metamaterial wedge Electromagnetic fields in the ...

Downloaded from rspa.royalsocietypublishing.org on March 12, 2014

Electromagnetic fields in the presence of an infinite metamaterial wedge Mohamed A Salem and Aladin H Kamel Proc. R. Soc. A 2008 464, doi: 10.1098/rspa.2008.0031, published 8 August 2008

References

This article cites 9 articles, 3 of which can be accessed free

Subject collections

Articles on similar topics can be found in the following collections

http://rspa.royalsocietypublishing.org/content/464/2096/2077.full. html#ref-list-1

mathematical physics (155 articles)

Email alerting service

Receive free email alerts when new articles cite this article - sign up in the box at the top right-hand corner of the article or click here

To subscribe to Proc. R. Soc. A go to: http://rspa.royalsocietypublishing.org/subscriptions


Proc. R. Soc. A (2008) 464, 2077–2089 doi:10.1098/rspa.2008.0031 Published online 10 April 2008

Electromagnetic fields in the presence of an infinite metamaterial wedge B Y M OHAMED A. S ALEM 1, *

AND

A LADIN H. K AMEL 2

1

Department of Electrical and Computer Engineering, New Jersey Institute of Technology, Newark, NJ 07192, USA 2 Advanced Industrial, Technological and Engineering Center, PO Box 433, Heliopolis Center, 11757 Cairo, Egypt Electromagnetic fields, excited by an electric line source in the presence of an infinite metamaterial wedge, are determined by application of the Kontorovich–Lebedev transform. Uncoupled singular integral equations for the spectral functions are derived and a numerical scheme is devised and implemented to solve them. Numerical results showing the influence of a metamaterial wedge presence on the directivity of a line source are presented and verified through finite-difference frequency-domain simulations. Keywords: metamaterial wedge; electromagnetic diffraction; Helmholtz equation; Kontorovich–Lebedev transform; Green’s function; perturbation technique

1. Introduction The concept of left-handed material (LHM) was proposed by Veselago in (1968), where both the dielectric constant and the magnetic permeability are negative. Much attention has been received recently on the study of LHM on theory, experiments and potential applications (IEEE 2003). To mention a few of their applications, LHMs can be used to focus electromagnetic energy; they have the potential to form highly efficient, low reflectance surfaces by cancelling the scattering properties of other materials. The methods based on the Kontorovich–Lebedev transform (Lebedev 1965) have been successfully applied in the past to a number of problems of diffraction by wedges (Oberhettinger 1954; Forristall & Ingram 1971, 1972; Rawlins 1972, 1999; Osipov 1993; Salem et al. 2006). A strategy has been presented by Salem et al. (2006) to solve the problem of diffraction by an ordinary material wedge. This paper extends the strategy of Salem et al. (2006) to metamaterial wedges of arbitrary negative electric and negative magnetic constants and with arbitrary source-observer configurations. It shows that the singular integral equation formulation is suitable for numerical analysis and develops numerical procedures that provide numerical results. In §2, the singular integral equations for the Kontorovich–Lebedev spectra are derived by analytically continuing those of an ordinary material wedge; a successive approximation scheme is implemented to solve them numerically; the * Author for correspondence ([email protected]). Received 25 January 2008 Accepted 14 March 2008

2077

This journal is q 2008 The Royal Society


2078

M. A. Salem and A. H. Kamel

Figure 1. The geometry of the problem.

