This article was downloaded by: [Quaid-i-azam University] On: 16 January 2015, At: 23:43 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Click for updates Journal of Electromagnetic Waves and Applications Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tewa20 Scattering of electromagnetic plane wave by a PEC strip in homogeneous isotropic chiral medium M. Afzaal a , A.A. Syed a , A. Imran a , Q.A. Naqvi a & K. Hongo a a Department of Electronics, Quaid-i-Azam University, Islamabad, Pakistan Published online: 24 Mar 2014. To cite this article: M. Afzaal, A.A. Syed, A. Imran, Q.A. Naqvi & K. Hongo (2014) Scattering of electromagnetic plane wave by a PEC strip in homogeneous isotropic chiral medium, Journal of Electromagnetic Waves and Applications, 28:8, 999-1010, DOI: 10.1080/09205071.2014.901195 To link to this article: http://dx.doi.org/10.1080/09205071.2014.901195 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &
Transcript
This article was downloaded by: [Quaid-i-azam University]On: 16 January 2015, At: 23:43Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
Click for updates
Journal of Electromagnetic Waves andApplicationsPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/tewa20
Scattering of electromagnetic planewave by a PEC strip in homogeneousisotropic chiral mediumM. Afzaala, A.A. Syeda, A. Imrana, Q.A. Naqvia & K. Hongoa
a Department of Electronics, Quaid-i-Azam University, Islamabad,PakistanPublished online: 24 Mar 2014.
To cite this article: M. Afzaal, A.A. Syed, A. Imran, Q.A. Naqvi & K. Hongo (2014) Scattering ofelectromagnetic plane wave by a PEC strip in homogeneous isotropic chiral medium, Journal ofElectromagnetic Waves and Applications, 28:8, 999-1010, DOI: 10.1080/09205071.2014.901195
To link to this article: http://dx.doi.org/10.1080/09205071.2014.901195
PLEASE SCROLL DOWN FOR ARTICLE
Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoever orhowsoever caused arising directly or indirectly in connection with, in relation to or arisingout of the use of the Content.
This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &
Scattering of electromagnetic plane wave by a PEC strip inhomogeneous isotropic chiral medium
M. Afzaal, A.A. Syed, A. Imran, Q.A. Naqvi* and K. Hongo
Department of Electronics, Quaid-i-Azam University, Islamabad, Pakistan
(Received 10 December 2013; accepted 28 February 2014)
Scattering of left and right circularly polarized electromagnetic plane waves from aperfect electrically conducting strip placed in homogenous isotropic chiral mediumis investigated using Kobayashi Potential (KP) method. The KP method is beingemployed for the first time to investigate the scattering phenomenon in homoge-neous isotropic chiral medium. In the problem formulation, we encounter dualintegral equations which are solved using discontinuous properties of Weber–Schafheitlin’s integral. Far zone scattered co- and cross-components are determinednumerically using steepest descent method. Dependence of scattered co- and cross-components on angle of incidence and the chirality parameter has been demon-strated through numerical results.
1. Introduction
A number of techniques have been used by the researchers to address the problems ofscattering of electromagnetic waves from strip.[1–16] The high-frequency techniquesare most often used in situation when the size of the object is large compared to thewavelength of the incident electromagnetic waves. They include the methods of Physi-cal Optics,[8,9] Physical Theory of Diffraction, and Geometrical Theory of Diffrac-tion.[10] When the size of a scatterer is not larger, Method of Moment (MoM),originally introduced by Harrington, works very well and is numerically exact.[11–13]Wiener–Hopf technique has also been used in scattering from strip by many research-ers.[14–16]
