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공학박사학위논문 Ka- 대역모노펄스반사경안테나 Ka-band Monopulse Reflector Antenna 충북대학교대학원 전기 전자 정보 컴퓨터학부전파통신공학과 Gombo Otgonbaatar 2014 년 2 월

공학박사학위논문 Ka- 대역모노펄스반사경안테나 Ka-band Monopulse Reflector Antenna 지도교수 안병철 전기 전자 정보 컴퓨터학부전파통신공학과 Gombo Otgonbaatar 이논문을공학박사학위논문으로제출함. 2014 년 2 월

본논문을검보오트건바타르의공학박사학위논문으로인정함. 심사위원장심사부위원장심사위원심사위원심사위원 안재형印김경석印안병철印노진입印방재훈印 충북대학교대학원 2014 년 2 월

Ka- 대역모노펄스반사경안테나 * 오트건바타르검보 충북대학교대학원전파통신공학과 지도교수안병철 요약문 본논문에서는유도탄표적탐색용 Ka-대역모노펄스반사경안테나의설계를제시하였다. 안테나는다중모드사각형혼, 도파관형모노펄스비교기및카세그레인반사경으로구성된다. 우선카세그레인반사경의광학설계를제시하였다. 주반사경크기는원하는값의합채널의이득이얻어지도록결정하였다. 부반사경의크기와초점거리를구현가능한다중모드혼의개구면크기와빔폭을고려하여반복적인방법으로구하였다. 카세그레인반사경광학구조와다중모드혼구조를동시에설계함으로써최상의결과를얻었다. 카세그레인반사경에필요한피드의빔폭이주어질경우다중대역혼피드의개구면크기가작을수록좋다. 그이유는피드의크기가작을수록피드에의한차폐와산란이적기때문이다. 다중대역혼의크기는합패턴과차패턴을방사하는사각형개구면의방사특성에의해결정된다. 다중모드혼의설계는안테나전체 * A dissertation for the degree of Doctor in February 2014. i

설계에있어서매우중요하다. 다음으로표적추적용으로적합한다중대역혼을설계하였다. 설계한혼은 4 개의급전포트, 모드발생용계단및단일개구면으로구성된다. 4 개의급전포트에의해도파관이여기될경우이로부터합패턴과차패턴을생성하는모드의조합을발생시키는전계면계단과자계면계단의설계가매우중요하다. 모드발생용계단의최종설계안은반복적인시뮬레이션과수동최적화를통해도출하였다. 전계면계단과자계면계단에연결된단일개구면에미라미드혼구조를추가하여다중대역혼의빔폭을줄일수있다. 피라미드혼의길이는합패턴과차패턴을생성하는모드의위상지연을결정하며이의최적값은혼의방사패턴을관측하면서결정하였다. 최종적으로다중모드혼의개구면크기는카세그레인반사경의광학설계와연계하여반복적인방법으로구하였다. 다중대역혼이정확한방사패턴을내려면 4 개의급전포트에서정확한크기와위상으로급전되어야하며이작업은모노펄스비교기에의해수행된다. 본논문에서는모노펄스비교기를이등분형태의도파관구조를이용하여구현하였다. 모노펄스비교기는 4 개의도파관형 180 링하이브리드와여러가지형태의도파관벤드와직선도파관으로구성된다. 단일 180 링하이브리드를설계한후 4 개의링하이브리드를단일평면에적절히배치하였다. 4 개링하이브리드의입력및출력포트는도파관직선부와벤드를이용하여각각다중대역혼의급전포트와안테나의합채널및차채널포트와연결된다. 도파관모노펄스비교기의구조를이등분가공기법으로가공하기에적합하도록설계하였다. 카세그레인반사경에적용하기전에다중대역혼과모노펄스비교기가연결되었을경우의특성을시뮬레이션하여접합성을검증하였다. 모노펄스비교기에의해급전되는다중모드혼피드가적용된카세그레인반사경의방사패턴을분석하여모노펄스추적안테나의성능을예측하였다. 설계된안테나를제작한후성능을측정하였다. 측정결과제작된안테나는합패턴의경우이득 34.75dB, 3-dB ii

빔폭 3.2, 부엽레벨 -20dB 등의양호한특성을보였다. 차패턴의경우최대이득 32.23dB, 영점깊이 -38dB 및부엽레벨 -21 db 등의양호한특성을보였다. iii

Ka-band Monopulse Reflector Antenna Otgonbaatar Gombo School of Electrical Engineering and Computer Science, Graduate School of Chungbuk National University, Cheongju, Korea Supervised by Professor Bierng-Chearl Ahn, Ph. D. Abstract This thesis presents the design of a Ka-band monopulse reflector antenna to be used for missile seeker applications. The antenna consists of a multimode rectangular horn, a waveguide monopulse comparator, and a Cassegrain reflector. First, the optical design of the Cassegrain reflector antenna is carried out. The main reflector size is determined for a desired sum channel gain. The subreflector size and its focal point are designed in an iterative procedure by considering the aperture size and beamwidth of the realizable multimode horn. Best results are achieved when the Cassegrain optics and the multimode horn are designed concurrently. For a given beamwidth required of the Cassegrain reflector's feed, the smaller the aperture size of the multimode horn, the better the antenna's performance, since there will be smaller blockage and scattering by the feed. * A dissertation for the degree of Doctor in February 2014. iv

The size of the multimode horn is dictated by the radiation properties of the rectangular aperture radiating the sum and difference patterns. Thus the crucial step in the overall antenna development is the design of the multimode horn. Next a multimode horn is designed to achieve characteristics suitable for target tracking applications. The horn has four excitation ports, mode generating steps and a single aperture. The most important part in the multimode horn design is E-plane and H- plane steps generating a required combination of modes that generate sum and difference patterns when properly excited by four input waveguides. A final design of mode generating steps is achieved by repeated numerical simulations and manual optimization. The beamwidth of the multimode horn is sharpened by adding a pyramidal horn structure to the common aperture connected to the E- and H-plane steps. The length of the pyramidal horn controls the phase delay of the modes used in forming the sum and difference patterns so that its optimum value is obtained by observing the horn's patterns. The aperture dimension is determined in an iterative procedure combined with the Cassegrain reflector's optical design. The multimode horn functions accurately only when it is excited in an exact magnitude and phase, the task of which is taken by the comparator. The monopulse comparator is realized using split-block rectangular waveguide techniques. It consists of four 180 ring hybrids and various forms of waveguide bends and runs. A single 180 ring hybrid is designed first. And then four ring hybrids are properly laid out on a single plane. Output and input ports of the four ring hybrids are routed to the multimode horn's excitation ports and the antenna's sum and difference channel ports using waveguide bends and runs. The structure of the waveguide monopulse comparator is designed in such a way that it can be easily fabricated using split-block waveguide techniques. The combined performance of the multimode horn and the comparator is simulated and verified before applying it to the Cassegrain reflector. Finally the monopulse tracking antenna performance is predicted by analyzing the patterns of the Cassegrain antenna fed by the multimode horn and the comparator. The designed antenna is fabricated and its performance is measured. Measurements v

show that the antenna has a sum channel gain of 34.75dB, a 3-dB beamwidth of 3.2, a sidelobe level of -20dB. The difference pattern has a maximum gain of 32.23dB, a null depth of -38dB, and side lobe level of -21 db. vi

