Microwave Orbital Angular Momentum Mode Generation and Multiplexing Using a Waveguide Butler Matrix
2017; Electronics and Telecommunications Research Institute; Volume: 39; Issue: 3 Linguagem: Inglês
10.4218/etrij.17.0115.1100
ISSN2233-7326
AutoresWangjoo Lee, Ju Hong, Min Soo Kang, Bong Su Kim, Kwang Seon Kim, Woo Jin Byun, Myung Sun Song, Yong Heui Cho,
Tópico(s)Antenna Design and Optimization
ResumoETRI JournalVolume 39, Issue 3 p. 336-344 ArticleFree Access Microwave Orbital Angular Momentum Mode Generation and Multiplexing Using a Waveguide Butler Matrix Wangjoo Lee, Corresponding Author Wangjoo Lee wjlee@etri.re.kr Corresponding Authorwjlee@etri.re.krSearch for more papers by this authorJu Yeon Hong, Ju Yeon Hong jyhong@etri.re.kr Search for more papers by this authorMin Soo Kang, Min Soo Kang mssong@etri.re.kr Search for more papers by this authorBong Su Kim, Bong Su Kim bskim1@etri.re.kr Search for more papers by this authorKwang Seon Kim, Kwang Seon Kim gskim@etri.re.kr Search for more papers by this authorWoo Jin Byun, Woo Jin Byun wjbyun@etri.re.kr Search for more papers by this authorMyung Sun Song, Myung Sun Song mssong@etri.re.kr Search for more papers by this authorYong Heui Cho, Yong Heui Cho yhcho@mokwon.ac.kr Search for more papers by this author Wangjoo Lee, Corresponding Author Wangjoo Lee wjlee@etri.re.kr Corresponding Authorwjlee@etri.re.krSearch for more papers by this authorJu Yeon Hong, Ju Yeon Hong jyhong@etri.re.kr Search for more papers by this authorMin Soo Kang, Min Soo Kang mssong@etri.re.kr Search for more papers by this authorBong Su Kim, Bong Su Kim bskim1@etri.re.kr Search for more papers by this authorKwang Seon Kim, Kwang Seon Kim gskim@etri.re.kr Search for more papers by this authorWoo Jin Byun, Woo Jin Byun wjbyun@etri.re.kr Search for more papers by this authorMyung Sun Song, Myung Sun Song mssong@etri.re.kr Search for more papers by this authorYong Heui Cho, Yong Heui Cho yhcho@mokwon.ac.kr Search for more papers by this author First published: 01 June 2017 https://doi.org/10.4218/etrij.17.0115.1100Citations: 11 Wangjoo Lee (corresponding author, wjlee@etri.re.kr), Ju Yeon Hong (jyhong@etri.re.kr), Min Soo Kang (mskang@etri.re.kr), Bong Su Kim (bskim1@etri.re.kr), Kwang Seon Kim (gskim@etri.re.kr), Woo Jin Byun (wjbyun@etri.re.kr), Myung Sun Song (mssong@etri.re.kr) are with the Broadcasting & Media Research Laboratory, ETRI, Daejeon, Rep. of Korea. Yong Heui Cho (yhcho@mokwon.ac.kr) is with the School of Information and Communication Engineering, Mokwon University, Daejeon, Rep. of Korea. This work was supported by the ICT R&D program of MSIP/IITP (B0101-16-0222, Development of core technologies to improve spectral efficiency for mobile big-bang). AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract In this paper, we propose a convenient microwave orbital angular momentum (OAM) mode generation and multiplexing method operating in the 18 GHz frequency band, based on a uniform circular array and a Butler matrix. The three OAM modes −1, 0, and +1 were generated and verified using spatial S-parameter measurements; the measured back-to-back mode isolation was greater than 17 dB in the full 17 GHz to 19 GHz range. However, the radiated OAM beam centers were slightly dislocated and varied with both frequency and the mode index, because of the non-ideal characteristics of the Butler matrix. This resulted in mode isolation degradation and transmission distance limitations. I. Introduction Electromagnetic propagation modes with an orbital angular momentum (OAM) were first studied in the context of Laguerre–Gaussian laser beam modes [1]. The OAM mode is characterized by an intensity vortex on the beam axis and a helical phase front. It has a ejlϕ phase factor, where l is an integer indicating the OAM degree, and ϕ is the azimuthal angle measured at any plane orthogonal to the beam axis. Optical OAM modes can be used as micro-particle rotating tweezers [2] and for quantum key distribution [3], and their use in high-capacity data communications has been attempted [4]–[6]. Moreover, in the area of microwave wireless communications, OAM modes are considered potential candidates for increasing capacity without requiring any extra frequency resources. The advantage of OAM modes when applied to communications lies in their spatial orthogonality, which allows sharing the same frequency band, time slot, polarization, and spatial channel with other OAM