Analysis of scattering of plane electromagnetic waves by stratified spheres of radially uniaxial anisotropic materials

Analysis of Scattering of Plane Electromagnetic Waves by Stratified Spheres of Radially Uniaxial Anisotropic Materials Liu Huizhe (B. Eng., National University of Singapore) A Thesis Submitted for the Joint Degree of Doctor of Philosophy ´ ´ by National University of Singapore and Ecole Sup´erieure d’Electricit´ e 2011 Acknowledgments First of all, I would like to express my heartfelt gratitude to my main supervisor, Prof. Koen Mouthaan. Without his kindness in monitoring my progress and his patience in proofreading my papers and thesis to perfection, I would not have completed this work in time. Many thanks also go to my previous main supervisor, Prof. Joshua Le-Wei Li, who left for UESTC, for initiating this interesting project and giving me the opportunity to travel to France. I am deeply grateful to my supervisor, Prof. Sa¨ıd Zouhdi, for his hospitality during my research attachment in LGEP-Sup´elec and the financial support to attend a series of enriching conferences and PhD schools in Europe and Africa. I am greatly indebted to my supervisor, Prof. Leong Mook Seng, for stimulating discussions and his unfailing support. I am grateful to the National University of Singapore for the research scholarship provided throughout my candidature. I am also grateful to the French government, in particular, the CROUS in France and the French Embassy in Singapore, for the financial support during my stay in France. I also owe my gratitude to teachers and friends I have met in Singpore, in i Acknowledgments ii particular, Prof. Chen Xudong, Prof. Qiu Cheng-Wei, Dr. Zhong Yu, Dr. Xiao Ke, Mr. Wan Chao, Mr. Han Tiancheng, Dr. Hu Li from NUS and Dr. Boris Luk’yanchuk from DSI for valuable discussions; Miss Zheng Xuhui, Miss Rao Hailu, Miss Ding Pingping, Miss Ye Xiuzhu, Dr. Wu Yuming, Ms. Liang Dandan, Ms. Wang Xuan, Ms. Li Yanan, Dr. Li Yanan, Mr. Tang Kai, Mr. Ye Huapeng, Dr. Liu Xiaofei, Dr. Fei Ting for their support to me in one way or another; lab officers Mr. Jack Ng Chin Hock, Mdm. Guo Lin, Mr. Sing Cheng Hiong for their kind assistance with lab facilities. I am also thankful to friends I have met in France, in particular, Prof. Alain Bossavit, Prof. Olivier Dubrunfaut, Mr. Roger Pereira, Mr. Lotfi Beghou and Prof. Alexey P. Vinogradov for engaging discussions; Mr. Laurent Santandrea for his kind assistance in coursework and in printing my first conference poster; Dr. Yu Peiqing, Dr. Zhu Yu, Dr. Liu Bing, Dr. Liu Xiaofeng, Dr. Song Li, Mr. Tang Qingshan, Dr. Hicham Belyamoun, Mr. Hassan Hariri, Prof. Ahachad Mohammed, Mr. Yves Bernard for making my attachment in France a memorable one. I am also grateful to my thesis committee members, Prof. Chen Xudong, Prof. Dominique Lesselier and Dr. Boris Luk’yanchuk for their valuable suggestions to improve my thesis. I would like to express my deepest gratitude to my beloved parents for always being supportive of my academic pursuit and for taking care of the preparation of my wedding ceremony so that I could concentrate on thesis writing. Last but not least, I owe plenty to my beloved husband Mr. Zhang Xiaomeng for his patience in editing papers and thesis, discussing ideas and practising presentations with me. Contents Acknowledgments i Contents iii Summary vii List of Figures viii List of Acronyms xii List of Publications xiii Introduction 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Original contributions and structure of the thesis . . . . . . . . . . iii Contents iv Preliminaries 11 2.1 Problem set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Fields and potentials in anisotropic media . . . . . . . . . . . . . . 