High-order approximations to the Mie series for electromagnetic scattering in three dimensions

Transcription

1 Proceedings of the 9th WSEAS Internationa Conference on Appied Mathematics Istanbu Turkey May (pp ) High-order approximations to the Mie series for eectromagnetic scattering in three dimensions M. GANESH Dept. of Math. and Comp. Sciences Coorado Schoo of Mines Goden C USA mganesh S. C. HAWKINS Schoo of Mathematics University of New South Waes Sydney NSW 2052 Austraia stuart Abstract: Approximations to the Mie series are fundamenta in many appications. In particuar for vaidation of efficient eectromagnetic scattering agorithms (to compute the radar cross section of three dimensiona targets) or to understand the eectromagnetic scattering by rain drops it is important to evauate the Mie series with accuracy cose to machine precision. In this paper we propose a quadrature based agorithm to approximate the Mie series to very high order accuracy and demonstrate that the approach eads to better approximations than obtained using a industria standard Mie series scattering code. Key Words: Mie series eectromagnetics scattering radar cross section quadrature 1 Introduction Approximations to the Mie series pay an important and indirect roe in understanding and deveoping physica processes in many appications ranging for exampe from the radar cross section simuations in steath technoogy to the backscattering of sunight by coud water dropets. The concept of trying to understand ight scattering through approximations started a few centuries ago and anaytica (Mie series) soutions for scattering by spherica partices were deveoped about one century ago. Geometrica optics and computationa approximations to the Mie series have been deveoped over the ast haf a century and industria standard agorithms for efficient evauation of the Mie series were deveoped over the ast three decades. In this work we deveop a quadrature based approximation to the Mie series with amost machine precision accuracy. The geometrica optics or asymptotic expansions do not yied good accuracy when the ight waveength is simiar to or arger than the partice diameter as the ight interaction area is arger than the geometric cross section of the partice. Efficient computation of the partia sums of the Mie series is essentia in such cases [4]. Computationa approximations to the Mie series have been deveoped by many researchers and a ceebrated industria agorithm and code is due to Wiscombe and coaborators [5 6]. Incuded in [5] is a widey accepted formua for the number of terms that shoud be used in the partia sum. This paper is motivated by the vaidation of our spectray accurate eectromagnetic scattering agorithm (currenty at the fina stage of deveopment). This agorithm is an eectromagnetic counterpart of our acoustic scattering agorithm [2]. Spectra (and amost machine) accuracy for acoustic pane wave scattering by the one waveength sphere is demonstrated in [2 Page 231] and the accuracy is shown to increase with increasing number of unknowns for ow to medium frequency scattering by the sphere and other obstaces [2 Page ]. For the vaidation of our agorithm for eectromagnetic scattering by the sphere as it is standard in iterature we used the Wiscombe Mie series approximations (and code) [5] as a benchmark. Despite choosing the recommended number terms (and even more terms) in the partia sum in the Wiscombe code [5] we observed stagnated error of the order O(10 10 ) between our spectray accurate soution and the Wiscombe Mie series soution. This ed to us investigate the approximations to the Mie series considered in this paper. In genera the Mie-series computation with accuracy of the order O(10 10 ) is considered to be sufficient as most eectromagnetic scattering agorithms give ony a few decima paces of accu-

