Electromagnetic fields in a circular waveguide with D'B ... - IOS Press

International Journal of Applied Electromagnetics and Mechanics 40 (2012) 301–308 DOI 10.3233/JAE-2012-1593 IOS Press

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Electromagnetic fields in a circular waveguide with D’B’-boundary conditions internally coated with chiral-nihility medium Muhammad Haseeb Hassan∗ , Muhammad Junaid Mughal and Abdul Razzaq Farooqi GIK Institute of Engineering Sciences and Technology, Khyber Pakhtunkhwa, Pakistan Abstract. The present study deals with the propagation of electromagnetic fields and power in a circular waveguide, with D’B’boundary conditions, internally coated with chiral-nihility medium. The electromagnetic fields inside the two layer circular waveguide defined by D’B’-boundary conditions, with one layer containing free space while other layer containing chiralnihility medium, is investigated. D’B’-boundary conditions require vanishing of derivatives of the normal field components of D and B vectors, hence named as D’B’-boundary conditions. It has been investigated that unlike to the chiral-nihility coated perfect electromagnetic conductor (PEC) waveguide, there is a non-zero power transfer in chiral-nihility region in waveguide with D’B’ boundary conditions. Keywords: D’B’ boundary conditions, chiral-nihility medium, circular waveguide

1. Introduction In electromagnetics, the boundary-value problems involve linear relations between electric and magnetic field components that are tangential to the boundary surface. The general form for such boundary conditions can be represented as [1]. ˆr × H = Y¯s · E ˆar × E = Z¯s · H, , a (1)

where ˆar is the unit vector normal to the boundary and Z¯s and Y¯s are the surface-impedance and surfaceadmittance dyadics, respectively. These boundary conditions are called impedance boundary conditions. Perfect electric conductor (PEC) and perfect magnetic conductor (PMC) are the typical examples of impedance boundary conditions and they can be represented as [1,2]. ˆr × E = 0, i.e., Z¯s = 0, PEC : a (2) PMC :

aˆr × H = 0, i.e., Y¯s = 0

(3)

A perfect electromagnetic conductor (PEMC) boundary is a more general interface and is defined as 1 ¯ i.e., a ˆr × (M E + H) = 0 PEMC : Z¯s = aˆr × I, (4) M ∗ Corresponding author: Muhammad Haseeb Hassan, GIK Institute of Engineering Sciences and Technology, FEE, GIKI, Topi, Swabi, KPK, 23640, Pakistan. Tel.: +92 00923348518233; E-mail: [email protected].

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

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M.H. Hassan et al. / Electromagnetic fields in a circular waveguide with D’B’-boundary conditions

where M is the (PEMC) admittance [3]. For M = 0 and 1/M = 0, the above relation reduces to the (PEC) and (PMC) cases, respectively. Lindell proposed boundary conditions which deal with the vanishing of derivatives of the normal components of electric and magnetic flux densities. ˆ r · B = 0 ˆar · D = 0 , a

(5)

These conditions are termed as D’B’-boundary conditions [4]. The corresponding conditions for E and H vectors depend on the medium in front of the boundary. It was also investigated that D’B’-boundary behaves as (PMC) for fields polarized T E z and (PEC) for fields polarized T M z with respect to the normal z direction [4]. D’B’-boundary conditions can be realized by a waveguiding medium with normal permittivity and permeability parameters to be zero [5]. The planar DB-boundary placed in chiral and chiral-nihility medium have been discussed by Naqvi et al. [6]. These boundary conditions proved invaluable for the construction of electromagnetic cloaking structures [7–9]. It has also been shown that objects defined by D’B’-boundary conditions and possessing certain symmetry cannot be seen by the radar i.e., they have zero backscattering [10]. Moreover, DBboundary finds applications as coupling between aperture antennas on a DB plane is smaller than on a (PEC) plane [7,8]. Chiral medium was discovered in the beginning of the 19th century and has attracted many researchers and scientists in recent years. Chiral medium is well-known for its circular dichroism and optical activity i.e., it can rotate the polarization vector of an incident linearly polarized wave when propagating through chiral medium [11–14]. Chiral media are characterized by two intrinsic eigenwaves, with one having lefthanded circular polarization while other having right-handed circular polarization [13,15,16]. Both the waves travel with different phase velocities and have different refractive indices. Chiral-nihility medium, which is an extension of Lakhtakia’s concept of nihility [17–21], is a special case of chiral medium in which the permittivity and permeability approaches zero simultaneously while the chirality parameter is non-zero for certain frequency called the nihility frequency [22,23]. Chiral-nihility made its mark as a material which supports negative reflection and negative refraction [23–30]. Chiral-nihility medium got particular attention when it was discovered that chiral-nihility medium backed by (PEC) interface produces a backward wave as reflected wave which cancels out the incident wave [24], hence, there is null power propagation in chiral-nihility medium backed by (PEC). Waveguides containing chiral and chiral-nihility media have been investigated extensively because of their potential applications in optics and microwaves [31–34]. The constitutive relations of G-chiral media and the realization of negative index media have also been discussed in literature [35]. Qiu et al. investigated realization of chiralnihility medium using advanced approaches i.e., nonreciprocity route and gyrotropic route [36–38]. Circular waveguide with DB boundary conditions has been investigated by Lindell et al. [39]. Hassan et al. investigated circular waveguide defined by DB bounday conditions internally coated with chiral-nihility medium [40]. In present communication circular waveguide, defined by D’B’-boundary conditions, internally coated with chiral-nihility medium is investigated. In Section 2, expressions for electromagnetic fields are derived in free space and chiral-nihility medium inside the waveguide. In Section 3, the unknown coefficients are calculated by imposing D’B’ boundary conditions . It is investigated that in a circular waveguide, internally coated with chiral-nihility medium, defined by D’B’-boundary conditions both the electric and magnetic fields exist in free space as well as in chiral-nihility medium. Hence, there is non-zero power propagation in nihility as well as in non-nihility regions inside the waveguide.

