Plane-wave expansion of elliptic cylindrical functions

Optics Communications 349 (2015) 185–192 Contents lists available at ScienceDirect Optics Communications journal homepage: www.elsevier.com/locate/optcom Plane-wave expansion of elliptic cylindrical functions Carlo Santini n, Fabrizio Frezza, Nicola Tedeschi Department of Information Engineering, Electronics and Telecommunications, “La Sapienza” University of Rome, Via Eudossiana 18, 00184 Rome, Italy art ic l e i nf o a b s t r a c t Article history: Received 14 November 2014 Received in revised form 20 March 2015 Accepted 24 March 2015 Available online 27 March 2015 Elliptic Cylindrical Waves (ECW), defined as the product of an angular Mathieu function by its corresponding radial Mathieu function, occur in the solution of scattering problems involving two-dimensional structures with elliptic cross sections. In this paper, we explicitly derive the expansion of ECW, along a plane surface, in terms of homogeneous and evanescent plane waves, showing the accuracy of the numerical implementation of the formulas and discussing possible applications of the result. & 2015 Elsevier B.V. All rights reserved. Keywords: Elliptic cylinder Plane-wave spectrum Mathieu functions Plane surface Fourier integral 1. Introduction The plane-wave representation of electromagnetic field, in its connection to Fourier analysis, is a fundamental tool in dealing with several aspects of the electromagnetic theory [1,2]. By expressing complex electromagnetic fields in terms of superpositions of very simple solutions of Maxwell's equations, it is capable of delivering a great simplification in the analytical treatment of several complex radiation, propagation and diffraction problems. In particular, the aforementioned technique may be used to express the electromagnetic field radiated by localized sources or scattered by localized obstacles with simple shapes, expressing the typical solution of Helmholtz equation in orthogonal curvilinear coordinates in terms of natural solutions of Maxwell's equations in Cartesian coordinates [2,3]. This approach has proved to be extremely fruitful in dealing with the reflection of complex electromagnetic fields by plane surfaces, e.g., when the fields are expanded in terms of cylindrical functions in circular coordinates [4– 8]. Since the reflection and transmission properties of surfaces are known, or at least easily expressible, just for incident plane waves, solutions of diffraction problems in the presence of a generally reflecting plane surface require an integral expansion of the diffracted field along the plane surface in terms of homogeneous and n Corresponding author. Fax: þ 39 06 44585918. E-mail addresses: [email protected], [email protected] (C. Santini). URL: http://151.100.120.244/personale/frezza/ (F. Frezza). http://dx.doi.org/10.1016/j.optcom.2015.03.057 0030-4018/& 2015 Elsevier B.V. All rights reserved. evanescent plane waves, for which the reflection behavior may be characterized by means of the Fresnel coefficients [9–12]. The solution of two-dimensional scattering problems in elliptic coordinates is pursued by expanding the diffracted field by means of Elliptic Cylindrical Waves (ECW), defined as the product of an angular Mathieu function by its corresponding radial Mathieu function. Integral plane-wave representations of ECW as a contour integral in the complex plane may be found in many fundamental works [2,13,14] but, to the best of our knowledge, none of the available forms is suitable for the straightforward application of the aforementioned analytical procedure. In this paper we show the explicit derivation of the plane-wave spectrum of ECW, whose final