Modal-based tomographic imaging from far-zone

Northeastern University Electrical Engineering Dissertations

Department of Electrical and Computer Engineering

January 01, 2009

Modal-based tomographic imaging from far-zone observations Ersel Karbeyaz Northeastern University

Recommended Citation Karbeyaz, Ersel, "Modal-based tomographic imaging from far-zone observations" (2009). Electrical Engineering Dissertations. Paper 13. http://hdl.handle.net/2047/d10019157

This work is available open access, hosted by Northeastern University.

Modal-Based Tomographic Imaging from Far-Zone Observations

A Thesis Presented by Ersel Karbeyaz to The Department of Electrical and Computer Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering in the field of Fields, Waves and Optics

Northeastern University Boston, Massachusetts April 2009

c Copyright by Ersel Karbeyaz ° All Rights Reserved

NORTHEASTERN UNIVERSITY Graduate School of Engineering Thesis Title: Modal-Based Tomographic Imaging from Far-Zone Observations. Author: Ersel Karbeyaz. Department: Electrical and Computer Engineering. Approved for Thesis Requirements of the Doctor of Philosophy Degree

Thesis Advisor: Prof. Carey M. Rappaport

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Thesis Reader: Prof. Edwin A. Marengo

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Thesis Reader: Dr. Jose Angel Martinez-Lorenzo

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Department Chair: Prof. Ali Abur

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Director of the Graduate School

Date

Abstract A novel method of optical diffraction tomography (ODT) to image weakly scattering, electrically large two dimensional (2D) objects using the far-zone scattered field data is presented. The proposed technique is based on the expansion of the target object function in terms of Fourier-Bessel basis functions, and an alternative approximation for the total electric field within the support of the investigated scatterer. Analytical (Mie) plane-wave scattering by a layered, circularly symmetric, lossy dielectric cylinder, and finite-difference time-domain simulations involving plane-wave scattering first by a more general, lossless dielectric phantom and then by embryo models involving various mitochondrial distributions are utilized to compare the performance of the proposed method with that of the standard ODT techniques which are based on the Born/Rytov approximations and Fourier diffraction theorem. Quantitative and qualitative superiority of the presented method is demonstrated. The proposed method can be used without being confined to far-zone observations with a proper (cylindrical-spherical) receiver configuration, and can be easily modified to handle multi-frequency data for a wideband reconstruction whenever applicable. The proposed 2D technique can be readily extended to more realistic three dimensional scenarios, and can be used in tomographic microscopy for various purposes, such as the cell-embryo health assessment for in vitro fertilization procedures. [This work was supported in part by Gordon-CenSSIS, the Bernard M. Gordon Center for Subsurface Sensing and Imaging Systems, under the Engineering Research Centers Program of the National Science Foundation (Award Number EEC-9986821).]

Acknowledgements I am deeply grateful to my advisor, Prof. Carey M. Rappaport, for his guidance, patience and support over the years we have worked together. His knowledge and ideas have shaped the course of my research. I would also like to thank the members of my dissertation committee, Prof. Edwin A. Marengo and Dr. Jose Angel Martinez-Lorenzo, for their insightful comments and contributions. I want to express my gratitude to Prof. Yaman Yener for giving me the opportunity to come to the United States for my Ph.D. study. I thank Prof. Michael B. Silevitch, Kristin L. Hicks, Deanna Beirne, Brian Loughlin, Anne Magrath and Mariah Nobrega for being always so kind and helpful. I also want to thank my parents, Tuncay and T¨ ulin Karbeyaz, whose endless love has helped me at my every step. Finally, I have been, am and always will be grateful to my beloved ¨ wife Ba¸sak Ulker Karbeyaz, without whom I could never find enough strength in myself to finish this study. Words are simply incapable of describing what she means to me. Her contribution to this dissertation is invaluable. This thesis is dedicated to her.

ii

Contents

1 Introduction

1

1.1

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Optical Diffraction Tomography . . . . . . . . . . . . . . . . . .

3

1.3

Proposed Technique . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.4

Radial Optimization Technique . . . . . . . . . . . . . . . . . .

7

1.5

Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.6

Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . .

8

2 Proposed Technique

10

2.1

Investigated Problem . . . . . . . . . . . . . . . . . . . . . . . .

11

2.2

Volume Equivalence Theorem . . . . . . . . . . . . . . . . . . .

12

2.3

Far-Zone Scattered Field Pattern . . . . . . . . . . . . . . . . .

12

2.4

Derivation of the Method . . . . . . . . . . . . . . . . . . . . . .

14

2.5

Approximations for Fνi (ρ0 ) Functions . . . . . . . . . . . . . . .

17

2.6

Handling Near/Middle-Zone Observations . . . . . . . . . . . .

23

2.7

Handling Non-planar Incident Fields . . . . . . . . . . . . . . .

23

2.8

Fusion of Multi-Frequency Data for Dispersionless Objects . . .

24

i

2.9

Iterative Use of the Technique . . . . . . . . . . . . . . . . . . .

3 Standard ODT Techniques

26 29

3.1

Fourier Diffraction Theorem and the FBP Algorithm . . . . . .

3.2

Equivalence of the Born and Rytov Approximations in the Far Zone 32

3.3

Handling Far-Zone Observations . . . . . . . . . . . . . . . . . .

4 Forward Problem Solution 4.1

4.2

29

33 37

Scattering by a Circularly Symmetric, Multilayered, Lossy Dielectric Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

FDTD Method . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

4.2.1

Maxwell’s Curl Equations . . . . . . . . . . . . . . . . .

41

4.2.2

The Yee Algorithm . . . . . . . . . . . . . . . . . . . . .

42

4.2.3

Numerical Stability . . . . . . . . . . . . . . . . . . . . .

44

4.2.4

Lattice Truncation . . . . . . . . . . . . . . . . . . . . .

44

4.2.5

Numerical Dispersion . . . . . . . . . . . . . . . . . . . .

47

4.2.6

Total Field-Scattered Field Formulation . . . . . . . . .

49

4.2.7

Near-to-Far Field Transformation . . . . . . . . . . . . .

51

5 Simulations

53

5.1

Circularly Symmetric, Multilayered, Lossy Dielectric Cylinder .

54

5.2

Lossless General Phantom . . . . . . . . . . . . . . . . . . . . .

59

5.2.1

Multi-frequency Reconstruction . . . . . . . . . . . . . .

67

5.2.2

Noise Considerations . . . . . . . . . . . . . . . . . . . .

68

Embryo Models with Miscellaneous Mitochondrial Distributions

74

5.3

6 Radial Optimizations

91

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6.1

Description of the Optimization Problem . . . . . . . . . . . . .

91

6.2

Analytical Evaluation of the Jacobian Matrix . . . . . . . . . .

93

6.3

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

6.3.1

Brute-Force (Direct) Inversion of Azimuthally Symmetric Cylindrical Objects . . . . . . . . . . . . . . . . . . . . .

6.3.2

6.3.3

98

Minimization of the Backscattering from Metallic Cylinders (Cloaking) . . . . . . . . . . . . . . . . . . . . . . .

99

Anechoic Chamber Design . . . . . . . . . . . . . . . . .

104

7 Conclusion and Future Work

107

7.1

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107

7.2

Significance . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

108

7.3

Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

110

Bibliography

113

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List of Tables

5.1

RMSE values for the circularly symmetric lossy cylinder. . . . .

58

5.2

RMSE values for the lossless general phantom. . . . . . . . . . .

65

5.3

RMSE values for the proposed technique at various wavelengths and SNR levels. . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

5.4

RMSE values for FBP at λ = 1.5 µm and various SNR levels. .

74

5.5

Embryo Modeling Parameters. . . . . . . . . . . . . . . . . . . .

76

5.6

RMSE values for various embryo models, wavelengths and recon-

6.1

struction techniques. . . . . . . . . . . . . . . . . . . . . . . . .

90

Values of the layer parameters for the designed absorber. . . . .

103

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List of Figures

2.1

Investigated problem and the ODT setup. . . . . . . . . . . . .

11

3.1

ODT setup and relevant parameters for the FBP technique. . .

31

3.2

Data and interpolation points for a set of 16 experiments. . . . .

35

4.1

Standard 3D Yee cell. . . . . . . . . . . . . . . . . . . . . . . . .

43

4.2

2D FDTD grid enclosed with PML ABC. . . . . . . . . . . . . .

46

5.1

Reconstructions of the real (a) and imaginary (b) parts of O(ρ) with FBP at λ = 0.75 µm (Extremely weak scatterer). . . . . .

5.2

Reconstructions of the real (a) and imaginary (b) parts of O(ρ) with FBP at λ = 0.75 µm (Weak scatterer). . . . . . . . . . . .

5.3

57

Reconstructions of the real (a) and imaginary (b) parts of O(ρ) with various algorithms at λ = 1.25 µm. . . . . . . . . . . . . .

5.5

55

Reconstructions of the real (a) and imaginary (b) parts of O(ρ) with various algorithms at λ = 1.5 µm. . . . . . . . . . . . . . .

5.4

55

57

Reconstructions of the real (a) and imaginary (b) parts of O(ρ) with various algorithms at λ = 1 µm. . . . . . . . . . . . . . . .

v

58

5.6

Reconstructions of the real (a) and imaginary (b) parts of O(ρ) with various algorithms at λ = 0.75 µm. . . . . . . . . . . . . .

58

5.7

Same RMSE values in bar graph format. . . . . . . . . . . . . .

59

5.8

Object function profile Opha for the lossless dielectric phantom. .

60

5.9

Reconstruction of Opha from far-zone observations at λ = 1.5 µm, using the proposed technique (a) and FBDT (b).

. . . . . . . .

61

5.10 Reconstruction of Opha from near-zone observations at λ = 1.5 µm with FBP, using Rytov (a) and Born (b) approximations. . . . .

61

5.11 Reconstruction of Opha from far-zone observations at λ = 1.25 µm, using the proposed technique (a) and FBDT (b).

. . . . . . . .

62


62

5.13 Reconstruction of Opha from far-zone observations at λ = 1 µm, using the proposed technique (a) and FBDT (b).

. . . . . . . .

63

5.14 Reconstruction of Opha from near-zone observations at λ = 1 µm with FBP, using Rytov (a) and Born (b) approximations. . . . .

63

5.15 Reconstruction of Opha from far-zone observations at λ = 0.75 µm, using the proposed technique (a) and FBDT (b).

. . . . . . . .

64


64

5.17 Same RMSE values in bar graph format. . . . . . . . . . . . . .

66

5.18 Optimum single-frequency reconstruction (a) at λ = 1.25 µm (I = 8, T = 1), versus multi-frequency reconstruction (b) at λ = 1.5, 1.25, 1 and 0.75 µm (I = 8, T = 4)

vi

. . . . . . . . . . . . .

68

5.19 Noisy (SNR = 5 dB) and noise-free data along the receiver array at λ = 1 µm: (a) Scattered electric field and (b) far-zone scattered field pattern.

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

5.20 Reconstruction with the proposed technique at λ = 1.5 µm from noisy data: SNR = (a) 20, (b) 10 and (c) 5 dB. . . . . . . . . .

70


70

5.22 Reconstruction with the proposed technique at λ = 1 µm from noisy data: SNR = (a) 20, (b) 10 and (c) 5 dB. . . . . . . . . .

71


71

5.24 Corresponding αu parameters for noisy and noise-free data. . . .

72

5.25 Reconstruction with FBP (Born) at λ = 1.5 µm from noisy data: SNR = (a) 20, (b) 10 and (c) 5 dB. . . . . . . . . . . . . . . . .

73

5.26 Reconstruction with FBP (Rytov) at λ = 1.5 µm from noisy data: SNR = (a) 20, (b) 10 and (c) 5 dB. . . . . . . . . . . . . . . . .

73

5.27 Object function profiles for the embryo models involving four typical mitochondrial distributions: (a) homogeneous (Ohom ), (b) aggregated (Oagg ), (c) perinuclear (Oper ) and (d) cortical (Ocor ).

75

5.28 Reconstruction of Ohom from far-zone observations at λ = 2 µm, using the proposed technique (a) and FBDT (b).

. . . . . . . .

77

5.29 Reconstruction of Ohom from near-zone observations at λ = 2 µm with FBP, using Rytov (a) and Born (b) approximations. . . . .

77

5.30 Reconstruction of Ohom from far-zone observations at λ = 1.5 µm, using the proposed technique (a) and FBDT (b).

vii

. . . . . . . .

78

5.31 Reconstruction of Ohom from near-zone observations at λ = 1.5 µm with FBP, using Rytov (a) and Born (b) approximations. . . . .

78

5.32 Reconstruction of Ohom from far-zone observations at λ = 1 µm, using the proposed technique (a) and FBDT (b).

. . . . . . . .

79

5.33 Reconstruction of Ohom from near-zone observations at λ = 1 µm with FBP, using Rytov (a) and Born (b) approximations. . . . .

79

5.34 Reconstruction of Oagg from far-zone observations at λ = 2 µm, using the proposed technique (a) and FBDT (b).

. . . . . . . .

80

5.35 Reconstruction of Oagg from near-zone observations at λ = 2 µm with FBP, using Rytov (a) and Born (b) approximations. . . . .

80

5.36 Reconstruction of Oagg from far-zone observations at λ = 1.5 µm, using the proposed technique (a) and FBDT (b).

. . . . . . . .

81

5.37 Reconstruction of Oagg from near-zone observations at λ = 1.5 µm with FBP, using Rytov (a) and Born (b) approximations. . . . .

81

5.38 Reconstruction of Oagg from far-zone observations at λ = 1 µm, using the proposed technique (a) and FBDT (b).

. . . . . . . .

82

5.39 Reconstruction of Oagg from near-zone observations at λ = 1 µm with FBP, using Rytov (a) and Born (b) approximations. . . . .

82

5.40 Reconstruction of Oper from far-zone observations at λ = 2 µm, using the proposed technique (a) and FBDT (b).

. . . . . . . .

83

5.41 Reconstruction of Oper from near-zone observations at λ = 2 µm with FBP, using Rytov (a) and Born (b) approximations. . . . .

83

5.42 Reconstruction of Oper from far-zone observations at λ = 1.5 µm, using the proposed technique (a) and FBDT (b).

viii

. . . . . . . .

84

5.43 Reconstruction of Oper from near-zone observations at λ = 1.5 µm with FBP, using Rytov (a) and Born (b) approximations. . . . .

84

5.44 Reconstruction of Oper from far-zone observations at λ = 1 µm, using the proposed technique (a) and FBDT (b).

. . . . . . . .

85

5.45 Reconstruction of Oper from near-zone observations at λ = 1 µm with FBP, using Rytov (a) and Born (b) approximations. . . . .

85

5.46 Reconstruction of Ocor from far-zone observations at λ = 2 µm, using the proposed technique (a) and FBDT (b).

. . . . . . . .

86

5.47 Reconstruction of Ocor from near-zone observations at λ = 2 µm with FBP, using Rytov (a) and Born (b) approximations. . . . .

86

5.48 Reconstruction of Ocor from far-zone observations at λ = 1.5 µm, using the proposed technique (a) and FBDT (b).

. . . . . . . .

87

5.49 Reconstruction of Ocor from near-zone observations at λ = 1.5 µm with FBP, using Rytov (a) and Born (b) approximations. . . . .

87

5.50 Reconstruction of Ocor from far-zone observations at λ = 1 µm, using the proposed technique (a) and FBDT (b).

. . . . . . . .

88

5.51 Reconstruction of Ocor from near-zone observations at λ = 1 µm with FBP, using Rytov (a) and Born (b) approximations. . . . .

88

5.52 Same RMSE values in bar graph format. . . . . . . . . . . . . .

90

6.1

Reconstructions of the real (a) and imaginary (b) parts of O(ρ) with brute-force inversion algorithm at λ = 0.75 µm. . . . . . .

99

6.2

Backscattering minimization with cloaking shells. . . . . . . . .

100

6.3

ECCOSORBr AN family microwave absorbers. . . . . . . . . .

