NO9800029 NEI-NO--890 VELAUTHAPILLAI DHAYALAN FOCUSING OF ELECTROMAGNETIC WAVES DEPARTMENT OF PHYSICS UNIVERSITY OF BERGEN NORWAY 1996 il FOCUSING OF ELECTROMAGNETIC WAVES Thesis submitted for the degree of Doctor Scientiarum by VELAUTHAPILLAI DHAYALAN Department of Physics University of Bergen Norway 1996 Velauthapillai Dhayalan Department of Physics University of Bergen Allegt. 55, N-5007 Bergen Norway Thesis submitted for the degree of Doctor Scientiarum. May 1996 ISBN 82-993933-0-2 Front cover:Time-averaged electric energy density in the focal plane for a converging electric-dipole wave (Kirchhoff solution). Fresnel number N — 10 and f—number — 0.5. TO APPA, AMMA, SIVANANTHY, DHANUSHAN AND MAURAN Preface and Acknowledgement The thesis consists of one published article and four articles submitted or to be submitted for publishing. The five articles are presented in chapters 2-6. Chapters 2-4 deal with focusing of electromagnetic waves in a single medium, and chapters 5 and 6 deal with focusing in a layered medium. The following chapters have appeared or will appear elsewhere: Chapter 2: Pure Appl. Opt. 5, 195-226 (1996) Chapter 3: Pure Appl. Opt., submitted Chapter 4: Pure Appl. Opt., submitted Chapter 5: J. Opt. Soc. Am. A, submitted Chapter 6: To be submitted Parts of the work covered by these papers are presented or will be presented at three conferences: Norwegian Electro-Optics Meeting, Norway (1995) Workshop on Diffractive Optics, Prague, Czech Republic (1995) Annual meeting of the Opt. Soc. Am., New York, USA (1996) Now I want to express my sincere gratitude to number of special persons. First of all, I will characterize myself as the luckiest person on earth to have Prof. Jakob J. Stamnes as supervisor. He has been there to help me and guide me whenever I needed his advice. The encouragement and inspiration he rendered me during this study is impossible to express in words. His comments, revisions and suggestions have been very valuable for the completion of the thesis. I thank him very much from the bottom of my heart. This work is carried out at the Department of Physics, University of Bergen under a grant given by the Research Council of Norway for which I will be allways thankful. All the support and encouragement from my collegues and staff at the Physics institute of University of Bergen is greatly appreciated. I would like to thank especially my friends in the optics group for all the kind assistance and encouragement. I thank sincerely for the blessings, support and encouragement from my parents, my wife's parents, brothers, sisters, in-laws and all the relatives and friends. I would like to thank especially my dear wife Sivananthy and my wonderful children Dhanushan and Mauran for their understanding and tolerance during the course of this study. Finally, but most importantly, I thank the allmighty god for helping me to complete this thesis. Bergen, May 1996 Velauthapillai Dhayalan Contents Introduction 1 1.1 Background 1 1.2 Summary 3 Focusing of electric-dipole waves 9 1 Introduction 10 2 Diffraction through an aperture in a plane screen 12 2.1 Exact solution of boundary-value problem 12 2.2 Kirchhoff solution of diffraction problem 14 3 Focusing through an aperture in a plane screen 16 3.1 Debye approximation of the focused field 17 3.2 Kirchhoff approximation of the focused field 18 4 Focusing of converging electric-dipole wave 19 4.1 Specification of incident field 19 4.2 Debye approximation of converging electric-dipole wave 20 4.3 Kirchhoff approximation of converging electric-dipole wave 23 5 Comparison with Debye theory for rotationally symmetric op- tical systems 25 5.1 Analytical comparisons 26 5.2 Numerical comparisons 29 6 Concluding remarks 38 Focusing of mixed-dipole waves 41 3.1 Introduction 42 3.2 Radiation from a magnetic source 43 3.2.1 Angular-spectrum representations 45 3.2.2 Field from a magnetic dipole 47 3.3 Focusing of converging electromagnetic waves 48 3.3.1 Focusing through an aperture in a plane screen .... 49 3.3.2 Focusing of converging magnetic-dipole waves 52 3.4 Focusing of converging mixed-dipole wa ve in the Debye ap- proximation 59 3.4.1 Polarization vector, strength factor, and energy pro- jection 60 3.4.2 Energy densities 64 3.5 Numerical Results 68 3.5.1 Energy densities 68 3.5.2 Poynting vector or direction of e nergy flow 72 3.5.3 Cross polarization 76 3.6 Summary and Conclusions 78 3.7 Aknowledgments 79 4 Focusing of electric-dipole waves in the Debye and Kirchhoff approximations 83 4.1 Introduction 84 4.2 Evaluation of the angular spectrum in the aperture plane . 85 4.2.1 Angular spectrum in the Kirchhoff approximation . 85 4.2.2 Angular spectrum in the Debye app roximation .... 