Contents 요약문 i Abstract iv List of Figures viii List of Tables xii I Introduction 1 II Monopulse Tracking Antenna 4 2.1 Monopulse Antenna... 4 2.2 Cassegrain Antenna... 7 III Design of Multimode Feed Horn 11 3.1 Multimode Feed Horn Design Requirements... 11 3.2 Multimode Feed Horn Design Theory... 11 3.3 Multimode Feed Horn Design... 26 IV Design of Monopulse Comparator 46 4.1 Monopulse Comparator Design Requirements... 46 4.2 Design of Monopulse Comparator... 46 V Monopulse Reflector Antena 71 5.1 Proposed Reflector Antenna Structure and Requirements... 71 5.2 Reflector Antenna Geometry... 72 5.3 Cassegrain Antenna Simulation... 76 VI Fabrication and Measurement 81 6.1 Multimode Horn Fabrication and Measurement... 82 6.2 Monopulse Comparator Fabrication and Measurement... 87 6.3 Reflector Antenna Fabrication and Measurement... 108 VII Conclusion 118 REFERENCES 120 ACKNOWLEDGEMENTS 126 vii

LIST OF FIGURES Fig. 2.1 Azimuth difference pattern beams... 5 Fig. 2.2 Signal amplitude response of each beam... 5 Fig. 2.3 Sum and difference signal response (a) sum (b) difference... 6 Fig. 2.4 Cassegrain reflector's structure and geometry... 8 Fig. 3.1 Plots of E y in (a) x and (b) y directions for m = 1, 2, 3 and n = 0, 1, 2, 3... 13 Fig. 3.2 Mode summation for the sum pattern... 16 Fig. 3.3 Mode summation for the azimuth difference pattern... 17 Fig. 3.4 Mode summation for the elevation difference pattern... 18 Fig. 3.5 Effect of amplitude ratio in sum pattern E-field distribution(a) x and (b) y direction... 19 Fig. 3.6 Effect of amplitude ratio in difference pattern s E-field distribution (a) x and (b) y direction... 21 Fig. 3.7 Calculated E-field distributions in (a) sum and (b) difference patterns.... 24 Fig. 3.8 Computed radiation patterns of the multimode horn... 26 Fig. 3.9 Composition of the proposed multimode feed horn... 27 Fig. 3.10 Structure of the proposed multimode feed horn (a) A side view and (b) 3D view... 27 Fig. 3.11 Electric fields of the input waveguides of the multimode horn for (a) the sum pattern, (b) the azimuth difference pattern, and (c) the elevation difference pattern... 28 Fig. 3.12 Polarities of the input waveguide excitation for (a) the sum pattern, (b) the azimuth difference pattern, and (c) the elevation difference pattern... 28 Fig. 3.13 Structure of a general H-moder... 29 Fig. 3.14 H-moder of the feed horn: (a) configuration and (b) simulation model... 30 Fig. 3.15 Structure of the E-moder... 32 Fig. 3.16 Structure and design parameters of the multimode feed horn... 33 Fig. 3.17 Ports setting of multimode feed horn... 34 Fig. 3.18 Feed horn reflection coefficients... 34 viii

Fig. 3.19 Simulated 2D aperture distributions of (a) sum, (b) azimuth and (c) elevation difference patterns... 35 Fig. 3.20 Normalized aperture distributions of the sum and difference patterns... 37 Fig. 3.21 3D radiation patterns of the multimode feed horn. (a) Sum channel absolute gain, (b) sum channel theta gain, (c) sum channel phi gain, (d) azimuth difference channel absolute gain, (e) azimuth difference channel theta gain, (f) azimuth difference channel phi gain, (f) elevation difference channel absolute gain, (h) elevation difference channel theta gain, and (i) elevation difference channel phi gain... 38 Fig. 3.22 Sum and difference radiation patterns... 43 Fig. 3.23 Phase pattern in the sum channel... 44 Fig. 3.24 Phase pattern in the difference channels (a) azimuth and (b) elevation... 44 Fig. 4.1 Monopulse comparator design structure... 48 Fig. 4.2 Hybrid ring coupler structure... 48 Fig. 4.3 Hybrid ring coupler simulation model... 49 Fig. 4.4 E-field propagation in hybrid ring coupler (a) sum port (b) difference port... 50 Fig. 4.5 Reflection coefficients of coupler ports... 51 Fig. 4.6 Transmission coefficients... 52 Fig. 4.7 Phase difference at two output ports... 53 Fig. 4.8 Four hybrid ring couplers combination... 54 Fig. 4.9 Round waveguide bend geometry... 55 Fig. 4.10 Final monopulse comparator design (a) front view and (b) back view... 56 Fig. 4.11 Waveguide bend geometry (a) round and (b) stepped... 57 Fig. 4.12 Connecting bend combinations... 58 Fig. 4.13 Reflection coefficients of monopulse comparator. (a) At output ports and (b) input ports... 59 Fig. 4.14 Sum pattern channel's (a) transmission coefficients and (b) phase differences. 60 Fig. 4.15 Azimuth differnce pattern channel's (a) transmission coefficients and (b) phase differences... 61 ix