singularities of the Kontorovich–Lebedev spectrum inside the metamaterial wedge are identified; and the residues are evaluated. In §3, the near fields inside and outside the metamaterial wedge are represented by residue series; the far field inside the metamaterial wedge is represented by direct numerical integration of the Kontorovich–Lebedev integrals; the far field outside the wedge is represented by residue series; and an extension to the plane wave illumination case is also given. In §4, the matched double negative wedge is discussed. In §5, numerical results are presented and compared with the results of finite-difference frequency-domain simulations. Conclusions are given in §6. 2. The singular integral equations (a ) Derivation The geometry of the problem is shown in figure 1. The parameters k 1, e1 and m1 are respectively the wavenumber, permittivity and permeability outside the metamaterial wedge. k 2, e2 and m 2 are corresponding quantities inside the metamaterial wedge and NZk 2/k 1. The line source is located at (r0, 40) outside the wedge and the wedge angle is 2b. It should be noted that a lossless, dispersionless doubly negative material is not physically realizable. Hence, for the metamaterial wedge, Re (k 2, e2, m2)!0 and Im (k 2, e2, m2)O0 for the assumed, and omitted throughout, time factor exp (Kiut). It should be noted that the analysis carried out here is for the case jN jO1. Modifications for the case jN j!1 are straightforward. Since the problem under consideration is one of scattering and diffraction, the wavenumbers k 1,2 are real (complex) for the lossless (lossy) media. However, initially we assume that k 1,2 are such that arg k1;2 Z

p : 2

ð2:1Þ

Under the conditions given by (2.1), the steps leading to the derivation of the singular integral equations are similar to those employed by Salem et al. (2006) and will only be summarized here. Fields (Ez, Hr , H4) are constructed from symmetric and antisymmetric parts (with respect to the planes 4Z0, Gp) and are represented by Kontorovich–Lebedev spectra. The continuity conditions of the fields Ez, Hr on the wedge face at 4Zb, together with the distribution given Proc. R. Soc. A (2008)


Electromagnetic fields in metamaterial wedge

2079

by Forristall & Ingram (1972), Osipov (1993) and Rawlins (1999) led, for the symmetric part of the field, to ð iN A2 ðsÞ Z QðsÞ C C ðsÞv:p: neipn sinðnpÞKðn; sÞA2 ðnÞ dn; Im sR 0; Re s Z 0; 0

ð2:2Þ QðsÞ Z

ð1Þ Hs ðk 1 r0 Þ

sinðs4 Þ h i 0 ; k s cosh s ln k 21 M ðsÞ

KiN2 ðsÞ h i ; k s cosh s ln k 21 M ðsÞ 1 Kisp=2 k 21 k 2 Ks N2 ðsÞ Z e 1K 2 ; 2 k2 k1 C ðsÞ Z

Kðn; sÞ Z I ðn; sÞM ðn; sÞ; k 21 sKn F 1 C sCn ; 1 C ; 2; 1K 2 2 k 22 I ðn; sÞ Z e Kinp=2 ; cosðnpÞKcos ðspÞ M ðn; sÞ Z s sinðn½pKbÞcosðsbÞ C

m1 n sinðsbÞ cosðn½pKbÞ m2

and M ðs; sÞ ; s where v.p. in front of the integral sign denotes that the Cauchy principal value is to be taken. In order to address the original scattering and diffraction problem, we continue (2.2) analytically with respect to k 1 and k 2 as we switch them back to real (complex) for the lossless (lossy) media. For the metamaterial wedge under consideration (jN jO1), one can note that the argument of the Gauss hypergeometric function, ZZ(1K(1/N 2)), will attain values in the range j1KZ j!1, possibly with jZ jO1 for which the series representation of the hypergeometric function in (2.2) is invalid. Therefore, analytic continuation of the hypergeometric function in (2.2) is required in order to obtain a singular integral equation (SIE) valid for all possible values of Z+1 in the range j1KZ j!1. To that end, we make use of Whittaker & Watson (1990) to obtain 1 GðcÞGðcKaKbÞ 1 F a; b; c; 1K 2 Z F a; b; a C bKc C 1; 2 GðcKaÞGðcKbÞ N N ðcKaKbÞ GðcÞGða C bKcÞ 1 C GðaÞGðbÞ N2 1 ð2:3Þ ; !F cKa; cKb; cKaKb C 1; 2 N M ðsÞ Z

Proc. R. Soc. A (2008)