102

6.4

RCS reduction with ECCOSORBr AN-79 and 16 optimized layers.103

6.5

A cylindrical metallic anechoic chamber with inner absorber layers.104 ix

Chapter 1

Introduction

1.1

Motivation

In vitro fertilization (IVF) is a billion-dollar industry and a widespread practice being performed in countless infertility clinics all around the world. More than a million babies conceived via IVF have been born worldwide since the introduction of the procedure in 1978. Although IVF is a relatively old practice, the success rate is still around 25%, which is pretty low compared to almost any medical technique of comparable past. Since it is very difficult to determine the viability of preimplantation embryos in vitro, generally all of them are implanted in the mother during the procedure to increase the probability of success. This approach often leads to pregnancies with undesired multiple babies, causes physiological, psychological and ethical problems which could be overcome if a noninvasive viability assessment technique for the oocytes and preimplantation embryos were devised. Noninvasive assessment of the health of an embryo or a single cell is an open issue of critical importance. To achieve this goal, a spatial understanding of the investigated structure, such as how tens of thousands of mitochondria are distributed within a cell, or how many cells reside in an embryo, could be 1

CHAPTER 1. Introduction

2

useful. It is believed that there exists a tight correlation between the cell-embryo viability and their inner organelle, especially mitochondrial distributions [1–10]. The most promising way to reveal the inner composition of cells and embryos in a noninvasive manner is to probe them with low intensity laser light within or near the visible spectrum, and process the collected scattered light to reconstruct the electrical properties of a region of interest containing the investigated biological structure, which is an optical diffraction tomography (ODT) problem by definition. Light scattering through cells and embryos has been extensively studied before, both computationally and experimentally [11–18]. Little has been done, however, to develop imaging techniques and diagnostic tools which can process the observed scattered light and reveal the inner structure of an interrogated specimen. IVF is not the only medical procedure which could benefit from a noninvasive cellular imaging technique. Some cellular organelles, especially mitochondria, are thought to play an important role in the emergence and progress of many human diseases, such as cancer [19], aging [20], Parkinsons and Alzheimers [21]. A reliable imaging technique which could characterize the mitochondria in live cells could be of great importance as a diagnostic tool. With soaring interest in highly debated stem cell research, and as the correlation between cellular organelles and human diseases is clarified, it is not difficult to predict that the need for a robust cellular imaging technique will not only continue, but rise.


1.2

3

Optical Diffraction Tomography

In ODT, a semitransparent object of unknown nature embedded in a known and preferably uniform background is probed with a set of incident optical waves, and the corresponding scattered light is recorded and processed to reconstruct its object function (relative permittivity contrast) profile. When the inhomogeneities in an investigated object are comparable in size to the wavelength of the incident signal, as is the case of an embryo being illuminated by a laser operating in or near the visible spectrum, techniques based on straight ray assumptions suffer from the effects of diffraction and refraction. In this case, diffraction tomography techniques considering wave propagation and diffraction phenomena must be employed [22]. Imaging techniques which are not based on straight ray assumptions can be broadly divided into two classes, namely qualitative and quantitative imaging techniques. Qualitative imaging techniques are signal-subspace reconstruction methods which generally work within exact scattering regime, considering multiple scattering. There is currently much interest in non-iterative qualitative methods which are suitable for real-time imaging applications. These techniques aim to estimate the shape and support of an investigated structure using either full scattered field data including phase information or its intensity only. Among these techniques are the linear sampling method, the factorization method, time-reversal multiple signal classification (MUSIC) method, the intensity-only signal-subspace-based imaging technique and so on [23–28]. Quantitative techniques, on the other hand, aim to reveal the object function variation (i.e., the scattering potential profile) within an investigated structure


4

in addition to its support information. Among these techniques are various iterative optimization algorithms dealing with the nonlinear nature of the problem, and linear diffraction tomography techniques such the well-known filtered backpropagation algorithm (FBP) [29]. Being a linear diffraction tomography technique, the imaging method presented in this study differs from the aforementioned qualitative algorithms in estimating the actual object function of the investigated structure as well, and therefore these qualitative techniques are beyond the scope of this study in terms of performance comparisons. The phase of the scattered fields observed in ODT experiments, which the classical diffraction tomography techniques like FBP require, can be pretty challenging to measure. To overcome this problem, intensity-only diffraction tomography techniques [30–33], phase retrieval algorithms [34] or alternative methods such as the phase-shifting holography [35] can be employed. In this study, we assume that the phase information of the measured scattered fields is available through such a phase retrieval algorithm, and therefore intensity-only ODT techniques are ignored while performing reconstructions and comparisons. Standard ODT techniques like FBP are generally based on either Born or Rytov approximations, and require plane waves to be used as probing signals and a homogeneous background due to the Fourier diffraction theorem employed in their derivation. The success of these techniques is subject to strict restrictions regarding the size and/or relative refractive index of the object of interest [22]. When the assumptions underlying these approximations are no longer valid for an investigated case, these conventional algorithms fail to provide satisfactory results. Higher order (nonlinear) diffraction tomography techniques based on


5

Born and Rytov series exist in the literature [36, 37], but they provide little improvement over the FBP algorithm. A recently developed inverse scattering algorithm [38] can handle some of the shortcomings of the standard approach through the distorted-wave Born approximation (DWBA), such as the use of any kind of incident field, inhomogeneous background and limited available data [39, 40]. However, the underlying Born approximation concept, even if it is “distorted” by considering the known features of an inhomogeneous background, still limits the applicability and the performance of the conventional algorithms. For the cases investigated in this study, we assume a homogeneous host background, plane-wave illumination and abundant available data, hence these relatively recent, DWBA-based methods have no apparent superiority over the conventional techniques, and therefore are not considered in this thesis. When it is already known that an object of interest being probed in an ODT experiment exhibits no frequency-dependent electrical properties over the range of operation frequencies employed, the collected multispectral data can be fused to perform a wideband reconstruction of the interrogated structure. First-order or distorted-wave Born-based wideband reconstruction techniques exploiting this concept exist in the literature [41–43]. Like their single-frequency counterparts [29, 38], however, the performance and applicability of these algorithms are rather limited.

1.3

Proposed Technique

The novel method presented in this dissertation is an alternative to conventional ODT techniques for two dimensional (2D) reconstruction problems. The


6

method makes use of the modal decomposition of the far-zone scattered field data corresponding to multiple experiments (multiple probing plane waves of distinct incidence angles), and reconstructs the object function of the interrogated scatterer using a new, alternative total field approximation within the object support. Compared to the standard Born and Rytov approximations, this novel approximation is more elaborate and successful in linearizing the actual nonlinear scattering mechanism, and therefore provides better reconstructions for a broader class of inverse scattering problems. The method is derived starting from basic concepts and tested with synthetic data for various 2D scenarios involving plane-wave scattering first by circularly symmetric, multilayered, lossy dielectric cylinders, then by a more general lossless dielectric phantom, and finally by embryo models involving various mitochondrial distributions. The same synthetic data are also processed with the conventional ODT algorithms for performance comparisons. Superiority of the proposed technique is demonstrated. This novel technique can be employed with near/middle-zone observations as well with cylindrical (spherical for 3D cases) configuration of the receiver array, and can be easily modified to handle multi-frequency data for a wideband reconstruction, provided that the probed specimen exhibits no dispersion within the operation frequency spectrum. Once extended to handle 3D cases, we believe that the proposed technique may find use in several areas of medical imaging and tomographic microscopy, such as the cell-embryo health assessment for IVF.


1.4

7

Radial Optimization Technique

A radial optimization technique which can rapidly manipulate the layer parameters of a multilayered, azimuthally symmetric, possibly lossy dielectric cylinder in order to match the plane-wave scattering through it with an observed, or desired scattered field pattern is developed. The technique is based on the fast analytical evaluation of the Jacobian matrix relating the change in layer parameters to the change in the scattered field pattern, and its use with a conjugate-gradient type optimization technique like the Levenberg-Marquardt algorithm. In addition to brute-force inversions for azimuthally symmetric structures being probed, the technique can be slightly altered and used for backscattering minimization (cloaking-radar cross section reduction) purposes, and in the design of anechoic chambers.

1.5

Publications

The research presented in this thesis has yielded/will soon yield the following journal and conference publications: • E. Karbeyaz and C. M. Rappaport, “Modal-based tomographic imaging from far-zone observations,” J. Opt. Soc. Am. A 26, pp. 19-29, January 2009 • E. Karbeyaz and C. M. Rappaport, “Modal-based tomographic imaging from far-zone observations: Multi-frequency case,” Opt. Lett. 34, May 2009 (In press)


8

• E. Karbeyaz and C. M. Rappaport, “Modeling and inversion of weakly scattering subcellular inhomogeneities in electrically large cells,” IEEE Antennas and Propag. Soc. Int. Symposium (Hawaii), pp. 4304-4307, 2007 • E. Karbeyaz and C. M. Rappaport, “Modeling and Inversion of Weakly Scattering Structure in Electrically Large Cells,” Annu. Rev. Prog. Appl. Comput. Electromagnetics (Verona, Italy), 2007 • E. Karbeyaz and C. M. Rappaport, “Modal-based tomographic imaging of embryos involving various mitochondrial distributions,” (To be submitted to IEEE Trans. Med. Imag.) • E. Karbeyaz and C. M. Rappaport, “Minimization of the backscattering from metallic cylindrical objects through optimized lossy dielectric layers,” (To be submitted to IEEE Trans. Antennas Propag.)

1.6

Outline of the Thesis

Motivation behind this work, brief background information about standard ODT techniques, and the outline of the new approach are presented in this introductory Chapter 1. The proposed technique is explained in detail and fully derived in Chapter 2. Born/Rytov-based conventional ODT techniques, including the FBP algorithm and a variant of theirs that the author has developed to deal with far-zone observations are discussed in Chapter 3. The forward solution techniques employed to provide the inversion routines with the required synthetic scattered field data are investigated in Chapter 4. Chapter 5 is dedicated


9

to various simulation scenarios designed to illustrate the proposed technique and compare its performance with the standard algorithms. The radial optimization technique and its potential applications are presented in Chapter 6. Finally, the conclusions drawn and the possible future work are discussed in Chapter 7.

Chapter 2

Proposed Technique

In this chapter, the theory behind the novel ODT technique is presented. The chapter starts with the description of the investigated problem, and continues with the detailed derivation of the proposed technique starting from very basic concepts. Among the subject covered in this chapter are: • How to approximate the total electric field within the object support to linearize the actual nonlinear scattering mechanism • How to process the near/middle-zone scattered field data in addition to far-zone observations • How to employ any kind of incident field instead of strict plane-wave probing • How to perform the fusion of multi-frequency data for a wideband reconstruction, in the presence of a dispersionless object. The chapter is concluded with the discussion of the iterative use of the proposed technique to improve the quality of reconstructions.

10

CHAPTER 2. Proposed Technique

2.1

11

Investigated Problem

For a 2D scattering problem where there is a TMz illumination and no structural variation along the z dimension, resulting scattered and total electric fields have only z components, which will be referred to as Es and Et respectively. Let us consider such a 2D scenario involving a scatterer of support radius a. Let this object be probed at a constant operation frequency with plane waves of I distinct incidence angles, and the scattered electric field data Esi corresponding to the ith incidence (view) angle φi be collected along a measurement line located in the far zone. A representation of the described scenario is shown in Fig. 2.1.

y

Measurement Line: E is ( ρ ,φ )

ρ l 0 >> 4a 2 λ

φ

a

x

x

φi

Incident Plane Wave: E iinc (x, y ) = exp[− jk( cosφi x + sinφi y)]

Figure 2.1: Investigated problem and the ODT setup.

The purpose of the inversion algorithm which will be developed in the rest of


12

this chapter is to process the I distinct Esi data sets collected during the ODT experiments to reconstruct the electrical properties of the probed object, such as its relative permittivity contrast, in order to reveal its inner structure. We note that the derivation and illustration of this novel technique can also be found in [44].

2.2

Volume Equivalence Theorem

The electric and magnetic fields scattered by a dielectric object when probed with some incident field can be generated using equivalent electric Jeq and magnetic Meq current densities, which are given by [45]

Jeq = jω(² − ²0 )E

(2.1a)

Meq = jω(µ − µ0 )H

(2.1b)

with E and H being the total fields of the initial problem with exp(jωt) time dependence. These equivalent sources are confined within the object boundaries and radiate the scattered fields of the original scenario in a uniform space with permittivity ²0 and permeability µ0 .

2.3

Far-Zone Scattered Field Pattern

We define the far-zone scattered field pattern f i (φ), a view-dependent, complexvalued function varying with the observation angle φ, to be the far-zone scattered electric field Esi measured at an observation point (ρ → ∞, φ), normalized for


13

the phase retardation and geometric attenuation: √ f i (φ) , lim {exp(jkρ) ρEsi (ρ, φ)}.

(2.2)

ρ→∞

The volume equivalence theorem can be employed to obtain f i (φ). Being proportional to the total electric field, the view-dependent Jieq has only a z component, which will be referred to as Jeqi . Assuming µ = µ0 for the object, Mieq vanishes and Esi can be computed in cylindrical coordinates as [45] Z Esi (ρ, φ)

2π

Z

a

= jkη 0

0

Jeqi (ρ0 , φ0 )G(ρ, φ; ρ0 , φ0 )ρ0 dρ0 dφ0

(2.3)

where G(ρ, φ; ρ0 , φ0 ) = −

1 (2) H (k|ρ − ρ0 |). 4j 0

(2.4)

Here, a is the radius of the smallest circle enclosing the object completely (i.e., the support radius), k is the wave number ω(µ0 ²0 )1/2 and η is the intrinsic impedance (µ0 /²0 )1/2 of the background medium, G is the 2D Green’s function (2)

for cylindrical coordinates and H0 is the Hankel function of the second kind of order zero, defining outgoing waves. (2)

The expression H0 (k|ρ − ρ0 |) can be expressed as a series summation using the addition theorem of Hankel functions for ρ ≥ a ≥ ρ0 : [45] (2) H0 (k|ρ

0

− ρ |) =

∞ X

Ju (kρ0 )Hu(2) (kρ) exp[ju(φ − φ0 )]

(2.5)

u=−∞

(2)

where Ju is the Bessel function of the first kind of order u and Hu

is the

Hankel function of the second kind of order u. Large argument approximation (2)

for the Hankel functions enables us to replace Hu (kρ) by a simpler expression


14

as ρ → ∞: [46]

r lim

ρ→∞

Hu(2) (kρ)

=j

u

2j exp(−jkρ) . √ πk ρ

(2.6)

Substituting Eqs. 2.3-2.6 in Eq. 2.2 and changing the order of integration and summation in the resulting expression, the far-zone scattered field pattern can be written as i

f (φ) = A

∞ X

αui exp(juφ)

(2.7)

u=−∞

where r

jk A = −η 8π Z 2π Z αui = j u 0

(2.8a) a 0

Jeqi (ρ0 , φ0 )Ju (kρ0 ) exp(−juφ0 )ρ0 dρ0 dφ0 .

(2.8b)

Modal expansion coefficients αui decay exponentially fast for |u| > ka [47], enabling us to approximate the far-zone scattered field pattern using ∼ (2 [ka] + 1) terms, where [ka] is the greatest integer less than ka:

i

f (φ) ≈ A

[ka] X

αui exp(juφ).

(2.9)

u=−[ka]

2.4

Derivation of the Method

For an ith experiment (view) which belongs to a set of I distinct experiments, corresponding modal expansion coefficients αui can be calculated by simply orthogonalizing f i (φ): αui

1 = 2πA

Z

2π 0

f i (φ) exp(−juφ)dφ.

(2.10)


15

Equating the forward (Eq. 2.8b) and inverse (Eq. 2.10) expressions for the modal expansion coefficients αui , we obtain Z j

2π

u 0

Z

a 0

Jeqi (ρ0 , φ0 )Ju (kρ0 ) exp(−juφ0 )ρ0 dρ0 dφ0 = αui Z 2π 1 = f i (φ) exp(−juφ)dφ. 2πA 0

(2.11)

Equivalent, view-dependent electric current density Jeqi can be more appropriately expressed as Jeqi (ρ0 , φ0 ) = jω(² − ²0 )Eti (ρ0 , φ0 ) =

jk O(ρ0 , φ0 )Eti (ρ0 , φ0 ) η

(2.12)

where the object function O(ρ0 , φ0 ) is defined in terms of the relative permittivity contrast ²r of the investigated object with respect to the background permittivity: O(ρ0 , φ0 ) , ²r (ρ0 , φ0 ) − 1.

(2.13)

The object function O(ρ0 , φ0 ) can be expanded as a weighted sum of 2D basis functions gmn being formed by the product of an M -term Fourier-Bessel series and (2N + 1) complex exponentials responsible for the radial and azimuthal representation sensitivity respectively, with the unknown weight coefficients cmn to be determined: 0

0

O(ρ , φ ) =

M X N X m=1 n=−N

cmn gmn (ρ0 , φ0 )

(2.14)


16

where gmn (ρ0 , φ0 ) = J0 (

χm 0 ρ ) exp(jnφ0 ). a

(2.15)

Here χm is the mth zero of J0 (x). We note that cm(−n) = c∗mn for a real-valued object function. These gmn basis functions satisfy the following orthogonality property [48]: Z 0

0

0

2π

0

Z

a

hgmn (ρ , φ ) · gm0 n0 (ρ , φ )i , 0

0

∗ 0 0 0 0 0 gmn (ρ0 , φ0 )gm 0 n0 (ρ , φ )ρ dρ dφ

= [πa2 J12 (χm )]δ(m − m0 )δ(n − n0 ).

