Focal region field of a two dimensional Gregorian system coated with chiral medium is analyzed at high frequency. Maslov's method is used because the Geometrical Optics approximation fails at focal points. Maslov’s method combines the simplicity of ray theory and the generality of Fourier transform. Fields patterns are calculated numerically and the results are plotted. The effects of thickness of chiral layer, chirality parameter of the chiral medium and permittivity of the medium are analyzed. The problem of simple dielectric layer is discussed as a special case of this problem.
Chiral medium Maslovsmethod Gregorian system focal region field1. Introduction
The knowledge of focal region field of focusing systems is useful for synthesizing feed arrays in imaging and design of multiple beam antennas in communication systems. The focusing of electromagnetic waves into material media is also a subject of considerable current interest due to applications in hyperthermia, microscopy, and optical data storage. Geometrical optics (GO) approximation is one of the well known method for evaluating high frequency field. It has been widely used to study various kinds of problems in different areas of electromagnetics, acoustics and seismology [1-3]. GO approximation for wave solution is important in electromagnetics because it provide insight into the behavior of wave front. GO is used only for high frequency approximation of a wave, provided the ray tube does not vanish. However, at caustic regions the ray tube shrinks to zero and GO show singularity at these regions. These regions are of great importance in all practical problems e.g. parabolic, paraboloidal and circular reflectors etc. To avoid these singularities Maslov proposed a method based on Maslov`s theory [4,5]. Maslov’s method has been used to find the field at caustic regions [6-20]. The idea in Maslov's method provides a systematic procedure for predicting the field in the caustic region by combining the simplicity of ray theory and generality of the transform method. High frequency field expressions has been derived around feed point of a two dimensional Gregorian system using the Maslov’s method in [19]. The same focussing system has been coated with isotropic and homogeneous chiral medium and field expressions are obtained. In Section 2 the plane wave reflection from a chiral slab backed by perfect electric conducting (PEC) plane is considered. In Section 3 high frequency expression for the field of a chiral coated Gregorian system excited by plane wave is derived. Numerical results and discussion are presented in Section 4 and the paper is concluded in Section 5.
2. Reflection of Plane Waves From a Chiral Slab Backed by Conductor
In this paper we want to find the reflected field around the focal region of a two dimensional Gregorian’s main parabolic reflector coated with chiral medium. To achieve this the reflection of plane waves from a chiral slab backed by perfect electric conducting (PEC) plane is discussed as in [15,21]. As shown in figure 1 the region is occupied by free space defined by following equations
and the region is occupied by the chiral medium defined by Drude-Born-Fadorov (DBF) constitutive relations [21] as follows
The PEC is placed at as shown in figure 1. The reflection coefficients for the parallel and perpendicular components of polarization is calculated in [11]
In the above equation, , and, are the parallel and perpendicular components of polarization of the incident and reflected fields respectively. and are matrices. Elements of the matrices, which are Fresnel coefficients, are given in [15]. and are also matrices and are given in [15]. Using these reflection coefficients, the focal region fields of a two dimensional Gregorian's system are derived.
3. Focal Region Field of Two Dimensional Gregorian Reflector
Gregorian system consists of two reflectors, one is parabolic main reflector and another is hyperbolic sub-reflector as shown in figure 2. This system has several advantages over a single parabolic reflector.In this paper we want to study the caustic region field of a two dimensional Gregorian system when the main reflector is coated with chiral medium using GO and Maslov’s method. The equations of each reflector are given as follows
In the above relations and are the Cartesian coordinates of the point on the parabolic and elliptical reflectors, respectively. The Incident wave traveling along the negative -axis is given by
where, , are the components along -axis and -axis of the initial amplitude of the incident field respectively. The wave reflected from the hyperboloidal surface is given by
(4)
The wave vector of the wave reflected by the parabolic reflector is given by
and the wave vector of the wave reflected by the hyperbolic reflector is
where, the angle and angle are given by the relation
In the above equations and are the distances from the point to the focal points and, respectively. The unit normal vectors to the parabolic and hyperbolic surfaces are given by and respectively. These normals can be written as
The general form of the geometrical optics solution for the wave is given by [19]
The Jacobian associated with the wave reflected by the parabolic reflector is given by [19]
The Cartesian coordinates of the ray reflected by the hyperbolic surface is given by
where,. In the above equation and are the rectangular components of reflected wave vectors and, respectively. Now we consider the field after the reflection from the hyperbolic cylinder. The transformation from the Cartesian coordinates to the ray coordinates is given by [19]
Thus the geometrical ray expression of each component of the reflected wave is
(12)
(14)
where, is the amplitude of the incident wave at the reflection point on the parabolic surface and
As indicated by (7), will become infinite for i.e. at the caustic points. The expression which is valid at the focal point according to Maslov’s method is given by [19]
The amplitude term in the above equation is given by [19]
The phase function is given by
where, is given by (15). The extra term is given by
We substitute (19)-(20) into (18) and taking, we can find the finite field around the caustic as given below
In the above equation, , , and are expressed in terms of and and are the subtention angles at the edges of the parabolic and hyperbolic surfaces. The limit of integration are calculated by the expression
4. Result and Discussion
Field pattern around the caustic region of a Gregorian system are determined using (21-23) by performing the integration numerically. The line plots of the field around the focal region located between the two reflectors, that point in figure 2. Simulation were done for , , , , and different values of, , and. We have considered two types of polarization for incident wave. One is, and the other is,. The results are plotted in Figures 3-20. In all figures horizontal axis is -axis and vertical axis corresponds to the absolute value of reflected field component. All plots are taken at. We have studied the effects of thickness of the the coated layer (), the chirality parameter () and the relative permittivity of the me-
dium () on the focal region field. Figures 3-5 show the effect of increase in the value of keeping and, that is ordinary dielectric case. We have shown in figure 3 and 5the and for. These figures show that increase in thickness of the coated layer () results an increase in and. Figure 4 shows the plot of for. This figure shows decrease in field strength as increases. Figures 6-11 show the effect of while keeping and for both types of polarization. Figures 7 and 11 show increase in the field strength of and for, respectively. Figure 6 and 10 show the cross polarization effect due to chiral medium for polarization, which also increases with increase in. For, cross polarization effect vanishes and field strength increases with increase in. Figure 8 shows the plot of for, which decreases with increase in, and Figure 9
shows the plot for. This figure shows that cross polarization effect are zero for, but increases with increase in layer thickness.