2 Proceedings of the 9th WSEAS Internationa Conference on Appied Mathematics Istanbu Turkey May (pp ) racy due to imitations of computer power. However as we moved from crude approximations to an incident pane wave E i H i with direction d 0 In case of eectromagnetic waves induced by Wiscombe code type accuracy over a haf century and poarization p 0 ago it is usefu to deveop the important Mieseries computation to amost machine accuracy E i (x) = i { } for vaidation of high order eectromagnetic scattering agorithms and in further understanding of = ik k cur cur p 0 e ikx bd 0 [( d 0 p 0 ) d ] 0 e ikx bd 0 (1) scattering by various targets and processes. } For vaidation of eectromagnetic scattering H i (x) = cur { p 0 e ikx bd 0 agorithms for benchmark radar targets that are ] not spherica [7] a standard approach is to pace = ik [ d0 p 0 e ikx bd 0 (2) an off center point source inside a three dimensiona target. In this case the exact far fied is known it is the fied created by the source itsef. The point source induced far fied requires a simpe function evauation without any series. Such an approach is usefu for vaidation because the partia differentia (Maxwe) equations and their discrete matrix approximations are independent of the incident waves. The anaytica (series) soution of the timeharmonic Maxwe equations for the eectric and magnetic far fied induced by any tangentia (incuding the point source and pane wave) incident fied on the sphere can be easiy derived provided an expansion of the incident fied in spherica waves is avaiabe. We start our approximations with a genera series representation and the pane wave scattering Mie series soution is a specia case of the series. We use quadrature rues on the sphere that are exact for poynomias of degree equa to the chosen number of terms in the partia sums of the genera series. In our numerica approximations we show that our approach yieds cose to machine accuracy for the eectric and magnetic dipoe point-source probems and aso that the error between our approximations and the Wiscombe code approximations remains stagnated with order O(10 10 ). 2 Probem Formuation The spatia components E H of the time harmonic eectromagnetic wave scattered by a sphere satisfy the harmonic Maxwe equations [1 Page 154] cur E(x) ikh(x) = 0 cur H(x) + ike(x) = 0 x R 3 \ D k = 2πω/c is the wavenumber w the frequency of the wave c is the speed of ight D = D D and D is a ba with perfecty conducting spherica surface D of diameter equa to some constant mutipe of the incident waveength λ = 2π/k = c/ω. the tota fieds E H are given by [1 Page 3] E(x) = E i (x) + E s (x) H(x) = H i (x) + H s (x) x R 3 \ D (3) the unknown scattered fieds E s H s comprise a radiating soution of the Maxwe equations (1) that satisfies the Siver-Müer radiation condition im [Hs (x) x E s (x)] = 0. The radiating soution E s H s has the asymptotic behavior of an outgoing spherica wave [1 Theorem 6.8 Page 164] E s (x) = eik H s (x) = eik { ( 1 E ( x) + O { H ( x) + O ( 1 )} )} (4) as uniformy in a directions x = x/. The vector fieds E H defined on the unit sphere (denoted throughout the paper by B) are the far fied patterns of E s H s. A boundary condition is required in order to sove the exterior Maxwe equations with the Siver-Müer radiation condition. The perfect conductor assumption on D [1 Page 155] eads to the Dirichet boundary condition n E = 0 on D n(y) denotes the unit outward norma at the point y D. Hence for the pane wave probem (with incident direction d 0 and poarization p 0 ) using (3) and (1) we get the boundary condition n(x) E s (x) = f(x) x D (5) f(x) = n(x) E i (x). (6) The unique anaytica soution of the spherica scattering probem (1) (6) is given by the Mieseries [4]. As described in the introduction to

3 Proceedings of the 9th WSEAS Internationa Conference on Appied Mathematics Istanbu Turkey May (pp ) check the accuracy of approximations to the Mie (infinitey continuousy differentiabe) boundary series it is usefu to consider the point source radiation probem with simpe exact soutions. To and hence can be represented using the Fourier data f is tangentia on the scattering sphere D this end we aso consider the boundary conditions (Lapace) series n(x) E ED (x) = f(x) n(x) E MD (x) = f(x) x D (7) the boundary data f is induced by the foowing eectric and magnetic dipoe soutions E ED H ED and E MD H MD of the Maxwe equations with radiation from a point source with poarization p src ocated at x src D [1 (6.20) (6.21) Page 163]: E ED (x) = 1 ik cur xcur x { p src Φ(x x src )} H ED (x) = cur x { p src Φ(x x src )} (8) and E MD (x) = cur x { p src Φ(x x src )} H MD (x) = 1 ik cur xcur x { p src Φ(x x src )}(9) Φ(x y) = 1 e ik x y 4π x y (10) is the fundamenta soution of the Hemhotz equation. The point source radiating fieds E ED H ED and E MD H MD satisfy the asymptotic expansions (4) eading to four far fied patterns. The far fied patterns can be computed by appying the foowing asymptotics to the Mie series and dipoe soutions (8) (9) and comparing with (4). For a continuous vector fied a on D the fundamenta soution in (10) satisfies the asymptotics cur x {a(y)φ(x y)} = ik e ik { e ikbx y x ( )} a(y) a(y) + O 4π cur x cur x {a(y)φ(x y)} = k2 e ik { e ikbx y x ( )} a(y) (a(y) x) + O 4π as uniformy for a y D [1 Page 164] x = x/ B. In the remaining sections of the paper E denotes the radiating eectric fied E s or E ED or E MD ; x denotes a point on D and points on the unit sphere are denoted by x. 3 Probem Soution f(x) = A j U j ( x) + B j V j ( x) =1 j =1 j x D x = x/ B (11) U V are the tangentia vector harmonics on the unit sphere U j ( x) = 1 ( + 1) Grad Y j ( x) V j ( x) = n( x) U j ( x) = j Grad is the surface gradient [1] and for x = p(θ φ) = (sin θ cos φ sin θ sin φ cos θ) T Y j ( x) = ( 1) (j+ j ) ( j )! 2 4π ( + j )! P j (cos θ)e ijφ (12) are the orthonorma scaar spherica harmonics. The Fourier coefficients in (11) are required to compute the Mie series. Using orthonormaity of U j and V j in (11) the Fourier coefficients are given by A j = f q 1 U j B j = f q 1 V j 1 j (13) q : x x and the orthonormaity of the tangentia vector harmonics is with respect to the inner product defined for two 3-vector vaued continuous function G H on the unit sphere by G H = I(H T G) = H T ( x)g( x) ds( x). Next we derive the far fied Mie series representation using the Fourier series representation (11) of the boundary data f. Using [1 Theorem 6.25 Page 181] there exist constants a j b j for = 1 2 j such that the eectric radiating fied E can be written as E(x) = B { aj cur {e(x)} =1 j +b j cur cur {e(x)} } (14) Using (6) for pane wave scattering and (7) for point source radiation it is cear that the smooth e(x) = xh (1) (k)y j ( x). (15)

