Single-feed Multi-Beam Reflectarray Antennas - IEEE Xplore

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 2, FEBRUARY 2012

Design and Experiment of a Single-Feed Quad-Beam Reflectarray Antenna Payam Nayeri, Fan Yang, and Atef Z. Elsherbeni

Abstract—Reflectarray antennas show momentous promise as a cost-effective high-gain antenna, capable of generating multiple simultaneous beams. A systematic study on various design methods of single-feed multi-beam reflectarray antennas is presented in this communication. Two direct design methods for multi-beam reflectarrays, geometrical method and superposition method, are investigated first. It is demonstrated that although both methods could generate a multi-beam radiation pattern, neither approach provides satisfactory performance, mainly due to high side-lobe levels and gain loss in these designs. The alternating projection method is then implemented to optimize the phase distribution on the reflectarray surface for multi-beam performance. Mask definition and convergence condition of the optimization are studied for multi-beam reflectarray designs. Finally a Ka-band reflectarray prototype is fabricated and tested which shows a good quad-beam performance. Index Terms—Alternating projection method, intersection approach, multi-beam, optimization, reflectarray.

I. INTRODUCTION Reflectarray antennas combine the advantages of both printed arrays and parabolic reflectors and create a high gain antenna with a low-profile, low-mass and low-cost [1], [2]. They have received considerable attention over the years and are quickly finding applications in satellite communications, cloud/precipitation radars, and commercial usages [3], [4]. In addition to these advantages that are mainly due to the use of printed circuit technology, the reflectarray allows for an individual control of the phase shift of each element in the array. As a result, the reflectarray can achieve contoured beam performance without any additional cost [5]. Similarly, multi-beam performance can also be realized by designing the phase shift of the elements appropriately. Multi-beam antennas have numerous applications, such as electronic countermeasures, satellite communications, and multiple-target radar systems [6]. These multi-beam antennas are typically based on reflectors with feed-horn clusters [7] or large phased arrays [8]. Horn array feeds for reflector antennas on communication satellites can provide multiple beams with tailored earth coverage patterns. For phased array antennas, multiple simultaneous beams can be generated by connecting the array to a beamforming network with multiple ports. Considering the complexity of fabricating these antennas and deployment for space applications, these multiple beam designs are relatively high cost. The numerous advantages of reflectarrays, in particular the low-mass and low-cost features, makes the multiple beam reflectarray a suitable antenna candidate. Manuscript received November 29, 2010; revised June 14, 2011; accepted August 26, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. This work was supported in part by the NASA EPSCoR program under contract number NNX09AP18A. P. Nayeri and A. Z. Elsherbeni are with the Electrical Engineering Department, The University of Mississippi, University, MS 38677 USA (e-mail: [email protected]; [email protected]). F. Yang is with the Electrical Engineering Department, The University of Mississippi, University, MS 38677 USA and also with the Electronic Engineering Department, Tsinghua University, Beijing, China (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173126

Reflectarrays can generate single or multiple beams with single or multiple feeds. A two-beam reflectarray prototype using a single feed was demonstrated in [9] while [10], [11] present a single-feed reflectarray generating four simultaneous beams. Multi-feed multi-beam reflectarrays with shaped patterns were also studied in [12]. In addition, multi-feed single-beam reflectarray antennas were investigated in [13]. In these papers, different design approaches have been introduced to achieve the multi-beam performance. The main objective of this communication is to provide a comprehensive and systematic comparison of various multi-beam design approaches, including both direct design methods and iterative optimization techniques, through a case study of a single-feed quad-beam reflectarray. II. DIRECT DESIGN METHODS FOR MULTI-BEAM REFLECTARRAYS A. Basics of Two Direct Design Methods Two direct design methods are available for multiple beam reflectarray antennas. The basic idea behind the first approach, geometrical sub-arrays method, is simply to divide the reflectarray surface into where each sub-array can then radiate a beam in the required direction [10]. Although the array division and beam allocation can be arbitrary, it is feasible to define them based on the directions of the beams they are designed to generate. It should be noted that with this approach each of the power from the feed horn while using 1 zone receives 1 of the aperture surface. Another approach for multi-beam reflectarray designs is by using the superposition of the aperture fields associated with each beam on the reflectarray aperture [2]. To generate beams with a single feed, the tangential field on the reflectarray surface can simply be written as

