Controlled resolution reconstruction of one-dimensional permittivity profiles [Elektronische Ressource] / von Jaleel Akhtar

Informations

• Publié par: otto-von-guericke-universitat_magdeburg
• Publié le: 01 janvier 2003
• Nombre de lectures: 24
• Langue: English
• Poids de l'ouvrage: 6 Mo

Extrait

Controlled Resolution Reconstruction of
One-Dimensional Permittivity Profiles
Dissertation
zur Erlangung des akademischen Grades
Doktoringenieur
(Dr.-Ing.)
von M.Eng. Md. Jaleel Akhtar
geb. am 26 M¨arz 1971 in Gaya, Indien
genehmigt durch
die Fakulta¨t fur¨ Elektrotechnik und Informationstechnik
der Otto-von-Guericke-Universit¨at Magdeburg
Gutachter:
Prof. Dr.-Ing. A. S. Omar
Prof. Dr.-Ing. habil. Bernd Michaelis
Prof. Dr. V. F. Hanna
Promotionskolloquium am: 03. September 2003Acknowledgements
Seldomisanythingaccomplishedwithouttheassistanceorencouragementofothers.
Thisthesiswouldalsonothavebeeninitspresentformwithoutthesupportofmany
peoplewho,inonewayorother,providedhelptome. Itismyhumbledutytothank
all those people at this stage.
First and foremost, I would like to thank my supervisor, Prof. Dr.-Ing. A. S.
Omar, for his kind guidance throughout this research work. Apart from being an
excellent supervisor, he has also been my mentor and a great source of inspiration.
His positive attitude towards younger researchers, and his unquenchable curiosity
and love for the subject are probably some of the things, which I really admired
in him. I am also grateful to Prof. Dr.-Ing. habil. Bernd Michaelis, University of
Magdeburg, and Prof. Dr. V. F. Hanna, University of Pierre et Marie Curie, Paris,
for kindly agreeing to be referees for this thesis in spite of their hectic schedules.
I would like to thank Deutsche Forschungsgemeinschaft for sponsoring a project
to carry out this research work.
IamalsothankfultomyhostinstitutionsinIndia,CentralElectronicsEngineer-
ing Research Instiute, Pilani, and the Council of Scientific and Industrial Research,
New Delhi, for granting me study leave to pursue the Dr.-Ing. degree.
Thanks are due to all colleague of my institute, who were always quite helpful
during my stay. I would specially like to thank Dr.-Ing. Jstingmeier and Dipl.-
Ing. T. Meyer for many thoughtful discussions and useful suggestions. I am also
thankful to Dr.-Ing. Holub for providing the logistic support throughout my stay
at this institute.
Finally, Iexpressmyappreciationtomyfamilyspeciallymyparents, whosekind
blessings and support have always been with me. I would also like to thank my wife
for being patient enough over the whole period.
(Md. Jaleel Akhtar)Table of Contents
Zusammenfassung V
Abstract VI
List of Symbols VII
1 Introduction 1
1.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The direct problem: electromagnetic scattering theory . . . . . . . . . 6
1.3 The inverse problem: microwave inverse scattering . . . . . . . . . . . 8
1.4 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Microwave Inverse Scattering Theory 18
2.1 Basic formulas and definitions . . . . . . . . . . . . . . . . . . . . . . 18
2.1.1 The electromagnetic field in free space . . . . . . . . . . . . . 18
2.1.2 Interaction of electromagnetic waves with dielectric materials . 19
2.1.2.1 The electric dipole moment . . . . . . . . . . . . . . 20
2.1.2.2 The macroscopic electric field due to polarized di-
electric molecules . . . . . . . . . . . . . . . . . . . . 21
2.1.2.3 The concept of general electric flux density and the
local electric field . . . . . . . . . . . . . . . . . . . . 22
2.1.2.4 The electronic polarization. . . . . . . . . . . . . . . 24
2.1.2.5 The ionic polarization . . . . . . . . . . . . . . . . . 25
2.1.2.6 The orientational polarization . . . . . . . . . . . . . 27
2.1.2.7 The concept of relative permittivity. . . . . . . . . . 28
2.1.2.8 The effect of time-varying electromagnetic field . . . 29
2.1.3 The scattering parameters . . . . . . . . . . . . . . . . . . . . 30
2.1.4 The multiple reflections from inhomogeneous transmission lines 33
2.1.5 The integral transform . . . . . . . . . . . . . . . . . . . . . . 37
2.2 The inverse problem from the mathematical point of view . . . . . . . 40
2.3 The common methods for solving inverse scattering problems . . . . . 45
2.3.1 The Born approximation . . . . . . . . . . . . . . . . . . . . . 45
2.3.2 The distorted Born approximation . . . . . . . . . . . . . . . 48
2.3.3 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . 49
2.3.4 The Riccati-equation approach . . . . . . . . . . . . . . . . . 52
3 The Direct Problem Formulation 54
3.1 The planar geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.1.1 TEM illumination . . . . . . . . . . . . . . . . . . . . . . . . . 54