86 4.2.3 Asymptotic evaluation of the angu lar spectrum in the Kirchhoff approximation 87 4.2.4 Numerical evaluation of the angul ar spectrum .... 89 4.3 Focusing of electric-dipole waves 92 4.3.1 Focusing of electric-dipole waves in Debye approxima- tion 93 4.3.2 Focusing of electric-dipole waves in Kirchhoff approx- imation 94 4.3.3 Numerical results 95 4.4 Aknowledgments 95 5 REFLECTION AND REFRACTION OF AN ARBITRARY ELECTROMAGNETIC WAVE IN A LAYERED MEDIUM 109 5.1 Introduction 110 5.2 The one-layer medium Ill 5.2.1 TE and TM plane-wave represen tations Ill 5.2.2 Whittaker potentials 113 5.2.3 Solutions for spectral-domain pot entials 114 5.2.4 Exact solutions for reflected, tr ansmitted, and slab fields 118 5.3 Matrix theory for a multilayered mediu m 122 5.3.1 Scalar theory 123 5.3.2 Electromagnetic theory 132 5.4 Dyadic Green's functions for layered m edium 138 5.4.1 Reflected and transmitted fields 138 5.4.2 Field inside the jth layer 139 5.5 Summary and Conclusions 142 5.6 Aknowledgments 143 A Spectral-domain dyadic Green's functions for a slab 145 B Spectral-domain dyadic Green's functions for a multilayered medium 147 in C Comparison between slab and the multilayered system re- duced to a one-layer system 149 6 Focusing of electromagnetic waves insidea dielectric slab 153 6.1 Introduction 154 6.2 Problem Definition 155 6.2.1 Exact solutions for reflected, transmitted fields and slab fields 156 6.2.2 Fields inside a slab 159 6.3 Focusing of electromagnetic waves inside a slab 160 6.4 Numerical Techniques 163 6.4.1 Reduction from double integral to single integral . 163 6.4.2 Dividing the integration area 165 6.4.1 Integration Algorithms 168 6.5 Asymptotic evaluation 171 6.6 Comparison between exact and asymptotic solutions 173 6.6.1 Analytical comparison 173 6.6.2 Numerical comparisons 176 6.7 Numerical Results 185 6.8 Conclusion 194 6.9 Aknowledgments 194 D Transmission and reflection coefficients 195 IV Chapter 1 Introduction 1.1 Background The first time diffraction phenomena ever were mentioned, was as early as in the 15th century. But the real understanding of diffraction phenomena, which are a consequence of the wave character of light, came after the work of Grimaldi 150 years later. He was the first to mention that light has wave characteristics. Without knowing Grimaldi's work, Huygens proposed a new wave theory, nowadays referred to as Huygens's principle, sug- gesting that every point on a wave front has to be regarded as the center of a secondary wave front which forms at a later instant an envelope of secondary wavefronts. Fresnel (1818) contributed his part to the so called 'Huygens-Fresnel principle' by accounting for the interference between the secondary wave fronts. The earliest theoretical studies of the diffraction theory of imaging have to be attributed to the work of Airy (1835). He established that the intensity distribution in the focal plane due to an aberration-free wave front from a circular aperture would have the form of [Ji(v)/v]2, where J\(v) is the first order Bessel function. Lommel (1885) presented both theoretical and experimental work, thus extending Airy's treatment to account for the intensities at all points in the viscinity of the focus. In 1909 Debye established the general features of the diffracted field near and far away from the focus [1], and the assumptions on which his work was based, are now commonly referred to as the Debye approximation. The classical theory of imaging, which the optical community has relied on for many years, depends on three approximations [2]: the assumption of a scalar field, the parax- ial approximation, and the Debye approximation. But owing to Maxwell's work on the electromagnetic theory (1873), it is well known that visible light is merely one form of elec- tromagnetic energy, usually described as electromagnetic waves. The complete spectrum of electromagnetic waves extend from radio waves to Gamma radiation. The classical scalar theory of imaging which neglects polarization effects can be applicable only to systems of moderate angular apertures. The second and third approximations mentioned above for the classical theory to be valid, cannot be fulfilled at the same time. The paraxial approx- imation is poor for large apertures, while the Debye approximation is best at large Fresnel numbers N defined by N = a2f\zi. Here a is the aperture radius, A is the wavelength, and zi is the distance between the aperture and the focal point. The range of validity of the classical theory is thus restricted. Although the vector treatment is complicated compared to the scalar treatement one must employ it to obtain a proper description of the polarization and the energy flow in the focal region.