13 2.3 Harmonic expansion of plane wave and formulation of Debye potentials 26 2.4 Formulation of boundary conditions and derivation of scattering co- 2.5 efficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Transparent spherical particles with radial anisotropy 42 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.2 Theoretical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2.1 Theoretical background . . . . . . . . . . . . . . . . . . . . . 45 3.2.2 Derivation of transparency relation . . . . . . . . . . . . . . 48 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.3.1 Far-field analysis . . . . . . . . . . . . . . . . . . . . . . . . 51 3.3.2 Near-field analysis . . . . . . . . . . . . . . . . . . . . . . . 55 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.3 3.4 Contents v Plasmon-resonant spherical particles with radial anisotropy 62 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.2 Theoretical Formulation . . . . . . . . . . . . . . . . . . . . . . . . 65 4.3 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.3.1 Effect of anisotropy on proportion of permissible regions . . 72 4.3.2 Effects of particle sizes, modes, dissipative losses and config- 4.4 urations on resonance trace . . . . . . . . . . . . . . . . . . 74 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Conclusions and recommendations 89 Bibliography 95 A Radar cross section 104 A.1 Scattered electric fields . . . . . . . . . . . . . . . . . . . . . . . . . 104 A.2 Far field approximations . . . . . . . . . . . . . . . . . . . . . . . . 106 Contents B Bessel Functions vi 111 B.1 Cylindrical Bessel functions . . . . . . . . . . . . . . . . . . . . . . 111 B.2 Spherical Bessel functions . . . . . . . . . . . . . . . . . . . . . . . 112 B.3 Riccati-Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 113 C Legendre Functions 116 Summary The aim of this thesis is to characterize electrically-small spheres with radial anisotropy and stratified structure for either minimizing or maximizing scattering by plane waves based on the extended Mie scattering model. This theory is applicable to the case of single-layer spheres as well. The original contributions of the thesis are twofold. First, for the case of a single-layer sphere with radial anisotropy, a transparency condition comprising of radial and tangential permittivity is analytically established. As such, particles with carefully engineered radial anisotropy are transparent without using any coating and are ideal for applications with space constraint and stringent transparency criteria. Second, for the case of a coated sphere with radial anisotropy, a resonance condition is analytically derived. The derived condition provides a guideline for choosing material and structural parameters in the design of plasmon-resonant particles at given frequencies of interest for applications in sensing. For both cases, the effective permittivity of the scatterers of interest is derived in comparison with pertinent expressions of scattering coefficients for isotropic spheres. Physical insights are also provided. In addition, full-wave numerical analysis is presented to validate the proposed conditions. vii List of Figures 1.1 Illustration of a metallic nanoparticle (a) with localized surface plasmonic resonance (b) and (c) [1]. . . . . . . . . . . . . . . . . . . . . 2.1 Configuration of scattering of an incident plane wave by a stratified radially uniaxial anisotropic spherical scatterer. . . . . . . . . . . . 3.1 12 Configuration of scattering of an incident plane wave by a radially uniaxial anisotropic spherical scatterer. . . . . . . . . . . . . . . . . 3.2 46 (a) Normalized total scattering cross section and (b) contribution of the first-order TM scattering coefficient, with respect to εr /ε0 , with εt = 3ε0 , µ = µ0 , and a = λ0 /100. . . . . . . . . . . . . . . . . . . . 3.3 52 (a) Normalized total scattering cross section and (b) contributions of several TM and TE scattering coefficients, with respect to εr /ε0 , with εt = 3ε0 , µ = µ0 , and a = λ0 /5. . . . . . . . . . . . . . . . . . viii 54 List of Figures 3.4 ix (a) Normalized total scattering cross section and (b) contributions of several TM and TE scattering coefficients, with respect to εt /ε0 , with εr = 3ε0 , µ = µ0 , and a = λ0 /5. . . . . . . . . . . . . . . . . . 3.5 56 Contour plots of magnitude of radial components of scattered electric fields in the x-z plane for (a) an isotropic sphere with ε = 3ε0 , µ = µ0 , and a = λ0 /5; (b) the same as (a), except for a coating of ac = 1.49a and εc = 0.5ε0 ; (c) the same as (a), except that εt = 3ε0 and εr = 0.25ε0 ; and (d) the same as (a), except that εt = 0.62ε0 and εr = 3ε0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 58 Distribution of magnitude of time-average Poynting vector and associated power flow lines in the E plane for the same four cases as in Fig. 3.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Configuration of scattering of an incident plane wave by a coated radially uniaxial anisotropic spherical scatterer. . . . . . . . . . . . 4.2 59 65 Contour plot of resonance condition over εt,1 –εt,2 (εr,1 –εr,2 ) space: colored region corresponds to < a2 /a1 < 1; blank region corresponds to forbidden region. . . . . . . . . . . . . . . . . . . . . . . . 4.3 70 Proportion of permissible area in the 4th quadrant of contour map against εt,2 /εr,2 with εt,1 /εr,1 = 1, εm = 1.7689ε0 and (a) |εt,2,max | = |ε1,max | = 15ε0 ; (b) |εt,2,max | = |ε1,max | = 100ε0 . . . . . . . . . . . . . 80 Bibliography 102 [55] S. J. Oldenburg, R. D. Averitt, S. L. Westcott, and N. J. Halas, “Nanoengineering of optical resonances”, Chem. Phys. Lett., vol. 288, pp. 243–247, 1998. [56] V. L. Sukhorukov, G. Meedt, M. K¨ urschner, and U. Zimmermann, “A singleshell model for biological cells extended to account for the dielectric anisotropy of the plasma membrane”, J. Electrost., vol. 50, pp. 191–204, 2001. [57] B. Lange and S. R. Arag´on, “Mie scattering from thin anisotropic spherical shells”, J. Chem. Phys., vol. 92, pp. 4643–4650, 1990. [58] H.-Z. Liu, K. Mouthaan, M. S. Leong, and S. Zouhdi, “Resonance relation for coated spheres with radial anisotropy”, Suzhou, China, 2011, PIERS 2011. [59] Y. R. Lin-Liu, H. Ikezi, and T. Ohkawa, “Radiation damping and resonance scattering”, Am. J. Phys, vol. 56, pp. 373, 1988. [60] C. A. Balanis, Advanced engineering electromagnetics, John Wiley & Sons, New York, 1989. [61] E. F. Knott, J. F. Shaeffer, and M. T. Tuley, Radar cross section: Its prediction, measurement and reduction, Artech House, Inc., Dedham, 1985. [62] H. C. van de Hulst, Light scattering by small particles, Dover Publications, New York, 1981. [63] L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of electromagnetic waves: theory and applications, Wiley series in remote sensing. John Wiley & Sons, New York, 2000. [64] M. Kerker, The scattering of light and other electromagnetic radiation, Academic Press, New York, 1969. [65] G. N. Watson, A treatise on the theory of bessel functions, The University Press, Cambridge, 2nd edition, 1944. [66] C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles, Wiley Science Paperback Series. John Wiley & Sons, New York, 1998. Appendix A Radar cross section Radar cross section (RCS) is defined as the area intercepting that amount of power which, when scattered isotropically, produces at the receiver a density which is equal to that scattered by the actual target [60] (also see [61, 62]). It is a fictitious area property of the target, characterizing the spatial distribution of scattered energy or power. Mathematically, RCS is given by σ = lim 4πr r→∞ |E | s , |E i |2 (A.1) where |E s |2 = |Ers |2 + |Eθs |2 + Eϕs (A.2) and r is the distance between the observer and scatterer. A.1 Scattered electric fields Substituting expansions of Debye potentials of scattered fields Eqs. (2.34a)–(2.34b) into field expressions Eqs. (2.36)–(2.38), three spherical components of scattered 104 electric fields are expressed in terms of Debye potentials as )[ ] ∞ cos ϕ ∑ (1) T M ˆ (1) = −an Tn Hn (km r) Pn (cos θ) km n=1 ]  [ (1)  ˆ n (km r) ∞   ∂2 H ∑ TM (1) ˆ + Hn (km r) Pn(1) (cos θ) −an Tn = cos ϕ   ∂ (km r)2 ( Ersca ∂2 + km ∂r n=1 = cos ϕ ∞ ∑ [ ] ˆ (1) (km r) P (1) (cos θ) ˆ (1)′′ (km r) + H −an TnT M H n n n n=1 = cos ϕ ∞ ∑ n=1 Eθsca [ −an TnT M ] n(n + 1) ˆ (1) Hn (km r) Pn(1) (cos θ). (km r(2 (A.3) ] [ ∞ ∂2 cos ϕ ∑ ˆ n(1) (km r) Pn(1) (cos θ) −an TnT M H = r ∂r∂θ km ] [ n=1 ∞ iωµm ∂ sin ϕ ∑ ˆ n(1) (km r) Pn(1) (cos θ) + −an TnT E H r sin θ ∂ϕ ηm km n=1 [ ] [ ] (1) (1) ˆ ∞ ∂ H (k r) ∂ P (cos θ) n n m cos ϕ ∑ = −an TnT M km r n=1 ∂ (km r) ∂θ ∂ sin ϕ ωµm ∑ ˆ (1) (km r) Pn (cos θ) −an TnT E H n ∂ϕ ηm km r n=1 sin θ ∞ ] cos ϕ ∑ [ T M ˆ (1)′ ˆ (1) (km r) πn (θ) . (A.4) = an Tn Hn (km r) τn (θ) + iTnT E H n km r n=1 ∞ (1) +i Eϕsca ] [ ∞ ∂2 cos ϕ ∑ ˆ n(1) (km r) Pn(1) (cos θ) −an TnT M H = r sin θ ∂r∂ϕ km n=1 [ ] ∞ ∑ iωµm ∂ sin ϕ ˆ n(1) (km r) Pn(1) (cos θ) − −an TnT E H r ∂θ ηm km n=1 [ ] (1) ˆ ∞ (1) ∂ H (k r) n m Pn (cos θ) ∂ (cos ϕ) ∑ −an TnT M = km r ∂ϕ ∂ (km r) sin θ n=1 [ ] (1) ∞ ∂ Pn (cos θ) ∑ ωµm T E ˆ (1) sin ϕ −a T −i H (k r) n m n n r ηm km ∂θ n=1 ∞ ] sin ϕ ∑ [ T M ˆ (1)′ ˆ (1) (km r) τn (θ) .(A.5) = − an Tn Hn (km r) πn (θ) + iTnT E H n km r n=1 where as defined by Kong [63] τn (cos θ) = − [ ] (1) d Pn (cos θ) dθ (1) Pn (cos θ) πn (cos θ) = − , sin θ , (A.6a) (A.6b) 2n+1 which differs from [62] by the − sign. It should be noted that an = in+1 n(n+1) . TnT M and TnT E are scattering coefficients determined by structural and material parameters, as well as the order number. With reference to [62, 63], at cos θ = ±1, we have πn (1) = τn (1) = n (n + 1) , −πn (−1) = τn (−1) = (−1)n A.2 (A.7a) n (n + 1) . (A.7b) Far field approximations When the observation point is in the far field of at least 2D λ2 where D is the largest dimension of the object [60], we can apply asymptotic expansions to the RiccatiBessel functions in Eq. (B.12) to simplify field expressions Eqs. (A.3)–(A.5) to lim Ers = 0, (A.8a) r→∞ lim r→∞ Eθs ∞ ] eikm r ∑ 2n + [ T M Tn τn (θ) + TnT E πn (θ) , = i cos ϕ km r n=1 n (n + 1) lim Eϕs = −i sin ϕ r→∞ (A.8b) ∞ ] eikm r ∑ 2n + [ T M Tn πn (θ) + TnT E τn (θ) . (A.8c) km r n=1 n (n + 1) For convenience, following [62] and [63], we define two functions of θ as ∞ ∑ ] 2n + [ T M S1 (θ) = Tn πn (θ) + TnT E τn (θ) , n (n + 1) n=1 (A.9a) ∞ ∑ ] 2n + [ T M S2 (θ) = Tn τn (θ) + TnT E πn (θ) . n (n + 1) n=1 (A.9b) Hence, Eqs. (A.8b) and (A.8c) are simplified as [63] lim Eθs = −i r→∞ lim Eϕs = i r→∞ eikm r S2 (θ) cos ϕ, km r (A.10a) eikm r S1 (θ) sin ϕ. km r (A.10b) Substituting spherical electric components Eqs. (A.8a), (A.10a) and (A.10b) and values of S functions Eq. (A.16) into Eq. (A.2), we have lim |E | s r→∞ eikm r eikm r = −i S2 (θ) cos ϕ + i S1 (θ) sin ϕ km r km r ] [ 2 2 = |S2 (θ)| cos ϕ + |S1 (θ)| sin ϕ . (km r) (A.11) Bistatic scattering Substituting Eq. (A.11) into Eq. (A.1), the bistatic scattering cross section (SCS) is given by σ(θ, ϕ) = lim 4πr r→∞ ] | 4π [ 2 2 = |S (θ)| cos ϕ + |S (θ)| sin ϕ . 2 km |E i |2 |E In the H-plane where ϕ = s π (A.12) and θ varies between and π, we have ( π ) 4π σ θ, = |S1 (θ)|2 . km (A.13) The angular efficiency / gain [64] is derived after normalization of SCS with respect to the geometric cross section of the sphere whose outer-most radius is denoted by a. We have σ |S1 (θ)|2 = . πa2 (km a)2 (A.14) SCS normalized with respect to wavelength is given by σ = |S1 (θ)|2 . λ π (A.15) Similarly, in the E-plane where ϕ = and θ varies between and π, we can simply replace |S1 (θ)|2 by |S2 (θ)|2 in Eqs. (A.13–A.15). Backscattering For backscattering, θ = π and ϕ can be any value. By substituting Eq. (A.7b) into Eqs. (A.9a) and (A.9b), we have ∞ ∑ ) 2n + ( T M n (n + 1) Tn − TnT E (−1)n −S1 (π) = S2 (π) = n (n + 1) n=1 = ∞ ∑ n=1 (−1)n ) 2n + ( T M Tn − TnT E . (A.16) Eq. (A.11) can be reduced to 1 2 2 |S2 (π)| cos ϕ + |S1 (π)| sin ϕ (km r) (km r) 1 2 2 = |−S1 (π)| cos ϕ + |S1 (π)| sin ϕ (km r) (km r) = (A.17) |S1 (π)| . (km r) lim |E s (θ = π)|2 = r→∞ Substituting |E i | = and Eqs. (A.16)–(A.17) into Eq. (A.1), the radar cross section becomes ∞ ) 2n + ( T M 4π ∑ 4π (−1)n σ = |S1 (π)|2 = Tn − TnT E . km km n=1 (A.18) The backscatter efficiency or gain is given by ∞ ∑ ) σ n 2n + ( T M TE (−1) = T − T n n πa2 (km a)2 n=1 ∞ ∑ ( ) = (−1)n (2n + 1) TnT M − TnT E . (km a) n=1 (A.19) Normalization with respect to wavelength square yields σ = |S1 (π)|2 . λ π (A.20) The cut-off number of terms to be included in the summation is given by ] [ ncut−off = Integer km a + 4.3 (km a) + . (A.21) Total scattering cross section The total scattering cross section σT is defined as the ratio of the time average total scattered power to the time averaged incident Poynting vector, and is related to the bistatic cross section by the equation σT = 4π ∫ π ∫ 2π σ (θ, ϕ) sin θdθdϕ, θ=0 (A.22) ϕ=0 which can be converted to an expression in terms of the scattering coefficient as ∞ [ 2π ∑ σT = (2n + 1) TnT M km n=1 + TnT E ] . (A.23) After normalization with respect to geometric cross section, we have ∞ [ σT ∑ TnT M (2n + 1) = a2 πa2 km n=1 + TnT E ] . (A.24) Normalization with respect to wavelength square yields ∞ ( ∑ σT = (2n + 1) TnT M λ 2π n=1 + TnT E ) . (A.25) Extinction cross section When the material of the scatterer is lossy, some incident power is absorbed within the particle in the form of heat. In this case, the total loss from the incident beam, scattering plus absorption, is defined as the efficiency for extinction as [64] σext = ∞ ( ) ∑ (2n + 1) Re TnT M + TnT E . 2 km a n=1 (A.26) Furthermore, we have σabs = σext − σsca . (A.27) Appendix B Bessel Functions B.1 Cylindrical Bessel functions With reference to Abramowitz [49], Bessel’s equation of order v is given by z2 ) d2 w dw ( + z − v w = 0. +z dz dz (B.1) Solutions are cylindrical Bessel functions of the first kind J±v (z), of the second kind (1) Yv (z) (also called Weber or Neumann functions) and of the third kind Hv (z), (2) Hv (z) (also called Hankel functions). Jv (z) (Re(v) ≥ 0) is bounded as z tends to in any bounded range of the (1) (2) argument z. The pairs Jv (z), Yv (z) and Hv (z), Hv (z) are linearly independent