Analysis Method

The numerical calculations begin with derivation of the Green’s function for stratified media. The vector potential Av

due to the presence of a current distribution Jv

in the source region is expressed as:

v

where the spatial domain Green’s function G~

are represented by the Sommerfeld integral as: with H0(2) being the Hankel function of the second kind and SIP the Sommerfeld

integration path. The spectral domain Green’s functions G~

for vector potential are derived in closed form in the source layer [6]-[7]. These expressions are then extended to an arbitrary layer through an iterative process individually for TE and TM components of the Green’s functions in the source layer. The coefficients appeared in the Green’s functions are functions of the generalized reflection coefficients, which can be obtained by applying appropriate boundary conditions for the proposed configuration. The main difficulty lies in the Hankel transform was resolved by employing the saddle point technique in this work, because the Green’s functions in the far field region are smoothly varying functions of distance.

For a planar infinite array, the periodic Green’s function G~_p

is expressed in terms of a sum of the spectral domain Green’s functions for layered structure [8]-[9]

as:

) wavenumbers associated with the phase shifted plane wave.

In this work, the commercial software IE3D [10] based on Method of Moments was utilized to calculate the induced current Jv

. For simplicity of analysis, the unit cell of a CPSS was duplicated to form a 3×3 array, with periodic boundary conditions applied on the four sides (see Figure 4.2). No significant differences in the calculated currents were observed by having any larger array from preliminary simulations.

This product, of the currents extracted from the center unit cell of the 3×3 array and the periodic Green’s function, was substituted into (4.1) and integrated over the source region to obtain the vector potential Av

. Then the far-field RHCP and LHCP components of the scattered field were derived for each illumination.

4.4 Results

The term “Isolation” in this chapter is defined as the amount of the designated polarization field blocked by the surface. The “Isolation” for a RHCPSS is found by applying a RHCP plane wave illumination, then measuring the difference of the RHCP field intensity received with and without the RHCPSS inserted in front of the receiver. On the other hand, the “Transmission Loss” is found by measuring the difference of the LHCP field received with and without the RHCPSS inserted into a LHCP plane wave illumination.

l l

Periodic Boundary

Figure 4.2 The 3×3 array used for analysis. The currents on the center unit cell are extracted for succeeding manipulations.

The dielectric constant and the thickness of the substrate would suggest an appropriate selection of the laminate. There are many commercially available, off-the-shelf products, which are suitable for the development of CPSSs. A laminate with a dielectric constant of 2.33 (Duroid 5870) and 31-mil thickness was chosen for this work, because, under such conditions, the distance the wave traveled in the substrate is the closest to λ/4 at the target frequency of 60 GHz.

According to the fabrication capability provided by PCB manufacturers, both the minimum achievable trace width and spacing are 4 mils, and the minimum achievable radius of a via is 2 mils. In this work, all trace widths and spacing are set to 0.1 mm (≈4 mils), and the lengths of both arms are identical. The periodicity of the array has an analogous effect on isolation and transmission loss. It is observed that the larger the periodicity is, the less the isolation and transmission loss appear.

The periodicity is set to 2.2 mm as it maintains a good isolation.

The optimization process starts from taking 3λ/8 as the initial value of the arm lengths (l). Then the arm lengths for different radii of the via are varied to achieve the optimal design. Figure 4.3 and Figure 4.4 display the simulated isolation and transmission loss against arm lengths at 60 GHz. The solid line and the dashed line respectively present the results obtained when the via radii (r) are 2 mils and 4 mils.

It is seen that the isolation is a fast-varying function of the arm length, while the transmission loss is a slow-varying one. For the 2-mil via, the optimal length for largest isolation occurs at 1.5 mm with a corresponding isolation of 26.4 dB. At this length, the transmission loss is –0.43 dB. For the 4-mil via, the optimal length for largest isolation occurs at 1.425 mm with a corresponding isolation of about 22 dB and the transmission loss is –0.44 dB.

0 5 10 15 20 25 30

1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7

Arm Length (l ) (mm)

RHCP Isolation (dB) .

r= 2 mil r= 4 mil

Figure 4.3 The simulated isolation for various lengths of arms. The radii (r) of the vias are 2 mils and 4 mils, respectively.

-1 -0.8 -0.6 -0.4 -0.2 0

1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7

Arm Length (l ) (mm)

LHCP Transmission Loss (dB) .

r= 2 mil r= 4 mil

Figure 4.4 The simulated transmission loss for various lengths of arms. The radii (r) of the vias are 2 mils and 4 mils, respectively.

As Figure 4.3 and Figure 4.4 indicate, the performance of the circular polarization selectivity is extremely sensitive to the lengths of arms. The radius of the via also has significant influence; thinner via will have longer optimum arm lengths and a better performance. Thus, the design with the via radius of 0.05 mm and the arm length of 1.5 mm was determined for realization.

The finished RHCPSS has 1256 unit elements, made on a disk-shape substrate, with a radius of about 45 mm. However, it apparently differed from what had been proposed. The manufacturer, based on its accumulated experience, achieved a via by a hole with copper plating on the walls and two additional annular ring pads encompassing the hole on each side of the substrate in order to prevent the thin metallic traces from peeling off. Figure 4.5 is the photograph of the finished RHCPSS. The schematic diagram of the realized deformed structure observed with a microscope is also sketched. By measuring, the lengths from the center of the ring to the ends of the arms are 1.5 mm, and the inner radii of the rings are about 0.075 mm.

However, the outer radii of the rings are inconsistent, ranging from 0.1 mm to 0.2 mm.

In order to better understand the deviation, simulations that set the models with outer radii of 0.1 mm, 0.15 mm, and 0.2 mm and the unchanged arm length of 1.5 mm were performed. Figure 4.6 and Figure 4.7 illustrate the simulated results.

Measurements are performed with an Agilent 8510C VNA equipped with an extra milli-meter wave controller and a milli-meter wave test set. Two LP horns are collimated and aligned with each other. For shielding, a metal plate with a hole having the same radius as the CPSS is placed in between the two horns. The CPSS under test is installed in place of the hole. The distances between the RHCPSS and the two horns must be large enough to ensure the phase of the wave to be uniformly distributed over the aperture of the RHCPSS. It is set to 1 m in this work.

Outer radius

Inner radius

l

Figure 4.5 Photograph of the finished RHCPSS, which has 1256 elements on a disk-shape substrate. The radius of the disk is about 45 mm.

Frequency (GHz)

55 57 59 61 63 65

RHCP Isolation (dB)

0 5 10 15 20 25 30 35 40

r= 0.1 mm r= 0.15 mm r= 0.2 mm Measured

Figure 4.6 The simulated isolations for outer radius (r) of 0.1 mm, 0.15 mm and 0.2 mm (arm length l= 1.5 mm, inner radius= 0.075 mm), and the measured isolation of the finished RHCPSS.

Frequency (GHz)

55 57 59 61 63 65

LHCP Transmission Loss (dB)

-20 -15 -10 -5 0

r= 0.1 mm r= 0.15 mm r= 0.2 mm Measured

Figure 4.7 The simulated transmission loss for outer radius (r) of 0.1 mm, 0.15 mm and 0.2 mm (arm length l= 1.5 mm, inner radius= 0.075 mm), and the measured data of the finished RHCPSS.

Two LP horns are installed on the two ends of the network analyzer on account of the lack of standard CP horn antenna. The cross-polarization levels of the LP horns are less than –40 dB, whose effects are neglected in the following analysis. LP wave can be considered as a combination of a pure LHCP and a pure RHCP wave, each contributing half the power.

Let EMajor and EMinor be the electric field magnitudes in respective major and minor axis of the polarization ellipse of a CP wave:

2

where EL0 and ER0 denote the intensities of left- and right-hand components.

By definition, the axial ratio (AR) of a CP wave is:

)

Rotate the receiving horn and RHCPSS under test until maximum and minimum S21 appear, the axial ratio can be calculated from (4.6) and the ratio of LHCP component to RHCP component can be derived as:

1

Suppose that the transmission loss of the LHCP component compared to the reduction in the RHCP component, caused by the RHCPSS, is negligible, i.e., EL0 >

ER0 at the receiving side, EL0 and ER0 can be deduced from the measured maximum and minimum S21.

Remove the RHCPSS, while keeping the shielding plate in place, and then re-perform the S21 measurement to figure out the referenced RHCP and LHCP components of the incident wave, so as to obtain the isolation and transmission loss.

Figure 4.6 shows the frequency response of the measured isolation for the realized RHCPSS. The simulation responses for outer radius of 0.1 mm, 0.15 mm and 0.2 mm (arm length l= 1.5 mm, inner radius= 0.075 mm) are also shown for comparison. As the figure indicates, the measured data is quite similar to the simulated result of the case with outer radius of 0.2 mm. The maximum measured isolation of 23.89 dB is obtained at 58.4 GHz, while the isolation at 60 GHz is only 7.82 dB. Figure 4.7 illustrates the corresponding transmission losses for measurements and simulations. The minimum transmission loss of about –2.25 dB is measured at 58.4 GHz, while at 60 GHz the transmission loss is –4.65 dB.

