# Use of microwave lenses in phase retrieval microwave holography of reflector antennas

код для вставкиСкачатьINFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Photographs included in the original manuscript have been reproduced xerographicaliy in this copy. Higher quality 6” x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. ProQuest Information and Learning 300 North Zeeb Road. Ann Arbor. Ml 48106-1346 USA 800-521-0600 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Use of Microwave Lenses in Phase Retrieval Microwave Holography of Reflector Antennas A Thesis Presented to The Academic Faculty by Wonchalerm Chalodhorn In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy School of Electrical and Computer Engineering Georgia Institute of Technology March 2002 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3046879 UMI* UMI Microform 3046879 Copyright 2002 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ii Use of Microwave Lenses in Phase Retrieval Microwave Holography of Reflector Antennas Approved: David ^ D e B o e r, Comimttee Chair Andrew F. Peterson Paul G. Steffes Date Approved J j M A X 2 0 0 7 . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. DEDICATION To my beloved grandmother Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENTS I am indebted to many people for their contributions in the completion of this thesis. In particular, I am indebted to my advisor, Dr. David DeBoer, for introducing me to this interesting topic and his guidance throughout my graduate career. I would also like to thank him for not only his technical advices, but also his friendship and assistance. I am indebted to Dr. Paul Steffes for his invaluable advice and participation in this work. I also thank him for introducing me to this project and for his tremendous guidance, especially during the past two years. I would like to thank the following individuals: Dr. Andrew Peterson for his invaluable input on this dissertation, Dr. Douglas Williams and Dr. Robert Roper for spending their time as committee members of this thesis, and Dr. Monson Hayes for serving as a committee member of my qualifying examination. I thank those who provided assistance and participated in this work including Tim Gilbert, Frederick Hidle, Dr. James Hoffman, Dr. Jeff Piepmeier, and other individuals who participated in the renovation of the Georgia Tech Woodbury Research Facility, which played a significant role in this thesis. I also thank my “fifth floor” inhabitants: Scott Borgsmiller, Priscilla Mohammed, Allen Petrin, and Bryan Karpowicz for their friendship and support. I am forever grateful to my father and mother, Chookiat and Niramol Chalodhom, for their love, understanding, and support that they have always given me. Special thanks go to my aunt, Ampha, my sister, Passaporn, and other relatives for their love and encouragement. My long-time roommates, Worayot Lertniphonphun and Isara Indra, also deserve special thanks for their friendship and assistance. I would like to thank several friends for their constant support and encouragement: Keerati Leungchookiat, Orapin Asarangchai, Chanin Nilubol, Pas an Kulvanit, Wichit Saiklao, Krit Athikulwongse, Weeraya Pantumanee, and Chulaphan Bunjongsat to name a few. This work was supported in part by the SETI Institute, Mountain View, California, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. V under subcontract 430-9601. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. vi TABLE OF CONTENTS DEDICATION iii ACKNOWLEDGEMENTS iv LIST OF TABLES ix LIST OF FIGURES x SUMMARY I xv INTRODUCTION 1 1.1 Introduction............................................................................................................. 1 1.2 Problem S tatem en t................................................................................................. 3 1.3 Parabolic Reflector A n te n n a s .............................................................................. 4 1.3.1 G eo m etry .................................................................................................... 4 1.3.2 Beam Pattern and Illumination Function 5 1.3.3 Antenna’s Gain and E ffic ie n c y ................................................... 6 1.3.4 Pathlength E r r o r ............................................................................ 7 ........................................... 1.4 Woodbury Research Facility ( W R F ) ................................................................. n MICROWAVE HOLOGRAPHY 2.1 2.2 14 Microwave H olography.......................................................................................... 2.1.1 Fourier Transform R elationship................................................... 14 2.1.2 Sampling, Resolution, and A c c u ra c y .......................................... 18 2.1.3 Az/El-to-U/V Coordinate Transformation and Interpolation . . . . 2.1.4 Least Square Error Best F i t .......................................................... 14 19 21 Phase Coherent Microwave Holography.............................................................. 2.2.1 Measurement System ...................................................................... 2.2.2 Existing Research on Phase Coherent Microwave Holography . . . . 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Further reproduction prohibited without permission. 8 21 21 22 vii 2.3 Phase Retrieval Algorithm for Microwave H o lo g ra p h y .................................. 23 2.3.1 Existing Research on Phase Retrieval Microwave Holography . . . . 24 2.3.2 The Misell Phase Retrieval A lg o rith m .................................................. 26 III MICROWAVE LENSES 28 3.1 Ray TVacing............................................................................................................. 28 3.2 Dielectric L e n s ...................................................................................................... 31 3.3 Fresnel Zone P l a t e ................................................................................................ 32 3.3.1 Standard FVesnel Zone Plate .................................................................. 32 3.3.2 Phase-corrected Fresnel Zone P la t e ........................................................ 33 3.3.3 Fresnel Zone Plate for Microwave H olography...................................... 34 Metal-plate L e n s ................................................................................................... 35 3.4.1 P a ra m e te rs ................................................................................................. 36 3.4.2 Construction Tolerances and Frequency S en sitiv ity ............................ 38 3.4.3 Specification and C o n stru c tio n .............................................................. 39 Circular Waveguide L ens........................................................................................ 43 3.5.1 P a ra m e te rs ................................................................................................. 44 3.5.2 Construction Tolerances and Frequency S en sitiv ity ............................ 46 3.5.3 Specification and C o n stru c tio n .............................................................. 46 Computer S im ulations.......................................................................................... 48 3.6.1 Finite Difference Time Domain(F D T D ).................................................. 48 3.6.2 FDTD of the FVesnel Zone P l a t e ............................................................. 54 3.6.3 FDTD of the Metal-plate L e n s ................................................................ 56 Criteria in Lens S e le c tio n .................................................................................... 56 3.4 3.5 3.6 3.7 IV MEASUREMENTS AND RESULTS 4.1 4.2 60 Measurement System and D ata C ollection........................................................ 60 4.1.1 Phase Coherent H olography...................................................................... 61 4.1.2 Phase Retrieval H olography...................................................................... 71 Measurement Results from P C H ........................................................................ 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. viii 4.3 Measurement Results from PRH using the Metal-plate L e n s .......................... 73 4.4 Measurement Results from PRH using the Circular Waveguide Lens . . . . 87 V ERROR ANALYSIS 92 5.1 Standard Propagation of E rro rs ........................................................................... 92 5.2 Empirical Error Analysis....................................................................................... 97 5.2.1 Analysis Procedure..................................................................................... 97 5.2.2 The Effect of the Measurement Noise .................................................. 97 5.2.3 The Effect of the Lens Transfer Function E r r o r .................................. 100 5.2.4 The Effect of the Amount of Phase V a ria tio n ........................................ 105 5.2.5 The Effect of the Phase Variation Profile ............................................ 110 VI CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH 114 6.1 C onclusions............................................................................................................. 114 6.2 Contributions from this W o rk .............................................................................. 116 6.2.1 116 6.3 Publications and P resen tatio n s............................................................... Future Research .................................................................................................... 117 6.3.1 Lens Construction Process......................................................................... 117 6.3.2 Determination of Lens Transfer Function ............................................. 118 6.3.3 Extension of the Empirical Error A nalysis............................................ 119 6.3.4 Optimal PRH L en s..................................................................................... 119 6.3.5 Demonstration of Technique on Small Antennas 119 ................................ REFERENCES 120 VITA 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ix LIST OF TABLES 4.1 Summary of mean square phase error (Rad?) for different phase retrievals utilizing a metal-plate lens (Numbers in parentheses rank the results from best to worst)........................................................................................................... 4.2 82 Summary of mean square phase error (Rad2) for different phase retrievals utilizing a circular waveguide lens (Numbers in parentheses rank the results from best to worst).................................................................................................. 5.1 91 Examples of mean square phase error (deg2) of retrievals using different phase variation profiles (empirical study)......................................................................... Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 113 X LIST OF FIGURES 1.1 Cassegrain configuration of parabolic reflector a n te n n a ................................... 5 1.2 (a) Predicted WRF antenna’s far-field pattern excluding subreflector block age and strut scattering, (b) Predicted far-field pattern of uniform magnitude and phase aperture distributions (O S U -N E C R E F )......................................... 10 1.3 (a) Predicted WRF antenna’s far-field pattern including subreflector blockage and strut scattering, (b) Predicted effect of WRF subreflector blockage and strut scattering (O SU -N EC R EF)........................................................................ 11 1.4 Predicted WRF antenna’s far-field pattern including subreflector blockage, strut scattering, and approximated surface deformation (4.2 GHz): (a) az imuth, (b) elevation. Measured WRF far-field pattern (4.2 GHz): (c) azimuth (solid line)(d) elevation (solid l i n e ) ..................................................................... 13 2.1 Geometry of reflector antenna for microwave holography’s derivation[l] . . . 15 2.2 Geometry of reflector antenna demonstrating surface distortionfl]................ 18 2.3 Schematic block diagram describing microwave holographic measurement 22 2.4 Block diagram of the Misell phase retrieval algorithm for microwave holog raphy ...................................................................................................................... 27 3.1 Paraboloid geometry used for Ray T racing......................................................... 29 3.2 Hyperboloid geometry used for Ray T rac in g ...................................................... 30 3.3 Illustration indicating quantities used in Ray T racing...................................... 31 3.4 Standard Fresnel zone plate[2]............................................................................... 33 3.5 Illustration of the design of the shape of the metal-plate lens’ profile. F is the focal point, and f is the focal length. A path length difference between two ray paths indicates a phase d e v ia tio n ........................................................ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 xi 3.6 Numerical calculation of the effects of frequency and metal-plate spacing on phase variation of the WRF metal-plate lens. The absolute amount of phase differences from the designed values are calculated at the antenna aperture edge according to deviations from the designed values of measurement fre quency and metal-plate spacing............................................................................. 40 3.7 Drawings of the WRF metal-plate le n s .............................................................. 41 3.8 (a) Picture of W RF metal-plate lens for use in this application, (b) Picture 3.9 of the lens placed in front of the WRF feed during the measurement . . . . 42 Illustration of wave components travelling through the metal-plate lens. . . 43 3.10 Numerical calculation of the effects of frequency and waveguide radius on phase variation of the WRF circular waveguide lens. The absolute amount of phase differences from the designed values are calculated at the antenna aperture edge according to deviations from the designed values of measure ment frequency and waveguide radii..................................................................... 47 3.11 Drawings of the WRF circular waveguide le n s ................................................. 49 3.12 (a) Picture of WRF circular waveguide lens for use in this application, (b) Picture of the lens placed in front of the WRF feed during the measurement 50 3.13 Illustration of FDTD c e ll[3 ]................................................................................ 52 3.14 FDTD simulation: Relative electric field magnitudes along the focal axis of the feed horn for the FVesnel zone plates with different focal distances ... 55 3.15 Comparison between the calculation from the design and FDTD simulation of the aperture phase changes of the feed horn due to the metal-plate lens . 57 3.16 FDTD simulation: (a) Magnitudes of the aperture field of the feed horn for three different cases. Overall shapes are unchanged among cases except for small magnitude variations and reductions, (b) Aperture phase difference of the feed horn between the metal-plate lens with and without polyethylene. . 58 4.1 Phase drift characteristic of the WRF holographic measurement system . . 62 4.2 Illustration of neighboring satellite interference during microwave holographic m e asu re m en t.......................................................................................................... Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 xii 4.3 Down-converted beacon frequency variation monitored at W RF over one measurement r u n .................................................................................................... 64 4.4 The effect of modulation rate variation on measured far-field magnitude dis tribution (dB). Labels on the horizontal and vertical axes indicate sampling points.......................................................................................................................... 65 4.5 Detailed hardware diagram of WRF microwave holographic measurement sy ste m ...................................................................................................................... 66 4.6 Picture illustrating WRF microwave holographic measurement system’s im plementation .......................................................................................................... 67 4.7 Schematic block diagram of software processing to obtain phase differences between signals from AUT and reference antenna during data collection . . 68 4.8 (a) Comparison between the simulated incoming noisy signal and the output signal from Steiglitz-McBride algorithm. Each period of the ’’cleaned” has one more iteration of the algorithm applied, (b) Simulation result demon strating the improvement from the use of the Steiglitz-McBride algorithm on phase m easurem ent................................................................................................. 69 4.9 (a) WRF microwave holographic measurement grid in Az-El coordinate sys tem, (b) Corresponding WRF measurement grid in u-v coordinate system . 72 4.10 Magnitude(dB) and phase(deg) of the WRF antenna’s far-field distributions 74 4.11 WRF antenna’s surface deformation map (mm) obtained from PCH. Mea surements were performed in (a) fall 1998 (b) spring 1999 (c) spring 2000 (d) spring 2001 ............................................................................................................. 76 4.12 Aperture phase distribution (deg) of the WRF antenna resulting from the phase coherent holography. A, B, C, D, E indicate significant features of the aperture phase d is tr ib u tio n ................................................................................. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 xiii 4.13 Magnitude and phase of the lens transfer function computed from PCH mea surement results of the WRF antenna with and without the metal-plate lens, (a) Magnitude of the PCH-deduced lens transfer function (dB). (b) Phase of the PCH-deduced lens transfer function (deg). (c) Average of one dimensional phase cuts of the PCH-deduced lens transfer function compared to the design calculation...................................................................................... 80 4.14 Aperture phase distributions (deg) of the WRF antenna resulting from the phase retrieval measurement employing a metal-plate lens: (a) Using uni form magnitude and PCH-deduced phase lens transfer function (ranked # 1 in Table 4.1), (b) Using PCH-deduced magnitude and phase lens transfer function (rank #4), (c) Using uniform magnitude and parabolic phase lens transfer function (rank #9), (d) Using uniform magnitude and FDTD simu lated phase lens transfer function (rank # 3 ) ..................................................... 86 4.15 Aperture phase distribution of the WRF antenna obtained from (a) PCH (b) PRH utilizing the circular waveguide lens (theoretical lens transfer function and OSU-NECREF initial function are used in the Misell algorithm) 90 5.1 Flow chart for propagation of errors in the Misell phase retrieval algorithm 94 5.2 Schematic block diagram illustrating the empirical study procedure 98 5.3 (a) Aperture phase distribution of the model used in the empirical analysis (b) Aperture phase distribution of one of the retrievals from the empirical a n a ly s is ................................................................................................................... 99 5.4 Characteristic of MSE versus the simulated noise level for a) quadratic phase transfer function profile (100 degrees) and b) WRF circular waveguide stepped p r o f ile ...................................................................................................................... 101 5.5 Characteristic of MSE versus the simulated lens transfer function error for a) quadratic phase transfer function profile (100 degrees), and b) WRF circular waveguide stepped p r o f ile .................................................................................... 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Further reproduction prohibited without permission. 103 xiv 5.6 Characteristic of RMSE versus the deterministic lens transfer function error for different locations (distance in meters from the center of the antenna aperture) a) quadratic phase transfer function profile (100 degrees), and b) WRF circular waveguide stepped p ro file ............................................................ 104 5.7 Characteristic of RMSE versus the location of the deterministic errors of different severity a) quadratic phase transfer function profile (100 degrees), and b) WRF circular waveguide stepped p ro file ............................................... 5.8 106 Characteristic of RMSE versus the location of a 1-mm d e te rm in istic lens transfer function error for different amount of phase variation (at the aperture edge) of the quadratic phase profile a) 60-140 degrees phase profile b) 160-220 degrees phase profile c) 240-320 degrees phase p ro file...................................... 108 5.9 Characteristic of MSE versus the amount of phase variation (at the aper ture edge) of the quadratic phase profile a) under the presence of constant measurement noise b) under the presence of constant lens transfer function e r r o r .......................................................................................................................... 109 5.10 Characteristic of MSE versus the order of the polynomial describing the phase variation profile a) under the presence of constant measurement noise b) under the presence of constant lens transfer function e r r o r ........................... Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 112 XV SUMMARY The objective of this research is to investigate and develop a novel measurement method of microwave holography for the situation where the subreflector or feed may not be easily displaced or when the system is desired to be measured in-place, th at is to characterize a particular alignment. Other potential advantages unique to this technique will also be discussed. Since the primary effect of the slight defocusing is a phase shift between ray paths reflecting on the antenna surface, a microwave lens may be employed as an alternative method of defocusing the antenna. As part of this study, different types of microwave lenses have been studied, designed, and implemented to achieve this primary purpose, and measurements have been performed to demonstrate and explore the potential and feasibility of the proposed method. This method is well suited to those facilities that are not able or willing to move the subreflector or feed, or cannot move them a sufficient amount, and do not wish to set up a coherent reference antenna. In addition, this method affords the possibility of examining the effects of a feed or subreflector at a specific position on the system as a whole. As another advantage, utilization of the lens also allows the freedom to select the profile and amount of the phase variation over the aperture, which according to the preliminary study may result in some advantages in obtaining accurate results or in probing specific effects. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 CHAPTER I INTRODUCTION Microwave holography has been a popular and powerful technique for measuring reflec tor antenna surface deviations. Standard phase coherent holography requires direct phase measurement of the far-field distribution, which presents some practical difficulties in im plementation. Of all the phase retrieval methods proposed for use in microwave holography, the Misell algorithm appears to be the most popular and suitable. The algorithm requires that the subreflector or feed of an antenna under test (AUT) be moved axially in order to obtain defocused far-field distributions. This method of defocusing however cannot be easily achieved at some facilities, including the Georgia Tech Woodbury Research Facility (WRF). The use of a microwave lens has therefore been shown as an alternative for antenna defocusing for phase retrieval microwave holography. Different types of microwave lenses have been studied, designed, and implemented to achieve this primary purpose, and mea surements have been performed to demonstrate and explore the potential and feasibility of the method. In addition, this method affords the possibility of examining the effects of a feed or subreflector at a specific position on the system as a whole. As another advantage, utilization of the lens also allows the freedom to select the profile and amount of the phase variation over the aperture, which may result in some advantages in obtaining accurate results or in probing specific effects. 1.1 Introduction The performance of a reflector antenna depends primarily upon accurately positioning and shaping the reflector surface, as well as properly locating the feed and subreflector. A number of techniques have been proposed and applied to measure the surface deformation of reflector antennas. These methods can be categorized into three types: mechanical, optical, and electromagnetic. The mechanical method can be performed by using devices Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 such as dial gauges or micrometers. The measurement results are normally compared with the ideal surface geometry. Typically, this method is restricted to small antennas, and its accuracy may not be as good as that of the others. Some optical methods such as the tape and theodolite [4], and analytic photogrammetry [5] have been successfully introduced and applied. However, most of them are proven to be either time-consuming, difficult to implement, or expensive. Microwave holography [I, 6, 7, 8, 9, 10, 11, 12) has been a popular technique for char acterizing reflector antenna surface distortion and feed/subreflector displacement. The technique exploits the Fourier transform relationship between the complex far-field radi ation pattern and the complex aperture distribution. Surface deviations from a reference paraboloid and feed/subreflector displacement can be derived from the phase distribution of the resulting aperture field. There is therefore a need to measure, or by some means obtain, the phase distribution of either the far-field or aperture field. IVaditional microwave holography, called phase coherent holography (PCH) [1, 6, 7, 8, 9, 10, 11, 12], directly measures the far-field magnitude and phase radiation pattern of the antenna under test (AUT). The relative phases are obtained from the phase differences between the AUT and a reference antenna, while the former scans over a specified grid and the latter points at the far-field transmitter. Even though this technique has been widely implemented and is very accurate, the requirements of a reference antenna and two coherent receiving channels may be difficult to arrange. There are also difficulties in accurately measuring phase at higher frequencies. An alternative to PCH is to take magnitude-only measurements and attem pt to retrieve the phase distributions, often referred to as phase retrieval holography (PRH). Phase re trieval has been considered in many applications such as electron microscopy [13], X-ray crystallography [14], and optics [15]. The technique attem pts to reconstruct the partic ular complex function out of various types of constraints, including its Fourier intensity and region of support. It has been extensively shown that for the two-dimensional case Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 the desired complex function can be uniquely determined from its Fourier transform in tensity [16, 17, 18]. For microwave holography, the desired complex function is a twodimensional antenna aperture distribution. Constraints that could be applied include a known far-field magnitude distribution, the aperture region of support, or in some cases, the magnitude of the aperture distribution itself. Microwave holographic measurements have been conducted at the Georgia Tech Wood bury Research Facility (WRF) to characterize the condition of the antenna surface of one of the large parabolic reflector antennas at the facility [19]. The Woodbury facility consists of a pair of 30-meter Cassegrain antennas located 65 miles south of Georgia Tech’s At lanta campus. The initial microwave holographic implementation employed phase coherent holography whereby a reference antenna was coherently linked to the AUT and the com plex far-field antenna pattern was directly measured. Two sets of measurements at C-band (4.2 GHz) were obtained using two different geostationary satellite beacons at slightly dif ferent elevation angles serving as signal sources [20]. The similar features shown in both measurements indicate the reproducibility and validity of the measurement system. To further investigate the properties of microwave holography, an experiment employing the phase retrieval technique proposed by Misell [21] was conducted. This technique requires the measurement of at least one defocused intensity pattern of the antenna under test for retrieving the aperture phase distribution. Typically, the defocusing can be done by axially moving the feed, or the subreflector for a Cassegrain antenna. Unfortunately, the Woodbury antennas are not amenable to any subreflector or feed displacement, which may be an issue that many facilities encounter in implementing phase retrieval holography. In addition, it may be desirable to measure system performance in a given configuration. 1.2 Problem Statement The objective of this research is to investigate and develop a novel measurement method of microwave holography for the situation where the subreflector or feed may not be easily displaced or when the system is desired to be measured in-place, so as to characterize a particular alignment. Other potential advantages unique to this technique will also be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 discussed. Since the primary effect of the slight defocusing is a phase shift between ray paths reflecting on the antenna surface, a microwave lens may be employed as an alternative method of defocusing the antenna. As part of this study, different types of microwave lenses have been studied, designed, and implemented to achieve this primary purpose, and measurements have been performed to demonstrate and explore the potential and feasibility of the proposed method. This method is well suited to those facilities th a t are not able or willing to move the subreflector or feed, or cannot move them a sufficient amount, and do not wish to set up a coherent reference antenna [22, 23]. In addition, this method affords the possibility of examining the effects of a feed or subreflector at a specific position on the system as a whole. As another advantage, utilization of the lens also allows the freedom to select the profile and amount of the phase variation over the aperture, which according to the preliminary study may result in some advantages in obtaining accurate results or in probing specific effects. 1.3 1.3.1 Parabolic Reflector Antennas Geometry Reflector antennas play important roles in many engineering and scientific applications. One of the most popular types of reflector antennas is the parabolic reflector, as this type of antenna has been used extensively in communications, radio astronomy, etc. The simplest configuration of this antenna is to have a parabolic shape main reflector, and locate a feed system at the focal point of the paraboloid, where the waves reflected from the reflector constructively combine. One disadvantage of this configuration is the requirement that the feed system be placed up above the main reflector, which may be impractical for cases where a dimension of the feed is large. The most popular configuration of the large parabolic reflector antenna is called the Cassegrain configuration, where an additional reflector is utilized. The antenna system consists of a parabolic main reflector, a secondary hyperbolic reflector (subreflector), and a feed system. Due to the addition of a subreflector, this configuration allows the feed system to be placed conveniently closer to the antenna’s surface. Fig 1.1 illustrates this Cassegrain configuration. Additional antenna Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 Reflected Wave Figure 1.1: Cassegrain configuration of parabolic reflector antenna gain over the focal-fed configuration may be obtained by using slightly “shaped" reflectors, where the antenna’s illumination efficiency is improved [24, 25]. It has also been shown that the effect of this “shaping” may be ignored in surface deformation and structural analysis [26]. Other advantages of the Cassegrain system include the reduction of spillover and minor lobe radiation as well as the capability to obtain a much greater equivalent focal length than the actual physical focal length [26]. 1.3.2 Beam Pattern and Illumination Function A reflector antenna’s beam pattern is primarily dictated by the illumination function of that antenna. An illumination function could simply be a uniform function across the antenna’s aperture. This type of illumination typically yields the Airy function of beam pattern. The level of the first sidelobes is 17.6 dB below the mainbeam magnitude level, which may be too high for some applications. In an attem pt to reduce the sidelobe level, the illu m in a tio n function is usually tapered. The resulting beamwidth for this illumination however is slightly Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 increased. Additionally, the illumination efficiency (the ratio of the on-axis directivity for the given illumination to that produced by a uniform aperture distribution with the same total radiated power) is decreased with tapering. In general, the smoother the illumination function, the wider the beamwidth and the lower the sidelobe level will be for the beam pattern. Beside an illumination function, a beam pattern may also be affected by physical struc ture of the antenna such as the presence of a subreflector and a surface distortion of the main reflector. The presence of a subreflector usually widens the beamwidth of the an tenna. A surface distortion can cause anomalies in a beam pattern, such as an asymmetry or distorted beam, depending on the severity and pattern of the distortion. 1.3.3 Antenna’s Gain and Efficiency One of the advantages of using the parabolic reflector antenna is the directional nature of its beam pattern. It also yields exceptional gain, which is especially important for radio astronomy and deep-space applications. Antenna gain may be described as the ratio of the maximum power density radiated by the antenna to the power density radiated by a lossless isotropic antenna. The reflector antenna’s gain is a function of the aperture area and frequency, and may be expressed as G=n r (i.i) where A is the antenna’s aperture area and Tfr is the aperture efficiency factor. The efficiency factor includes a number of contributions from different types of losses. They include effects such as illumination, spillover, leakage, as well as cross-polarization [26]. Other sources of efficiency reduction include aperture blockages, surface manufacturing tolerance, and surface deformation. The most important factor in efficiency reduction is perhaps the surface deformation. A reasonably good approximation of the efficiency due to this factor is called the Ruze equation [27] and may be expressed as 17 = exp )2}, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (1.2) 7 where a is the zero-mean root-mean-square (rms) half pathlength error of rays reflected on the antenna’s surface. This rms pathlength error can be readily obtained from microwave holography. The Ruze equation is derived based on the assumptions th a t the surface errors have a Gaussian distribution and possess limited correlation. Even though practically the surface errors tend to be more systematic, which would seem to violate these assumptions, the equation still provides a good quick approximation of the resulting efficiency. It may be seen that for antennas with mediocre surfaces, the efficiency will be sharply reduced as the operating frequency increases. For high-efficiency antennas, the value of illumination efficiency is usually larger than 85%. Spillover loss may be considered insignificant for a large antenna. The antenna’s surface tolerance can be controlled by precise manufacturing techniques so that a very small loss in efficiency may be experienced. The effect of other contributing factors usually ranges from 95 to 99%, which is insignificant compared to the efficiency reduction due to the antenna’s surface roughness. From equation (1.1), it appears that as the operating frequency of the antenna increases, more gain can be obtained. However, the effect of roughness from the Ruze equation decreases the efficiency as the frequency gets higher; thus, limiting the increase in antenna’s gain. A gain limit is defined as the point where the increase in antenna’s gain due to higher operating frequency is offset by the reduction in its efficiency. The rms half pathlength error at this gain limit is ' - c - (1J> This value is often used to indicate the upper bound of the frequency the antenna can be used [26]. 1.3.4 Pathlength Error Pathlength errors of an antenna can be measured directly with a microwave holographic measurement. For a perfect parabolic reflector antenna, the pathlengths between the an tenna’s focal point and different locations on the antenna’s aperture are equal so that the waves constructively combine at the feed. Different pathlengths due primarily to surface deformation of the antenna thus cause an imperfect combination and reduce the antenna’s Reproduced with permission of the copyright owner. Further reproduction prohibited w ithout permission. 8 achievable efficiency. The total pathlength error may be described as the difference between the amount of electrical length for a given ray path reflecting on the antenna’s surface and an ideal pathlength. Total pathlength errors for different locations on the antenna’s surface may be obtained horn microwave holographic measurement, and half pathlength errors are simply half of those total pathlength errors. The half pathlength errors can be converted into physical lengths, and the rms value is calculated. This rms value can then be applied to the Ruze equation to evaluate the antenna’s efficiency as described earlier. 1.4 Woodbury Research Facility (WRF) The Georgia Tech Woodbury Research Facility (WRF), built in mid-1970’s for use as a satellite communications station, consists of a pair of 30-meter Cassegrain antennas located 65 miles south of the Georgia Tech’s Atlanta campus [19]. Though abandoned in the early 1980’s the facility provided a resource for radio astronomy and satellite communications research and was acquired by Georgia Tech in 1992. One of the two antennas was fully upgraded and was used as part of SGTI Institute’s Project Phoenix for two years. Between 1998 to 2001, the facility was used to perform all the measurements which appear in this research. The far-field pattern of the WRF antenna has been predicted using the Ohio State University Reflector Antenna code (OSU-NECREF code) [28, 29]. This code employs the method of aperture field integration (AI) for the first few sidelobes, then switches to Geo metrical Theory of Diffraction (GTD) for the further sidelobes, where diffraction starts to take effect. In order to predict the pattern of the antenna, the feed pattern used for that antenna has to be specified. Although the feed pattern of the WRF antenna is not readily available, the physical shape of the feed horn including its flare angle and aperture size has been approximated, and its pattern was calculated. For Cassegrain antennas, the pattern determination is a two-step process. First, the scattered field from the subreflector has to be computed using the available feed pattern. This scattered field was then used along with the geometry of the antenna to calculate the predicted far-field pattern of the antenna’s Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 main reflector. For WRF antenna, the predicted far-field pattern calculated using this procedure is illustrated in Fig 1.2. Besides slight differences at the first null and in the far sidelobes, the pattern is comparable to that computed using uniform magnitude and uniform phase aperture field. Note that this prediction does not incorporate some effects encountered in practice such as the antenna surface deformation as well as the subreflector blockage and strut scattering. It is possible however to include those effects into the prediction. A subreflector blockage may be modelled employing Physical Optics (PO), which should yield accurate results at least within the mainlobe and near-sidelcbe region. Scattered fields from struts can be calculated by separating each strut into smaller segments. A strut is assumed to have circular cross-section, and its segments to scatter in the same way as does an infinite cylinder. More details in dealing with subreflector blockage and strut scattering for the NECREF code may be found in the literature [30]. The fields due to these structures as predicted by the NECREF code are illustrated in Fig 1.3. The total field of the WRF antenna with subreflector and struts is also shown. The width of the mainlobe is slightly widened due to the effect of the subreflector blockage. Some differences in the far sidelobes may be noticed. W ith the measured surface distortion profile obtained from PCH (See Chapter 4), the attem pt has been made to predict the WRF far-field pattern with the effects of those surface distortions included. Unfortunately, due to the limitations of the NECREF code in creating surface distortions, only a rough estimate of the WRF surface distortion profile can be used. Plots of the predicted azimuth and elevation pattern as well as the measured patterns are illustrated in Fig 1.4. Even though close predictions may not be obtained, the result however shows some similarities to the measured patterns. The sidelobe level is noticeably increased in both plots. Asymmetry in the first sidelobes of both azimuth and elevation patterns is clearly visible in the measured patterns as well. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 PredcMWRFpattanvwtobioctagM -10 -20 •30 •40 -SO -00 -70 •SO Off-boraigN mglt (dig) (a) Pradktid (Nftim: urtfofm ip « tu re nw gndudi^tiM e -10 -20 -30 •40 •SO •00 -70 -SO -90 -100 OfFborafght ingli (dig) (b) Figure 1.2: (a) Predicted WRF antenna’s far-field pattern excluding subreflector block age and strut scattering, (b) Predicted far-field pattern of uniform magnitude and phase aperture distributions (OSU-NECREF) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 Pradktad WRF pdMnu « / Meekae* m d Kdtadng -10 -20 -30 I •40 •SO -60 -70 -00 -4 -3 -2 -1 0 1 Off-6orwJ(7* angto (dag) 2 3 4 (a) Predated WRF utxaflador blockaga md strut scattaring 0 subreflector blockage strut scattering -10 -20 -30 1-40 -50 -60 -70 •00 •90 - 4 - 3 - 2 1 0 Off-berttitf* and* (dag) 1 2 3 4 (b) Figure 1.3: (a) Predicted WRF antenna’s far-field pattern including subreflector blockage and strut scattering, (b) Predicted effect of W RF subreflector blockage and strut scattering (OSU-NECREF) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Pradctad WRF uJnutft paflam; w/ jurfaea deformation -10 -30 -40 -50 -70 -00 Off-borasfght angla (dag) (a) Preddad WRF alevatien paBam: m tturfaea dafonnatlon A/\ -40 -60 ■70 - •00 OfcbowdgN w tfa «fcg) (b) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 -to -IS -2* -30 -s j -a« -03 04 00 (c) ia<i -10• -29 02 00 (d) Figure 1.4: Predicted WRF antenna's far-field pattern including subreflector blockage, strut scattering, and approximated surface deformation (4.2 GHz): (a) azimuth, (b) elevation. Measured WRF far-field pattern (4.2 GHz): (c) azimuth (solid line)(d) elevation (solid line) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 CHAPTER II MICROWAVE HOLOGRAPHY In this chapter, the technique of microwave holography in clu d in g a theoretical development, important parameters, implementation, and measurement accuracy is presented. Then an introduction to the phase retrieval problem and the phase retrieval algorithm applied to microwave holography is presented and discussed. 2.1 Microwave Holography The technique of microwave holography was first introduced in 1970s [6, 7], and has been an effective and popular method for determining the surface distortions of reflector antennas ever since. As previously mentioned, the technique relies on the Fourier transform rela tionship between the far-field and the aperture field distribution of the antenna. Since the antenna's surface distortion relates directly to the phase information of the aperture dis tribution, one can take advantage of the simplicity in measuring the far-field distribution, and obtain the antenna’s surface deformation from post-processing. Apart from the surface deformation, by noticing specific features of the aperture phase distribution, one can also differentiate other characteristics of the antenna such as a feed translation, or a subreflector movement. 