4 Proceedings of the 9th WSEAS Internationa Conference on Appied Mathematics Istanbu Turkey May (pp ) Here h (1) denotes the th degree spherica Hanke function of the first kind. The correspondtering sphere and λ is the incident waveength. and x = 2πr/λ r is the radius of the scating eectric far fied is given by [1 Theorem 6.26 For some specia cases (for exampe an incident pane wave [3 Page 419]) the Fourier co- Page 182] efficients of the boundary data are known anayticay. In genera we require a quadrature E ( x) = 1 1 { k i +1 ikb j ( + 1)U j ( x) rue on the sphere to compute the Fourier coefficients in (13). To this end we use an m-point =1 j } a j ( + 1)V j ( x). (16) quadrature rue with points x m q B and weights R q = 1 m of the form Using [1 Equations (6.64) and (6.65) Page 180] and x (U( x) x) = U( x) we get x cur {e(x)} = h (1) (k) ( + 1)U j ( x) and x cur cur {e(x)} = 1 {h(1) (k) + kh (1) (k)} ( + 1)V j ( x) h (1) is the derivative of the th degree spherica Hanke function of the first kind. Let r be the radius of the scattering sphere. Writing x = r x and taking the vector product of x with (14) gives x E(r x) = a j h (1) (kr) ( + 1)U j ( x) + =1 j =1 j 1 b j r {h(1) (kr) + krh (1) (kr)} ( + 1)V j ( x). Hence using the boundary condition n E = f and equating coefficients with (11) we get E ( x) = 1 1 k i +1 =1 j { ikrb j h (1) (kr) + krh (1) U j ( x) (kr) } A j h (1) (kr) V j( x). (17) In practica Mie series computations the series (17) must be truncated. An empirica study in [5 Section V] recommended truncation after the first N max terms for pane wave probems x + 4x 1/ x 8 N max = x x 1/ < x < 4200 x + 4x 1/ x (18) ζ m q (G H) m = Q m (H T G) = m ζq m H( x m q ) T G( x m q ). q=1 (19) In our agorithm for approximations to the Mie series by its partia sum containing at most the first n terms we choose the number of quadrature points m such that (G H) m = G H if G and H are tangentia vector harmonics of degree at most n. Hence in the rest of the paper we use the notation m(n) to identify the connection between the number of quadrature points and the number of terms in the partia sums. One such quadrature rue that we use in our computations is the 2(n + 1) (n + 1)-point rectange-gauss rue with m(n) = 2(n + 1) 2 : Q m G = 2n+1 r=0 n+1 µ n r νs n G(p(θs n φ n r )) (20) s=1 θs n = cos 1 z s z s are the zeros of the Legendre poynomia of degree n + 1 νs n are the corresponding Gauss-Legendre weights (s = 1 n + 1) and µ n r = π n + 1 φn r = rπ r = n + 1. n + 1 (21) Our computabe approximations to the Fourier coefficients A j B j in (13) are denoted by à m(n) j m(n) B j and these are defined using (19) as à m(n) j = ( f q 1 U j )m(n) 1 n B m(n) j = ( f q 1 V j j. (22) )m(n) Using the derivation above and quadrature approximation of the Fourier coefficients of the boundary data induced by the incident fied a natura n-term computabe quadrature approximation to the far fied E in (17) is denoted by E m(n) n and is defined for every observed direction