2080


where jarg (1KZ )j!2p and j1KZ j!1. In what follows, we will term the r.h.s. of (2.3) Fc(s, n, N ). The equation (2.2) with the hypergeometric function in the kernel replaced by Fc(s, n, N ), and with [k 2/k 1]G1 continued to [N ]G1, is the one satisfied by the spectral function A2(s) of the electric field inside the metamaterial wedge. The spectral function A1(s) of the scattered electric field outside the metamaterial wedge is found from A2(s) (see Salem et al. 2006) with the same analytic continuation procedure as given above. The singular integral equations satisfied by the Kontorovich–Lebedev spectra of the antisymmetric part is derived by replacing sin(s40) by Kcos(s40) and the remaining sin($), cos($) by cos($), Ksin($), respectively. (b ) Numerical scheme to solve the singular integral equation We apply the scheme devised, detailed by Salem et al. (2006) and inspired by the Neumann series expansion approach introduced by Rawlins (1999), to solve (2.2) numerically; now with k 1,2 switched back to real (complex) for the lossless (lossy) diffraction and scattering problem. A summary of the scheme is as follows: (i) Multiply both sides of (2.2) by coshðs ln N Þ. (ii) Expand all functions of N in the Neumann series in powers of (1K(1/N 2 )), i.e. N X 1 n ðnÞ A2 ðsÞ 1K 2 ; ð2:4Þ A2 ðsÞ Z N nZ0 1 s 1 Ks 1 coshðs ln N Þ Z F ; 1; 1; 1K 2 C F ; 1; 1; 1K 2 ; ð2:5Þ 2 2 2 N N N m X 1 coshðs ln N Þ Z cm 1K 2 ; ð2:6Þ N mZ0 X N Ks 1 1 l Ks ; 1; 1; 1K 2 Z N ZF nl 1K 2 ; ð2:7Þ 2 N N lZ0 X N s Cn sKn 1 1 j fj 1K 2 ; ð2:8Þ ;1C ; 2; 1K 2 Z F 1C 2 2 N N jZ0 "

# G mK s2 G m C s2 1 C s

; ð2:9Þ cm Z 2ðm!Þ G Ks G 2 2

G l K s2 nl Z Ks

ð2:10Þ G 2 ðl!Þ and

G j C 1 C sCn G j C 1 C sKn Gð2Þ 2 2

fj Z : ð2:11Þ sCn sKn G 1 C 2 G 1 C 2 Gðj C 2Þðj !Þ (iii) Equate equal powers of (1K(1/N 2)) from both sides of the equation. Proc. R. Soc. A (2008)



2081

Hence, ð1Þ

ð0Þ

A2 ðsÞ Z

ð1Þ

A2 ðsÞ Z C2 ðsÞv:p:

ðnÞ

A2 ðsÞ Z C2 ðsÞv:p:

Hs ðk 1 r0 Þ sinðsf0 Þ ; sM ðsÞ

ð iN h i neipn=2 sinðnpÞM ðn; sÞ ð0Þ ð0Þ A2 ðnÞn 0 f0 dnKA2 c1 ; cosðnpÞKcosðspÞ 0

ð iN "X q nK1 X 0

qZ0

!# ðnKqK1Þ

A2

nqKm fm

mZ0

ð2:12Þ

ð2:13Þ

neipn=2 sin ðnpÞM ðn; sÞ dn cosðnpÞKcosðspÞ

nK1 X ðj Þ K A2 cnKj

ð2:14Þ

jZ0

and C2 ðsÞ Z

Ki e Kisðp=2Þ : 2 sM ðsÞ

ð2:15Þ

The above equation defines an iterative scheme in which one starts with the ð0Þ known A2 ðsÞ and generates the rest of the Neumann series coefficients from (2.14).