Fig. 4.16 Elevation differnce pattern channel's (a) transmission coefficients and (b) phase differences... 62 Fig. 4.17 Transition between horn and comparator (a) 3D simulation model (b) insertion and return losses... 64 Fig. 4.18 Combined horn and comparator... 65 Fig. 4.19 Combined monopulse feed horn (a) reflection and (b) isolation coefficients... 65 Fig. 4.20 Far-field plots (a) sum E-plane and elevation difference (b) sum H-plane and azimuth difference... 67 Fig. 5.1 Structure of the monopulse antenna... 70 Fig. 5.2 Cassegrain antenna geometry and design parameter... 72 Fig. 5.3 Relation between blockage efficiency and Ds / Dm... 73 Fig. 5.4 Relation between the subtended angle and Fm / Dm... 73 Fig. 5.5 Cassegrain antenna simulation model with feed horn far-field source (a) sum (b) azimuth (c) elevation pattern (d) reflector itself.... 76 Fig. 5.6 Simulated far-field pattern of Cassegrain antenna (a) azimuth and sum H-plane (b) elevation and sum E-plane (c) sum 3D pattern... 78 Fig. 6.1 Fabricated multimode horn (a) disassembled and (b) assembled... 81 Fig. 6.2 Round waveguide corners for fabrication... 82 Fig. 6.3 Measured reflection coefficients of the multimode horn... Fig. 6.4 Measured far-field patterns of the multimode horn. (a) E-plane pattern of the sum channel, (b) the H-plane pattern of the sum channel, (c) elevation difference pattern, and (d) azimuth difference pattern... 84 Fig. 6.5 Splitting diagram for the fabrication of the monopulse comparator... 87 Fig. 6.6 Fabricated monopulse comparator. (a) block 1, (b) block 2, (c) block 3, and (d) assembled comparator... 88 Fig. 6.7 Measurement setup for the monopulse comparator... 90 Fig. 6.8 Waveguide matched load used for in the monopulse comparator measurements (a) Rectangular waveguide and (b) tapered ferrite absorber... 90 x

Fig. 6.9 Reflection coefficients at the output of the fabricated monopulse comparator. (a) At port 1, (b) at port 2, (c) at port 3, and (d) at port 4... 91 Fig. 6.10 Reflection coefficients at the input of the fabricated monopulse comparator. (a) At port 5, (b) at port 6, and (c) at port 7... 93 Fig. 6.11 Measured transmission coefficient of the sum port. (a) S51 (b) S52 (c) S53 (d) S54... 95 Fig. 6.12 Azimuth difference port transmission coefficient measurements (a) S61 (b) S62 (c) S63 (d) S64... 97 Fig. 6.13 Elevation difference port transmission coefficient measurements (a) S71 (b) S72 (c) S73 (d) S74... 99 Fig. 6.14 Sum channel phase difference (a) at port 1, (b) at port 2,and (c) port 3... 101 Fig. 6.15 Azimuth difference channel phase difference (a) at port 1, (b) at port 2, and (c) at port 3... 103 Fig. 6.16 Elevation difference channel phase difference (a) at port 1, (b) at port 2, and (c) at port 3... 104 Fig. 6.17 Comparator input port isolations (a) S56, (b) S75, (c) S76... 106 Fig. 6. 18 Fabricated main and sub reflectors. (a) Main reflector back side and (b) front side of the main and sub reflectors... 109 Fig. 6.19 Sub reflector on the radome... 110 Fig. 6.20 Radome (a) geometry and (b) design parameter... 111 Fig. 6.21 Overall reflector antenna... 111 Fig. 6.22 Reflection coefficient versus feed horn displacement (a) sum port, (b) azimuth difference port, and (c) elevation difference port... 112 Fig. 6.23 E-field distribution (a) abs value and (b) phase in sum pattern (c) abs value (d) phase in azimuth difference pattern (e) abs value (f) phase in elevation difference pattern... 115 Fig. 6.24 Far-field pattern of the fabricated monopulse reflector antenna (a) E-plane pattern of the sum and azimuth difference channel and (b) H-plane pattern of the sum and elevation difference channel... 117 xi

LIST OF TABLES Table 3.1 Design requirements of the multimode horn... 11 Table 3.2 Mode summary... 14 Table 3.3 Optimum realizable mode amplitudes... 22 Table 3.4 Step size ratio versus amplitude and phase... 31 Table 3.5 Optimized dimensions of the multimode feed horn (mm)... 33 Table 3.6 Ports excitation settings... 33 Table 3.7 Important parameters of the sum and difference radiation patterns... 43 Table 4.1 Design requirements of the monopulse comparator... 46 Table 4.2 Simulation results summery... 68 Table 5.1 Reflector antenna specifications... 71 Table 5.2 Determined parameters... 75 Table 5.3 Determined parameters... 77 Table 5.4 Summarized simulation results of proposed Cassegrain antenna... 80 Table 6.1 Comparator measurement summery... 108 xii

Chapter I Introduction The monopulse antennas are used in the tracking systems, which usually track the aircrafts, missiles or satellites. The tracking system measures its target coordinates, which may be used to calculate the target trajectory and the future position. The target coordinate information may include the elevation angle, the azimuth angle, the distance, and the Doppler frequency shift. The tracking systems can be divided into two types, the continuous tracking system and the track-while-scan system. The continuous tracking system provides continuous tracking data (coordinate data) on a particular target, whereas the track-while-scan system provides sampled data on one or more targets [1]. There are several techniques used in the continuous tracking system such as the sequential lobing, the conical scan, and the monopulse. The monopulse tracking technique uses the monopulse antenna [2]. The monopulse tracking technique uses a resulting single pulse and derives angular error information on the basis of a single pulse. In the monopulse technique, more than one beam are formed simultaneously and then the echo signals are received from respective beams. The received echo signal's amplitude and phase are used to extract the angular error. The monopulse tracking technique can be divided into two forms, the amplitude-comparison monopulse and the phase-comparison monopulse [3]. The monopulse antennas are usually in reflector [4]-[7], lens [8]-[10] or array [11]- [18] forms. The monopulse reflector antennas are usually implemented in prime-focus [4] and Cassegrain forms [5]-[7]. The monopulse array antennas are usually implemented in printed [11]-[16] and slotted [17]-[18] forms. Also there are several different feed networks for the monopulse array antenna [12], [19]-[20]. In the monopulse reflector antenna, the reflector antenna is fed by a feed, which radiates sum and difference patterns. 1

