author copy

International Journal of Applied Electromagnetics and Mechanics 38 (2012) 181–193 DOI 10.3233/JAE-2012-1422 IOS Press

181

Focusing of electromagnetic field by a circular reflector coated with chiral medium Abdul Razzaq Farooqi∗ , Muhammad Junaid Mughal and Muhammad Qasim Mehmood

Y

GIK Institute of Engineering Sciences and Technology, Topi, District Swabi, Khyber Pakhtunkhwa, Pakistan

OP

Abstract. High frequency electromagnetic field expressions are analyzed for the focal region of a circular reflector coated with chiral medium. Maslov’s method has been used for the study of focal region fields because geometrical optics is an approximate method and fails at the focal points. The effects of chirality parameter, thickness of chiral coating and permittivity of the medium on the focused fields are observed. It is concluded from the results that focused field strength increases with increase in chirality parameter, relative permittivity and thickness of the chiral coating.

1. Introduction

OR C

Keywords: Circular reflector, chiral medium, perfect electric conductor, Maslov’s method, geometrical optics

AU

TH

The metallic and dielectric reflectors are widely used in imaging as optical devices, in laser cavities and for solar energy collection. Metal reflectors have the property to reflect light over a wide frequency range. However, due to certain amount of absorption, small amount of incident field is lost at high frequencies. These losses can be minimized by coating the reflectors with chiral medium as they have the property to control the losses via chirality parameter [1,2]. Chiral coatings have many potential applications in biomedicine, biochemistry and for obtaining data in the remote sensing of vegetation layers [3,4]. Chiral medium consists of geometrical structures that cannot be mapped onto their mirror images by any kind of rotation and translation. When an electromagnetic wave passes through chiral medium, it adapts to the handedness of chiral medium molecules. Such medium can be characterized by two intrinsic waves with left circular polarization (LCP) and right circular polarization (RCP), having different refraction indices and phase velocities. Scientists have made extensive efforts to study chiral medium, such as the wave properties and interaction, reflection and transmission through chiral interfaces, cross coupling of electric and magnetic fields and radiation characteristics of antennas placed in the chiral medium [5–13]. To analyze the focal region fields, Maslov’s method is used as Geometrical Optics (GO) approximation fails at focal points. This method has been used by many research groups for various focusing systems [14–21]. It uses the transformation of GO representation from space coordinates to a mixed coordinates system having position and wave vectors. The expression obtained is then converted to space coordinates by using Fourier transform. There have been a few chiral focusing studies [22–25]. Analysis ∗ Corresponding author: Abdul Razzaq Farooqi, GIK Institute of Engineering Sciences and Technology, FEE, GIKI, Topi, Swabi, KPK, 23640, Pakistan. Tel.: +92 3336396892; E-mail: [email protected].

1383-5416/12/$27.50  2012 – IOS Press and the authors. All rights reserved

182

A.R. Farooqi et al. / Focusing of electromagnetic field by a circular reflector coated with chiral medium

of focal region fields of chiral coated parabolic and the paraboloidal reflector has been done by [1,2]. In this paper, focal region fields of circular reflector coated with chiral medium are analyzed. This paper is organized in five sections. GO and Maslov’s method are discussed in Section 2. Section 3 covers the relation between the amplitudes of incident and reflected waves from the chiral coated perfect electric conductor (PEC) plane. In Section 4, GO fields for circular reflector coated with chiral medium are derived. The numerical results and the conclusions are presented in Sections 5 and 6 respectively. 2. Geometrical optics (GO) and Maslov’s method In order to study the geometrical optics (GO) and Maslov’s method, consider a wave equation (∇2 + ko2 )w(r) = 0

(1)

w(r) =

∞

S m (r)(jko )−m exp(−jko ϑ)

m=0

OP

Y

√ where r = (x, z), ∇2 = ∂ 2 /∂ 2 x + ∂ 2 /∂ 2 z and ko = ω o μo is the wavenumber of free space. The solution of Eq. (1) can be obtained in the form of Luneberg-Kline series as,

(2)

OR C

If we assume large values of ko , the higher order terms can be ignored, only retaining the first term. By substituting Eq. (2) in Eq. (1) and comparing the coefficients of ko2 and ko , the eikonal equation and transport equations are obtained as given by [26], (∇ϑ)2 = 1 2∇S.∇ϑ + S∇2 ϑ = 0

(3) (4)

dz = pz dt

dpx = 0, dt

dpz =0 dt

AU

dx = px , dt

TH

only S o has been retained and represented with S . Wave vector is defined as p = ∇ϑ and Hamiltonian H(r, p) = (p.p − 1)/2. So, the eikonal equation becomes H(r, p) = 0 and can be solved by the method of characteristic as, (5)

(6)

The solution of Eqs (5) and (6) are, x = ξ + px t, px = pxo ,

z = ζ + pz t

p z = p zo

(7) (8)

where (ξ, ζ ) is initial value of (x, z ) and (px0 , pz0 ) is the initial value of (px , pz ). The phase function φ is given as, t φ = φo + dt = φo + τ. (9) 0

183

OP

Y


Fig. 1. Reflection of electromagnetic waves from chiral coated PEC plane.