(2.16)

Similar to the object function, the total electric field inside the object support corresponding to the ith experiment can be expanded in terms of complex exponentials and unknown radial functions Fνi (ρ0 ) to be approximated: Eti (ρ0 , φ0 )

=

∞ X

Fνi (ρ0 ) exp(jνφ0 ). ρ0 ≤ a.

(2.17)

ν=−∞

Substituting Eq. 2.12, Eq. 2.14 and Eq. 2.17 in Eq. 2.11, after changing the order of integration and summation in the resulting expression and considering the orthogonality of complex exponentials over the [0,2π] interval, one can obtain M X N X m=1 n=−N

· cmn

k 2π j u+1 η

Z

a 0

χm i Fu−n (ρ0 )J0 ( ρ0 )Ju (kρ0 )ρ0 dρ0

= αui , |u| ≤ [ka] , 1 ≤ i ≤ I.

¸

a

(2.18)

Once suitable approximations for Fνi (ρ0 ) functions are derived, Eq. 2.18 represents a complex linear system of I (2 [ka] + 1) equations as u and i vary within [− [ka] , [ka]] and [1, I] respectively, which can be solved for M (2N +1) unknown


17

cmn coefficients via standard inversion techniques such as truncated singularvalue decomposition (TSVD). The line integral appearing in Eq. 2.18 can be evaluated using a numerical integration algorithm such as Gauss-Legendre quadrature, while forming the equation system. We note that the integrand can be very oscillatory for certain combination of m, n and u values, requiring either a fixed high number of nodes or an adaptive scheme such as Gauss-Kronrod quadrature to be used in the numerical integration process. Unlike many conventional ODT algorithms which allow the use of each additional view data set in the reconstruction phase as they become available, the proposed technique requires all available data to be used at once and a constant M -N pair high enough to capture the essence of the object function within the limits the available computational resources permit. As will be shown, however, this disadvantage of the new scheme is compensated for by the superior quality of the reconstructions it provides In the case of a real-valued object function corresponding to a lossless object, the cm(−n) = c∗mn property can be incorporated into the resulting equation system to force the reconstruction to be real-valued.

2.5

Approximations for Fνi (ρ0) Functions

Adequate approximations for Fνi (ρ0 ) functions are necessary to be able to use Eq. 2.18 for a successful reconstruction of the object function. Several approaches are tested for this purpose.


18

Born Approximation The Born approximation assumes that the total field inside the object can be approximated by the incident field Einc , neglecting the scattered field within the object support. This approximation holds for limited refractive index-object size combinations. Any a priori information about the scenario, such as the size and average refractive index of the investigated structure or the observed scattered field are ignored with this approximation, limiting its performance. Using the cylindrical wave expansion of a unit-amplitude plane wave of incidence angle φi , the total field (and hence Fνi (ρ0 ) functions) can be expressed as [45] Eti (ρ0 , φ0 )

=

i Einc (ρ0 , φ0 )

=

∞ X ν=−∞

exp(−jνφi )j −ν Jν (kρ0 ) exp(jνφ0 ). | {z }

(2.19)

Fνi (ρ0 )

Average Cylinder The a priori information about the support radius a and average refractive index na of the investigated object can be incorporated to the determination of Fνi (ρ0 ) functions. The total field inside a dielectric cylinder of radius a and refractive index na under a unit-amplitude plane-wave illumination of incidence angle φi is given by [45] Eti (ρ0 , φ0 )

=

∞ X ν=−∞

biν Jν (ka ρ0 ) exp(jνφ0 ), ρ0 ≤ a | {z }

(2.20)

Fνi (ρ0 )

where 0 (2)

biν

= exp(−jνφi )j

−ν

0

(2)

Jν (ka)Hν (ka) − Jν (ka)Hν (ka) 0 (2)

(2)

Jν (ka a)Hν (ka) − na Jν0 (ka a)Hν (ka)

, ka = na k. (2.21)


19

Here and in the rest of this thesis, the prime indicates the derivative of the related cylindrical function with respect to its entire argument. With the inclusion of the available information regarding the object of interest, one expects this average cylinder approach to be more successful than the Born approximation in representing the total electric field within the object support. Effective Cylinder Instead of using the a priori average refractive index information na about the object, one can attempt to determine the view-dependent effective refractive index nie which matches best with the far-zone scattered field pattern observed in the ith experiment. Once this goal is achieved, na in Eq. 2.21 can be replaced by nie to provide Fνi (ρ0 ) functions through Eq. 2.20. The field scattered by a dielectric cylinder of radius a and refractive index nie under a unit-amplitude plane-wave illumination of incidence angle φi is given by [45] Esi (ρ, φ) =

∞ X

ciu Hu(2) (kρ) exp(juφ), ρ ≥ a

(2.22)

u=−∞

where 0

ciu

= exp(−juφi )j

−u

0

nie Ju (ka)Ju (kei a) − Ju (ka)Ju (kei a) 0 (2) Ju (kei a)Hu (ka)

−

(2) nie Ju0 (kei a)Hu (ka)

, kei = nie k. (2.23)

The corresponding far-zone scattered field pattern fei (φ) for this effective cylinder can be obtained substituting Eq. 2.6 in Eq. 2.22:

fei (φ)

≈A

[ka] X u=−[ka]

γui exp(juφ)

(2.24)


20

4 u i where γui = − kη j cu and A is as given in Eq. 2.8a.

The effective refractive index nie , for which the cost function

i

C (z) =

[ka] X

|γui (z) − αui |2

(2.25)

u=−[ka]

takes its minimum value near the initial guess z = na , can be easily determined. In addition to the a priori information used in the previous average cylinder method, this approach also incorporates the observed field data (αui coefficients) into the process, hence is expected to perform slightly better. Circular Mode Matching Like the previously discussed average and effective refractive index techniques, the circular mode matching (CMM) approach assumes the circular object support region to be occupied by a homogeneous cylinder of the same radius. However, unlike the previous techniques which correspond to physically realizable scenarios, the CMM technique assumes a non-physical homogeneous cylinder, which reacts differently to each νth circular mode of the incident field due to its hypothetical mode-dependent refractive index niν , resulting in the νth circular mode of the scattered field observed in an ith experiment. With the additional degree of freedom denoted by the subscript ν in niν , the total field profile that this new scheme provides satisfies the continuity of the total tangential fields at the object support boundary, which is a highly desirable property that all previously discussed techniques lack. With slight modifications to Eq. 2.20 to simplify some algebraic manipulations and replacing na by niν , the total electric field Eti inside the object support


21

is assumed to be ∞ X

Eti (ρ0 , φ0 ) =

ν=−∞

j −ν diν Jν (niν kρ0 ) exp(jνφ0 ), ρ0 ≤ a. {z } |

(2.26)

Fνi (ρ0 )

The scattered electric field outside the object support can be expressed in terms of the modal expansion coefficients ανi given in Eq. 2.10 as Esi (ρ, φ)

∞ kη X −ν i (2) =− j αν Hν (kρ) exp(jνφ), ρ ≥ a. 4 ν=−∞

(2.27)

The incident electric field everywhere, a unit-amplitude plane wave of incidence angle φi , was given in Eq. 2.19 and is repeated here for convenience: i Einc (ρ, φ)

=

∞ X

j −ν exp(−jνφi )Jν (kρ) exp(jνφ).

(2.28)

ν=−∞ i For each of Eti , Esi and Einc , the azimuthal components of the corresponding

magnetic field can be computed using the Faraday’s law in cylindrical coordinates: Hφ =

1 ∂Ez . jωµ ∂ρ

(2.29)

Along the circle ρ = a, continuity of the tangential field components (Ez and Hφ ) and the orthogonality of complex exponentials over the [0,2π] interval result in kη i (2) α H (ka) 4 ν ν kη 0 0 0 diν niν Jν (niν ka) = exp(−jνφi )Jν (ka) − ανi Hν(2) (ka) 4 diν Jν (niν ka) = exp(−jνφi )Jν (ka) −

(2.30a) (2.30b)

for the ν th circular mode, where the derivatives of Bessel and Hankel functions


22

can be evaluated via the recurrence relation [46]: ν Cν0 (x) = Cν−1 (x) − Cν (x). x

(2.31)

Eq. 2.30a, Eq. 2.30b and Eq. 2.31 can be combined to obtain a transcendental equation involving only niν : (2)

βνi Hν−1 (ka) + Jν−1 (ka) niν Jν−1 (niν ka) = (2) Jν (niν ka) βνi Hν (ka) + Jν (ka)

(2.32)

1 βνi = − kηανi exp(jνφi ). 4

(2.33)

where

Eq. 2.32 can be easily solved for niν using Newton’s method. Generally, Eq. 2.32 has infinitely many complex roots, and it must be solved for as many distinct roots as possible within a designated search region, among which the one closest to na (or preferably nie ) can be selected as niν . Once niν is obtained, diν can be easily computed via Eq. 2.30a. Repeating this procedure for all integer ν values such that |ν| ≤ ka, one can obtain the necessary Fνi (ρ0 ) functions through Eq. 2.26. The CMM method utilizes every piece of a priori information about the nature of the unknown object such as its support radius a and average refractive index na , as well as the experimental data (ανi coefficients) while computing nie values as initial guesses and then niν parameters. If available, the lower and upper limits within which the actual refractive index varies can be imposed as a constraint in the computation of niν values. The inner total electric field profile the CMM technique comes up with satisfies the continuity of the tangential


23

components of the total electric and magnetic fields at the boundary of the object support. With these superior properties, one expects the CMM approximation to perform best among the investigated methods.

2.6

Handling Near/Middle-Zone Observations

Although the proposed technique is aimed to process far-zone field measurements, it can be generalized to handle a data set obtained at any distance from the investigated object, provided that the observation points lie on a circle whose center coincides with the origin of the reconstruction space. If the observation points are located on such a circle of radius ρ0 , one can obtain the αui parameters which the described method requires by simply using αui

2.7

=−

Z

2j u (2)

πkηHu (kρ0 )

2π 0

Esi (ρ0 , φ) exp(−juφ)dφ.

(2.34)

Handling Non-planar Incident Fields

The proposed method is not limited to the use of plane waves as probing signals, although it is derived so. Any kind of incident signal which can be expressed as a summation of cylindrical waves originating from the center of the object support can be given by i (ρ, φ) Einc

=

∞ X

j −ν γνi Jν (kρ) exp(jνφ).

(2.35)

ν=−∞

Once the γνi coefficients are obtained with proper cylindrical wave transformations and integral evaluations along sources, their inverses substitute exp(jνφi )


24

terms appearing in Eq. 2.33, yielding in the new βνi parameters that the CMM approximation requires: 1 βνi = − kηανi /γνi . 4

2.8

(2.36)

Fusion of Multi-Frequency Data for Dispersionless Objects

The reconstruction technique developed so far in this chapter is a constantfrequency (monochromatic) technique, and can therefore handle any type of object, including those which exhibit dispersion as well. It can be modified, however, to handle the fusion of available multi-frequency (multispectral) data, provided that the object of interest exhibits no frequency-dependent electrical properties over the range of operation frequencies employed. Let us consider such a 2D scenario involving a dispersionless scatterer of support radius a. Let this object be probed with plane waves of I distinct incidence angles, and T distinct frequencies per incidence angle. Let the scattered electric field data Esi,t corresponding to the ith incidence angle φi and tth angular frequency ω t be collected along a measurement line located in the far zone. Similar to the monochromatic case, one can use the I × T distinct Esi,t data sets collected during the ODT experiments to reconstruct the object function of the interrogated dispersionless scatterer once again via Eq. 2.13: O(ρ0 , φ0 ) , ²r (ρ0 , φ0 ) − 1 =

M X N X m=1 n=−N

cmn J0 (

χm 0 ρ ) exp(jnφ0 ). a

(2.37)


25

The required cmn coefficients can be obtained by solving the following modified version of Eq. 2.18: · ¸ Z k t u+1 a i,t 0 χm 0 t 0 0 0 cmn 2π j Fu−n (ρ )J0 ( ρ )Ju (k ρ )ρ dρ η a 0 m=1 n=−N M X N X

£ ¤ = αui,t , |u| ≤ k t a , 1 ≤ i ≤ I, 1 ≤ t ≤ T

(2.38)

where αui,t

1 = 2πAt

Z

2π

f i,t (φ) exp(−juφ)dφ

(2.39)

0

√ f i,t (φ) = lim {exp(jk t ρ) ρEsi,t (ρ, φ)} ρ→∞ r jk t At = −η . 8π

(2.40) (2.41)

Here, k t is the wave number ω t (µ0 ²0 )1/2 corresponding the tth probing frequency. Fνi,t (ρ0 ) functions appearing in Eq. 2.38 can be determined using the CMM approximation as follows: £ t ¤ i,t t 0 Fνi,t (ρ0 ) = j −ν di,t J (n k ρ ), |ν| ≤ ka . ν ν ν

(2.42)

ni,t ν can be computed by solving the following transcendental equation (2)

i,t t βνi,t Hν−1 (k t a) + Jν−1 (k t a) ni,t ν Jν−1 (nν k a) = (2) t Jν (ni,t βνi,t Hν (k t a) + Jν (k t a) ν k a)

(2.43)

1 βνi,t = − k t ηανi,t exp(jνφi ). 4

(2.44)

where


26

i,t Once ni,t ν is obtained, dν can be easily computed via (2)

di,t ν =

Jν (k t a) + βνi,t Hν (k t a) . t Jν (ni,t ν k a) exp(jνφi )

(2.45)

Once all relevant parameters are computed through Eqs. 2.39-2.45 and substituted, Eq. 2.38 represents a complex, generally ill-posed linear system of P I Tt=1 (2 [k t a] + 1) equations as i, t and u vary within [1, I], [1, T ] and [− [k t a] , [k t a]], respectively, which can be solved for M (2N + 1) unknown cmn coefficients via TSVD. Once again, in the special case of a lossless object, the cm(−n) = c∗mn property can be incorporated into the resulting equation system to force the reconstruction to be real-valued. We note that the same derivation details and an illustrative example, showing the superiority of this multifrequency approach over the ordinary monochromatic one, can be found in [49].

2.9

Iterative Use of the Technique

The proposed method can be used iteratively to improve the accuracy of the total field approximation at each step, improving the quality of the successive reconstructions. Once an initial reconstruction of the investigated object is performed with the proposed inversion technique and the CMM approximation, this initial object profile can be computationally probed with the same known incident signals using forward solution techniques such as the finite-difference frequency-domain (FDFD) method, and the resulting total field profile within the object support can be obtained, generally along a cartesian grid. Then, the view-dependent,


27

total electric field values Eti (ρ0 , φ0 ) at points lying on a circle of radius ρ0 (ρ0 ≤ a) centered at the designated support origin can be approximated with 2D bilinear interpolation, and Fνi (ρ0 ) values can be computed by evaluating the Fourier transform of these interpolated fields: Fνi (ρ0 )

1 = 2π

Z

2π 0

Eti (ρ0 , φ0 ) exp(−jνφ0 )dφ0 .

(2.46)

These new field profiles (Fνi (ρ0 ) values) can then replace the CMM approximation in the proposed technique to perform a second reconstruction which is expected to be of higher quality, and the iteration continues for a third reconstruction. The iterations can be terminated when the difference between two successive reconstructions is less than a desired tolerance value. At each iteration step, the new Fνi (ρ0 ) profiles computed with the forward solver can be further refined before being substituted into Eq. 2.18 as follows: At each iteration, for each ith view, the corresponding far-zone scattered field pattern f i (φ) is computed using the near-zone forward solver simulation results, and then subtracted from the actually observed pattern to obtain ∆f i (φ) = f i (φ) − f i (φ). ∆f i (φ) is decomposed into its corresponding ∆αui coefficients, from which one can calculate the minimum-energy (i.e., minimum L2 norm) i within the object support which radiates current density distribution ∆Jmin i into the support radius ∆f i (φ), as explained in [47]. The field radiated by ∆Jmin

is then computed via various variants of the Lommel integral formulae [50]. This field is added to the inner total field computed with the forward solver before being processed through Eq. 2.46 to obtain the refined Feνi (ρ0 ) profiles, which would satisfy to total tangential field continuity along the support circle. This extra refinement step should improve the convergence rate of the iterative


28

approach. For a general 2D case, the implementation of this iterative scheme is pretty involved and therefore left out of the scope of this dissertation. The technique could be relatively easily implemented for an azimuthally symmetric 2D case. However, especially considering the success of the brute-force inversion technique introduced in Chapter 6 for such scenarios, this case is omitted as well. As the last remark, we note that the first reconstruction with the CMM approximation can also be used as a good initial guess for a standard iterative distorted-wave Born method.

Chapter 3

Standard ODT Techniques

In this chapter, the Fourier diffraction theorem and the FBP algorithm are briefly discussed. The derivation of the FBP algorithm is omitted since it is not relevant to this dissertation. Only the final mathematical expressions used in the reconstructions are given. A Born-based ODT algorithm capable of processing far-zone observations is fully derived and explained. Dealing with such special cases as circularly symmetric or lossless objects is outlined.

3.1

Fourier Diffraction Theorem and the FBP Algorithm

The Fourier diffraction theorem, also known as the generalized projection-slice theorem, is the basis of virtually all standard ODT reconstruction algorithms. The theorem states that the 1D spatial Fourier transform of the observed scattered field (or, that of the phase disturbance of the observed total field) along a measurement line is proportional to the 2D spatial Fourier transform of the object function of the investigated scatterer along a semicircular arc, provided that the incident field is a plane wave and the Born (or Rytov) approximation 29

CHAPTER 3. Standard ODT Techniques

30

is valid [22]. Like all inversion algorithms based on the Fourier diffraction theorem, the FBP algorithm produces a low-pass filtered reconstruction of the actual object function. In mathematical terms, the technique can be summarized as 1 O(x, y) = − 2π

Z

2π 0

Π∗φi (x sin φi − y cos φi , x cos φi + y sin φi )dφi

(3.1)

where Z k 1 e φ (κ, η) exp(jκξ)dκ Π Πφi (ξ, η) = 2π −k i e φ (κ, η) = Γ eφ (κ)|κ| exp[j(k − γ)(l0 − η)] Π i i √ γ = k 2 − κ2 Z ∞ eφ (κ) = ΓB,R Γ i φi (ξ) exp(−jκξ)dξ −∞ · ¸∗ j Esi (ξ) B (with Born Approximation) Γφi (ξ) = i k Einc (ξ) · i ¸∗ j Et (ξ) R Γφi (ξ) = (with Rytov Approximation). ln i k Einc (ξ)

(3.2a) (3.2b) (3.2c) (3.2d) (3.2e) (3.2f)

Here, l0 is the distance between the measurement line and the origin of the reconstruction space, φi is the incidence angle of the ith probing plane wave, i Esi , Eti and Einc are the view-dependent scattered, total and incident electric

fields respectively, recorded along the measurement line. A representation of the ODT setup for the FBP algorithm, including relevant parameters and coordinate systems, is shown in Fig. 3.1. Various complex conjugate operations appearing in the FBP formulation above, but not in the original references [29, 51], arise from the exp(jωt) time dependence adopted in this dissertation. If it is known that the investigated object is lossless (i.e., the object function


31

y

Measurement Line: E is (ξ ,η = l0 )

l0 η x

ξ

φi

Incident Plane Wave: E iinc (x, y ) = exp[− jk ( cosφi x + sinφi y)]

Figure 3.1: ODT setup and relevant parameters for the FBP technique.

is real-valued), this a priori information can be taken into account in the recone φ (−κ, η) = Π e ∗ (κ, η) condition before evaluating struction process by imposing Π i φi Πφi (ξ, η) through Eq. 3.2a. For a circularly symmetric object with no azimuthal dependence, the FBP algorithm simplifies to [30, 39] 1 O(ρ) = − π

Z

k 0

p e exp[j(k − γ)l0 ]}∗ J0 ( 2k(k − γ)ρ)κdκ. {Γ(ξ)

(3.3)


3.2

32

Equivalence of the Born and Rytov Approximations in the Far Zone

ΓR φi (ξ) given by Eq. 3.2f can be algebraically manipulated as ΓR φi (ξ)

¸∗ · i j Et (ξ) = ln i k Einc (ξ) · ¸∗ j Esi (ξ) = ln 1 + i . k Einc (ξ)

(3.4)

For a far-zone observation point (l0 → ∞), under plane-wave illumination, we can safely state that

¯ i ¯ ¯ Es (ξ) ¯ ¯ ¯ ¯ E i (ξ) ¯ ¿ 1.

(3.5)

inc

Using Eqs. 3.4-3.5 and the first order Taylor approximation ln(1 + z) ≈ z for |z| ¿ 1, we obtain ΓR φi (ξ)

· ¸∗ j Esi (ξ) ≈ i k Einc (ξ) ≈ ΓB φi (ξ).

(3.6)

Hence, we conclude that the Rytov and Born approximations are equivalent for a scattering problem where the probing field is a plane wave and the observations are made in the far zone. The Fourier diffraction theorem and FBP algorithm are not suitable to deal with the scattered field data measured in the far zone, since an impractically high number of observation points are required along a relatively longer measurement eφ (κ) through Eq. 3.2d. How line to satisfy the Nyquist rate while evaluating Γ i to handle far-zone observations with a Born-based ODT technique is discussed


33

in the following section.

3.3

Handling Far-Zone Observations

Substituting Eq. 2.12 in Eq. 2.3, one can derive the Lippmann-Schwinger integral equation relating the scattered field observed in an ith experiment to the object function distribution and the total electric field profile within the object support as follows: ZZ Esi (ρ, φ)

= −k

2

O(ρ0 , φ0 )Eti (ρ0 , φ0 )G(ρ, φ; ρ0 , φ0 )ds0 .

(3.7)

object

In primed cartesian coordinates, Green’s function G can be approximated for a far-zone observation point as [45] 1 G(ρ, φ; x , y ) = − 4j 0

r

0

2j exp(−jkρ) exp[jk(x0 cos φ + y 0 sin φ)] √ πk ρ

(3.8)

i and the incident electric field Einc is given by

i Einc (x0 , y 0 ) = exp[−jk(x0 cos φi + y 0 sin φi )].

(3.9)

i approximating Eti and the definition of Substituting Eq. 3.8, Eq. 3.9 with Einc

the far zone scattered field pattern (Eq. 2.2) in Eq. 3.7, after extending the integration limits to ∞ and replacing x0 -y 0 by x-y for clarity, one obtains Z

∞

˜ v) , O(u,

∞

O(x, y) exp(−jux) exp(−jvy)dxdy −∞

r =

Z

−∞

8πj i f (φ) k3

(3.10)


34

where

u = k(cos φi − cos φ)

(3.11a)

v = k(sin φi − sin φ).

(3.11b)

The object function O(x, y) can be readily reconstructed via the two dimensional inverse Fourier transform µ O(x, y) =

1 2π

¶2 Z

∞ −∞

Z

∞

˜ v) exp(jux) exp(jvy)dudv O(u,

(3.12)

−∞

˜ v) valwhich can be easily evaluated using existing IFFT algorithms, once O(u, ues are computed and interpolated along a cartesian u-v grid using the nearestneighbor point approach. As an illustration, the data points lying on circular arcs and the cartesian interpolation points are shown in Fig. 3.2, for a set of 16 experiments and a forward-facing field of view of 1200 . In the rest of this thesis, the inversion scheme which has just been derived will be referred to as the far-zone Born-based diffraction tomography (FBDT). The a priori information that O(x, y) is real-valued can be taken into ac˜ ˜ ∗ (u, v) condition before count in the reconstruction by imposing O(−u, −v) = O performing the inverse Fourier transform. For a circularly symmetric object with no azimuthal dependence, the 2D spatial Fourier transform of its circularly symmetric object function is circularly symmetric as well. For such a case, rewriting the integral in Eq. 3.12 in polar (κ, α) coordinates with u = κ cos α, substituting y = 0 and x = ρ for simplicity,


35

1 0.8 0.6 0.4

v/k

0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1

−0.8

−0.6

−0.4

−0.2

0 u/k

0.2

0.4

0.6

0.8

1

Figure 3.2: Data and interpolation points for a set of 16 experiments.

and assuming an infinitely long measurement line, we obtain: µ O(ρ) =

1 2π

¶2 Z

√

2k

·Z

¸

2π

˜ O(κ)κ

exp(jκ cos αρ)dα dκ.

(3.13)

0

0

The inner integral appearing in Eq. 3.13 can be evaluated as [45] Z

2π

exp(jκ cos αρ)dα = 2πJ0 (κρ),

(3.14)

0

which yields 1 O(ρ) = 2π

√

Z

2k

˜ O(κ)J 0 (κρ)κdκ.

(3.15)

0

Assuming φi = 0 for simplicity, for a given observation angle φ such that 0 ≤


36

φ < 900 , the corresponding κ can be computed using the cosine theorem as

κ=k

p

2(1 − cos φ),

(3.16)

˜ and the corresponding O(κ) value is given by r ˜ O(κ) =

8πj f (φ). k3

(3.17)

One expects the FBDT algorithm to perform slightly worse than FBP due to the far-zone approximations and the nearest-neighbor point interpolation involved in the derivation.

Chapter 4

Forward Problem Solution

Simulation programs solving various electromagnetic scattering problems are required to provide synthetic data to the inversion routines implementing the proposed and conventional ODT techniques. The analytical and computational techniques implemented in these forward solvers (Mie scattering by a layered cylinder and the finite-difference time-domain (FDTD) method for more general scattering problems) are discussed in this chapter.

4.1

Scattering by a Circularly Symmetric, Multilayered, Lossy Dielectric Cylinder

The first scenario being investigated is the scattering of a T M z plane wave by an infinitely long, z-directed, multilayered, lossy dielectric cylinder with azimuthal symmetry. A fast, iterative analytical forward solution to this type of scattering problem is available [52, 53]. Main steps of this solution are summarized below. For an incident unit-amplitude plane wave of zero incidence angle, the total

37

CHAPTER 4. Forward Problem Solution

38

electric field outside an M-layered cylinder of radius a is given by

Et (ρ, φ) =

∞ X £

¤ j −n Jn (kρ) + Cn Hn(2) (kρ) exp(jnφ)

n=−∞

≈

[ka] X

£ ¤ en j −n Jn (kρ) + Cn Hn(2) (kρ) cos(nφ)

(4.1)

n=0

where e0 = 1 and en = 2 for n ≥ 1. The total electric field and the azimuthal component of the total magnetic field inside the mth layer of the cylinder can be written as Etm (ρ, φ)

≈

[ka] X

en [Amn Jn (km ρ) + Bmn Yn (km ρ)] cos(nφ)

(4.2a)

n=0

1 ∂Etm jωµm ∂ρ

Htφ,m (ρ, φ) =

[ka] i km X h 0 0 ≈ en Amn Jn (km ρ) + Bmn Yn (km ρ) cos(nφ) (4.2b) jωµm n=0

where km = ω(µm ²m )1/2 , and µm and ²m are the material parameters of layer m. Without loss of generality, one may begin with the definite unnormalized amplitudes (indicated with primes) in the first (innermost) layer: 0

0

A1,n = 1 and B1,n = 0, for all n.

(4.3)

Continuity of the total tangential field components at the boundary between the mth and (m + 1)th layers, and the orthogonality of complex exponentials over the [0,2π] interval result in 

 0





 0

 Am+1,n   Umn Wmn   Amn   0   0  = Tm  Vmn Xmn Bmn Bm+1,n

(4.4)


39

where

Tm =

πρm 2µm

(4.5a) 0

0

Umn = µm km+1 Jn (km ρm )Yn (km+1 ρm ) − µm+1 km Jn (km ρm )Yn (km+1 ρm ) (4.5b) 0

0

Vmn = µm+1 km Jn (km+1 ρm )Jn (km ρm ) − µm km+1 Jn (km+1 ρm )Jn (km ρm ) 0

(4.5c)

0

Wmn = µm km+1 Yn (km ρm )Yn (km+1 ρm ) − µm+1 km Yn (km ρm )Yn (km+1 ρm ) (4.5d) 0

0

Xmn = µm+1 km Yn (km ρm )Jn (km+1 ρm ) − µm km+1 Yn (km ρm )Jn (km+1 ρm ). (4.5e) Starting with the assumed first layer field amplitudes, one can iteratively obtain the field amplitudes in each layer up to m = M + 1 (i.e., background) via Eq. 4.4. A normalization constant Kn may be defined for the primed amplitudes such that 0

0

Amn = Kn Amn and Bmn = Kn Bmn .

(4.6)

Equating Eq. 4.1 to Eq. 4.2a (with m = M + 1) outside the cylinder and using Eq. 4.6, Cn and Kn are found to be 0

BM +1,n Cn = −j 0 0 BM +1,n + jAM +1,n 1 . Kn = j −n 0 0 AM +1,n − jBM +1,n −n

(4.7a) (4.7b)

The αn coefficients that the proposed technique requires can be obtained in terms of the computed Cn coefficients as

αn = −

4 n j Cn . kη

(4.8)


40

In practice, Tm and Kn coefficients given by Eq. 4.5a and Eq. 4.7b respectively, are omitted while computing Cn parameters through Eq. 4.7a. They should be taken into account, however, while computing the total electric and magnetic fields within the layers of the cylinder.

4.2

FDTD Method

The FDTD method, first proposed by Yee [54], is a time domain numerical technique based on the spatial and temporal discretization of the Maxwell’s curl equations with central finite differences. The fields are marched in time according to the resulting finite difference equations and the frequency-domain fields can be computed on-the-fly using discrete Fourier transformation at the desired locations and frequencies. Being an explicit time-domain technique, FDTD provides simulation results at multiple frequencies in a single run with proper probing signal arrangement, and requires no costly solution to linear equation systems, making it ideal to deal with large scattering problems. Basic concepts of the FDTD method, such as the perfectly matched layer absorbing boundary condition to simulate the open boundaries, the near-to-far field transformation and the total field-scattered field formulation are implemented in the simulation programs. Details of these techniques, which are summarized in the following sections, can be found in the related FDTD literature [55, 56].


4.2.1

41

Maxwell’s Curl Equations

Consider a region of space where there are no electric or magnetic current sources, but there may be materials absorbing electric or magnetic field energy. The time-dependent Maxwell’s curl equations for this type of scenario are given by ∂B = −∇ × E − Jm ∂t ∂D = ∇ × H − Je ∂t

(4.9a) (4.9b)

where E is the electric field intensity (V /m), D is the electric flux density (C/m2 ), H is the magnetic field intensity (A/m), B is the magnetic flux density (W eber/m2 ), Je is the electric conduction current density (A/m2 ) and Jm is the equivalent magnetic conduction current density (V /m2 ). In linear, isotropic and nondispersive materials, following constitutive relations hold:

D = ²E

&

B = µH

(4.10a)

Je = σE

&

Jm = σ ∗ H

(4.10b)

where ² is the electric permittivity (F arad/m), µ is the magnetic permeability (Henry/m), σ is the electric conductivity (Siemens/m) and σ ∗ is an equivalent magnetic resistivity (Ω/m). When Eqs. 4.10a-4.10b are substituted in Eqs. 4.9a-4.9b and the curl operations are performed in 3D cartesian (x, y, z) coordinate system, we obtain the


42

following system of six coupled, scalar, partial differential equations: ∂Hx ∂t ∂Hy ∂t ∂Hz ∂t ∂Ex ∂t ∂Ey ∂t ∂Ez ∂t

= = = = = =

µ 1 ∂Ey µ ∂z µ 1 ∂Ez µ ∂x µ 1 ∂Ex µ ∂y µ 1 ∂Hz ² ∂y µ 1 ∂Hx ² ∂z µ 1 ∂Hy ² ∂x

¶ ∂Ez ∗ − − σ Hx ∂y ¶ ∂Ex ∗ − − σ Hy ∂z ¶ ∂Ey ∗ − − σ Hz ∂x ¶ ∂Hy − − σEx ∂z ¶ ∂Hz − − σEy ∂x ¶ ∂Hx − − σEz . ∂y

(4.11a) (4.11b) (4.11c) (4.11d) (4.11e) (4.11f)

For the 2D, T M z scenarios being investigated in this work, the preceding system of six partial differential equations reduces to µ ¶ ∂Hx 1 ∂Ez ∗ = − − σ Hx ∂t µ ∂y µ ¶ ∂Hy 1 ∂Ez ∗ = − σ Hy ∂t µ ∂x µ ¶ ∂Ez 1 ∂Hy ∂Hx = − − σEz . ∂t ² ∂x ∂y

4.2.2

(4.12a) (4.12b) (4.12c)

The Yee Algorithm

In 1966, Kane Yee published a set of finite-difference equations for the time domain Maxwell’s curl equations, based on a space lattice structure he proposed. The building block of this lattice structure is called the Yee cell, and shown in Fig. 4.1. Due to the placement of the electric and magnetic field components within the Yee cell, the resulting finite-difference expressions for the space derivatives


43

Figure 4.1: Standard 3D Yee cell.

used in the curl operators are central in nature and therefore second order accurate. The algorithm also centers these field components in time in what is termed a leapfrog arrangement, which results in second order accurate, central finite difference expressions for the time derivatives as well. For the 2D, T M z case involving nonpermable (µ = µ0 and σ ∗ = 0) but possibly lossy (σ 6= 0) media, Eqs. 4.12a-4.12c can be discretized according to the Yee algorithm as follows, using a directionally uniform space step ∆ (∆ = ∆x = ∆y) and a time step ∆t: ¡ ¢ n+ 1 n− 1 Hx |i,j+21 = Da Hx |i,j+21 + Db Ez |ni,j − Ez |ni,j+1 2

(4.13a)

2

¢ ¡ n+ 1 n− 1 Hy |i+ 12,j = Da Hy |i+ 12,j + Db Ez |ni+1,j − Ez |ni,j (4.13b) 2 2 ³ ´ n+ 12 n+ 12 n+ 12 n+ 12 n Ez |n+1 = C E | + C H | (4.13c) − H | + H | − H | a z i,j b y i+ 1 ,j y i− 1 ,j x i,j− 1 x i,j+ 1 i,j 2

2

2

2


44

where σi,j ∆t 2²i,j σi,j ∆t 1 + 2²i,j Ã ∆t 1 ²i,j ∆ 1 + σi,j ∆t 2²i,j

1−

Ca = Cb =

Da = 1 ∆t . µ0 ∆

Db =

4.2.3

(4.14a) ! (4.14b) (4.14c) (4.14d)

Numerical Stability

For numerical stability of this 2D FDTD algorithm, ∆t and ∆ should satisfy the following condition: ∆ ∆t ≤ √ . c 2

(4.15)

Here, c is the maximum speed of light in the region of interest. This condition is known as Courant stability condition.

