DEVELOPMENT OF HYBRID PSTD METHODS AND THEIR APPLICATION TO THE ANALYSIS OF FRESNEL ZONE PLATES FAN YIJING (B.S.), PEKING UNIVERSITY A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2008 Acknowledgements I would like to express my utmost gratitude to my project supervisor Associate Professor Ooi Ban Leong, for being so approachable and his numerous suggestions on my research topic. I would like to express my sincere thanks to my other project supervisor Professor Leong Mook Seng, for teaching me so much about fundamental Electromagnetics, and being extremely supportive of my research. I would like to thank all the staffs of RF/Microwave laboratory and ECE department, especially Mr. Sing Cheng Hiong, Mr. Teo Tham Chai, Mdm Lee Siew Choo, Ms Guo Lin, Mr. Neo Hong Keem, Mr Jalul and Mr. Chan for their very professional help in fabrication, measurement and other technical, and administrative support. In addition, all my friends around me played a no less important role in making my research life much more enjoyable. Tham Jing-Yao is my most loyal companion, having gone through thick and thin with me. Ng Tiong Huat has a sea of knowledge and experience, which he does not hesitate to share with me. Zhang Yaqiong is a great friend whom I always had engaging conversations with. The numerous interesting emails Ewe Wei Bin sent always lightened my day. I would like to thank all of them, and all other friends I got to know along the way - for being there. Last but not least, I am grateful to my parents for their patience and love. Without them this work would never have come into existence. Singapore Fan Yijing Jan 2008 i Table of Contents Acknowledgements i Table of Contents ii Summary vi List of Tables ix List of Figures x List of Symbols xv List of Acronyms xvi Introduction 1.1 Large-Scale Fresnel Zone Problem and Pseudo-Spectral Time Domain Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Hybrid Method of Time Domain Finite Element Method (TDFEM) and PSTD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Hybrid Method of Finite difference Time Domain Method (FDTD) and PSTD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 FMM-based PSTD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Fresnel Zone Plate Design . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Objectives and Significance of the Study . . . . . . . . . . . . . . . . . 1.6.1 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Major Contributions . . . . . . . . . . . . . . . . . . . . . . . 1.6.3 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 11 12 13 Pseudo-Spectral Time Domain Method (PSTD) 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Pseudo-Spectral Method . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Implementation of Fourier-PS Method with FFT algorithm . . 2.2.2 Implementation of Chebyshev-PS Method with FFT algorithm 2.3 Pseudo-Spectral Time Domain Method Formulations . . . . . . . . . 2.4 Dispersion and Stability Analysis . . . . . . . . . . . . . . . . . . . . 16 16 19 19 21 21 22 ii . . . . . . 1 2.5 2.6 2.7 2.8 UPML Implementation in PSTD . . . . . . . . . . . . . . . . . . . . . 2.5.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 2.5.2.1 1D Propagation Problem . . . . . . . . . . . . . . . 2.5.2.2 2D Propagation Problem . . . . . . . . . . . . . . . Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Hard Source Excitation . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Plane Wave Excitation Using Total Field/Scattering Field Scheme. Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 2D Scattering Problem With Two Metal Square Cylinders. . . . 2.7.2 2D Scattering Problem With Two Metal Circular Cylinders. . . 2.7.3 2D Scattering Problem With Two Dielectric Square Cylinders. . Comparison of PSTD with FDTD . . . . . . . . . . . . . . . . . . . . 24 25 27 27 29 31 31 32 37 37 41 41 42 Hybrid Method of TDFEM-PSTD 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 TDFEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Absorbing Boundary Condition . . . . . . . . . . . . . . . . . 3.2.3 Stability Issue . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 3.2.4.1 A Simple Radiation Problem . . . . . . . . . . . . . 3.2.4.2 Stability Analysis . . . . . . . . . . . . . . . . . . . 3.2.4.3 Scattering From a Metallic Circular Cylinder . . . . . 