OR C

By the application of Gauss’s theorem to a paraxial ray tube, the solution of Eq. (4) is given by [26], w(r) = S(ξ)J −1/2 exp{−jko (φo + τ )}

(10)

TH

where S (ξ ) is initial value of field amplitude and J is the Jacobian of transformation from ray coordinates (ξ, ζ ) to the cartesian coordinate system (x, z ). The GO solution is not valid at focal points where J = 0, so Maslov’s method is used to find the fields around the focal region of a focusing system as analyzed in [15–25]. Thus, equation at the focal region of a circular reflector is given as [17], ∞ 1 ∂(x, pz ) −1/2 ko w(r) = So (ξ) j2π −∞ D(0) ∂(ξ, τ ) × exp(−jko φo + τ − zo pz + zpz )

AU

(11)

3. Reflection of electromagnetic waves from chiral coated PEC plane In order to obtain the focused field around the focal region of a circular reflector coated with chiral medium, consider the reflection of plane electromagnetic wave from chiral coated PEC plane as in [1,2, 6]. As Fig. 1 shows, the region z 0 consists of free space characterized by, D = o E ,

B = μo H

(12) √

o and μo are the permittivity and permeability of free space, with wavenumber ko = ω o μo and intrinsic impedance ηo = 377, while the region 0 z d consists of chiral coated PEC plane. Chiral medium is characterized by Drude-Born-Federov (DBF) constitutive relations as [6], D = (E + β∇ × E),

B = μ(H + β∇ × H)

(13)

184


where , μ and β are the permittivity, permeability and chirality parameter of the chiral medium respectively. The incident electric field may be written in free space as,

kox koz Einc = A⊥ ay + A − ax + az (14) exp(jkoz z + jkox x) ko ko and the reflected field can be written as,

kox koz Eref = B⊥ ay + B ax + az exp(−jkoz z + jkox x) ko ko

(15)

the field in chiral layer is expressed in the form of Beltrami fields as, Echiral = QL − jη QR

(16)

OP

where

Y

Hchiral = QR − jη −1 QL

k1z kox QL = A1 ay + j ax + az exp(−jk1z z + jkox x) k1 k1

k1z kox ax + az exp(jk1z z + jkox x) +B1 ay + j − k1 k1

OR C

kox k2z ax + az exp(−jk2z z + jkox x) k2 k2

k2z kox ax + az exp(jk2z z + jkox x) +B2 ay − j − k2 k2

QR = A2

ay − j

(17)

(18)

(19)

AU

TH

2 + k2 = k2 , k2 + k2 = k2 , k where k1z oz = ko cos γ and kox = ko sin γ . γ is the angle of incident ox ox 1 2z 2 and reflected waves in free space with z -axis. Applying the boundary conditions at z = 0 yields the following equations, B⊥ r11 r12 A⊥ T11 T12 A1 = + B r21 r22 A T21 T22 A2 R11 R12 A1 t11 t12 A⊥ D1 (20) = + D2 R21 R22 A2 t21 t22 A

Matrices elements are given in terms of Fresnel coefficients as, r11 = −[(ηo2 − η 2 )(ζ1 + ζ2 ) + 2ηo η(ζ1 ζ2 − 1)]/Δ r12 = 2jηo η(ζ1 − ζ2 )/Δ r21 = −r12 r22 = (ηo2 − η 2 )(ζ1 + ζ2 ) − 2ηo η(ζ1 ζ2 − 1)/Δ t11 = 2η(ηζ2 + ηo )/Δ


