A study of the transformation bandwidth and the thickness sensitivity of the anisotropic-slab LP to CP polarizer with several media

Authors


Abstract

[1] In this paper, we investigate the transformation bandwidth and the thickness sensitivity of the anisotropic-slab linearly polarized (LP) to circularly polarized (CP) polarizer. The effect is represented graphically on the complex polarization ratio plane. We define a transformation bandwidth and the thickness sensitivity by using a specified axial ratio, e.g., 3 dB, as a basis. It is shown that the polarization locus for a given axial ratio leads to a circle in the polarization state diagram. When combined with the graphical description of the change in the polarization state, the transformation bandwidth and the thickness sensitivity from an initial LP wave to a desired CP wave can be obtained easily. Under the small reflection approximation, i.e., only the forward waves are considered, this method can be applied to the design of the anisotropic-slab LP to CP polarizer. This method eliminates the lengthy derivation and gives deeper physical insight to the problem. The transformation bandwidths for the anisotropic-slab polarizer with a lossless and nondispersive biaxial medium are studied. The thickness sensitivities for the anisotropic-slab polarizer with several lossless media are also investigated. The results are discussed and illustrated.

1. Introduction

[2] In satellite communications a CP wave is often employed to combat the Faraday rotation introduced by the ionosphere [Richharia, 1999]. An example is the right-hand circularly polarized (RHCP) used in the GPS reception. Because of this, how to obtain a CP wave for a given incident polarization finds some applications. An anisotropic medium can change the polarization state of the wave. Usually a perfect CP transformation can be achieved only at a particular frequency using a suitable anisotropic slab with a particular slab thickness. However, the same slab may not be able to achieve a perfect CP at all frequencies in a given band. Similarly, a perfect CP cannot be obtained using an inaccurate slab thickness. We study the transformation bandwidth and the thickness sensitivity of the anisotropic slab for LP to CP polarizers.

[3] The polarization locus of an electromagnetic wave is in general an ellipse, which can be characterized by the axial ratio, tilt angle and sense of rotation [Kelso, 1964; Institute of Electrical and Electronics Engineers (IEEE), 1983; Mott, 1992]. The axial ratio of a perfect CP wave is equal to one. In general, the polarization state of the wave transmitted through an anisotropic slab depends on the frequency and the slab thickness. When an exact CP cannot be obtained, an approximation may be acceptable if its axial ratio is below a specified value, e.g., 3 dB. For this reason, a transformation bandwidth and the thickness sensitivity of the anisotropic-slab LP to CP polarizer based on the axial ratio can be defined. A given axial ratio will thus correspond to the transformation bandwidth or the thickness sensitivity.

[4] On the complex polarization ratio (R) plane, the contours of constant axial ratio are circles [e.g., Booker et al., 1951; Kelso, 1964], and the loci of the polarization as the wave propagates in the lossless anisotropic media follow a circle, too [Yeh et al., 1999a]. On the complex R plane, the transformation bandwidth and the thickness sensitivity can be determined from the relations between these two sets of circles. The formulation is described in section 2. As examples, in section 3, we study the case of a linearly polarized wave incident normally to a slab corresponding to one of three different media: biaxial, chiral, and magnetoionic. The results are used to explain how to obtain the transformation bandwidth or the thickness sensitivity. Properties of the bandwidth or the thickness sensitivity and design issues to obtain the maximum bandwidth or thickness sensitivity are investigated. Discussion on some advantages of this approach over analytical methods is given in section 4.

2. Formulation

[5] The complex polarization ratio can be used to describe the polarization state of a wave. As a result, the axial ratio of the wave can be determined once the complex polarization ratio is known. The complex polarization ratio (R) is defined as REy/Ex for a wave propagates in the z direction [Mott, 1992]. After some derivations the relation between the axial ratio and the complex polarization ratio is

equation image

where AR is the axial ratio, Rre and Rim denote the real part and the imaginary part of the complex polarization ratio, respectively. Equation (1) describes a circle, which we call an AR circle. The radius b and the coordinates of the center C of the AR circle are

equation image

and

equation image

[6] On the complex R plane, we can draw two circles for a given value of axial ratio. In Figure 1, circles corresponding to some values of axial ratio are shown. Points on an AR circle have the same axial ratio, but they correspond to different tilt angles. Since the axial ratio is real, all circle centers are on the imaginary axis, as indicated by (3). From (2) and (3), since b is smaller than the modulus of C, these circles never cross each other as they never cross the real axis. The circles actually form mirror images relative to the real axis. Circles in the upper half plane represent elliptical polarization rotating in the left-handed (LH) sense, corresponding to the lower sign in (1). Similarly, circles in the lower half plane represent right-handed (RH) elliptical polarization, corresponding to the upper sign in (1). When the value of the axial ratio is equal to one, the circles degenerate into two points at ±j. When the value of axial ratio increases, the centers will move from ±j to ±∞. A polarization state inside a given circle implies that its axial ratio is smaller than that of the given AR circle.

Figure 1.

The contours of constant axial ratio, with ratios equal to 1, 2, 3, 4, and 5 dB in the complex R plane.

[7] The change of polarization when the wave propagates a characteristic distance in the lossless anisotropic medium is a complete circle and we call it an R circle [e.g., Yeh et al., 1999a, 1999b]. The geometry of the R circle depends on the incident wave and the characteristics of the medium. The sense of rotation of the polarization locus on the R circle depends on the characteristics of the medium. Once the center and the radius are known, the circle can be drawn in the complex R plane.

[8] For an anisotropic slab LP to CP polarizer, the slab thickness dc is designed to obtain the smallest axial ratio of the transmitted wave at the operating frequency fc. Consider an anisotropic slab that is used as an LP to CP polarizer. The slab has a thickness d. Under the small reflection approximation, the R circle of the transmitted wave can be drawn. When the wave propagates in the z direction, the complex polarization ratio as a function of the slab thickness can be written as [Yeh et al., 1999a]

equation image

where the complex constant coefficients a1x, a1y, a2x, a2y, E1, and E2 depend on the incident wave and the characteristic waves of the medium. In (4), z = exp[−j(k2k1)d], the exponential part, is a periodic function of slab thickness at a given operating frequency, i.e., fc. We can write ∣k2k1∣ = (2πN/c) · f, where c is the light speed, f is the frequency, and N is the dimensionless difference of dielectric constants for two polarizations. We denote the period of the slab thickness as dH = c/Nfc. For a nondispersive medium, since characteristic phase constants k1 and k2 are proportional to the frequency, z is a periodic function of frequency for a fixed slab thickness, i.e., dc. We denote the period of the frequency as fH = c/Ndc.

[9] Equation (4) can be treated as the bilinear transformation of complex variable [e.g., Brown and Churchill, 1996]. A unit circle in the complex z plane maps to a circle, an R circle, in the complex R plane. From (4), the relation between z and the radius ρ and the coordinates of the center W of the R circle can be written as

equation image

where θR is the angle between the point having maximum Rre, i.e., point m in Figure 2, and other point on the R circle from the center of R circle. The point m corresponds to θR = 0. Thus, the argument of z, which we call za, is a function of the angle θR. In (4), the polarization state of the initial wave can be found at za = −(k2k1)d = 0. We can demonstrate that za equal to zero implies θR equal to zero from (4) and Yeh et al. [1999a].

Figure 2.

Illustration of finding the bandwidth or thickness sensitivity of the anisotropic slab CP polarizer.

[10] To illustrate how to find the transformation bandwidth or the thickness sensitivity of the anisotropic-slab LP to CP polarizer, we use LP to left-hand circularly polarized (LHCP) polarizer as an example. This is shown in Figure 2 where the center of the R circle is marked as W. The following assumptions are made: the polarization locus on the R circle rotates counterclockwise as frequency or slab thickness increases, the center of the R circle is on the imaginary axis, the R circle crosses the real axis, and the incident polarization state is at the point s, an LP wave. Also drawn in Figure 2 is an AR circle for a given axial ratio value. The two intersections L and U between the AR circle and the R circle cover an angle of 2θ from the center of the R circle. The angle 2θ can be related to the transformation bandwidth or the thickness sensitivity for an acceptable axial ratio. We define an angle difference ADa as

equation image

where θo, equal to π/2, is the angle between g and m from the center of the R circle, g is the point on the R circle having the smallest axial ratio, when the slab thickness is equal to dc and the operating frequency is equal to fc. Thus, ADa can be used to find the transformation bandwidth BWa = ADa · (c/2πNdc) for a suitable slab thickness dc or the thickness sensitivity from STa = ADa · (c/2πNfc) at the operating frequency fc. It can be seen in Figure 2 that the angle difference ADa becomes maximum if the lines equation image and equation image are tangential to the AR circle. The maximum angle difference ADa,max can be used to find the maximum transformation bandwidth BWa,max or the maximum thickness sensitivity STa,max.

[11] In the study of the bandwidth of the quarter-wave transformer, the fractional bandwidth is used to represent the transformation bandwidth [e.g., Collin, 1992; Pozar, 1993]. Thus, we define a percentage angle difference ADf for finding the fractional transformation bandwidth from BWf = BWa/(fc + nfH) = ADf, n = 0,1,2,…, for the nth cycle under the slab thickness dc or the percentage thickness sensitivity STf = STa/(dc + ndH) = ADf, n = 0,1,2,…, for the nth cycle at the operating frequency fc. The percentage angle difference ADf is given by

equation image

[12] A value of n equal to zero yields the largest ADf, which we call the fundamental percentage angle difference ADfo. Only this case is considered below. The maximum percentage angle difference ADfo,max can be found from (7), which in turn can be used to find the maximum fundamental fractional transformation bandwidth BWfo,max and the maximum fundamental percentage thickness sensitivity STfo,max.

[13] In the next section the method to obtain the angle difference ADa, the maximum angle difference ADa,max, the fundamental percentage angle difference ADfo and the maximum percentage angle difference ADfo,max will be applied to several examples.

3. Several Examples

[14] There has been discussion for the polarization effect of three anisotropic media: biaxial, chiral, and magnetoionic by Yeh et al. [1999a]. Under the small reflection approximation, when a uniform plane wave incident normally onto these medium slabs, the bandwidth or thickness sensitivity of such slab LP to CP polarizers can be found. Following Yeh et al. [1999a], such examples are selected for our investigation and discussion.

3.1. Biaxial Slab

[15] On the complex R plane, the R circle of a biaxial slab is centered at the origin, i.e., W = 0 in Figure 2, and its radius ρ is equal to the modulus of the complex polarization ratio Ro of the incident wave [Yeh et al., 1999a]. Consider a linearly polarized wave with polarization ratio Ro = 1 incident normally to a biaxial slab with a thickness dc, and the locus of the R circle rotates counterclockwise as the frequency of the wave increases. That is, the R circle of the slab is a unit circle. When the operating frequency of incident wave is fc, the CP transmitted wave is obtained. Assume the acceptable axial ratio is AR = 3dB = 1.4. The corresponding R circle and AR circle are shown in Figure 3. The included angle θ is found to be θ = 18.9°. In this case, ADa is equal to 37.8° and ADfo is equal to 42%.

Figure 3.

Determination of the bandwidth or thickness sensitivity of the biaxial slab CP polarizer.

[16] In Figure 3, if we would like to obtain ADa,max, the maximum angle difference, the line equation image and equation image must be tangential to the AR circle. Then both triangles ΔWCL and ΔWCU become the right triangle. It is easy to see the two triangles ΔWCL and ΔWCU are right triangles if lengths of the line segments equation image and equation image are both equal to one. It means that ADa,max can be obtained when the radius of the R circle is equal to one. In this case we have θmax = cos−1(1/a), where a = (AR2 + 1)/(2AR) is the length of the line segments equation image. The maximum percentage angle difference ADfo, max corresponding to the acceptable axial ratio is given by

equation image

[17] The above conclusion is reached for any value of acceptable axial ratios. As a result, for the case of normal incidence of the linearly polarized waves to a biaxial slab, an initial polarization ratio of Ro = 1 will give the maximum angle difference ADa,max regardless of the acceptable axial ratio. If the required angle difference AD is less than ADa,max, the corresponding angle θ is less than θmax, and the tilt angle needs not to be exactly equal to π/4. Thus, by letting r = acosza(θ) the allowable polarization ratio Ro of the linearly incident wave lies in the range of

equation image

This can be further translated into a range of the tilt angle τ as

equation image

[18] This has the implication in the sensitivity of the polarization direction of the incident LP wave. Equations (9) and (10) allow us to choose the incident wave when the required bandwidth or thickness sensitivity is smaller than the maximum bandwidth or thickness sensitivity.

