Mathieu functions approach to bidimensional scattering by dielectric elliptical cylinders
MMathieu functions approach to bidimensional scattering bydielectric elliptical cylinders
E. Cojocaru
Department of Theoretical Physics, IFIN HH,Bucharest-Magurele MG-6, Romania ∗ (Dated: today) Abstract
Two-dimensional scattering by homogeneous and layered dielectric elliptical cylinders is analyzedfollowing an analytical approach using Mathieu functions. Closed-form relations for the expansioncoefficients of the resulting electric field in the vicinity of the scatterer are provided. Numericalexamples show the focalizing effect of dielectric elliptical cylinders illuminated normally to the axis.The influence of the confocal dielectric cover on the resulting scattered field is envisaged.
PACS numbers: 41.20.Jb, 42.25.Fx, 42.25.Gy ∗ Electronic address: [email protected] a r X i v : . [ phy s i c s . c o m p - ph ] A ug . INTRODUCTION The problem of wave scattering by an elliptical cylinder has been known for some time[1, 2]. Exact solutions can be obtained by separating the wave equation in elliptic cylindricalcoordinates and constructing exact eigenfunction expansions. But these exact solutions havebeen of limited practical value due to problems associated with the computation of thecorresponding eigenfunctions, i.e. angular and radial Mathieu functions.Recently, a new efficient method for computing these functions was presented [3], themethod being implemented in matlab [4]. To extend the applicability of the new computa-tional method, we derive in this paper the eigenfunctions for two-dimensional (2D) scatteringby dielectric elliptical cylinders. Closed-form relations for the normally incident plane wavescattering by homogeneous and layered elliptical cylinders are provided.
II. WAVE EQUATION IN ELLIPTIC CYLINDRICAL COORDINATES
Firstly, the elliptic cylindrical coordinates ( u, v, z ), as shown in Fig. 1, are introduced interms of rectangular coordinates ( x, y, z ) x = f cosh u cos v, y = f sinh u sin v, z = z, (1)with 0 ≤ u < ∞ , 0 ≤ v ≤ π , and f the semifocal length of the ellipse. The contoursof constant u are confocal ellipses, and those of constant v are confocal hyperbolas. The z axis coincides with the cylinder axis. In the following we consider the transverse-electric(TE) polarized electromagnetic field (i.e., the electric field only exists in the z direction).By expressing the Laplacian in elliptic cylindrical coordinates [1], the scalar wave equationin a homogeneous dielectric medium of permittivity (cid:15) is given by2 µ f (cosh 2 u − cos 2 v ) (cid:16) ∂ E z ∂u + ∂ E z ∂v (cid:17) + (cid:15) (cid:16) (cid:15) µ ∂ E z ∂z − ∂ E z ∂t (cid:17) = 0 . (2)An exp ( − iωt ) time dependence is assumed, where ω is the circular frequency. Separation ofvariables implies that we assume a solution E z of the form E z = Z ( z ) S ( v ) R ( u ) (3)1n substituting Eq. (3) in Eq. (2) we obtain (cid:16) d d z + k z (cid:17) Z ( z ) = 0 , (4a) (cid:104) d d v + ( a − q cos 2 v ) (cid:105) S ( v ) = 0 , (4b) (cid:104) d d u − ( a − q cosh 2 u ) (cid:105) R ( u ) = 0 , (4c)where k z and a are separation constants, q = f k t /