4.2.4

Lattice Truncation

To mimic the unbounded nature of the real life scenarios being simulated with limited storage capabilities, the computational grid must be truncated in such a way that the outgoing waves are absorbed and not reflected back into the region of interest. This task is achieved with what is called an absorbing boundary condition (ABC). In this work, perfectly matched layer (PML) ABC proposed by Berenger [57] is used for that purpose, as is the case for most FDTD solvers of modern era. For the 2D T M z case, the theory of the PML ABC can be summarized as


45

follows: In the PML medium surrounding the computational domain, the Ez component is split into two hypothetical subcomponents, namely Ezx and Ezy . Maxwell’s equations for the 2D T M z case then take the following form: ∂Ezx + σx Ezx ∂t ∂Ezy ²0 + σy Ezy ∂t ∂Hx µ0 + σy∗ Hx ∂t ∂Hy µ0 + σx∗ Hy ∂t

²0

∂Hy ∂x ∂Hx = − ∂y ∂(Ezx + Ezy ) = − ∂y ∂(Ezx + Ezy ) = ∂x =

(4.16a) (4.16b) (4.16c) (4.16d)

∗ where σx,y and σx,y are the direction-dependent electric and magnetic conductiv-

ities (losses), respectively. It is known that if the following condition is satisfied σ σ∗ = , ²0 µ0

(4.17)

the wave impedance of the lossy free-space medium matches that of the lossless vacuum, and no reflection occurs when a plane wave propagates normally across the interface between them. In Fig. 4.2, a vacuum computational domain is surrounded by a PML layer, consisting of 8 different regions and terminated with a perfect electric conductor ∗ parameters are (PEC) wall. The values used in each region for σx,y and σx,y

given below: • PML Regions B, F: (σx , σx∗ , σy , σy∗ ) = (0, 0, σy , σy∗ ) • PML Regions D, H: (σx , σx∗ , σy , σy∗ ) = (σx , σx∗ , 0, 0) • PML Regions A, C, E, G: (σx , σx∗ , σy , σy∗ ) = (σx , σx∗ , σy , σy∗ ).


46

A

B

C

H

Vacuum

D

G

F

E

y

x

Figure 4.2: 2D FDTD grid enclosed with PML ABC.

In each region, the pairs (σx , σx∗ ) and (σy , σy∗ ) should be chosen in such a way that Eq. 4.17 is satisfied. Berenger proposed that the loss should increase gradually from zero with depth ρ within each PML region:

σ(ρ) = σmax

³ ρ ´n δ

(4.18)

where δ is the PML thickness and σ is either σx or σy . Once the profile of the loss, i.e. n in Eq. 4.18, and the PML thickness δ are selected, σmax can be calculated for a desired PML reflection coefficient at normal incidence, which is given by:

· ¸ 2σmax δ R0 = exp − . (n + 1)²0 c

(4.19)

Following PML ABC parameters are used for the FDTD implementation


47

employed in this dissertation: R0 = 10−6 , n = 2 and δ = 10 Yee cells. Berenger suggests the use of exponential time-stepping in the finite-difference equations within the PML region because of the extremely rapid decay the traveling wave is exposed to, which can be implemented simply by replacing Ca,b and Da,b in Eqs. 4.14a-4.14d with µ

Ca0

=

Cb0 = Da0 = Db0 =

4.2.5

¶ σi,j ∆t exp − ² · 0 µ ¶¸ 1 σi,j ∆t 1 − exp − σi,j ∆ ²0 µ ∗ ¶ σi,j ∆t exp − µ0 · µ ∗ ¶¸ σi,j ∆t 1 1 − exp − . ∗ σi,j ∆ µ0

(4.20a) (4.20b) (4.20c) (4.20d)

Numerical Dispersion

One of the artifacts inherent in the FDTD method is the fact that the phase velocity of the numerical wave modes in the computational grid differs from the speed of light in the actual medium being modeled, which is called the numerical dispersion. This difference depends on the modal wavelength (λ), the direction of propagation in the grid (φi ), the space step (∆) and the time step (∆t). Especially for electrically large structures, numerical dispersion should be taken into account in the programs implementing the FDTD algorithm to increase the accuracy of the results, since phase errors can accumulate and distort the results in these cases. The numerical wavenumber e k for a plane wave traveling in a 2D FDTD grid with a propagation angle φi satisfies the following dispersion relation (∆x =


48

∆y = ∆): µ

∆ c∆t

¶2

µ sin

2

ω∆t 2

Ã

¶ = sin

2

e k∆ cos φi 2

!

Ã + sin

2

e k∆ sin φi 2

! .

(4.21)

The numerical wavenumber e k satisfying this relation can be computed with Newton’s method in just a few iteration steps, starting from e k0 = ω/c: sin2 (Ae ki ) + sin2 (B e ki ) − C e ki+1 = e ki − A sin(2Ae ki ) + B sin(2B e ki )

(4.22)

where ∆ cos φi 2 ∆ sin φi B = 2 µ ¶2 µ ¶ ∆ ω∆t 2 C = sin . c∆t 2 A =

(4.23a) (4.23b) (4.23c)

The phase delay α experienced along a distance l by a numerical plane wave traveling in an FDTD grid with respect to the actual plane wave being simulated can be computed via α=

³ω c

´ −e k l.

(4.24)

For an electrically large simulation scenario, prior to the FDTD simulation, the phase delay α for the largest frequency of interest and along the largest distance l the incident plane wave will travel within the grid should be computed. If it is larger than ∼ π/8, proper measures should be taken in the light of the discussion below. Errors due to numerical dispersion in a 2D FDTD simulation decrease as


•

λ ∆

•

c∆t ∆

49

increases approaches

√1 2

• φi approaches ±450 or ±1350 . To minimize the numerical dispersion related errors as much as possible in an FDTD simulation when solving a challenging, electrically very large problem, one should use the smallest ∆ the computational resources allow and the largest ∆t the Courant stability condition permits. If this does not suffice, as the last resort, a constant illumination angle of 450 should be used. A custom incidence angle φi can then be realized by rotating the investigated model by an angle of 450 − φi in the positive direction. Dispersion-optimized FDTD formulations based on 4th order spatial finite differences exist in the literature, but these techniques are rarely preferred in practice due to various problems.

4.2.6

Total Field-Scattered Field Formulation

The total field-scattered field formulation, first proposed by Mur [58], is the first and most common compact wave source for use in simulations of plane-wave illumination. The technique is based on a source generator wave traveling along a 1D auxiliary grid, and the interpolation of the incident field values at the grid points along a connection rectangle totally enclosing the region of interest. This rectangle splits the computational domain into to regions called the total field and scattered field regions. Electric field values at grid points lying on the connection rectangle, and magnetic field values at adjacent grid points within the scattered field region


50

should be carefully updated in time, in order to avoid distorting the scattered field outside the rectangle and to inject the incident field into the total field region as intended. A common problem encountered while implementing this technique is the wave leakage into the scattered field region, which arises from the fact that the numerical phase velocities within the 1D auxiliary and 2D simulation grids are different when a common space step is used as the common sense dictates, unless the angle of incidence is an integer multiple of 900 . This wave leakage problem can be overcome by adjusting the space step ∆r of the 1D auxiliary grid in such a way that the numerical phase velocity in this grid is equal to the numerical phase velocity in the direction of propagation within the 2D simulation grid. For a 3D FDTD simulation environment with space step values ∆x, ∆y, ∆z, and the polar coordinate angles φi and θi for an incident plane wave, the required ∆r can be approximately computed with [59]: q ∆r ≈ ∆x2 (cos φi sin θi )4 + ∆y 2 (sin φi sin θi )4 + ∆z 2 (cos θi )4 .

(4.25)

Surprisingly, ∆r is independent of the operation frequency and the common time step ∆t. As reported in [59], for ∆x = ∆y = ∆z = ∆, the above expression provides the equality of the 1D-3D phase velocities unless λ < 5∆, a constraint of no practical importance. The general expression given in Eq. 4.25 can be adapted to the 2D case employed in this study by setting θi = 900 and ∆x = ∆y = ∆: q ∆r = ∆

cos4 φi + sin4 φi .

(4.26)


4.2.7

51

Near-to-Far Field Transformation

This technique allows the far-zone scattered field data to be computed using the near-zone scattered field values that the FDTD simulations provide. The technique is based on the surface equivalence theorem, Green’s function theory and large argument approximations for the special functions involved. It is a frequency-domain technique, hence the phasors of the tangential near-zone scattered electric and magnetic fields should be computed on-the-fly via discrete Fourier transformation at the desired frequencies, along a rectangle C fully enclosing the region of interest. For an investigated 2D, TMz electromagnetic scattering problem, the farzone scattered field pattern f (φ) defined by Eq. 2.2 can be computed in terms of the equivalent surface currents Jeq and Meq along the rectangle C via I b] exp(jkb [ωµ0 b z0 · Jeq (ρ0 ) + k b z0 × Meq (ρ0 ) · ρ ρ · ρ0 )dl0

f (φ) = K

(4.27)

C

where b 0 × Hs Jeq = n

(4.28a)

Meq = −b n0 × E s

(4.28b)

exp[j(5π/4)] √ 8πk b = cosφ x b + sinφ y b ρ

K =

(4.28c) (4.28d)

b 0 is the outward unit vector along the rectangle C. Here, n The rectangle C along which the line integral is evaluated should be placed in the scattered-field region of the computational domain. This rectangle generally


52

passes through the electric field grid points, and consequently does not include the tangential magnetic field grid points, due to the structure of the Yee cell. The tangential magnetic field components along the rectangle should be computed by averaging the values at neighboring grid points.

Chapter 5

Simulations

To illustrate the proposed method and compare its performance with the FBP and FBDT algorithms, three simulation scenarios involving plane-wave scattering first by a circularly symmetric, multilayered, lossy dielectric cylinder, then by another, more general 2D lossless dielectric phantom, and finally by four two-cell embryo models involving aggregated, cortical, homogeneous and perinuclear distributions are investigated. Synthetic scattered field data provided by these simulations are used to perform reconstructions via both the standard ODT algorithms and the proposed technique for performance comparisons. The impact of noise on the performance of the proposed technique is examined. The modified wideband version of the novel algorithm, which can handle the fusion of available multi-frequency data in the special case of a dispersionless scatterer, is exemplified. In all simulation scenarios, the object of interest is probed with plane waves. Corresponding scattered fields are measured with two distinct CCD cameras. The first camera is located 120 µm from the center of the object support and has a pixel resolution of 0.4 µm. The near/middle-zone data collected with this camera are used to perform reconstructions with the FBP algorithm, under the

53

CHAPTER 5. Simulations

54

Born and Rytov approximations. The second camera is located 3 cm from the center of the object support and has a pixel resolution of 100 µm. The far-zone data collected with this camera are used to perform reconstructions with both the FBDT algorithm and the proposed technique. Both cameras consist of 1001 pixels and subtend a total forward-view angle of ∼ 1180 . For quantitative comparisons, relative mean-squared error (RMSE) is defined in terms of the actual and reconstructed object functions Oa and Or : RR RMSE =

object

RR

|Or − Oa |2 ds

|Oa |2 ds object

.

(5.1)

RMSE values are computed and tabulated for all scenarios, probing frequencies and investigated inversion algorithms.

5.1

Circularly Symmetric, Multilayered, Lossy Dielectric Cylinder

In the first test scenario, five-layered, circularly symmetric, lossy dielectric cylinders under plane-wave illumination are considered. The forward solution to this type of scattering problem was summarized in Chapter 4. Before the real simulation scenario, the implementation of the FBP algorithm is validated first with an extremely weak, then with a stronger but relatively still weak, two five-layered cylinders. Refractive indices of the layers of the weaker cylinder are 1.0002 − 0.00002j, 1.0001 − 0.0001j, 1.00015 − 0.00004j, 1.00005 − 0.00008j and 1.0001 − 0.00006j, from the innermost to the outermost layer. The stronger cylinder exhibits ten times more contrast with respect to the common,


55

free-space background. Both cylinders have the same spatial dimensions. The radius of the inner core is 4 micrometers (µm), and the successive annular shells are 5, 3, 5 and 3 µm thick, which results in an object support radius of 20 µm. Reconstruction results for these validation cases are shown in Figs. 5.1-5.2. −4

5

−4

x 10

0 Actual FBP−Rytov FBP−Born

3 2

Actual FBP−Rytov FBP−Born

−0.5 Imag{O(ρ)}

Real{O(ρ)}

4

x 10

1

−1 −1.5 −2

0 0

5

10 ρ (µm)

15

−2.5

20

0

5

(a)

10 ρ (µm)

15

20

(b)

Figure 5.1: Reconstructions of the real (a) and imaginary (b) parts of O(ρ) with FBP at λ = 0.75 µm (Extremely weak scatterer).

−3

5

−3

x 10

0 Actual FBP−Rytov FBP−Born

3 2 1 0 0

Actual FBP−Rytov FBP−Born

−0.5 Imag{O(ρ)}

Real{O(ρ)}

4

x 10

−1 −1.5 −2

5

10 ρ (µm)

(a)

15

20

−2.5

0

5

10 ρ (µm)

15

20

(b)

Figure 5.2: Reconstructions of the real (a) and imaginary (b) parts of O(ρ) with FBP at λ = 0.75 µm (Weak scatterer).


56

The cylinder used in the real simulation scenario has the same spatial dimensions as the previous validation cases, but it is much stronger. Refractive indices of the layers for this case are 1.02 − 0.002j, 1.01 − 0.01j, 1.015 − 0.004j, 1.005−0.008j and 1.01−0.006j, from the innermost to the outermost layer. It is probed in free-space background, with plane waves at 4 distinct frequencies corresponding to the free-space wavelengths of 1.5 µm, 1.25 µm, 1 µm and 0.75 µm. With the a priori information that the investigated structure has no azimuthal variation, only a single plane wave is used to illuminate the layered cylinder at each frequency, as was the case for the validation scenarios. The recorded near-zone and far-zone scattered fields are utilized to perform the reconstructions. For the proposed method, M and N parameters appearing in Eq. 2.18 are chosen to be [ka] + 1 and 0 respectively. Reconstructions with all four ODT techniques and at all four probing frequencies are shown in Figs. 5.3-5.6. The qualitative superiority of the proposed technique over the standard ones is clearly seen in these figures. Layers are well resolved with fairly correct thickness and object function values. The quality of the reconstructions deteriorates as the probing frequency is increased, starting from the center of the cylinder, as the underlying CMM approximation starts to fail. However, due to the failure of the underlying Born and Rytov approximations, standard ODT algorithms fail to provide any useful information about the investigated structure, especially as the frequency is increased. The difference between the reconstructions performed with FBDT and Born-based FBP algorithms is infinitesimal as expected.


