Near Field Focusing and Radar Cross Section for a Finite Paraboloidal Screen

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1 EuCAP - Convened Papers Near Field Focusing and Radar Cross Section or a Finite Paraboloidal Screen Vitaliy S. ulygin, Yuriy V. Gandel, Alexander I. Nosich * A.Y. Usikov Institute o Radio-Physics and Electronics NASU, Kharkov, Ukraine. Vitaliy_ulygin@ieee.org Abstract A problem o monochromatic electromagnetic wave diraction by a perectly electrically conducting (PEC surace o rotation located in a ree space is investigated. The problem is reduced to sets o hypersingular and singular integral equations, which are solved by the method o discrete singularities, using interpolation type quadrature ormulas. From the solutions o these sets the electric ield expression and the ar zone patterns are obtained. The presented method has a guaranteed convergence or none axially symmetric primary ield. I. INTRODUCTION ased on solving o integral equations methods are used in numerical modeling o three-dimensional arbitrary electromagnetic wave diraction by an arbitrary PEC surace or a long time [,]. However, such methods are in need with considerable computer resources, because deals with big-size matrices, and still the question o their convergence is open. The Nystrom method means that unknown smooth unctions approximate by polynomials. ecause o this, the numerical algorithms, using this method, have high-order convergence. However, they are used only in two-dimensional diraction problems [3-]. Using the Nystrom method, we developed a ast numerical algorithm with guaranteed convergence or three-dimensional diraction by the rotation surace, deals with the small-size matrices. Among the works on the o electromagnetic waves diraction by perectly conducting rotation suraces should mark the papers [,7,]. ecause o analytical taking into account o axial symmetry o perectly conducting rotation surace, three-dimensional diraction problem is reduced to the one-dimensional integro-dierential equations systems solving (although or three-dimensional diraction problems is typical the two-dimensional ones, that signiicantly decreases the order o discrete model matrix. The methods developed only or some speciic threedimensional problems with relectors o simple shape (disc, sphere, cone, circular cylinder and only in the case o primary ields small set is rigorously justiied and have guaranteed convergence. Let list them below. The electromagnetic wave diraction by PEC circular disk is considered or example in [9,], by PEC circular cylinder is consider in []. In [] the axial-symmetric electromagnetic wave diraction by PEC inite cone is solved, using the method o analytical regularization. The plane electromagnetic wave diraction by PEC sphere with a hole is solved in [3] in the case o plane wave propagate parallel to its rotation axis. In the paper the arbitrary electromagnetic wave diraction by arbitrary PEC rotation surace reduce to systems o onedimensional hypersingular and singular integral equations with variable coeicients or the irst time. Numerically these equations are solved, using interpolation-type quadrature ormulas that provide a high convergence order. II. DIFFRACTION Y ROTATION SURFACE A. Problem ormulation A problem o arbitrary monochromatic electromagnetic wave ( E, H diraction by a PEC surace o rotation S located in a ree space is investigated. The total tot tot electromagnetic ield ( E, H is presented in the orm o primary ( E, H and scattering ield ( EH, sum. The scattering electrical and magnetic ield satisy axwell equations, Sommereld radiation condition, eixner edge condition and PEC boundary condition on the surace o rotation. The dependence o the ields on time is e iϖ t. Choose cylindrical coordinates ρ,,z. The surace S is created by rotation some contour C around the axis z. It is convenient to choose curvilinear orthogonal coordinates q,, in which the surace S has the parameterization S : q = q, [,], [, π ]. ( Dependence o cylindrical coordinates rom the introduced curvilinear coordinates is expressed, using the ormula ρ = ρ( q,, z = z( q, ( The Lame coeicients o the coordinate system q,, is l = ( ρ + ( z, l q q q = ( ρ + ( z, l φ = ρ. The curvilinear coordinate orts is ( q,, : = ( x ρ cos+ y ρ sin + z z / l, = q, (3 = x sin+ y cos ( 33