modes. The key technologies of OAM-mode-based communications are associated with the mode generation and multiplexing steps. Demultiplexing is not an independent technology in itself, given that it is simply the reverse of multiplexing. There are two basic approaches to OAM beam generation and multiplexing in the microwave region. The first one is to first generate single OAM modes, and then multiplex them. Spiral phase plates (SPPs) [7] and helical reflectors [8] can be used to transform the zero OAM Hermite–Gaussian mode into finite OAM Laguerre–Gaussian modes, and vice-versa. However, a single SPP or helical reflector can only generate one OAM mode, and the several different OAM modes should be optically multiplexed—with a beam splitter, for example [9]—before transmission or transmitted individually without multiplexing [10]. The other approach is to generate all the multiplexed OAM modes simultaneously, using a Butler matrix and a uniform circular array (UCA) [11], [12]. A UCA is an array of N sub-antennas located along a circumference at equal spacing. The ith element is driven by a signal with a i · θl phase delay, where is an integer smaller than N/2 [13], [14]. In these conditions, the wave front will contain the desired ejlϕ phase factor. Figure 1 shows various simulated UCA-generated OAM beam patterns (with a 10λ UCA diameter) at a distance of 1,000λ. The element antennas are modeled as micro vertical dipole antennas. The intensity and phase vortices are clearly shown at the beam center in the column. In the and box, the individual intensities are combined as in [15]. Figure 1.Open in figure viewerPowerPoint Simulation of UCA generated OAM beam patterns (with a 10λ antenna diameter) at a distance of 1,000λ, for different values of N and l. One advantage of the UCA approach is that each antenna element can be driven by a compound signal. Therefore, a single UCA can generate multiple OAM modes simultaneously, if driven by a suitably synthesized signal, as shown in Fig. 2. Figure 2.Open in figure viewerPowerPoint Simulated compound OAM beam patterns generated by a single UCA with the same dimensions considered in Fig. 1. A Butler matrix is an matrix circuit widely used in beamforming networks. For each input signal, there are N output signals, with phase delays proportional to the output port number. The phase delay increment is determined by the input port; in the case of simultaneous excitation of multiple input ports, the Butler matrix outputs the necessary compound signals, which can then be fed to a UCA to generate multiplexed OAM beams. The remainder of this paper is organized as follows. Section II reviews the operations of Butler matrix and the S-parameter characteristic of our Butler matrix. In Section III, the measured field distribution of triple OAM modes are presented and the beam center fluctuation is analyzed. In Section IV, the transmission characteristics of our triple OAM modes are described with an indoor TV signal transmission experiment. Finally, Section V presents our conclusions. II. Butler Matrix for Triple OAM Mode Generation Figure 3 shows the Butler matrix circuit used in this paper, which is composed of a 90° hybrid coupler, a crossover, and a 180° phase shifter. In the 90° hybrid coupler, the coupled output is delayed by 90° than the through output. The 90° phase delay is expressed as , neglecting the coupling magnitude factor without affecting the relative magnitudes on each path. We then have that, for integer . In Fig. 3(a), signal Z0 is input to Port 1 and the next step signals are identified in the Butler matrix circuit. The final outputs are Z8, Z6, Z4, and Z2, each one of these phase delay by −90° relative to the previous one in the list. When signal Z0 is input to Port 3 (Fig. 3(b)) all outputs are "," which means that they all have the same phase, and when input signal Z4 is fed to Port 4 (Fig. 3(c)) the outputs are Z0, Z2, Z4, and Z6, each one of these delayed in phase by +90° relative to the previous one in the list. Figure 3.Open in figure viewerPowerPoint Butler matrix circuit to generate a triple OAM mode signal: (a) Port 1 input (successive −90° phase delays), (b) Port 3 input (zero phase delays), and (c) Port 4 input (successive +90° phase delays). The signals delayed in phase by −90°, 0, and +90° are fed to a UCA to generate the −1, 0 and +1 OAM modes, respectively. When the three signals are simultaneously applied to the UCA, superposed