M.H. Hassan et al. / Electromagnetic fields in a circular waveguide with D’B’-boundary conditions

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2. Formulations Consider a circular D’B’ waveguide of radius b, which is of infinite extent. The space inside the waveguide is divided into two circular regions, i.e., r  a filled with free space and a  r  b filled with chiral-nihility medium. The z -axis of the cylindrical co-ordinate system coincide with the axis of the circular waveguide. The waveguide is shown in Fig. 1. 2.1. Fields in free space The free space region is defined by constitutive parameters μ0 and 0 . The wave is assumed to propagate in the positive z direction and the field component varies as exp (−jγz). The time dependency exp (jωt) is suppressed through out the discussion. Assume a T E z mode is excited in region r  a, the solution to the wave equation in this region is written as Ez0 = 0

(6)

Hz0 = Fm Jm (k0r r) exp (jmθ) (7) jω0  Eθ0 = Fm Jm (k0r r) exp (jmθ) (8) k0r γm Hθ0 = Fm Jm (k0r r) exp (jmθ) (9) k0r ω0 Er0 = Fm Jm (k0r r) exp (jmθ) (10) k0r −jω  Hr0 = Fm Jm (k0r r) exp (jmθ) (11) k0r  √ where, k0r = k02 − γ 2 and k0 = ω μ0 0 . Jm (.) is Bessel function of order m and prime denotes the derivative with respect to the argument.

2.2. Fields in chiral-nihility medium The chiral-nihility medium is defined by constitutive parameters ( = 0, μ = 0, κ = 0) having following constitutive relations √ (12) D = −jκ 0 μ0 H √ B = jκ 0 μ0 E (13) In chiral-nihility medium, one right circularly polarized (RCP+) and other left circularly (LCP−) polarized waves propagate with wavenumbers k± = ±κk0 , respectively. The fields in chiral-nihility medium can be decomposed as E = E+ + E− j H = (E+ − E− ) η

(14) (15) 

μ is the wave impedance of chiral-nihility medium. In the chiral-nihility  medium, the longitudinal components of the electromagnetic fields E± can be written as

where, η = lim→0, μ→0

E+z1 = A1m Jm (kr+ r) exp(jmθ) + A2m Ym (kr+ r) exp(jmθ)

(16)

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M.H. Hassan et al. / Electromagnetic fields in a circular waveguide with D’B’-boundary conditions

E−z1 = B1m Jm (kr− r) exp(jmθ) + B2m Ym (kr− r) exp(jmθ) (17)   2 − γ2 = k 2 2 where, kr± = k± 0 κ − (γ/k0 ) = kr . The transverse components can be written as   jγ  jmκk0 Er1 = (A1m + B1m ) Jm (kr r) − Jm (kr r) exp(jmθ) kr2 r kr   jγ  jmκk0 +(A2m + B2m ) Ym (kr r) − Ym (kr r) exp(jmθ) (18) kr2 r kr   jmκk0 j jγ  Jm (kr r) − Jm (kr r) exp(jmθ) Hr1 = (A1m − B1m ) η kr2 r kr   jγ  jmκk0 j Ym (kr r) − Ym (kr r) exp(jmθ) (19) + (A2m − B2m ) η kr2 r kr   κk0  γm Eθ1 = (A1m + B1m ) J (kr r) exp(jmθ) Jm (kr r) − kr2 r kr m   κk0  γm +(A2m + B2m ) Ym (kr r) − Y (kr r) exp(jmθ) (20) kr2 r kr m   j κk0  γm Jm (kr r) − Hθ1 = (A1m − B1m ) J (kr r) exp(jmθ) η kr2 r kr m   κk0  γm j Ym (kr r) − Y (kr r) exp(jmθ) (21) + (A2m − B2m ) η kr2 r kr m