The monopulse reflector antenna feed can be in different forms such as a multimode horn [21]-[29], a dielectric rod [30], a twelve-horn feed [31]-[32], and an array [33]. Multimode horns are usually powered by the four-waveguide feed and its monopulse comparator. Monopulse comparators are usually implemented in waveguide [34]-[36] and microstrip [37]-[41] forms and it is composed of 180 hybrid couplers [42]. Researchers Qian Song-song, Li Xing-guo, Wang Ben-qing designed a Ka-band monopulse antenna [5]. They implemented their design with a Cassegrain reflector, the four dielectric tapered rod antennas as the feed, and a monopulse comparator. In this design the dielectric tapered rod antennas form the sum and the difference patterns just like other types of feed horns. In order to form the sum and difference patterns the rod antennas have to be excited with various phase settings. Thus they designed waveguide monopulse comparator which is composed of four magic-t couplers and a few waveguide bends. Using the dielectric rod antennas as the feed, the blockage caused by the feed is reduced, and the wideband performance is achieved. Also the dielectric rod antennas can be fabricated precisely compared to the horn antennas. The sub reflector is supported by a tripod. Using a tripod has advantages such as low cost, simple fabrication, and less effort in the antenna performance calculation compared to other feed supporting techniques. In this thesis, a Ka-band monopulse Cassegrain antenna is developed for missile seeker applications. The development of the proposed antenna involves designing a multimode feed horn, a monopulse comparator and a Cassegrain reflector. In the multimode feed horn design, the E-field distribution transforming method is implemented. The implementation is done by using H- and E- plane moders (steps) with particular phasing sections. Also the desired radiation pattern beamwidth and gain are achieved by properly adjusting the aperture size in a pyramidal horn structure. In the monopulse comparator design, 180 degree hybrid ring couplers are employed. The proposed monopulse comparator is composed of four 180 hybrid ring couplers and various types of the waveguide bends. The Cassegrain antenna's initial dimensions are calculated from 2

the overall antenna gain and the multimode feed horn specifications such as aperture diameter and the edge taper angle. The Cassegrain antenna design is optimized using the multimode feed horn radiation pattern data. The Microwave Studio 2012 by CST is used during the all components simulation and optimization. This dissertation is organized as follows. Chapter II introduces the structure of the monopulse antenna and its operating principle, and a general Cassegrain antenna structure. Chapter III discusses the multimode feed horn design theory, and its design and design procedure. Chapter IV discusses the monopulse comparator's operating principle and its design including the 180 hybrid ring coupler. Chapter V discusses the design of the Cassegrain reflector antenna or its geometry calculation and the reflector antenna simulation. Chapter VI gives the fabrication and measurement information of each component. Finally Chapter VII provides the conclusion and discussion. 3

Chapter II Monopulse Tracking Antenna 2.1 Monopulse Antenna There are several radio signal tracking methods such as the sequential lobing technique, the conical scan and the monopulse tracking. In the sequential lobing technique, the antenna beam is switched between two beams in the horizontal or vertical direction. The received signal amplitudes from each beam are the key information for determining the received signal direction. The conical scan method is an extension of the sequential lobing technique. Instead of switching the beams, the antenna itself rotates around its boresight axis. The monopulse technique is similar to the conical scanning method in concept, but the monopulse antenna splits its radiating beam into more than one beam in slightly different directions instead of rotating and then sends a resulting pulse signals out of the antenna. The signals from the monopulse antenna are reflected if there is any target in their traveling paths. The monopulse system picks the reflected signals up in each of the split beams and amplifies separately and then compares their amplitude or phase to each other so that the target direction can be determined. Since the direction where the target moving is determined, tracking the target would be possible. The monopulse tracking method which compares the received signal's amplitude is called "amplitude-comparison monopulse", whereas the monopulse tracking method which compares the phases is called "phase-comparison monopulse". The amplitude comparison monopulse method uses two overlapping radiation patterns in azimuth and elevation direction to obtain the azimuth or elevation angular error, and those patterns are called the sum, azimuth and elevation difference patterns. For instance, one-directional monopulse measurement process is explained as follows. The direction 4

could be either the azimuth or elevation. As mentioned above, the monopulse tracking antenna has more than one radiating beam. To measure angular errors in the azimuth direction, the azimuth difference pattern is used, which includes two beams as shown in Fig. 2.1. Reflected or received signal's response of each beam can be defined as functions f 1 (θ) and f 2 (θ) given by f ( q ) = f ( q ) = f ( q - q ) (2.1) 1 1 0 2 2 0 k f ( q ) = f ( q ) = f ( q + q ) (2.2) k Fig. 2.2 shows the signal amplitude response of each beam (amplitude versus angle). Fig. 2.1 Azimuth difference pattern beams Fig. 2.2 Signal amplitude response of each beam Received signals from each beam are subtracted to form a difference response or error signal as shown in Fig. 2.3(a). This error signal or difference response is given by: D ( q ) = f ( q ) - f ( q ) (2.3) 1 2 5

This difference response is used as a feedback signal in the closed-loop system of monopulse tracking. A null is formed in the middle of the two beams. The monopulse tracking system keeps its target within the null of the difference pattern. When the target is within the null, the error signal becomes very small due to the subtraction of signal responses. This type of the system is called a "null tracker". The error signal becomes very small when the target enters the null region or gets out from the radar range or enters the null region that is not in the tracking direction. These conditions could lead to a wrong tracking direction. In order to overcome this situation, one more signal response is used which is the sum response. The sum response is a summation of signal responses of each beam. The sum response is given by S ( q ) = f ( q ) + f ( q ) (2.4) 1 2 The sum response function is shown in Fig. 2.3 (b). (a) (b) Fig. 2.3 Sum and difference signal response (a) difference and (b) sum The sum response is actually used for target detection and to avoid unambiguous tracking conditions. If the offset q k is very small, the difference pattern expression can also be written as follows. ( ) D ( z)» 2 q f ' q = 2q d f / dq (2.5) k 0 k 6

The difference response normalized by the sum response is given by: 1( ) 2( ) d / d / f q - f q qk f q D å =» f ( q ) + f ( q ) f ( q ) 1 2 0 (2.6) In the difference response shown in Fig. 2.3(a), the slope crossing the zero point on the measurement axis is called the difference slope of the monopulse measurement. The rate of change in the slope of the curve at this point expresses the relative measurement sensitivity of the system. A sharply rising slope indicates a high sensitivity, and a slow rising slope indicates a low sensitivity. The normalized difference slope as a differential function is given by: k m d( D / S) = - (2.7) d( q / q ) 3 q = q 0 where q3 is the 3-dB beamwidth. The equation (2.8) expresses the fundamental relationship of the RMS position error of the monopulse estimate in a thermal noise environment. q3 q3 q3 s q = =» (2.8) k 2 E / N k 2( S / N) 2 ( S / N) m 0 m n n 2.2 Cassegrain Antenna The Cassegrain antennas are widely used in telecommunication and radar systems. A Cassegrain antenna is a reflector antenna that a feed antenna is mounted at or near the surface of a concave main reflector and is aimed at a convex sub-reflector. Energy from 7

the feed antenna illuminates the sub-reflector, which reflects it back to the main reflector, which then forms the desired forward beam. The Cassegrain antenna has many advantages. The feed antenna is easily supported at the back side of the main reflector and it makes the antenna geometry compact. A receiver unit or circuit is attached directly to the feed antenna thus loss is low. The sub-reflector illuminates the main reflector more uniformly thus the gain is maximized. The focal length is longer than the prime focus antenna and it improves the cross polarization discrimination. Another advantage is that the feed antenna is directed forward so the spill-over sidelobes are directed to the sky which prevents the ground noise. The Cassegrain reflector antenna generally consists of a main reflector, a sub-reflector and a feed. Fig. 2.4 shows the Cassegrain antenna structure and geometry. From the Fig. 2.4 it can be seen that reflected wave from the main reflector travels to +z direction. The point where the two reflectors focal points exist is called the virtual focal point. x Actual Focal Point D m D b f r D s f n Virtual Focal Point z Feed Horn L r L v F S F C F m Fig. 2.4 Cassegrain reflector's structure and geometry 8