5 Proceedings of the 9th WSEAS Internationa Conference on Appied Mathematics Istanbu Turkey May (pp ) as Scattering by sphere of diameter 0.5λ E m(n) N n ( x) max (0.5λ) = 7 = 1 n 1 m(n) ikr B j k i +1 =1 j h (1) (kr) + krh (1) U j ( x) e e e-02 (kr) e e e-12 Ã m(n) e e e-10 j h (1) (kr) V j( x). (23) e e e e e e-10 4 Numerica Experiments We verify the accuracy of our approximations by comparison with benchmark soutions for the eectric and magnetic dipoe point source radiation probems (denoted ED and MD respectivey) and for the pane wave scattering probem (denoted PW). The scattering object is a perfecty conducting sphere of radius 0.5 centered at the origin. For the point source radiation probems our benchmarks are the exact soutions computed using (7) (10). In the radiation probems considered the point source is ocated at x src which is at a distance 0.1 away from the origin in the direction θ = 30 φ = 90. The point source has poarization vector p src = (1 1 0) T / 2. For the pane wave scattering probem our benchmark is the (truncated) Mie series soution computed using Wiscombe s code. In the scattering probem considered the incident wave has direction e 3 = (0 0 1) T and poarization e 1 = (1 0 0) T. We compare our computed far fied En m(n) with the benchmark far fied E using the reative maximum error { [ ] } e max bx B 3 k=1 E T k ( x) En m(n) ( x) { } e max bx B 3 k=1 E T k ( x) approximated using more than 1300 observed directions. In Tabes 1 5 our soution is compared against the benchmark soution for spheres of eectromagnetic size λ/2 to 24λ. In the pane wave case the benchmark soution is computed using Wiscombe s code with highest order term N max N max is given by (18). Our soution is computed using (23) for a range of n bracketing N max. The resuts show that our approximations are very accurate for the radiation probem. Choosing n N max is sufficient to obtain O(10 13 ) accuracy in the far fied approximations. The resuts aso show very good agreement with the Wiscombe code for the pane wave scattering probem. However the approximatey Tabe 1: Scattering from perfecty conducting sphere of size 0.5λ with k = O(10 10 ) error obtained in this case is severa orders of magnitude arger than the error obtained for the radiation probems. The errors in the pane wave probem stagnate for n N max + 5 and cannot be reduced by increasing n. We remark that these errors cannot be reduced by using more than N max terms in the Wiscombe code. The high accuracy of our approximation has been demonstrated for the radiation probems. We therefore beieve that the arger error observed for the pane wave scattering probem is due to a restriction of the accuracy of the Wiscombe code to about O(10 10 ). Comparisons with our spectray accurate scheme for pane wave scattering have shown that we can obtain errors of O(10 14 ) between the approximations described in this paper and our spectray accurate soution compared with errors of about O(10 10 ) between the benchmark and our spectray accurate soution. In Tabe 6 we give AMD Opteron (2 GHz) CPU timings for computing three approximate far fieds (for the two radiation and the pane wave scattering probems) using our agorithm and code and demonstrate that ony a fraction of a minute of CPU time is required to compute the quadrature based approximations to the Mie series at 1352 observed directions for Mie scattering by spheres of various eectromagnetic size. 5 Concusion The approximate Mie series using numericay computed expansion of the incident fied as described in this paper has been shown to produce soutions accurate amost to machine precision. We concude that the quadrature based approximation to the Mie series is a suitabe benchmark and note that it has wider appicabiity and is not restricted to pane waves and certain forms of poarizations.

6 Proceedings of the 9th WSEAS Internationa Conference on Appied Mathematics Istanbu Turkey May (pp ) Scattering by sphere of diameter λ CPU time to compute three far-fied approximations at 1352 directions N max (λ) = 10 n sphere size x time 4 0.1λ secs 7 0.5λ secs e e e λ secs e e e λ secs e e e λ secs e e e λ secs e e e-09 Tabe 2: Scattering from perfecty conducting sphere of size λ with k = Scattering by sphere of diameter 8λ N max (8λ) = e e e e e e e e e e e e e e e-10 Tabe 3: Scattering from perfecty conducting sphere of size 8λ with k = Scattering by sphere of diameter 16λ N max (16λ) = e e e e e e e e e e e e e e e-10 Tabe 4: Scattering from perfecty conducting sphere of size 16λ with k = Scattering by sphere of diameter 24λ N max (24λ) = e e e e e e e e e e e e e e e-10 Tabe 5: Scattering from perfecty conducting sphere of size 24λ with k = Tabe 6: Time taken to compute three far-fieds (MD ED and PW) at 1352 directions for perfecty conducting spheres of size λ to 24λ with n = N max. Acknowedgement Support of the Austraian Research Counci is gratefuy acknowedged. References: [1] D. Coton and R. Kress. Inverse Acoustic and Eectromagnetic Scattering Theory. Springer [2] M. Ganesh and I. G. Graham. A high-order agorithm for obstace scattering in three dimensions. J. Comput. Phys. 198: [3] J. A. Stratton. Eectromagnetic Theory. McGraw Hi [4] H. C. van de Hust. Light Scattering by Sma Partices. Dover Pubications Inc [5] W. J. Wiscombe. Improved Mie scattering agorithms. Appied Optics [6] W. J. Wiscombe. Mie scattering cacuations: Advances in technique and fast vector-speed computer codes. Technica Report NCAR/TN-140+STR [7] A. C. Woo H. T. Wang M. J. Schuh and M. L. Sanders. Benchmark radar targets for the vaidation of computationa eectromagnetics programs. IEEE Antennas Propag. Mag. 35: