12th International Conference on Mathematical Methods in Electromagnetic Theory June 29 – July 02, 2008, Odesa, Ukraine
FRACTIONAL DUAL PARABOLIC CYLINDRICAL REFLECTOR Akhtar Hussain, M Faryad and Q. A. Naqvi Department of Electronics, Quaid-i-Azam University, Islamabad, Pakistan E-mail: akhtar
[email protected] Abstract – Geometrical optics (GO) fields of a fractional dual parabolic cylindrical reflector have been studied around the focusing region using Maslov’s method. GO fields contain both co- and cross-polarized components. Considering perfect electric conductor as original case, magnitudes of the GO fields do not depend upon the fractional parameter. For impedance boundary reflector as original case, the field components depend upon the fractional parameter. Dependence of co-polarized and cross-polarized field components in different ranges of surface impedance has been studied. I. INTRODUCTION Fractional curl operator which is represented as curlα , where α denotes the order of fractional curl operator, may be used to fractionalize the principle of duality in electromagnetics [1]. In present work, fractional curl operator has been utilized to obtain the GO fields of a fractional dual parabolic reflector having impedance boundary. Maslov’s method has been used to remedy the problem of GO around the focal region of the reflector. In section II, the reflection coefficients from a plane fractional dual impedance boundary are recalled and utilized in section III to find the GO fields of a parabolic reflector. Results are discussed in section IV. II. FRACTIONAL DUAL PLANE REFLECTOR Application of fractional curl operator on a planer boundary having surface impedance η reduces the surface to fractional dual surface and the reflected fields may be written in tensor form as [2] i r,α R11 R12 E1 E1 = (1) r,α E2 R21 R22 E2i i,r where E1,2 means magnitudes of the transverse electric and transverse magnetic components of the incident and reflected field and R11 , R21 are the reflection coefficients of the transverse electric and transverse magnetic reflected fields when the incident wave is transverse electric. Similarly R12 , R22 are the reflection coefficients of the transverse electric and transverse magnetic reflected fields when the incident wave is transverse magnetic. These coefficients may be written as π
π
R11 = −ejα 2 (B 2 RH − A2 RE ), jα π 2
R21 = −e
R12 = −ejα 2 AB cos θ(RH + RE ) jα π 2
AB cos−1 θ(RH + RE ),
R22 = −e
(A2 RH − B 2 RE )
(2a) (2b)
where RE =
1 − (η/η0 ) cos θ , 1 + (η/η0 ) cos θ
−1
RH =
1 − (η/η0 )
cos θ
1 + (η/η0 )
cos θ
−1
,
π A = sin(α ), 2
π B = cos(α ) 2
here θ is the angle that incident ray makes with normal to the reflector at the point of incidence and α is the fractional parameter having any value in the range 0 ≤ α ≤ 1. III. GEOMETRICAL OPTICS FIELD OF A PARABOLIC REFLECTOR Consider a parabolic reflector defined by ζ = f −ξ 2 /4f , where f is focal length of parabola. Consider a transverse electric plane wave of unit amplitude that is travelling in the positive z-direction hits the reflector having surface impedance η. Taking time dependence as e−jωt , the incident wave is Ei = yˆejkz 978-1-4244-2284-5/08/$25.00 © 2008 IEEE
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12th International Conference on Mathematical Methods in Electromagnetic Theory June 29 – July 02, 2008, Odesa, Ukraine
For a parabolic cylindrical reflector, solutions for the reflected field can be approximated by Maslov’s method [3] as E(x, z) =
2kf π exp(−j2kf − j ) π 4
γ 2
− γ2
A0ξ sec θ exp{−jk
x x2 + z 2 cos(2θ − tan−1 ( ))}dθ z
(3)
where γ = tan−1 (a/2f ), a is aperture of reflector, and A0ξ is initial amplitude of the reflected wave. In case of fractional dual boundary, the initial amplitude is a function of incidence angle θ and fractional parameter α and hence field components can be written as E11 = E21 =
2kf π exp(−j2kf − j ) π 4 2kf π exp(−j2kf − j ) π 4
γ 2
− γ2
γ 2
− γ2
A11 (θ, α) sec θ exp{−jk
x x2 + z 2 cos(2θ − tan−1 ( ))}dθ (4a) z
A21 (θ, α) sec θ exp{−jk
x x2 + z 2 cos(2θ − tan−1 ( ))}dθ (4b) z
where A11 (θ, α) and A21 (θ, α) are the initial values of the co-polarized and cross-polarized reflected fields and has been found as
π 1 − { ηη0 }−1 cos θ 1 − ηη0 cos θ π π 2 2 cos (α ) − sin (α ) A11 (θ, α) = −A0ξ exp jα 2 2 1 + { ηη0 }−1 cos θ 2 1 + ηη0 cos θ
π 1 − ηη0 cos θ 1 − { ηη0 }−1 cos θ π π −1 cos(α ) sin(α ) cos θ + A21 (θ, α) = −A0ξ exp jα 2 2 2 1 + { ηη0 }−1 cos θ 1 + ηη0 cos θ Plots of (4) for different combinations of α and η are given in Fig. 1-2. IV. RESULTS AND DISCUSSION Plots of co-polarized and cross-polarized components of the reflected field along the reflector axis are shown in Fig. 1, taking surface impedance η = 1.5η0 . As can be seen from Fig. 1, fractional parameter α only affects the amplitude of the reflected field while pattern remains the same. (a)
(b)
Fig. 1. Reflected fields versus kz for (a) co-polarized (b) cross-polarized components. Fig. 2 shows magnitude plots of the reflected field components versus fractional parameter α for different values of η. The first pair of plots (co and cross) in Fig. 2 shows that for η = 0, that is PEC reflector as original case, field magnitudes become independent of fractional parameter α. Second pair of plots in this figure shows that for η = η0 , co-polarized and cross polarized components are symmetric around α = 0.5. The third pair of plots shows that cross-polarized component has similar dependence 978-1-4244-2284-5/08/$25.00 © 2008 IEEE
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12th International Conference on Mathematical Methods in Electromagnetic Theory June 29 – July 02, 2008, Odesa, Ukraine
as in second pair while dependence of co-polarized field on α is different. Each case satisfies the duality principle. That is, taking (η/η0 = 0.1, α = 0) gives same result as (η/η0 = 10, α = 1) and vice versa. (a)
(b)
Fig. 2. Variation of reflected fields versus α for (a) co-polarized (b) cross-polarized components. REFRENCES [1] N. Engheta, Fractional curl operator in electromagnetics, Microwave and Optical Technology Letters, vol. 17,, 2, pp: 86–91, 1998
[2] M. V. Ivakhnychenko and E. I. Veliev, Fractional operator approach in electromagnetic reflection problem J. of Electromagn.,vol. 21, No. 13, 1787-1802, 2007.
[3] Kohei Hongo, Yu Ji and Fiji Nakajima, High frequency expression for the field in the caustic region of a reflector using Maslov’s method, radio science, vol. 21,no. 6, 911-919,1986.
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