3.3 Hybrid Method of TDFEM-PSTD . . . . . . . . . . . . . . . . . . . . 3.3.1 The Bounded Domain TDFEM Model With Special Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 The Entire Domain PSTD . . . . . . . . . . . . . . . . . . . . 3.3.3 Result Exchange at the Interface . . . . . . . . . . . . . . . . . 3.3.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 3.3.5.1 Scattering from Two Perfectly Conductive Cylinders . 3.3.5.2 Reflection Analysis at the TDFEM-PSTD Interface . 3.3.5.3 Stability Analysis . . . . . . . . . . . . . . . . . . . 48 48 49 51 57 58 59 59 61 62 63 Hybrid Method of PSTD-FDTD 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 4.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Dispersion and Stability Analysis . . . . . . . . . . . . 4.4 Numerical Examples . . . . . . . . . . . . . . . . . . 4.4.1 Scattering Problem With a Circular Cylinder . 4.4.2 Scattering Problem With a Square cylinder . . 4.4.3 3D Scattering Problem With a Metallic Sphere 4.5 Interpolation and Fitting Scheme at the Interface . . . . 77 77 79 82 86 86 87 89 91 iii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 66 67 69 70 70 72 73 4.5.1 4.5.2 Line Interface in 2D Scattering Problem . . . . . . . . . . . . . Surface Interface in 3D Scattering Problem . . . . . . . . . . . 92 93 FMM-based PSTD 100 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.2 Pseudo-Spectral Method and Cardinal Functions . . . . . . . . . . . . . 101 5.2.1 Pseudo-Spectral Method . . . . . . . . . . . . . . . . . . . . . 101 5.2.2 Cardinal Functions . . . . . . . . . . . . . . . . . . . . . . . . 104 5.3 FMM-based PSTD Method . . . . . . . . . . . . . . . . . . . . . . . . 106 5.3.1 2D Fast Multipole Method . . . . . . . . . . . . . . . . . . . . 110 5.3.2 Multipole Expansion . . . . . . . . . . . . . . . . . . . . . . . 111 5.3.3 Shifting the Center of the Multipole Expansion . . . . . . . . . 113 5.3.4 Converting the Multipole Expansion to Local Expansion . . . . 114 5.3.5 Shifting the Center of Local Expansion . . . . . . . . . . . . . 116 5.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.4.1 Comparison of Different Cardinal Functions: A Simple Radiation Transient Analysis. . . . . . . . . . . . . . . . . . . . . . 118 5.4.2 Accuracy Comparison: Scattering from Circular Metallic Cylinder120 Application of the PSTD-FDTD Method for Fresnel Zone Plates Analysis and Design 123 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.2 Traditional Analytical Methods for Analyzing Fresnel Zone Plate . . . . 125 6.2.1 Empirical Prediction . . . . . . . . . . . . . . . . . . . . . . . 125 6.2.2 Kirchhoff’s Diffraction Integral Method . . . . . . . . . . . . . 126 6.2.2.1 Formulation . . . . . . . . . . . . . . . . . . . . . . 126 6.2.2.2 Complexity Analysis . . . . . . . . . . . . . . . . . 128 6.3 Implementation of PSTD-FDTD Method in the FZP Analysis. . . . . . 129 6.3.1 Data Exchange at the Interface of PSTD-FDTD . . . . . . . . . 129 6.3.2 Interpolation Scheme . . . . . . . . . . . . . . . . . . . . . . . 131 6.4 Implementation of PSTD-FDTD to Analyze Some Classical Fresnel Zone Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6.4.1 Classical Ring type Soret Fresnel Zone Plate . . . . . . . . . . 135 6.4.2 2D Cross Fresnel Zone Plate . . . . . . . . . . . . . . . . . . . 136 6.5 Implementation of PSTD-FDTD to Design New Fresnel Zone Plates . . 138 6.5.1 Two-Layer Ring Type Fresnel Zone Plate . . . . . . . . . . . . 139 6.5.2 FSS-FZP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 6.5.3 Fabrication and Measurement of Fresnel Zone Plates . . . . . . 146 Conclusion and Future Work 7.1 Hybrid PSTD Method . . . 7.2 FMM-PSTD . . . . . . . . 7.3 Fresnel Zone Plate Design 7.4 Future Works . . . . . . . . . . . . . . . . . . . . . . . iv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 150 152 153 154 References 158 v Summary The objective of this thesis is to develop hybrid Pseudo-Spectral Time Domain (PSTD) methods, for effective simulation of large scale scattering problems with complex scatterers. This is achieved by combining PSTD with other numerical methods to develop hybrid methods. The newly proposed PSTD method [32] is well known for its great efficiency for simulation of large-scale problems. The coarse grids of PSTD method make it much more efficient than traditional numerical methods