N

=N

=N

N

ER (xi ; yi ) =

N n=1

An;i (xi ; yi )ej

(x ;y ) :

(1)

Here n;i and n;i are the required amplitude and phase of the th element which will radiate the th beam. In reflectarrays the amplitude of each element is fixed by the feed position and element location, which are independent of the beam direction, therefore

8

A

i

n

ER (xi ; yi ) = AFeed i (xi ; yi )

N 1

n=1

ej

(x ;y ) :

(2)

The summation of the complex field distributions in (2) will give the overall required amplitude and phase distributions. A basic problem exists here which is due to the fixed amplitude distribution imposed by the feed in reflectarray antennas. Although the required phase in (2) can be satisfied by proper element designs, the amplitude requirement cannot be satisfied in reflectarray antennas. The reason is that in (2)

N n=1

ej

(x ;y ) = 6 1:

(3)

As a result of this difference in the amplitude distribution on the aperture, reflectarrays designed using the superposition approach may show a deteriorated performance. B. Comparison of Direct Design Methods To demonstrate the multi-beam design capabilities of these approaches, we study a quad-beam reflectarray antenna. The antenna is designed for the operating frequency of 32 GHz and has a circular aperture with a diameter of 17 at the design frequency. The element periodicity is 2 and ideal phasing elements are used here to

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=


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TABLE I CALCULATED RADIATION CHARACTERISTICS OF THE SINGLE-BEAM AND MULTI-BEAM REFLECTARRAYS

Fig. 1. Normalized radiation patterns of the single- and quad-beam designs.

compare different design approaches. A centered prime focus left hand circularly polarized (LHCP) feed is used for this design and positioned with an F=D ratio of 0.735 that gives an edge taper of 012.39 dB. The reflectarray elements are dual-linear, which reflect the LHCP incident wave to a right hand circularly polarized (RHCP) wave. The radiation patterns of the reflectarrays are calculated using spectral transformation of the aperture fields as described in [2]. This quad-beam reflectarray is designed to generate four beams in the directions of (1;2;3;4 = 30 , '1 = 0 , '2 = 90 , '3 = 180 , '4 = 270 ). Here is the angle between the axe normal to the reflectarray plane and the beam direction. To compare the performances of these two reflectarrays, we also design a single-beam reflectarray antenna as a reference whose beam is in the direction of ( = 30 , ' = 0 ). In Fig. 1 the normalized radiation pattern of this single beam design is compared with the multi-beam reflectarrays. Since the reference beam is in the ' = 0 plane, only this plane is given here, and a similar pattern was observed in the orthogonal plane. From these results it can be seen that both design methods can realize a quad-beam performance with a single-feed horn. In both designs the four beams are generated in the required directions. Comparison of the radiation patterns shows small beam deviations in the multi-beam reflectarray designs. The radiation patterns also show that the side-lobe level is below 011 dB for the geometrical design and below 017 dB for the superposition design. The side-lobe levels (SLL) are much higher than the reference single-beam design. Furthermore, the quad-beam antenna designed using the superposition method shows a beamwidth identical to the reference single-beam design, while in the geometrical design the beamwidth is much wider. The antenna directivity is an important measure to compare the radiation performance of these multi-beam design methods. For multi-beam reflectors using a single-feed, theoretically the power of each beam will be reduced by 1=N . Therefore ideally it is expected that generating four beams will reduce the antenna directivity by 6 dB. However, the geometrical design has a directivity reduction of 11.73 dB and the superposition design exhibits a directivity reduction of 7.02 dB. Some important results of these three reflectarrays are summarized in Table I. For the multi-beam reflectarray designed with the geometrical approach, the amplitude distribution in each zone is maximum at the corner of that zone (near the array center) and minimum at the outer edge, which results in a significant increase in the side-lobe level. The wider beamwidth and lower directivity in the geometrical designs however requires further attention. The reduction in antenna directivity is the result of using one fourth of the array surface and one fourth of the power from the feed horn to generate each beam. This reduction of array surface is also the reason for the increase in beamwidth. For the