Kobayashi proposed another method using characteristic functions satisfyingboundary and edge conditions simultaneously by making use of the properties ofWeber–Schafheitlin’s discontinuous Integrals. Kobayashi used this method to solve theelectrostatic problem of an electrified conducting disk. The method was later named asKobayashi Potential (KP) by Sneddon in [17]. It is based on eigen function expansionapproach and has similarities to the MoM in the spectral domain with different formu-lation. The MoM is based on integral equation while the KP method is based on dualintegral equations. Also, for the KP method, the mathematical formulation satisfies boththe edge and boundary conditions and leads to faster convergent solutions. The disad-vantage of the KP method is that it is applicable to limited geometries such as circularand rectangular plates and apertures.[18] The KP method has been employed for anumber of applications.[19–32]
A chiral medium is an optically active medium that has been the subject of researchfor years. A number of applications exist in the real world, such as its use as a coatingmaterial to reduce the radar cross-section.[33] As a matter of fact, it is one of the reci-procal media with peculiar properties like existence of different phase velocities for theright circularly polarized (RCP) and left circularly polarized (LCP) electromagneticplane waves. When the chiral medium becomes lossless, there is rotation of the polari-zation of the linearly polarized wave that passes through such medium. An isotropicchiral medium in Drude–Born–Fedorov representation is characterized by the followingconstitutive relations [34]
D ¼ eEþ ebr� E (1a)
B ¼ lHþ lbr�H (1b)
Here, ɛ and μ are the permittivity and the permeability of the isotropic chiral medium,respectively, and β is the chirality parameter. Electric and magnetic fields in chiral med-ium may be expressed in terms of Beltrami fields as
E ¼ Q1 � jgQ2 (2a)
H ¼ � j
gQ1 þQ2 (2b)
where Q1 and Q2 are LCP and RCP Beltrami fields, respectively, and g ¼ ffiffile
pis
impedance of chiral medium. Beltrami fields are known as fields whose divergence iszero and curl is parallel to itself and they satisfy the following set of equations
r�Q 1
2
¼ �c 1
2
Q 1
2
(3a)
r �Q 1
2
¼ 0 (3b)
where upper signs are for LCP Beltrami field and lower signs are for RCP Beltramifield. Wave numbers γ1 = k/(1 − kβ) and γ2 = k/(1 + kβ) are associated with Q1 and Q2,respectively, and k ¼ x
ffiffiffiffiffiel
pis the wave number of corresponding achiral medium.
The analysis of the scattering from a conducting strip placed in bi-isotropic mediumis done using the Wiener–Hopf method.[14–16] The scattering phenomenon from anobstacle placed in homogenous isotropic chiral medium is performed by making use ofthe surface integral equations.[35] However, the KP method has not yet been deployedin finding the scattering from any type of geometry placed in homogenous isotropicchiral medium. In this paper, we investigate the scattering of LCP and RCP planewaves from Perfect Electric Conducting (PEC) strip placed in the homogenous isotropicchiral medium. For the geometry taken, the curl equations are obtained by taking intoaccount only the longitudinal components of the LCP and RCP wave as other compo-nents could be derived from them. The boundary conditions are used to obtain the dualintegral equations which are solved with the help of Weber–Schafheitlin’s discontinuousintegrals. The solution to the matrix equations, obtained using orthogonal property of
Jacobi’s polynomials, is obtained numerically which leads to the determination of theexpansion co-efficients. Finally, the scattered far zone fields are calculated using thesteepest descent method and plotted for the corresponding analysis.
2. Formulation of the problem and KP method
Consider a PEC strip of negligible small thickness having width 2a and of infiniteextent along z-axis is placed in an isotropic chiral medium. Longitudinal componentsof incident LCP and RCP Beltrami fields are written below
Qinc1z ¼ P1e
�jc1aðcos/0xaþsin/0yaÞ (4a)
Qinc2z ¼ P2e
�jc2aðcos/0xaþsin/0yaÞ (4b)