185

t12 = −2jη(ηo ζ2 + η)/Δ t21 = 2j(ηζ1 + ηo )/Δ t22 = −2(ηo ζ1 + η)/Δ R11 = (ηo2 + η 2 )(ζ1 − ζ2 ) + 2ηo η(ζ1 ζ2 − 1)/Δ R12 = −2jηζ2 (ηo2 − η 2 )/Δ R21 = 2jζ1 (ηo2 − η 2 )/ηΔ R22 = −(ηo2 − η 2 )(ζ1 − ζ2 ) + 2ηo η(ζ1 ζ2 − 1)/Δ T11 = 4ηo ζ1 (ηζ2 + ηo )/Δ T12 = −4jηηo ζ2 (ηζ1 + ηo )/Δ T22 = −4ηηo ζ2 (ηζ1 + ηo )/Δ

where Δ = (ηo2 + η 2 )(ζ1 + ζ2 ) + 2ηo η(ζ1 ζ2 + 1), 2

ko k1

1−

sin2 γ and ζ2 = secγ

1−

2 ko k2

B1 B2

where

A1 = [F ] R [F ] A2

exp (−jk1z d) [F ] = 0

and

R11 R = R21

where R11

=

1− 1−

R12 R22

ko k1

2 ko k1

0 exp (−jk2z d)

2

sin2 γ − 2

sin γ +

1− 1−

2jηk1 k2z k1 k2z − k2 k1z

(21)

(22)

AU

TH

z = d yields the following equations, 1 k2 k1z − k1 k2z A1 exp (−jk1z d) = −2j k2ηk1z A2 exp (−jk2z d) k2 k1z + k1 k2z B1 exp (jk1z d) × B2 exp (jk2z d)

or

sin2 γ . Applying boundary conditions at

OR C

ζ1 = secγ

OP

Y

T21 = 4jηo ζ1 (ηo ζ2 + η)/Δ

ko k2 ko k2

2

2

(23)

(24)

sin2 γ sin2 γ


OP

Y

186

Fig. 2. Circular reflector with chiral layer define by ζ = g(ξ).

= 1−

2 ko k1

1−

2

sin γ + 1−

R22

=

2

= R21

ko k2

sin2 γ

1−

ko k1

2

ko k2

2

sin2 γ

sin2 γ

−2j η 2 2 1 − kko1 sin2 γ + 1 − kko2 sin2 γ 1− 1−

2 ko k2

2 ko k1

2

sin γ − 2

sin γ +

1−

ko k1

2

sin2 γ

TH

R12

2jη

OR C

1−

ko k2

2

sin2 γ

AU

From Eqs (20) and (22), the reflection coefficients are found to be, −1 A⊥ B⊥ [t] = [r] + [T ] [F ] R [F ] − [R] B A

(25)

Using these reflection coefficients, the initial amplitude and initial phase are calculated for circular reflector coated with chiral medium in the next section. 4. Geometric optics field for the circular reflector coated with chiral medium Consider the reflection of an electromagnetic wave traveling along positive z -axis which is incident on a circular reflector as shown in Fig. 2. The circular reflector is characterized by the equation, ζ = a2 − ξ 2 (26)


187

Consider incident wave is traveling in z -axis which is given by, Einc = (Ay iy − Ax ix ) exp(−jko z)

(27)

and making angle γ with unit surface normal given by, an = sin γ ix + cos γ iz

(28)

where sin γ =

ξ , a

cos γ =

ζ a

Eref = {By iy + Bxz (− cos 2γ ix − sin 2γ iz )}

× exp{jko (xsin2γ + z cos 2γ)}

OP

The reflected wave vector is given as,

Y

The wave reflected from the reflector coated with chiral medium is given by,

p = − sin 2γ ix − cos 2γ iz

(29)

(30)

The Jacobian of transformation is given by J(τ ) = D(τ )/D(0), where 2τ a cos γ

OR C

D(τ ) = −1 +

The expression for Jacobian is, J(τ ) =

2τ D(τ ) =1− D(0) a cos γ

Φ = a cos γ + τ

(33)

exp(−jko a cos γ − jko τ )

(35)

z Eref

= By

2τ 1− a cos γ

= − sin 2γBxz

AU

The GO is also written in the form of rectangular components as follows, −1/2 2τ x Eref = − cos 2γBxz 1 − exp(−jko a cos γ − jko τ ) a cos γ y Eref

−1/2

(32)

TH

The phase function is,

(31)

2τ 1− a cos γ

−1/2

exp(−jko a cos γ − jko τ )

(34)

(36)

To find the field by Eq. (11), we need the expression given as, J

2 sin2 2γ ∂pz =− ∂z a cos γ

(37)

188


The phase function Ψ is given as, Ψ = 2a cos γ − x 1 − cos2 2γ

(38)

The field around focal region is, π/4 ko a √ x exp(jπ/4) Eref = cos 2γBxz cos γ π −π/4 × exp{jko ρ cos(2γ − θ) − 2jko a cos γ}dγ =

ko a exp(−jπ/4) π

π/4 −π/4

√ By cos γ

× exp{jko ρ cos(2γ − θ) − 2jko a cos γ}dγ z Eref

=

ko a exp(jπ/4) π

π/4 −π/4

√ sin 2γBxz cos γ

OP

Y

y Eref

(39)

× exp{jko ρ cos(2γ − θ) − 2jko a cos γ}dγ

5. Results and discussion

OR C

where x = ρ sin θ and z = ρ cos θ .