57

RMSE values for this scenario, computed and given in Table 5.1 and Fig. 5.7, demonstrate the clear quantitative superiority of the novel technique over the standard ones, among which the FBP algorithm under the Rytov approximation performs best. 0.05

0.01 Actual Proposed FBP−Rytov FBP−Born FBDT

0.03

0 Imag{O(ρ)}

Real{O(ρ)}

0.04

0.02 0.01

Actual Proposed FBP−Rytov FBP−Born FBDT

−0.01

−0.02 0 −0.01 0

5

10 ρ (µm)

15

−0.03

20

0

5

(a)

10 ρ (µm)

15

20

(b)

Figure 5.3: Reconstructions of the real (a) and imaginary (b) parts of O(ρ) with various algorithms at λ = 1.5 µm.

0.05


0.03

0 Imag{O(ρ)}

Real{O(ρ)}

0.04

0.02 0.01


−0.01

−0.02 0 −0.01 0

5

10 ρ (µm)

(a)

15

20

−0.03

0

5

10 ρ (µm)

15

20

(b)



58

0.05


0.03

0 Imag{O(ρ)}

Real{O(ρ)}

0.04

0.02 0.01


−0.01

−0.02 0 −0.01 0

5

10 ρ (µm)

15

−0.03

20

0

5

(a)

10 ρ (µm)

15

20

(b)

Figure 5.5: Reconstructions of the real (a) and imaginary (b) parts of O(ρ) with various algorithms at λ = 1 µm. 0.05


0.03

0 Imag{O(ρ)}

Real{O(ρ)}

0.04

0.02 0.01


−0.01

−0.02 0 −0.01 0

5

10 ρ (µm)

15

20

−0.03

0

5

(a)

10 ρ (µm)

15

20

(b)


Proposed FBP (Rytov) FBP (Born) FBDT

1.5 µm 0.0196 0.1909 0.4714 0.4718

1.25 µm 0.0208 0.3142 0.5615 0.5641

1 µm 0.0284 0.4056 0.6682 0.6724

0.75 µm 0.0617 1.1162 0.7747 0.7811

Table 5.1: RMSE values for the circularly symmetric lossy cylinder.


59

1.4 Proposed 1.2

FBP (Rytov) FBP (Born) FBDT

RMSE

1

0.8

0.6

0.4

0.2

0

1.5

1.25

1

0.75

Wavelength (µm)

Figure 5.7: Same RMSE values in bar graph format.

5.2

Lossless General Phantom

In the second test scenario, a more general 2D lossless dielectric object under plane-wave illumination is considered. The refractive indices of the various components within this object vary between 1 and 1.02, and the object support has a radius of 20 µm. The object function profile for this scatterer is shown in Fig. 5.8. Employed probing frequencies are the same as the previous layered cylinder case. This time, however, the investigated object is probed with 24 distinct plane waves of incidence angles 150 apart at each frequency as opposed to the single incident plane wave of the previous case, since an azimuthal symmetry is no longer present. The recorded scattered fields are utilized to perform the reconstructions with the proposed, FBP and FBDT techniques. For the proposed method, M and N parameters are chosen to be 40 and 60 respectively, satisfying


60

20

0.04

Transverse Distance ( µm)

10

0.03

5 0

0.02

−5 −10

0.01

Relative Permittivity Contrast

15

−15 −20 −20

−15

−10

−5 0 5 10 Transverse Distance ( µm)

15

20

Figure 5.8: Object function profile Opha for the lossless dielectric phantom.

N ≈ π2 M . The a priori information that the probed object is lossless is taken into account during the inversions with all the investigated techniques. Reconstructions with all four ODT algorithms and at all four probing frequencies are shown in Figs. 5.9-5.16. The forward solution to this scattering problem is implemented with the FDTD method. A brief description of the FDTD method and details concerning its implementation can be found in Chapter 4. The qualitative superiority of the proposed technique over the standard ones at each probing frequency is clearly seen in these figures. The correct size, shape, location and object function value of each component within the investigated region are determined with the proposed technique at all frequencies.


61

0.04

0.03

0.02

0.01

(a)

(b)

Figure 5.9: Reconstruction of Opha from far-zone observations at λ = 1.5 µm, using the proposed technique (a) and FBDT (b).

0.04

0.03

0.02

0.01

(a)

(b)

Figure 5.10: Reconstruction of Opha from near-zone observations at λ = 1.5 µm with FBP, using Rytov (a) and Born (b) approximations.


62

0.04

0.03

0.02

0.01

(a)

(b)


0.04

0.03

0.02

0.01

(a)

(b)



63

0.04

0.03

0.02

0.01

(a)

(b)

Figure 5.13: Reconstruction of Opha from far-zone observations at λ = 1 µm, using the proposed technique (a) and FBDT (b).

0.04

0.03

0.02

0.01

(a)

(b)

Figure 5.14: Reconstruction of Opha from near-zone observations at λ = 1 µm with FBP, using Rytov (a) and Born (b) approximations.


64

0.04

0.03

0.02

0.01

(a)

(b)


0.04

0.03

0.02

0.01

(a)

(b)



65

Although the reconstructions obtained with the standard techniques are slightly informative at λ = 1.5 µm, they quickly deteriorate as the frequency is increased. What we witness with the Born-based techniques in these figures as the frequency is increased is actually one of the most severe limitations of the Born approximation, called the real-imaginary part mixing [22]. This problem exhibits itself as total blurring at the transition stage in Fig. 5.13(b) and Fig. 5.14(b), and as several white regions in Fig. 5.15(b) and Fig. 5.16(b), which are informative for the size, shape and location of some, but not all, components with totally wrong object function values. The proposed technique based on the CMM approximation, on the other hand, fails gradually as the frequency is increased and always performs better than the FBP and FBDT algorithms. RMSE values for this scenario, computed and given in Table 5.2 and Fig. 5.17, demonstrate the clear quantitative superiority of the proposed technique over the standard ODT algorithms among which, once again, the FBP algorithm under the Rytov approximation performs best. The FBP algorithm under the Born approximation performs slightly better than the FBDT algorithm as expected. Proposed FBP (Rytov) FBP (Born) FBDT

1.5 µm 0.1834 0.2256 0.2887 0.3092

1.25 µm 0.1709 0.2701 0.4002 0.4195

1 µm 0.1802 0.3565 0.6642 0.6667

0.75 µm 0.2287 0.5795 1.2517 1.1927

Table 5.2: RMSE values for the lossless general phantom.


66

1.4 Proposed 1.2

FBP (Rytov) FBP (Born) FBDT

RMSE

1

0.8

0.6

0.4

0.2

0

1.5

1.25

1

0.75

Wavelength (µm)


For a given object, a finite-length receiver array (CCD camera) located a constant distance from the center of the object support, and a set of I probing monochromatic plane waves of the same frequency but distinct incidence angles, the quality of the reconstruction performed with this novel technique depends solely on the frequency of operation. As the frequency is increased, the negative effect of the finite length of the receiver array on the computation of αu parameters decreases. Moreover, the total number of equations in the final equation system increases, allowing a larger number of singular values satisfying a given cut-off criterion to be used while solving the ill-posed equation system with TSVD. However, the underlying CMM approximation fails gradually as the frequency is increased, limiting the accuracy of the proposed model. The trade-off between these factors leads to the existence of an optimum frequency


67

of reconstruction. One can clearly verify this conclusion examining the reconstruction results presented in this section, which indicates that λ = 1.25 µm is the optimum reconstruction wavelength among the ones employed.

5.2.1

Multi-frequency Reconstruction

As explained in Chapter 2, the modified version of the proposed technique can handle the fusion of available multi-frequency data, provided that the object of interest exhibits no dispersion within the investigation spectrum. This wideband inversion scheme is illustrated below, using the lossless, dispersionless phantom considered in this section. For this multi-frequency reconstruction example, we use a subset of the available simulation results. We assume that the same object is probed at the same 4 frequencies. At each frequency, a reduced set of 8 distinct plane waves of incidence angles separated by 450 are used to illuminate the object. Then these multispectral measurements are combined according to the described technique and a reconstruction of the probed object is obtained. This result is compared with the optimum monochromatic reconstruction at λ = 1.25 µm. Both inversion results are shown in Fig. 5.18. First n singular values S1 , S2 , ...Sn such that S1 Sn

≤ 50 are used in the computation of pseudo-inverses while performing both

reconstructions. Reconstruction sensitivity parameters M and N are chosen to be 40 and 60, respectively, for both cases. RMSE values for the wideband and monochromatic reconstructions are computed and found to be 0.2150 and 0.2872, respectively, indicating a 25% improvement. The wideband reconstruction of the investigated structure is superior to the


68

monochromatic one in every aspect. All components of the of the phantom, especially the ellipses, are well resolved with fairly correct size, shape, location and object function values in the wideband reconstruction result, whereas they are rather smeared and indistinguishable in the monochromatic inversion. 0.04

0.03

0.02

0.01

(a)

(b)

Figure 5.18: Optimum single-frequency reconstruction (a) at λ = 1.25 µm (I = 8, T = 1), versus multi-frequency reconstruction (b) at λ = 1.5, 1.25, 1 and 0.75 µm (I = 8, T = 4)

Whenever applicable, computationally and experimentally feasible, the wideband reconstruction technique should be preferred since it performs better than the monochromatic one, even if the monochromatic technique is employed at the optimum inversion frequency among the available ones. The wideband technique also relieves the burden of performing separate monochromatic reconstructions at each available frequency and trying to compare them to choose the best one.

5.2.2

Noise Considerations

Robustness of the proposed technique is tested at various noise levels using the available simulation results employed in the preceding sections. The amplitude


69

of the noise (|X(ξ)|) added to the noise-free scattered electric field Es (ξ) at each observation point (pixel) location ξ along the camera is modeled as the absolute value of a zero-mean Gaussian random variable, with a variance proportional to |Es (ξ)|2 . The phase of the noise is modeled as another independent random variable with uniform distribution in [0,2π], indicating no preference. The proportionality constant between |Es (ξ)|2 and the variance of |X(ξ)| determines the ratio of the noise-free signal power to the interfering noise power, integrated (summed up) along the camera [39]: |X(ξ)| ∼ |N (0, 100.1×SNR × |Es (ξ)|2 )| ∼ 100.05×SNR × |Es (ξ)| × |N (0, 1)|.

(5.2)

Here, N (µ, σ 2 ) is a normally distributed random variable with mean µ and variance σ 2 , and SNR is the signal-to-noise ratio in decibels (dB), defined by ·R ¸ |Es (ξ)|2 dξ camera SNR = 10 log R (dB). |X(ξ)|2 dξ camera

(5.3)

As an example, synthetic, noise-free far-zone scattered electric field, its distorted version contaminated with noise of 5 dB SNR, and corresponding far-zone scattered field patterns along the far-zone CCD camera for the λ = 1 µm and φi = 0 case are given in Fig. 5.19. 20, 10 and 5 dB-SNR noise signals are generated and used to distort the synthetic pure far-zone observations at all four probing frequencies and for all 24 distinct views. These noisy data are then processed with the proposed technique, and the inversion results shown in Figs. 5.20-5.23 are obtained.


70

0.4

0.07 Noisy Noise−free Noise

0.3

0.05 0.04 |f|

|Es|

Noisy Noise−free Noise

0.06

0.2

0.03 0.02

0.1

0.01 0 400

450

500 CCD Pixel

550

600

0 400

450

500 CCD Pixel

(a)

550

600

(b)

Figure 5.19: Noisy (SNR = 5 dB) and noise-free data along the receiver array at λ = 1 µm: (a) Scattered electric field and (b) far-zone scattered field pattern. 0.04 0.03 0.02 0.01

(a)

(b)

(c)

Figure 5.20: Reconstruction with the proposed technique at λ = 1.5 µm from noisy data: SNR = (a) 20, (b) 10 and (c) 5 dB. 0.04 0.03 0.02 0.01

(a)

(b)

(c)

Figure 5.21: Reconstruction with the proposed technique at λ = 1.25 µm from noisy data: SNR = (a) 20, (b) 10 and (c) 5 dB.


71

0.04 0.03 0.02 0.01

(a)

(b)

(c)

Figure 5.22: Reconstruction with the proposed technique at λ = 1 µm from noisy data: SNR = (a) 20, (b) 10 and (c) 5 dB.

0.04 0.03 0.02 0.01

(a)

(b)

(c)

Figure 5.23: Reconstruction with the proposed technique at λ = 0.75 µm from noisy data: SNR = (a) 20, (b) 10 and (c) 5 dB.

RMSE values for all three SNR levels and four probing frequencies are computed and given in Table 5.3. RMSE values for the noise-free case are repeated for convenience. ∞ 20 dB 10 dB 5 dB

1.5 µm 0.1834 0.1858 0.1863 0.2004

1.25 µm 0.1709 0.1752 0.1914 0.2048

1 µm 0.1802 0.1845 0.1953 0.2192

0.75 µm 0.2287 0.2387 0.2510 0.2612

Table 5.3: RMSE values for the proposed technique at various wavelengths and SNR levels.


72

When these reconstruction results and RMSE values are examined, one notices the extraordinary noise rejection capability of the proposed technique, even in the presence of noise as high as 5 dB SNR. The reason for this surprising property lies in the fact the angular Fourier transform of the noise-free far-zone scattered field pattern is band limited (αu ≈ 0 for |u| > ka), and this property is exploited in the novel technique by discarding αu parameters for |u| > ka. However, no such constraint applies to the noise. While all the power content of the noise-free signal is concentrated on the [−ka,ka] interval, that of the noise is often distributed evenly along all its possible α eu values, limiting the actual data distortion experienced by the proposed technique. This effect can be clearly seen when the αu parameters corresponding to the noise-free and noisy data displayed in Fig. 5.19 are computed and examined. These coefficients are shown in Fig. 5.24. Noisy Noise−free Noise

1

|αu|

0.8

0.6

0.4

0.2

0 −300

−200

−100

0

100

200

300

u

Figure 5.24: Corresponding αu parameters for noisy and noise-free data.


73

³ P∞ ´ |α2u | We note that, while 10 log Pu=−∞ is still 5 dB for this case, the effective ∞ 2 |e αu | u=−∞ µ P[ka] ¶ |α2u | u=−[ka] SNR given by 10 log P[ka] is ∼16 dB. 2 u=−[ka]

|e αu |

To a lesser degree, the FBP algorithm possesses some noise rejection capability as well, mainly due to the filtering and backpropagation steps involved (Eqs. 3.2b-3.2a). As an example, reconstruction results obtained with FBP from 5 dB-SNR noisy data for the λ = 1.5 µm case are shown in Figs. 5.25-5.26. Corresponding RMSE values are given in Table 5.4 0.04 0.03 0.02 0.01

(a)

(b)

(c)

Figure 5.25: Reconstruction with FBP (Born) at λ = 1.5 µm from noisy data: SNR = (a) 20, (b) 10 and (c) 5 dB.

0.04 0.03 0.02 0.01

(a)

(b)

(c)

Figure 5.26: Reconstruction with FBP (Rytov) at λ = 1.5 µm from noisy data: SNR = (a) 20, (b) 10 and (c) 5 dB.

The FBDT algorithm introduced in Chapter 2 has no noise rejection capability at all, since it directly processes the noisy far-zone scattered field pattern,


74

without any filtering stage involved. It is therefore omitted in this section. ∞ 20 dB FBP (Born) 0.2887 0.2812 FBP (Rytov) 0.2256 0.2269

10 dB 0.3025 0.2524

5 dB 0.3445 0.3227

Table 5.4: RMSE values for FBP at λ = 1.5 µm and various SNR levels.

5.3

Embryo Models with Miscellaneous Mitochondrial Distributions

Two-cell embryo models involving four typical types of mitochondrial distributions (homogeneous, aggregated, perinuclear and cortical distributions) are prepared and used in the simulations. Embryos are modeled as rotated ellipses including two cells with irregular boundaries and an extracellular region. Each cell consists of cytoplasm, a circular nucleus and specifically distributed mitochondria. The mitochondria are modeled as randomly oriented and sized, nonoverlapping ellipses. Highly refractive, but extremely thin embryonic, cellular and nuclear membranes are added. Four embryo models differ from each other only in their mitochondrial distributions. They are all assumed to be embedded in a background medium with dielectric characteristics of the cytoplasm. Approximate size and refractive index values used in the models are selected from the related literature [18, 60] and summarized in Table 5.5. The object function profiles of the prepared embryo models are given in Fig. 5.27.


75

0.07

0.06

0.05

(a)

(b)

0.04

0.03

0.02

0.01

(c)

(d)

Figure 5.27: Object function profiles for the embryo models involving four typical mitochondrial distributions: (a) homogeneous (Ohom ), (b) aggregated (Oagg ), (c) perinuclear (Oper ) and (d) cortical (Ocor ).