In above equations γ1a = aγ1, γ2a = aγ2, xa ¼ xa, ya ¼ y
a. P1 and P2 are amplitudes of lon-gitudinal component of incident LCP and RCP Beltrami fields, respectively, and bothare making angle ϕ0 with positive x-axis. As strip is infinite along z-axis, curl Equa-tions (3a) reduce to following set of equations
Q1x ¼ 1
c1a
@Q1z
@ya(5a)
Q1y ¼ � 1
c1a
@Q1z
@xa(5b)
Q2x ¼ � 1
c2a
@Q2z
@ya(5c)
Q2y ¼ 1
c2a
@Q2z
@xa(5d)
From above equations, it may be noted that with the help of longitudinal component ofLCP and RCP Beltrami fields other components can be determined using these compo-nents. So we only need to determine scattering of longitudinal components. ScatteredBeltrami fields from strip with unknown weighting functions may be assumed as
Qsca1z ¼
Z 1
0g1cðnÞ cosðxanÞ þ g1sðnÞ sinðxanÞð Þe
ffiffiffiffiffiffiffiffiffiffiffin2�c2
1a
p� �yadn ya [ 0 (6a)
Qsca2z ¼
Z 1
0ðg2cðnÞ cosðxanÞ þ g2sðnÞ sinðxanÞÞe
ffiffiffiffiffiffiffiffiffiffiffin2�c2
2a
p� �yadn ya [ 0 (6b)
Journal of Electromagnetic Waves and Applications 1001
Dow
nloa
ded
by [
Qua
id-i
-aza
m U
nive
rsity
] at
23:
43 1
6 Ja
nuar
y 20
15
Qsca1z ¼
Z 1
0ðh1cðnÞ cosðxanÞ þ h1sðnÞ sinðxanÞÞe
�ffiffiffiffiffiffiffiffiffiffiffin2�c2
1a
p� �yadn ya\0 (6c)
Qsca2z ¼
Z 1
0ðh2cðnÞ cosðxanÞ þ h2sðnÞ sinðxanÞÞe
�ffiffiffiffiffiffiffiffiffiffiffin2�c2
2a
p� �yadn ya\0 (6d)
where unknown weighting functions g1c(ξ), g1s(ξ), h1c(ξ), h1s(ξ) and g2c(ξ), g2s(ξ),h2c(ξ), h2s(ξ) will be determined using boundary conditions. Boundary conditions ofelectric and magnetic fields are
Etzjya¼0 ¼ 0 for jxaj � 1 (7a)
Etzjya¼0þ ¼ Et
zjya¼0� for all xa (7b)
Etxjya¼0 ¼ 0 for jxaj � 1 (7c)
Etxjya¼0þ ¼ Et
xjya¼0� for all xa (7d)
H scaz jya¼0þ ¼ Hsca
z jya¼0� for jxaj � 1 (7e)
Htxjya¼0þ ¼ Ht
xjya¼0� for jxaj � 1 (7f)
whereas electric and magnetic fields have following edge conditions for PEC strip
Etzjya¼0; Ht
z jya¼0 �ð1� x2aÞ1=2 for jxaj ! 1 (8a)
Etxjya¼0; Ht
xjya¼0 �ð1� x2aÞ�1=2 for jxaj ! 1 (8b)
Using boundary condition (7b), we get
g1cðnÞ � jgg2cðnÞ ¼ h1cðnÞ � jgh2cðnÞ (9a)
g1sðnÞ � jgg2sðnÞ ¼ h1sðnÞ � jgh2sðnÞ (9b)
Similarly from boundary condition (7d), we get following set of equationsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2 � c21a
Equations (11)–(14) are dual integral equations. Equations (13) and (14) will be satis-fied using discontinuous properties of Weber–Schafheitlin’s integral while other twoequations will help us to determine the expansion co-efficients. Equation (13) is satis-fied, if we assume the following relations
½g1cðnÞ � h1cðnÞ þ jg½g2cðnÞ � h2cðnÞ ¼X1m¼0
AmJ2mþ1ðnÞ
n(15a)
½g1sðnÞ � h1sðnÞ þ jg½g2sðnÞ � h2sðnÞ ¼X1m¼0
BmJ2mþ2ðnÞ
n(15b)
By choosing above relation, we have satisfied edge condition of Htz . Similarly Equation
(14) along with edge condition of Htx is satisfied using discontinuous properties of
Weber–Schafheitlin’s integral by choosing following relations.
Journal of Electromagnetic Waves and Applications 1003
Now separation of even and odd functions of Equation (11) is accomplished and thenthe trigonometric functions are expanded in terms of Jacobi’s polynomial u�1=2
n ðx2aÞ.Here, Jacobi’s polynomial u�1=2
n ðx2aÞ is used to satisfy edge condition of Etz. Orthogonal
property of Jacobi’s polynomial is then used to get following matrix equations
Similarly, the separation of even and odd functions of Equation (12) is accomplishedand trigonometric functions are expanded in terms of Jacobi’s polynomial v�1=2
n ðx2aÞ.
1004 M. Afzaal et al.
Dow
nloa
ded
by [
Qua
id-i
-aza
m U
nive
rsity
] at
23:
43 1
6 Ja
nuar
y 20
15
Here, Jacobi’s polynomial v�1=2n ðx2aÞ is used to satisfy edge condition of Et
x. Orthogonalproperty of Jacobi’s polynomial is then used to get following matrix equations
The expansion co-efficients Am, Bm, Cm, and Dm can be determined using Equations(18) and (19).The far zone scattered fields are determined from Equations (6a) and (6b)using steepest descent method and are written below
In above equations, ϕ is an observation angle in cylindrical coordinate.