3.2. Chiral Slab

[19] When a linearly polarized wave is incident normally onto a lossless chiral slab, the transmitted wave will remain linearly polarized. This conclusion has been presented by Yeh et al. [1999a]. For the LP to CP polarizer design, therefore, the lossless chiral media should be excluded. However, this feature of chiral media can also be viewed from the relation between the AR circle and the R circle of such a medium. The radius ρ and the coordinates of the center W of the R circle of such a medium are given by Yeh et al. [1999a]:

equation image

and

equation image

where Ro and Θ denotes the modulus and argument of the complex wave polarization ratio of the incident wave, respectively. If the Ro and Θ are given as Ro = AR or Ro = AR−1 and Θ = ±π/2, the equations (11) and (12) are equivalent to (2) and (3), respectively. That is, all of the polarization states on the R circle of the lossless chiral medium have the same axial ratio, and this axial ratio can be found from the two intersections between the R circle and the imaginary axis on the complex R plane. Therefore, if the polarization state of the incident wave locates on the AR circle, the AR circle and the R circle will overlap completely on the complex R plane. From (2) and (3), when the incident wave is polarized linearly, i.e., an infinity axial ratio, the AR circle degenerates into a line along the real axis. Thus, the chiral media will not transform the LP wave into the CP wave.

3.3. Magnetoionic Slab

[20] A magnetoionic medium is a cold plasma permeated by a steady magnetic field. Such a medium is dispersive [e.g., Yeh and Liu, 1972; Budden, 1985; Davies, 1989]. The variable N is now frequency dependent. Hence, it is more meaningful to investigate the tolerance of the slab thickness. That is, ADa represents the thickness sensitivity STa and ADfo denotes the fundamental percentage thickness sensitivity STfo. At a fixed operating frequency fc, the characteristic waves of the magnetoionic medium can change from linear polarization wave to circular polarization wave depending on the relationship between the direction of the external magnetic field and the propagation direction of the incident wave. When the external magnetic field is perpendicular to the direction of wave propagation, the characteristics of the magnetoionic medium are similar to those of biaxial media. When the external magnetic field is parallel to the direction of wave propagation, the characteristics of the magnetoionic medium are similar to those of chiral media. For the calculation of the thickness sensitivity of the anisotropic slab polarizer, therefore, the biaxial medium and the chiral media are the special cases of the magnetoionic medium.

[21] On the complex R plane, the coordinates of the center of the R circle of the magnetoionic medium lie on the imaginary axis [Yeh et al., 1999a, 1999b] and it changes with the direction of external magnetic field. In Figure 2, let the coordinates of L be (x, y), and the center of the R circle be W = ju. Since the length of the line segment equation image corresponds to the modulus of the complex polarization ratio Ro of the linearly incident wave, the radius of the R circle can be written as ρ = (u2 + Ro2)1/2. Consider the tilt angle τ of the linearly incident wave lying in the range of 0° ≤ τ ≤ 90°, i.e., 0 ≤ Ro ≤ ∞. After some manipulation, the angle difference ADa and the fundamental percentage angle difference ADfo can be written as

equation image

and

equation image

where

equation image

[22] In the following, one special case is considered first, before the consideration of the general case. We will assume that the acceptable axial ratio to be 3 dB.

3.3.1. Case 1: τ = 0°

[23] In this case, τ = 0° means that the linearly incident wave is polarized along the x axis. Substituting τ = 0° into (13), (14), and (15), ADa and ADfo are both function of u based on a specified axial ratio. The variable u is constrained in the following range to give the positive ADa and ADfo:

equation image

[24] In Figure 4, ADa and ADfo are depicted as function of u. The range of the variable u is 0.36 < u < 0.7. In this case, the maximum angle difference ADa,max = 53.78° and the maximum percentage angle difference ADfo,max = 29.88% at u = 0.5.

Figure 4.

The angle difference ADa (solid line) and fundamental percentage angle difference ADfo (dashed line) of the specified axial ratio AR = 3dB as a function of u.

3.3.2. Case 2: 0° ≤ τ ≤ 90°

[25] In this case, to give a positive ADa and ADfo, the variable u is constrained in the following range:

equation image

[26] Four examples, including the tilt angle τ = 0°, τ = 15°, τ = 30°, and τ = 45°, of the fundamental percentage angle difference ADfo are depicted as function of u in Figure 5. Figure 5 shows that the symmetrical contour of ADfo only occurs in the case of τ = 45° and the maximum fundamental percentage angle difference ADfo,max actually depends on the tilt angle of the linearly incident wave.

Figure 5.

The fundamental percentage angle difference ADfo of the specified axial ratio AR = 3dB as a function of u with the tilt angle τ = 0°, τ = 15°, τ = 30°, and τ = 45° of the linearly incident wave.

[27] Figure 6 depicts ADa,max and ADfo,max as function of τ at a corresponding u. Figure 6 shows that the different values of τ give different results of ADa,max and ADfo,max, the maximum values of the ADfo,max occurs at τ = 45°, the results of ADa,max and ADfo,max of magnetoionic slab are equal to the results of biaxial slab at τ = 45°, both of the two contours are actually symmetric relative to the line τ = 45°, and the trend of variation of the two contours is opposite.