analytical form is directly applicable to the study of the reflection of ECW by a planar discontinuity between propagation media. Such result is significant in many fields of applied optics, since it constitutes a basilar step in the construction of full-wave solutions of scattering problems regarding cylindrical diffracting structures with elliptic cross sections. The paper is organized as follows: in Section 2 we resume some fundamental concepts about ECW, we define the notations used in this paper and we show explicit analytical derivation of the planewave expansion. In Section 3 we present numerical results, pointing out the accuracy and reliability of the proposed planewave spectrum representation. In Section 4 we discuss relevant applications of the proposed expansion. Finally, conclusions are given in Section 5, where further developments are outlined too. 186 C. Santini et al. / Optics Communications 349 (2015) 185–192 2. Plane wave expansion of ECW respectively, give rise to corresponding ECW representing outgoing (first kind, ECWp (1) ) and ingoing (second kind, ECWp (2) ) fields 2.1. Elliptic Cylindrical Waves when a time factor exp( − iωt) is assumed. In this paper, for relevant applications to diffraction theory, we will focus on ECWp (1) functions corresponding to Mathieu–Hankel functions of n the first kind: n With reference to the notation used in [3,15], we will denote with symbols Spn (v, q) the angular Mathieu functions (AMF), and with symbols Jp (u, q), Npn (u, q) the radial Mathieu functions n (RMF) of the first and second kind, respectively: (u, v) being the elliptic cylindrical coordinates, q being the elliptic parameter, the index p = {e, o} and index n ∈  denoting functions of even or odd ECWp (1) (u, v, q) = H p (1) (u, q)·Spn (v, q); n n n (3) for this reason, in the following sections, the superscript “(1)” will be dropped. type p and integer order n, respectively. With symbols H p (m) (u, q), n m = {1, 2} , we will denote radial Mathieu functions of the third kind, analogous to the Hankel functions in circular coordinates, defined as H p (1) (u, q) = Jp (u, q) + iNpn (u, q), (1) H p (2) (u, q) = Jp (u, q) − iNpn (u, q). (2) n n n n By means of such notation, the basic solutions of the Helmholtz equation in elliptic coordinates are of the form H p (m) (u, q)·Spn (v, q); n the elliptic parameter q is connected to the wavenumber k in the Helmholtz equation since q = k 2ρ2 /4 , k = 2π /λ , λ being the wavelength, ρ = d/2, d being the interfocal distance of the reference ellipses. For the sake of simplicity and readability in the rest of this paper, we will refer to these basic solutions as Elliptic Cylindrical Waves (ECW), denoted by symbols ECWp (m) (u, v, q), to focus on the n analogies with the results of plane-wave expansion of circular cylindrical waves in [4]. We point out that the two different forms of Mathieu functions of the third kind in (1) and (2), also called Mathieu–Hankel functions of the first and second kind, 2.2. Integral representation of ECW The geometric layout of the problem is shown in Fig. 1, where the axes and coordinates of the Cartesian reference frame are visualized together with the corresponding elliptic coordinates. We will refer to dimensionless Cartesian coordinates ξ = kx and η = ky . Our aim is to express the ECW field distribution (3), across a plane η = η0 > 0, as a superposition of plane waves, following an analytical approach similar to the one used in [4]. We start from the integral representation in [14], reported as “Integral Representation with Elementary Kernel” in [16,17] at Section 28.28.7 1 π ∫3 exp(2ihw)meν (t, h2) dt π = exp(iν )meν (α, h2) Mν(3) (z, h), 2 (4) where      variable w is defined as w = cosh z cos t cos α + sinh z sin t sin α , w, z, α, t represent complex variables, h2 = q , ν represents a complex index, the complex variable t must follow an integration path 3 Fig. 1. Geometry of the problem and reference frame; ξ and η are Cartesian dimensionless coordinates, defined as ξ = kx and η = ky , where k = 2π /λ , λ being the wavelength; (u, v) are elliptic coordinates; F1 and F2 are the foci of the elliptic reference frame, ρ = F1F2 /2. 