that requires fine grids. However, the coarse grids also result in large staircase errors when dealing with curved boundary. Moreover, PSTD method is not capable of modeling small scatterers whose dimension is smaller than the grid size. In order to overcome these limitations and expand the scope of PSTD’s applications, two novel hybrid pseudo-spectral time domain methods are proposed. They are the hybrid method of PSTD and Time-Domain Finite-Element Method (TDFEM), and the hybrid method of PSTD and Finite-Difference Time-Domain Method (FDTD). The finite element method(FEM) [5] has been well developed in the frequency domain and in the time domain for the past years. It is a great tool to analyze curved boundary and complex objects. However, the high computation burden of FEM method limits its application in large scale simulations. In this thesis, a novel hybrid method of (PSTD) [32] and TDFEM is proposed, in order to simulate large scale scattering problems with complex scatterers. The formulation and combining schemes are developed. The stability issue of the hybrid method is investigated and an unconditionally stable vi vii scheme is proposed. In addition, absorbing boundary condition and excitation issues are also investigated. Moreover, some numerical experiments are conducted. The performance of the hybrid method TDFEM-PSTD is compared with traditional TDFEM and PSTD methods. The advantage of the hybrid method is validated by a number of numerical examples. Compared with PSTD, the TDFEM-PSTD can deal with metallic or unstructured objects more accurately. Compared with TDFEM, the TDFEM-PSTD greatly alleviates the computation burden, as only cells per wavelength are needed for PSTD mesh. Finite-Difference Time-Domain method (FDTD) [38] is another widely used time domain method. For structured scatterers, it can achieve similar accuracy as FEM and with better efficiency. However, the computation burden of FDTD for simulating large scale problem is also quite high. The Courant limit of FDTD requires more than 10 cells per wavelength to ensure the stability. The fine grids result in large number of unknowns and influences the efficiency. In this thesis, a new hybrid scheme of PSTD and FDTD is also proposed, in order to simulate large scale scattering problems with small scatterers. The combination scheme of PSTD and FDTD is developed. The reflection at the interface between two grids is investigated. In addition, the dispersion and stability issues of the hybrid method PSTD-FDTD are discussed. The required stability criteria is next derived. In addition, some numerical examples are conducted to examine the performance of PSTD-FDTD. The computation results and computation complexity of the hybrid method are compared with FDTD and PSTD methods. Compared to PSTD, better accuracy is achieved for small scatterers. Compared to FDTD, less memory and CPU time is required for the hybrid method PSTD-FDTD. Both improved accuracy and efficiency are achieved. In the proposed hybrid methods TDFEM-PSTD and PSTD-FDTD, the well-known wraparound effect and Gibbs phenomenon also exist [32]-[37]. These problems are caused by the FFT scheme employed by the traditional PSTD. They influence the accuracy of the PSTD methods. In this thesis, the Fast Multipole Method (FMM) [42]-[44] is viii employed to combine with the Pseudo-Spectral method. A new FMM-PSTD method is proposed to reduce the wraparound effect and Gibbs phenomenon. The 2D FMM-PSTD formulation is developed and the combination scheme is explained. Different collocation points and cardinal functions for developing FMM-PSTD methods are investigated and compared. In addition, some numerical examples are provided. The performance of FMM-PSTD is compared with traditional PSTD. For large-scale problems with large number of collocation points (grid points), the FMM-PSTD achieved similar efficiency as the traditional PSTD. After developing these hybrid methods, a practical implementation of the hybrid method is carried out in this thesis. Due to the time and resource limitation, only the PSTD-FDTD hybrid method is explored to analyze the practical problem of the Fresnel Zone Plates [51]. Nowadays, some complex structures like frequency selective surface (FSS) are employed to improve gain and directivity performance of Fresnel Zone Plates (FZP) [51]. Due