4, with k t = k − k z , where k = k √ (cid:15) , k = 2 π/λ , and λ is the wavelength of the incident light in vacuum. By taking the advantageof the elliptical cylinder geometry, we focus our analysis on the 2D scattering of normallyincident plane waves ( k z = 0). Equation (4b) is known as the angular Mathieu equation.The solution is denoted by S p m ( q, v, n ) where p, m denote even ( e ) or odd ( o ), and n denotesthe order [3, 4]. Equation (4c) is known as the radial Mathieu equation. Similarly to thecircular cylindrical coordinates where the radial solution is expressed in terms of Besselfunctions J n ( · ), Y n ( · ), H (1) n ( · ), and H (2) n ( · ), the radial Mathieu equation has four kinds ofsolutions: J p m , Y p m , H p m , and H p m , where p, m = e, o [1, 3, 4].Let a scalar plane wave of unit amplitude be normally incident on the axis of the ellipticalcylinder in air, so that the propagation direction forms an angle φ with the x -axis in the x − y plane. Hence the electric field of the incident plane wave is given by E inz = e ik ( x cos φ + y sin φ ) = e ik f (cosh u cos v cos φ +sinh u sin v sin φ ) (5)To obtain exact solutions for the scattered wave, we expand the incident field in Eq. (5) interms of the eigenfunctions of the elliptical cylinder [1, 3] E inz = √ π (cid:88) n i n J p m ( q, u, n ) S p m ( q, v, n ) S p m ( q, φ, n ) /N p m ( q, n ) , (6)where N p m is normalization constant. III. SCATTERING BY HOMOGENEOUS ELLIPTICAL CYLINDERS
Consider a homogeneous dielectric elliptical cylinder of permittivity (cid:15) with the boundarylocated at u , in air. The electric field E z outside and inside the elliptical cylinder is given2y [2, 3]( u > u ) E z = √ π (cid:88) n i n J p m ( q , u, n ) S p m ( q , v, n ) S p m ( q , φ, n ) /N p m ( q , n )+ i n α ( sc ) p m ( n ) H p m ( q , u, n ) S p m ( q , v, n ) S p m ( q , φ, n ) , ( u < u ) E z = √ π (cid:88) n i n α (1) p m ( n ) J p m ( q , u, n ) S p m ( q , v, n ) S p m ( q , φ, n ) , (7)where α (1) p m and α ( sc ) p m are coefficients to be determined from the boundary conditions. Thetangential magnetic field H v in each region is obtained from E z [1, 2, 3],( u > u ) H v = 1 k p (cid:16) − iω(cid:15) ∂E z ∂u (cid:17) , ( u < u ) H v = 1 k (cid:15) p (cid:16) − iω(cid:15) ∂E z ∂u (cid:17) , (8)where p = f (sinh u + sin v ) / . The boundary conditions require the continuity of thetangential components of the electric and magnetic field at the boundary u = u . Oneobtains (cid:88) n i n [ J p m ( q , u , n ) /N p m ( q , n ) + α ( sc ) p m ( n ) H p m ( q , u , n )] S p m ( q , v, n ) S p m ( q , φ, n )= (cid:88) n i n α (1) p m ( n ) J p m ( q , u , n ) S p m ( q , v, n ) S p m ( q , φ, n ) , (cid:88) n i n [ J (cid:48) p m ( q , u , n ) /N p m ( q , n ) + α ( sc ) p m ( n ) H (cid:48) p m ( q , u , n )] S p m ( q , v, n ) S p m ( q , φ, n )= (cid:88) n i n α (1) p m ( n ) J (cid:48) p m ( q , u , n ) S p m ( q , v, n ) S p m ( q , φ, n ) , (9)where the prime denotes differentiation with respect to u . To determine the coefficients α (1) p m ( n ) and α ( sc ) p m ( n ), p, m = e, o , we multiply Eq. (9) by S p m (cid:48) ( q , v, n ), integrate the resultingequation over v from 0 to 2 π , and use the orthogonality relations of the angular Mathieufunctions [1, 3, 4]. Applying those orthogonality relations allows waves in each order n , andeach combination p, m = e, o to decouple. One obtains J p m ( q , u , n ) /N p m ( q , n ) + α ( sc ) p m ( n ) H p m ( q , u , n )= α (1) p m ( n ) γ p m ( n ) J p m ( q , u , n ) /N p m ( q , n ) ,J (cid:48) p m ( q , u , n ) /N p m ( q , n ) + α ( sc ) p m ( n ) H (cid:48) p m ( q , u , n )= α (1) p m ( n ) γ p m ( n ) J (cid:48) p m ( q , u , n ) /N p m ( q , n ) , (10)3here correlation factors γ p m ( n ), p, m = e, o are defined by (cid:90) π S p m (cid:48) ( q , v, n ) S p m ( q , v, n ) d v = γ p m ( n ) δ m m (cid:48) . (11)Expressions for the correlation factors in terms of expansion coefficients of Mathieu functionsare given in [4]. From Eq. (10) we find α (1) p m ( n ) = J p m ( q , u , n ) γ p m ( n ) J p m ( q , u , n ) J p m ( q , u , n ) − H p m ( q , u , n ) J p m ( q , u , n ) − H p m ( q , u , n ) ,α ( sc ) p m ( n ) = J p m ( q , u , n ) N p m ( q , n ) H p m ( q , u , n ) J p m ( q , u , n ) − J p m ( q , u , n ) J p m ( q , u , n ) − H p m ( q , u , n ) , (12)where we introduced the log-derivative function F = F (cid:48) /F , F being J p m or H p m , theprime denoting differentiation with respect to u .As an example we consider an elliptical cylinder of permittivity (cid:15) = 2 [Figs. 2(a) and (b)]and (cid:15) = 3 [Figs. 2(c) and (d)]. The semifocal length is f = 0 . u = 1 . d x = 0 . d y = 0 . − ˆ x direction, perpendicularly to axis z .The incident light wavelength is λ = 0 . n = 0 ÷ E z )] distribution in the vicinity of the scatterer are shown in Figs. 2(a) and (c)for (cid:15) = 2 and 3, respectively, normalization being against the maximum absolute real value.Figures 2(b) and (d) show the distribution of | E z | , normalized to the maximum value, for (cid:15) = 2 and 3, respectively. One can see the focalizing effect of the perpendicularly illuminateddielectric elliptical cylinder, the greater the permittivity value, the higher the intensity onthe focalizing direction. IV. SCATTERING BY LAYERED ELLIPTICAL CYLINDERS
Consider a homogeneous dielectric elliptical cylinder of permittivity (cid:15) and boundary at u covered by a confocal dielectric elliptical layer of permittivity (cid:15) and outer boundaryat u , in air. Let q and q be the respective elliptical parameters, q = f k (cid:15) / = f k (cid:15) /
4. The electric field E z in each region is given by( u > u ) E z = √ π (cid:88) n i n [ J p m ( q , u, n ) /N p m ( q , n )+ α ( sc ) p m ( n ) H p m ( q , u, n )] S p m ( q , v, n ) S p m ( q , φ, n ) , ( u < u < u ) E z = √ π (cid:88) n i n [ α (2) p m ( n ) J p m ( q , u, n ) (13)+ α (3) p m ( n ) H p m ( q , u, n )] S p m ( q , v, n ) S p m ( q , φ, n ) , ( u < u ) E z = √ π (cid:88) n i n α (1) p m ( n ) J p m ( q , u, n ) S p m ( q , v, n ) S p m ( q , φ, n ) . The tangential magnetic field H v in each layer is H v = 1 k (cid:15) j p (cid:16) − iω(cid:15) j ∂E z ∂u (cid:17) , j = 1 , . (14)The boundary conditions require the continuity of the tangential components of the electricand magnetic field at the boundaries u = u j , j = 1 ,
2. Following the same procedure as in theprevious section we obtain four equations for each order n . (For simplicity, the dependenceon order n is skipped.) J p m ( q , u ) /N p m ( q ) + α ( sc ) p m H p m ( q , u )= γ (2) p m /N p m ( q )[ α (2) p m J p m ( q , u ) + α (3) p m H p m ( q , u )] ,J (cid:48) p m ( q , u ) /N p m ( q ) + α ( sc ) p m H (cid:48) p m ( q , u ) (15)= γ (2) p m /N p m ( q )[ α (2) p m J (cid:48) p m ( q , u ) + α (3) p m H (cid:48) p m ( q , u )] ,γ (2) p m [ α (2) p m J p m ( q , u ) + α (3) p m H p m ( q , u )] = α (1) p m γ (1) p m J p m ( q , u ) ,γ (2) p m [ α (2) p m J (cid:48) p m ( q , u ) + α (3) p m H (cid:48) p m ( q , u )] = α (1) p m γ (1) p m J (cid:48) p m ( q , u ) , where correlation factors γ ( j ) p m , p, m = e, o , j = 1 ,