multi-beam reflectarray designed using the superposition method, the high side-lobes are due to the amplitude error in (2) which alters the required illumination taper on the aperture. Also in comparison of the calculated antenna directivity with the reference, this design approach shows a directivity loss about 1 dB higher than the ideal directivity reduction (6 dB), which is mainly due to the high side-lobe levels of this design. In summary the shortcomings of both these direct design approaches are tabulated here. Geometrical design method: 1) High side-lobe due to illumination taper; 2) Gain loss and beam broadening due to dividing the array surface into sub-arrays; 3) Small beam deviation. Superposition design method: 1) High side-lobe due to amplitude error; 2) Gain loss due to the increase in side-lobe level; 3) Small beam deviation. As a result of the above problems associated with the direct methods of multi-beam reflectarray design, it is necessary to implement some form of optimization routine to achieve desirable performance. III. ALTERNATING PROJECTION METHOD FOR MULTI-BEAM REFLECTARRAY DESIGN A. The Alternating Projection Method Another approach in multi-beam reflectarray design is to view this as a general array synthesis problem. In reflectarrays however the synthesis of radiation patterns is restricted by the fact that the amplitude of each reflectarray element is fixed by the feed properties and element location. As a result, design of multi-beam reflectarrays requires a phase only synthesis approach. The alternating projection method (APM) also known as the intersection approach [14] has been applied successfully to the phase synthesis of antenna arrays. A simple example of a two-beam reflectarray has been demonstrated using this method [9]. This method is basically an iterative process that searches for the intersection between two sets, i.e. the set of possible radiation patterns that can be obtained with the reflectarray antenna and the set of radiation patterns that satisfy the mask requirements (set M ). Comparing to other phase synthesis methods developed for array antennas, the main advantage of the alternating projection method is the significantly reduced computational time for convergence of the solution [14], which makes it suitable for large reflectarray antennas. The pattern requirements for the design are usually defined by a mask, i.e., two sets of bound values, between which the pattern must lie. The general form of the radiation patterns that satisfy the mask requirements is

set M fF (u; v) : M (u; v) jF (u; v)j M (u; v)g ; u = sin cos '; v = sin sin ' L

U

(4)

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Fig. 3. Radiation pattern of the optimized design at 32 GHz.

Fig. 2. 2D view of the mask model for the quad-beam reflectarray.

where F is the far-field radiation pattern of the array and (u; v ) are the angular coordinates. MU and ML set the upper and lower bound values of the desired pattern in the entire angular range. With set M defined, the alternating projection method can be implemented to obtain the desired radiation pattern. Implementing the alternating projection method requires definition of two projection operators: the mask projector (PM ) and the inverse projector (PI ) [15]. The mask projector uses the upper and lower bounds of the mask to correct the radiation pattern. The inverse projection (PI ) consists of a series of functions which projects the pattern back to the array excitation coefficients. It calculates new phase values for the reflectarray elements while the elements amplitude remain unchanged. B. Implementing APM for Multi-Beam Reflectarray Design The first step is to define the mask for multi-beam operation. Typical masks for different contour beams can be found in the literature [16]; however for multi-beam designs the mask definition is different. The required masks for multi-beam radiation patterns are typically circular contours defined in the direction of each beam. Since in this quad-beam design we don’t want to change the beamwidth, which is directly related to the aperture size and illumination, the mask upper and lower bounds in the beam area were defined according to the reference single-beam design. This upper and lower bounds are defined as

If If

(u; v )

(u; v )

2 main beam : MU (u; v) = 0 dB 2 0 3 dB beamwidth : ML (u; v) = 03 dB:

(5)

The main objective of this optimization is to minimize the side-lobe level. While it is possible to control the side-lobe level by defining an upper bound (MU ) at certain values, in order to further minimize the side-lobe level, both upper and lower bounds in the side-lobe area were set to zero. A 2D figure of this beam mask model for the quad-beam reflectarray is plotted in Fig. 2 using dashed lines. It should be noted that in practice it was found that for this quad-beam reflectarray design, defining mask levels to zero or to an achievable level showed almost similar results. In all optimization routines, it is necessary to define a cost function that should be minimized and can also control the number of iterations required for the convergence of the solution. Since in this optimization the requirements in the main beam will be satisfied by the projection operators with the bounds set in (5), the cost function need only to take into account the side-lobe performance of the array [17]. Thus, the cost

is evaluated over every point in the (u; v ) space which does not belong to the main beams using the following equation: If

jF (u; v)j > MU (u; v) 2 (jF (u; v )j 0 MU (u; v )) : Cost = u +v 1

(6)