triple OAM beams are generated (as shown in Fig. 2). The multiplexed triple OAM beams can be separated by passing the signals through the Butler matrix in the backward direction. For example, Fig. 4 shows the arrangement for multiplexing and demultiplexing the −1 mode using Butler matrices connected back-to-back. In this figure, the consecutively connected 180° phase shifter and crossover between the multiplexer (MUX) and demultiplexer (DeMUX) cancel each other, and are therefore not represented. Experimental results will be also described later. Figure 4.Open in figure viewerPowerPoint MUX/DeMUX circuit with reduced Butler matrices connected back-to-back. We fabricated a WR51-waveguide-based Butler matrix to operate in the 18 GHz band. Figure 5 shows a drawing of the resulting waveguide-type Butler matrix. The input and output signals are connected to the input and output ports by subminiature version A (SMA) connectors, whose inner axis pierced the waveguide wall from bottom or top. To simplify the Butler matrix structure, the 180° phase shifter in Fig. 3 is embodied by using an upward SMA in the respective output port, given that all the other three are in the downward direction. The output port crossover in Fig. 3 is implemented by crossing the coaxial cables connecting the Butler matrix and the UCA. Figure 5.Open in figure viewerPowerPoint Drawing of the WR51-waveguide-based Butler matrix. Figure 6 shows the S-parameter characteristics of the fabricated Butler matrix. For each input signal, there are four output signals, as in Fig. 3. Figures 6(a) and (b) show the output signals for the ±1 modes with a small and flat loss. The phase variations appear to be relatively uniform within the 17 GHz to 19 GHz band. However, the remaining (albeit small) amplitude and phase non-uniformity introduce OAM beam center fluctuations that result in degraded mode isolation in the wireless channel. Figure 6.Open in figure viewerPowerPoint S-parameter characteristics of the Butler matrix between 17 GHz and 19 GHz, corresponding to input signals at (a) Port 1, (b) Port 4, and (c) Port 3. As previously mentioned, a backward-connected Butler matrix operates as an OAM mode demultiplexer. Figure 7(a) shows the MUX/DeMUX circuit implemented with Butler matrices connected back-to-back, in the reduced form of Fig. 4. The Butler matrix on the left side operates as an OAM-mode MUX, and the Butler matrix on the right side—which is connected in the backward direction—operates as a DeMUX. The internal circuits of both matrices are the same, except for the folded/not folded formats. As shown in Fig. 7(b), in the 18 GHz band the −1 mode isolation is larger than 25 dB, but the 0 and +1 modes have a slight mutual leakage, and therefore exhibit only 17 dB of isolation, which may limit the OAM mode communication distance. Figure 7.Open in figure viewerPowerPoint Butler matrices connected back-to-back for MUX/DeMUX testing: (a) physical arrangement, (b) Port 1 input, (c) Port 3 input, and (d) Port 4 input. III. Triple OAM Mode Generation and Beam Center Fluctuation Orbital angular momentum modes have a peculiar intensity and phase vortex at the beam center; a direct field distribution measurement is therefore a good method to verify OAM modes. Figure 8 shows the test setup. Each input port of the Butler matrix is driven using a vector network analyzer, and the Butler matrix outputs are fed to a feeder, to radiate an electromagnetic wave. The used feeder is a four-element UCA with a open-ended rectangular waveguide with a cross section [12]; a small probing antenna is used to measure the spatial field distribution by scanning a area (in 1 cm steps) 25 cm away from the feeder. Figure 8.Open in figure viewerPowerPoint Field distribution measurement setup, cm. Figure 9(a) shows the electric field pattern 25 cm away from the feeder at 18 GHz, as simulated with the CST Microwave Studio. Assuming that the effective diameter of the feeder is approximately 4 cm, the Fraunhofer distance is approximately 19 cm; the simulated patterns are therefore far-field patterns. Figure 9(b) shows the two-dimensional S21 measurement results at the same 25 cm distance. Figure 9.Open in figure viewerPowerPoint Simulated and measured OAM beam (b) patterns 25 cm away from the feeder, (b. c.: beam center position; cm): (a) CST-simulated OAM patterns; left: 0 mode, right: ±1 mode, and (b) measured OAM mode patterns. Well-formed intensity