2.1.1 Fourier Transform Relationship In this section, the Fourier transform relationship between the far-field distribution of the antenna and aperture field distribution as well as surface displacement will be discussed. The derivation closely follows Rahmat-Samii’s paper [1]. The geometry of the parabolic reflector antenna with diameter of D and focal length of F is illustrated in Fig 2.1. The origin of the coordinate system is located at the focal point of the parabolic surface as is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 Figure 2.1: Geometry of reflector antenna for microwave holography’s derivation[l] the feed. It is known that a far-field distribution can be written as S = - 3 kVt Xp[ ~ Jr - - ( T e e + T*0), (2.1) where k is the wave propagation constant, t) is the wave impedance in free space, and f{9,4>) = J J( ? ) exp { j k r • f)dS'. (2.2) In (2.2), S represents the reflector surface, while J is the induced surface current defined as J = 2 h x H i, (2.3) where n is the surface normal unit vector, and f f l is the incident magnetic field generated from the feed. By employing the concept of the surface projection Jacobian, the integral in (2.2) can be performed in terms of the aperture plane coordinates (x',y'). Equation (2.2) then becomes f ( 9 , <f>) = J J( ? ) exp (jkr- t ) J adx'dy', Reproduced with permission of the copyright owner. Further reproduction prohibited w ithout permission. (2.4) 16 where s represents the area of the projection of the reflector surface onto the aperture plane. In (2.4), the Jacobian transformation, J3 , can be expressed as • '. = [ i + ( J £ ) 2 -k J £ > t 2. m > where / describes the reflector surface z = f{x,y). By introducing the following notations: J{x',y') = J { 7 ) J s, (2.6) f ' • r = z'cosO 4- ux' + vy', (2.7) where u = sin 9 cos <f>and v = sindsin 0, (2.4) can be rewritten as 7*(u, v) = J J(x', y') exp (jkz') exp {—jkz'{ 1 —cos 9)) exp (jk(uxr + vy'))dx'dy'. (2.8) Expanding (2.8) using Taylor series results in f( « , v) = f ; h - j k ( l - cos 9 )] % , p=o P* (2.9) where 'fp = J z'pJ(x', y') exp {jkz') exp (jk{ux' + vy'))dx'dy'. (2.10) For a small observational angle (small 6) and small amount of lateral feed displacement, only the dominant term contributes significantly to the series. Therefore, the far-field component can be expressed as r ( u ,u ) = J J(x', y') exp {jkz') exp {jk(ux' + vy'))dx'dy'. (2.11) It has been shown [1] that, for predominantly y-polarized radiated field, the co-polarized and cross-polarized components of the far-field pattern can be expressed in terms of the Cartesian components as Tx sin <j) COS < t> Tcroas—pol COS <(> —sin <(> cos 6 cos <f> cos 0 sin <j> —sin 9 —sm<f) costp 0 Ty * Tcopol Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • ( 2 - 12) 17 For a small observation angle, Tcopol T cross—pol . T* . According to the reciprocity theorem, if the reflector antenna is impinged by a plane wave,the output at the feed port will be proportional to the gain of the receiving antenna in that direction of incidence and plane wave polarization. For the y-dominant polarization, T(u, v) = j J {x', y') exp {jkz') exp {jk{ux' + vy'))dx'dy', (2.14) where only one polarization is considered. Note th at (2.14) represents the two-dimensional Fourier transform relationship between the far-field distribution and the function related to the induced surface current of the reflector. In order to relate the surface deformation to this relationship, s(x, y) is defined as the surface distortion in the surface normal direction on the reflector. The geometry for this surface distortion and relating quantity is illustrated in Fig 2.2. For a parabolic reflector, the angle between the surface normal and the z-direction can be obtained by cosS H l + ^ ^ r 1/2. (2.15) It can be seen that, for a small surface displacement, f P + PQ = 2ecos£. (2.16) Note th at since the feed and its phase center is located at the focal point, the argument of the Fourier transform of (2.14) can be expressed as J ( x \ y') exp {jkz') = |J{x', y)| exp {-j k r') exp {jkz’). (2.17) From the geometry - r ' + z' = - 2 F + 2 £ co s£ , Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.18) 18 z DISTORTED SURFACE — ORIGINAL SURFACE Figure 2.2: Geometry of reflector antenna demonstrating surface distortion[l] and (2.14) becomes T(u, v ) = exp (~j2kF) |J(x ', y ' ) l exp (jS) exp (,jk(u x' + vy'))dx'dy', (2.19) where S = 4w(e/X) cos£ denotes the full pathlength phase error due to surface distortion. As a result, by solving the inverse Fourier transform of the far-field component, or farfield distribution, the surface distortion at a given point on the antenna aperture can be obtained using \ 2 2 e(x,y) = — (1 + ^ ± f ) - W p h aSe { e x p ( j 2 k F ) F - l [T(u,v)}}. 2.1.2 (2.20) Sam pling, R eso lu tio n , a n d A ccu racy Given that the surface information of the reflector antenna can be obtained from the phase of the inverse Fourier transform of the antenna’s far-field distribution, the far-field data is typically obtained by sampling the continuous far-field distribution. The Discrete-time Fourier Transform (DFT) can then be applied to the problem. In determining the sam pling interval, an aliasing problem which could occur in the aperture field domain must be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 considered. It can be shown from the sampling theorem that, for a reflector antenna of diameter D, the largest sampling interval possible is X/D. In general, the sampling interval is typically given by Au = Av —k \ / D , (2-21) where 0.5 < k < 1. If the continuous far-field distribution is known to an infinite extent with acceptable level of the signal-to-noise ratio (SNR), the surface information of the corresponding antenna can be exactly determined. Practically, however, apart from the fact that only a limited number of samples of the far-field distribution can be obtained, the range of the measurement along with the SNR is restricted. Therefore, the measurement resolution and accuracy are also limited. Consider the measurement with sampling interval of A u and Av. Let N, designate the number of samples in both u and v direction, and Np designate the number of points for DFT. It is seen that even though the surface distortion can apparently be determined at the intervals A l = A» = ^ v r n 5 ' (222) the actual resolution however depends primarily upon Na, and can be given by = Ayres = K{N^ _ i y (2-23) In practice, it is difficult and time-consuming to collect the far-field data in the uniform u-v coordinate system. Typically, the data collection is performed in a more conventional coordinate system such as the Azimuth-Elevation one. The data has then to be interpolated into a uniform u-v grid before processing. 2.1.3 Az/El-to-U/V Coordinate Transformation and Interpolation Since the Fourier transform relationship holds on a u-v coordinate and Fast Fourier Trans form (FFT) may be conveniently utilized on a uniform u-v grid,it would be best to measure the far-field pattern in the uniform u-v grid system. Practically however, most facilities uti lize either an azimuth/elevation (az/el) or an hour angle/declination (HA/dec) coordinate Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 system when scanning the antennas. A data collection with a unifo rm grid in the u-v co ordinate system is realizable by mapping the u-v coordinates to az/el or HA/dec. In order to do so, however, requires a non-uniform scanning, which presents some inconvenience and may consume significant data collection time. As a result, a more conventional coordinate system is typically used in the measurement, and the desired uniform u-v grid far-field pattern may be obtained by interpolating the measured data. Measurements at WRF were performed using an elevation and cross-elevation (el/X-el) coordinate system, which resembles that of az/el. The difference is th at changes in the arc length on the sphere due to changes in elevation angle are taken into account. The relationship between the cross-elevation angle and azimuth angle is simply (2.24) where Aaz, A ( X —El), and El are the difference between two adjacent azimuth angles, the difference between two adjacent cross-elevation angles, and the elevation angle, respectively. For the case where a source is very far from the antenna-under-test (AUT), relationships between az/el and u-v coordinate has been readily derived, and may be expressed as [31] u = cos (Els) sin(AAz), (2.25) v = cos A A z cos Els sin El —sin Els cos El, (2.26) where Els, El, and A A z are the elevation angle of the source, the elevation angle of the antenna, and the azimuthal difference between the source and the antenna, respectively. By utilizing (2.25) and (2.26), one may relate the two coordinate systems and per form the interpolation of the far-field data using any existing interpolation methods. Note that alternate relationships between az/el and u-v are also available for a compact range measurement [10]. In summary, by using the appropriate relationships for the measurement scenario, one obtains the interpolated far-field pattern in the uniform u-v coordinate system for use in further processing. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 2.1.4 Least Square Error Best Fit It can be seen that the surface distortion obtained from (2.20) is the deviation from the perfect parabolic surface, whose vertex is located at the origin of the system (fig 1.1). How ever, with appropriate pointing, another parabolic surface, which is translated or rotated from the original, may perform equally well. Therefore, after the aperture phase distri bution and thus the resulting surface profile of the antenna has been retrieved, the least square error, best fit process may be performed to fit the paraboloid to the retrieved surface profile. The possible fitting parameters consist of five rigid body movement parameters and an additional change in focal distance. For a paraboloid revolving around the z-axis, those five rigid body parameters include translations in three principal axes and two rotations around x and y-axes. The least square best fitting process is performed by setting up the equation for the pathlength error as a function of those six fitting parameters. The best fit parameters and the new RMS pathlength error (after best fitting) may simply be obtained by means of solving the normal equations. Further detail of the best fitting process for reflector antennas has been described extensively by Levy [26]. 2.2 Phase Coherent Microwave Holography Phase coherent microwave holography (PCH) relates closely to the above theoretical de velopment in that it requires a measurement of the complex far-field distribution of the antenna-under-test (AUT). The complex far-field is processed to obtain the surface distor tion of the antenna. The flow diagram which summarizes PCH is illustrated in Fig 2.3. 2.2.1 Measurement System Since the measurement of the complex far-field distribution is required, the measurement system for microwave holography has to possess the ability to accurately measure both the magnitude and phase of the received signals. The system in general consists of two phasecoherent channels connected to the AUT and the reference antenna. Typically, it utilizes a heterodyne receiver, where the incoming high-frequency signals are down-converted to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 AUT far-field magnitude-phase extraction Az-El to u-v transformation Interpolation to uniform u-v grid 2-D Inverse Fourier Transform Reference Surface displacement map Aperture phase distribution and surface displacement relationship Complex aperture distribution Figure 2.3: Schematic block diagram describing microwave holographic measurement the baseband frequency before processing. The phase coherency can be ensured by the use of common local oscillator at every frequency down-conversion. The magnitude is either measured individually by each channel or obtained as the ratio of the two signals. In order to measure the phase between signals from two channels, a correlation between the two signals is typically performed. 2.2.2 Existing Research on Phase Coherent Microwave Holography Several results have been shown over the years providing proof of the successful use of microwave holography in measuring the surface deformation of the reflector antennas [8,32]. Apart from the standard measurement technique mentioned above, some improvements have been done to increase the accuracy and validity of the measurement. One of the proposed methods utilizes the so-called ‘iterative scheme,’ which is a technique that tries to extend the measured pattern beyond the measurement range [1]. The method is essentially the minimization of the error function of the far-field distribution. The idea of pattern subtraction has also been proposed to better predict the surface deformation in the subreflector and support structure blockage region [33]. The method is accomplished by predicting the patterns or scattering caused by the subreflector and support structure, and subtracting them from the measured pattern. Another idea used to infer the surface in those areas is to make measurements at a specified range of frequencies, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 subsequently filtering in the time-domain the signal from the blockages region [34]. However, this method requires an enormous amount of data collection and may not be very practical. 2.3 Phase Retrieval Algorithm for Microwave Holography A phase retrieval problem is generally a problem of determining the phase information of a function solely from its magnitude and Fourier magnitude, or intensity, as well as some other available information about the function such as a region of support [35, 36]. First consider the signal in one-dimensional case. /(x ) denotes a one-dimensional func tion in the object domain. Its Fourier transform and the inverse, F(u) and F - 1(u), can be uniquely determined, which means that the particular object function can also be uniquely determined from its corresponding Fourier transform. Unfortunately, such a relationship does not hold for the intensity [35]. However, for the case when /(x ) has finite support (/(x ) is non-zero only for a lim ited number of x), the real and imaginary part of its Fourier transform, F(u), satisfy the dispersion relation ReF(u) = - H { I m F { u ) } , (2.27) ImF(u) = H{ReF{u)}, (2.28) where H denotes the Hilbert transform. It has been shown that F(u) can be considered as a projection on the real axis of the entire function F(z), z = u + jv, which is encoded by its zeros 2*. It has also been proven that if Zkis a zero of an entire function F(z) (a function that is analytic1at each point in the entire finite plane), then F(z)/( 1 - z/zk) is also an entire functionas is Fl (z) = F(z) = F(z)B(z). (2.29) .( * - * * ) . where B(z) is called a Blaschke factor. Since |f?(2)| = 1, then \F(u)\ = |F i(u)|, i.e., the intensities of the Fourier function F(u) and Fi(u) are identical. Moreover, /i(x ), the corresponding object function of Fi(u), is also shown to have the same finite support as 1A function is analytic in an open set if it has a derivative at each point in that set. A function is analytic at a point if it is analytic in a neighborhood of that particular point [37] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 f{x) [35]. Therefore, a unique phase solution is difficult to obtain without some additional information to remove the phase ambiguity. For a two-dimensional function, it has been shown th at almost all functions are irre ducible [17]. This can be proven by showing that the subset of all multidimensional reducible polynomials is of measure zero. Hayes [18] showed that two-dimensional irreducible func tions that have the same Fourier magnitude are related trivially to each other. Therefore, it can be concluded that almost all two-dimensional functions can be uniquely determined from their Fourier transform intensity. A similar conclusion has also been confirmed by Bates [16]. Since the objective of microwave holography is to achieve the complex aperture field distribution of the antenna, especially the phase, solutions to phase retrieval problem can be applied to microwave holography. Several methods and algorithms have been investi gated for use in microwave holography including the Gerchberg and Saxton algorithm [38], the Misell algorithm [21], the plane-to-plane diffraction algorithm [39], and the method of simulated annealing [40, 41, 42]. 2.3.1 Existing Research on Phase Retrieval Microwave Holography Existing research on phase retrieval holography exhibits that, apart from occasional locking onto the local minima, the overall performance and the suitability to microwave holography of the Misell algorithm is very good [43, 44]. Therefore, the Misell algorithm has become the most widely used phase retrieval method in microwave holography. As for the others, the plane-to-plane diffraction technique provides a comparable performance to that of the Misell. However, instead of defocusing, the intensity measurements must be done at differ ent measurement planes in the Fresnel region. Difficulties may arise in practice with the requirement of having transmitter(s) /receiver(s) in the Fresnel region. Accurate distances between those transmitters and the AUT, which are required to calculate the diffraction filters, may be difficult to determine as well. The Gerchberg and Saxton algorithm exhibits slower convergence rate than the previous two, and therefore attains an inferior solution after the same number of iterations. Its requirement of a priori knowledge of the aperture Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 magnitude distribution is sometimes impractical. A computer simulation shows that, by incorporating the simulated annealing idea into the Misell algorithm, a better final solution can be achieved. Simulated annealing is a technique to find a good solution to an opti mization problem by trying random variations of the current solution. A worse variation is accepted as the new solution with a probability which decreases as the computation proceeds.(This definition is obtained from the World Wide Web at http://www.nist.gov/dads). The computational complexity of this method, however, is at least an order of magnitude higher than that of the others. Additionally, an appropriate “cooling schedule” (that is, the rate of reduction of the probability of accepting the uphill climbing of the error profile) for microwave holography has not been determined yet. One problem in using the Misell algorithm is that the solution may get trapped in a local minimum, which may be far from the global solution resulting in severe artifacts or totally incorrect retrieval results [45]. This problem may occur when the correspondence between the phase correction term (or in this case the lens transfer) and the measured magnitude distribution is incorrect, especially in the presence of noise. This also makes the algorithm sensitive to the initial function chosen, as will be seen later. An algorithm proposed by Isernia et.al. [46] minimizes the error formed by the difference of the field intensity (squared magnitude) instead of the magnitude as in the Misell algorithm to alleviate this problem of local trapping. The idea of quadratic inverse helps in understanding the problem of local minima and in determining the means of avoiding the trapping problem [46]. Additionally, the idea of using only one far-field magnitude pattern has been proposed [47]. The algorithm however has to deal with a highly non-linear problem, and requires an accurate initial function to converge to the correct solution. 2.3.2 The Misell Phase Retrieval Algorithm The Misell algorithm is arguably the most popular phase retrieval method applied to mi crowave holography applications. The algorithm requires that at least two intensities be measured under different focusing conditions to achieve phase retrieval. For microwave holography, the algorithm must be supplied with an initial trial function of the antenna Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 aperture distribution and two different measured far-field magnitude distributions. The ini tial function is typically obtained from ray tracing, or by simply using an ideal magnitude distribution and a random phase distribution. Generally, one of the far-field magnitude dis tributions is measured while properly focused, while the other measurement is performed while defocused. However, two defocused magnitude constraints perform equally well [48]. The phase shift induced by the defocusing structure or device is also required in order to retrieve the aperture fields. This algorithm has proven successful in retrieving surface features of a reflector antenna [43]. Fig 2.4 depicts the block diagram of the Misell algorithm. To start the algorithm, an appropriate initial function is chosen, then Fourier transformed into the far-field domain. The far-field magnitude is then replaced by the measured far-field magnitude while retain ing the calculated phase distribution. The newly obtained infocus far-field distribution is then inverse-Fourier transformed back to the aperture domain, where the aperture region of support constraint (i.e., the physical size of the antenna) may be imposed. The aperture distribution is then defocused by adding the corresponding phase correction, and the result is then Fourier transformed back to the far-field domain, where its corresponding far-field magnitude distribution is replaced by the defocused far-field magnitude from the measure ment. The result is then transformed back to the aperture domain, where the aperture constraint region, if any, and the phase shift due to defocusing is again applied to yield the first iteration in the estimate of the complex aperture field distribution. The algorithm then repeats until some convergence constraint is satisfied. During each iteration, in either the infocus or defocused plane, the sum of the squared difference between the measured magnitude distribution and the current estimate is taken. This value is monitored for every iteration and used as the condition to terminate the algorithm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 Initnl g u m Apartura A partura domain tra n ita r tunctlon O alo cu iad -to -F scu aad D afocuiad Apartura (laid DFT M agnitude Substitution Ir No M aasurad dalocuaad far-Hald magnituda IDFT t C onvarganca Daciaion Critaria To next stage processing O atoeuiad Apartura flald A partura domain tra n ita r (unction F o cuiad-to-D atocuaar F o c u n d Apartura Halt DFT M agnitude Substitution M d iiu r td fo cu iad fir-liild magnituda Foeuaad Far-flald IDFT Focuiad Apartura Halt ■ Figure 2.4: Block diagram of the Misell phase retrieval algorithm for microwave holography Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 CHAPTER III MICROWAVE LENSES Microwave lenses can be employed to defocus an antenna or alter the phase of the different ray paths reflecting on the antenna surface to emulate the effect of antenna defocusing required in the Misell phase retrieval algorithm. Different types of microwave lenses that are applicable for use in microwave holography measurement are introduced in this chapter. 3.1 Ray Tracing It has been mentioned that for WRF the antenna and its feed horn are considered to be electrically large at the frequency of interest (4.2 GHz). The method of ray tracing can be applied in designing the defocusing lens for microwave holography. The technique treats an electromagnetic wave by ways of ray optics, and neglects a presumably small effect of the diffraction of the field. The goal of this ray tracing design is to determine the location on the aperture of the antenna’s feed that corresponds to a specific location on the antenna aperture from where the same electromagnetic wave has been reflected. In order to do so for Cassegrain antenna, the geometry of the antenna’s main reflector as well as its subreflector and the location of the feed have to be known. Typically, a main reflector surface has a parabolic profile, while a subreflector has a hyperbolic one. The main reflector can be described as x2 (31) where x and y are the referred to as in Fig 3.1, and F is the focal distance of the parabola. It is required to determine the angle of reflection, 0, of an incident wave reflected on this main reflector at a specific distance x from the center of the antenna. Since, for a parabola, a normal vector to the curve always bisects the angle between a line parallel to the focal axis and the focal radius, the determination of 0 can be achieved simply by differentiating Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 y Figure 3.1: Paraboloid geometry used for Ray Tracing (3.1). The desired angle of reflection is 0 = arctan(— ). (3.2) Once the ray of the wave is traced to the subreflector, the hyperbolic geometry shown in Fig 3.2 requires that (!)’ - ( ! ) - ' <-> where b = y/<^ - a2 on the subreflector. The ray with the angle of reflection 0 can be represented in the same x-y coordinate by a linear equation y =- ta n 20 + c. (3.4) By solving (3.3) and (3.4), a location where this ray intercepts the subreflector surface can be found. The radial (horizontal) position of this reflection point is the solution of 1 X2 a2 tan2 20 — jy r 2c x+ .a2 tan 20. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.5) 30 y Figure 3.2: Hyperboloid geometry used for Ray Tracing A vertical position, y, can be obtained from (3.3) once (3.5) is properly solved. Considering the geometry in Fig 3.3, an angle a between the x-axis and the tangent of the hyperbola at the reflection point is simply dy a?x a = arctan (— ) = arctan(-^r-), (3.6) where x and y are obtained from solving (3.3) and (3.4). Additionally, an angle 7 between the y-axis and the ray path travelling from the reflection point to the feed can be found using 7 = 2(q —0). (3.7) Let / be the distance from the phase center of the feed horn to the feed’s aperture. The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 Figure 3.3: Illustration indicating quantities used in Ray Tracing distance from the center of the feed aperture where the aforementioned ray crosses is p = f tan 7 . (3.8) Utilizing these steps, it is possible to trace the ray from the aperture of the antenna (x ) to the feed horn’s aperture (p), or vice versa. 3.2 Dielectric Lens Among different types of microwave lenses, a dielectric lens is probably the most common and popular of all. This type of lens has been used in many electromagnetic applications [49, 50]. It modifies the electromagnetic wavefront by utilizing the fact that the phase velocities in different types of media are usually different. The lens consists of a material with known dielectric constant and loss tangent. Its surface(s) is machined to the desired shape to give the proper wavefront modification. Since the shape of the lens’ surface as well as the lens’ thickness depends upon the desired focal distance, the operating frequency, and the relative permittivity of the lens, the selection of the lens’ material has to be considered carefully. Intuitively, the loss tangent Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 of the material has to be small. Its refractive index or, equivalently, relative permittivity cannot be too large or too small. A small relative permittivity reduces an undesirable reflection coefficient at the lens surface, but increases the lens’ thickness. On the other hand, a large permittivity results in a higher reflection coefficient, but keeps the thickness of the lens small. Beside these electrical properties, the material is required to possess reasonable physical strength to retain the desired shape of the lens throughout the measurement. 3.3 Fresnel Zone Plate A Fresnel Zone Plate (FZP) utilizes a concept of Fresnel zones [51, 2] in order to focus the intensity of the wave passing through it. The most typical and basic type of zone plates, referred to as a standard FZP, consists of planar concentric rings, whose radii are those of corresponding fresnel zones. According to the FVesnel zone concept, waves from two consecutive zones add destructively. Therefore, a wave radiating through alternate zones is blocked by rings of metal conductor, while a radiation from the other zones is left undisturbed. This type of zone plate has long been used in electromagnetic and antenna applications [52]. 3.3.1 Standard FVesnel Zone Plate It is well known th at a maximum field intensity at the desired focal distance along the axis can be attained by blocking radiation from alternate zones. The easiest way to realize such a situation is to use rings of metal to block desired alternate FVesnel zones, leading to a standard type of FVesnel zone plate previously described. Radii of the zones, pn, can be calculated from k o W f i + £ - z ) = n7r! n = 1,2,3,... (3.9) where k„ is a free space wave propagation constant, and z is the distance from the focal point to the center of the aperture of the zone plate. The structure of this zone plate is shown in Fig 3.4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 ZONE I ZONE 2 Figure 3.4: Standard Fresnel zone plate[2] 3.3.2 Phase-corrected Fresnel Zone Plate It has been shown that the aforementioned type of FZP performs well for some applica tions [52, 53]. Nevertheless, it is necessary in most applications to make the zone plate competitive in focusing efficiency to the ordinary lens [54, 55]. Thus, the idea of phasecorrected FZP has been employed. The basic idea is very similar to th at of the standard FZP in th at it tries to combine constructively the wave radiated from different regions of the aperture. Instead of only dividing regions in the aperture to half-period Fresnel zones, each zone is also subdivided into a number of subzones. It is obvious th at if the size of subzones or, equivalently, the amounts of phase difference between adjacent zones are small, the wave radiated through each subzone can be approximated to have constant phase within th at zone. In order to have constructive addition for waves from different subzones, a phase-corrected device is required to correct or adjust the phases of waves radiated through subzones such th at all the phases are roughly equal. This phase-corrected device is referred to as a phase-corrected FZP. Mathematically, the design equation for this zone plate is very similar to that of the standard zone plate. The difference is that, for the adjacent zone radii being calculated, the phase difference is not ir but 2it/ M , where Af is the number of subzones in each full-period Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 fresnel zone. The equation similar to (3.9) can be expressed as + (3.10) where n = 1,2,3,... and m = 1,2,3, ...,M denote the number of full-period Fresnel zone and the subzone, respectively. 3.3.3 Fresnel Zone Plate for Microwave Holography It is important to notice that the above design equations are derived based on the assumption that the incoming wave is a plane wave. Most reflector antennas use feed horns, which instead create a spherical or curvature wave at the horns’ aperture [25]. Therefore, a spherical or curvature phase term is required to be included into those equations so that they can be correctly applied to this specific problem. New design equations for standard and phase-corrected FZP derived as part of this research are M v ^ - F)= m ' + (3.11) and M ) / A . + F 2 - F) = 2* ((n - 1) + y ) + (3.12) where F is the desired focal distance from the zone plate, and R is the radius of the curvature centering at the imaginary apex of the horn in consideration. After some simple mathematical manipulation, (3.11) and (3.12) become pi (2R? and Pim (2/ 2)2 ( l R{{* l) + M) fl)p n m + {(n-)+S}+AF{(n~)+S} 1 2 2 1 = 0, (3.14) respectively. The radius of each zone and subzone can then be solved quadratically. For WRF, the antenna’s feed is an ultra-wideband stepped circular horn with diameter of 1.3 m at the aperture [43]. The focal distance from the aperture of the horn is approximately 2.1 m along the horn axis. The calculation for Fresnel zone radii for a standard FZP using Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 (3.13) are performed using a desired focal distance of 1.9 m. For this particular configuration of focal distance and feed horn diameter, the horn aperture contains less than one half of one Fresnel zone period. Thus the focusing intensity and accuracy would not be very good, and they might adversely affect the accuracy of the phase retrieval. An alternative of using a phase-corrected FZP has therefore been considered. The zone plate radii of the 128subzone plate for the previously mentioned parameters have been calculated. In this case, the aperture is filled with 14 subzones, which will improve the focusing property relative to the standard zone plate. There are two different ways of implementing this FZP. One is to build a zone plate with uniform thickness, and use different types of materials, thus different dielectric constants, for each subzone to correct the phase. The difficulty for this implementation lies in obtaining the appropriate permittivity/material for each subzones. The other method is to use the same material, but varying the thickness of the lens in each subzones. For M-subzone zone plate, the difference in thickness of the material for adjacent subzones, , can be determined using It is interesting that, for the latter implementation with enough subzones, this phasecorrected FZP actually becomes an approximation of the dielectric lens. As will be seen later, the FDTD simulation shows th at this phase-corrected FVesnel zone plate works well and can potentially be used as a defocusing lens for microwave holography. However, its phase deviation profile is limited to the quadratic shape. Its properties also depend on the dielectric material, which does not provide much advantage over the dielectric lens. Moreover, a large number of subzones, which are necessary to improve the focusing ability, require a higher accuracy of m ac h in in g and construction for this type of zone plate. As a result, this zone plate was not constructed or used in any W RF measurement. 3.4 Metal-plate Lens The concept of a metal plate lens has long been considered an effective implementation of a microwave lens and has found use in various systems [56, 57]. This type of lens has been Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 chosen over a dielectric lens because it is lighter and easier to machine to the designed shape. The lens is formed by aligning shaped metal plates on the focal plane at a given spacing, to achieve a desired equivalent refractive index. Each metal plate must be machined to the shape required to provide the necessary phase change, then properly aligned and secured. (See Figure 3.8) 3.4.1 Parameters Since the phase variation is caused by the refraction of the lens, the first important param eter to be considered is the lens’ equivalent refractive index. 3.4-1.1 Refractive Index There are two distinct sets of propagating waves through the lens depending on whether the electric field orientation is perpendicular or parallel to the plates. The former would results in TEM wave propagation; thus, the phase velocity and therefore the equivalent refractive index of the lens would be that of the free space. In the latter (where the electric field orientation is parallel to the metal plates) the dominant mode of propagation is very similar to the T E i or TEio mode in a parallel-plate or rectangular waveguides respectively. The refractive index is shown to be <3 i 6 > where f}\ is propagation constant through the lens, kQ is the propagation constant in free space, and a is the plate separation. Note that the refractive index is always less than unity, in contrast with the typical dielectric lens, and is explicitly dependent on the operating frequency. The selection of the appropriate value of the refractive index depends upon the reflection loss and the lens thickness [56]. A small refractive index will increase the reflection loss a t the lens’ surface, while a value close to unity can in turn increase the lens thickness, making it difficult to manufacture and manipulate. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 3.4-1.2 Phase Variation Profile The shape of the lens’ surface is determined by the desired amount and spatial distribution of the phase variation. The design procedure, based on ray tracing, is shown in Fig 3.5. A pathlength difference is taken at an arbitrary point on the y-axis between two cases with and without the lens. It is then expressed as phase difference and equated to the desired amount of phase variation at that point. It is known that axially displacing the feed/subreflector of an antenna produces a phase change varying approximately quadratically with radius across the aperture. Consequently, to simulate the effect of such subreflector movement, the desired phase variation is set to vary quadratically as well. The starting equation of the lens’ shape design can then be expressed as k o \ j ( f - x)2 + y2 + nk0x - k0\ / f 2 + y2 = Dy 2, (3.17) where / is a distance between the phase center of the feed horn and the center of the lens aperture, and D is a constant dictating a maximum phase deviation a t the aperture edge of the antenna. After simple mathematical manipulation, the shape of the lens’ surface can be deter mined by the following equation: (n2 - l ) i 2 + (2/ - - 2n v//2 + y2) i + ^ y / P + y2V2) = 0. (3.18) It is important to note that the parabolic profile of the phase variation is not necessary for phase retrieval holography and other shaping may be used. However, in an attempt to simulate the phase variation of the standard defocusing method, the shape of the lens’ surface has been designed and built accordingly. For this lens, constructed with a refractive index of 0.6 and using small incidence angles characteristic of its installation on the WRF antenna, both transmission coefficient and phase discontinuity can be considered constants over the lens aperture [58]. For different scenarios, both of these parameters would vary, but could be simply evaluated. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 y Figure 3.5: Illustration of the design of the shape of the metal-plate lens’ profile. F is the focal point, and f is the focal length. A path length difference between two ray paths indicates a phase deviation 3.4.2 Construction Tolerances and Frequency Sensitivity The performance of the lens clearly depends upon the accuracy of the lens’ construction. Given a phase aberration A^(radian), the corresponding surface tolerance (At) can be easily found using A t = A<t>/{k0(l - n)}, where is the equivalent refractive index of the lens, ka is the propagation constant in free space, and A t is the positional aberration in meters. Because the equivalent refractive index and therefore the whole metal plate lens design process is dependent on frequency and metal plate spacing, it is important to study and estimate the effect of any frequency variation and plate spacing accuracy on the lens prop erties, especially the phase variation. The resulting changes of the refractive index due to small changes of frequency and plate spacing are An = and A n = —1~r*2^ ° , respectively. Since the above linear expressions are only accurate for small AA and A a, numerical calculations have been performed specifically on the WRF metal plate profile. Different variations in frequency and spacing were added to the designed values, and the resulting Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 changes in refractive index and phase variation have been calculated. For frequency sensi tivity, the phase deviations from the designed values of the WRF lens were calculated across the feed aperture for each frequency in the specified range. The final result, a relationship between the maximum phase deviations occurring at the aperture edge as a function of fre quency, is illustrated in Fig 3.6. It can be seen that although the phase deviation increases monotonically with frequency, the value of the phase deviation is still small at the edge of the frequency range. Since the frequency variation in the actual measurement due to fre quency drifts in the receiving system as well as drifts in the signal source (a satellite beacon) are relatively small (orders of kHz), its effect is negligible for this application. Similarly, a study of the tolerance of plate spacing was also performed. The result, a relationship between the maximum phase deviations occurring at the aperture edge as a function of plate spacing accuracy, is also shown in Fig 3.6. 3.4.3 Specification and Construction The metal plate lens for WRF holographic measurement was designed to operate at 4.1995 GHz. The lens diameter of 1.395 m was dictated by the aperture diameter of the antenna feed horn [59]. The desired equivalent refractive index was chosen to be approximately 0.6, so that the reflection loss at the lens surface can be kept small, while still maintaining an acceptable lens thickness. According to (3.16), the metal plate spacing of 4.38 cm was thus required. As previously mentioned, for this measurement the shape of the lens was designed to simulate the effect of the quadratic aperture phase variation resulting from axially moving the antenna subreflector. Choice of the phase variation depends on the signal-to-noise ratio of the signal source. The amount of phase variation at the aperture edge was chosen to be 1.3 rad (75 degree) for this measurement. Once the design was complete, thirty-two aluminum plates were individually prepared according to their position in the lens. The plate thickness is 1/16 inch (0.159 cm or A/45) in order to minimize the effect of the metal thickness while still maintaining adequate physical stability. The outline of these metal plates as well as lens are depicted in Fig 3.7. Pieces of polyethylene (PE) foam, with relative permittivity of 1.55, were then machined to the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 14 — Spacing effcict I 12- 3 is £ E 3 E I 3.9 4.0 4.1 4.2 4.3 4.4 4.5 Frequency (GHz) i i i i i i i 4.1 4.2 4.3 4.4 4.5 4.6 4 .7 Metal-plate spacing (cm) Figure 3.6: Numerical calculation of the effects of frequency and metal-plate spacing on phase variation of the WRF metal-plate lens. The absolute amount of phase differences from the designed values are calculated at the antenna aperture edge according to deviations from the designed values of measurement frequency and metal-plate spacing. designed plate spacing thickness and used as spacers between the metal plates. All of the spacers are of the same thickness; therefore, they do not affect the positional shape of the phase variation of the lens. The relative permittivity of the foam spacers is such that only the dominant mode can propagate through the lens. These spacers were also cut into a wedge shape in an attem pt to minimize reflections. A picture of the WRF metal plate lens is shown in Fig 3.8, where its parabolic surface shape can be clearly seen. The accuracy of the machining of the metal plate, At, is approximately 1 mm, which causes a phase aberration of 2 degrees. The accuracy of the metal plate spacing depends upon the accuracy of the thickness of PE spacers, which is 1.6 mm for the W RF lens. The maximum phase aberration caused by this variation is approximately 9 degrees, as per the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 ^995343954 1.4 meters Figure 3.7: Drawings of the WRF metal-plate lens Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 Figure 3.8: (a) Picture of WRF metal-plate lens for use in this application, (b) Picture of the lens placed in front of the WRF feed during the measurement Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 Direction perpendicular to the metal plates Electric field TEM componen TEt component Direction parallel to the metal Figure 3.9: Illustration of wave components travelling through the metal-plate lens, previously mentioned numerical calculation. 3.5 Circular Waveguide Lens One of the problems of using the metal-plate lens is the requirement to align the metal plates with the polarization of the incoming electric field. For imperfect alignment, part of the wave experiences a TEM-mode propagation as illustrated in Fig 3.9. Therefore, its phase does not change comparing to the other part experiencing a TE-mode propagation. An error therefore occurs when these two wave components combine at the lens’ output, and as a result causes an additional inaccuracy in determining the phase variation of the metalplate lens. Depending on the situation, a perfect alignment may be practically difficult to achieve. There is therefore a motivation to identify a different type of microwave lens without orientation preference, but still retains similar advantages of the metal-plate lens over the dielectric lens and the Fresnel zone plate. One of the solutions is a lens composed of square waveguides. In this case, both compo nents of the wave in Fig 3.9 will experience the same TE-mode propagation; therefore, the misalignment error at the lens’ output will not occur. The design procedure is identical to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 that of the metal-plate lens. Dimension of the waveguide as well as the equivalent refractive index is also the same as those of the metal plate one. However, the construction might be more difficult. Another alternative is to use circular waveguides. Similar to a parallel plate and rectan gular waveguide, the expressions for electromagnetic fields as well as properties of modes of propagation can be obtained by solving the Maxwell’s equations with appropriate boundary conditions. Since the derivations and some of the results are lengthy and can be found in appropriate textbooks (see, e.g. Collin [60]), only the related results are presented here. 3.5.1 3.5.1.1 Parameters Equivalent Refractive Index As anticipated, a variety of T M and T E modes can propagate through the circular waveg uide depending upon the waveguide’s diameter. As usual, let ka be the wave propagation constant in free space. The propagation constants of TM nm and T E nm mode of propagation can be given by ___________ 0TMnm = ) j ~ ' (3-19) ~ ( ~ ' ) 2> (3-20) and__________________________________ ___________ IhEnm = where a is a waveguide radius, and pnm and Pnm are the m th root of the n th order Bessel’s function and derivative of the Bessel’s function, respectively. From values of those pnm and p'nm, it can be found th at the first and dominant mode to propagate in any circular waveguide is the T E \\. Since j/n is 1.841, from (3.20), the propagation constant of this mode is 0 n m = \Jk2o ~ ( ^ r ) 2- (3-2D The next mode to propagate is the TAToi, whose corresponding pm is 2.405. Therefore, in order to have the dominant mode propagation, a waveguide radius has to satisfy the following condition: 0.293A < a < 0.383A, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.22) 45 where A is the operating wavelength. W ith the selection of the waveguide radius such that only the dominant T E u mode can propagate, the equivalent refractive index is found to be 0.293AN2 (3.23) As with the metal-plate lens, this refractive index is one of the most important parameters to consider in designing the lens since it dictates the amount of the phase or wavefront variation. The selection of its value is also dependent on the lens’ reflection loss as well as the lens’ thickness. 