International Journal of Applied Electromagnetics and Mechanics 38 (2012) 39–45 DOI 10.3233/JAE-2011-1407 IOS Press
39
Analysis of caustic region fields of a cassegrain system having PEMC reflectors embedded in homogeneous chiral medium Muhammad Qasim Mehmood∗, Muhammad Junaid Mughal and Tarim Rahim
Y
GIK Institute of Engineering Sciences and Technology, Khyber Pakhtunkhwa, Pakistan
OR C
OP
Abstract. Analysis of high frequency field expressions for the Cassegrain system having perfect electromagnetic conducting (PEMC) boundaries are studied. Both the reflectors of Cassegrain system are PEMC and are embedded in chiral medium. Two different cases have been analyzed depending upon the values of chirality parameter (kβ). In the first case, chiral medium supports positive phase velocity (PPV) for both the left circularly polarized (LCP) and right circularly polarized (RCP) modes. In the second case, chiral medium supporting PPV for one mode and negative phase velocity (NPV) for the other mode is taken into account. Mathematical recipe proposed by the Maslov is used to obtain the field expressions in the focal region because geometrical optics (GO) causes non-realistic singularity in this region. Field patterns of focal plane are given in the paper for different values of admittance (M ) of the PEMC boundary and the chirality parameter (kβ). Keywords: Cassegrain system, chiral medium, perfect electromagnetic conductor, Maslov’s method and geometrical optics
1. Introduction
TH
Perfect electric conductor (PEC) and perfect magnetic conductor (PMC) medium can be generalized to a PEMC medium. The boundary conditions for the PEC and PMC are given by the following expressions [1,2] n·B=0
(P EC)
(1a)
n × H = 0,
n·D=0
(P M C)
(1b)
AU
n × E = 0,
where, n denotes the unit vector normal to the surface of boundary. General form of the PEMC boundary conditions is as following n × (H + M E) = 0,
n · (D − M B) = 0
(P EM C)
(2)
where M denotes the admittance of the PEMC boundary. PMC corresponds to M = 0, while PEC is obtained as the limit M → ±∞ [3,4]. In this paper our goal is to find the field expressions in the focal region of Cassegrain system having PEMC boundary, while the whole system is embedded in a chiral medium. Chiral medium is a composite of uniformly distributed and randomly oriented chiral objects. ∗ Corresponding author: Muhammad Qasim Mehmood, GIK Institute of Engineering Sciences and Technology, FEE, GIKI, Topi, Swabi, KPK, 23640, Pakistan. Tel.: +92 6594694338; E-mail:
[email protected].
1383-5416/12/$27.50 2012 – IOS Press and the authors. All rights reserved
M.Q. Mehmood et al. / Analysis of caustic region fields of a cassegrain system having PEMC reflectors
OP
Y
40
OR C
Fig. 1. PEMC Cassegrain reflector in chiral medium, kβ < 1 and kβ > 1.
AU
TH
Chiral object is a three-dimensional body that cannot be brought into congruence with its mirror image by any translation or rotation. Collection of such chiral objects will form a medium that is characterized by a right or a left handedness. A linearly polarized wave incident upon such a medium will split into two waves, a left-circularly polarized wave and a right circularly polarized one [5–7]. Chiral medium has many advantages over an ordinary medium like polarization control, impedance matching and cross coupling of electric and magnetic fields. By changing the chiral medium parameters , μ and kβ the desirable values of the wave impedance and