to the overall large size (up to 10 λ) and complex structure of the design, 3D full-wave analysis has not been attempted before. In this thesis, efficient PSTD-FDTD method is employed to analyze and design Fresnel zone plates (FZP) [49]. The PSTDFDTD scheme is modified and adapted to the FZP structure. Interpolation schemes at the interface between PSTD and FDTD are investigated for the specific FZP problem. Computation complexities of PSTD-FDTD and traditional Kirchhoff’s Diffraction Integral (KDI) [49] method are compared. The superior efficiency of PSTD-FDTD is demonstrated. Subsequently, some classical FZPs are analyzed with PSTD-FDTD and traditional KDI method. The results are compared with the measured result. The PSTDFDTD method achieves good accuracy with much better efficiency. In addition, some novel FZPs designed using PSTD-FDTD are also proposed in this thesis. List of Tables 2.1 Empirical design formulas . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Comparison of computation complexities of different methods for the scattering problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 72 Comparison of complexities of different hybrid methods for the scattering problem B.(2000 time steps) . . . . . . . . . . . . . . . . . . . . . 4.2 47 88 Comparison of complexities of different methods for the 3D scattering problem.(600 time steps) . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.3 Mean errors for different interpolation/fitting methods . . . . . . . . . . 96 4.4 Mean errors for different interpolation/fitting methods . . . . . . . . . . 97 6.1 Empirical design formulas . . . . . . . . . . . . . . . . . . . . . . . . 126 6.2 Complexity comparison for KDI and PSTD-FDTD . . . . . . . . . . . 129 6.3 List of error means for different interfacing schemes and different interpolation/fitting methods . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6.4 Radius of rings from inner circle to outer circle . . . . . . . . . . . . . 135 6.5 Perpendicular distance from center to strips (near to far) . . . . . . . . . 136 6.6 Distance D (m) between focal point and Fresnel Zone Plate calculated from different methods . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.7 Complexity comparison of KDI and PSTD-FDTD for 60 degree cross Fresnel Zone Plate analysis . . . . . . . . . . . . . . . . . . . . . . . . 138 6.8 Radius of rings at both layers from inner circle to outer circle . . . . . . 139 6.9 Focal point position D (m) from FZP plate calculated by different methods140 6.10 Complexity comparison for KDI and PSTD-FDTD for FSS-FZP analysis 146 ix Chapter Conclusion and Future Work 7.1 Hybrid PSTD Method In order to improve the accuracy of the PSTD method when dealing with curved or fine objects, this thesis developed hybrid methods TDFEM-PSTD and PSTD-FDTD in the time domain. TDFEM/FDTD was applied near the curved or fine scatterers, while coarse PSTD grids overlapped with fine FDTD grids and distributed over the whole domain. Excitation scheme and absorbing boundary conditions of different algorithms were developed individually and combined properly at the interface. In addition, interpolation scheme between coarse PSTD grids and fine TDFEM/FDTD grids were investigated and chosen to minimize the reflection at the interface. Some typical numerical experiments were conducted and the result of hybrid methods were compared with other numerical methods. The comparison showed that for curved or fine scatterers, TDFEM-PSTD/FDTD-PSTD had similar accuracy as TDFEM/FDTD and agreed well with analytical solutions. The good accuracy of the hybrid method is probably attributed to three factors. First, fine grids TDFEM/FDTD near the curved or fine scatterer is able to adapt well to the scatterer shape and provide accurate modeling. Secondly, the entire domain PSTD proposed for new hybrid methods helps to retain the high order accuracy of the PSTD method. Finally, the good compatibility of the new transition scheme at the interface improves the continuity of field distribution in the entire domain. In addition, the properly chosen interpolation scheme between 150 151 coarse/fine grids helps to reduce the interpolation error. In the numerical experiments, the computation complexity of TDFEM-PSTD/FDTDPSTD were also compared with other numerical methods. TDFEM-PSTD