Total axial fields in the chiral-nihility region are Ez1 = (A1m + B1m )Jm (kr r) exp(jmθ) +(A2m + B2m )Ym (kr r) exp(jmθ) j Hz1 = (A1m − B1m )Jm (kr r) exp(jmθ) η j − (A2m − B2m )Ym (kr r) exp(jmθ) η

(22)

(23)

3. D’B’ waveguide As the wall of the guide located at r = b is a D’B’ boundary. By applying the boundary conditions at the interfaces r = a and r = b, unknown coefficients can be obtained. The boundary conditions are  Dr1 = 0,

 Br1 = 0,

r=b

(24)

Dr0 = Dr1 , Br0 = Br1 r = a

(25)

Now at the interface of D’B’ boundary and chiral-nihility medium i.e., at r = b  =0 Dr1

√  −jκ 0 μ0 Hr1 =0

(26) (27)

M.H. Hassan et al. / Electromagnetic fields in a circular waveguide with D’B’-boundary conditions

Fig. 1. Circular waveguide, internally coated with chiral-nihility medium, defined by D’B’ boundary conditions.

305

Fig. 2. T E01 mode in circular waveguide.

 Hr1 =0

(28)

similarly,  Er1 =0

(29)

Moreover, at the free space and chiral nihility interface i.e., at r = a Dr0 = Dr1

(30)

√ 0 Er0 = −jκ 0 μ0 Hr1

(31)

similarly,

√ μ0 Hr0 = jκ 0 μ0 Er1

(32)

Using Eqs (18) and (19) in Eqs (28) and (29) we get following equations c12 (A1m + B1m ) = −(A2m + B2m ) c11 c12 (A1m − B1m ) = −(A2m − B2m ) c11 where,  jγ  jmκk0   c11 = 2 2 bJm (kr b) − Jm (kr b) − Jm (kr b) kr b kr  jγ jmκk0  c12 = 2 2 bYm (kr b) − Ym (kr b) − Ym (kr b) kr b kr Similarly, using Eqs (10), (11), (18) and (19) in Eqs (31) and (32) we get −ωμ0  (A1m + B1m )d11 + (A2m + B2m )d12 = Fm Jm (k0r a) √ k0r κ 0 μ0 (A1m − B1m )d11 + (A2m − B2m)d12 =

ηω20 Fm Jm (k0r a) √ k0r κ 0 μ0

(33) (34)

(35) (36)

(37) (38)

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where, jmκk0 Jm (kr a) − kr2 a jmκk0 d12 = Ym (kr a) − kr2 a d11 =

jγ  J (kr a) kr m jγ  Y (kr a) kr m

(39) (40)

By solving Eqs (33), (34), (37) and (38) simultaneously yield the following results for the unknown coefficients   d00 c12 ω(μ0 − η20 ) A1m = Fm (41) √ 2κk0r μ0 0 d12 c11 − c12 d11   ω(μ0 + η20 ) d00 c12 B1m = Fm (42) √ 2κk0r μ0 0 d12 c11 − c12 d11   ω(η20 − μ0 ) d00 c11 A2m = Fm (43) √ 2κk0r μ0 0 d12 c11 − c12 d11   −ω(η20 + μ0 ) d00 c11 B2m = Fm (44) √ 2κk0r μ0 0 d12 c11 − c12 d11 where,  d00 = Jm (k0r a) − Jm (k0r a)

(45)

Relationship between unknown coefficients shows that both the electric and magnetic fields are nonzero in both free space and chiral-nihility regions inside the waveguide. The Fig. 2 shows the T E01 mode of electric field inside the waveguide at frequency 5 GHz. It can be observed that in free space region the magnitude of electric field intensity is small as compared to that of chiral-nihility medium. The interesting result is that electric field intensity in chiral-nihility medium with D’B’ waveguide is non-zero unlike to that of chiral nihility coated waveguide with PEC walls in which electric field intensity is zero in chiral-nihility medium. The energy flux along z -axis, inside the waveguide, may be calculated using the following expression 1 Sz = Re(E × H∗ ) · zˆ 2 (46) 1 = Re(Er Hφ∗ − Eφ Hr∗ ) 2 Similarly, power can be calculated using the integration of energy flux. It is obvious that there is non-zero power propagation in chiral-nihility region. 4. Conclusions The expressions for electromagnetic fields inside a circular waveguide, internally coated with chiralnihility medium, defined by D’B’ boundary conditions have been derived. It has been shown that the electric and magnetic fields inside the waveguide in both free space and chiral-nihility regions are nonzero. Hence, it is concluded that there is non-zero power propagation both in the nihility and non-nihility regions.

M.H. Hassan et al. / Electromagnetic fields in a circular waveguide with D’B’-boundary conditions

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