# Analysis of the elliptic-profile cylindrical reflector with a non-uniform resistivity using the complex source and dual-series approach: H-polarization case

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## Abstract

An elliptic-profile reflector with varying resistivity is analyzed under the illumination by an H-polarized beam generated by a complex-source-point (CSP) feed. The emphasis is done on the focusing ability that is potentially important in the applications in the optical range related to the partially transparent mirrors. We formulate the corresponding electromagnetic boundary-value problem and derive a singular integral equation from the resistive-surface boundary conditions. This equation is treated with the aid of the regularization technique called Riemann Hilbert Problem approach, which inverts the stronger singular part analytically, and converted to an infinite-matrix equation of the Fredholm 2nd kind. The resulting numerical algorithm has guaranteed convergence. This type of solution provides more accurate and faster results compared to the known method of moments. In the computations, a CSP feed is placed into a more distant geometrical focus of the elliptic reflector, and the near-field values at the closer focus are plotted and discussed. Various far-field radiation patterns including those for the non-uniform resistive variation on the reflector are also presented.

### Keywords

Electromagnetic scattering Method of analytical regularization Optical devices## 1 Introduction

The scattering of electromagnetic waves by the partially transparent thin curved scatterers, made up of lossless and lossy dielectric materials or imperfect metals, occupies a remarkable place in the electromagnetic wave theory. This can be understood because of, at least, two circumstances. Firstly it can be said that the well-known microwave reflector antenna systems are sometimes realized using thin resistive films, which cannot be simulated as perfectly electrically conducting (PEC) surfaces. This calls for a modified formulation of the corresponding wave-scattering problem. Secondly the same problem can be thought as generated by the micro-mirrors design in the optical systems because at optical wavelengths the PEC conditions are not valid at all.

Here, parabolic-profile thin screens and reflectors are most frequent because of the collimation and focusing properties. Still hyperbolic and elliptic sub-reflectors are used as the primary-feed illuminated parts of the large Cassegrain and Gregorian dual-reflector antennas with parabolic main dishes (Scott 1990). In the optical applications, sometimes it is necessary to provide highly directional radiation of laser that can be achieved using the micro-lenses and micro-mirrors, which can have elliptic. Besides, the optical or terahertz-wave beam can be guided along a finite chain of elliptic reflectors or micro-mirrors. Reliable and economic numerical simulation of these structures can be performed using the electromagnetic boundary-value problem formulations with modified boundary conditions.

In the full-wave modeling of the electromagnetic systems from the microwaves to optical range, finite-difference time-domain (FDTD) method is the one of the oldest techniques. It needs huge number of unknowns due to the discretization of the all physical domain and it also has a disadvantage in failing to satisfy the far-field radiation condition explicitly. In terms of the accuracy, FDTD may have some deficiency even in application to some simple circular-shell structures (Hower et al. 1993). Well-known alternative way is to obtain singular integral equation (SIE) from the corresponding boundary condition on the reflector. In this approach, the radiation condition is satisfied automatically and the numbers of unknowns are reduced drastically. However some problems still remain and they are associated with mathematical convergence and accuracy. For instance, a simple and well-known way is the method of moments (MoM) with local and global basis and testing functions (Jenn et al. 1995; Barclay and Rusch 1991; Heldring et al. 2004). In the application of the MoM to solve a SIE of Cauchy type or higher, the convergence of the solution is not guaranteed and strongly depends on the implementation. As a result, non realistic computation time may occur. Generally it is known that the MoM can be applied to small and medium size reflectors (up to 10 wavelengths); the achievable accuracy is frequently only a few digits for the surface current function.

The mathematically exact solution of certain reflector problems can be derived by using the modified Wiener–Hopf technique (Mittra and Lee 1971). For instance, a 2D curved strip scattering was studied in Idemen and Büyükaksoy (1984). This technique is important because it enables one to obtain the asymptotic high frequency expressions for the scattering characteristics. These alternatives are known as the high frequency techniques such as geometrical optics (GO), physical optics (PO) and physical theory of diffraction (PTD) especially for the larger PEC reflectors (Hasselmann and Felsen 1982; Suedan and Jull 1991; Martinez-Burdalo et al. 1993). Lately the PO has been combined with the Gaussian beam field decomposition aimed at the faster analysis of larger reflectors (Anastassiu and Pathak 1995; Chou et al. 2003; Rieckmann 2002), however this is still an approximation of the actual solution.