saved a lot of memory and CPU time comparing to the pure TDFEM method, although it was not as efficient as PSTD method. For a scattering problem with a circular cylinder (R = λ) and truncated with 6λ × 6λ boundary, pure TDFEM needed over 100MB memory and nearly a million seconds of computation time. In contrast, the hybrid TDFEM-PSTD method needed only 20MB memory and a few thousand seconds of computation time. A lot of computation resource and time were saved. Although PSTD needed only 0.2 MB memory and around 10 seconds for the same scattering problem, it could not provide an accurate solution. Hence, TDFEM-PSTD hybrid method is the best method in terms of efficiency and accuracy. For FDTD-PSTD method, the computation consumption of the hybrid method was nearly the same as the pure PSTD method. As shown in the scattering example with a square cylinder in chapter 4, both the FDTD-PSTD method and the pure PSTD method needed less than 1MB memory and only a few seconds of computation time. The high efficiency of the hybrid method is because only a very small region of the computation domain is meshed with TDFEM/FDTD fine grids. The large homogenous region is still covered with coarse PSTD grids. The efficiency of PSTD is preserved. Only a small extra computation burden is imposed by TDFEM/FDTD. Since TDFEM generates more unknowns than FDTD, and needs to solve a matrix equation at the each time step, the memory and CPU time requirements of TDFEM are higher than FDTD. The stability issue of the TDFEM-PSTD was investigated and the stablility criterion was given. The stability of TFEM-PSTD method was examined in a scattering example and compared with the traditional method. Late time instability was observed in the traditional method with Dirichlet boundary condition. By contrast, the TDFEM-PSTD method developed in this thesis was free of instability and converged to a constant in the late time. The improved stability of the new hybrid method may be due to the boundary 152 integral interfacing scheme employed in TDFEM-PSTD. For the PSTD-FDTD method, the dispersion and stability analysis were also conducted. PSTD-FDTD had good dispersion continuity across the interface. It converged to some small errors compared to the standard solution and no late time instability was observed. Due to the good performance in both efficiency and accuracy, the hybrid PSTD method may provide a tool for transient analysis of electrically large problems with small or complex scatterers. This suggests that the fast simulations of complex Fresnel zone plates diffraction problems are possible. Moreover, unlike the traditional analytical method which was employed for Fresnel zone plate analysis in the past, hybrid PSTD method is capable of time domain simulation. It can provide a clear picture of waveform transformation for evaluating the performance of the design. 7.2 FMM-PSTD In order to reduce the wraparound effect and Gibbs phenomenon in traditional PSTD method, this thesis developed a new FMM-based PSTD method. FMM algorithm instead of FFD/FCT algorithm was employed to evaluate the Differential Matrix Multiplication (DMM) in PSTD method. 2D FMM formulation was derived to update partial derivatives with respect to different directions in the PSTD method. Different collocation points and corresponding cardinal functions were used for FMM calculation and the results were compared. The Gergerber function was chosen for non-periodical scattering domain to reduce vibration near end points. The resulted FMM-PSTD was examined in some scattering examples. The result of FMM-PSTD was compared with traditional FFT-PSTD and the exact analytical solution. The waveform curve of the FMM-PSTD was closer to the exact analytical solution. Some ripples caused by the wraparound effect and the Gibbs phenomenon were observed in the early and the late time region of the FFT-PSTD waveform curve. However, in the waveform curve of the FMM-PSTD, nearly no ripple was observed. Wraparound 153 effect and Gibbs phenomenon were alleviated by the FMM-PSTD. The accuracy of the PSTD method was improved by incorporating it with FMM. This may be due to different characteristics of FFT/FCT scheme and FMM scheme. The memory and CPU time consumption of FMM-PSTD was also compared to the traditional FFT-PSTD for a particular propagation problem. When the sampling points number N was larger than a certain number, similar efficiency Nlogx (N) as FFT/FCT was achieved by FMM. Hence, for large scale problems with big number of sampling points, the FMM-PSTD has similar efficiency as FFT-PSTD. Moreover, FMM is applicable to different kinds of collocation points and different cardinal functions. Good accuracies are achieved by different cardinal functions in the propagation example. Hence, the application scope of the FMM-PSTD method is expanded compared to the traditional FFT-PSTD method which is only applicable to uniform or Chebyshev collocation points. 