(c ) The pole singularities of the spectra The processes of identifying the pole singularities, their order and the quantification of their residues have been detailed by Salem et al. (2006); a summary is given here. Making use of the analytic continuation process (see Salem et al. 2006), one identifies, for A2(s), the set of poles spl where spl Z sp 0 C 2l;

l Z 0; 1; 2; .;N;

p Z 1; 2; 3; .;N

ð2:16Þ

and M ðsp 0 Þ Z 0:

ð2:17Þ

The order of the spl poles is analysed next. With the observation that the sp 0 poles are simple (first-order poles), one can note that the wedge angle, 2b, being a rational multiple of p (i.e. 2bZr1p/r, where r1 and r are positive integers, r1!2r) is the condition for higher order poles to take place. With Sq fRe s 2 ð2q; 2qC 2Þ; Im s 2 ðKN;NÞg; qZ 1; 2; . denoting the q th strip, one can note that the poles in the strips Sq, 2rOqR0, are of first order; the poles in the strips Sq, 4rOqR2r, are of second order and the order increases by one every band of 2r strips. When the wedge angle is not a rational multiple of p, the spl poles are of first order. Proc. R. Soc. A (2008)


2082


(i) Residues computation (i) Poles of the type sp 0 in the Sq strip Res ½A2 ðsp 0 Þ Z

1 M 0 ðsp 0 ÞN sp 0 q X

K

ð iN ½s 0 ðsÞK Rðn; sÞLðn; sÞA2 ðnÞ dn KiN

K Cj ðsÞA2 ðsK2jÞsp 0 ;

ð2:18Þ

jZ1 ð1Þ

Hs ðk 1 r0 Þ sinðs40 Þ ; s 1 Lðn; sÞ Z ; cosðnpÞKcosðspÞ s 0 ðsÞ Z

N2 ðsÞ Rðn; sÞ Z n M ðs; nÞe Kipn=2 Fc ðs; n; N Þ; 2s

ð2:19Þ ð2:20Þ

K Cj ðsÞ Z

and

ðK1Þj GðsÞ M ðs; sK2jÞN s ðsK2jÞ 2sGð1 C jÞGð1 C sKjÞ 1 !Fc Ks C j;Kj;Ks C 1; 2 N dM ðsÞ ; M ðsp0 Þ Z ds sp 0 0

where (2.14) is used to find A2(n) under the integral sign. (ii) Poles of the type spl, lZ1, 2, .,N K1 Cl Res½A2 ðspl Þ Z K ðsÞ Res½A2 ðsp 0 Þ; l Z 1; 2; 3; .;N: M ðsÞN s spl

ð2:21Þ

ð2:22Þ

ð2:23Þ

ð2:24Þ

The residues of higher order poles, if available, are not detailed here but are straightforward and require the use of the higher order residue formula instead of the first-order formula used in this analysis. The spectral amplitude A1(s) has an extra set of poles ss with ss Z s;

s Z 1; 2; 3; .;N:

ð2:25Þ

There are simple poles with ð1Þ ðK1Þs m1 Hs ðk 1 r0 Þ sinðs40 Þ sinðs½pKbÞ cosðs½pKbÞ Res½A1 ðss Þ Z : 1K sM ðsÞ p m2 ð2:26Þ A more detailed presentation of the residue computation is given by Salem et al. (2006). Proc. R. Soc. A (2008)


2083


3. Field representations inside and outside the wedge (a ) Near field inside and outside the metamaterial wedge ð2Þ

For r!r0, we represent the electric field inside the wedge, E z , and the electric ð1Þ field scattered outside the wedge, E z , by the Kontorovich–Lebedev integrals ð iN 1 nJ ðk rÞA2 ðnÞ sin½nðpK4Þ dn ð3:1Þ E zð2Þ Z 2 KiN n 2 and ð 1 iN ð1Þ Ez Z nJ ðk rÞA1 ðnÞ sinðn4Þ dn: ð3:2Þ 2 KiN n 1 From Gradshteyn & Ryzhik (1980) ð3:3Þ Jn ðeip zÞ Z eipn Jn ðzÞ and Kn 1 2n Jn ðzÞ w pffiffiffiffiffiffiffiffi ; jarg nj! p; jarg zj! p=2; ð3:4Þ 2pn ez we obtain h p 1=2 i =jnj ; n/CiN; ð3:5Þ Jn ðk 2 rÞ Z O exp in arg ðk 2 ÞK 2 5p ð3:6Þ Jn ðk 2 rÞ Z O exp in arg ðk 2 ÞK =jnj1=2 ; n/KiN 2 and p h i ð3:7Þ Jn ðk 1 rÞ Z O exp jnj Karg ðk 1 Þ =jnj1=2 ; jnj/N: 2 Next, the behaviour of A2(s), as s/iN is estimated. (i) Write (2.2) in the form A2 ðsÞ Z QðsÞ C K 0 ðn; sÞ A2 ðnÞ:

ð3:8Þ

(ii) Express K 0 ðn; sÞ A2 ðnÞ as the sum of three integrals, with d very small, ð sKd ð sCd ðN 0 K ðn; sÞ A2 ðnÞ Z C C ð3:9Þ K 0 ðn; sÞA2 ðnÞ dn: 0

sKd

sCd

(iii) Let A2 ðnÞZ O½expðinjÞ and, making use of the asymptotic formulae by Jones (2001), we obtain 3p 1 1 3p 0 2 K ðn; sÞ Z O exp is C argðN ÞK xKb exp in bK ; 2 2 2 2 ð3:10Þ s/ iN; n finite and x Z Im ½cosh K1 ð2N 2 K 1Þ; to reach ð sKd 1 1 0 2 K ðn; sÞA2 ðnÞ dn Z O exp is arg ðN ÞK x C j : 2 2 0 Proc. R. Soc. A (2008)

ð3:11Þ


2084


(iv) Make use of

1 1 s 1K 2 F 1 C s; 1; 2; 1K 2 Z N 2s K1; N N

ð3:12Þ

to obtain ð sCd N 2s K1 cosðpðs C dÞÞKcos ðpsÞ 0 i log K ðn; sÞA2 ðnÞ dn wexpðisjÞ h cosðpsÞKcos ðpðsKdÞÞ sKd s 1K N12 Z O½expðisðarg ðN 2 Þ C jÞÞ:

ð3:13Þ

(v) From Jones (2001), we obtain h p p h i K 0 ðn; sÞ Z O exp in bK K exp is Kb ; 2 2 2 2 K1 K1 ; ð3:14Þ n/ iN; s finite and h Z Im cosh 2 N to reach

ðN

h h i : K 0 ðn; sÞA2 ðnÞ dn Z O exp is K C j 2 sCd

Hence,

h KargðN 2 ÞCx 2 A2 ðsÞ Z QðsÞO exp is max ;Karg ðN Þ; ; s/iN; 2 2 ð3:15Þ i.e. the behaviour of A2(s) as s/iN is bounded by that of Q(s). Similar analysis led to the same conclusion for s/KiN, as well as for the behaviour of A1(s) as s/GiN, (see Salem et al. (2006) for the SIE corresponding to A1(s)) with the inhomogeneous term of that SIE given by ð1Þ m1 Hs ðk 1 r0 Þ sinðs40 Þ sinðsðpKbÞÞ cosðsðpKbÞÞ : Q1 ðsÞ Z 1K s sinðpsÞM ðsÞ m2 The above conclusion is confirmed by the numerical results of §2b. Hence, one concludes that the integral in (3.1) converges and that in (3.2) diverges, which was also confirmed numerically. Following Salem et al. (2006), closing contours in the r.h.s. of the complex n-plane and collecting residue contributions, we express X E ð2Þ npl Jnpl ðk 2 rÞ Res ½A2 ðnpl Þ sinðnpl ½pK4Þ ð3:16Þ z ðr; 4Þ ZKpi npl

and E zð1Þ ðr; 4Þ ZKpi

X

npl Jnpl ðk 1 rÞ Res½A1 ðnpl Þsinðnpl 4Þ

npl

Kpi

N X sZ1


sJs ðk 1 rÞ Res½A1 ðsÞ sinðs4Þ:

ð3:17Þ


2085


The behaviour of the npl and ns series summands are as reported by Salem et al. (2006). One can note that, since (3.1) converges, an alternative representation for the near field in the metamaterial wedge exists in terms of the direct numerical integration of (3.1). (b ) Far field inside and outside the metamaterial wedge ð2Þ