IITABLE OF CONTENTS III
3.1.1.1 The discontinuous structure . . . . . . . . . . . . . . 56
3.1.1.2 The continuous media . . . . . . . . . . . . . . . . . 59
3.1.2 TE-mode illumination . . . . . . . . . . . . . . . . . . . . . . 61
3.1.3 TM-moden . . . . . . . . . . . . . . . . . . . . . . 63
3.2 The cylindrical geometry . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2.1 Radial transmission line approach . . . . . . . . . . . . . . . . 66
3.2.1.1 The stratified media . . . . . . . . . . . . . . . . . . 72
3.2.1.2 The continuous media . . . . . . . . . . . . . . . . . 75
3.2.2 Higher order mode illuminations . . . . . . . . . . . . . . . . . 78
3.3 The spherical geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.3.1 Higher order mode illuminations . . . . . . . . . . . . . . . . . 87
3.3.2 Lower order mode illuminations . . . . . . . . . . . . . . . . . 93
4 The Inverse Solution 95
4.1 The planar geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.1.1 TEM illumination . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.1.1.1 Thespecialcasewithacontinuityattheair-dielectric
interface . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.1.1.2 The general case with a discontinuity at the air-
dielectric interface . . . . . . . . . . . . . . . . . . . 100
4.1.1.3 Numericalalgorithmfortransformingthevirtualspace
variable into the actual physical distance . . . . . . . 103
4.1.1.4 Determinationofthepermittivityattheair-dielectric
interface . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.1.2 TE-mode illumination . . . . . . . . . . . . . . . . . . . . . . 105
4.1.3 TM-modetion . . . . . . . . . . . . . . . . . . . . . . 106
4.2 The RDE technique applied to non-planar structures . . . . . . . . . 107
4.2.1 Generalization of the one dimensional Fourier transform tech-
nique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.2.2 Extendingthevalidityoftheanalysistoageneralone-dimensional
case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.3 The cylindrical geometry . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.3.1 Radial transmission line approach . . . . . . . . . . . . . . . . 114
4.3.2 Higher order mode illuminations . . . . . . . . . . . . . . . . . 122
4.4 The spherical geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5 Reconstructed Examples 137
5.1 Simulated results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.1.1 Planar geometry . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.1.2 Cylindrical geometry . . . . . . . . . . . . . . . . . . . . . . . 146
5.1.3 Spherical geometry . . . . . . . . . . . . . . . . . . . . . . . . 158
5.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5.2.1 Coaxial airline measurements . . . . . . . . . . . . . . . . . . 165
5.2.2 Free-space measurements . . . . . . . . . . . . . . . . . . . . . 174
6 Summary 187IV TABLE OF CONTENTS
A The solution of two simultaneous differential equations having a
common factor 190
B The derivatives of the spherical Hankel functions 191
B.1 The first order derivative . . . . . . . . . . . . . . . . . . . . . . . . . 191
B.2 The second order derivative . . . . . . . . . . . . . . . . . . . . . . . 191
C Integral containing two Hankel functions 193
C.1 The cylindrical Hankel functions . . . . . . . . . . . . . . . . . . . . . 193
C.2 The spherical Hankel functions . . . . . . . . . . . . . . . . . . . . . 193
D The spherical Fourier-Bessel transform 195
Bibliography 197
Curriculum Vitae 206Zusammenfassung
Diese Arbeit beschreibt einen neuen Ansatz fur¨ die Rekonstruktion von eindimen-
sionalentiefenabh¨angigenPermittivit¨atsprofilenvonplanen,zylindrischenundkugelf¨ormi-
gen Objekten, die von einer beliebigen TE oder TM polarisierten Welle beleuchtet
werden. Die Rekonstruktion von Permittivit¨atsprofilen von planen und nicht pla-
nen Objekten bildet die Basis fu¨r die Darstellung von beliebig geformten dielek-
trischen K¨orpern, wenn Antennen hoher Direktivit¨at fur¨ die laterale Abtastung ver-
wendetwerden. DieHauptvorteilederneuenvonunsvorgeschlagenenTechniksind,
dass eine eindeutige L¨osung erreicht wird und das die Darstellung stark streuen-
der Objekte m¨oglich wird. Ein weiterer Vorteil ist, dass die Aufl¨osung des Bildes
im Raumbereich durch die Wahl der Bandbreite der gemessenen Reflektionsdaten
exakt eingestellt werden kann.
Fur¨ einplanesObjektwerdenzun¨achstdieRiccati-Differentialgleichungen,welche
das direkte Problem fur¨ den Fall von TE, TM und TEM Beleuchtung beschreiben,
untersucht. Diese Differentialgleichungen werden mit der von uns vorgeschlagenen
nichtlinearen Normierung inver