(40)

(41)

AU

TH

In this section, Eq. (39) to Eq. (41) are solved numerically for ko = 1, a = 1, μ = μo and different values of ε, d and β . The limits of integration are from γ = −π/4 to γ = π/4. Two type of polarization is considered for incident wave i.e., (Ay = 0, Ax = 1) and other is (Ay = 1, Ax = 0). The effect of chirality parameter β , relative permittivity ε and thickness d of chiral layer on the focal region fields of circular reflector is analyzed by the variation of these parameters. Figures 3(a)–(c) shows the effect of increase in chirality parameter β keeping d = 0.8 and ε = 3. It shows that increase in the value of β increases the focused field strength because the concentration of chiral molecules is increasing which results in the increased focused field strength. Figures 3(d)–(f) shows the effect of increase in relative permittivity ε while keeping β = 0.3 and d = 0.2. The figure indicates that increase in the value√of ε increases the focused field intensity. It is due to the fact that impedance η is proportional to 1/ ε. As ε increases, the value of η decreases and it results in the increased focused field strength. Figure 4 shows the effect of chiral layer thickness d on the focused field intensity. Both type of polarization of incident wave is considered to observe the cross polarization effect. It is evident that increasing the value of d keeping β = 0.5 and ε = 1, focused field intensity increases gradually due to x | and |E z | are zero when thickness d increased number of chiral molecules. The focused fields |Eref ref y of chiral layer is zero for (Ay = 1, Ax = 0) and |Eref | is also zero for (Ay = 0, Ax = 1). Next considering the case when chirality parameter β = 0 i.e., circular reflector is coated with dielectric medium. Figures 5(a)–(c) shows the effect when value of d is increased keeping β = 0 and ε = 3. It indicates that increase in the value of d increases the focused field strength. Figure 5(d)–(f) shows the effect when ε of dielectric layer is increased keeping β = 0 and d = 0.5. It is evident that focused field strength increases with increase in value of relative permittivity ε of the dielectric coating.

189

AU

TH

OR C

OP

Y


x Fig. 3. Chirality parameter β and relative permittivity ε effect on the focused EM fields (a) Effect of β on |Eref | (b) Effect of y y z x z | (c) Effect of β on |Eref | (d) Effect of ε on |Eref | (e) Effect of ε on |Eref | (f) Effect of ε on |Eref |. β on |Eref


AU

TH

OR C

OP

Y

190

Fig. 4. Chiral layer thickness d effect on the focused EM fields (a) y-polarized incident wave (b)x-polarized incident wave (c) x-polarized incident wave (d)y-polarized incident wave (e) y-polarized incident wave (f) x-polarized incident wave.

191

AU

TH

OR C

OP

Y


x Fig. 5. Dielectric layer thickness d and relative permittivity ε effect on the focused EM fields (a) Effect of d on |Eref | (b) Effect y y z x z | (c) Effect of d on |Eref | (d) Effect of ε on |Eref | (e) Effect of ε on |Eref | (f) Effect of ε on |Eref |. of d on |Eref

192


6. Conclusions The geometrical optics (GO) fields focused from a circular reflector coated with chiral medium are obtained using Maslov’s method. It is observed that increasing the chirality parameter β and relative permittivity ε of chiral layer, the focused field strength increases. Cross polarization effect is observed for both type of polarization of incident wave and field strength increases with increase in the value of d. As a special case, dielectric layer coating is discussed by assuming chirality parameter β = 0. It is observed that focused field intensity increases with increase in thickness d and relative permittivity ε of the dielectric layer. References

[5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

Y

OP

[4]

OR C

[3]

TH

[2]