nulls and phase vortex distributions are observable for the ±1 modes, whereas the 0 mode exhibits the typical circular intensity and phase distributions. At a 50 cm distance, the shapes did not change, but were twice enlarged. The measured combination of intensities is partially consistent with the simulation results. The "b. c." abbreviation in Fig. 9(b) represents the beam centers of the OAM modes—the maximum intensity position in the 0-mode case, and the minimum intensity positions in the ±1-mode cases. The beam centers vary with frequency and mode index; therefore, the beam center variation was checked in detail. To achieve that, the OAM beams were radiated by a 24 cm diameter Cassegrain antenna fed by the previously mentioned feeder and measured by the probing antenna of Fig. 8, placed 2 m away from the Cassegrain antenna. Figure 10 shows the beam center variation of the three modes between 17.5 GHz and 18.5 GHz, with 1 cm resolution. The variation areas are approximately 5 cm in diameter, but can reach 2.5 m at a 100 m distance. Figure 10.Open in figure viewerPowerPoint OAM beam center variation in the 17.5 GHz to 18.5 GHz frequency range, 2 m away from the transmitting Cassegrain antenna. A more serious difficulty is the fact that the beam centers of the three modes do not coincide at most frequencies. Figure 11 shows the inter OAM beam center separations as a function of frequency—within the 17.5 GHz to 18.5 GHz band, with a 2.5 MHz spacing—as red (0 to −1 mode), green (0 to +1 mode) and blue (−1 to +1 mode) lines. The average distances are 1.6 cm (red), 1.2 cm (green), and 1.4 cm (blue). These distances correspond (respectively) to 79 cm, 62 cm, and 72 cm of average separation at a 100 m transmission distance. Figure 11.Open in figure viewerPowerPoint Triple OAM beam center distances in the 17.5 GHz to 18.5 GHz band, 2 m away from the transmitting Cassegrain antenna; red lines: 0 to −1 mode; green lines: 0 to +1 mode; and blue lines: −1 to +1 mode. The variation in the beams centers of the different OAM modes is mainly due to the non-ideal characteristics of the fabricated Butler matrix. Figure 12 shows the constellation of the four Butler matrix outputs; in this figure, the first Butler matrix output port signals are set to be 45° (located in the first quadrant) for each measured frequency. Figure 12.Open in figure viewerPowerPoint Constellation of the four Butler matrix outputs: (a) 17.5 GHz to 18.5 GHz and (b) 17 GHz to 19 GHz. Given that the UCA-generated OAM modes are the interference patterns of the individual antenna elements, the field distribution depends on the signals fed to the UCA. Figure 13 shows the simulated far-field pattern radiated from an UCA ( feeder) using ideal and real feeding signals from the fabricated Butler matrix, at 18 GHz. In the ideal case, all the OAM beam patterns are symmetric about the z-axis (the blue crossing point in the center). However, in the non-ideal case, the beam patterns show asymmetries about the z-axis and, as a result, they propagate to slightly different directions, thus introducing the beam center mismatch and some mode isolation degradation. Figure 13.Open in figure viewerPowerPoint Simulated OAM far-field patterns using ideal and non-ideal feeding signals with an UCA. IV. OAM Mode Transmission Characteristics To use microwave OAM modes in wireless communications, a certain degree of signal to interference and noise ratio (SNIR) is required. For example, nearly 12 dB of SNIR are required for a 16 QAM, 1 GBd signal [16]. Given that the crosstalk between OAM modes is usually the dominant source of noise, it is important to control both the received power and the mode isolation level. In the OAM mode transmission experiment, we used the previously referred 24 cm diameter Cassegrain antenna, which was fed by the Butler matrix and the feeder. We also used a camera tripod and a pointing laser for easy antenna angle alignment, both at the transmitting and receiving ends. Figure 14 shows the Tx/Rx antenna (including the Butler matrix), and Fig. 15 shows the transmission test chamber. Figure 14.Open in figure viewerPowerPoint OAM mode Tx/Rx antenna with the Butler matrix (BM in the figure). Figure 15.Open in figure viewerPowerPoint Transmission test chamber. Figure 16 shows the test setup configuration. A E8267D signal generator was used for powering the transmission Butler matrix at 0 dBm, and a 8565EC spectrum analyzer was used to