In the numerical modeling of the above mentioned geometries with the aid of SIEs a remarkable alternative is the method of analytical regularization (MAR) (Nosich 1999). In MAR, the kernel of the SIE is separated into two parts, the more singular part (usually static) and the remainder. In the H-polarization scattering by a PEC screen, the SIE involves the second derivative of the 2-D Green’s function and hence is a hyper-singular equation. Then, the choice of the global basis functions that are orthogonal eigenfunctions of the most singular part enables one to perform analytical inversion by using the Riemann Hilbert problem (RHP) technique. The remainder leads to the Fredholm second-kind matrix equation that provides a convergent numerical solution. This technique, combined with the dual-series equations, was presented in detail in Nosich (1993) for a PEC 2-D circular screen and then applied to the modeling of various 2-D circular and non-circular reflector antenna systems in free space and in more complicated near-field environments (Oğuzer et al. 1995; Yurchenko et al. 1999; Oğuzer 2001; Oğuzer and Altıntaş 2005, 2007; Boriskina et al. 2000; Oğuzer et al. 2001, 2004); here, the CSP field was adopted to simulate a feed. In either case, the Green’s functions of the host media have logarithmic singularities, and therefore the SIE-MAR techniques exploit the same ideas when performing the partial inversion.

As for the imperfect 2-D reflectors, one can firstly mention about the PO approximate treatment of the problem. In Senior (1978), the resistive half plane illuminated by a plane wave and then in Umul (2007), a beam excitation case of the same geometry has been studied by using the Modified-PO technique. Later a 2-D impedance parabolic reflector fed by a line source was studied with the same technique (Umul 2008). As mentioned, imperfect-reflector problems can be efficiently analyzed with the SIE-MAR technique. In Nosich et al. (1996), a circular 2-D reflector with a uniform resistivity surface was studied, illuminated by the plane waves. In Nosich et al. (1997), the same geometry but with non-uniform resistivity was modeled under the CSP-beam illumination. In that study, a non-uniform resistivity was realized as an almost-PEC reflector edge-loaded with a linearly increasing resistivity region. This edge-loaded case reduces the diffraction effects that are caused by the sharp edges of the PEC reflector, especially at the penumbra region and back side lobes of the radiation patterns. Lately in Oğuzer et al. (2009), we have studied the 2-D scattering problem assuming a noncircular contour with uniformly and non-uniformly resistive (i.e. edge-loaded) surface. Our aim was to simulate a parabolic reflector fed by a directive beam and therefore we used the CSP field as the incident wave. In the mentioned study, however, only the E-polarization case was treated with SIE-MAR that meant the inversion of the logarithmic singularity using the discrete Fourier inversion procedure.

In the present study, we consider the alternative case of H-polarization of the same problem as in Oğuzer et al. (2009). In this case, the SIE is a hyper-type one due to the derivatives of the Green’s function in the kernel however non-zero resistivity is a simple perturbation of the PEC case. Therefore the same RHP as in the PEC case technique is used to invert the most singular part and reduce the problem to a Fredholm second-kind matrix equation, whose numerical solution guarantees convergence. Our derivations and final algorithm are valid for arbitrary smooth contour of reflector with non-uniform resistivity; however unlike (Oğuzer et al. 2009) we concentrate our numerical study on the elliptic-profile reflectors. In the optical frequency range, this study has applications in the design of micro-size metallic mirrors used in the pump-radiation focusers for the semiconductor lasers. Finite chains of elliptic reflectors can be also used to build low-loss beam waveguides met in the heating of plasma with millimeter waves in controlled nuclear-fusion machines (Thumm and Kasparek 2002), in the tomography of the same plasma with sub-millimeter waves, and in the front-end circuits of millimeter and far-infrared wave receivers of radio astronomy antennas (Goldsmith 1998).