M. Faryad and Q.A. Naqvi, High frequency expression for the field in the caustic region of a parabolic reflector coated with isotropic chiral medium, J of Electromagn Waves and Appl 22 (2008), 965–986. T. Rahim, M.J. Mughal, Q.A. Naqvi and M. Faryad, Focal region field of a paraboloidal reflector coated with isotropic chiral medium, Progress In Electromagnetics Research 94 (2009), 351–366. E. Bahar, Muller matrices for waves reflected and transmitted through chiral materials: waveguide modal solutions and applications, J Opt Soc Am B 24 (2007), 1610–1619. H. Cory and I. Rosenhouse, Electromgnetic wave reflection and transmission at a chiral-dielectric interface, Journal of Modern Optics 38 (1991), 1229–1241. S. Bassiri, C.H. Papas and N. Engheta, Electromagnetic wave propagation through a dielectric-chiral interface and through a chiral slab, J Opt Soc Am A 5 (1988), 1450–1459. A. Lakhtakia, Beltrami fields in chiral media, Contemporary Chemical Physics, World Scientific Series, 1994. A. Lakhtakia, V.K. Varadan and V.V. Varadan, Eds, Time Harmonic Electromagnetic Fields in Chiral Media, Springer, Berlin, 1989. D.L. Jaggard, X. Sun and N. Engheta, Canonical sources and duality in chiral media, IEEE Trans Antennas Propag 36 (1988), 1007–1013. D.L. Jaggard A.R. Mickelson and C.H. Papas, On electromagnetic waves in chiral media, Appl Phys 18 (1978), 211–216. I.V. Lindell, A.H. Sihvola, S.A. Tretyakov and A.J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media, Artech House, MA, 1994. C.-W. Qiu and H.Y. Yao, N. Burokur, S. Zouhdi and L.W. Li, Electromagnetic scattering properties in a multilayered metamaterial cylinder, IEICE Trans Commun E 90-B(9) (2007), 2423–2429. C.-W. Qiu, H.Y. Yao, L. Li, T.S. Yeo and S. Zouhdi, Routes to left-handed media by magnetoelectric couplings, Phys Rev B 75 (2007), 245214. C.-W. Qiu, L.-W. Li, H.-Y. Yao and S. Zouhdi, Properties of Faraday chiral media: Green dyadics and negative refraction, Phys Rev B 74 (2006), 115110. G.A. Dechamps, Ray techniques in electromagnetics, Proc IEEE 60 (1972), 1022–1035. R.W. Ziolkowski and G.A. Deschamps, Asymptotic evaluation of high frequency field near a caustic: An introduction to Maslov’s method, Radio Sci 19 (1984), 1001–1025. A. Ghaffar, Q.A. Naqvi and K. Hongo, Analysis of the fields in three dimensional Cassegrain system, Progress In Electromagnetics Research 72 (2007), 215–240. K. Hongo, Y. Ji and E. Nakajima, High frequency expression for the field in the caustic region of a reflector using Maslov’s method, Radio Sci 21 (1986), 911–919. K. Hongo and Y. Ji, High frequency expression for the field in the caustic region of a cylindrical reflector using Maslov’s method, Radio Sci 22 (1987), 357–366. A. Ghaffar, A. Hussain, Q.A. Naqvi and K. Hongo, Radiation characteristics of an inhomogeneous slab using Maslov’s method, J of Electromagn Waves and Appl 22 (2008), 301–312. A. Aziz, A. Ghaffar, Q.A. Naqvi and K. Hongo, Analysis of the fields in two dimensional Gregorian system, J of Electromagn Waves and Appl 22 (2008), 85–97. A. Ghaffar, A. Rizvi and Q.A. Naqvi, Field in the focal space of symmetrical hyperboloidal focusing lens, Progress In Electromagnetics Research 89 (2009), 255–273. T.M. Kayani, M.Q. Mehmood, M.J. Mughal and T. Rahim, Analysis of the field focused by hyperbolic lens embedded in chiral medium, Progress In Electromagnetics Research M 20 (2011), 43-56. M.Q. Mehmood and M.J. Mughal, Analysis of focal region fields of PEMC gregorian system embedded in homogenous chiral medium, Progress In Electromagnetics Research Letters 18 (2010), 155–163.

AU

[1]


Y OP OR C

[26]

TH

[25]

M.Q. Mehmood, M.J. Mughal and T. Rahim, Focal region fields of gregorian system placed in homogenous chiral medium, Progress In Electromagnetics Research M 11 (2010), 241–256. M.Q. Mehmood, M.J. Mughal and T. Rahim, Focal region fields of cassegrain system placed in homogenous chiral medium, Progress In Electromagnetics Research B 21 (2010), 329–346. C.A. Balanis, Advanced Engineering Electomagnetics, John Wiley and Sons, 1989.

AU

[24]

193