187 C. Santini et al. / Optics Communications 349 (2015) 185–192   belonging to the strip in the complex plane delimited by ( − r1 + i∞) and (r2 − i∞), (r1, r2 ∈ ) , meν (in plain roman font) and Mν(3) denote a Mathieu Function and a Modified Mathieu Function of the third kind, respectively, the integral representation, as stated in [14], is convergent if −r1 < arg{h [cosh(z + iα)] ± 1} < π − r1, − r2 < arg{h [cosh(z − iα)] ± 1} < π − r2. (5) To express equations in the desired notation, we write (4) in the case ν ∈ , separate the two cases ν = ± n (n ∈ ) and apply the following relations, available in [18] Mn(3) (z, h) = 2 He (z, q), π n n M−(3) n (z, h) = ( − 1) men (α, h2) = n ≥ 1, (6) n ≥ 0, 2π So (α, q), Nno n me−n (α, h2) = − i Nne 2 Ho (z, q), π n 2π Se (α, q), Nne n Fig. 2. Integration path 3 on the complex plane, 3 = *1 ∪ *2 ∪ *3. n ≥ 0, n ≥ 1, (7) Nno where and are normalization factors for AMF. With simple algebra and by referring only to real values of variables z and α, representing elliptic coordinates u, v, respectively, we get to the following form: ECWpn (u, v, q) = Hpn (u, q)·Spn (v, q) = ( − i)n 2π ∫3 exp(2ihw) Spn (t, q) dt, (8) where w = cosh u cos v cos t + sinh u sin v sin t . (9) In (9) the explicit dependence of the exponential factor on variables ξ, η0, t and parameters k, q may be shown considering that h = q = kρ /2 2ihw = ikρ cosh u cos v cos t + ikρ sinh u sin v sin t , (10) thus, by using relations x = ρ cosh u cos v and y = ρ sinh u sin v , between elliptic cylindrical (u, v) and Cartesian (x,y) coordinates 2ihw = ikx cos t + iky sin t . (11) By introducing Cartesian dimensionless coordinates, a point (x, y0 ) on the plane y = y0 is identified by (ξ, η0 ), with ξ = kx and η0 = ky0 , thus 2ihw = i (ξ cos t + η0 sin t). (12) ECWpn (u, v) = 1 2π +∞ ∫−∞ ⎛ ⎞ Fpn ⎜η0, β⎟ exp(iξβ) dβ, ⎝ ⎠ in which the elliptic coordinates are functions of the Cartesian dimensionless coordinates ξ = kx , η = ky , thus u = u (ξ, η0 ), v = v (ξ, η0 ), and the field produced by an ECW, evaluated across a plane η = η0 , is explicitly expressed as a superposition of plane waves. 2.3.1. Choice of the integration path As stated before, the integration path 3 of the complex variable t = (r + is) ∈  may be chosen arbitrarily in the strip of the complex plane delimited by (−r1 + i∞) and (r2 − i∞) , (r1, r2 ∈ ). By choosing values r1 = 0 and r2 = π relations (5) ensure convergence of the integral representation (13) for u ∈ [0, + ∞), v ∈ [0, π]. Under these assumptions, we choose the integration path as composed of three linear portions *1, *2, *3, shown in Fig. 2,  *1: (0 + i∞) → (0 + i0),  *2: (0 + i0) → (π + i0),  *3: (π + i0) → (π − i∞), each one giving rise to a separate contribution to the integral (13). Such choice allows us to derive a plane-wave integral representation of ECW explicitly taking into account, by means of path *2, all homogeneous plane waves propagating along directions v ∈ [0, π] and by means of path *1 and *3, all evanescent plane waves propagating on the surface along directions v = 0 and v = π . ECWpn (ξ , η0, q) = ( − i)n 2π ∫3= * ∪ * ∪ * 1 2 3 exp[ i (ξ cos t + η0 sin t)] Spn (t , q) dt , ECWpn (ξ , η0, q) = 01 + 02 + 03, ECWpn (ξ , η0, q) where ( − i)n 