In Cassegrain antenna design the main reflector surface geometry can be calculated in the same way as in the prime-focus reflector case and it is expressed as follows. 2 m m m m x = 4 F ( z + F ) (2.9) The main reflector diameter, the actual focal point, the sub-reflector diameter, and the virtual focal point satisfy the following relations. -1 Dm fv = 2tan ( ) (2.10) 4F m 1 1 Fc 2 tanf + tanf = D (2.11) v r s 1 sin ( fv -fr ) 1 2 L - = 2 1 sin ( fv + fr ) F 2 v c (2.12) The sub-reflector illuminated from the actual focal point acts like that it is illuminating the main reflector from the virtual focal point. The sub-reflector's surface geometry is given by x s æ zs + a + Lr ö = b ç -1 a è ø 2 (2.13) where a and b are the eccentricities of the hyperbolic function and can be obtained from following equations. é1 ù sin ê ( fv -fr ) 2 ú e = ë û é 1 ù sin ê ( fv + fr ) ë 2 ú û (2.14) 9

F c a = (2.15) e 2 b = a e - 1 (2.16) In a typical Cassegrain antenna design, the antenna gain, the half power beamwidth and the side lobe level are important characteristics. The electrical performances of the main reflector and the sub-reflector design are calculated from the feed horn edge taper level and the reflector's aperture diameter. 10

Chapter III Design of Multimode Feed Horn 3.1 Multimode Feed Horn Design Requirements There are several essential requirements in designing a multimode monopulse feed horn such as the reflection coefficient, the radiation pattern symmetry, the edge taper level, the horn aperture size, and the side lobe level. The horn aperture should be as small as possible so that the center blockage caused by the feed horn is small. The center blockage of the reflector radiation pattern directly depends on the feed horn aperture size. But the small horn aperture gives a broad beamwidth and the big horn aperture gives a narrow beamwidth. The horn aperture size is an important consideration in meeting the design requirements. Table 3.1 shows the multimode feed horn antenna requirements. The edge taper is specified at an angle where the feed sees the sub-reflector edge. Table 3.1 Design requirements of the multimode horn Items Edge taper Sidelobe at 32.8 level Σ 8-20 db -20 db AZ-Δ 3-12 db -15 db EL-Δ 3-12 db -15 db Polarization Vertical and linear Maximum aperture size 3.5λ 0 x 3.5λ 0 Operating frequency 34.75-35.25 GHz 3.2 Multimode Feed Horn Design Theory Designing a multimode feed horn involves forming the E-field distribution at the horn aperture to get the desired radiation pattern. In order to obtain a suitable E-field 11

distribution, we use more than one higher-order propagating modes. The desired E-field distributions are obtained in the following steps. First we choose high-order modes and next make a summation of chosen higher-order modes and finally determine the amplitude of each higher-order mode. After doing all these steps, the design implementation begins. This section describes the E-field distribution forming steps. A. Choosing High-order Modes In order to obtain a desired E-field distribution, we use a sum or combination of more than one higher-order propagating mode. Higher-order modes are chosen as follows. General equations for the E-field of the TE mn and TM mn propagating modes in a rectangular waveguide are given by following equations [43], [44] for TE mn modes mp x np y - jb cos cos mnz H z = Amn e (3.1) a b jwmnp mp x np y - jb cos sin mnz Ex = Amn e (3.2) k b a b 2 c - jwmmp mp x np y - jb E sin cos mn y = Amn e k a a b 2 c z (3.3) and for TM mn modes with mp x np y - jb sin cos mnz Ez = Bmn e (3.4) a b - jb mp mp x np y - jb cos sin mnz Ex = Bmn e (3.5) k a a b mn 2 c - jb np mp x np y - jb sin cos mnz Ey = Bmn e (3.6) k b a b mn 2 c 12

2 2 2 2 2 2 2 mn k kc, k, kc ( m / a) ( n / b) b = - = w me = p + p (3.7) where a and b are the width (in x direction) and the height (in y direction) of the waveguide cross section, A mn and B mn are modal amplitudes, k c is the cutoff wave number and β mn is the propagation constant. Since the feed is required to radiate a vertically polarized wave, we consider the y component of the electric field (E y ) of various modes. Different propagating modes of the E y component plots in x and y directions are shown in Fig. 3.1. It is obvious that we should use m = 1, 3, 5, and n = 0, 2, 4, modes for the sum pattern, since these E- field distributions should have an even symmetry in both x and y directions. Similarly, for the azimuth difference pattern, we use m = 2, 4, 6, and n = 0, 2, 4,..., while for the elevation difference pattern we use m = 1, 3, 5,...and n = 1, 3, 5, (a) (b) Fig. 3.1 Plots of E y in (a) x and (b) y directions for m = 1, 2, 3 and n = 0, 1, 2, 3. 13

Another consideration is the number of higher-order modes to be used in the multimode horn antenna design. Using many higher-order modes increases difficulties in implementing a multimode horn antenna. For the sum pattern, we use the TE 10, TE 30, TE 12, and TM 12 modes to obtain a bell-shaped distribution in both x and y directions. When n is not zero, both TE mn and TM mn modes have a non-zero E x, which is the cross polarized component. From (3.1)-(3.6), it follows that the E x of the TE mn mode is cancelled by that of the TM mn mode when the following condition is met. wmn A b mn bmnm = B mn (3.8) a With n 0, we use both TE mn and TM mn modes to have the E x cancelled. We denote a proper combination of the TE mn and TM mn modes as the HE mn mode. The modes for the azimuth and elevation difference patterns can be chosen in the similar way. In the azimuth difference pattern, we may use only one mode which is the TE 20 mode to simplify the design. In this case the E-field distribution along the vertical axis will be uniform. Therefore, a better choice is to utilize the TE 20 and HE 22 modes together. In the elevation difference pattern, HE 11, HE 13 and HE 31 modes were chosen. Finally, Table 3.2 shows the modes used the sum and difference patterns. Table 3.2 Mode summary. Channel H-moder E-moder TE Sum TE 10 TE 10, TE 10 TE 10, HE 12 30 TE 30 TE 30 Az. diff. TE 10 TE 20 TE 20 TE 20, HE 22 TE El. diff. TE 10 TE 10, TE 10 HE 11, HE 13 30 TE 30 HE 31 14