propagation constants can be achieved by which reflections can be adjusted (decreased or increased). In this respect, a chiral medium can be controlled by the variation of three parameters , μ, kβ , whereas an achiral medium has only two parameters, , μ, that can be varied. Moreover, in the case of negative reflection caused by NPV, it also gives the advantage of invisibility [8,9]. Due to these unique characteristics of chiral medium, we have embedded the PEMC Cassegrain system in chiral medium. Maslov’s method is used to study the fields at the focal regions. It combines the simplicity of asymptotic ray theory and the generality of the Fourier transform method. This is achieved by representing the geometrical optics fields in hybrid coordinates consisting of space coordinates, and wave vector coordinates, that is by representing the field in terms of six coordinates [10,11]. Analysis of focusing systems has been worked out by various authors using Maslov’s method [12–22]. 2. Geometrical optics fields of two dimensional PEMC Cassegrain system embedded in chiral medium Reflection of plane waves from PEMC boundary placed in chiral medium is considered. PEMC Cassegrain system placed in chiral medium is shown in Fig. 1 (gray lines show the negative reflected waves). When both RCP and LCP waves will hit on main PEMC parabolic reflector, it will cause four
M.Q. Mehmood et al. / Analysis of caustic region fields of a cassegrain system having PEMC reflectors
41
reflected waves designated as LL, RR, LR and RL [12]. These four waves are then incident on the PEMC hyperbolic subreflector and will cause eight reflected waves. Only four of these rays (LLL, RRR, LLR, RRL) converge in the focal region. These rays are shown in Fig. 1. For, kβ > 1, LCP wave travels with NPV and RCP wave with PPV. Reflected rays in this case are shown in Fig. 1 by gray lines. Only two rays (LLL and RRR) rays are contributing to the focal plane while RRL and LLR are divergent (shown by gray lines). Geometrical optics fields of these reflected waves have been calculated for PEC Cassegrain system in [8]. For the case of PEMC, LLR and RRL waves will have different initial amplitudes as compared with PEC, while LLL and RRR waves have the same initial amplitudes. Expressions for the initial amplitudes of all the four converged rays are given as following. cos α − cos α2 cos γ − cos γ2 A0LLL = (3a) cos α + cos α2 cos γ + cos γ2
A0RRL = A0LLR =
cos α − cos α1 cos α + cos α1 cos α − cos α2 cos α + cos α2
cos γ − cos γ1 cos γ + cos γ1 Mη − j Mη + j
Y
OP
A0RRR
cos α − cos α1 = cos α + cos α1
2 cos γ cos γ + cos γ1
2 cos γ 1 − jM η − 1 + jM η cos γ + cos γ2
and the corresponding initial phases are cos 2α −c S0LLL = −n1 ζ1 = n1 2f 1 + cos 2α
(3c)
(3d)
(4a)
(4b)
cos 2α −c 2f 1 + cos 2α
(4c)
cos 2α −c 2f 1 + cos 2α
(4d)
S0RRL = −n2 ζ1 = n2
AU
S0RRR = −n2 ζ1 = n2
TH
cos 2α −c 2f 1 + cos 2α
S0LLR = −n1 ζ1 = n1
(3b)
OR C
while the extra terms of the phase for all four rays are as following SexLLL = −n1 [x sin(2α − 2ψ) + z cos(2α − 2ψ)] + n1 [ξ2 sin(2α − 2ψ) + ζ2 cos(2α − 2ψ)] SexRRR = −n2 [x sin(2α − 2ψ) + z cos(2α − 2ψ)] +n2 [ξ2 sin(2α − 2ψ) + ζ2 cos(2α − 2ψ)]
(5a)
(5b)
M.Q. Mehmood et al. / Analysis of caustic region fields of a cassegrain system having PEMC reflectors
AU
TH
OR C
OP
Y
42
Fig. 2. |URRL | and |ULLR | of PEMC Cassegrain system at kx = 0 for kβ = 0, 0.001, 0.005, and 0.01 for: (a,e) M η = 0 (b,f) M η = 1 (c,g) M η = 5 (d,h) M η = 10.