For rOr0, we represent the electric field inside the wedge, E z , and the electric ð1Þ field scattered outside the wedge, E z , by the Kontorovich–Lebedev integrals ð i iN ipn E ð2Þ Z K ne sinðnpÞHnð1Þ ðk 2 rÞA2 ðnÞ sin½nðpK4Þ dn ð3:18Þ z 2 0 and ð i iN ipn E ð1Þ ZK ne sinðnpÞHnð1Þ ðk 1 rÞA1 ðnÞ sinðn4Þ dn: ð3:19Þ z 2 0 From Gradshteyn & Ryzhik (1980) Hnð1Þ ðeip zÞ ZKe Kipn Hnð2Þ ðzÞ; rffiffiffiffiffiffi n n 2 2n ð2Þ ; arg O0 Hn ðzÞ wi pn ez z and Hnð1Þ ðzÞ w we obtain

rffiffiffiffiffiffi Kn 2 2n ; pn ez

arg

n z

ð3:21Þ

ð3:22Þ

"

Hnð1Þ ðk 2 rÞ

# exp Kin argðk 2 ÞK p2 ; ZO jnj1=2

O 0;

ð3:20Þ

and

" Hnð1Þ ðk 1 rÞ Z O

# exp Kin p2 Kargðk 1 Þ jnj1=2

;

n/CiN

ð3:23Þ

n/CiN;

ð3:24Þ

which, together with the corresponding behaviour of A1,2(n), show that the integral in (3.18) converges while that in (3.19) diverges. Hence, we represent the far field inside the metamaterial wedge by direct numerical integration of (3.18). The far field outside the wedge is found, similar to Salem et al. (2006), by invoking reciprocity to obtain X E ð1Þ npl Jnpl ðk 1 r0 Þ Res ½A1 ðnpl Þ sinðnpl 40 Þ z ðr; 4Þ ZKpi npl

Kpi

N X

sJs ðk 1 r0 Þ Res ½A1 ðsÞ sinðs40 Þ;

ð3:25Þ

sZ1

where the sums are on the residues of A1(n) with the source located at (r, 4) and the observer located at (r0, 40), both outside the wedge. Proc. R. Soc. A (2008)


2086


(c ) The plane wave illumination case The fields due to a normally (with respect to Z ) incident plane wave are recovered ð1Þ by replacing H n ðk 1 r0 Þ by expðKipn=2Þ in the line source results; hence, the expressions are similar to (3.17) and (3.16) with the residues replaced by those for plane wave illumination. One can note that the field inside the metamaterial wedge could, alternatively, be obtained by direct numerical integration of (3.18) with the Kontorovich–Lebedev spectrum A2(n) replaced with that of the plane wave illumination. Remark. Similar to Salem et al. (2006), the truncated residue sums on npl should be understood as giving some asymptotic approximation in terms of first identified poles; the antisymmetric part of the field, the field structure when higher order poles exist and the suitability of the Bessel series for field computations in the plane wave illumination case is as reported by Salem et al. (2006). 4. The matched double negative wedge To recover the case of the lossless double negative wedge, ðe2 ; m2 Þ ZKðe1 ; m1 Þ; ð4:1Þ one can start with (2.2), with k 2/k 1 replaced by N, which is valid (i.e. the series representation of the hypergeometric function converges) in the vicinity of (and at) NwK1, continue the SIE into the right half s-plane, identify the poles and quantify the residues of A2(s) as detailed by Salem et al. (2006). This is followed by taking the limit of the field as N/K1 and m2/Km1. We obtain, for the total field (the symmetric and antisymmetric parts) of a line source located at (r0, 40), Kpi X E ð2Þ ðr; 4Þ Z e J ðk r ÞH ð1Þ ðk r Þ cosðnn ½ð2bK 40 ÞK4Þ ð4:2Þ z pK2b n n nn 1 ! nn 1 O n