B. Mode Summation Since the modes utilized in the sum and difference patterns have been chosen, the mode summation for each pattern can be expressed easily. Assuming that A 10, A 30, A 12, A 20, A 22, A 11, A 13, and A 31 are mode amplitudes for each mode utilized in the sum and difference patterns, their values will be determined in a later section so that the E-field distributions in the horizontal and vertical directions are as close as possible to the desired shape. First, the aperture distribution of the sum pattern shown in Fig. 3.2 can be expressed by following equations p x 3p x p x 2p y E A A A a a a b y å = 10 sin + 30 sin + 12 sin cos (3.9) At y = b/2 (maximum amplitude), we have p x 3p x p x E A A A a a a y å = 10 sin + 30 sin - 12 sin (3.10) and at x = a/2 (maximum amplitude), we have E A A A 2p y b y å = 10-30 + 12 cos (3.11) Similarly for the azimuth difference pattern, E y is given by, 2p x 2p x 2p y E A A a a b yd = 20 sin + 22 sin cos (3.12) az At y = b/2, we have 2 p x 2 p x E A A a a yd = 20 sin - 22 sin (3.13) az 15

and at x = a/4, we have E A A 2p y b yd = 20 + 22 cos (3.14) az The mode summation for the azimuth difference pattern is shown in Fig. 3.3. A 10 sin( p x / a) A 30 sin(3 p x / a) A 12 sin( p x / a) A10 30 A A 12 cos(2 p x / b) Fig. 3.2 Mode summation for the sum pattern 16

TE 20 HE 22 x A 20 sin(2 p x / a) A 22 sin(2 p x / a) A 20 A 22 cos(2 p y / b) y Fig. 3.3 Mode summation for the azimuth difference pattern Finally, E y for the elevation difference pattern is given by, p x p y 3p x p y p x 3p y E A A A a b a b a b yd = 11 sin cos + 31 sin cos - 13 sin cos (3.15) el At y = b/4, we have 1 æ p x 3p x p x ö EyD = A11 sin A31 sin A13 sin el ç + - 2 è a a a ø (3.16) and at x = a/2, we have p y p y 3p y E A A A b b b yd = 11 cos - 31 cos + 13 cos (3.17) el Fig. 3.4 shows the mode summation for the elevation difference pattern. 17

HE 11 HE 31 HE 13 x A 11 sin( p x / a) A 31 sin(3 p x / a) A 13 sin( p x / a) y A 11 cos( p y / b) A 31 cos( p y / b) A 13 cos(3 p y / b) Fig. 3.4 Mode summation for the elevation difference pattern C. Determining Mode Amplitudes By properly adjusting the mode ratios (amplitude ratios), the E-field distribution can be obtained to satisfy the beamwidth and sidelobe requirements in Table 3.1. First for the sum pattern, from equations (3.9)-(3.11) and Fig. 3.2, we should have A 10 = 1 (reference amplitude) (3.18) A + A = A (3.19) 10 30 12 to let E y = 0 at y = 0 and y = b. Here we have only one equation and two unknowns A 30 and A 12. Defining r = A 30 /A 12, we adjust r until E y in the vertical and horizontal distributions get symmetric (in x and y directions). Fig. 3.5 shows the plots of E y in x and y directions depending on r. For each case we calculate the sum pattern and check how closely the design requirements of Table 3.1 are met. The result is r = 0.3. Since r is known A 12 and A 30 can be determined. A = - 0.43, A = - 1.43 (3.20) 30 12 18

(a) (b) Fig. 3.5 Effect of the amplitude ratio in the sum pattern E-field distribution. (a) x and (b) y direction For the azimuth difference pattern, we use equations (3.12)-(3.14) and Fig. 3.6 to 19

obtain the following condition. A = A (3.19) 20 22 so that E y vanishes at y = 0 and y = b. As shown in Fig. 3.3, the E-field distribution in the horizontal and vertical directions for the azimuth difference pattern is neatly realized using only two modes TE 20 and HE 22. Finally for the elevation difference pattern, we should have A 11 = 1 (reference amplitude) (3.20) A + A = A (3.21) 11 31 13 to let E y = 0 at y = 0 and y = b. Here again we have only one equation and two unknowns A 31 and A 13. Defining s = A 31 / A 13, we adjust s until the E y 's horizontal distribution becomes same as the sum pattern distribution and the vertical distribution becomes same as the azimuth difference pattern's horizontal distribution. Fig. 3.6 shows the plots of E y in the x and y directions versus the mode amplitude ratio s. From these graphs we can see the optimum value of s = 0.3. Since s is known, A 13 and A 31 can be determined as follows. A = - 0.43, A = - 1.43 (3.22) 31 13 In the above analysis, amplitudes of higher-order modes A 12 and A 13 are greater than the fundamental TE 10 mode, which is impossible to realize in reality using the step junction in a rectangular waveguide. In practice there is a maximum realizable amplitude level of the higher-order modes using the E- or H-plane steps and therefore we have to determine mode amplitudes with this constraint [44]. Table 3.3 shows a realizable set of mode ratios. For the optimum mode amplitudes derived above, we let them as close as possible to realizable values, which inevitably leads to sub-optimum aperture distributions. 20

(a) (b) Fig. 3.6 Effect of the amplitude ratio in the difference pattern's E-field distribution. (a) x and (b) y direction 21