2π ∫3 exp[i (ξ cos t + η0 sin t)] Spn (t, q) dt. (13) (15) thus, Finally, we obtain the desired form of integral representation of the ECW: = (14) 0m = ( − i)n 2π ∫* m = 1, 2, 3. m (16) exp[ i (ξ cos t + η0 sin t)] Spn (t , q) dt , (17) 2.3. Fourier integral representation Each contribution (17) will be treated individually to show the analytical derivation. Our intention is to express the integral (15) in the form of a spatial Fourier integral: 2.3.2. First contribution: portion *1 From the contribution related to portion *1 of the integration 188 C. Santini et al. / Optics Communications 349 (2015) 185–192 path it is possible to derive the explicit expression of the right part of the plane-wave spectrum of ECW in (14), corresponding to evanescent plane waves with β > 1: ( − i)n 01 = 2π ∫* 1 exp[i (ξ cos t + η0 sin t)] Spn (t , q) dt , (18) cos r = β, β ∈ [ − 1 ;+ 1], sin r = 1 − β2 , r = arccosβ, dr = − dβ/ 1 − β 2 , by posing t = 0 + is with s ∈ (+∞ ; 0] and dt ¼i ds on path *1, we obtain ( − i)n 01 = 2π 02 = 0 ∫+∞ exp[i (ξ cos is + η0 sin is)] × Spn (is , q) i ds ( − i)n + 1 = 2π ∫0 +∞ (19) ( − i)n 2π (26) 1 ∫−1 exp(iξβ)exp(iη0 × Spn (arccosβ, q) dβ 1 − β2 1 − β2 ) . (27) exp(iξ cosh s)exp( − η0 sinh s) By comparing (27) with (14), we get to the expression of the × Spn (is , q) ds . (20) In the last passage thus sin is = i sinh s , exp(iη0 sin is) = exp( − η0 sinh s), so in (20) as s → + ∞, exp( − η0 sinh s) → 0, providing convergence. By posing ( ) homogeneous plane-wave spectrum of ECW, Fpn η0 , β , for |β| < 1, ( 2π ( − i)n exp iη0 1 − β 2 Fpn (η0, β) = 1− × Spn (arccosβ, q), cosh s = β, ) β2 |β| < 1. (28) β ∈ [1 ;+∞], β2 − 1 , sinh s = s = ln ( ) β2 − 1 + β , ds = dβ/ β 2 − 1 , (21) 2.3.4. Third contribution: portion *3 From the contribution relevant to the portion *3 of the integration path it is possible to derive the explicit expression of the left part of the plane-wave spectrum of ECW in (14), corresponding to evanescent plane waves with β < − 1: we obtain 01 = (− i)n + 1 ∫1 +∞ 03 = exp(iξβ)exp(−η0 β 2 − 1 ) 2π ⎡ × Spn ⎢i ln( β 2 − 1 + β), ⎢⎣ (22) ( ) portion of the plane-wave spectrum of ECW, Fpn η0 , β , for β > 1, 2π ( − i)n + 1 exp(−η0 β 2 − 1 ) ⎤ q⎥, ⎥ ⎦ 3 exp[i (ξ cos t + η0 sin t)] Spn (t , q) dt , 03 = ( − i)n 2π ∫0 −∞ (29) exp[iξ cos(π + is)]exp[iη0 sin(π + is)] × Spn (π + is , q) i ds = ( − i)n + 1 2π (30) 0 ∫−∞ exp( − iξ cosh s)exp( + η0 sinh s) × Spn (π + is , q) ds . β2 − 1 ⎡ × Spn ⎢i ln( β 2 − 1 + β), ⎢ ⎣ ∫* by posing t = π + is with s ∈ (−∞ ; 0] and dt = i ds on path *3, ⎤ dβ q⎥ . ⎥⎦ β 2 − 1 By comparing (22) with (14), we get to the expression of the Fpn (η0, β) = ( − i)n 2π β > 1. (23) (31) In the last passage thus sin is = i sinh s , exp (−iη0 sin is) = exp (+η0 sinh s), so in (31) as s → − ∞, exp( + η0 sinh s) → 0, providing convergence. By posing cosh s = − β, 2.3.3. Second contribution: portion *2 From the contribution relevant to portion *2 of the integration path it is possible to derive the explicit expression of the part of the plane-wave spectrum of ECW in (14) corresponding to homogeneous plane waves with |β| < 1: 02 = ( − i)n 2π ( − i)n 2π By posing sinh s = − β2 − 1 , ( ) s = ln − β 2 − 1 − β , ds = dβ/ β2 − 1, (32) we obtain ∫* 2 exp[i (ξ cos t + η0 sin t)] Spn (t , q) dt , (24) by posing t = r + i0 with r ∈ [0; π] and dt ¼dr, on path *2, 02 = β ∈ [ − ∞; − 1], ∫0 π exp(iξ cos r)exp(iη0 sin r) Spn (r , q) dr . 