3. Numerical results and discussion
The determination of the expansion co-efficients Am, Bm, Cm, and Dm for LCP andRCP, has been accomplished by solving the Equations (18) and (19). The integralsappeared in Equations (18) and (19) have been written in series forms and orders ofmatrices were taken ð2c1a þ 1Þð2c1a þ 1Þ in computation. After computing, they havebeen used for the computation of the far field scattered patterns using Equations (20a)and (20b). The plots have been taken separately for the LCP and RCP to have ananalysis of the scattered fields deeply. Figure 1(a) and (b) shows the scattered co-polar-ized component and the cross-polarized component, respectively, when LCP wave is
Journal of Electromagnetic Waves and Applications 1005
Dow
nloa
ded
by [
Qua
id-i
-aza
m U
nive
rsity
] at
23:
43 1
6 Ja
nuar
y 20
15
incident. It is observed that the main lob of the co-component happens at ϕ = π − ϕ0,while that of the cross-component is at / ¼ p=2þ sin�1ððc1a=c2aÞ cosð/0ÞÞ. Thescattered field dependence on the chirality has also been analyzed for both types ofincidence. Figure 2(a) and (b) shows that for LCP, the main lobes of co- and cross-components occur at ϕ = π − ϕ0 and / ¼ p=2þ sin�1ððc1a=c2aÞ cosð/0ÞÞ, respectively.
Figure 3(a) and (b) shows the scattering pattern of cross- and co-polarized compo-nents of the RCP, respectively. Here, the main lobes occur approximately at ϕ = π − ϕ0
0 20 40 60 80 100 120 140 160 1800
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
|Qsc
a1
z| P1= 1
P2= 0
0 20 40 60 80 100 120 140 160 1800
0.5
1
1.5
2
2.5
3
3.5
4
| Q
sca
2z
| P1= 1
P2= 0
0=60o
0=70o
0=90o
0=60o
0=70o
0=90o
(a) (b)
Figure 1. (a) Effect of incidence angle of LCP plane wave on scattered co-component for kβ = 0.2,(b). Effect of incidence angle of LCP plane wave on scattered cross-component for kβ = 0.2.
0 20 40 60 80 100 120 140 160 1800
0.5
1
1.5
2
2.5
3
3.5
4
|Q
sca
2z
|
P1= 1
P2= 0
k =0.001
k =0.09k =0.20
0 20 40 60 80 100 120 140 160 1800
0.5
1
1.5
2
2.5
3
3.5
4
4.5
|Qsc
a1
z|
P1= 1
P2= 0
k =0.001
k =0.09k =0.20
(a) (b)
Figure 2. (a) Effect of chirality kβ on scattered co-component when LCP plane wave is incidentat ϕ0 = 70°, (b) Effect of chirality kβ on scattered cross-component when LCP plane wave is inci-dent at ϕ0 = 70°.
1006 M. Afzaal et al.
Dow
nloa
ded
by [
Qua
id-i
-aza
m U
nive
rsity
] at
23:
43 1
6 Ja
nuar
y 20
15
and / ¼ p=2þ sin�1ððc2a=c1aÞ cosð/0ÞÞ for the co- and cross-polarized components,respectively. Also, there is a respective increase and decrease in the amplitude of themain lobes for the scattered LCP and RCP components, respectively, for increase inchirality and it is shown in Figure 4(a) and (b).
When the incident fields have amplitudes P1 = 1/2, P2 = j/2η and kβ→ 0, the inci-dent circularly polarized plane wave becomes Ez polarized. However, it becomes Hz
0 20 40 60 80 100 120 140 160 1800
1
2
3
4
5
P1= 0
P2= 1
0 20 40 60 80 100 120 140 160 1800
0.5
1
1.5
2
2.5
3
3.5
|Qsc
a2z
|
P1= 0
P2= 1
0=60o
0=70o
0=90o
0=60o
0=70o
0=90o
| -1
Qsc
a1z
|(a) (b)
Figure 3. (a) Effect of incidence angle of RCP plane wave on scattered cross-component forkβ = 0.2, (b) Effect of incidence angle of RCP plane wave on scattered co-component for kβ = 0.2.