Table 3.3 Optimum realizable mode amplitudes Σ AZ-Δ EL-Δ A 10 A 30 A 12 A 20 A 22 A 11 A 31 A 13 1.0-0.2-0.9 1.0-1.0 1.0-0.6-0.9 D. Determination of Lengths for Phase Sections and Horn In obtaining a good E-field distribution at the horn aperture, it is very important to make the modes used to form a desired aperture distribution be all in the same phase. The phase of each mode at the aperture is determined by the lengths of the phase sections A, B, and C, the length of the radiator horn D, and the average phase velocity (phase constant) b mn E H of the higher-order modes. b mn and b mn are phase velocites of TE mn and TM mn. E H a mn and a mn are initial phase velocities at the E-moder and H-moder. Let's denote A and B, C, D with l, l 1, l 2 respectively. In the sum pattern case, to keep the modes in the same phase, it is required that l, l 1, and l 2 satisfy the following equations: E E E 10 12 1 10 12 12 ( b - b ) l + ( b - b ) l - a = 2 pp (3.23) H H E E H 10 30 2 10 30 1 10 30 30 ( b - b ) l + ( b - b ) l + ( b - b ) l - a = 2qp (3.24) Similarly in the azimuth and elevation difference patterns cases, it is required that l, l 1, and l 2 satisfy the following equations: E E E 20 22 1 20 22 22 ( b - b ) l + ( b - b ) l - a = 2rp (3.25) E E E 11 13 1 11 13 13 ( b - b ) l + ( b - b ) l - a = 2tp (3.26) 22

H H E E H E E 10 30 2 11 31 1 11 31 22 11 31 ( b - b ) l + ( b - b ) l + ( b - b ) l -a -a - a = 2kp (3.27) b Here, p, q, r, t, k are all integers. mn is the propagation constant of TE mn or TM mn in the moders which is given by: b mn 2 2 2 æ mp ö æ np ö = k - ç - ç è a ø è b ø (3.28) b mn is the average value of propagation constant TE mn and TM mn mode, which is given by: b mn l 2 2 2 æ mp l ö æ np l ö k ç ç dz (3.29) ò 0 a1l + ( a - a1) z b1 l + ( b - b1 ) z = - - è ø è ø The E-field distributions of the sum and the difference patterns obtained from the realizable mode ratio are shown in Fig. 3.7. Using software simulation instead of mathematically calculating the lengths of the horn and the phasing sections is a simpler way to find the optimum lengths. In this study the higher-order modes phases were determined by using CAD software. In Fig. 3.7(a), the aperture distributions for the sum pattern in the x and y directions closely match each other for x/a or y/b from 0.13 to 0.87. The aperture distribution in the y direction for the azimuth difference pattern nicely resembles a bell shape with a taper higher than that for the sum pattern. The aperture distribution in the x direction for the elevation pattern is bell-shaped with a highest taper. In Fig. 3.7(b), the anti-symmetric aperture distributions for the azimuth and elevation difference patterns closely match each other. The aperture distribution for the elevation pattern does not go to zero at edges due to the limitations in the realizable amplitudes of higher-order modes. 23

The sum and the difference radiation patterns can be computed from their calculated E-field distributions by integrating the aperture field. The computed radiation patterns are shown in Fig. 3.8. Fig. 3.7 Calculated E-field distributions in (a) sum and (b) difference patterns. (a) 24

(b) Fig. 3.8 Continued Radiation patterns in Fig. 3.8 show good performances for the sum and difference patterns. The symmetries in E- and H-plane sum and difference patterns are excellent. The sum pattern shows an edge taper of 17 db at 32.8, a sidelobe level of -25 db. The azimuth difference pattern shows a null depth greater than 45 db relative to the sum pattern maximum, an edge taper of 11 db at 32.8, and a sidelobe level of -21.5 db. The elevation difference pattern shows a null depth greater than 45 db relative to the sum pattern maximum, an edge taper of 11 db at 32.8, and a sidelobe level of -25 db. 25

Fig. 3.9 Computed radiation patterns of the multimode horn 3.3 Multimode Feed Horn Design The composition and structure of the proposed multimode feed horn are shown in Fig. 3.9 and Fig. 3.10 respectively. The proposed feed horn antenna consists of four input feed waveguides, H-moders with phasing sections A and B, an E-moder with phasing section C. The pyramidal horn has a length of D. The input feed waveguides are used to transmit the waves from the monopulse comparator to the horn moders. The moders convert part of the propagating mode TE 10 's energy into those of higher-order modes. The phasing sections guide the fundamental and higher-order propagating modes to the horn. The phasing sections are simple rectangular waveguides with a certain length that gives a desired phase delay. 26

Fig. 3.10 Composition of the proposed multimode feed horn A C D Feed waveguides B H-moders E-moder (a) Horn (b) Fig. 3.10 Structure of the proposed multimode feed horn. (a) A side view and (b) 3D view A. Operating Principles It is well-known that the multimode monopulse feed horn has three different types of radiation patterns the sum pattern, the azimuth difference pattern and the elevation 27

difference pattern. In the proposed multimode horn design, those three different radiation patterns can be obtained by exciting TE 10 modes in four input waveguides with correct phase combinations. Fig. 3.11 shows the electric fields of the input waveguides for the sum pattern, the azimuth difference pattern, and the elevation difference pattern. Polarities of the input waveguide excitation are shown in Fig. 3.12. (a) (b) (c) Fig. 3.11 Electric fields of the input waveguides of the multimode horn for (a) the sum pattern, (b) the azimuth difference pattern, and (c) the elevation difference pattern + + - + + + + + - + - - (a) (b) (c) Fig. 3.12 Polarities of the input waveguide excitation for (a) the sum pattern, (b) the azimuth difference pattern, and (c) the elevation difference pattern Four input waveguides are excited in the fundamental TE 10 mode by the monopulse comparator. Higher-order modes, which characterize the multimode horn antenna radiation pattern together with the fundamental TE 10 mode, are generated in the H- and E- moders due to abrupt steps in H- and E-planes. The pyramidal horn is used for controlling 28

the beamwidth of the radiation pattern. Individual higher-order mode's phases are adjusted by changing the lengths of the H-moders, E-moder and the horn (A, B, C and D parameters in Fig. 3.10). B. Design of the H- and E-moders Fig. 3.13 shows the structure of of a general H-moder. As shown in Fig. 3.13, the H- moder consists of a step discontinuity in the x direction at z = 0. Guided waves travel toward the +z direction. Starting from z = 0, TE 10 and TE 30 modes propagate together through the phasing sections following the H-moder. Fig. 3.13 Structure of a general H-moder The H-moders used in proposed multimode feed horn are not same as a general H- moder shown in Fig. 3.13. In proposed multimode horn design, we use a two H-moder combination as shown in Fig. 3.14. In the sum and elevation difference pattern case, the two H-moders are excited with the TE 10 mode and the TE 30 mode is generated by the discontinuity at z = 0. In the following phasing section, both TE 10 and TE 30 modes propagate. In the azimuth difference pattern case, the two H-moders are excited with the TE 10 mode with 180-degree phase difference and the TE 20 mode is generated at the discontinuity. In this case, there is only one mode which is the TE 20 mode propagating in the phasing section. 29