7.3 Fresnel Zone Plate Design In order to examine the performance of hybrid PSTD methods in practical problems, the 3D PSTD-FDTD method developed in the Chapter was implemented to simulate Fresnel Zone Plate diffraction problems in this thesis. Some classical Fresnel Zone Plates were analyzed with PSTD-FDTD. The results were compared with traditional KDI method and measured results. The half open Soret zone plate designed at 9.375GHz was simulated with the PSTD-FDTD method. The on-axis curve was extracted to compare with the result of traditional KDI method as reported in the reference [49]. The good agreement between two results showed that PSTD-FDTD achieved similar accuracy as traditional KDI method for Fresnel Zone Plate analysis. Another implementation was cross plates with different crossing angles. The focal region field distributions of cross plates were calculated with PSTD-FDTD and KDI methods. The mean difference between two results was within 0.1dB. At the 154 same time, PSTD-FDTD method consumed much less CPU time than KDI method. These two experiments showed the advantages of PSTD-FDTD method over the traditional KDI method in the Fresnel Zone Plate analysis. It provided similar accuracy with much less CPU times. By using PSTD-FDTD simulation, some novel Fresnel Zone Plates were designed. A 2-layer Fresnel Zone Plate was designed by using complementary ring-layer. Comparing to the performance of the single-layer ring type fresnel zone plate, the gain of the 2-layer Fresnel zone plate increased while the side lobes outside the focal region decreased. The other design was the FSS-FZP plate which consisted of one Soret ring type Fresnel Zone Plate and one four-leg FSS plate. The frequency response was obtained from the time domain result. By incorporating the FSS plate, the main beam of the Fresnel Zone Plate was narrowed, and side lobes were lowered. The Q factor of the Fresnel Zone Plate was improved. In addition, FSS was able to increase the focusing gain by 5dB while reducing the side lobes. The 2-layer Fresnel Zone Plate and FSS-FZP were both fabricated and measured. The relative error between the simulated result and the measured result was within 0.1 deviation. Reasonable agreement was achieved except for some differences caused by measurement and fabrication errors. 7.4 Future Works However, there are still certain limitations in the TDFEM-PSTD method and the FMMPSTD method proposed in this thesis. For the TDFEM-PSTD method, there are two major limitations. Firstly, only 2D model was considered in this thesis. The more general 3D case was not explored. This may restrict the application of TDFEM-PSTD to some 3D problems, such as radar scattering and airplane simulation. The other limitation is that the TDFEM employed in the hybrid method is only second order accurate. This may affect the performance of the TDFEM-PSTD. However, the accuracy is acceptable for most cases, and upgrading 155 the FEM to higher order may not be desirable as it would result in large computational burden. There are several interesting directions for future work in the TDFEM-PSTD method development. One possible future work is the extension of 2D TDFEM-PSTD to 3D for a larger scope of applications. The TDFEM-PSTD may find further applications, such as radar and airplane. There are already well-developed 3D PSTD method [59]-[60] and FEM method [5]. Moreover, a pyramidal vector element has been reported recently to connect the FEM tetrahedral element and the PSTD cube element [9]. Hence, the construction of the 3D TDFEM-PSTD should be feasible. Another direct extension of the work would be upgrading second order FEM to a higher order FEM for application in some accuracy sensitive