 90°. The results presented in Fig. 4 demonstrate that the scattering echowidth generally increases with an increase in the incident angle. However, in the case of the large cavity, two local minimums are observed at around ϕi = 60° and 80°, where most of the energy is scattered in a direction that is different from that of the incident. Considering MoM as the reference, the results shown in Fig. 4 indicate that the proposed method is applicable to cases pertaining to both small and large cavities. Furthermore, Fig. 5 displays the TM bistatic σTM2D (where the incident angle ϕi is fixed and the observation angle ϕd changes) in terms of the observation angle ϕd when ϕi = 60° and 90° for the cavity depicted in Fig. 1, with a/λ = 0.5, b/λ = 0.25, and d/λ = 0.125. Therefore, this method exhibits good agreement with the MoM used in FEKO software.
The simulation time of the proposed method and MoM (for a truncated semi-elliptic cavity with specifications as shown in Fig. 2) were calculated to conduct a comparison of the time consumed. The computing time for the proposed procedure was 7.23 seconds, while the time consumed for MoM was about 18 minutes. This indicates that the solution devised in this study enables rapid computation while also ensuring accuracy for electrically large cavities. However, computational efficiency becomes a significant problem when scattering evaluation is required for frequent simulations, such as inverse problems where a large amount of data is required to estimate the shape of a cavity or an object from scattering patterns.
In another example, the effect of cavity depth on echowidth at normal incidence was investigated. The results of this examination are presented in Fig. 6. The specifications of the truncated semi-elliptic cavity in this example were a/λ = 2, b/λ = 1.5. Furthermore, the cavity depth was intially kept at d = 0.15λ, which was then increased by steps of 0.05λ until = 1.5λ was reached. Fig. 6 depicts three dips at a = 0.5λ, 1λ, and 1.5λ, indicating that less energy is scattered in the direction of the incidence and more is scattered in other directions in these depths.
In addition, this study examined the influence of cavity depth d on the bistatic echowidth pattern. In the previous example, the bistatic echowidth of the cavity ( σTM2D) was computed for nine values of d at ϕi = 90°. These values were plotted in the polar coordinate system as functions of the observation angle ϕd in three charts (a), (b) and (c) as displayed in Fig. 7. The scattering patterns shown in Fig. 6 demonstrate that as the depth of the cavity increases, the number of side lobes decreases. In Fig. 7(a), when d increases to 0.5λ, the width of the side lobes increases, and the pattern becomes wider and smoother. This trend can be observed for the other charts as well, where d increases from 0.75λ to 1λ in Fig. 7(b) and from 1.05λ to 1.5λ in Fig. 7(c). As also noted in Fig. 5, an unexpected decrease in echowidth occurs at some depths. Fig. 7 clearly reveals that at these depths, the main lobe is weakened, and the side lobes are joined together, creating a unified pattern. Furthermore, at these depths, the waves are roughly but uniformly scattered.
As observed in Fig. 7, the effect of the depth of the truncated semi-elliptic cavity on the scattering pattern is significant. In practice, the depth of the truncated cavity can also be considered an important parameter for controlling and shaping scattering patterns. Moreover, the results obtained in this study can be employed to design appropriate electromagnetic structures for various applications.

IV. CONCLUSION

This study semi-analytically examined the scattering waves generated by a truncated semi-elliptic cavity in a PEC using decomposition and region-point matching techniques. The described method follows a simple procedure that can be applied to address problems pertaining to elliptic boundaries, for which the Mathieu function addition theorem, which is characterized by difficult formulations, has been conventionally used. The results obtained on implementing the proposed process were compared to those obtained for the MoM used in the FEKO software. The comparison revealed that the proposed procedure is efficient and in good agreement with the results of the time-consuming and entirely numerical MoM. This study also examined the effects of cavity depth and incident angle on scattering patterns by calculating numerical results for a few typical cases.