M.Q. Mehmood et al. / Analysis of caustic region fields of a cassegrain system having PEMC reflectors
SexRRL = −n1 [x sin(γ1 − ψ) + z cos(γ1 − ψ)]
(5c)
+n1 [ξ2 sin(γ1 − ψ) + ζ2 cos(γ1 − ψ)] SexLLR = −n2 [x sin(γ2 − ψ) + z cos(γ2 − ψ)]
(5d)
+n2 [ξ2 sin(γ2 − ψ) + ζ2 cos(γ2 − ψ)]
and t1 =
(ξ2 − ξ1 )2 + (ζ2 − ζ1 )2 ,
t=
43
(x − ξ2 )2 + (z − ζ2 )2
(6)
k 2jπ
U (r)RRR =
A2
A1
+
−A2
−A1
OR C
OP
Y
Where (ξ1 , ζ1 ) and (ξ2 , ζ2 ) are the cartesian coordinates of the points on the parabolic and hyperbolic reflectors, respectively. Finite GO fields of these reflected rays around the focal point, not repeating the calculations [8], are as following. A2 −A2 k U (r)LLL = + A0LLL R1 2jπ A1 −A1 (7a) × exp[−jk{S0LLL + n1 t1 + SexLLL }]d(2α)
A0RRR
R1
(7b)
× exp[−jk{S0RRR + n2 t1 + SexRRR }]d(2α) k 2jπ
A2 A1
+
−A1
⎡ ×⎣
−A2
A0RRL (ξ)
TH
U (r)RRL =
1
n21 − n22 sin2 γ ⎤−1/2
R1 R2 bn2 cos γ1
⎦ abn2 + a (R1 R2 )(n21 − n22 sin2 γ) − bn2 R1
(7c)
k 2jπ
U (r)LLR =
AU
× exp[−jk{S0RRL + n1 t1 + SexRRL }]d(2α) A2
A1
+
−A2
−A1
A0LLR (ξ)
1 n22 − n21 sin2 γ
⎡ ×⎣
⎤−1/2
R1 R2 bn1 cos γ2 ⎦ 2 2 2 abn1 + a (R1 R2 )(n2 − n1 sin γ) − bn1 R1
× exp[−jk{S0LLR + n2 t1 + SexLLR }]d(2α)
while S0 , Sex and t1 are given in Eqs (4a–4d), Eqs (5a–5d) and Eq. (6) respectively.
(7d)
44
M.Q. Mehmood et al. / Analysis of caustic region fields of a cassegrain system having PEMC reflectors
3. Results and discussions Magnitude variations of fields are shown in Fig. 2(a-h). Values for different parameters of PEMC Cassegrain reflector are: kf = 170, ka = 40, kb = 60, kd = 50, kD = 150. Plots for ULLL and URRR are exactly same as that of PEC Cassegrain system. Plots of URRL and ULLR for kβ = 0,0.001,0.005,0.01 and M η = 0,1,5,10 are given in Fig. 2(a-h). It shows that the trend of the plots is same as in the case of PEC Cassegrain system, i.e., by increasing the value of kβ , gap between the focal points of RRL and LLR rays increases. For kβ > 1, LLL and RRR rays are similar to that of PEC Cassegrain sysytem, while RRL and LLR rays diverge. 4. Conclusion
References
[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
TH
[2]
I.V. Lindell and A.H. Sihola, Transformation method for problems involing PEMC structures, IEEE Trans on Ant and Propag 53 (2005), 3005–3011. I.V. Lindell and A.H. Sihvola, Perfect electromagnetic conductor, Journal of Electromagnetic Waves and Applications 19(7) (2005), 861–869. M.A. Fiaz, A. Aziz, A. Ghaffar and Q.A. Naqvi, High frequency expression for the field in the caustic region of a PEMC Gregorian system using Maslov’s method, Progress In Electromagnetics Research, PIER 81 (2008), 135–148. M.Q. Mehmood and M.J. Mughal, Analysis of focal region fields of PEMC gregorian system embedded in homogeneous chiral medium, Progress In Electromagnetic Research Letters 18 (2010), 155–163. H. Cory and I. Rosenhouse, Electromagnetic Wave Reflection and Transmission at a Chiral-dielectric Interface, Journal of Modern Optics 38(7) (1991), 1229–1241. A. Lakhtakia, M.W. McCall and W.S. Weiglhofer, Brief Overview of Recent Developments on Negative Phase-Velocity Mediums (alias Left-Handed Materials), AEU – International Journal of Electronics and Communications 56(Issue 6) (2002), 407–410. A. Lakhtakia, V.V. Varadan and V.K. Varadan, What happens to plane waves at the planar interfaces of mirror conjugated chiral media, Journal of the Optical Society of America A: Optics, Image Science, and Vision 6(1) (January 1989), 2326. M.Q. Mehmood, M.J. Mughal and T. Rahim, Focal region fields of Cassegrain system embedded in chiral medium, Progress In Electromagnetic Research B 21 (2010), 329–346. M.Q. Mehmood, M.J. Mughal and T. Rahim, Focal region fields of Gregorian system embedded in chiral medium, Progress In Electromagnetic Research M 11 (2010), 241–256. V.P. Maslov, Perturbation theory and asymptotic methods, Izdat Moskov Gos Univ, Moscow, 1965 (in Russian). Translated into Japanese by Ouchi et al., Iwanami. V.P. Maslov and V.E. Nazaikinski, Asymptotics of operator and pseudo-differential equations, Consultants Bureau, N.Y., 1988. M. Faryad and Q. A. Naqvi, High frequency expression for the field in the caustic region of cylindrical reflector placed in chiral medium, Progress In Electromagnetics Research, PIER 76 (2007), 153–182. A. Ghaffar, Q.A. Naqvi and K. Hongo, Analysis of the fields in three dimensional Cassegrain system, Progress In Electromagnetics Research, PIER 72 (2007), 215–240.