problems. Higher order FEM [14] has already been reported and PSTD is infinite order accurate, so the combination of higher order FEM with PSTD would be feasible and promising. For the FMM-PSTD method proposed in this thesis, there are also some limitations. Firstly, this hybrid method employs the most primitive hierarchical method of FMM. The efficiency is not good enough. In the past, a lot of work has been published on the improvement of FMM algorithm. Some multilevel and fast algorithms have been proposed. These algorithms have not been explored to combine with the PSTD method. Secondly, only 2D algorithm and formulation are given in this thesis for illustration of the idea and concept. More general 3D case is not explored. This may restrict the application of the FMM-based PSTD to some 3D problems, such as airplane scattering and ground penetrating radar application. Thirdly, the arbitrary domain needs to be transformed into the square domain before FMM-PSTD calculation. This influences the efficiency of the algorithm. Moreover, transformation and inverse transformation may introduce some errors, which may eventually influence the solution accuracy. Finally, only regular distributed collocation points are employed to distribute in a square domain. Arbitrary collocation distribution according to the arbitrary shape domain is not explored. Cardinal functions for arbitrary points distribution are also not explored. 156 There are some suggestions for future work in the development and improvement of the FMM-PSTD method. One possible direction for future work is the extension of 2D model to 3D for broader application. The FMM-based PSTD may find further applications, such as radar and airplane. There are already well-developed 3D PSTD method [59]-[60] and FMM method [56]. The construction of the 3D FMM-PSTD should be feasible. Another direction for the future work is to explore arbitrary shape computation domain with arbitrary points distribution. FMM algorithm is applicable to irregularly distributed points. Only near-far grouping needs some modifications. If special cardinal function for the given points distribution can be derived, the development of FMMPSTD with arbitrary shape domain should be possible. One possible extension of the work would be upgrading simple FMM algorithm to its multilevel or fast algorithms. The efficiency of the FMM-PSTD can be further improved. Since the multilevel FMM [56] and some fast algorithms like FastCap [46] have already been reported, the combination of these improved algorithms with the PSTD would be feasible and promising. In conclusion, this thesis develops some improved numerical methods based on PSTD method: TDFEM-PSTD, PSTD-FDTD, and FMM-PSTD. The concept and development of these improved methods are explained. The performance of these methods are examined in numerical examples. The advantages of these methods are validated by comparing with the traditional PSTD method and analytical solutions. One of the improved method PSTD-FDTD is expanded to 3D model and implemented to analyze Fresnel zone plate diffraction problems. Similar accuracy and better efficiency are achieved as compared to the traditional method. This is the first time that a numerical method can perform fast full wave analysis of Fresnel Zone Plates. Although some differences between simulation result and measured result is observed, the relative error is within 0.1. The deviation observed is probably caused by measurement setup and fabrication error. Useful prediction can still be provided by PSTD-FDTD simulation 157 for Fresnel Zone Plate design and optimization. Moreover, PSTD-FDTD method would not only be a useful tool for Fresnel Zone Plate designs, but also a good approach for analyzing a lot of electrically large problems with complex structures, such as ground penetrating radar and electromagnetic interference. References [1] J. P. 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[...]... 