Fig. 1
Geometry of a truncated semi-elliptic cavity embedded in a PEC.
jees-2024-1-r-203f1.jpg
Fig. 2
Distribution of electromagnetic fields as a function of x position at y = 0 for the cavity in Fig. 1, a/λ = 1.5, b/λ = 1, d/λ = 0.75 when ϕi = 90° (a) Amplitude of the z-component of the electric field |Ez|, (b) amplitude of the x-component of the magnetic field |Hx|, (c) amplitude of the x-component of the magnetic field |Hy|, and (d) normalized error versus the number of M samples.
jees-2024-1-r-203f2.jpg
Fig. 3
Distribution of electromagnetic fields as a function of x position at y = 0 for the cavity in Fig. 1, a/λ = 1.5, b/λ = 1, d/λ = 0.75 when ϕi = 60°: (a) Amplitude of the z-component of the electric field |Ez|, (b) amplitude of wthiteh x-component of the magnetic field |Hx|, (c) amplitude of the x-component of the magnetic field |Hy|.
jees-2024-1-r-203f3.jpg
Fig. 4
Echowidths σTM2D versus the incident angle ϕi for two different cavity sizes pertaining to Fig. 1, a/λ = 1.5, b/λ = 1, and d/λ = 0.75; (b) a/λ = 0.5, b/λ = 0.25, and d/λ = 0.2.
jees-2024-1-r-203f4.jpg
Fig. 5
Bistatic 2-D RCS ( σTM2D) versus the observation angle ϕi for a truncated sem-elliptic cavity with a/λ = 0.5, b/λ = 0.25, and d/λ = 0.125 at two different incident angle (ϕi = 60° and 90°). The results obtained using the proposed method are compared to those obtained using the MoM.
jees-2024-1-r-203f5.jpg
Fig. 6
Normal incidence echowidth σTM2D for a truncated semie-elliptic cavity as a function of depth when a/λ = 2, b/λ = 1.5.
jees-2024-1-r-203f6.jpg
Fig. 7
Bistatic scattering pattern of a truncated semi-elliptic cavity ( σTM2D) in the polar coordinate system as a function of the observation angle ϕd, with aλ=2 and b/λ 1.5 at ϕi = 90°, for different values of d :(a) dλ=0.15,0.35 and 0.5, (b) dλ=0.55,0.75 and 1, and (c) dλ=1.05,1.35 and 1.5.
jees-2024-1-r-203f7.jpg

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Appendices

Appendix
The coefficients Znm, Lnm, Znm and Inm can be determined as follows:
(A1)
Znm=0πcen(η,q)sem(η,q)dη,
(A2)
Ynm=0π+η0sen(η,q)sem(η,q)dη,
(A3)
Knm=ππ+η0cen(η,q)sem(η,q)dη,
(A4)
Lnm=-η02πsen(η,q)sem(η,q)dη,
(A5)
Inm=-η02πcen(η,q)sem(η,q)dη.
The Mathieu functions in the above expressions can be computed using Fourier and Bessel expansion methods [19, 23, 24]. Furthermore, some recurrence relations were obtained for the Fourier expansion coefficients to compute them rapidly. To avoid unnecessary repetition of the same calculations and to decrease computing time, a data bank containing the expansion coefficients and Bessel functions was created. Once the Fourier coefficients of the angular Mathieu functions were known, integrals (A1)–(A5) could be calculated analytically. For example, integral (A3) can be expressed as follows:
(A6)
Knm=ππ+η0cen(η,q)sem(η,q)dη=Σk=0Σk=0A2p2n(q)B2p+22m+2(q)ππ+η0cos (2pη)sin ((2k+2)η)dη
where A2p2n(q) and B2p+22m+2 are the Fourier coefficients of the angular Mathieu functions when n and m are even. A similar expression can be obtained for the odd values of n and m. It is worth mentioning that the integral in (A6) has an analytic solution [20].

Biography

jees-2024-1-r-203i1.jpg
Mehdi Bozorgi, https://orcid.org/0000-0003-0694-5390 was born in Isfahan, Iran, on September 4, 1977. In 2018, he joined Arak University, Arak, Iran, where he is currently an assistant professor in the Department of Electrical Engineering. His current research interests include the scattering of electromagnetic waves, electromagnetic nondestructive testing, and optics.
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