2, are defined by (cid:90) π S p m (cid:48) ( q , v ) S p m ( q j , v ) d v = γ ( j ) p m δ m m (cid:48) , j = 1 , . (16)5elations (15) can be written in the form A · X = B , where A = γ (2) p m J p m ( q , u ) γ (2) p m H p m ( q , u ) − H p m ( q , u ) N p m ( q )0 γ (2) p m J (cid:48) p m ( q , u ) γ (2) p m H (cid:48) p m ( q , u ) − H (cid:48) p m ( q , u ) N p m ( q ) − γ (1) p m J p m ( q , u ) γ (2) p m J p m ( q , u ) γ (2) p m H p m ( q , u ) 0 − γ (1) p m J (cid:48) p m ( q , u ) γ (2) p m J (cid:48) p m ( q , u ) γ (2) p m H (cid:48) p m ( q , u ) 0 ,X = α (1) p m α (2) p m α (3) p m α ( sc ) p m , and B = J p m ( q , u ) J (cid:48) p m ( q , u )00 . (17)The unknown coefficients α ( j ) p m ( j = 1 , , , sc ) are determined at each order n by solving thematricial equation. In matlab , an equation of this form is solved simply with command X = A \ B .As an example, we show the distribution of | E z | normalized to the maximum value for alayered elliptical cylinder with permittivities (cid:15) = 2, (cid:15) = 3 in Fig. 3(a), and (cid:15) = 3, (cid:15) = 2in Fig. 3(b). The semifocal length is f = 0 . u = 0 . d x = 0 . d y = 0 . u = 1 . d x = 0 . d y = 0 . λ = 0 . − ˆ x direction, normally to the z axis. One can see that a confocal cover influenceson the scattered field distribution; a lower (higher) permittivity cover decreases (increases)the intensity of the scattered field. The focalizing effect of the perpendicularly illuminatedelliptical cylinder is increased by a lower permittivity confocal cover. V. SUMMARY
In this contribution we have studied the plane wave scattering by homogeneous andlayered elliptical dielectric cylinders following an analytical approach that is based on Math-ieu functions. By taking the advantage of the elliptical cylinder geometry, we focused ouranalysis on the 2D scattering of normally incident plane waves. We have provided closed-form relations for the expansion coefficients of the resulting electric field in the vicinity ofthe homogeneous and layered dielectric elliptical cylinders. Numerical examples have been6rovided showing the focalizing effect of a homogeneous elliptical cylinder illuminated per-pendicularly to the axis, and the influence of a confocal cover layer on the resulting electricfield distribution. The results presented could be a valuable contribution to the enlargementof the Mathieu functions applicability. [1] J. A. Stratton,
Electromagnetic Theory (Mc-Graw Hill, New York, 1941).[2] C. Yeh, J. Math. Phys. , 65 (1963).[3] J. J. Stamnes and B. Spjelkavik, Pure Appl. Opt. , 251 (1995); J. J. Stamnes ibid. , 841(1995).[4] E. Cojocaru, matlab free available computer code Mathieu Functions Toolbox v. 1.0. ↓ u=const ↓ v=const2f < >^ > xy F F FIG. 1: The elliptic cylindrical coordinates. F and F are the foci of the ellipse; f is the semifocallength. IG. 2: (a) Snapshot and (b) norm of the resulting electric field distribution in the vicinity of anelliptical cylinder with permittivity (cid:15) = 2; (c) and (d), the same with permittivity (cid:15) = 3. The lightis incident from the − ˆ x direction, in the x − y plane, normally to the z axis. IG. 3: (a) Norm of the resulting electric field distribution in the vicinity of a layered ellipticalcylinder with inner (outer) permittivity (cid:15) = 2( (cid:15) = 3) and (b) with inner (outer) permittivity (cid:15) = 3( (cid:15) = 2). The light is incident from the − ˆ x direction, in the x − y plane, normally to the z axis.axis.