measure the received power. Figure 16.Open in figure viewerPowerPoint OAM mode transmission test configuration. Figure 17 shows the measured received power and crosstalk power at distances up to 14 m (in 2 m steps), when single-tone signals for the 0 and −1 modes are individually transmitted at frequencies near 18 GHz. The results for the +1 mode were similar to the ones of the −1 mode, and are therefore omitted. At each distance, the angles of the transmitting and receiving antennas were carefully adjusted, so as to attain both maximum signal and minimum crosstalk power. It should be noted that the obtained results are very sensitive to antenna alignment [17], and therefore our results constitute merely one reasonable case; other experiments could result in better or worse results. Figure 17(a) shows the 0-channel characteristics and Fig. 17(b) shows the −1-channel characteristics. In the 0-channel case, the 0-mode power is approximately 20 dB above the crosstalk with the other two modes throughout the whole region. In contrast, in the −1 channel case, the −1-mode power reduces 10 dB per every doubling of distance, and even though it is 10 dB and 40 dB over the received 0-mode and +1-mode powers at 2 m, respectively, these margins reduce to 5 dB and 25 dB at 10 m, respectively. Therefore, to reach the required 12 dB of SINR, the 0-mode transmitted power should be appropriately reduced, as in [10]. Table 1 shows the calculated triple OAM mode isolation, assuming that the 0-mode transmitted power is optimally reduced. In this context, mode isolation refers the difference between the signal mode power and the larger crosstalk power. Figure 17.Open in figure viewerPowerPoint Received power when only one OAM mode is transmitted: (a) 0 mode channel and (b) −1 mode channel. Table 1. Calculated mode isolation with optimized 0-mode power. Transmission distance (m) 2 4 6 8 10 12 14 Mode isolation (dB) 13.6 13.9 12.3 12.9 12.2 11.2 10.5 For wideband OAM signals the mode isolation level may become even worse, because of the frequency-dependent OAM beam center mismatch. In such conditions, a Butler matrix with better performance is required, to enhance the mode isolation level and the transmission distance. Figure 18 shows a TV signal transmission experiment using the ±1 modes in our laboratory. Figure 18.Open in figure viewerPowerPoint TV signal transmission experiment using the ±1 modes. V. Conclusion The generation and transmission of data in triple OAM mode at a distance of 14 m is experimentally demonstrated using a 24 cm diameter Cassegrain antenna fed by a waveguide Butler matrix and a UCA and operating in the 18 GHz band. The generated OAM modes were verified by spatial field measurements, and demultiplexed by a reversely connected Butler matrix. The radiated OAM beam centers varied with both frequency and mode index, which resulted in mode isolation degradation; this effect is mainly due to the non-ideal characteristics of the fabricated Butler matrix. Higher levels of performance are therefore required from the Butler matrix to achieve longer communication distances. In the presented transmission experiment, the received ±1-mode power exhibited a reduction of approximately 10 dB for each doubling of distance; therefore, the mode isolation was calculated assuming an adequate reduction of the 0-mode transmitted power, and the required value of 12 dB isolation was achieved for narrow-band OAM signals at transmission distances of 10 m. For longer distances and wideband OAM mode wireless communications, high antenna gains and uniform frequency characteristics are very important issues. However, the proposed scheme constitutes a very convenient and compact method for multiple microwave OAM-mode-based communications. Biographies Wangjoo Lee received his BS degree in physics from the Seoul National University, Rep. of Korea, in 1986, and his MS and PhD degrees in physics from the Korea Advanced Institute of Science and Technology, Daejeon, Rep. of Korea, in 1988 and 1999, respectively. From 1988 to 1993, he was with Hyundai Electronics Industry Co., Icheon, Rep. of Korea, where he developed CMOS ACISs and SRAMs. In 2000, he joined ETRI, Daejeon, Rep. of Korea, where he worked for the Microwave Technology Research Section as a research staff principal member. His current interests include microwave circuits, system design, and spectrum engineering. Ju Yeon Hong received her BS and MS degrees