76

Constituent Mitochondria Nucleus Cytoplasm Membranes Extracellular

Refractive Index 1.42 1.40 1.37 1.46 1.39

Dimensions and Ratios Embryo (full-axis) 40.5×45 µm Nuclei (diameter) 10.8 µm Mitochondria (mean full-axis) 0.4×1 µm Mitochondria-Embryo Ratio ∼15% Membranes (thickness) 10 nm Table 5.5: Embryo Modeling Parameters. Each embryo model is computationally probed at 3 distinct frequencies corresponding to the free-space wavelengths of 2, 1.5 and 1 µm, using the FDTD method. At each frequency, the investigated model is probed with 24 distinct plane waves of incidence angles 150 apart. The collected synthetic scattered field data are used to reconstruct the sub-cellular structure with the proposed, FBP and FBDT techniques. For the proposed technique, reconstruction parameters M and N are chosen to be 50 and 75, satisfying N ≈ values S1 , S2 , ...Sn such that

S1 Sn

π M. 2

First n singular

≤ 45, 35 and 25 are used in the computation

of pseudo-inverses while performing the reconstructions via TSVD at λ = 2, 1.5 and 1 µm, respectively. Reconstructions for all four embryo models and at all three probing frequencies are shown in Figs. 5.28-5.51.


77

0.07 0.06 0.05 0.04 0.03 0.02 0.01

(a)

(b)

0

Figure 5.28: Reconstruction of Ohom from far-zone observations at λ = 2 µm, using the proposed technique (a) and FBDT (b).

0.07 0.06 0.05 0.04 0.03 0.02 0.01

(a)

(b)

0

Figure 5.29: Reconstruction of Ohom from near-zone observations at λ = 2 µm with FBP, using Rytov (a) and Born (b) approximations.


78

0.06 0.05 0.04 0.03 0.02 0.01

(a)

(b)

Figure 5.30: Reconstruction of Ohom from far-zone observations at λ = 1.5 µm, using the proposed technique (a) and FBDT (b).

0.06 0.05 0.04 0.03 0.02 0.01

(a)

(b)

Figure 5.31: Reconstruction of Ohom from near-zone observations at λ = 1.5 µm with FBP, using Rytov (a) and Born (b) approximations.


79

0.05

0.04

0.03

0.02

0.01

(a)

(b)

0

Figure 5.32: Reconstruction of Ohom from far-zone observations at λ = 1 µm, using the proposed technique (a) and FBDT (b).

0.05

0.04

0.03

0.02

0.01

(a)

(b)

0

Figure 5.33: Reconstruction of Ohom from near-zone observations at λ = 1 µm with FBP, using Rytov (a) and Born (b) approximations.


80

0.07 0.06 0.05 0.04 0.03 0.02 0.01

(a)

(b)

0

Figure 5.34: Reconstruction of Oagg from far-zone observations at λ = 2 µm, using the proposed technique (a) and FBDT (b).

0.07 0.06 0.05 0.04 0.03 0.02 0.01

(a)

(b)

0

Figure 5.35: Reconstruction of Oagg from near-zone observations at λ = 2 µm with FBP, using Rytov (a) and Born (b) approximations.


81

0.06 0.05 0.04 0.03 0.02 0.01

(a)

(b)

Figure 5.36: Reconstruction of Oagg from far-zone observations at λ = 1.5 µm, using the proposed technique (a) and FBDT (b).

0.06 0.05 0.04 0.03 0.02 0.01

(a)

(b)

Figure 5.37: Reconstruction of Oagg from near-zone observations at λ = 1.5 µm with FBP, using Rytov (a) and Born (b) approximations.


82

0.05

0.04

0.03

0.02

0.01

(a)

(b)

0

Figure 5.38: Reconstruction of Oagg from far-zone observations at λ = 1 µm, using the proposed technique (a) and FBDT (b).

0.05

0.04

0.03

0.02

0.01

(a)

(b)

0

Figure 5.39: Reconstruction of Oagg from near-zone observations at λ = 1 µm with FBP, using Rytov (a) and Born (b) approximations.


83

0.07 0.06 0.05 0.04 0.03 0.02 0.01

(a)

(b)

0

Figure 5.40: Reconstruction of Oper from far-zone observations at λ = 2 µm, using the proposed technique (a) and FBDT (b).

0.07 0.06 0.05 0.04 0.03 0.02 0.01

(a)

(b)

0

Figure 5.41: Reconstruction of Oper from near-zone observations at λ = 2 µm with FBP, using Rytov (a) and Born (b) approximations.


84

0.06 0.05 0.04 0.03 0.02 0.01

(a)

(b)

Figure 5.42: Reconstruction of Oper from far-zone observations at λ = 1.5 µm, using the proposed technique (a) and FBDT (b).

0.06 0.05 0.04 0.03 0.02 0.01

(a)

(b)

Figure 5.43: Reconstruction of Oper from near-zone observations at λ = 1.5 µm with FBP, using Rytov (a) and Born (b) approximations.


85

0.05

0.04

0.03

0.02

0.01

(a)

(b)

0

Figure 5.44: Reconstruction of Oper from far-zone observations at λ = 1 µm, using the proposed technique (a) and FBDT (b).

0.05

0.04

0.03

0.02

0.01

(a)

(b)

0

Figure 5.45: Reconstruction of Oper from near-zone observations at λ = 1 µm with FBP, using Rytov (a) and Born (b) approximations.


86

0.07 0.06 0.05 0.04 0.03 0.02 0.01

(a)

(b)

0

Figure 5.46: Reconstruction of Ocor from far-zone observations at λ = 2 µm, using the proposed technique (a) and FBDT (b).

0.07 0.06 0.05 0.04 0.03 0.02 0.01

(a)

(b)

0

Figure 5.47: Reconstruction of Ocor from near-zone observations at λ = 2 µm with FBP, using Rytov (a) and Born (b) approximations.


87

0.06 0.05 0.04 0.03 0.02 0.01

(a)

(b)

Figure 5.48: Reconstruction of Ocor from far-zone observations at λ = 1.5 µm, using the proposed technique (a) and FBDT (b).

0.06 0.05 0.04 0.03 0.02 0.01

(a)

(b)

Figure 5.49: Reconstruction of Ocor from near-zone observations at λ = 1.5 µm with FBP, using Rytov (a) and Born (b) approximations.


88

0.05

0.04

0.03

0.02

0.01

(a)

(b)

0

Figure 5.50: Reconstruction of Ocor from far-zone observations at λ = 1 µm, using the proposed technique (a) and FBDT (b).

0.05

0.04

0.03

0.02

0.01

(a)

(b)

0

Figure 5.51: Reconstruction of Ocor from near-zone observations at λ = 1 µm with FBP, using Rytov (a) and Born (b) approximations.


89

RMSE values for all four distinct embryo models, four distinct inversion algorithms being tested, and three probing frequencies employed, are computed and given in Table 5.6 and Fig. 5.52. Examining the reconstructions shown in Figs. 5.28-5.51, one can clearly notice the outline of the embryo, distinguish the cell nuclei and determine the type of mitochondrial distribution using the proposed technique, with the exception of the perinuclear one at λ = 1 µm. The reconstructions of the embryo models involving homogeneous and aggregated mitochondrial distributions are especially remarkable. One can easily distinguish between these two most challenging distributions with the proposed technique, whereas such a differentiation is hard to do with the standard algorithms. We also note that the quality and accuracy of the reconstructions provided by the novel algorithm at λ = 2 µm are so high that the object function values can be used to determine the refractive indices of the cellular constituents, like that of the nucleus. Performance of the employed technique deteriorates as the probing frequency increases, due to the gradual failure of the underlying CMM approximation. Among the conventional techniques, once again the FBP algorithm under the Rytov approximation is the closest rival of the proposed algorithm. However, based on these reconstruction results, on can clearly state that the novel technique is superior to its three competitors both qualitatively and quantitatively.


90

λ = 2.0 µm Homogeneous Aggregated Proposed 0.5626 0.4779 FBP (Rytov) 0.6222 0.6238 FBP (Born) 0.7541 0.7472 FBDT 0.7767 0.7681

Perinuclear Cortical 0.5514 0.4117 0.5529 0.5529 0.7836 0.7836 0.7779 0.7779





Table 5.6: RMSE values for various embryo models, wavelengths and reconstruction techniques. FBP (Rytov)

FBP (Born)

FBDT

0.8 0.6 0.4 0.2

RMSE (λ = 1.5 µm)

0 1.6

RMSE (λ = 1.0 µm)

RMSE (λ = 2.0 µm)

Proposed 1

3 2.5 2 1.5 1 0.5 0

1.2 0.8 0.4 0

Homogeneous

Aggregated

Perinuclear

Cortical


Chapter 6

Radial Optimizations

Analytical evaluation of the electric field scattered by an azimuthally symmetric, multilayered, lossy dielectric cylinder under plane-wave illumination was discussed at the beginning of Chapter 4. Based on this discussion, a radial optimization technique has been developed, where one can modify the layer parameters of such a multilayered cylinder to fit the cylinder’s plane-wave scattered field pattern to an observed or desired one. This technique has many potential application areas, ranging from brute-force inversion of azimuthally symmetric objects to the design of lossy coating layers, which can reduce the radar cross section (RCS) of cylindrical objects or increase the fidelity of anechoic chambers used in test facilities. This chapter is dedicated to the derivation and illustration of this novel optimization method.

6.1

Description of the Optimization Problem

For the case of a 2D, z-directed, infinitely long, azimuthally symmetric, M layered dielectric cylinder of radius a under the illumination of a T M z , unitamplitude plane wave with wavenumber k, the scattered electric field outside

91

CHAPTER 6. Radial Optimizations

92

the cylinder and the corresponding far-zone scattered field pattern can be determined with

Es (ρ, φ) =

∞ X

Cn Hn(2) (kρ) exp(jnφ)

n=−∞

≈

[ka] X

en Cn Hn(2) (kρ) cos(nφ)

(6.1)

n=0

and

r f (φ) ≈

[ka] 2j X en j n Cn cos(nφ) πk n=0

(6.2)

where e0 = 1 and en = 2 for n ≥ 1, and the Cn coefficients can be computed via Eq. 4.7a using Richmond’s iterative technique, as explained in Chapter 4. Computation of these Cn coefficients is the solution to the forward problem for a given multilayered cylinder under known illumination. The term “solution” can be misleading here, since no parameters are actually solved for during this effort. They are just evaluated. On the other hand, under the same known plane-wave illumination, the effort to estimate the layer parameters of an azimuthally symmetric, M -layered dielectric cylinder which will yield a set of Cn coefficients matching best with en coefficients in a least-squares sense, is a nonlinear the observed (or desired) C least-squares optimization problem. During this optimization effort, 5M real parameters (real and imaginary parts of the relative layer permittivities and permeabilities, in addition to the layer thicknesses) are manipulated and tried to be estimated to minimize the sum of 2([ka] + 1) real parameters (squares of en − Cn ) for [ka] ≥ n ≥ 0). In mathematical the real and imaginary parts of (C


93

terms, this optimization problem can be expressed with e − C(²1 , ...²m , ....²M ; µ1 , ...µm , ...µM ; ∆1 , ...∆m , ...∆M )||2 min ||C 2

(6.3)

arg{²1 , ...²m , ....²M ; µ1 , ...µm , ...µM ; ∆1 , ...∆m , ...∆M },

where ²m , µm and ∆m are the relative permittivity, relative permeability and the thickness of the mth layer within the cylinder, respectively. Among many nonlinear optimization schemes, gradient-type optimization techniques like the Levenberg-Marquardt or Gauss-Newton algorithms step up with their robustness and efficiency. However, at each iteration step, these techniques involve the evaluation of a Jacobian matrix which, in this case, requires the evaluation of the partial derivatives

∂Cn ∂Cn , ∂²m ∂µm

and

∂Cn . ∂∆m

For small values

of M and [ka], these partial derivatives can be evaluated numerically with finite differences at each iteration step using Richmond’s method directly. As M and [ka] increase, however, this computation becomes prohibitive fast, forcing the use of other techniques requiring no derivative evaluations at the expense of speed, reliability and accuracy. Analytical evaluation of these partial derivatives is therefore a must if one were to use these gradient optimization techniques for real-life problems.

6.2

Analytical Evaluation of the Jacobian Matrix

According to Richmond’s iterative method summarized in Chapter 4, the Cn coefficients can be computed via Eq. 4.7a, which is simplified in notation before


94

being repeated here for convenience: Cn = −j −n

Bn . Bn + jAn

(6.4)

An and Bn can be computed using cascaded matrix multiplications given in Eq. 4.4, which are written explicitly as follows: 



 An  n n n n   = ZM ZM −1 ...Z1 R Bn

(6.5)

where 



 1  Rn =   0    Umn Wmn  n Zm =  Vmn Xmn

(6.6a)

(6.6b) 0

0

Umn = µm km+1 Jn (km ρm )Yn (km+1 ρm ) − µm+1 km Jn (km ρm )Yn (km+1 ρm ) 0

(6.6c)

0

Vmn = µm+1 km Jn (km+1 ρm )Jn (km ρm ) − µm km+1 Jn (km+1 ρm )Jn (km ρm ) (6.6d) 0

0

Wmn = µm km+1 Yn (km ρm )Yn (km+1 ρm ) − µm+1 km Yn (km ρm )Yn (km+1 ρm ) (6.6e) 0

0

Xmn = µm+1 km Yn (km ρm )Jn (km+1 ρm ) − µm km+1 Yn (km ρm )Jn (km+1 ρm ). (6.6f) The partial derivatives

∂Cn ∂Cn , ∂²m ∂µm

and

∂Cn ∂ρm

can be evaluated with

¶ Bn Cn = −j Bn + jAn An Bn − An Bn = j −n+1 (Bn + jAn )2 µ

−n

(6.7)


95

where the overbar operator denotes the partial derivative with respect to any layer parameter. Inspecting Eqs. 6.6c-6.6f, one draws the following conclusions: n . 1. ρm appears only in Zm n n and Zm−1 for m > 1. 2. ²m and µm appear only in Zm

3. ²1 and µ1 appear only in Z1n . Consequently, An and Bn can be evaluated with 



 An  n n n n ZM   = ZM −1 ...Z1 R Bn   n n ...Z n Rn , for ρ  ZM ...Zm  m 1      n  ZM ...Z2n Z1n Rn , for ²1 and µ1       (Z n ...Z n )(Z n Z n + Z n Z n )Rn , for ²2 and µ2 2 1 3 2 1 M =  n n n n n Zn  (ZM ...Zm+1 )(Zm  m−1 + Zm Zm−1 )×      n  (Zm−2 ...Z1n Rn ), for ²m and µm (M > m > 2)       (Z n Z n + Z n Z n )(Z n ...Z n Rn ), for ²M and µM . 1 M −2 M M −1 M M −1 ∂Cn ∂∆m

can be expressed in term of the computed

∂Cn ∂ρm

(6.8)

values as

M

X ∂Cn ∂Cn = . ∂∆m t=m ∂ρt

(6.9)

First and second derivatives of the cylindrical Bessel functions involved in


96

the computations can be evaluated using dXn (υ) n = Xn−1 (υ) − Xn (υ) dυ υ µ ¶ 2 1 d Xn (υ) n2 + n = − Xn−1 (υ) + − 1 Xn (υ) dυ 2 υ υ2

(6.10a) (6.10b)

where Xn is either Jn or Yn with possibly complex argument υ. Instead of dealing with intermediate Cn parameters, one is generally more interested in generating the scattered electric field Es (ρ, φ) or the far-zone scates (ρ, φ) or fe(φ) in tered field pattern f (φ) which matches best with the intended E a least-squares sense at selected (ρ, φ) points or φ directions. These optimization problems can be expressed with e s − Es (²1 , ...²m , ....²M ; µ1 , ...µm , ...µM ; ∆1 , ...∆m , ...∆M )||2 min ||E 2

(6.11a)

min ||e f − f (²1 , ...²m , ....²M ; µ1 , ...µm , ...µM ; ∆1 , ...∆m , ...∆M )||22

(6.11b)

arg{²1 , ...²m , ....²M ; µ1 , ...µm , ...µM ; ∆1 , ...∆m , ...∆M }.

Evaluation of the partial derivatives E s (ρ, φ) and f (φ) with respect to the layer parameters ²m , µm and ∆m can be accomplished with

E s (ρ, φ) =

[ka] X

en C n Hn(2) (kρ) cos(nφ)

(6.12a)

n=0

r f (φ) =

[ka] 2j X n j en C n cos(nφ) πk n=0

while forming the corresponding Jacobian matrices.