(a) (b) Fig. 3.14 H-moder of the feed horn. (a) Configuration and (b) simulation model In the design shown in Fig. 3.14, the parameter a 0 is chosen to be as small as possible for the TE 10 mode and the parameter a s is chosen to excite the TE 30 mode. This means that the TE 10 mode cut-off frequency of the input waveguide with a width of a 0 is slightly below the operating frequency. The TE 30 mode cut-off frequency of the H-moder having a width of 2a s is also slightly below the operating frequency. The mode amplitude ratios mentioned in the previous section (A 11, A 20, A 30...) are obtained from simulation using the Microwave Studio TM 2012 by CST. The mode ratios including amplitudes and phases are extracted from the simulated S parameters of the H- moder. The simulation model is shown in Fig. 3.14(b). Table 3.4 shows the step size ratio of the simulation model versus the corresponding amplitude and phase. 30

Step-size ratio a 1 /a 0 Table 3.4 Step size ratio versus amplitude and phase. TE 10 mode TE 20 mode TE 30 mode Amplitude Phase Amplitude Phase Amplitude Phase 1.05 0.6820 261.0 0.70685 251.4 0.35330-83.58 1.1 0.683 261.4 0.70610 251.8 0.35820-84.6 1.15 0.6835 261.7 0.70510 252.2 0.35960-85.5 1.2 0.6845 261.8 0.70400 252.5 0.35825-86.4 1.3 0.6870 261.7 0.70170 253.4 0.34655-88.07 1.4 0.6895 261.2 0.69945 254.4 0.32045-89.89 1.5 0.6940 259.9 0.69735 255.3 0.25685-92.8 Judging from Table 3.4, the amplitudes and phases of the TE 10 and TE 20 modes do not change much as the step ratio changes compared to those of the TE 30 mode. It means that the TE 30 mode is more sensitive than the other two modes. Designing the E-moder can be done in the exactly same way as in the H-moder case. Fig. 3.15 shows the E-moder configuration. The E-moder is excited by the two H- moder phasing sections A and B. In the sum pattern case, the E-moder is excited with the TE 10 and TE 30 modes from the two H-moder phasing sections, then the higher-order mode HE 12 is generated at the discontinuity (z=0) and then the TE 10, TE 30 and HE 12 mode waves propagate through the E-moder phasing section. In the azimuth difference pattern case, the E-moder is excited with the TE 20 mode and then the HE 22 mode is generated at the discontinuity. In the elevation difference pattern case, the E-moder is excited by the TE 10 and TE 30 modes in 180-degree phase and then the HE 11, HE 13 and HE 31 modes are generated at the discontinuity. The E-moder has been optimized in the same way as the H-moders. 31

Fig. 3.15 Structure of the E-moder In this design step, the critical design parameters are a 1, a 2, b 1, and b 2. The next step is to determine the waveguide dimensions a 1 and b 1 at the horn throat. From the waveguide transmission theory [44], we know that, if the conditions l / a1» 0.54 and l / b1» 0.62 are satisfied, only selected modes could be propagated though the waveguide. Thus, the dimensions a 1 and b 1 are determined, and then, a 2 and b 2 could be determined automatically from the step-size ratios. C. Optimized Feed Horn Design The design parameters of the proposed multimode horn have been obtained using the methods described above and a CAD simulation tool. The structure of the multimode feed horn is shown in Fig. 3.16. The optimized parameters of the multimode feed horn are shown in Table 3.5. 32

Fig. 3.16 Structure and design parameters of the multimode feed horn Table 3.5 Optimized dimensions of the multimode feed horn (mm) a a 1 a 2 a th b b 1 b 2 b th l l 1 l 2 19.51 11.07 4.6 1.5 16.51 9.48 2.63 1.5 20 2.3 18.3 D. Simulation and Results The simulation of the proposed horn antenna with dimension given in Table 3.5 has been done in the following way. Waveguide ports are defined at the four input waveguides with appropriate phase relations for the sum, azimuth and elevation difference patterns as shown in Table 3.6. Fig. 3.17 shows the simulation model and its port attachment. Table 3.6 Port excitation settings Port Sum Azimuth difference Elevation difference number Amp. Phase Amp. Phase Amp. Phase 1 1 0 1 180 1 180 2 1 0 1 0 1 180 3 1 0 1 180 1 0 4 1 0 1 0 1 0 33

Fig. 3.17 Ports setting of multimode feed horn The simulation was performed in the frequency range of 33 GHz to 37 GHz. Fig. 3.18 shows the reflection coefficients of different port excitations. The reflection coefficients for the sum and difference pattern configurations are less than -10 db over 34.2-35.7 GHz. The sum pattern configuration shows the widest bandwidth (34.2-37.0 GHz) although the impedance matching is slightly inferior to other cases. Due to the non-symmetry in the structure, azimuth and elevation difference configurations show slightly different reflection coefficients. The elevation difference case shows the best impedance matching. Reflection Coefficient (db) 0-10 -20-30 -40 Sum AZ EL 33 34 35 36 37 Frequency (GHz) Fig. 3.18 Feed horn reflection coefficients 34

The simulated 2D E-field distribution at the horn aperture is shown in Fig. 3.19. The magnitude of the aperture electric field for the sum pattern is of a circular symmetric shape although not perfect. Due to limitations in the realizable amplitudes of higher-order modes, the distribution in the y direction is not exactly same as that in the x direction. It is, however, far superior than the uniform distribution of the TE 10 mode alone. The same remarks apply to the aperture distributions in the y direction for the azimuth and elevation difference patterns. The anti-symmetric aperture distributions for the azimuth and elevation patterns do not match each other closely. A detailed quantitative analysis can be done using the 1D aperture field distributions shown in Fig. 3.20. First, one can notice that the realized aperture distributions are significantly different from the sub-optimal distributions shown in Fig. 3.7. In the sum pattern case, the aperture distribution in the y direction does not closely match that in the x direction. The field at the top and bottom aperture edges is as large as 0.4 compared to 0 at the left and right edges. The anti-symmetric aperture distribution for the elevation difference pattern is significantly different from that for the azimuth difference pattern. All of the above mismatches in the aperture distributions are due to the limitations inherent in the multimode horn itself. (a) Fig. 3.19 Simulated 2D aperture distributions of (a) sum, (b) azimuth and (c) elevation difference patterns 35