in electronic engineering from Dongguk University, Seoul, Rep. of Korea, in 1999 and 2001, respectively. In 2001, she joined ETRI, Daejeon, Rep. of Korea, where she was a design/modeling engineer for wireless and millimeter-wave circuits, such as microwave monolithic integrated circuits (MMIC), for nine years. She is currently working in the Radio Resource Research Group of ETRI and is currently pursuing her PhD in electronic engineering from the Korea Advanced Institute of Science and Technology, Daejeon, Rep. of Korea. Her research interests include RF/microwave/ millimeter-wave circuits, antennas, and electromagnetic analysis. Min Soo Kang received his BS and MS degrees in electric engineering from the Sogang University, Seoul, Rep. of Korea, in 1996 and 1998, respectively, and his PhD degree in electrical engineering from the Chungnam National University, Daejeon, Rep. of Korea, in 2011. From 1998 to 2000, he was with Hyundai Electronics Industry Co., Icheon, Rep. of Korea, where he developed RF transceivers for mobile communication base stations. In 2000, he joined ETRI, Daejeon, Rep. of Korea, and worked for the Microwave Technology Research Section as a research staff principal member. His current interests include microwave circuits, system design, and spectrum engineering. Bong Su Kim received his BS and MS degrees in information communication engineering from the Chungnam National University, Daejeon, Rep. of Korea, in 1999 and 2001, respectively. In 2000, he joined ETRI, Daejeon, Rep. of Korea, where he is currently with the Radio Resource Research Group as a senior member of the engineering staff. His research interests include design and analysis of millimeter-wave active/passive components and application systems. Kwang Seon Kim received his BS and MS degrees in electric engineering from the Kyungpook National University, Daegu, Rep. of Korea, in 1998 and 2000, respectively. In 2000, he joined ETRI, Daejeon, Rep. of Korea, and worked for the Mobile RF Research Section as a principal researcher. His research interests are in RF transceiver technology and millimeter-wave systems. Woo Jin Byun received his BS degree in electronic engineering from the Kyungpook National University, Daegu, Rep. of Korea, in 1992, and his MS and PhD degrees in electrical engineering from the Korea Advanced Institute of Science and Technology, Daejeon, Rep. of Korea, in 1995 and 2000, respectively. In 1999, he joined the Samsung Electro-Mechanics Company, Suwon, Rep. of Korea, where he developed mobile communication devices such as power amplifiers and radio modules from 1999 to 2004. He was with the ATHENA group at the Georgia Institute of Technology, as a visiting scholar, from 2015 to 2016. He is currently a managing director for the Radio Resource Research Group at ETRI, Daejeon, Rep. of Korea. His current research areas include RF/millimeter-wave/THz integrated circuits, system design, planar reflector antennas, and electromagnetic scattering analysis. Myung Sun Song received his BS and MS degrees in electronics engineering from the Chungnam National University, Daejeon, Rep. of Korea, in 1984 and 1986, respectively. Since then, he has worked for the Radio Technology Department of ETRI, Daejeon, Rep. of Korea as a principal member of the engineering staff. He developed microwave and millimeter-wave communication systems, cognitive radio, and WLAN technologies. His research interests include system engineering for cognitive radio, microwave and millimeter wave communication systems, radar, and jammers. Yong Heui Cho received his BS degree in electronics engineering from the Kyungpook National University, Daegu, Rep. of Korea, in 1998, and his MS and PhD degrees in electrical engineering from the Korea Advanced Institute of Science and Technology, Daejeon, Rep. of Korea, in 2000 and 2002, respectively. From 2002 to 2003, he was a senior researcher with the ETRI, Daejeon, Rep. of Korea. In 2003, he joined the School of Information and Communication Engineering, Mokwon University, Daejeon, Rep. of Korea, where he is currently a professor. In 2011, he was on sabbatical leave with the Department of Electrical and Computer Engineering, University of Massachusetts Amherst, MA, USA. His main research interests include electromagnetic wave theory and scattering, design of reflectarrays, and dispersion characteristics of waveguides. References 1L. 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