With the mask and cost function defined, the optimized phase distribution of the reflectarray elements can be obtained with an iterative procedure. It should be noted here that the optimization is considered to be converged when the cost function becomes stable. In most cases the optimization converges with only a few iterations; however a suitable starting point can reduce the number of iterations. In this design the phase distribution obtained by the superposition method in Section II is used as the starting point for the optimization. The alternating projection method is then implemented to improve the reflectarray performance by optimizing the phase distribution on the reflectarray aperture. A far-field pattern of 400 2 400 points evenly spaced in the angular coordinates was computed for each candidate reflectarray at each cost evaluation. For this quad-beam design, the solution converges after 23 iterations. Although the number of iterations required for the optimization generally depend on the problem at hand, in most cases the APM will converge with just a few iterations [14]–[16]. The radiation pattern of the optimized design is given in Fig. 3. A quad-beam performance is obtained for the reflectarray with side-lobes below 026 dB. It can be seen that implementing the optimization here has corrected the amplitude problems associated with the initial superposition design ((3)). As a result, the side-lobe level has been reduced by about 9 dB and all four beams are exactly scanned to 30 off broadside. Also the calculated directivity for this antenna is 26.95 dB, which is about 1 dB higher than the initial superposition design (Table I). These results clearly demonstrate the effectiveness of the phase optimization process. The computational time for the APM optimization with 30 iterations was 456 seconds on a 2.2 GHz Intel core Duo CPU with 4 GB RAM. This is to be compared with 16.5 seconds for both direct design methods. It is important to point out that although in some cases the APM optimization might converge to local minima, for this symmetric quadbeam design the optimization did not get trapped and the solution converged smoothly. Similar radiation pattern results were observed when the phase distribution obtained by the geometrical approach was used for the starting point. For non-symmetric multi-beam designs however the problem with local minima is more challenging and some approaches that can circumvent the local minima problem such as [9] may be required.


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Fig. 4. (a) Optimized phase distribution of the reflectarray elements, (b) fabricated quad-beam reflectarray.

IV. KA-BAND QUAD-BEAM REFLECTARRAY PROTOTYPE A. Prototype Fabrication The optimized quad-beam prototype is fabricated on a 20 mil Rogers 5880 substrate. The reflectarray has a circular aperture with a diameter of 15.94 cm. The phasing elements are variable size square patches with a unit-cell periodicity of =2 at the design frequency of 32 GHz. The unit-cell simulations are carried out using the commercial software Ansoft Designer [18], where the fabrication limit of our LPKF ProtoMat S62 milling machine is also taken into account by enforcing the minimum gap size between the elements and the achievable fabrication tolerance. It is worthwhile to point out that in general the reflection characteristics of the phasing elements are angle dependent and oblique incidence needs to be considered. Our simulations showed that for these elements normal incidence can present good approximations for oblique incidence angles up to 35 ; thus the prototype was designed based on the simulated reflection coefficients obtained with normal incidence. The optimized phase distribution for the reflectarray elements and the photograph of the fabricated array with 848 square patch elements are shown in Fig. 4. The centered prime focus LHCP feed horn is mounted on a mechanical alignment system and positioned with an F/D ratio of 0.735. To avoid blockage from the supporting strut of the feed horn, the array is rotated 45 in the reflectarray plane so the main beams are in the directions of (1;2;3;4 = 30 , '1 = 45 , '2 = 135 , '3 = 225 , '4 = 315 ). Since dual-linear square patch elements are used in this design, the reflected co-polarized radiation of the reflectarray system is RHCP.