03 = ( − i)n + 1 − 1 exp(iξβ)exp(−η0 β 2 − 1 ) −∞ 2π ⎡ ⎤ dβ . × Spn ⎢π + i ln(− β 2 − 1 − β), q⎥ ⎢⎣ ⎥⎦ β 2 − 1 ∫ (33) (25) By comparing (33) with (14) we get to the expression of the left ( ) part of the plane-wave spectrum of ECW, Fpn η0 , β , for β < − 1, 189 C. Santini et al. / Optics Communications 349 (2015) 185–192 Fpn (η0, β) = 2π ( − i)n + 1 exp(−η0 β 2 − 1 ) β2 − 1 ⎡ × Spn ⎢π + i ln(− β 2 − 1 − β), ⎢ ⎣ ⎤ q⎥, ⎥ ⎦ β < − 1. (34) We may summarize results (23), (28) and (34) in the following expression: n Fpn (η0, β) ⎧ 2π ( − i)n + 1 exp(−η γ) 0 ⎪ Spn [i ln(γ + β), q], ⎪ γ =⎨ ⎪ 2π ( − i)n exp(iη0 γ) Spn (arccosβ, q), ⎪ γ ⎩ values computed by applying numerical quadrature formulas to the plane-wave expansion (14) (named ECWp int ). To avoid nun |β| > 1, |β| < 1, (35) 2 where γ = |1−β | . Finally, by means of the definition of the arccos function in the complex domain, as described in [9], we can express the plane wave spectrum function Fpn in a compact form Fpn (η0, β) = plotted versus β for values of η0 = 1, 5, 10, 20. The evanescent plane-wave portion of the ECW spectrum is strongly affected by the variation of the η0 parameter, compared to the homogeneous plane-wave portion. Numerical tests involving expression (35) have been carried out to estimate its suitability for computational purposes. To perform convergence and accuracy validation, values of ECW computed by means of reliable numerical methods available in the literature [15,19] (named ECWp lib ) have been compared to ECW function ( 2π ( − i)n exp iη0 1 − β 2 1− β2 ∀ β ∈ \{ ± 1}. )S pn (arccosβ, q), (36) 3. Numerical results In Fig. 3, curves of the modulus of Fe n (η0 , β) for η0 = 20 , q¼ 4 are plotted versus β for values n = 4, 6, 8, 10 (Fig. 3a) and n = 5, 7, 9, 11 (Fig. 3b). The curves, showing a singularity for |β| = 1, may be compared to the plane-wave spectrum of cylindrical functions, related to a circular cylinder, presented in [4]. These singularities disappear when the constant k is complex, i.e., when the losses are considered in the physical problem, similar to the case of circular cylindrical waves [8]. In Fig. 4a, curves of the modulus of Fe 4 (η0 , β) for η0 = 20 are plotted versus β for values of q = 1, 2, 5, 10. The homogeneous plane-wave portion of the ECW spectrum is strongly affected by the variation of the q parameter, compared to the evanescent plane-wave portion. In Fig. 4b, curves of the modulus of Fe 4 (η0 , β) for q ¼10 are merical integration issues due to the presence of singularities in the plane-wave spectrum, it is possible to resort to a suitable change of variable, as stated in [9]. By bringing integrals (20), (27) and (31) back to their previous form (19), (25) and (30), respectively, it is possible to remove the presence of the singularity at the cost of keeping an oscillating term in the integrand. We point out that the oscillating factor in the form exp(iξ cosh s) in (30) and (31) or exp(iξ cos r) in (25) may behave as a highly oscillating kernel for large values of |ξ|: in this case, to avoid loss of accuracy due to numerical cancellation, the development of specific numericalquadrature strategies and algorithms [23,24] might turn out to be necessary. For our validation purposes, the adaptive recursive algorithm based on Gauss–Kronrod quadrature formulas provided by MATLABs libraries has proved to rely on sufficient precision. In Figs. 5–8 the real and imaginary part of ECWpn , respectively, is plotted versus ξ variable, for values q ¼10, η0 = 20, p = {e, o} , for different even or odd values of n. To estimate numerical reliability of the integral representation, we define a computational relative error function Erelpn (ξ, η0 , q) as Erelpn (ξ , η0, q) = ∣ECWp lib (ξ , η0, q) − ECWp int (ξ , η0, q) ∣ n n ∣ECWp lib (ξ , η0, q) ∣ . n (37) In Fig. 9 computational relative error function Erele 4 , for values q¼ 10, η0 = 20, is plotted versus ξ variable in two different intervals ξ ∈ [−10 ;+ 10] and ξ ∈ [+10 ;+ 1000]. Typical relative error values for plots in Figs. 5–8 have proved to be smaller than 10−7. 