In Sect. 2, we formulate the boundary-value problem and derive hyper-singular IE. Section 3 is concerned with analytical regularization of SIE and its conversion to the matrix equation. In Sect. 4, we derive radiation characteristics. Section 5 contains the numerical results obtained for various reflector surfaces and resistivity cases. The conclusions are formulated in Sect. 6.

## 2 Formulation and sie

## 3 Regularization and discretization

Assume that the curve \(M\) can be characterized with the aid of the parametric equations \(x=x(\varphi )\) and \(y=y(\varphi )\), with \(0\le |\varphi |\le \theta \), in terms of the polar angle, \(\varphi \). Besides, define the differential lengths in the tangential direction at any point on \(M\) as \(\partial l=a\rho (\varphi )\partial \varphi \) and \(\partial l^{\prime }=a\rho (\varphi ^{\prime })\partial \varphi ^{\prime }\), respectively, where \(\rho (\varphi )=r(\varphi )/[a\cos \gamma (\varphi )], \xi (\varphi )\) is the angle between the normal on \(M\) and the \(x\)-direction, \(\gamma (\varphi )\) is the angle between the normal and the radial direction, and a is the radius of the circular arc S.

## 4 Radiation characteristics

Here, the current \(\tilde{J}_t ({\varphi }^{\prime })\) is expanded in terms of FS defined in Eq. (9) where the coefficients \(x_n\) are obtained by the solution of the matrix Eq. (16). Besides, \(x(\varphi ^{\prime })\) and \(y(\varphi ^{\prime })\) are the coordinates of the source point on the reflector surface and \(x\) and \(y\) are the coordinates of the observation point. In the presented below numerical results, we have computed the magnetic field in the second geometrical focus and normalized the total field magnitude value by the incident field value at the middle point of the CSP beam aperture.

## 5 Numerical results

The presented formulation has been verified using various characteristics of the elliptic reflector illuminated by a CSP source as shown in Fig. 1, i.e. with \(\varphi _0 =\pi \). For the sake of performance comparison, we have also applied the method of moments (MoM) to the same problem. Here we have used Galerkin approach choosing the triangular subdomain basis and testing functions. The main matrix in the MoM formulation contains the double integrals which have been computed using the optimized routines of Matlab 6.9. To generate all these numerical results we have used a desktop PC core i-5 computer with 4GB RAM and Windows 7 operating system.

In Fig. 2b, we see that making the reflector size larger leads to the necessity of working with larger matrices but do not slow the convergence rate. This is also an expected result because a larger aperture dimension requires more terms in (9) to approximate the surface current (the same takes place for a PEC reflector).

In the further figures, we assume that the CSP feed is located in the first focus of the elliptic reflector (further from the reflector) and observe the field amplitude at the second focus (nearer to the reflector). This will provide us an understanding of the performance of the elliptic-reflector focuser.

Figure 7 presents the variation of the field amplitude in the second focus with the increasing aperture dimension \(d\) of the reflector at the fixed \(f/d\) ratio. The plots are obtained for two different \(R/{Z_0 }\) values in (a) and (b). As one can see, the part (b) curves are lower than the corresponding part (a) ones. This is expected result because the higher-resistivity reflector in (b) is more transparent than the lower-resistivity one in (a). The increasing distance \(d\) also increases the focus distance f under the fixed \(f/d\) ratio. Then due to the varying distance, the interference produces the curves presented in Fig. 7a, b. In the same figure, two different edge illumination cases are presented. It is seen that the more directive feed illumination produces higher focusing effect. In this case the edges are illuminated weaker and the feed radiation bypassing the reflector is smaller, so more power is reflected from the reflector surface and gathers in the second focus point.

## 6 Conclusions

In this study, the H-polarized wave scattering from the arbitrary-profile 2-D reflector with varying resistivity has been analyzed using the SIE, dual series equations and the analytical regularization technique. Therefore the obtained results can be considered as accurate data for the validation of the other numerical approximations. As an example of implementation, we have studied the re-focusing of the complex-source-point beam from one focus of the elliptic reflector to the other. These results provide the information necessary for a better understating of various optical systems like elliptic focusers of laser pumping circuits. They demonstrate the importance of the proper choice of the edge illumination and show the effect of the partial transparency of reflector on the focusing ability.

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