2 The points on the surace S have such cylindrical coordinates: ρ = ρ( t = ρ( q,, t z = z( t = z( q, t. I t is the integration variable, we will use notations like ρ = ρ(, t z = z( t, h = l( q, t, t [,], while or the observation point we will use notations as ρ: = ρ(, z: = z(, [,]. The rotation surace, curvilinear orts, an integration point X, an observation point Y and the primary ield ( E, H are demonstrated in Fig.: Introduce unknown smooth unctions ( conormity with the edge condition: ρ ( t j ( t = u ( t t, u t and w ( t in j ( t h ( t = w (/ t t (3 For going to the limit in (7, ( we need to introduce the ollowing integral operators: hypersingular integral operator understood in the sense o Hadamard inite part ( ( ( ( Au = ( π u t t /( t dt the singular integral operators with the dierent weights: Fig. Problem ormulation In term o introduced notations, the PEC boundary condition is equal to: E + E = E + E = ( S S Our purpose is to ind the scattering ield ( EH,.. The systems o hypersingular and singular integral equations The electric ield vector E can be expressed in terms o the electromagnetic scalar and vector potentials as: E =gradφiϖ A ( Using expression or electrical ield throw scalar and vector potentials (, the PEC boundary condition ( are written as: lim[ iωa( X (/ l / ( X] E ( Y X Y + Ψ =, Y S, (7 lim[ i ϖ A ( X (/ / ( X ] E ( Y ρ X Y where A( Y + Ψ =, Y S, ( is the vector electromagnetic potential and Ψ is the scalar electromagnetic potential. The current density components j and j components and the primary ield E components on surace S are represented in terms o the Fourier series on the surace S as ollows: (, ψ ( = (, ψ i j t j t e ψ ( i ( = =, =, ( E t = E t e ψ, =, ( ( ( ( ( ( π ( /(( ( π ( /( Γ u = u t t t dt, ( Γ u = u t t t dt, ( integral operators with logarithm kernels: I ( ( π ( II ( ( π Lu = ( ln tu t t dt, (7 Lu = ( ln t ut ( / tdt, ( and integral operators with smooth kernels: I ( ( π ( ( II ( ( π ( ( K u = ( K, t u t t dt (9 K u = ( K, t u t / t dt. ( We will notate the integral operators with smooth kernels (9, and kernels o this operators, using the same symbol. I we go to the limit in (7-( then we obtain the set o onedimensional hypersingular and singular integral equations (HSIE and SIE systems with variable coeicients: a A+ b Γ + c L + K b Γ+ c L + K I ( I II ( II I ( I II ( II b Γ + c L + K c L + K 3 ( w ( ikρ E ( h( / Z = 3 ( ( u ikρ E ( / Z i, ( where the variable coeicients are a ( ρ =, ( c ( = ( k ρ + ( ρ + z ( z + 3 ρ /, (33 ( b = iρ, ( c = iρρ, b ( = iρ, ( ( 3 c = iρ, c ( = ρ + k ρ ( i 337

3 the smooth kernels are K (, t = ρ [ S / t k ( ρ ρ S + z zs ] ( 3 + a( /( t b( /( t c ( ln t 3 K (, t = ρ [ k ρρ S i S / ] b ( /( t c ( ln t 3 K (, t = ρ ( i / ρ S / tk ρ S b ( /( t c ( ln t 3 + ρ ρ ρ (7 ( K (, t = ( / S k S c (ln t (9 ikl S q,, t = S = cos ψ e / Ldψ, (3 ( ( cos ( L = ρ + ρ ρρ ψ + z z, Z = µ ε. (3 Successively satisying the irst integral equation in the system ( at the zeros o the second kind Chebyshev polynomial { n } n {cos( / } n t j j= = j+ π n j= and the second integral equation in ( at the zeros o the irst kind Chebyshev polynomial { n } n {cos( /( } n tk k= = k + π n k=, using the interpolation type quadrature ormulas [] we obtain a set o linear algebraic equations with unknowns ( n n ( tl u and w ( n n ( tk. Convergence o approximate solutions to exact one i n is guaranteed according to []. C. Electric ield and ar-ield scattering patterns expressions Representing in terms o the Fourier series the scalar and vector potentials we obtain expressions or electrical ield using the solutions un ( t and w ( n t o the system (: ( = E ( ρ, z = ( iz / πk[ Q (,, ( ρ z t u t t dt + Q (, z, t w ( t/ t dt],,, z, ρ = ρ S + S Qρ = k ρ S, Qρ = ρk S i (,, i E ρ z = E ρ, z e, = ρ,,z (3 ρ t (33, (3 ρ i S + Q = k ρ S, Q = S k ρs, (3 ρ t ρ Qz = S / z t k z S, Q = z ( i S / z. (3 The ar-ield scattering pattern components are deined as ollows: ikr F (, = lim E ( R,, Re, η =, (37 η R η Using the expressions (3-(3 or electric ield, we obtain the ar-ield scattering pattern components expressions: + i (, ( = π (, ( / ], F = e F, =, (3 F ( = ( iz / k[ U (, t u ( t t dt + (39 + U t w t t dt U = ( k ρ / i(, ( = +, ( U = k [ z sin cos ( + /], ( =. (3 + + U ( k ρ /( + ρ + U (, t k ρ cos ( /i Denote the ar-ield scattering pattern: F(, = F (, + F (, ( III. PARAOLIC REFLECTOR ILLUINATED Y A PLANE ELECTROAGNETIC WAVE. Consider a paraboloidal relector with the ocal distance and the diameter D illuminated by a plane electromagnetic wave (Fig.. z k z y H O O x E y x D Fig. Parabolic relector illuminated by the electromagnetic plane wave The plane wave electric ield expression is ikr E z = Zme ( ρ,, (,, ( where R = ( ρcos, ρsin, z, k = k(sin,,cos, m = (cos,, sin.using (3, and ( we obtain E and E azimuth harmonics: ( ikz cos( + ρ ( ikz cos( E = e Zcos( (/ i E = e Z[(/ cos( [( i J + ( T +, ( + ( i J ( T] z sin( ( i J ( T]/ l [( i J ( T ( i J ( T] + +, T = kρsin( (7 In Fig.3 and Fig. we show the normalized backscattering cross section σ / π D, σ (, / = π F π + Z and total scattering cross section σ / π D, π π σ = F(, sin( dd/ Z (respectively as a unction o the normalized requency in the case o the orthogonal incidence ( =, / D =. and / D =.. 33

4 . D =. D = 3 Fig.3 Paraboloidal relector / D=. and / D =. normalized backscattering cross section in the case o the orthogonal incidence ( = D =. D = 7., 7.., Fig. Paraboloidal relector / D=. and / D =. normalized backscattering cross section as a unction o the angle In Fig. 7 we show total electric ield module in the nearzone o the paraboloidal relector / D=. illuminated by the orthogonal incident plane wave in = Fig. Paraboloidal relector / D=. and / D =. normalized total scattering cross section in the case o the orthogonal incidence ( =. In Fig. 3 we mark backscattering cross section local minimum and maximum. In the case o / D =. σ / π D =.7 in ka =.9 and σ / π D =. in ka = 7.. In the case o / D =. / D. σ π = in = 9. and σ / π D =.73 in ka =. Let consider the normalized backscattering cross section as a unction o the plane wave incident angle in the marked in Fig. 3 the local σ / π D maximums =.9, / D =. and = 9., / D =. (Fig. and the local / D σ π minimums 7. / D =. ( Fig.,. =, / D =. and =., Fig. 7. The near ields o the paraboloidal relector / D=. illuminated by the plane wave in = 9..in the E-plane (let and the H-plane (right In Fig. we show total electric ield in the near-zone o the paraboloidal relector / D=. illuminated by the orthogonal incident plane wave in = ,.9., Fig.. Paraboloidal relector / D=. and / D =. normalized backscattering cross section as a unction o the angle. Fig.. The near ields o the paraboloidal relector / D=. illuminated by the plane wave in =.9.in the E-plane (let and the H- plane (right Let consider the ield behavior in the paraboloid geometric ocus. The electric ield module in the ocus is oc E = E (,,. The oc E / E as a unction o / shown in Fig. 9 in the case o D = λ, λ, λ, = D is 339

5 D = λ D = λ D = λ D / Fig. 9. The electric ield module in the paraboloid geometric ocus as a unction o / D in the case o D = λ, λ, λ, = One can see that the biggest electric ield module in the ocus we obtain in the case o / D.. oc The E / E as a unction o is shown in Fig.. in the case o / D =.,., = / D=. / D=. 3 Fig.. The electric ield module in the paraboloid geometric ocus as a unction o normalized requency in the case o / D =.,., =. oc The E / E as a unction o the angle is shown in Fig. in the case o D = λ, / D =.,. in logarithm scale. / D=. / D= Fig.. The electric ield module in the paraboloid geometric ocus as a unction o the angle in the case o / D=.,., D = λ. most eicient in the case o Fourier series primary ield rapidly converges or consists o a inite number o harmonics. For example in the case o considered in the paper PEC surace illumination by plane electromagnetic wave propagating parallel to rotation axis problem, it is necessary to solve only one system o one-dimensional hypersingular and singular integral equations. ecause o current density singularity on the surace edge is taken in account analytically and unknown smooth unctions approximate by polynomials, the method have high-order convergence. REFERENCES [] S.. Rao, D. R. Wilton, A. W. Glisson, Electromagnetic Scattering by Suraces o Arbitrary Shape IEEE Trans. Antennas Propagat., 9, vol. 3, no 3, pp. 9-. [] A. G. Davydov, E. V. Zakharov, Y. V. Pimenov,. Numerical decision method o the electromagnetic wave diraction by unclosed suraces problem Dokl. Akad. Nauk SSSR, 9, 7, no., pp.9-. [3] A. A. Nosich and Y. V. Gandel, Numerical analysis o quasiopticalmultirelector antennas in -D with the method o discrete singularities:e-wave case, IEEE Trans. Antennas Propagat., vol., no., pp. 399-, Febr. 7. [] J. L. Tsalamengas, Exponentially converging Nystrom methods in scattering rom ininite curved smooth strips. Part : T-Case IEEE Trans. Antennas Propagat. vol., no, pp [] J. L. Tsalamengas, Exponentially converging Nystrom methods in scattering rom ininite curved smooth strips. Part : TH-Case IEEE Trans. Antennas Propagat., vol., no, pp 37-3 [] A. G. Davydov, E. V. Zakharov, Y.V. Pimenov, Numerical analysis o ields in the case o electromagnetic excitation o unclosed suraces. Journal o Communications Technology and Electronics, vol.., Suppl.,, pp.s7-s9. [7] E. N. Vasilyev, Vozbuzhdenie tel vrashcheniya(excitation o odies o Rotation, oscow:radio I Svyaz, 97. [] A. erthon, R. ills, Integral analysis o radiating structures o revolution IEEE Trans. Antennas Propagat. 99, vol. 37, no 3,. pp [9] H. Hoenl, A. W. aue, K. Westpal, The diraction theory, Springer Verlag, 9 [] A. V. Lugovoy, V. G. Sologub, Scattering o electromagnetic waves by a disk at the interace between two media. Soviet Physics Techn. Physics,, 3, 973, pp 7-9. [] L. A. Puzynin, V. G. Sologub, Diraction radiation o a point charge moving along axis o segment o a circular waveguide, Radiophisics Quantum Electronics, 7,, 9, pp 9-9. [] D.. Kuryliak, Z. T. Nazarchuk, Illumination o a inite cone by an axial-symmetric electromagnetic wave. Radio Phys. Radio Astron (in Russian,,, pp [3] S. S. Vinogradov, Relectivity o spherical shield, Radiophysics and Quantum Electronics, vol., no, pp 7, 93. [] Y. V. Gandel, Introduction to the ethods o Computation o Singular and Hypersingular Integrals, KhNU Press, (in Russian. [] Y. V. Gandel, S. V. Eremenko, T. S. Polyanskaya, athematical problems o the ethod o Discrete Currents, KhNU Press,99 (in Rissian. IV. CONCLUSIONS Summarizing, we have presented an eicient and convergent numerical method or the analysis o an arbitrary PEC rotationally symmetric screen illuminated by an arbitrary incident electromagnetic ield. The presented method is the 3