4. Applications of the proposed plane-wave expansion Relevant applications are targeted to the extension of the Fig. 3. Curves of the modulus of Fen (η0 , β , q) for η0 = 20 , q ¼10 are plotted versus β for values of n = 4, 6, 8, 10 (a) and n = 5, 7, 9, 11 (b). 190 C. Santini et al. / Optics Communications 349 (2015) 185–192 Fig. 4. (a) Curves of the modulus of Fe 4 (η0 , β , q) for η0 = 20 are plotted versus β for values of q = 1, 2, 5, 10 . (b) Curves of the modulus of Fe 4 (η0 , β , q) for q¼ 10 are plotted versus β for values of η0 = 1, 5, 10, 20 . Fig. 5. Curves of real part (a) and imaginary part (b) of ECWen are plotted versus ξ for n = 4, 6, 8, 10 , η0 = 20 , q ¼10. Lines refer to ECW values computed by means of integral representation. Markers refer to ECW values computed by means of library numerical routines. Fig. 6. Curves of real part (a) and imaginary part (b) of ECWen are plotted versus ξ for n = 5, 7, 9, 11, η0 = 20 , q ¼10. Lines refer to ECW values computed by means of integral representation. Markers refer to ECW values computed by means of library numerical routines. analytical and numerical methods shown in [9,10] to the scattering of electromagnetic waves by a perfectly conducting or dielectric elliptic cylinder near a plane discontinuity between different propagation media. A solution for such problems is available in the special cases of a half-cylinder in the presence of a ground plane [20] or a single cylinder placed near a perfectly reflecting mirror [21]: these results have been obtained by applying the method of images and the addition formulas for ECW. By means of the proposed expansion, we could provide a full-wave solution to the problem of plane-wave scattering by a perfectly conducting elliptic C. Santini et al. / Optics Communications 349 (2015) 185–192 191 Fig. 7. Curves of real part (a) and imaginary part (b) of ECWon are plotted versus ξ for n = 6, 8, 10, 12, η0 = 20 , q ¼10. Lines refer to ECW values computed by means of integral representation. Markers refer to ECW values computed by means of library numerical routines. Fig. 8. Curves of real part (a) and imaginary part (b) of ECWon are plotted versus ξ for n = 5, 7, 9, 11, η0 = 20 , q ¼10. Lines refer to ECW values computed by means of integral representation. Markers refer to ECW values computed by means of library numerical routines. Fig. 9. Semilogarithmic plots of Erele 4 (ξ, η0 , q) , for η0 = 20 , q¼ 10, are drawn versus ξ in two different intervals: (a) ξ ∈ [−10 ;+ 10] and (b) ξ ∈ [+10 ;+ 1000]. cylinder in the presence of a generally reflecting plane surface [22]. The quest for the solution is pursued by separating the total electromagnetic field in fourcontributions: incident field Vi, reflected field Vr, diffracted field Vd, diffracted–reflected field Vdr. As shown in [22], while the incident plane wave Vi , reflected plane wave Vr, and the diffracted field Vd may be directly expanded in terms of AMF and RMF, the evaluation of the diffracted–reflected component Vdr may be expanded in terms of Reflected Elliptic Cylindrical Waves, obtained by a Fourier integral involving the proposed plane-wave spectrum of ECW functions and the reflection coefficient of the plane discontinuity. The resulting expression for Vdr may be given in a simple form, representing the 192 C. Santini et al. / Optics Communications 349 (2015) 185–192 generalization to the elliptic case of results available in [9]: such form proves to be suitable for the determination of the total electromagnetic field by imposing boundary conditions on the cylinder surface. 5. 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