and Ez ðr; 4Þ Z

Kpi X e J ðk r ÞH ð1Þ ðk r Þ cosðnn ½4K 40 Þ; pK2b n n nn 1 ! nn 1 O

ð4:3Þ

n

where nn Z np=ðpK2bÞ, nZ0, 1, 2, ., N, enZ1 for nZ0 and enZ2 for ns0 and r!jrO is the lesserjgreater of r and r0. The above is in agreement with Monzon et al. (2005). Equation (4.2) reveals, in agreement with the predictions of geometrical optics, that when j40 jR Remð3b; 2pÞ; ð4:4Þ where Rem (a, b) is the remainder of the division of a by b, an image of the field at the source is formed at the location (r0, 2bK40) inside the metamaterial wedge (when the inequality holds) or on the boundary (when the equality holds). For an odd geometry (metamaterial wedge of angle b on top of a ground plane), the symmetric part of (4.2) reveals that when 40 Z Remð3b; 2pÞ; ð4:5Þ an image of the field at the source is formed at the boundary (r0, b). The symmetric part of (4.3) shows a zero electric field along the plane 4Z(40Cb)/2 outside the wedge. Proc. R. Soc. A (2008)


2087

Electromagnetic fields in metamaterial wedge (a) 120

(b)

90

90

0.4 0.3

0.5

120

60

0.4

60

0.3 0.2

150

150

30

0.1 0

180

210

330 240

300

30

0.2 0.1

180

0

210

330 240

270

300 270

Figure 2. jEz j due to a p line metamaterial and the ordinary material wedges, ffiffiffi source for the p ffiffiffi respectively. (a) k 1 r Z 1= 2 and (b) k 1 r Z 2 2 (solid curve, metamaterial (calculated); dots, metamaterial (finite-difference frequency-domain); dashed curve, ordinary material).

One should note that, contrary to an infinite planar interface, a metamaterial wedge geometry, which is formed of truncated planar interfaces, is incapable of forming a faithful image of the source. This can clearly be seen from (4.2) where, even though the field at the image location is infinite and has a logarithmic nature as the observation point approaches the image location, the expression in ð1Þ (4.2) is not equal to the field of a line source, namely ði=4ÞH 0 ðk 1 jr Kr 0 jÞ.

5. Numerical results Using the developed numerical scheme, the electric field modulus jEz j is calculated for line source excitation and plane wave illumination in the near- and far-field pffiffiffi regions, with bZ2.521, 40Zp/4 and k 1 r0 Z 2 (for the line source case). Figure 2 shows the plots jEz j due to a unit strength line source when e2 =e1 ZK2C 0:001i and m2 =m1 ZK2C 0:001i, along with jEz j due to a unit strength line source when e2 =e1 Z 2C 0:001i and m2 =m1 Z 2C 0:001i (ordinary material wedge). For the results in figure 2, eight terms from the Neumann series of A2(n) and subsequently residues with n pl%5.02 for the symmetric part and the antisymmetric parts were required to produce fields within 1.1% accuracy in comparison with the finite-difference frequency-domain simulation. Figure 3 shows the same plots when e2/e1 and m2/m1 are changed to e2 =e1Z K1C 0:001i and m2 =m1 ZK1C 0:001i. Six terms from the Neumann series of A2(n) and subsequently residues with npl%5.02 for the symmetric part and the antisymmetric parts were sufficient to produce results within 1.4% accuracy in comparison with the finite-difference frequency-domain simulation. The magnitude of Ez duepto ffiffiffi a normally incident unit strength plane wave illumination, when k 1 r Z 2 2, e2 =e1 ZK2C 0:001i and m2 =m1 ZK2C 0:001i; along with jEz j due to the same excitation when e2 =e1 Z 2C 0:001i and m2 =m1Z 2C 0:001i (ordinary material wedge) are shown in figure 4. Proc. R. Soc. A (2008)


2088

M. A. Salem and A. H. Kamel (a)

90

(b)

0.3

120

90

120

60

0.4 0.3

0.2 150 0.1

0.2

150

30

60 30

0.1

180

180

0

210

330

210 240

0

330 240

300

270

270

300

Figure 3. jEz j due to a p line metamaterial and the ordinary material wedges, ffiffiffi source for the p ffiffiffi respectively. (a) k 1 r Z 1= 2 and (b) k 1 r Z 2 2 (solid curve, metamaterial (calculated); dots, metamaterial (finite-difference frequency-domain); dashed curve, ordinary material).