Fig. 5. Measured and simulated co-polarized radiation patterns of the reflecplane, (b) ' plane. tarray antenna: (a) '

= 45

= 135

B. Measured Radiation Patterns The radiation pattern is measured using our planar near-field measurement system. Comparisons of the simulated and measured co-polarized radiation patterns at 32 GHz are shown in Fig. 5. The simulated radiation pattern of the single beam reference design in Section II is also plotted here for comparison. For the quad-beam design, the simulated radiation patterns here also include the aperture blockage caused by the horn and the alignment system, which is calculated using the approach given in [19]. Note that four beams are generated in the required directions which are correctly scanned to 30 and the side-lobe levels are below 018 dB. The measured and simulated results show good agreements in the main lobes where the measured 03 dB beamwidth is 4.35 for both vertical and horizontal planes. Some discrepancies exist in the side-lobe regions, which are mainly due to fabrication errors and element design approximations. The beam level reduction is primarily due to the alignment errors of the measurement setup and the azimuth non-symmetry of the feed horn radiation pattern; however this reduction is less than 2.15 dB

Fig. 6. Measured and simulated cross-polarized radiation patterns of the reflectarray antenna in ' plane.

= 45

for any beams. Both simulated and measured cross-polarized radiation patterns of the reflectarray are given in Fig. 6 for the ' = 45 plane. Almost similar results were observed in the orthogonal plane (' = 135 ). The relatively high cross-polarization level of the reflectarray here is due to the high cross polarization of the feed horn. To reduce the cross polarization, one approach is to use a better feed

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Fig. 7. Measured gain and efficiency of the quad-beam reflectarray antenna.

horn with lower cross-polarization level. This was confirmed by repeating the simulations with an idealized horn antenna model [2]. For this case the cross-polarization level of the reflectarray was reduced to 023.1 dB. Another approach is to use the element rotation technique [2] for phase compensation, since it will only compensate the phase of the co-polarized CP components to form a focused beam. C. Gain and Efficiency The measured gain and aperture efficiency vs. frequency are given in Fig. 7. It should be noted that although all beams showed a similar gain performance vs. frequency, the gain results presented here are for the beam in the direction of = 30 , ' = 45 . At 32 GHz, the measured gain is 25.3 dB, and the 1 dB gain bandwidth is 8.6%. For multi-beam antennas, the classical definition of aperture efficiency might not be appropriate; therefore a modified definition is used here to calculate the aperture efficiency, i.e. a

=

N

G

2 i i=1 A

4

Fig. 8. Measured radiation patterns of the reflectarray antenna across the 1 dB plane. gain band in the '

= 45

The radiation patterns show quad-beam patterns across the entire band with a slight increase in side-lobe level at the extreme frequencies. The beam squint across the 1 dB gain bandwidth of the antenna is about 2.5 . Considering that the beamwidth of the antenna is more than 4 , the effect of this beam squint is acceptable for this quad-beam prototype. It is interesting to point out that in comparison between the multi-beam design methods, the direct geometrical approach shows the smallest beam squint. While the quad-beam prototype here did not show a very large beam squint across the band, it should be pointed out that in general for multi-beam reflectarrays, beam squint could limit the operating band of the antenna. Thus, for multi-beam designs where a minimum beam squint requirement is specified, additional constraints over the frequency bandwidth needs to be imposed when optimizing the phase distribution of the elements. V. CONCLUSION

(7)

where N represents the number of beams and A is the aperture area. This definition takes into account the measured gains of all four beams, and the aperture efficiency is calculated to be 35.26%. Besides the spillover and illumination effects, the loss in the aperture efficiency comes from the cross-polarization effect, the element loss, and the feed blockage. D. Beam Squint The bandwidth of a reflectarray antenna is usually defined by the 1 dB gain bandwidth [20], [21]. For multi-beam reflectarrays however, the practical bandwidth of the antenna is also limited by the fact that the beams shift with frequency. This is due to the fact that the main beam direction depends on the progressive total phase on the aperture (including all time delay effects from feed radiation and total reflection phase of the elements). It was shown in [22] that beam squint can be minimized in reflectarray antennas by enforcing the condition o = i , where i is defined as the angle from the phase center of the feed to the center of the array and o is the main beam direction. It is clear that this condition cannot be satisfied for multi-beam designs and beam squint would become a limiting factor. The measured radiation patterns at the center and extreme frequencies are given in Fig. 8 for the ' = 45 plane. Similar results were observed in the orthogonal plane.