2007 The hybrid methods TDFEM -PSTD, PSTD- FDTD and FMM -PSTD developed in this thesis improve the accuracy of the traditional PSTD method and preserve the efficiency of the PSTD algorithm They help to extend the application scope of the PSTD method to some practical large scale simulations with complex scatterers 3D PSTDFDTD method developed in this thesis provides an approach to perform full-wave analysis. .. requirement of the traditional full-wave methods is prohibitively large Considering the good efficiency and accuracy of the PSTD- FDTD method, it would be a powerful tool to perform the full-wave simulation of the focusing phenomenon in the Fresnel zone In this thesis, PSTD- FDTD algorithm is implemented to analyze some classical planar Fresnel Zone Plates Some simplifications and adaption of the 3D PSTD- FDTD... of time in analysis of the same problem and proved to be much more efficient than the traditional method Efficient full-wave simulation of large-scale FZP problem is accomplished by the PSTD- FDTD method After examining the accuracy and efficiency of the PSTD- FDTD method in the 11 analysis of classical Fresnel Zone Plates, some novel Fresnel Zone Plates with complex shapes are proposed in this thesis These... These Fresnel Zone Plates are simulated and designed by the PSTD- FDTD method The results of PSTD- FDTD and measurement are compared The ability of PSTD- FDTD to accurately simulate complex structures is shown The results of the new designs are compared with traditional Fresnel Zone Plates, and better gain and directivity are achieved 1.6 Objectives and Significance of the Study 1.6.1 Organization This thesis... schemes between coarse PSTD grids and fine FEM grids are investigated and compared GPOF interpolation scheme has proven to be the most accurate and can ensure the accuracy and stability of the hybrid method After explaining the combination scheme of the TDFEM and PSTD, some numerical examples are given The accuracy of the TDFEM -PSTD hybrid method is compared to the analytical solution and other numerical method... applicable Hence, integral methods are slow and cumbersome for large-scale Fresnel zone analysis For differential methods, there are two approaches to deal with large-scale Fresnel zone problems One approach is to truncate the domain outside the Fresnel zone The electric and magnetic (E/H) field can be obtained directly from the calculation which is similar to the calculation of the near field points However,... accuracy of the PSTD method The other limitation of PSTD is that it is only applicable to uniform or Chebyshev collocation points (grid points) This is because FFT scheme in PSTD is only able to handle uniform or Chebyshev sampling points For other collocation points like Gergerber and Legerdre collocation points, PSTD method cannot be used To reduce the staircase error, the hybrid method TDFEM -PSTD and PSTD- FDTD... FMM method are explained and shown The performances of FMM-based PSTD with different cardinal functions are compared with the traditional FFT-based PSTD in terms of efficiency and accuracy 12 • Chapter 6 implements the hybrid 3D PSTD- FDTD method for Fresnel Zone Plate diffraction analysis and design Hybrid PSTD- FDTD method is applied to analyze some classical Fresnel Zone Plates Its advantage is validated... traditional methods and references In addition, some novel Fresnel Zone Plate designs are proposed • Chapter 7 concludes the work in this thesis The limitations are pointed out, and the directions for future work are given 1.6.2 Major Contributions The major contribution of this thesis is developing fast and accurate numerical methods based on the PSTD method, and using these methods for large-scale Fresnel zone. .. FFT -PSTD 1.5 Fresnel Zone Plate Design Fresnel zone plate is a typical and important device used in the Fresnel zone region Its focusing effect in the Fresnel zone region has been widely used in radio wave propagation and antenna designs [49]-[51] Traditionally, the diffraction and focusing phenomenon of the Fresnel Zone plate (FZP) are examined using analytical methods However, for some complex Fresnel . DEVELOPMENT OF HYBRID PSTD METHODS AND THEIR APPLICATION TO THE ANALYSIS OF FRESNEL ZONE PLATES FAN YIJING (B.S.), PEKING UNIVERSITY A THESIS SUBMITTED. implementation of the hybrid method is carried out in this thesis. Due to the time and resource limitation, only the PSTD- FDTD hybrid method is explored to analyze the practical problem of the Fresnel Zone. applications, two novel hybrid pseudo-spectral time domain methods are proposed. They are the hybrid method of PSTD and Time-Domain Finite-Element Method (TDFEM), and the hybrid method of PSTD and Finite-Difference