90 0.15 60

120 0.1 150

30 0.05

180

0

330

210 240

300 270

Figure 4. jEz j due to applane wave for the metamaterial and the ordinary material wedges, ffiffiffi respectively, for k 1 r Z 2 2 (solid curve, metamaterial (calculated); dots, metamaterial (finitedifference frequency-domain); dashed curve, ordinary material).

Eight terms from the Neumann series of A2(n) and subsequently residues with npl%6.23 for the symmetric part and npl%7.40 for the antisymmetric part were sufficient to produce results within 2.3% accuracy in comparison with the finitedifference frequency-domain simulation. The developed finite-difference frequency-domain algorithm was detailed by Salem et al. (2006) and will not be repeated here. 6. Conclusions The application presented here, together with that by Salem et al. (2006), establishes the Kontorovich–Lebedev formulation as a viable solution strategy for diffraction problems in wedges with field continuity-type boundary conditions on Proc. R. Soc. A (2008)



2089

their faces. The proposed problem solving strategy is applicable for two- and threedimensional problems of thermal conductivity, acoustics and elastodynamics wedges and cones. The work presented here could be extended to the problems of monochromatic as well as transient diffraction by moving wedges and cones and by wedges and cones composed of single-negative materials. References Forristall, G. Z. & Ingram, J. D. 1971 Elastodynamics of a wedge. Bull. Seismol. Soc. Am. 61, 275–287. Forristall, G. Z. & Ingram, J. D. 1972 Evaluation of distributions useful in Kontorovich–Lebedev transform theory. SIAM J. Math. Anal. 3, 561–566. (doi:10.1137/0503055) Gradshteyn, I. S. & Ryzhik, I. M. 1980 Tables of integrals, series, and products. New York, NY: Academic Press. IEEE 2003 IEEE Transactions on Antennas and Propagation, 51, 2546–2750. Jones, D. S. 2001 Asymptotics of the hypergeometric function. Math. Methods Appl. Sci. 24, 369–389. (doi:10.1002/mma.208) Lebedev, N. N. 1965 Special functions and their applications. Englewood Cliffs, NJ: Prentice-Hall. Monzon, C., Forester, D. W. & Loschialpo, P. L. 2005 Exact solution to line source scattering by an ideal left-handed wedge. Phys. Rev. E 72, 056 606. (doi:10.1103/PhysRevE.72.056606) Oberhettinger, F. 1954 Diffraction of waves by a wedge. Commun. Pure Appl. Math. 7, 551–563. (doi:10.1002/cpa.3160070306) Osipov, A. V. 1993 Harmonic wave diffraction problems in sectored media. Vestnik St. Petersburg. Univ.: Fiz. Khim. 2, 10–21. [In Russian.] Rawlins, A. D. 1972 Electromagnetic diffraction by wedge shaped obstacles. PhD thesis, University of Surrey. Rawlins, A. D. 1999 Diffraction by, or diffusion into, a penetrable wedge. Proc. R. Soc. A 455, 2655–2686. (doi:10.1098/rspa.1999.0421) Salem, M. A., Kamel, A. H. & Osipov, A. V. 2006 Electromagnetic fields in the presence of an infinite dielectric wedge. Proc. R. Soc. A 462, 2503–2522. (doi:10.1098/rspa.2006.1691) Veselago, V. G. 1968 The electrodynamics of substances with simultaneously negative values of e and m. Sov. Phys. Usp. 10, 509–514. (doi:10.1070/PU1968v010n04ABEH003699) Whittaker, E. T. & Watson, G. N. 1990 A course in modern analysis, p. 291, 4th edn. Cambridge, UK: Cambridge University Press.