Figure 6.

The maximum angle difference ADa,max (solid line) and the maximum fundamental percentage angle difference ADfo,max (dashed line) of the specified axial ratio AR = 3 dB as a function of the tilt angle τ of the linearly incident wave. Here, τ lies in the range of 0° ≤ τ ≤ 90°.

[28] In Figure 6, all of the results of ADa,max and ADfo,max occur at a corresponding u. Figure 7 shows ADfo,max as function of τ, indicated by the solid curve, and the corresponding u for obtaining ADfo,max, shown by the dashed line. In Figure 7, since u will increase to infinity as τ increases, we only show the results of the range of 0° ≤. τ ≤ 45°. Actually, after some derivations, we find that the maxima ADa,max and ADfo,max occur when the R circle passes through the point j on the complex R plane. For the different polarization ratio Ro of the linearly incident wave, the umax, which represents the corresponding u giving rise to the maximum percentage angle difference ADa,max and the maximum fundamental percentage angle difference ADfo,max, can be found to be

equation image
Figure 7.

The maximum fundamental percentage angle difference ADfo,max of the specified axial ratio AR = 3dB as function of the tilt angle τ of the linearly incident wave (solid line), and the corresponding u for obtaining the ADfo,max (dashed line). Here, τ lies in the range of 0° ≤ τ ≤ 45°.

[29] Once the polarization ratio of incident wave is known, the radius and the center of the R circle which passes through the point j on the complex R plane can be determined. Then we can adjust the direction of external magnetic field to obtain the maximum thickness sensitivity.

4. Conclusions

[30] In this paper, we investigate the transformation bandwidth and the thickness sensitivity of the anisotropic-slab LP to CP polarizer, where the bandwidth and thickness sensitivity are based on a specified axial ratio, e.g., 3dB. Our method to study the transformation bandwidth and the thickness sensitivity of a CP polarizer based on graphic analyses is easy to apply when the center of the R circle is along the imaginary axis. Although only the results for the case of LP to LHCP polarizer are shown, the results for the case of LP to RHCP polarizer can be found by the same method. In our discussion, if the anisotropic medium is nondispersive, the thickness of the anisotropic slab is kept constant and in turn the bandwidth of the CP polarizer is determined. If instead the medium is dispersive, we fix the operating frequency of the incident wave and the tolerance of the thickness of the slab can be found.

[31] In a biaxial slab, the structure of the biaxial slab we considered is actually equivalent to the quarter-wave plate investigated by Shen and Kong [1995]. We are using our approach to determine the suitable parameters to give the maximum transformation bandwidth and thickness sensitivity. We show clearly that the maximum bandwidth and thickness sensitivity occurs when the polarization ratio of incident wave is equal to one.

[32] The second slab medium considered is a chiral medium. Such a slab will result in the same axial ratio independent of the slab thickness or the operating frequency. Consider the multilayer slab with mixing biaxial and chiral slab. The bandwidth can probably be increased further compared to what was obtained for a biaxial slab alone, and this requires further investigation.

[33] The third slab medium considered is a magnetoionic slab. As a dispersive medium, we discuss the tolerance of the slab thickness for this case. When the tilt angle τ of the linearly incident wave lies in the range of 0° ≤ τ ≤ 90°, the center of the R circle W = ju giving rise to the maximum thickness sensitivity can be found. Since u and τ are related to the coordinates of the center and the radius of the R circle, our results are useful for adjusting the direction of the external magnetic field to obtain the maximum fundamental percentage thickness sensitivity. As a matter of fact, the maximum fundamental percentage thickness sensitivity occurs when the direction of external magnetic field is perpendicular to the propagation direction of the incident wave. The results of the maximum fundamental percentage thickness sensitivity of such a slab are identical to the results of a biaxial slab. The reason for the above result is that the characteristics of the magnetoionic slab are similar to those of biaxial slab when the external magnetic field is perpendicular to the propagation direction of the incident wave.

[34] When the medium is lossy, the two characteristic waves will in general experience different attenuations. The locus of the wave polarization will no longer be a circle. After propagating a sufficiently long distance, the locus will eventually shrink into a point on the complex R plane, corresponding to the characteristic wave that has less attenuation. From the point of view of the polarizer design, the applications then become limited.

Acknowledgments

[35] This work is partially supported by the grants from the National Science Council, ROC, through NSC89-2213-E-110-078 and NSC90-2213-E-110-019.

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