A systematic analysis of single-feed multi-beam reflectarray antennas is presented in this communication through a case study of a quad-beam design. Two direct design approaches for multi-beam reflectarrays are investigated first, and their performance suggests that an optimization procedure on the elements phase distribution is necessary to achieve a satisfactory performance for complicated multi-beam reflectarrays. The alternating projection method is then implemented to optimize the performance of the multi-beam reflectarray antennas. Required masks, cost definition, and convergence conditions are discussed for multi-beam reflectarrays. Based on the optimization results, a Ka-band reflectarray antenna is designed and measured, which shows a good quad-beam performance using a single feed. ACKNOWLEDGMENT The authors want to thank the reviewers whose thoughtful comments added to the quality and efficacy of this communication. The authors also acknowledge Rogers Corporation for providing free dielectric substrates.

REFERENCES [1] D. M. Pozar, S. D. Targonski, and H. D. Syrigos, “Design of millimeter wave microstrip reflectarrays,” IEEE Trans. Antennas Propag., vol. 45, no. 2, pp. 287–296, Feb. 1997.


[2] J. Huang and J. A. Encinar, Reflectarray Antennas. Hoboken, NJ: Wiley, 2008, Institute of Electrical and Electronics Engineers. [3] R. E. Munson and H. Haddad, “Microstrip Reflectarray for Satellite Communication and RCS Enhancement and Reduction,” U.S. patent 4,684,952, Aug. 1987, Washington DC. [4] H. Shih-Hsun, H. Chulmin, J. Huang, and K. Chang, “An offset lineararray-fed Ku/Ka dual-band reflectarray for planet cloud/precipitation radar,” IEEE Trans. Antennas Propag., vol. 55, no. 11, pp. 3144–3122, Nov. 2007. [5] D. M. Pozar, S. D. Targonski, and R. Pokuls, “A shaped-beam microstrip patch reflectarray,” IEEE Trans. Antennas Propag., vol. 47, pp. 1167–1173, Jul. 1999. [6] R. C. Hansen, Phased Array Antennas, Wiley Series in Microwave and Optical Engineering. New York: Wiley, 1998. [7] P. Balling, K. Tjonneland, L. Yi, and A. Lindley, “Multiple contoured beam reflector antenna systems,” presented at the IEEE Antennas and Propagation Society Int. Symp., Michigan, Jun. 28–Jul. 2 1993. [8] L. Schulwitz and A. Mortazawi, “A compact dual-polarized multibeam phased-array architecture for millimeter-wave radar,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 11, pp. 3588–3594, Nov. 2005. [9] J. A. Encinar and J. A. Zornoza, “Three-layer printed reflectarrays for contoured beam space applications,” IEEE Trans. Antennas Propag., vol. 52, no. 5, pp. 1138–1148, May 2004. [10] J. Lanteri, C. Migliaccio, J. Ala-Laurinaho, M. Vaaja, J. Mallat, and A. V. Raisanen, “Four-beam reflectarray antenna for Mm-waves: Design and tests in far-field and near-field ranges,” in Proc. EuCAP, Berlin, Germany, Mar. 2009, pp. 2532–2535. [11] P. Nayeri, F. Yang, and A. Z. Elsherbeni, “Single-feed multi-beam reflectarray antennas,” presented at the IEEE Antennas and Propagation Society Int. Symp., Toronto, Canada, Jul. 2010. [12] M. Arrebola, J. A. Encinar, and M. Barba, “Multifed printed reflectarray with three simultaneous shaped beams for LMDS central station antenna,” IEEE Trans. Antennas Propag., vol. 56, no. 6, pp. 1518–1527, Jun. 2008. [13] F. Arpin, J. Shaker, and D. A. McNamara, “Multi-feed single-beam power-combining reflectarray antenna,” Electron. Lett., vol. 40, no. 17, pp. 1035–1037, Aug. 2004. [14] R. Vescovo, “Reconfigurability and beam scanning with phase-only control for antenna arrays,” IEEE Trans. Antennas Propag., vol. 56, no. 6, pp. 1555–1565, Jun. 2008. [15] O. M. Bucci, G. Franceschetti, G. Mazzarella, and G. Panariello, “Intersection approach to array pattern synthesis,” IEE Proc., vol. 137, no. 6, pp. 349–357, Dec. 1990. [16] O. M. Bucci, G. Mazzarella, and G. Panariello, “Reconfigurable arrays by phase-only control,” IEEE Trans. Antennas Propag., vol. 39, no. 7, pp. 919–925, Jul. 1991. [17] D. W. Boeringer and D. H. Werner, “Particle swarm optimization versus genetic algorithms for phased array synthesis,” IEEE Trans. Antennas Propag., vol. 52, no. 3, pp. 771–779, Mar. 2004. [18] Ansoft Designer v 4.1 Ansoft Corporation, 2009. [19] L. Diaz and T. Milligan, Antenna Engineering Using Physical Optics. Norwood, MA: Artech House, 1996. [20] J. Huang, “Bandwidth study of microstrip reflectarray and a novel phased reflectarray concept,” presented at the IEEE Antennas and Propagation Society Int. Symp., CA, Jun. 1995. [21] D. M. Pozar, “Bandwidth of reflectarrays,” Electron. Lett., vol. 39, no. 21, Oct. 2003. [22] S. D. Targonski and D. M. Pozar, “Minimization of beam squint in microstrip reflectarrays using an offset feed,” presented at the IEEE Antennas and Propagation Society Int. Symp., MD, Jul. 1996.