0 20 40 60 80 100 120 140 160 1800
1
2
3
4
5
P1=0
P2=1
0 20 40 60 80 100 120 140 160 1800
0.5
1
1.5
2
2.5
3
3.5
4
|Qsc
a2
z|
P1=0
P2=1
k =0.001
k =0.09k =0.20
k =0.001
k =0.09k =0.20
| -1
Qsc
a1
z|
(a) (b)
Figure 4. (a) Effect of chirality kβ on scattered cross-component when RCP plane wave is inci-dent at ϕ0 = 70°, (b) Effect of chirality kβ on scattered co-component when RCP plane wave isincident at ϕ0 = 70°.
Journal of Electromagnetic Waves and Applications 1007
Dow
nloa
ded
by [
Qua
id-i
-aza
m U
nive
rsity
] at
23:
43 1
6 Ja
nuar
y 20
15
polarized under the conditions P1 = jη, P2 = 1 and kβ→ 0. These results coincide withthe scattering from PEC strip in free space, and thus validate our formulation.
4. Conclusions
The scattering of the LCP and RCP plane electromagnetic wave is accomplished froma perfect electrically conducting strip placed in homogenous isotropic chiral mediumusing the KP method. The KP method allows the satisfaction of edge and boundaryconditions simultaneously leading to a simpler formulation. The results show that forthe LCP incident plane wave, an increase in the chirality of the medium increases theamplitude of the main lobe of the co-scattered field and decreases the amplitude of thecross-scattered field. However, the scattering angle of the main lobe of the co-scatteredfield does not vary while that of the cross-scattered field increases. For the RCP inci-dent wave, an increase in the chirality of the medium decreases the amplitude of themain lobe of the co-scattered field while the scattering angle remains the same. How-ever, the amplitude of the main lobe of the cross-scattered field increases while its scat-tering angle decreases.
References[1] Grinberg GA. Diffraction of electromagnetic waves by a strip of finite width. Dokl. Akad.
Nauk SSSR. 1959;129:295–298.[2] Hansen EB. Scalar diffraction by infinite strip and circular disk. J. Math. Phys.
1962;41:229–245.[3] Jones DS, Noble B. The low-frequency scattering by a perfectly conducting strip. Math.
Proc. Cambridge Philos. Soc. 1961;57:364–366.[4] Fialkovskiy AT. Diffraction of planar electromagnetic waves by a slot and a strip. Radio
Eng. Electron. 1966;2:150–157.[5] Herman MI, Volakis JL. High-frequency scattering by a resistive strip and extensions to con-
ductive and impedance strips. Radio Science. 1987;22:335–349.[6] Buyukaksoy A, Bicakci O, Serbest AH. Diffraction of an E-polarized plane wave by a resis-
tive strip located on a dielectric interface. J. Electromagnet. Wave. Appl. 1994;8:575–590.[7] Büyükaksoy A, Uzgören G. Secondary diffraction of a plane wave by a metallic wide strip
residing on the plane interface of two dielectric media. Radio Science. 1987;22:183–191.[8] Silver S. Microwave antenna theory and design. New York (NY): McGraw-Hill Book; 1949.[9] Balanis CA. Antenna theory analysis and design. New York (NY): Wiley; 1982.[10] Keller JB. A geometric theory of diffraction, in calculus of variations and its applications.
Proc. Sympos. Appl. Math. 1962;8:27–52.[11] Harrington RF. Field computation by moment methods. Malabar (FL): Krieger; 1968.[12] Barkeshli K, Volakis JL. Electromagnetic scattering from thin strips part I: analytical solu-
tions for wide and narrow strips. IEEE Trans. Educ. 2004;47:100–106.[13] Barkeshli K, Volakis JL. Electromagnetic scattering from thin strips part II: numerical solu-
tion for strips of arbitrary size. IEEE Trans. Educ. 2004;47:107–113.[14] Asghar S, Hayat T. Plane wave diffraction by a perfectly conducting strip in a homogeneous
biisotropic medium. Int. J. Appl. Electromagnet. Mech. 1998;9:39–51.[15] Asghar S, Hayat T, Asghar B. Cylindrical wave diffraction by a perfectly conducting strip
in a homogeneous bi-isotropic medium. J. Mod. Optic. 1998;45:515–528.[16] Ayub M, Ramzan M, Mann AB. A note on spherical electromagnetic wave diffraction by a
perfectly conducting strip in a homogeneous bi-isotropic medium. Prog. Electromagnet. Res.2008;85:169–194.
[17] Sneddon IN. Mixed boundary value Problems in potential theory. Amsterdam: North-Hol-land; 1966.