It is considered that the deviation between the measured and simulated transmission loss was caused by the variation in loss tangent. In the simulations, the loss tangent of Duroid 5870 was set to 0.0009 in the light of the data provided by the laminate vender. Nevertheless, this value is for 10 GHz, and the loss tangent at 60 GHz must be higher. Another likely reason for this could be the imperfection in workmanship, including the inconsistencies of the thickness of the copper plating for each via and the outer radius of each annular ring pad. Poor conduction of the vias can lead to geometrical resonance phenomenon on the whole structure by undesired CP illumination, causing an increase in the transmission loss.

4.5 Conclusions

This chapter reports theoretical and experimental investigations into a new CPSS. The proposed design is well suited for PCB manufacturing process. The closed-form spectral domain Green’s function for this CPSS configuration was first derived and transformed into the periodic Green’s function. Then the scattered fields of a planar infinite CPSS were obtained by the induced currents together with the

periodic Green’s function. Lastly, with the scattered fields, the isolation and transmission loss can be calculated.

The design of a 60GHz RHCPSS shows good performance, with the respective simulated isolation and transmission loss at 60 GHz being 26.4 dB and –0.43 dB.

However, the manufacturer added unprompted pairs of annular ring pads in the fabrication process to avoid any peeling effect. The sizes of the pads are even inconsistent. These variations make the finished RHCPSS disagree with the optimized design.

For the finished RHCPSS, the maximum measured isolation of 23.89 dB and the minimum measured transmission loss of –2.25 dB occur at 58.4 GHz, which is a little lower than the desired frequency. The deformations resulted from fabrication and the defects in workmanship indeed affect the performance. Simulations on various degrees of deformed structures reveal that the isolations are greater than 25 dB while the transmission losses are better than –1 dB. In addition, one of the results quite resembles the measured data. Therefore, as long as the deformations are arranged during the design process, an exceptional outcome can be reached.

With the findings in this chapter, the feasibility of developing a millimeter-wave CPSS with printed circuit technology is confirmed. This study can serve as a foundation for further research, and opens new possibilities for the beneficial application in this field.

References

1. D. G. Berry, R. G. Malech, and W. A. Kennedy, “The reflectarray antenna,” IEEE Trans. Antennas Propagat., vol. 11, pp. 645 - 651, Nov. 1963

2. C-P Chiu; and S-J Chung, “A new millimeter-wave folded microstrip reflectarray antenna with beam steering,” IEEE AP-S Int. Symp.Digest, vol. 3, pp. 140 - 143, June 2002

3. W. V. Tilson, T. Tralman, and S. M. Khanna, “A polarization selective surface for circular polarization,” IEEE AP-S Int. Symp.Digest, vol. 2, pp. 762 - 765, June 1988.

4. R. Pierrot, “Éléments résonants en polarization circulaire et réflecteur semi-transparent composé de ces éléments,” French Republic Patent # 89.609, No.

1.512.598, Dec 30, 1966.

5. G. A. Morin, “A simple circular polarization selective surface,” IEEE AP-S Int.

Symp.Digest, vol. 1, pp. 100 - 103, May 1990

6. G. Dural, and M. I. Aksun, “Closed-form Green's functions for general sources and stratified media,” IEEE Trans. Microwave Theory Tech., vol. 43, pp. 1545 - 1552, July 1995

7. D. G. Fang, J. J. Yang, and G. Y. Delisle, “Discrete image theory for horizontal electric dipoles in a multilayered medium,” in Proc. IEE Microwaves Antennas Propagat., vol. 135, pp. 297 - 303, Oct. 1988

8. R. M. Shubair, and Y. L. Chow, “A rapidly convergent summation of the periodic Green's function in layered media,” IEEE AP-S Int. Symp.Digest, vol. 1, pp. 200 - 203, July 1992

9. M. J. Park, and Sangwook Nam, “Efficient calculation of the Green's function for multilayered planar periodic structures,” IEEE Trans. Antennas Propagat., vol. 46, pp. 1582 - 1583, Oct. 1998

10. http://www.zeland.com/ie3d.html

5 A New Advance in Circular Polarization