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Oblique Diffraction of Arbitrarily Polarized Waves by an Array of Coplanar Slots Loaded by Dielectric Semi-Cylinders John L. Tsalamengas and Ioannis O. Vardiambasis

Abstract—We study oblique diffraction of arbitrarily polarized planewaves by a finite array of slots of infinite length on a common ground plane, backed by an array of dielectric semi-cylinders. The formulation is based on a combined eigenfunctions expansion and integral equation approach. For the diffracted field, series expansions in cylindrical wave functions are used to which several singular integral terms are superimposed that fully account for the presence of each of the slots. The relevant system of singular integral equations is discretized by an exponentially convergent Nyström method. Noticeably, all matrix elements take simple closed-form expressions. Numerical examples and case studies illustrate the convergence of the algorithm and bring to light the influence of the dielectric loads on the characteristics of the structure. Index Terms—Dielectric cylinders, electromagnetic diffraction, integral equations, Nyström method, slot arrays.

I. INTRODUCTION Fig. 1 shows an array of S slots of infinite length on the ground plane y = 0, loaded by dielectric semi-cylinders ("i ; i ) of radii Ri whose axes are located at y = 0, x = hi , i = 1; 2; . . . ; S . Without loss of generality it is taken that h1 = 0. Region 0 (y > 0) has the properties of vacuum ("0 ; 0 ), and the background medium (region a), taken to be lossless, has parameters ("a ; a ). Dielectric losses in the cylinders may be accounted for via complex constitutive parameters. The primary excitation is an arbitrarily polarized plane wave obliquely incident from region 0. This structure may be used as a polarization selector, a radiation suppressor over a given frequency bandwidth, and a multi-slot/multi-dielectric open waveguide. Previous related studies [1]–[8] only concern certain special (or limiting), definitely simpler, two-dimensional cases wherein a single slot is illuminated by a z -invariant primary excitation in presence of either dielectric or perfectly conducting cylindrical loads. In such cases TEz and TMz waves decouple and can be treated separately [9]. The case S = 1 of the present quasi three-dimensional problem has been treated in [10] via a combined Green’s function and singular integral equation approach. Nevertheless, when S (S > 1) semi-cylinders coexist in region y < 0, as in Fig. 1, the method of [10] breaks down due to insuperable difficulties in obtaining the Green’s function of the structure. To overcome such difficulties, in the present communication a new hybrid formulation technique, flexible and easily implemented, is proposed (Section II). The main idea is to use series expansions in cylindrical wave functions, in addition to properly selected singular integral Manuscript received December 18, 2010; revised April 08, 2011; accepted August 31, 2011. Date of publication October 21, 2011; date of current version February 03, 2012. This work was supported in part by the “THALES” Project ANEMOS, funded by the ESPA Program of the Ministry of Education of Greece. J. L. Tsalamengas is with the School of Electrical and Computer Engineering, National Technical University of Athens, GR-15773 Zografou, Athens, Greece (e-mail: [email protected]). I. O. Vardiambasis is with the Department of Electronics, Technological Educational Institute (TEI) of Crete, 73133 Chania Crete, Greece (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2173129

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