(6.12b)


97

Richmond’s iterative technique can be easily modified to handle a metallic inner core of radius ρ0 . The only necessary modifications to the technique itself, and to the above discussions and formulations about the evaluation of the partial derivatives are   Rn =  and



 1 1 ρ0 ) − YJnn(k (k1 ρ0 )

 



 An  n n n n  = (ZM ...Z2 )(Z1n R + Z1 Rn ), for ²1 and µ1 .  Bn

6.3

(6.13)

(6.14)

Applications

The radial optimization technique derived in this chapter can be applied to the direct (brute-force) inversion of azimuthally symmetric cylindrical scatterers. Its variant which is capable of handling a metallic inner core (or outer wall) can be used to design microwave absorber layers exhibiting electric and/or magnetic loss. One can utilize these absorber layers to minimize the backscattering either from a cylindrical target (RCS reduction) or from the metallic walls of a cylindrical test chamber (anechoic chamber design), with proper coating. Brute-force inversion and RCS reduction (cloaking) applications are illustrated with simulations in this section. Designing an anechoic chamber is explained conceptually. In all the simulation examples below, results of the employed radial optimization technique are subject to the following standard constraints, imposing the use of lossy, naturally available materials only. Note that the conditions on the imaginary parts of the layer parameters would be the opposite, if exp(−iωt)


98

time dependence were used in this study. • ∆m ≥ 0 • Re{²m } ≥ 1 • Im{²m } ≤ 0 • Re{µm } ≥ 1 • Im{µm } ≤ 0

6.3.1

Brute-Force (Direct) Inversion of Azimuthally Symmetric Cylindrical Objects

If it is already known (either as an a priori information, or as an experimental deduction) that an object whose structure is investigated exhibits no azimuthal dependence, it can be modeled as a multilayered cylinder. The layer parameters can then be efficiently estimated in such a way that the resulting scattered field matches the observed one. As an example, the azimuthally symmetric, multilayered, lossy dielectric cylinder considered in Chapter 5 is revisited. For the λ = 0.75 µm case, no investigated ODT algorithms, including the one presented in this dissertation, could reconstruct the probed cylinder with satisfactory results as shown in Fig. 5.6. The scattered field data collected with the far-zone CCD camera are now processed with the new optimization method. Note that the type of the optimization problem encountered in this application is defined by Eq. 6.11a. The object of interest is modeled as a 43-layered cylinder of radius 21 µm. 1.025 − 0.01j is used as the relative permittivity of the initial layers. Additional Re{µm } = 1


99

and Im{µm } = 0 conditions are imposed, assuming it is already known that a nonmagnetic object is being dealt with. The almost-perfect reconstruction result shown in Fig. 6.1 is obtained. 0.05

0.01 Proposed ODT Brute−Force Initial Guess Actual

Proposed ODT Brute−Force Initial Guess Actual

0

0.03

Imag{O(ρ)}

Real{O(ρ)}

0.04

0.02 0.01

−0.01

−0.02 0 −0.01 0

5

10 ρ (µm)

15

20

(a)

−0.03

0

5

10 ρ (µm)

15

20

(b)

Figure 6.1: Reconstructions of the real (a) and imaginary (b) parts of O(ρ) with brute-force inversion algorithm at λ = 0.75 µm.

Although this brute-force inversion scheme is interesting from a theoretical point of view, it has little practical value since it applies only to azimuthally symmetric objects. The application illustrated in the next section, however, might have concrete practical results.

6.3.2

Minimization of the Backscattering from Metallic Cylinders (Cloaking)

As mentioned before, the slightly modified version of the new optimization algorithm can be used to design electrically and/or magnetically lossy microwave absorber layers which can be produced at relatively low cost from naturally


100

available materials, under the right optimization constraints. The backscattering from a metallic cylinder can then be minimized at selected far-zone observation points, by coating with these absorber layers. A representation of the described scenario is shown in Fig. 6.2.

Observation Points (Uof,I)

y

x PEC Cylinder Incident Plane Wave

Figure 6.2: Backscattering minimization with cloaking shells.

We note that the type of the optimization problem encountered in this application is defined by Eq. 6.11b, with e f = 0 at the selected backscattering angles φ distributed within [900 , 1800 ] (or identically, [1800 , 2700 ]), assuming φinc = 0. Unlike the inversion case investigated in the previous section, there exists no sensible initial guess for this cloaking optimization, from which the algorithm could easily converge to an optimum solution. To overcome this problem, an extensive initial search with a single, thick layer must be carried out. Since there is only one layer involved in this initial stage, the search will quickly yield


101

the optimum single layer within the search region. Then this layer is divided into two layers of equal thickness, which are used as the initial guesses of a two-layered optimization effort. At the end of the nth such stage, the algorithm yields 2n optimized absorber layers minimizing a quantity proportional to the sum of the backscattered power at the selected observation directions. We define the RCS reduction R(φ), a positive real-valued function of the observation angle φ, to be the reduction of the power backscattered by a cloaked metallic cylinder with respect to the power backscattered by the same uncloaked cylinder, in the direction of an observation angle φ. In terms of the far-zone scattered field pattern f (φ), R(φ) can be computed with ¯ ¯ ¯ fc (φ) ¯ ¯ (dB) ¯ R(φ) = 20 log ¯ fu (φ) ¯

(6.15)

where fc (φ) and fu (φ) are the far-zone scattered field patterns generated by the cloaked and uncloaked cylinders, respectively, under the same plane-wave illumination. To illustrate the technique, a metallic cylinder of radius 1 m which is probed with a plane wave at 600 MHz is considered. The electromagnetic backscattering from this cylinder is minimized with a 16-layer microwave absorber designed with the presented technique. Only the use of lossy materials is allowed during the optimization process. The performance of the designed layered cloak is compared with that of ECCOSORBr AN-79, a commercial off-the-shelf (COTS) microwave absorber suitable for the employed operation frequency. Specifications of the ECCOSORBr AN family products are shown in Fig. 6.3.


102

Figure 6.3: ECCOSORBr AN family microwave absorbers.

RCS reduction R(φ) provided by both coatings are given in Fig. 6.4. The values of the layer parameters optimized with the new algorithm are given in Table 6.1 As shown clearly in Fig. 6.4, the designed layered absorber is vastly superior to the COTS product. The optimized layers are especially successful in minimizing R(φ) within [1700 , 1900 ] interval, practically rendering this metallic cylinder invisible to all monostatic radars operating at 600 MHz. As this last remark implies, the developed technique may easily find use in military applications once extended to handle realistic 2.5D and 3D scenarios.


103

−10

−20

RCS Reduction (dB)

−30

−40

−50

−60 ECCOSORB AN−79

−70

16 Optimized Layers −80 180

190

200

210

220

230

240

Observation Angle (degrees)

Figure 6.4: RCS reduction with ECCOSORBr AN-79 and 16 optimized layers. m ∆m (cm) 1 1.4566 2 1.4601 3 1.4900 4 1.6129 5 1.4870 6 1.4885 7 1.6800 8 1.6410 9 1.4098 10 1.4098 11 1.5827 12 1.2902 13 1.4114 14 1.4522 15 1.4720 16 1.4886

Re{²m } 3.5390 3.5046 3.2919 4.1140 3.3063 3.3252 3.6006 4.7034 8.8022 8.8013 5.1743 4.0845 2.4557 2.5124 2.6733 3.0157

Im{²m } -1.6985 -1.7784 -2.0283 -2.2779 -1.7402 -0.9695 -0.4184 -0.2084 -0.2388 -0.2378 -3.2309 -1.3464 -0.8428 -0.5910 -0.5459 -0.5235

Re{µm } 1.0702 1.0704 1.0713 1.0725 1.0699 1.0698 1.0699 1.0702 1.0722 1.0741 1.0719 1.0707 1.0701 1.0698 1.0698 1.0700

Im{µm } × 103 0 0 0 -0.8454 -0.8358 -0.0016 0 0 0 -0.3994 -1.8410 -2.5148 -1.2774 -0.5040 -0.0213 0

Table 6.1: Values of the layer parameters for the designed absorber.


6.3.3

104

Anechoic Chamber Design

With slight modifications to the described algorithm, cascaded circular lossy absorber layers can be designed and used to coat the inner surface of a metallic cylindrical chamber, in order to reduce the electromagnetic backscattering from its walls. Such a structure would find use in any application where anechoic chambers are of interest. A representation of the described scenario is shown in Fig. 6.5.

y

E t (U ,I )

x Support of Sources and Scatterers

Figure 6.5: A cylindrical metallic anechoic chamber with inner absorber layers.

The total electric field at an observation point (ρ, φ) inside such a metallic cylindrical chamber of radius ρ0 , but outside the concentric circular support of


105

the sources and scatterers of radius a, can be expressed with

Et (ρ, φ) ≈

[ka] X £

¤ Cn Hn(2) (kρ) + Dn Hn(1) (kρ) exp(jnφ)

(6.16)

n=−[ka]

(1)

(2)

where Hn (kρ) and Hn (kρ) represents the incoming (backscattered from the wall) and outgoing cylindrical waves, respectively. This expression can be reorganized as

Et (ρ, φ) ≈

[ka] X

[(Cn + Dn )Jn (kρ) − j(Cn − Dn )Yn (kρ)] exp(jnφ)

(6.17)

n=−[ka]

which yields Bn Cn − Dn = −j An Cn + Dn n j+B Dn An ⇒ γn , = . n Cn j−B An

(6.18)

Since γn = γ−n , one can define Γn = en γn (e0 = 1 and en = 2 for [ka] ≥ n ≥ 1), and then attempt to solve the nonlinear least-squares optimization problem given by min ||Γ(²1 , ...²m , ....²M ; µ1 , ...µm , ...µM ; ∆1 , ...∆m , ...∆M )||22

(6.19)

arg{²1 , ...²m , ....²M ; µ1 , ...µm , ...µM ; ∆1 , ...∆m , ...∆M }.

The partial derivatives

∂Γn ∂Γn , ∂²m ∂µm

and

∂Γn ∂ρm

which are required to construct the


106

corresponding Jacobian matrix can be evaluated with Ã Γn = en

j+ j−

= 2jen

Bn An Bn An

!

An Bn − An Bn , (Bn − jAn )2

(6.20)

where the overbar operator denotes the partial derivative with respect to any layer parameter. Evaluation technique of An and Bn with respect to ²m , µm and ρm are exactly the same as the case of cascaded layers with a metallic inner core of radius ρ0 , given by Eq. 6.8 and modified by Eqs. 6.13-6.14. Only the transition from the ∂Cn values to the evaluation of ∂∆ , which is given by Eq. 6.9, no m P Pm longer applies since ρm 6= ρ0 + m t=1 ∆m , but ρm = ρ0 − t=1 ∆m for this

computed

∂Cn ∂ρm

geometry: M X ∂Γn ∂Γn =− . ∂∆m ∂ρt t=m

Minimization of

P[ka] n=0

(6.21)

|Γn |2 by manipulating layer parameters of the wall

coating of the chamber will surely, and considerably decrease the total amount of power backscattered into the source/scatterer support region. The actual performance of the absorber coating will be scenario-dependent though, since the layers cannot distinguish between the power content of the distinct cylindrical modes, and try to attenuate each mode at the same level of priority.

Chapter 7

Conclusion and Future Work

7.1

Summary

In this dissertation, a novel 2D ODT technique is proposed and compared with the existing ODT algorithms, and a radial optimization technique is introduced and illustrated. The new ODT method is derived starting from the volume equivalence theorem. The far-zone scattered field pattern is first defined and then expressed as a series summation of complex exponentials with corresponding expansion coefficients using equivalent current densities, Green’s function theory and large argument approximations. The relation between these coefficients and the observed fields are clarified. The forward formulation for the modal expansion coefficients is redesigned by expanding the target object function in terms of Fourier-Bessel basis functions with unknown weights to be ultimately determined. The final expression for the resulting linear system of equations is derived. To linearize the actual nonlinear scattering mechanism, various candidates for an approximation to the total field inside the object support are stated. The method is tested with plane-wave scattering first by a multilayered, lossy dielectric cylinder, then by

107

CHAPTER 7. Conclusion and Future Work

108

a more general, lossless, 2D dielectric object with azimuthal structural dependence, and finally by two-cell embryo models involving various mitochondrial distributions. The performance of the proposed method is compared with those of the conventional ODT techniques based on the Born and Rytov approximations, and found to be superior qualitatively and quantitatively. Exceptional noise rejection capability of the new method is demonstrated, and the underlying reasons are explained. A modified version of the technique which can process multispectral data for a wideband reconstruction of a dispersionless object is developed and illustrated with an example. The developed radial optimization technique is the surprising product of a simple observation: Under a known illumination, the rate of change in the field scattered by a layered cylinder due to the rate change in the cylinder layer parameters can be analytically evaluated. This allows the pointwise fast linearization of the actual nonlinear scattering mechanism, fast formation of a Jacobian matrix and hence the use of effective gradient optimization techniques to estimate the layer parameter combinations yielding an observed, or desired scattered field profile. The technique is adapted to, and illustrated with various applications including brute-force inversion, backscattering (RCS) reduction and anechoic chamber designed.

7.2

Significance

To the author’s best knowledge: • The CMM approximation developed in accordance with the new ODT


109

technique is the first single-step linearization scheme which has managed to surpass the performance of the Rytov approximation in diffraction tomography problems. • The introduced ODT technique is superior to all existing non-iterative ODT techniques in reconstruction quality, provided that: 1. The observed data is abundant enough to correctly evaluate αu parameters. 2. The phase of the observations can somehow be obtained. 3. The support radius of the probed object is either known or can be determined. 4. No real-time imaging is requested. • The introduced ODT technique has an outstanding, unmatched noise rejection capability mainly due to the support radius information it exploits, a parameter standard ODT algorithms generally neglect. • The analytical evaluations of the derivatives presented in Chapter 6 paving the way to the introduced radial optimization technique have never been accomplished before. • Although it has little applicability, the brute-force radial reconstruction technique introduced in Chapter 6 is the first example of its kind. Not based on any approximation, the technique can in theory invert any radial object if a good initial guess is available. However, as the azimuthally symmetric cylinder being inverted grows larger and stronger, the initial


110

guess should be closer and closer to the actual object for a successful reconstruction, in order to avoid useless local maxima and minima. • The RCS reduction and anechoic chamber design techniques presented in Chapter 6 are original. Upon being extended to handle realistic 3D scenarios, the novel ODT scheme may find use in various fields of medical imaging and tomographic microscopy, such as the assessment of oocyte quality and embryo viability for IVF as well as other cellular imaging applications. The underlying CMM approximation can be “distorted” just like the Born approximation, to take the properties of a known inhomogeneous background into account. The radial optimization algorithm introduced in Chapter 6 can be adapted to military stealth applications and can be applied to the fidelity improvement of the electromagnetic test chambers. Another interesting application of the developed optimization scheme would be the realization of a layered L¨ uneburg lens, a spherical (cylindrical in 2D) structure with a radially varying refractive index, having two distinct foci: One on the sphere (cylinder) surface and the other at infinity.

7.3

Future Work

The introduced diffraction tomography technique can and should be extended to handle 3D scenarios. The extension is relatively easy for a scalar 3D case, where the conversion from complex exponentials, cylindrical Bessel functions and 2D Green’s function to spherical harmonics, spherical Bessel functions and 3D Green’s function would suffice. The 3D scalar version of the algorithm can


111

be used for acoustical diffraction tomography problems. The extension to the full 3D vectorial case, however, requires substantially more effort. As explained in Chapter 2, the proposed ODT method can be used iteratively to improve the accuracy of the total field approximation (i.e., the accuracy of the Fνi (ρ0 ) values) at each step, improving the quality of the successive reconstructions. The FDFD method is the strongest candidate for the forward solution technique which should be employed at each iteration. The real problem this idea would face in practice would be its slowness. At each iteration, the resulting equation system which needs to be solved for the cmn parameters will be ill-posed, just as the initial system is. The size of the system renders the use of a fast Gram-Schmidt type orthogonalization scheme impractical, because of the accumulated round-off errors arising from the finite-accuracy computer arithmetic. TSVD seems to be the only option to orthogonalize the system and eliminate the equations carrying redundant and unreliable information, and unfortunately SVD is a computationally costly algorithm. Since most real-life ODT experiments involve a known inhomogeneous background such as a glass tube containing the interrogated specimen, the distortedwave version of the introduced CMM approximation should be developed, just like the extension of the first-order Born approximation to DWBA. Presented reconstruction technique should also be modified accordingly to employ this novel linearization scheme. The optimization technique introduced in Chapter 6 should be generalized to handle 2.5D and 3D realistic scenarios. The technique works fine when dealing with materials exhibiting no or slight loss. As the amount of the electric and magnetic losses considered in the optimization layers rises, in other words as the


112

imaginary part of the argument km ρ on which Jn and Yn operate becomes dominant, the technique starts to suffer from 00 , ∞ − ∞ and 0 × ∞ type ambiguities. This problem should be addressed properly in order not to remain limited to the use of low-loss materials in the design of optimized absorber layers.

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