Figures 12 and 14 show the effect of chirality parameter on field components while keeping and. Figures 12 and 14 show the plots of and for respectively. These figures show that field strength increases with increase in chirality parameter. Figure 13 shows that decreases with increase in. Cross polarization effects have not been shown because they have same trends as discussed above. Figures 15-17 show the effect of while keeping and, that is for ordinary dielectric case. Figures 15 and 17 show the plot of and for polarization respectively and show the increase in the field strength if we increase the relative permittivity of dielectric medium coated on the surface of reflector. Figure 16 shows that decreases with increase in for polarization. Figures 18-20, which are for chiral coating, show the same trends as in Figures 15-17.
5. Conclusions
Focal region fields of a two dimensional Gregorian reflector is analyzed using GO and Maslov’s method. Two types of polarization are discussed. The reflected field components are analyzed numerically and the results are given in the focal plane. It is concluded that increase in chirality parameter (), increase in thickness () and relative permittivity (), results in increase in of and for. While decreases for. Cross polarized fields exist when and increase with increase in value of.
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is occupied by free space defined by following equations
is occupied by the chiral medium defined by Drude-Born-Fadorov (DBF) constitutive relations [
as shown in figure 1. The reflection coefficients for the parallel and perpendicular components of polarization is calculated in [
, 
and
, 
are the parallel and perpendicular components of polarization of the incident and reflected fields respectively. 
and 
are 
matrices. Elements of the matrices, which are Fresnel coefficients, are given in [
and 
are also 
matrices and are given in [
and 
are the Cartesian coordinates of the point on the parabolic and elliptical reflectors, respectively. The Incident wave traveling along the negative 
-axis is given by
, 
are the components along 
-axis and 
-axis of the initial amplitude of the incident field respectively. The wave reflected from the hyperboloidal surface is given by
(4)
and angle 
are given by the relation
and 
are the distances from the point 
to the focal points 
and
, respectively. The unit normal vectors to the parabolic and hyperbolic surfaces are given by 
and 
respectively. These normals can be written as
. In the above equation 
and 
are the rectangular components of reflected wave vectors 
and
, respectively. Now we consider the field after the reflection from the hyperbolic cylinder. The transformation from the Cartesian coordinates 
to the ray coordinates 
is given by [
(12)
(14)
is the amplitude of the incident wave at the reflection point on the parabolic surface and
will become infinite for 
i.e. at the caustic points. The expression which is valid at the focal point according to Maslov’s method is given by [
in the above equation is given by [
is given by (15). The extra term is given by
, we can find the finite field around the caustic as given below
, 
, 
and 
are expressed in terms of 
and 
and 
are the subtention angles 
at the edges of the parabolic and hyperbolic surfaces. The limit of integration are calculated by the expression
in figure 2. Simulation were done for 
, 
, 
, 
, 
and different values of
, 
, and
. We have considered two types of polarization for incident wave. One is
, 
and the other is
,
. The results are plotted in Figures 3-20. In all figures horizontal axis is 
-axis and vertical axis corresponds to the absolute value of reflected field component. All plots are taken at
. We have studied the effects of thickness of the the coated layer (
), the chirality parameter (
) and the relative permittivity of the me-
) on the focal region field. Figures 3-5 show the effect of increase in the value of 
keeping 
and
, that is ordinary dielectric case. We have shown in figure 3 and 5the 
and 
for
. These figures show that increase in thickness of the coated layer (
) results an increase in 
and
. 
for
. This figure shows decrease in field strength as 
increases. Figures 6-11 show the effect of 
while keeping 
and 
for both types of polarization. Figures 7 and 11 show increase in the field strength of 
and 
for
, respectively. 
, which also increases with increase in
. For
, cross polarization effect vanishes and field strength increases with increase in
. 
for
, which decreases with increase in
, and 
. This figure shows that cross polarization effect are zero for
, but increases with increase in layer thickness
.
on field components while keeping 
and
. Figures 12 and 14 show the plots of 
and 
for 
respectively. These figures show that field strength increases with increase in chirality parameter
. 
decreases with increase in
. Cross polarization effects have not been shown because they have same trends as discussed above. Figures 15-17 show the effect of 
while keeping 
and
, that is for ordinary dielectric case. Figures 15 and 17 show the plot of 
and 
for polarization 
respectively and show the increase in the field strength if we increase the relative permittivity of dielectric medium coated on the surface of reflector. 
decreases with increase in 
for polarization
. Figures 18-20, which are for chiral coating, show the same trends as in Figures 15-17.
), increase in thickness (
) and relative permittivity (
), results in increase in of 
and 
for
. While 
decreases for
. Cross polarized fields exist when 
and increase with increase in value of
.