3.5.1.2 Phase Variation Profile Once the equivalent refractive index and the desired amount and shape of the phase variation are determined, the shape of the surface of the circular waveguide lens can be obtained using the same equation, (3.17), as that of the metal-plate lens. 3.5.1.3 Waveguide Placement In order to place those circular waveguides as close as possible, hexagons are first filled into the proposed aperture of the lens. Then, those waveguides with the determined diameter are simply placed at the vertices of those hexagons. Practically, where possibly hundreds of waveguides have to be placed, a relatively simple computer code can be written to effectively assign the position of each waveguide on the lens’ aperture. 3.5.1.4 Mutual Coupling Since the waveguides sire closely placed, the problem of mutual coupling between modes in waveguides appears to be inevitable. This problem is similar to that occurring in the array feed application and has been studied in the past [61, 62]. The expressions of the coupling admittance and thus mutual coupling coefficient for a variety of modes of propagation were given. However, even though the coupling is inevitable, by choosing the thickness of the lens thick enough (5-cm thick for the WRF lens) such that the higher-order modes are suppressed at the output of the lens, the effect of this coupling can be considered in significan t. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 3.5.2 Construction Tolerances and frequency Sensitivity It is easy to see that the precision of the machining the lens’ surface affects directly the amount and shape of the phase variation of the lens. The relationship between the amount of phase error, or pathlength aberration, and the surface tolerance is identical to that of the metal-plate lens. Besides the surface profile, the size and shape of the lens surface also determined the equivalent refractive index of the lens. As seen from (3.23), the refractive index depends primarily upon the radius of the waveguides and measurement frequency. Consequently, the construction tolerance in waveguide radius and the frequency variation clearly have an impact on the lens’ operation as well. Similar to the metal-plate lens case, for small changes in waveguide radius and frequency, changes in refractive index can be obtained by taking derivatives of (3.23) with respect to a and A. The results can be expressed as A" - An 2 (3 -2 4 > 1 —v? A a -------— . n a (3.25) Note that these turn out to be the same as the metal-plate lens case, even though the expression for the refractive index is different. Again, to avoid the inaccuracy when AA and A a are not sufficiently small, numerical calculations of the change in phase variation of the lens due to AA and Aa have also been performed. The procedure is very similar to that described earlier for the metal-plate lens. A range of waveguide radius and frequency was chosen. These values were then incorporated with the lens profile, and the changes in maximum phase variation at the aperture edge were computed. The plots of these absolute values of phase deviations versus waveguide radius and frequency are depicted in Fig 3.10. Although the values of the phase deviation are different, the effect is shown to have similar characteristic to that of the metal-plate lens (Fig 3.6). 3.5.3 Specification and Construction The circular waveguide lens was designed for use in antenna defocusing for PRH using the same transm itting source as the previous measurement, a geostationary satellite beacon of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 Frequency effect “♦ “ "RaaiirBtTBcr-f' 5 . 20 3.9 4.0 4.1 4.2 4.3 4.4 4.5 Frequency (GHz) i i i i i i 2.50 2.55 2.60 2.65 2.70 2.75 Waveguide radius (cm) Figure 3.10: Numerical calculation of the effects of frequency and waveguide radius on phase variation of the WRF circular waveguide lens. The absolute amount of phase differences from the designed values are calculated at the antenna aperture edge according to deviations from the designed values of measurement frequency and waveguide radii. Telstar 5. The measurement frequency was 4.1995 GHz. The amount of phase variation was slightly increased over that of the metal-plate lens implemented previously to overcome the limited signal-to-noise ratio [63]. A refractive index of 0.6 was chosen, leading to a waveguide radius of 2.62 cm, which also satisfied the condition for dominant-mode-only propagation. The construction of the WRF lens utilized a sheet of Extruded PolyStyrene (EPS) foam as a substrate of the lens. The sheet was machined to the designed lens’ surface shape. Then, holes with diameter of the waveguide were cut into the foam. A conductive coating was later applied inside of those holes to form the circular waveguides. (A data sheet for this conductive coating is available via the World Wide Web at the Sandstrom Products company at http://www.sandstromproducts.com/Tigs/a905.pdf). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 In an attem pt to investigate the PRH measurement using a non-quadratic phase vari ation profile, a three-stepped surface profile was implemented instead of a parabolic one. Preliminary results from the simulation suggest that given an accurate determination of the lens transfer function, the quality of the retrieval resulting from the use of this profile may outperform that using the quadratic phase. This phase variation profile also substan tially reduces the difficulty, cost, and time spent in accurately machining the lens’ surface. Furthermore, besides having no orientation preference, this lens is designed to have slightly higher maximum phase variation than that of the metal plate lens. The values of the phase variation at the middle of each step are 35, 80, and 140 degrees, respectively. The outline of the three-stepped profile is shown in Fig 3.11 along with the placement of the waveguides. An actual photograph of this lens may be seen in Fig 3.12. 3.6 3.6.1 Computer Simulations Finite Difference Time Domain (FDTD) 3.6.1.1 Basic FDTD Algorithm The Finite Difference Time Domain (FDTD) technique for solving Maxwell’s equations in electromagnetics was first introduced in 1960’s by Yee [64]. Since then, various ideas and developments involving the technique have been continuously proposed, improving its capa bility significantly. Yee’s algorithm uses the second-order accurate central finite differences to achieve a numerical approximation of Maxwell’s curl equation. Yee’s grid represents the arrangement of the electric and magnetic field components as illustrated in Fig 3.13. In Cartesian coordinate, the x-, y-, and z-component of the electric field are located at position (i + 1/2,j,k ) , (i , j + 1/ 2, k), and ( i,j,k + 1/2) respectively. Meanwhile, the x-, y-, and z-component of the magnetic field are evaluated at position ( i , j + 1/ 2, * + 1/ 2), ( i+ 1/ 2, j, k-1-1/2), and ( i+ 1/ 2, j + 1/ 2, k) respectively. Both electric field and magnetic field are also evaluated a half time step apart creating a so-called “leapfrog” arrangement. The computational domain of FDTD is discretized into several Yee’s grids. The grid spacing Ax, Ay, and A z may be determined from the wavelength of the highest frequency component Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.11: Drawings of the WRF circular waveguide lens Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.12: (a) Picture of WRF circular waveguide lens for use in this application. Picture of the lens placed in front of the WRF feed during the measurement Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 of the fields. Typically, the grid spacing has to be on the order of A/20 or smaller in order to reduce the amount of numerical dispersion error [65]. The interval between time steps is directly related to the grid spacing. According to the Courant stability condition [66], the interval between time steps, A t, has to satisfy At < ■ . — 1 (3.26) + ( s ?)2 + ( s i )2 where c is the velocity of light, in order to maintain the numerical stability of the algorithm. Once the grid spacing and the interval between time steps are determined, Yee’s grids are placed over the computational domain. Appropriate boundary conditions have to be imposed along the boundary of the object(s) of interest as well as the boundary of the computational domain. The algorithm then proceeds by iteratively updating the electric and magnetic fields according to their values from the previous time step. Update equations are shown as follows: H2+l/2( i,j + 1/ 2,* + 1/ 2) = H r 1/2( i,j + 1/ 2, k + 1/ 2) + ( l i t ) W * i + 1/2,* + 1) - + 1/2,*)] (3.27) + ( A ) l£ ?(*'»**+ ! / 2) - £?(*'. i + 1, * + 1/ 2)], ffy"+l/2(i + 1/ 2, j, k + 1/ 2) = H r l/2(i + 1/ 2, j, k + 1/ 2) + (?& ) ^ k + V 2) " E W ' i ' k + V 2)l (3.28) + (sS fe ) [£?(*' + 1/ 2,J,* ) - £?(*• + 1/ 2, j,fc + 1)1, H7+1/2(i + 1/ 2, j + 1/ 2, k) = H z~ l,2{i + 1/ 2, j + 1/ 2, k) + (A ) ^ + V2, J + l , k) - £ ? (i + 1/ 2, j,* )] + ( s S * ) W * '. i + 1/ 2, k) ~ (3.29) + 1/ 2, k + 1)], £ ? +1(i + 1/2 ,j, k) = £5 (i + 1/ 2, j, k) + ( z £ b ) [H?+1/2(i + 1/2,7 + 1/2, k) - m +l/2(i + l / 2 , j - 1/2, *)] + ( « & ) [Hy+1/2(i + 1/2,7,* - 1/2) - H F 1/2(i + 1 /2 ,7 ,* + 1 /2 )], Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3-30) 52 r A k + k +j o e, a e , ■ « , A A X , t, 1 i k / i / / i + i / j • x« +l Figure 3.13: Illustration of FDTD cell[3j Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 ££+1(*',i + 1/2, k) = + 1/2, k) + ( i f e ) (^x+l/2(i, j + 1/2, k + 1/2) - m +l/2( i,j + 1/2, k - 1/2)] + ( « f e ) [H"+1/2(i - 1/2, j + 1 /2 , fc) - m +U2(i + 1/2, j + 1/2, *)], (3-31) ££+ l(i,i,* + 1/2) = E?(i,j,fc + 1/2) + ( A ) (^ y +1/2(*' + 1/2, i , * + 1/2) - tf„n+l/2(i - 1 /2 , j , & + 1/2)1 3 ~ 1/2. * + 1/2) - f l ? + l/2 ( i , i + 1/2, * + 1/2)1, + (« & ) (3-32) where (m0 and ea are the free space permeability and permittivity, respectively. Consider the magnetic field at time, n + 1 /2 . Each of its components may be evaluated from its own value at the previous time step, n — 1/ 2, and four surrounding electric field components at time step n. Later on at time n + 1, each electric field component is updated using its own value from the nth time step and four surrounding magnetic field components at the previous ban time step, n + 1/2. The process is repeated and therefore the values of the electric and magnetic field are constantly updated until the appropriate time is reached, and the algorithm terminated. Different kinds of excitation ranging from a single frequency sinusoid to various shapes of pulse covering a wide range of frequencies may be used in FDTD models depending upon the applications [3]. 3.6.1.2 Sources of Errors in FDTD Different types of error may occur in implementing an FDTD model including numerical dispersion error, modelling error, as well as reflection from the boundary condition of the computational domain. A numerical dispersion error results from an inability of the numerical fields to propagate with the exact free space velocity of light in the computational domain. It can cause the accumulation of phase error especially in an electrically large problem th at may results in several non-physical phenomena [65]. The error however may be limited to an acceptable level by selecting a small enough grid spacing as mentioned in the previous section. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 Modelling error results from inaccurately sampling the fields in the complex regions. This type of error may be reduced by utilizing smaller grid size in such regions. The other type of error occurs when the geometry of the object (s) does not conform to the FDTD grid. In this case, the appropriate boundary conditions cannot be imposed on the actual boundary but instead on the staircased approximation of the physical one. Even with a finer grid, this sometimes leads to a formulation that does not converge to the correct solution [65]. This however is not considered a serious problem in this work since the inner wall of WRF feed horn has the staircase-type structure as well. Another source of error comes from the reflection of the numerical waves from the computational boundary. An Absorbing Boundary Condition (ABC) is required to confine the computational domain since the outer boundary of the computational domain cannot be updated via Yee’s algorithm. The important requirement is that the ABC minimizes the numerical reflections from the boundary so that they do not interfere with the actual results. Different methods have been developed for this purpose including those of Higdon [67], Liao [68], and an improvement made by Mei and Fang [69]. One of the most promising ABCs is the Perfectly Matched Layer (PML) [70]. The idea behind the PML ABC is to split the electric and magnetic field in the absorbing regions into components and subcomponents, and then assign the losses to individual split field components. Depending on a number of layers, the PML ABC is reported to achieve a very small fraction of reflection coefficient, which are several orders of magnitude better compared to the values of 0.005 to 0.05 attained from previous ABCs. Details in the derivation of the method as well as its results may be found in Berenger [70]. 3.6.2 FDTD of the Fresnel Zone Plate A two-dimensional Finite Difference Time Domain (FDTD) computer simulation was per formed in order to study the Fresnel zone plate design. The simulation was done in twodimensions by approximating the feed system and placing the Fresnel zone plate profile in front of it. A sinusoidal excitation was generated at 4.1995 GHz, the desired operating frequency. A standard rectangular grid, which is well suited for the WRF stepped horn Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 o.» 0.1 0.7 0.1 0.5 0.4 1.4 Figure 3.14: FDTD simulation: Relative electric field magnitudes along the focal axis of the feed horn for the Fresnel zone plates with different focal distances structure [59], has been implemented with the cell size of 3.5x3.5 mm. Zone plates with different focusing distances were also computed and simulated for comparison purpose. A simulation without the zone plate was also performed. Due to memory limitation, every simulated electromagnetic field cannot be stored. How ever, the fields along the horn axis were recorded for every time step. After the simulations, those recorded fields were then processed, using the Fast Fourier TVansform (FFT), to pro vide the magnitudes of the fields along the horn axis. Fig 3.14 illustrates the plot of these magnitudes for different Fresnel zone plates. These zone plates are of the phase-corTected type with 128 subzones. The distinct focusing distance of each zone plate can be observed at 1.75 and 1.9 m. These distances correspond to the designed values showing th at the results from the design and simulation agree well. The focusing distance of the simulation without the zone plate occurs roughly at 2.1 m, which is the focal point of the horn itself. As expected, the focusing appears to be ‘sharper’ as the focusing distance gets smaller. This trend occurs since the closer the focal point to the zone plate, the higher number of zones can be placed into the horn aperture. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 56 3.6.3 FDTD of the Metal-plate Lens A two-dimensional Finite Difference Time Domain (FDTD) computer simulation was also performed for the metal-plate lens. Using the same setup as that for the Fresnel zone plate, the simulation ensures the validity of the design as well as providing a numerical study on the effect of utilizing the polyethylene foam as spacers. For comparison, a simulation without the metal plate lens was also performed. After both simulations, the magnitudes and phases of the aperture fields were calculated. The phase difference of the aperture field between the two simulations was computed and compared with the calculation from the design. A plot of these differences is illustrated in Fig 3.15. A very good agreement between the design calculation and simulation can be seen across the aperture. In contrast to the phase, the magnitude does not experience any significant change resulting from the lens except for a slight drop in value as seen in Fig 3.16a. Note that, since only an approximation of the WRF feed structure was used in the simulation, these magnitudes are not identical to those of the actual WRF feed. The effect of the PE spacers has also been studied. The FDTD simulations were done with and without the PE spacers between the metal plates. As seen from Fig 3.16b, a small phase deviation ( 2-3 degrees) caused by the inclusion of PE spacers is visible, with increasing deviation at the aperture edge, which may add some small uncertainty to the phase distribution of the lens transfer function. In addition, a slight magnitude reduction occurs for the outer part of the aperture when the spacers are added. The magnitude obtained from the simulation with spacers is slightly higher than th at of the lens alone for the middle portion of the feed horn suggesting that the addition of the PE spacers improve the match of the metal plate lens in the middle of the aperture. The total magnitude drop caused by the lens and PE spacers is approximately 1 dB across the aperture. 3.7 Criteria in Lens Selection The dielectric lens is probably the first logical choice to consider due to its simplicity. However, the selection of the appropriate material may be difficult to achieve, considering the reflection loss and lens’ thickness as well as its strength. The implementation also Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 180 Theoreticil FbTD Sinrjulation 160 -*• I g 140 f £ 100 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 Apartura radius (m) Figure 3.15: Comparison between the calculation from the design and FDTD simulation of the aperture phase changes of the feed horn due to the metal-plate lens Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 I •» •20 IPS- •25 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 Apartuis radius (m) (a) -100 -106 -110 -115 -125 -130 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 Apartura radius (m) (b) Figure 3.16: FDTD simulation: (a) Magnitudes of the aperture field of the feed horn for three different cases. Overall shapes are unchanged among cases except for small magnitude variations and reductions, (b) Aperture phase difference of the feed horn between the metalplate lens with and without polyethylene. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 requires a precise shaping of the material to the specific shape. Weight of the lens can also be a cause of concern depending on the type of material chosen. The standard FZP is simple to implement. However, depending on the size of the feed aperture and the focal distance, the required focusing efficiency may not be achievable. Similar to the dielectric lens, the use of the phase-corrected FZP requires the appropriate selection of the dielectric material. Apart from a high reflection, the high-permittivity material makes the difference between the thickness of the adjacent subzones smaller, which could be a problem in implementing the FZP with large number of subzones. Moreover, for the FZP, the shape of the phase variation will be th at of the defocusing and cannot be varied. One of the advantages of using the metal-plate lens is that, depending on the shape of the lens surface, it can have a variety of phase variations, which may help improving the property of the Misell phase retrieval algorithm. The lens is also easy to implement and light compared to the dielectric lens. A drawback is that this type of lens requires a proper alignment with the incoming electric field orientation. Use of the circular waveguide lens solves the problem of lens misalignment, while the freedom of selecting the shape and amount of the phase variation is still retained, as is the ease and low-cost of construction and the ease of handling due to its light-weight construction. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 CHAPTER IV MEASUREMENTS AND RESULTS 4.1 Measurement System and Data Collection A total of five sets of microwave holographic measurements have been performed at the YVRF. The first two measurements utilized phase coherent holography. The third measure ment utilized a metal plate lens for defocusing, and two full data sets were measured, one with and one without the lens. Even though the intention of the third measurement was to perform phase retrieval holography, where the far-field phase measurement is not required, both magnitude and phase were however recorded utilizing the reference antenna in order to experimentally study and examine the effect of the metal plate lens on the antenna aperture field. The fourth set of data, where a circular waveguide lens was utilized, was collected in the same way: with and without the lens. The last set of measurement was performed only with the circular waveguide lens in place. All five data collections were done on a 91x91 elevation/cross-elevation scan with a spacing of 0.075 degree between points. During the measurements, the antenna under test was periodically pointed back to the transmitting source, a geostationary satellite beacon, in order to correct for receiver gain and phase drift, as well as to compensate for movement of the satellite and atmospheric effects. After data processing, the final holographic maps from these measurements achieved a spatial resolu tion of 0.65 m. To ensure the validity of the measurement, a square metallic test target, 1.5x1.5 m, was placed onto the antenna under test (AUT)’s main reflector surface over the last three measurement sets. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 4.1.1 4.1.1.1 Phase Coherent Holography Practical Issues There are a few practical problems one may encounter in performing microwave holographic measurements. First, magnitude and phase drifts may become significant, especially over long measurement times. Magnitude variation, which may result in an inconsistent magni tude level over the measurement grid, can be caused by gain drift in the receiving system, variation in the atmosphere, or variation of the source itself. This problem may be solved by repointing the antenna towards the source periodically in order to calibrate the magnitude level. Phase variation may also occur. Fig 4.1, for example, exhibits the phase variation characteristic of the WRF system. The antennas, both AUT and reference, were left point ing at the GE-2 satellite beacon overnight, and the relative phase between the two signals was continuously measured. This variation is likely due to transmission line length varia tions resulting from temperature change, since portions of the cables used in the system are outdoors. The problem of phase variation may also be corrected by repointing the antenna. Also, further mitigation has been attempted at WRF by burying the cable underground. Typically, a larger spatial measurement range, with a reasonable grid size, provides improved aperture resolution. However, a signal from a neighboring satellite may interfere with the measured patterns, especially at the far sidelobes. Therefore, this practical aspect must be considered in selecting the signal source. An example of such interference is shown in Fig 4.2, where two peaks, one representing the signal source and the other representing an interference source, are clearly visible. Attempts may be made however to filter out such interference if it is of slightly different frequency. Frequency drift may also be of concern for some measurement systems. At WRF, frequency variations up to 12 kHz have been experienced. This level of variation is typically insignificant for most hardware components. However, for our measurement system, a post sampling digital filter has a very narrow 3-dB bandwidth (900 Hz). As a result, the system was designed to monitor the received frequency in real-time and adaptively re-center this filter according to am updated frequency estimate. Fig 4.3 shows the frequency variation Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 P ti r a dlWifnct b iin w n two chw U. both pointing at OE2 b—con 135 130 125 s I 115 110 105 200 400 600 800 1000 1200 1400 Numbor of point (tailing from 10 pm to 9 9 am of Aug 18,1098 1800 1800 Figure 4.1: Phase drift characteristic of the WRF holographic measurement system monitored at WRF over one measurement run. The variation in beacon modulation rate may cause improper normalization of the mea sured magnitude pattern. The effect of modulation is the spreading of the signal energy over a wider spectrum width. Since our system measures only the peak signal level, the carrier-to-noise ratio of the signal is decreased compared to the non-modulated case. The effect of this variation in modulation rate is shown in Fig 4.4, where strips of differing mag nitude levels represent different beacon modulation rates. Integrating the signal power over frequencies surrounding the center frequency of the beacon may help alleviate this problem. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 Maprtud* of i n data (dB) O', Figure 4.2: Illustration of neighboring satellite interference during microwave holographic measurement 4.1.1.2 Hardware The phase coherent holography system for WRF is shown in Fig 4.5 [19]. This receiver sup ports two identical phase-coherent heterodyne channels. The narrowband satellite telemetry beacon, at a nominal frequency of 4.1995 GHz, is received by both the AUT (a 30-meter Cassegrain antenna), and the reference antenna (a 3-meter prime focus antenna). The outputs from each antenna are fed to the two receiver channels, where they are amplified, and then down-converted to an intermediate frequency (IF) of 180 MHz. Two identical bandpass filters with a 230-kHz bandwidth are applied to both signals to filter out the excess noise and the signals are again amplified. The signal from the AUT is also connected to an HP8562B spectrum analyzer to provide an independent monitor of the signal level and frequency. Both signals are then coherently down-converted to baseband (10 kHz) and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 Frequency variation ovw WRF hotograpNc mnturanMni <pan 14 13 12 11 1 10 I ’ 1 . 7 6 5 4, 0 2000 4000 6000 8000 10000 12000 Measurement point Figure 4.3: Down-converted beacon frequency variation monitored at WRF over one mea surement run digitized. It should be noted that common local oscillators are required at every down conversion in order to retain the phase coherency between signals in both channels. Both signals are then sampled and digitized at a 200 kHz sampling rate using a two-channel analog/digital converter for further processing. Before sampling, two identical op-amp lowpass filters with a cutoff frequency of 100 kHz are applied to the signals to prevent aliasing during the sampling process. Those samples are then stored on a computer for magnitude and phase estimation. A picture showing part of this measurement system is illustrated in Fig 4.6. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 90 80 70 60 90 40 30 20 10 Figure 4.4: The effect of modulation rate variation on measured far-field magnitude distri bution (dB). Labels on the horizontal and vertical axes indicate sampling points. 4.1.1.3 Software The signal-processing algorithm is shown schematically in Fig 4.7. The digitized signals are both digitally-filtered to eliminate noise and sidebands as well as interfering signals from neighboring satellites. Since the frequency of the signal can slowly change over time, the system is designed to monitor the frequency of the incoming signal and calculate the new digital filter designed for that center frequency. Thus, the bandwidth of these adaptive filters can then be reduced to m inim ize the noise. However, a narrower bandwidth typically requires a higher order filter; thus, increasing the computational cost of the processing. For these measurements, a bandwidth of 900 Hz was used. The two signals are then processed using the Steiglitz-McBride algorithm to further reduce the noise level, and finally the phase detection is performed. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 N X S N s o 'C VI l o g v .« ° ~ ll 3 N ®„ N 06 I I_ c t; , l v!V IoV to as B ^ Al’ a N <® < ® 1 y g ^ “ 2 S" ? £ a -V 2I * Ji 3A l 8«“• B gS B A. 2 — 2 ><®< ® T3 § ■8 —U H * ’ S-J I .* i/i x a 9 n 2 a i l £» Ii -Cl s"3 ■5 co ^ a Qua Qua oo Figure 4.5: Detailed hardware diagram of WRF microwave holographic measurement system Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 Figure 4.6: Picture illustrating WRF microwave holographic measurement system’s imple mentation The Steiglitz-McBride Algorithm The signal modelling technique of Steiglitz-McBride [71] is applied to further reduce the noise level. The Steiglitz-McBride algorithm is based on iteratively m in im iz in g the least square error between the calculated and input signals. A general form of the update equation during each iteration may be written W A ? -% ) ' where EW(z) is an error in the z-transform domain for the kth iteration, Bqk^ and ( ’ are the estimates of the numerator and denominator of transfer function of the model to be calculated, respectively, at the kth iteration, and X( z ) is the z-transform of the modelled signal, q and p are the number of zeros and poles. W ith adequate numbers of poles and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 Baseband signal felO kH z AUT channel A/D sampling at 200 kHz Digital bandpass Alter BWaflOOHz Steiglitz-Mcbride Algorithm X-correlation o extract Phase phi se data difference Identical ----------------------- p A/D sampling Baseband signal at 200 kHz f=!0 kHz -P Digital bandpass Alter BW=900 Hz Steiglitz-Mcbride Algorithm Figure 4.7: Schematic block diagram of software processing to obtain phase differences between signals from AUT and reference antenna during data collection zeros specified, the amplitude, phase, and frequency of the signal are accurately modelled. The random noise however is not and is therefore eliminated. A visual improvement, shown by the comparison between the simulated noisy signal and the regenerated one, can be clearly seen in Fig 4.8a. A simulation result illustrating the effect of the Steiglitz-McBride algorithm on phase measurement is also shown in Fig 4.8b. A pair of sinusoidal signals was generated for each noise level. A phase shift, ranging from 1 to 360 degrees, was applied to one of the signals. The phase differences between the two signals were then retrieved both with and without the use of the Steiglitz-McBride algorithm. These detected phase differences were then compared to the actual values to obtain the phase measurement errors for the phase shift from 1 to 360 degrees. These phase errors were then averaged for each noise level. It is obvious from the figure th at significant improvement in average phase measurement error is realizable over a wide range of noise levels when the algorithm is applied. The WRF implementation utilized a model of two zeros and three poles with five iterations. Phase Detection The final process of the data collection is to measure the signals’ magnitude and phase difference from the two sampled signals. After the signals’ magnitudes are obtained, the phase measurement may be done using one of the following methods. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 3 2 !as 1 o ! ■1 ■2 -3 0 20 40 60 60 100 Number of u m p in (a) S mutation nmM of tha offiKt of Stoiolti Mcbrldo algorithm on photo maaauramarf 29 Withstmcb Without etmcl I 19 i 110 9 0. 0 0.1 0.2 0.3 0.4 0.9 0.0 Notoo ampnudo of thnialad Nomial diotitMtan notoo «Mh «aro1 0.7 (b) Figure 4.8: (a) Comparison between the simulated incoming noisy signal and the output signal from Steiglitz-McBride algorithm. Each period of the "cleaned” has one more iter ation of the algorithm applied, (b) Simulation result demonstrating the improvement from the use of the Steiglitz-McBride algorithm on phase measurement Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 Let us consider two coherent signals iq(£) = A \ cos (at0t + 0i) (4.2) V2 (t) = A 2 cos (u 0t + 02). (4.3) The goal is to detect the phase difference between these signals. The obvious method is to cross-correlate those two signals over multiple of periods, which yields Ca(T) = f Jo V i( t) V 2 { t + T ) d t = f A 1 A 2 cos (wo t + 01) cos (u>o{t + t ) + 02) d t . Jo (4.4) After simple trigonometric manipulation and integration, (4.3) becomes c a(r) Thus, the = ^ J ^ c o s ( w 07- + 0 2 - 0 1 )• phase difference can be readily obtained (4.5) from the zero-lag term C«(0) = Al 2 *T c o s (02 —0i). However, there exists a phase ambiguity problem. To solve for the correct phase, one of the signals, 1/2(t) in this case, can be phase shifted by 7r / 2. The cross-correlation between this phase-shifted signal and the other original signal is calculated as rT £&(t ) = I M M cos (u)0 t + 01) sin (ua(t + r) + Jo Again the value at zero lag, C(,(0) = A1A2I s,n (fa _ < fo ) d t. (4.6) ,s computed. By combining these two relations for zero lag terms, the phase ambiguity can be resolved. Rather than cross-correlating the two signals, a second approach is to employ the idea of I/Q (In-phase and Quadrature phase) detection. Consider the signal v\ (t); its in-phase component v\r can be obtained by multiplying it with cosine function as follows. vi(£) cos (u/0t) = Ai cos 0 i + Ai cos (2u at + 0 1). (4.7) By measuring only the DC component, ui/ = A ico s0 i. (4.8) Similarly, by multiplying the incoming signal by a sine function of the same frequency, and keeping only the DC term, the quadrature-phase component v \q can also be obtained and expressed as v \q = A i sin 0i- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.9) 71 Thus, ri = v u + j v i Q = Ai expO'^i), (4.10) and the phase of vi (t) is determined. The same can be done with the second signal. Finally, by forming the ratio of the output r\ and r2 of the two channels, (4.11) the required phase difference can now be retrieved. Both of the methods have been successfully implemented for use in microwave hologra phy [8, 11]. The cross-correlation idea was implemented for our WRF measurement system. F ar-field C o o rd in a te T ra n sfo rm a tio n a n d In te rp o la tio n Since all data collection at WRF was done in X-el/el coordinate, relationships between X-el/el and u-v coordinates expressed in (2.24), (2.25), and (2.26) were used to locate the collected data points on the u-v coordinate. The mapping from az/el grid to corresponding u-v grid for WRF is shown in Fig 4.9. The d ata were finally interpolated into the uniform u-v grid so that the FFT could be readily applied. For WRF measurement, interpolations were performed via the Delaunay triangle-based [72] cubic interpolation method. 4.1.2 P h a s e R e trie v a l H o lo g rap h y The hardware for the phase retrieval measurement was identical to th at of the PCH mea surement, however only the AUT magnitude data were used. All the data were stored however, in order to compare and understand the phase retrieval result for test purposes. In addition, the measured phase data were used to compute the lens transfer function, which is used as one possible selection in the retrieval process. Use of the measured transfer func tion is practical, since it is likely th at one would use a characterized lens (e.g. one measured in a near-field range) with a known transfer function when employing this method. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 0.96 0.96 0.94 0.92 I 0 0.9 0.66 0.66 °'M3 3.09 3.1 3.19 3.2 3.29 13 Az(rad) (a) 0.06 0.06 0.04 0.02 a 0 • 0.02 -0.04 -0.06 **06 1 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.06 V (b) Figure 4.9: (a) WRF microwave holographic measurement grid in Az-El coordinate system, (b) Corresponding WRF measurement grid in u-v coordinate system Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 4.2 Measurement Results from PCH In this thesis, microwave holographic results obtained via PCH are used primarily as a reference result as well as to derive PCH-deduced lens transfer functions (the effect of the microwave lens on the antenna's aperture distribution) for PRH. As previously mentioned, four PCH measurements have been performed at WRF. Although their result illustrates some correct features of the antenna, the data from the first set of measurement was rendered incorrect since it contained significant interference from the neighboring satellite (see section 4.1.1). The data also possessed bad scan lines, possibly as a result of a beacon modulation. Three other data sets however have been cleanly collected, and their results show good agreement. The “ring” structure of the magnitude and phase far-field distribution, as shown in Fig 4.10a and 4.10b, from the three measurements is clearly visible. Distinctive features of the support structures are observed as well. Least-square error best-fit surface distortion maps of these four measurements are illus trated in Fig 4.11a to 4.11d. Artifacts due to bad scan lines and interference are clearly visible for the surface map from the first measurement (Fig 4.11a). Similar features as marked may however be observed from the other surface maps, especially in Fig 4.11c and 4.1 Id, thus demonstrating the repeatability of the system. The test metallic box, which was placed onto the antenna surface during the last two measurements, has also been retrieved properly in both measurements (Fig 4.11c and 4.11d). This proper retrieval and repeata bility along with the similarity of the predicted antenna pattern to the measured pattern validate the PCH results and the proper operation of measurement system. 4.3 M easurement Results from PRH using the M etal-plate Lens Data obtained as previously described have been processed by both phase coherent and phase retrieval holography algorithms. The aperture phase distribution obtained using PCH is shown in Fig 4.12. The result is in very good agreement with previous results [23]. The test metallic box as well as the quadripod struts and subrefiector blockage are seen, demonstrating this to be a valid reference for the other methods. Each far-field magnitude and phase measurement both with and without the metal Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 (a) (b) Figure 4.10: Magnitude(dB) and phase(deg) of the W RF antenna’s far-field distributions Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 Figure 4.11: WRF antenna’s surface deformation map (mm) obtained from PCH. Measure ments were performed in (a) fall 1998 (b) spring 1999 (c) spring 2000 (d) spring 2001 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 Figure 4.12: Aperture phase distribution (deg) of the WRF antenna resulting from the phase coherent holography. A, B, C, D, E indicate significant features of the aperture phase distribution plate lens has been independently processed using the phase coherent technique to obtain the corresponding aperture field magnitude and phase. The effect of the metal plate lens on the antenna aperture field may be viewed as an aperture field transfer function of the lens. This transfer function was determined by simply forming the ratio of the measured complex aperture field distributions with and without the lens. The magnitude and phase of this transfer function are illustrated in Fig 4.13a and 4.13b respectively. According to the FDTD simulation, its magnitude ideally would have a fiat distribution with the value of approximately 0.9 over the antenna aperture region, and its phase would have a parabolic profile with the designed maximum phase variation at the aperture edge. Due to measurement noise as well as imperfections in lens construction, the phase turns out to be noisy over the aperture. Furthermore, the presence of the bump in the area of the metallic Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 box indicates different magnitude levels in the two measurements in th at area. This likely occurs from the depolarizing effect of the surface displacement of the box since this type of lens modifies the polarization property of the feed. The second implementation of this technique uses a circular waveguide lens that does not exhibit this effect. The average value of the magnitude over the antenna aperture is close to 0.9, however, as expected from simulation. An average of one-dimensional cuts of the transfer function’s phase compared to that calculated from the design is shown in Fig 4.13c. It does have the desired overall parabolic trend as anticipated, but with significant deviations across the aperture. In phase retrieval processing, the measured far-field magnitude distributions are used as the measurement constraint. The lens transfer function is also needed and typically would be calculated or measured independently a priori. However, to study the properties of this technique, several choices of lens transfer functions are examined. Its magnitude may be chosen assuming an ideal distribution, or by using the distribution obtained from PCH (Fig 4.13a). As for the phase, the ideal parabolic phase, the phase from FDTD simulation, or the phase of the transfer function computed from PCH (Fig 4.13b) may be used. Several initial trial functions may also be used. Some examples are uniform magnitude and random phase, and uniform magnitude and fixed value phase. Support structure and subreflector blockages, modelled by the Ohio State University reflector antenna code [28], and retrieved from PCH as well as constant-value guesses, have also been included in the initial function in an attem pt to make it resemble the actual system. By using various combinations of the above selections, phase retrieval holographic results have been obtained and compared. In comparing the results, the aperture phase distribu tions themselves have been used directly instead of the retrieved surface deformation maps (i.e. deviations from the best-fit paraboloidal surface), which include the effect from the least square best-fit process. Table 4.1 summarizes mean square phase differences from the PCH result of some of the retrievals after 1,000 iterations. The closest result to the reference obtained from PCH is retrieved using the combination of uniform magnitude and PCH-deduced phase as a lens transfer function with the initial function modelled by the OSU reflector antenna code. This is to be expected since good a priori d ata should find the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 (b) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 60 D esign C alculation ; AWaraga o* im aaaurem ants 40 20 0 -20 -40 -15 •10 0 •5 5 10 15 Antenna apartura (m) (c) Figure 4.13: Magnitude and phase of the lens transfer function computed from PCH mear surement results of the WRF antenna with and without the metal-plate lens, (a) Magnitude of the PCH-deduced lens transfer function (dB). (b) Phase of the PCH-deduced lens transfer function (deg). (c) Average of one-dimensional phase cuts of the PCH-deduced lens transfer function compared to the design calculation best global solution. Therefore, by accurately predicting the starting function (that is the expected far-field pattern for the actual antenna), the problem is partially alleviated. Even though distinct differences may be noticed in some areas across the aperture, especially the artifact area over the upper part of the antenna, it can be seen, in Fig 4.14a, that the large-scale features of the antenna are properly retrieved, compared to the standard result shown in Fig 4.12. Those features include the severe deviations marked A, B, C, and E. Some smaller features, including the test metallic box (marked D) located just above the subreflector in this projection, may also be resolved. The choice of using both magnitude and phase from the PCH-deduced transfer function was also tried and as expected performed fairly well. However, mean square phase errors, from Thble 4.1, indicate that it may not Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 be as good as using uniform magnitude for some initial functions. The best retrieval using this transfer function occurs when the PCH-deduced support structure and subreflector blockages are included in the initial function. Important features were retrieved properly, as seen in Fig 4.14b. These results show that, given a good correspondence between the lens transfer function and measured far-field magnitude data, the idea of using the microwave lens to defocus the antenna works reasonably well. In performing the retrieval using only the a priori predicted uniform magnitude and parabolic phase transfer function, only some of the large-scale features and the metallic test box are properly retrieved, as shown in Fig 4.14c, while the other features appear to be lost. One of the reasons is that the actual transfer function has been heavily “implementationcorrupted” , as seen in the phase of the transfer function in Fig 4.13b and 4.13c, such that an agreement between the ideal lens transfer function and the measured far-field magnitude data has been lost. Severe distortions of the phase transfer function around the subreflector and the antenna’s edge may result from the effect of edge diffraction. The solution is likely being trapped in a local minimum. Also since the metal plate lens has been designed to operate with an electric field orientation parallel to the metal plates, any imperfection in aligning the lens with the beacon polarization causes errors as could improper placement of the lens. Nevertheless, by designing the lens with higher phase variations to overcome the measurement noise, accurate phase retrieval should be achieved. An improvement in accuracy of construction as well as better lens alignment can also help in further improving the retrieval quality. An improvement in mean square phase error relative to those used parabolic phase transfer function may be obtained using the phase calculated from the FDTD simulation as a phase of the lens transfer function. The retrieval result is shown in Fig 4.14d. Due to the fact th at the qualities of the retrieval results vary with the selection of the lens transfer function, it should be stressed that the determination of the lens transfer function plays an important role in phase retrieval. A priori measurement is the most useful; however, improved modelling and construction should be adequate. A lens th at is less sensitive to alignment should also improve the result. As for the selection of the initial trial function, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 Table 4.1: Summary of mean square phase error (Rad2) for different phase retrievals utilizing a metal-plate lens (Numbers in parentheses rank the results from best to worst). Initial Function UM/CP/GS1 UM/RP/GS1 osu UM/RP/PS1 L ens^^N w ^ transfer function Reflector Code v. Uniform magnitude Estimation 0.7481 PCH-deduced phase PCH-deduced (5) 1.4507 magnitude/phase Uniform magnitude FDTD phase 0.7765 (2) 0.9297 (16) 0.9530 Parabolic phase Uniform magnitude 0.6396 0.7480 0.8392 (15) 0.7656 0.9364 (7) 0.9685 (12) 0.8523 (6) (1) (4) (9) 0.7627 1.0231 (8) (U ) (13) 0.5051 (14) 0.7108 (10) (3) 1 UM=Uniform Magnitude/ RP=Random Phase/ GS=Constant value guessed of strut and subreflector blockages/ CP=Constant phase of zero value/ PS=PCH-deduced strut and subreflector blockages. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 150 100 50 1° : r t V'£ H H -5 0 ^ H -1 0 0 ^ H -1 5 0 Fig 4.14 (b) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85 Fig 4.14 (c) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 E so B D -o v- Fig 4.14 (d) Figure 4.14: Aperture phase distributions (deg) of the WRF antenna resulting from the phase retrieval measurement employing a metal-plate lens: (a) Using uniform magnitude and PCH-deduced phase lens transfer function (ranked # 1 in Table 4.1), (b) Using PCHdeduced magnitude and phase lens transfer function (rank # 4 ), (c) Using uniform magni tude and parabolic phase lens transfer function (rank # 9 ), (d) Using uniform magnitude and FDTD simulated phase lens transfer function (rank #3 ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 slight differences among each case occur since different initial functions often lead to different local minima around the actual solutions. Estimation of the initial function using the OSU antenna code appears to perform better than the other choices since this estimate is closer to the actual antenna. Since the retrieval algorithm has not been modified with the use of the lens, the convergence property as well as computation cost of the algorithm is rather similar to that of the standard phase retrieval measurement. 4.4 Measurement Results from PRH using the Circular Waveguide Lens Two PRH measurements utilizing the aforementioned circular waveguide lens have been performed using the WRF holography measurement system [19]. Even though the phase measurement is not required, the far-field phase d ata was recorded for comparison purposes. This phase information along with magnitude is processed to provide the PCH-deduced lens transfer function. To accomplish this, the measured far-field magnitudes were used in the Misell algorithm along with different choices of lens transfer function and initial trial function. Choices of lens transfer function included the aforementioned PCH-deduced function as well as those derived from theoretical calculation. Different types of initial function have also been used including that modelled by the OSU (Ohio State University) reflector antenna code [28]. Fig 4.15a and 4.15b compare the aperture phase distribution of the AUT obtained via PCH with the retrieved aperture phase distribution from one of the measurements using the theoretical lens transfer function and an initial function computed using the OSU code. Artifacts are still visible due to the trapping problem of the Misell algorithm and lack of correspondence between the lens transfer function and measured far-field magnitudes caused by the measurement noise and lens’ construction error. However, some of the obvious features such as the test box indicate the validity of the technique. Large-scale deformations are also visible. These results illustrate the possibility of using different distributions of the phase profile of the lens transfer function other than the quadratic one. Moreover, the use of the circular waveguide lens helps eliminate the depolarization effect present in the metalplate lens configuration. The mean square phase differences (MSPD) from the reference Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 (PCH result) for different retrievals are shown in Table 4.2. The fact th a t the best result is obtained using an initial function computed using the OSU reflector antenna code further indicates the trapping problem. During the second measurement, the possible change in modulation rate on the received beacon caused improper normalization in the far-field magnitude pattern, where sudden changes between scan lines were exhibited. The measured far-field phase pattern also con tained some artifacts, which produced a noisy aperture phase distribution even when re trieved using PCH. Consequently, the data set is considered unreliable and the results are not presented. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 Fig 4.15 (a) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 Fig 4.15 (b) Figure 4.15: Aperture phase distribution of the WRF antenna obtained from (a) PCH (b) PRH utilizing the circular waveguide lens (theoretical lens transfer function and OSUNECREF initial function are used in the Misell algorithm). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 Table 4.2: Summary of mean square phase error (RacP) for different phase retrievals utilizing a circular waveguide lens (Numbers in parentheses rank the results from best to worst). ' . Initial Function UM/RP/GS* UM/CP/GS* OSU UM/RP/PS* Lens Reflector Code transfer fu n c tio n 's. Estimation Uniform magnitude 1.4162 PCH-deduced phase (10) 1.2901 Uniform magnitude (8) 0.9474 A priori calculated phase Uniform magnitude Average PCH-deduced (12) 1.5014 (13) (14) 1.3569 0.4537 (1) 1.4505 0.8985 (5) 05 ) (9) (2) 1.1432 (7) 1.8831 1.6751 0.4663 (4) 1.1807 (6) £ magnitude/phase 1.1498 00 3 rn PCH-deduced 1.4544 (3) (ID phase" * UM=Uniform Magnitude/ RP=Random Phase/ GS=Conslant value guessed of strut and subreflector blockages/ CP=Constant phase of zero value/ PS=PCH-deduced strut and subreflector blockages. b Cuts of the PCH-deduced phase of the lens transfer function are averaged. A new profile is created by rotating the average cut. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 CHAPTER V ERROR ANALYSIS 5.1 Standard Propagation of Errors A method of standard propagation of errors has been proposed and widely used in error analysis for various measurements [73, 74]. The method is based on the fact th at a measure ment error can be propagated through stages of data processing and calculated at the end. Therefore, the effect of the measurement error on the final form of the data after processing can be realized. First of all, let us consider an N x 1 measured quantity vector x, and an N x 1 desired vector y, where y can be related to x via a linear transform Tyx, i.e., y = Tyxx. The covariance matrix of x can be expressed as Cx = E {(x - x)(x - x)T }, (5.1) where x isthe vector of true values, x is the vector of measured values, and £{} is the expectation operator. The error or variance of each element of x can be obtained from the diagonal element of this covariance matrix. The covariance matrix of y can be related to the covariance m atrix of x via Cy = (5.2) This equation expresses the well-known law of error propagation. It is seen that not only the error in x, but also the covariances contribute to the error in y. For the phase retrieval algorithm however, the process includes magnitude substitutions, which is a non linear operation. The output vector y in this case can be re-expressed by expanding a non-linear Tyx using a Taylor series: y = y(x) + T i(x —x) + higher order term s, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.3) 93 where 9<Ji OX2 2sa. 0X2 ... 5xs . .. Sis Q X jL (5.4) flyn . . . ! £ 9xn 9x2 For a small fluctuation, the higher-order term can be neglected resulting in .S8J u l Cy = TiCxT j . (5.5) Thus, by determining the first-order transformation matrix and the covariance matrix of the measured quantities, the covariance matrix and errors of the desired or processed quantity can readily be attained. Consider the flow chart for propagation of errors in the Misell phase retrieval algorithm shown in Fig 5.1. Aai and Aa.2 denote the fluctuations in defocused and focused magnitude distributions. Ac indicates a lens’ construction error or an uncertainty in the lens transfer function, while A a n d A<j>^ represent uncertainties in the magnitude and phase of the retrieved complex far-field distributions after the ith iteration. The algorithm keeps iterating until the pre-determined ntfl iteration, when the resulting far-field magnitude and phase is inverse-Fourier transformed to obtain the desired aperture phase distribution, <p, with a corresponding uncertainty, A<p. From Fig 5.1, AmW and A c a n be expressed as = TmaA ai + TmcA c + T S A m f ' 11 + = T0aA al + T ^ A c + r £ )A r4 i- 1) + r <£ )A 4 <- 1) (5.7) Am? = TmaA a 2 + TmcAc + T ^ A m f 1 + T ^ A * ? (5.8) *4° = T ^A a 2 + T ^A c + T g A m f + T ^ A ^ (5.9) Amf A ^ ' 1} (5.6) where Tma, Tmc, Tmn, Tm$ , ... represent respectively the first-order transform from quantity a t o m , c to m, and so on. Since the magnitude of the retrieved far-field distribution is exactly that of the measured one, Tmc, Tmn, and Tm<$ are all zeros, and therefore (5.6) and (5.8) simply become Am? = A ai, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.10) 94 IFT-DsfocusingFT-Magnituds substitution IFT Figure 5.1: Flow chart for propagation of errors in the Misell phase retrieval algorithm Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 Am^ = A a2, (5.11) for every iteration. As for the A<f>terms, the expressions after the first iteration can simply be written as A *il) = T ^ A a i+ T ^ A c , (5.12) A 4 l) = T ^ A a 2 + TfcA c + i f f >Am<l) +Tg,A4>{l) = T ^ A a 2 + (T&Tm* + T & ^ A a x + (T * + T ^ T ^ A c . (5.13) Similarly, for the second iteration, A 0(t2) = T^aAfl! + T ^A c + T ^ ’Am*11 + T ^ A ^ = (T* + T {^ ]T ^ ]Tma 4 T ^ T ^ T ^ A a , - H T ^ T m a + T ^ T ^ ) A a 2 + (T,c + + T fflffT ^ A c , (5.14) A<42) = r*A a2 + T^Ac + T ^ A m ™ + T ^ A ^ = ( Z j y r - + i f f }r * + i f f i f f > i f f ’r , , , . + i f f }i f f > i f f > r * ) a « i +(T*, + T f f ’i f f )Tmo + T £ 2)T<i2)Tp/i:a)Aa2 + (T * + i f f ^ + i f f > iff ^ + r f f }r f f > r ff ^ A c . (5.15) It can be seen that these phase uncertainties can always be written as a function of A ai, Aa2, and Ac. Thus, for the final iteration (nth), where T ^ , A m 2{ ] = A a2 (5.16) A4>(2 ] = (5.17) T ^ A a ^ T ^ A a a + T ^A c, , and T^. are the combination of linear transforms of previous iterations from quantities aj, a 2, and c to fa , respectively. An uncertainty in aperture phase distribution, Atp, can be written as A<fi = r ^ + T ^A m S "’ = ( T ^ T ^ J A a i + ( T * ! ^ , + TpmTma) A a 2 -(- ( T ^ T ^ A c = FaiA a i + Fo2Aa2 + FcAc, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.18) 96 where and T^m is the linear transform from quantities <f>and m to <p, respectively. The covariance of the resulting aperture phase distribution can be expressed as Cv = E{A<p&<pT } = FaiCaiF ^ + FaiCajF^ 2 + FcCcF j + F0lCaia2F ^ + FaiCaiCF j FajCajaiF^ + Fa2 Ca2CF j + FcCcaxF jv + FcCca2 Fj2, (5.19) where Cai, Ca2, Cc represent the covariance matrices of the measured far-field magnitude distributions and the lens transfer function. Their cross-variance matrices are also denoted as shown. An error or variance of tp can be attained from the diagonal entries of its covariance matrix. Also since every element of ai, <12, and c are independently measured, calculated, and constructed, it is reasonable to assume that their covariance matrices contain only diagonal entries. It is also reasonable to assume that the errors and variances of each entry may be similar, since they were all performed using the same equipment under similar conditions. Additionally, it is reasonable to assume very small correlation between these three parameters. In this case, (5.19) may be simplified to C 9 = (Sa i )2 FaiF l + (5a2 )2 Fa2 Fj 2 + {6 c)2 FcF j , (5.20) where <fai, 8 a 2 , and Sc represent the uncertainties in far-field magnitude distributions and lens construction error. It is necessary however to determine the aforementioned transformation matrices. The transformation matrices T<sc, T^m, and are unfortunately difficult to obtain, since the expression for <t> must be explicitly written. The whole process also requires intensive computations, partially because the transformation matrices must be updated for every iteration. These reasons make it difficult to develop such an analysis. Consequently, an empirical analysis is additionally performed to provide an idea as to how the retrieval process reacts to the aforementioned effects. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 5.2 Empirical Error Analysis Due to the aforementioned difficulty in performing the standard propagation of errors, an empirical error analysis procedure has been designed and performed to observe the effect of the measurement noise, lens transfer function error, as well as choices of phase variation profiles on this novel PRH measurement. 5.2.1 Analysis Procedure The block diagram of the analysis is illustrated in Fig 5.2. A modelled antenna aperture distribution with known surface distortion is first created. For this study, three distorted regions with different error magnitudes are numerically located in the aperture. This aper ture model can then be combined with the simulated lens transfer function with a specified amount and profile of construction inaccuracy, and a systematic error resulting in the de focused aperture distribution. The magnitude of far-field distributions corresponding to those two apertures can be obtained by utilizing the Fourier transform relationship. Sim ulated measurement noise is added into these far-field data. This information along with the perfect lens transfer function is fed into the Misell retrieval algorithm. The retrieval result, particularly the antenna aperture phase distribution, is later compared with the model. Different types of error and noise can be independently included or excluded from the study to observe the effect of each of them separately. A visual comparison between the model and one of the retrievals may be observed in Fig 5.3a and 5.3b. Distorted regions are properly retrieved ensuring the validity of the analysis procedure. 5.2.2 The Effect of the Measurement Noise The effect of the measurement noise on PRH can be empirically studied by inserting the sim ulated random measurement noise of varying magnitude level into both far-field magnitude distributions. It is assumed that the amount of the noise that both focused and defocused patterns experience is the same since the two distributions are normally measured under the same conditions. These noise-corrupted far-field magnitude distributions are used to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 -----A perture distribution C o n stru ctio n an d analysis arror S pacitiad Ians tra n sla r lunction 6 — ►© M agnituda ot th a tarliald distribution ml Ians M sgnituda ol th a farllald distribution w/o Ians R an d o m m a a a u ra m a m n o iia Spacitiad Ians tra n sla r lunction R an d o m m a a tu ra m a n t n o ise Misell p h a sa retrieval algorithm il ratu lt Figure 5.2: Schematic block diagram illustrating the empirical study procedure Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 A k 30 r (a) 30 (b) Figure 5.3: (a) Aperture phase distribution of the model used in the empirical analysis (b) Aperture phase distribution of one of the retrievals from the empirical analysis Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 retrieve the aperture phase distribution of the antenna. Initially, the quadratic lens trans fer function with maximum phase variation of 100 degrees was used. For the worst case, a totally incorrect and noisy aperture distribution is obtained. However, as anticipated, the result improves as the inserted noise level decreases. The surface distortions are also retrieved properly indicating the validity of the solution and the simulation procedure. Quantitatively, the mean square error (MSE) between the retrieved and actual (model) aperture phase distribution is computed for each case. The nature of the variation of these MSE’s versus the noise levels is illustrated in Fig 5.4a. To reduce the uncertainty, the values of these MSE’s are obtained from the average of MSE’s from several simulations with different sets of simulated noise. As the noise level increases, the MSE also increases, as expected. This occurs since more noise has corrupted the far-field magnitude distributions; thus, reducing the correspondence between those far-field patterns and the lens transfer function in the Misell algorithm. The same study was also performed for the three-stepped phase distribution of the WRF circular waveguide lens. Similar characteristics are obtained and shown in Fig 5.4b. 5.2.3 The Effect of the Lens Transfer Function Error As with the study of measurement noise, simulated lens transfer function errors can be injected into the algorithm. This error represents the lens construction error in practice. Its magnitude level is varied to indicate differences in the severity of the error. This error is added to the lens transfer function, which will then be used to calculate the defocused far-field magnitude pattern. As before, the aperture phase distribution for each case is retrieved and then compared with th at of the model antenna. In Fig 5.5, MSE’s are plotted versus the severity of transfer function errors, whose values represent their corresponding standard deviations. As anticipated, more severe error in the transfer function causes higher MSE. This happens because the error in the lens transfer function affects the defocused far-field magnitude pattern, while the ideal lens transfer function is used in the algorithm. Consequently, the non-correspondence between the lens transfer function and the magnitude distribution is created. Similar results are obtained for the WRF circular waveguide profile Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 10000 8000 6000 4000 e 1 2000 -70 -60 •SO -40 -30 -20 -10 -20 -10 Relative nolaa level (dfl) (a) 10000 8000 f 6000 4000 | 2000 -70 •60 -SO •40 -30 Relative notea teval (dB) (b) Figure 5.4: Characteristic of MSE versus the simulated noise level for a) quadratic phase transfer function profile (100 degrees) and b) WRF circular waveguide stepped profile Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 as well. The aforementioned random error may result from limited assembly precision. It is pos sible however th at the error systematically occurs at a specific location of the lens aperture. A similar numerical study on the effect of the construction error at the specific location of the aperture has been performed using the previous procedure. However, instead of adding the random construction error onto the lens transfer function, only a deterministic error has been added to the location of interest on the aperture. Apart from illustrating how the retrieval algorithm would react to different errors from different locations, the results should also provide insight as to which area of the aperture the transfer function determination needs to be more accurate. Initially, the quadratic lens transfer function was used. As before, the phase retrieval algorithm is employed, and the results are compared with the model, where the MSE’s can be obtained. The study has been conducted by varying the amount of error, location of the error, as well as the amount of phase variation of the lens. The effect of the magnitudes of error may be viewed in Fig 5.6, where the abscissa and ordinate represent the level of error and the change in a root mean square error (RMSE) from an error-free case, respectively. Different curves indicate, as marked, different locations at which the errors occur. All the results were obtained using the quadratic phase transfer function with 100 deg of maximum phase variation. As anticipated, as the magnitude of transfer function error increases, so does the change in RMSE. However, the change in RMSE is not linear. Additionally, for this 100-deg maximum phase variation profile, the effect of the error is more significant when it occurs in the middle of the aperture, which is blocked by the subreflector and/or feed in an actual system. Nevertheless, the trend in RMSE with increasing transfer function error exhibits the same characteristics regardless of the location of error. Fig 5.7 shows a plot between changes in RMSE, for different levels of transfer function error, versus a location of the error when the 100-deg quadratic phase variation profile is used. Again for this phase variation profile, the effect of error is more significant at the middle of the aperture. The change in RMSE is rather constant for the area around the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 50 • 0 2 6 4 8 12 10 Lara translar function a m r (mm) (a) 100 S I I I o 0 0 2 4 3 5 6 Lana transfer function anor (mm) 7 8 (b) Figure 5.5: Characteristic of MSE versus the simulated lens transfer function error for a) quadratic phase transfer function profile (100 degrees), and b) W RF circular waveguide stepped profile Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 104 Quadratic 100-dcg 0.10 • 0.0 m B -B 3.325 m 4 —4 6.65 m 9.975 m • 13.3 m 0.08 I 0.06 0.04 0.02 0.00 0.0 0.5 1.0 1.5 2.0 3.0 as Magnitude of tha dafarmtniatic lana trarafar function anor (mm) (a) WRF Circular WQ 0.030 • —« 0.0 m B -B 3.325 m a —4 6.65 m V—» 9.975 m ♦ ♦ 13.3 m £ 0.025 0.020 0.015 i £ o.oio I 0.005 10.000 0.0 0.5 1.5 1.0 2.0 ZS Magnituda of tha datafmintatic lana tranafar function anor (mm) (b) Figure 5.6: Characteristic of RMSE versus the deterministic lens transfer function error for different locations (distance in meters from the center of the antenna aperture) a) quadratic phase transfer function profile (100 degrees), and b) WRF circular waveguide stepped profile Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 105 middle, and tapers off around the outer edge of the aperture. The study has been expanded to the other phase variation values as well. In Fig 5.8, similar plots are illustrated for 60- to 320-deg quadratic phase variation. Similar charac teristics to that of a 100-deg variation can be seen when the range of variation between 60 to 140 deg is used. The changes in RMSE are almost constant regardless of the location of error for 160 and 180 deg. For a 200-deg phase variation however, the outer edge and the middle of aperture are more sensitive to the error, while the effect is the least significant in the area in between. For 220 to 320 deg, the middle of the aperture again appears to be the area most sensitive to error. An interesting feature however may be noticed in the area around 8 meters (out of 15-meter radius) from the middle of the aperture, where a crossover in RMSE at 270 deg is seen. The changes of RMSE for every case are nevertheless the same at the outer edge. 5.2.4 The Effect of the Amount of Phase Variation An intuitive shape of the lens, or the phase variation resulting from it, is that of the paraboloid since it closely imitates the effect of the traditional antenna defocusing. Under the absence of measurement noise and all other imperfections, the amount of phase vari ation should not be of significant concern as long as two independent sets of data under different focusing conditions are provided. Nevertheless, under the presence of noise and imperfections, its effect on phase retrieval is significant. A study has been performed using the procedure developed in previous sections. The value of the phase variation at the aperture edge of the quadratic lens transfer function is varied from 20 to 360 degrees at intervals of 20 degrees. Aperture phase distributions are retrieved from the simulated far-field magnitude distributions under the presence of certain amounts of noise. The MSE is again computed for each case. The plot of the MSE versus these phase variations of the quadratic phase profile is shown in Fig 5.9a. The simulated measurement noise is kept constant for every simulation. Again the average value of MSE is used. It can be seen that in the first part of the plot, the MSE decreases as the amount of phase variation increases, since for the small value of the phase variation, the distinction Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 Quadratic 100-deg o.io • 0.5 0.08 *-« 1.0 A—A 15 a —▼ 2.0 3.0 0.06 2 0.04 0.02 0.00 0 2 6 4 Location of arrora (dttanca In im m 8 10 12 14 from tlM cantor of Ha a ilw n i apartura) (a) WRF Circular WQ 0.030 a - a o.5 8 - 8 1.0 I 0.025 a - a 2.o I 0.020 I 0.015 I * 0.010 0.005 0.000 0 2 4 e 8 10 12 14 Location of ttio anon (dlatanco in matarfrom tha cantor of tha antanna apartura) (b) Figure 5.7: Characteristic of RMSE versus the location of the deterministic errors of different severity a) quadratic phase transfer function profile (100 degrees), and b) WRF circular waveguide stepped profile Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 107 Quadratic 1-mm 60-140 dag 0.012 a —a 60 deg a —a so deg s 0.010 a - a ioo deg a - * 120 deg • • 140 deg 0.006 2 0.006 0.004 0.002 0.000 0 2 8 6 4 10 12 14 Location of errore (dietancee in meter* from the center ot the antenna aperture) (a) Quadratic 1-mm 160-220 dag 0.006 160 deg 180 deg 200 deg 220 deg f l 0.007 | 0.006 | 0.005 2 | 0.004 0.003 Iec 0.002 0.001 0 2 4 6 8 10 12 14 Location at errore (dietancee In matere from the center of the entenna aperture) (b) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 108 Quadratic 1-nun 240-320 dag 0.011 £ 0.010 | 0.009 | 0.008 240 dag 260 dag A—* 260 dag • —a 300 dag ♦ 320 dag a-a I 0.007 I | 0.006 8 0.005 S 0.004 at | 0.003 0.002 o 2 4 6 a 10 12 14 Location of arrora (diatancaa In malara from tha cantor of tha anlanna apartura) (c) Figure 5.8: Characteristic of RMSE versus the location of a 1-mm deterministic lens transfer function error for different amount of phase variation (at the aperture edge) of the quadratic phase profile a) 60-140 degrees phase profile b) 160-220 degrees phase profile c) 240-320 degrees phase profile between the two magnitude distributions is too small and therefore is easily corrupted by the presence of measurement noise, causing an inaccurate solution. When the amount of phase variation increases, the difference between the two magnitude patterns is increased leading to a better retrieval. At values greater than where the MSE appears to be at minimum (around 100 degrees), it is expected that the MSE will increase again since a large amount of phase variation usually results in a highly distorted magnitude distribution such that its features can be easily corrupted by the measurement noise. As seen, the MSE appears to increase as anticipated until the phase variation value reaches about 180 degrees. Subsequently, the MSE seems to stagnate and even decrease until 300 degrees, where it starts to rise again. This is possibly the contribution from the wrap-around effect of the phase at the edge of the antenna. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 109 92 90 88 86 84 82 i 78 76 74 0 50 100 150 200 250 300 350 400 Maximum phase variation (degrees) (a) 23 22 21 I 20 19 I 18 i 17 16 15 14 0 50 100 150 200 250 300 350 400 Maximum phaae variation (degrees) (b) Figure 5.9: Characteristic of MSE versus the amount of phase variation (at the aperture edge) of the quadratic phase profile a) under the presence of constant measurement noise b) under the presence of constant lens transfer function error Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 110 The effect of the amount of phase variation on the retrieved surface accuracy or aperture phase under the presence of a lens transfer function error may be determined utilizing a similar approach. The characteristic of the MSE under such conditions is shown in Fig 5.9b. For the small amount of phase variation, the difference between the two patterns is not distinctive; thus causing significant error. However, the MSE improves with increasing amount of phase variation. The MSE reaches a minimum at about 50 degrees and then increases, possibly because the construction error starts to dominate the highly distorted magnitude distribution. The “stagnation region” of the MSE between the phase variation of 180 and 300 degrees, which is likely the same feature th at has been seen on Fig 5.9a, is also visible. 5.2.5 The Effect of the Phase Variation Profile One of the advantages of using the microwave lens for microwave holography is that it presents the opportunity of using different types of spatial distributions of the phase varia tion rather than being restricted to that of traditional defocusing. The first logical choice in varying the profile’s shape is to change the order of the polynomial of the phase profile from two, which is the case for quadratic variation, to some other value. In this study, a range of the order from 0.5 to 18 for the phase variation has been used under different measurement noise and construction errors. It is found that, for a certain value of measurement noise less than approximately -40 dB, the increasing order of phase variation profile generally improves the MSE value. However, the improvement gets smaller as the order gets higher and the MSE finally appears to converge to a certain value. An example of this behavior of MSE due to the change in the order of phase variation profile is shown in Fig 5.10a for the noise level of -44.5 dB. Note that for the case where the transfer function error is present, the MSE increases as the order gets higher. The value then remains constant and may even slightly decrease at the higher polynomial order, as seen in Fig 5.10b. The magnitude of the MSE under the presence of the lens transfer function error is smaller with less variation than that under the presence of measurement noise (Fig 5.10a). Therefore, the use of higher order lens transfer function is probably still Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ill advantageous. A preliminary study employing different types of the phase variation profiles has been performed under a constant level of noise. Although only moderately significant, some of the profiles offer MSE’s that are smaller compared to that of the quadratic one indicat ing the potential benefit of utilizing the microwave lens method. Table 5.1 compares the mean square differences from the modelled retrievals using different phase variation profiles. Almost 8% of the reduction in MSE from the quadratic profile may be achieved. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 112 84 82 80 78 76 74 72 70 68 66 4 0 2 6 a 10 12 14 16 Order of tte pfias* variation profile's polynomial (a) 21 20 19 18 17 16 15 0 2 4 8 8 10 12 14 18 18 20 Order of the phase variation profHe's polynomial (b) Figure 5.10: Characteristic of MSE versus the order of the polynomial describing the phase variation profile a) under the presence of constant measurement noise b) under the presence of constant lens transfer function error Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 113 Table 5.1: Examples of mean square phase error (deg1) of retrievals using different phase variation profiles (empirical study). Percentage change Phase variation profile description1 MSE from the quadratic (deg2) profile retrieval (%) Quadratic with 100-deg maximum variation 76.9006 0 WRF circular waveguide lens profile 76.1880 -0.71 83.5121 >6.61 75.8437 -1.06 77.4367 >0.54 69.1972 -7.70 74.0369 -2.86 69.1238 -7.78 70.9170 -5.98 73.2514 -3.65 70.3500 -6.55 100cost-—) deg; V 0 Ha 100cos2(— ) deg; V 0 Ho 100C— -— )2 deg;r<0.17/?„, V0 0.17fio R “T “7 1O0C----- ---------r deg; r * 0 .1 7 /?, V 0 Ho - 0 M R o 100(— -— )2 deg; r <0.9/? , V0 0.9R0 R *T 7 100(-----2------- ) deg; r 2 0.9 R„, V0 H0 - 0 9 K o Odeg ;r< 0 .8 3 /?„, V 0 100 deg; r 2 0.83 Ra, V 0 A U nn Odeg ; r<0.94/?o, V 0 L J 100 deg; r 2 0.94/?0, V 0 Odeg ; r <0.87 R0 , t > 0.94Ra , V0 100 deg; 0.87 Ra S r S 0.94 R a , V0 Odeg ; r < 0.66/ J „ , r > 0 . 9 4 V0 SO deg ;0.66 R „ S r < 0.87Ra, V0 100 deg; 0.87 R0 S r S 0.94 R a . V 0 100deg;r20.94 Ra, 0 dee ; otherwise -J tl2 £ 0 < J tl2 11 U q ____ c Descriptions are given in terms of parameter r and 0 in a cylindrical coordinate, where r=0 is at the center of the antenna's aperture and R„ is the aperture’s radius. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 114 CHAPTER VI CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH 6.1 Conclusions Microwave holography has been a popular and powerful technique for measuring reflec tor antenna surface deviations. Standard phase coherent holography requires direct phase measurement of the far-field distribution, which presents some practical difficulties in im plementation. Of all the phase retrieval methods proposed for use in microwave holography, the Misell algorithm appears to be the most popular and suitable. The algorithm requires that the subreflector or feed of an AUT be moved axially in order to obtain defocused far-field distributions. This method of defocusing however cannot, be easily achieved at some facilities, including WRF. The use of a microwave lens has therefore been shown as an alternative for antenna defocusing for phase retrieval microwave holography. For initial measurements, a metal-plate lens was chosen and designed to have a parabolic phase vari ation in order to imitate the effect of defocusing by translating the antenna’s subreflector. A 2D FDTD simulation was performed to characterize not only the lens itself but also the effect of PE spacers placed between the metal plates. This FDTD simulated phase vari ation and that computed theoretically were shown to be in good agreement. During the defocused measurement, the lens was placed in front of the feed with metal plates aligned with incoming electric field polarization. Several choices of the lens transfer function and initial trial function were examined. The resulting antenna aperture phase distribution obtained using the PCH-derived lens transfer function agreed well with the standard result from PCH, therefore, giving a hint that this novel defocusing method works fairly well. However, only the large scale and very distinct features were retrieved when the theoretical lens transfer function was used. It Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 115 is believed that the measurement noise, lens imperfection, and slight misalignments were responsible for this lack of performance. By designing the lens with a higher phase variation to overcome the phase noise as well as by further rigorously characterizing the lens to obtain a more accurate transfer function, an accurate holographic result may be achieved using this method. A circular waveguide lens was designed for use in latter PRH measurements. This lens was designed to possess a non-quadratic three-stepped phase variation profile to show that a traditional quadratic phase variation was unnecessary. The amount of the maximum phase variation for this lens was chosen to be higher than that of its metal-plate counterpart. Furthermore, this circular waveguide lens also eliminated imperfect alignment and depolar ization effects encountered when using a metal-plate lens. Its PRH result illustrated similar features to those of the PCH including the test metallic box. A comparison between PRH using these two types of lenses shows that both measurements perform reasonably well. The best result in the sense of mean square phase error is obtained from the circular waveguide lens measurement, which implies the advantage of non-quadratic phase variation profile as well as the elimination of polarization problems. While PRH error analysis may be performed theoretically using the standard method of error propagation, elements of covariance matrices may not be easily determined. The method is also computationally intensive. Thus, an empirical study was instead performed. By varying different parameters of interest and comparing the retrieval result with the model, effects of measurement noise, lens transfer function error, and the amount of phase variation can be studied separately, and the results have been illustrated and discussed. The use of microwave lenses also presents the opportunity to select different spatial distributions of the phase variation profile. The results of the study indicate that by using non-quadratic profiles, moderately improved retrievals compared to th at of the traditional quadratic profile may be achieved. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 116 6 .2 Contributions fr o m t h i s W o rk This work investigated and implemented a novel practical method for phase retrieval mi crowave holography. The method was designed as an alternative for a facility possessing an inability or limited range of the subreflector/feed movement to perform the traditional measurement, as well as to allow in-place measurements of a particular alignment. The use of a microwave lens also presents a freedom in designing the shape of the lens transfer function, which according to the empirical analysis may result in an improvement in the phase retrieval process. The measurement utilizing the circular waveguide lens is believed to be the first in microwave holography to use a different phase variation profile. The imple mentation of this method not only demonstrates the validity and feasibility of the proposed idea, but also highlights the critical points on utilizing the method effectively. Additionally, implementing a microwave lens on a large reflector antenna may provide an alternative for some other related applications such as those attempting to shape the antenna pattern, or as a temporary correction of a large-scale antenna’s surface deformation. Additionally, the results of the error analysis provide an important information on how the Misell algorithm behaves under the presence of error in lens transfer function. They also illustrate the effect of the amount and spatial distribution of the lens’ phase variation required for accurate microwave holography. 6.2.1 Publications and Presentations Publications W. Chalodhom and D. R. DeBoer, “Use of a circular waveguide lens for phase retrieval microwave holography and empirical error analysis,” Submitted to IEEE Transactions on Antennas and Propagation, January 2002. W. Chalodhom and D. R. DeBoer, “Use of microwave lenses in phase retrieval microwave holography of reflector antennas,” IEEE Transactions on Antennas and Propagation, in press, to appear June 2002. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 117 Presentations W. Chalodhom and D. R. DeBoer, “Use of a circular waveguide lens for phase retrieval microwave holography and empirical error analysis,” International Union of Radio Science Programs and Abstracts: 2002 National Radio Science Meeting Abstract, pp. 71, 2002. Presented at the National Radio Science Meeting, Boulder, CO, January 2002. W. Chalodhom and D. R. DeBoer, “Novel method for practical microwave holography of antennas,” International Union of Radio Science Programs and Abstracts: 2000 National Radio Science Meeting Abstract, pp. 24, 2000. Presented at the National Radio Science Meeting, Boulder, CO, January 2000. W. Chalodhom, D. R. DeBoer, and P. G. Steffes, “Microwave holographic measurement of the Georgia Tech Woodbury Research Facility,” International Union o f Radio Science Programs and Abstracts: 1999 National Radio Science Meeting Abstract, pp. 119, 1999. Presented at the National Radio Science Meeting, Boulder, CO, January 1999. 6.3 Future Research While the practical possibility of using microwave lenses for PRH has been illustrated and an extensive error analysis empirically performed, the technique can be further refined, and a number of tasks still remain to be done. Possible future research may range from simply improving the lens construction to determining an optimal type and spatial distribution of a microwave lens for PRH. A more accurate determination of lens transfer functions can improve the retrieval quality. The empirical error analysis may be extended to a broader range of parameters, and its results formulated. Demonstration of the technique on smaller antennas is also useful, and likely of vast interest. 6.3.1 Lens Construction Process The simplest way to obtain a more accurate antenna’s profile retrieval is to construct a microwave lens as closely as possible to its design. The accurately built lens yields more correspondence between measured patterns and the lens transfer function, and therefore Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 118 improves the accuracy of phase retrieval (see Chapter 5). For a metal-plate lens, an im provement may be made in machining each metal plate’s profile with the use of a more accurate machine. Spacers with a more precise thickness also result in a more accurate spacing between metal plates. As for the circular waveguide lens, the diameter of its waveg uides as well as their placement is an important factor. The accuracy in machining lens profile may not be as important an issue for the stepped profile lens as th at for the quadratic or other complex profile types. For the WRF circular waveguide lens, it is also important to have a “clean” cut through the foam to create the waveguide with minimal surface rough ness. The type and property of the conductive coating used should be carefully studied. However, these two factors may not be of concern should another construction method be chosen. 6.3.2 Determination of Lens Transfer Function As mentioned earlier, an accurate determination of lens transfer function is crucial in ob taining accurate phase retrieval. Since the design of the lens was primarily performed using ray tracing, where the effect of diffraction is not included, a more accurate lens transfer function determination may be obtained via the use of some computational techniques. For this work, the FDTD method is an example of a technique which could be used. As observed in Chapter 4, the lens transfer function of the WRF metal-plate lens obtained from FDTD generally yields superior retrieval results both visually and quantitatively than those using the lens transfer function from the ray tracing design. More accurate results may still be attained by utilizing a finer FDTD grid, reducing the time step, as well as improving the absorbing boundary condition. However, these improvements are typically achieved at the cost of computational complexity and computation time. Another computational technique may be viable as well. Alternatively, the lens transfer function may also be obtained from measurements. This could be done directly within a measurement range, or as in this work, the lens transfer functions could be obtained from two PCH measurements (see Chapter 4). The improve ment in the accuracy of the measurement system thus has a direct impact on the accuracy Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 119 of lens transfer function. 6.3.3 Extension of the Empirical Error Analysis An empirical error analysis proves to be useful in observing bow the phase retrieval would perform under different conditions. The parameters in the analysis however can still be extended to a broader range. Moreover, analytical formulation of the results may be possible depending on its complexity. 6.3.4 Optimal PRH Lens One of the advantages of using a microwave lens for PRH over the standard measurement is that it offers an opportunity to achieve superior phase retrievals from the opportunity to select the spatial distribution of the lens transfer function. In Chapter 5, some profiles have been shown via empirical analysis to produce superior results especially in the mean square sense. However, the “optimal” choice of the spatial distribution has yet to be determined. In this thesis, four types of microwave lenses have been explored for PRH use. Other types of lenses nevertheless may also be considered, and the selection criteria may depend upon specific implementations. 6.3.5 Demonstration of Technique on Small Antennas The technique of utilizing a microwave lens for PRH has been successfully demonstrated on a large antenna (WRF antenna). With some modifications, the method can be applied for use with smaller antennas as well. Since the antenna is no longer electrically large, computational techniques employed in electromagnetics may be used in the design procedure of the microwave lens. An iterative process of the lens design may involve determining the first design by employing ray tracing and further refine the shape of the lens iteratively via numerical calculation. The selection of lens’ type and its construction process may also differ from the large antenna case. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 120 REFERENCES [1] Y. Rahmat-Samii. Surface diagnosis of large reflector antennas using microwave holo graphic metrology-an iterative approach. Radio Sci., 13:1205-1217, 1984. [2] G. S. Smith. An Introduction to Classical Electromagnetic Radiation. Cambridge, New York, 1997. [3] K. L. Shlager. The analysis and optimization of Bow-tie and TEM horn antennas for pulse radiation using the finite-difference time-domain method. PhD dissertation, Georgia Institute of Technology, Atlanta, GA, 1995. [4] J. W. Findlay. Filled-aperture antennas for radio astronomy. Annu. Rev. Astron. Astrophys., 9(6):271-292, 1971. [5] C. S. Fraser. Photogrammetric measurement of antennas reflectors. Antenna Measure ment Techniques Proceedings, pages 374-379, 1987. [6] J. C. Bennett, A. P. Anderson, P. A. Mclnnes, and A. J. T. Whitaker. Microwave holographic metrology for large reflector antennas. IEEE Trans. Antennas Propagat., 24:295-303, 1976. [7] P. F. Scott and M. Ryle. A rapid method for measuring the figure of a radio telescope reflector. 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Contr., 10:461-464, 1965. [72] D. F. Watson and G. M. Phillip. Systematic triangulations. Computer Vision, Graph ics, and Image Processing, 26:217-223, 1984. [73] S. Brandt. Data Analysis. Springer-Verlag, New York, 1998. [74] J. M. Jenkins. Variations in the 13 cm opacity below the main cloud layer in the atmosphere of Venus inferred from Pioneer-Venus radio occultation studies 1978-1987. PhD dissertation, Georgia Institute of Technology, Atlanta, GA, 1992. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 125 VITA Wonchalerm Chalodhorn was born in Bangkok, Thailand, on July 28, 1974. Growing up in Bangkok, he attended Chulalongkorn university and obtained a bachelor’s degree with the second-class honor in electrical engineering in May of 1995. After spending a year and a half working for the Jasmine International company in Bangkok, he attended in January of 1997 the Georgia Institute of Technology, where he graduated with a master’s and a doctoral degree, both in electrical and computer engineering, in 1998 and 2002, respectively. As a PhD student, Wonchalerm Chalodhorn studied and developed a novel measure ment method for phase retrieval microwave holography, where he incorporated the use of microwave lenses into the technique. His work related closely to the Georgia Tech Woodbury Research Facility. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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