[18] Hongo K, Jafri ADU, Naqvi QA. Scattering of electromagnetic spherical wave by aperfectly conducting disk. Prog. Electromagnet. Res. 2012;129:315–343.
1008 M. Afzaal et al.
Dow
nloa
ded
by [
Qua
id-i
-aza
m U
nive
rsity
] at
23:
43 1
6 Ja
nuar
y 20
15
[19] Nomura Y. The electrostatic problems of two equal parallel circular plates. Pro. Phys. Math.Soc. Japan. 1941;23:168–180.
[20] Nomura Y, Kawai N. On the acoustic field by a vibrating source arbitrarily distributed on aplane circular plate. Sci. Rep. 1949;33:197–207, Tohoku Imperial Univ., Sendai, Japan.
[21] Takahashi M, Hongo K. Capacitance of coupled circular microstrip disks. IEEE Trans. Mi-crow. Theory Tech. 1982;30:1881–1888.
[22] Ôtsuki T. Diffraction of an acoustic wave by a rigid rectangular plate. J. Phys. Soc. Jpn.1964;19:1733–1741.
[23] Hongo K, Serizawa H. Diffraction of electromagnetic plane wave by a rectangular plate anda rectangular hole in the conducting plate. IEEE Trans. Antenn. Propag. 1999;47:1029–1041.
[24] Imran A, Naqvi QA, Hongo K. Diffraction of plane wave by two parallel slits in an infi-nitely long impedance plane using the method of Kobayashi potential. Prog. Electromagnet.Res. 2006;63:107–123.
[25] Imran A, Naqvi QA, Hongo K. Diffraction of electromagnetic plane wave by an impedancestrip. Prog. Electromagnet. Res. 2007;75:303–318.
[26] Imran A, Naqvi QA, Hongo K. Diffraction of electromagnetic plane wave by an infinitelylong conducting strip on dielectric slab. Optic. Commun. 2009;282:443–450.
[27] Imran A, Naqvi QA, Hongo K. Diffraction of electromagnetic plane wave from a PEMCstrip. Int. J. Appl. Electromagnet. Mech. 2010;34:181–193.
[28] Hongo K, Naqvi QA. Diffraction of electromagnetic wave by disk and circular hole in aperfectly conducting plane. Prog. Electromagnet. Res. 2007;68:113–150.
[29] Jafri ADU, Naqvi QA, Hongo K. Scattering of electromagnetic plane wave by a circulardisk with surface impedance. Prog. Electromagnet. Res. 2012;127:501–522.
[30] Jafri ADU, Naqvi QA, Syed AA. Scattering of an arbitrarily oriented dipole field by a circu-lar disk with surface impedance. Prog. Electromagnet. Res. 2012;132:403–424.
[31] Jafri ADU, Naqvi QA, Syed AA, Hongo K. Scattering of electromagnetic plane wave by acircular disk on a grounded substrate. Can. J. Phys. 2012;90:1007–1015.
[32] Khanfir A, Faiz A, Ducourneau J, Chatillon J, Skali Lami SS. Scattered acoustic field abovea grating of parallel rectangular cavities. J. Sound Vibration. 2013;332:1047–1060.
[33] Krowne CM. Electromagnetic properties of nonreciprocal composite chiral-ferrite media.IEEE Trans. Antenn. Propag. 1993;41:1289–1295.
[34] Lakhtakia A. Beltrami fields in chiral media. Singapore: World Scientific; 1994.[35] Lakhtakia A, Varadan VK, Varadan VV. Time harmonic electromagnetic fields in chiral
media. Heidelberg: Springer; 1989.
Appendix 1Jacobi polynomial umn ðxÞ and vmn ðxÞ.[27]
umn ðxÞ ¼ffiffiffi2
pCðnþ 1ÞCðmþ 1ÞCðnþ mþ 1=2Þ x�m=2
Z 1
0
Jmðffiffiffix
pnÞJ2nþmþ1=2ðnÞffiffiffi
np dn
vmn ðxÞ ¼Cðnþ 1ÞCðmþ 1Þffiffiffi2
pCðnþ mþ 3=2Þ x
�m=2
Z 1
0
ffiffiffin
pJmð
ffiffiffix
pnÞJ2nþmþ3=2ðnÞdn
Orthogonal properties of Jacobi polynomial umn ðxÞ and vmn ðxÞ