AU
[1]
OR C
OP
Y
Geometrical optics fields focused by the Cassegrain system having PEMC boundary and embedded in homogeneous chiral medium are analyzed. It is seen that all the reflected and focused rays (LLL, RRR, RRL and LLR) are same as that of PEC Cassegrain system [8]. However, RRL and LLR rays are different in amplitude than PEC case as given in Eqs (3c) and (3d). Both the cases of strong (kβ > 1) and week (kβ < 1) chiral medium are taken into account. It is found that for both the strong and weak chiral medium LLL and RRR rays do not change their caustic point. However, for (kβ < 1) gap between the focal points of LLR and RRL rays increases by increasing the chirality parameter, while for (kβ > 1) the focal points of LLR and RRL rays diverge and do not from a real focuss.
M.Q. Mehmood et al. / Analysis of caustic region fields of a cassegrain system having PEMC reflectors
[19] [20] [21] [22]
Y
[18]
OP
[17]
OR C
[16]
TH
[15]
A. Hussain, Q.A. Naqvi and K. Hongo, Radiation characteristics of the wood lens using Maslov’s method, Progress In Electromagnetics Research, PIER 73 (2007), 107–129. M. Waqas, M. Faryad and Q.A. Naqvi, Analysis of high frequency field of Perfect Electromagnetic Conductor (PEMC) parabolic reflector placed in homogeneous chiral medium, Journal of Electromagnetic Waves and Applications 22 (2008), 1931–1941. T. Rahim, M.J. Mughal, Q.A. Naqvi and M. Faryad, Paraboloidal reflector in chiral medium supporting simultaneously positive phase velocity and negative phase velocity, Progress In Electromagnetics Research, PIER 92 (2009), 223–234. T. Rahim, M.J. Mughal, M. Faryad and Q.A. Naqvi, Fields around the focal region of a paraboloidal reflector placed in isotropic chiral medium, Progress In Electromagnetics Research B 15 (2009), 57–76. T. Rahim, M.J. Mughal, M. Faryad and Q.A. Naqvi, Focal region field of a paraboloidal reflector coated with isotropic chiral medium, Progress In Electromagnetics Research, PIER 94 (2009), 351–366. T. Rahim, M.J. Mughal and Q.A. Naqvi, Focal region field of Perfect Electromagnetic Conductor (PEMC) paraboloidal reflector placed in homogeneous chiral medium, Progress In Electromagnetics Research M 8 (2009), 143–152. T. Rahim, M.J. Mughal and Q.A. Naqvi, PEMC paraboloidal reflector in chiral medium supporting simultaneously positive phase velocity and negative phase velocity simultaneously, Progress In Electromagnetics Research Letters 10 (2009), 77–86. M. Faryad and Q.A. Naqvi, Cylindrical reflector in chiral medium supporting simultaneously positive phase velocity and negative phase velocity, Journal of Electromagnetic Waves and Applications 22(Issue: 4) (2008), 563–572. T. Rahim and M.J. Mughal, Spherical reflector in chiral medium supporting positive phase velocity and negative phase velocity simultaneously, Journal of Electromagnetic Waves and Applications 23(11–12) (2009), 1665–1673.
AU
[14]
45