for all values of v. The relations between the two pairs of solutions are given by Hv(1) (z) = Jv (z) + iYv (z) , (B.2a) Hv(2) (z) = Jv (z) − iYv (z) . (B.2b) Recurrent relations, where B represents any function of J, Y , H (1) , H (2) , or 111 any linear combination of these functions, are given below ′ Bv (z) = v v [Bv−1 (z) − Bv+1 (z)] = Bv−1 (z) − Bv (z) = −Bv+1 (z) + Bv (z) , z z (B.3) where the second definition is used in the computer programs in this thesis. Furthermore, the Wronskians between J and Y are defined as ′ W {J, Y } = JY − J ′ Y. (B.4) According to Abramowitz [49], Eq. (B.4) can equivalently be expressed as W {Jv (z) , Yv (z)} = Jv+1 (z) Yv+1 (z) − Jv (z) Yv+1 (z) = . πz (B.5) The Bessel function for small arguments applies when v is fixed and z → 0, i.e. 0 0. (B.6b) (z) ∼ − Γ (v) π Asymptotic expansions of the Hankel function of order v for large values of |z| are given by Watson [65]. B.2 Spherical Bessel functions With reference to Abramowitz [49], the differential equation is given by [ ] ′′ ′ z w + 2zw + z − n (n + 1) w = 0, (n = 0, ±1, ±2, .) . (B.7) Particular solutions are the spherical Bessel functions of the first kind √ jn (z) = π J (z) , 2z n+ (B.8a) spherical Bessel functions of the second kind √ yn (z) = π Y (z) , 2z n+ (B.8b) and the spherical Bessel functions of the third kind √ π (1) H (z) , 2z n+ √ π (2) H (z) . h(2) n (z) = jn (z) − iyn (z) = 2z n+ h(1) n (z) = jn (z) + iyn (z) = (1) (B.8c) (B.8d) (2) The pairs jn (z), yn (z) and hn (z), hn (z) are linearly independent for all values of n. Asymptotic expansions of spherical Hankel function are given by (−i)n eiz , (z) ∼ iz in e−iz h(2) . n (z) ∼ − iz h(1) n B.3 (B.9a) (B.9b) Riccati-Bessel functions With reference to Abramowitz [49], the differential equation is given by [ ] z w′′ + z − n (n + 1) w = 0, (n = 0, ±1, ±2, .) . (B.10) Particular solutions are (also applicable solutions with real order according to Gao. [24]) √ πz J (z), n+ √ πz = zyn (z) = Y (z), n+ √ πz (1) (1) = zhn (z) = H (z) , n+ √ πz (2) (2) H (z) . = zhn (z) = n+ Jˆn = zjn (z) = (B.11a) Yˆn (B.11b) ˆ (1) H n ˆ (2) H n (B.11c) (B.11d) All properties of these functions follow directly from those of Spherical Bessel functions. Asymptotic expansions of Riccati-Hankel function and their derivatives are given by ˆ (1) ∼ (−i)n+1 eiz , H n (B.12a) ˆ (1)′ ∼ (−i)n eiz , H n (B.12b) ˆ (1)′′ ∼ −H ˆ (1) . H n n (B.12c) Similarly, ˆ (2) ∼ in+1 e−iz , H n (B.13a) ˆ (2)′ ∼ in e−iz , H n (B.13b) ˆ (2) . ˆ (2)′′ ∼ −H H n n (B.13c) Recurrence relations for Riccati-Bessel functions are given by ˆv (z) − B ˆv+1 (z) = − v B ˆv (z) + B ˆv−1 (z) , ˆv′ (z) = v + B B z z (B.14) ˆ can be any one among J, ˆ Yˆ , H ˆ n(1) and H ˆ n(2) . Furthermore, the Wronskians where B for Riccati-Bessel functions can be derived, with reference to Eq. (B.5) and the third equation in Eq. (B.3), as { } ˆ ˆ W J, Y = 1, (B.15a) { } (1) ˆ ˆ W J, H = i. (B.15b) By making use of recurrence relation, the second order derivative of Bessel function can be rewritten as ˆ ′′ (z) = B v = = = = ]′ [ ′ ]′ [ ˆ (z) + B ˆ ˆ (z) = − v B (z) B v−1 v z v v ˆ′ v ˆ ˆ′ Bv (z) − B v (z) + Bv−1 (z) z z ] [v ] v ˆ v[ vˆ ˆ ˆ ˆ (z) B (z) + B (z) + B (z) − B − B (z) − v−1 v z2 v z z v z v−1 v ˆ v2 ˆ ˆ (z) B (z) + B (z) − B v z2 v z2 v [ ] v (v + 1) ˆ (z) . (B.16) −1 B v z2 As such, we have ] v (v + 1) ˆ v (z) , ˆ v (z) + B ˆ v (z) = −1 B B z2 ′′ [ (B.17) which can be used for the simplification of field expressions in Appendix A. Appendix C Legendre Functions With reference to Abramowitz [49], Legendre equation is given by ( [ ] ) d2 w dw m2 − 2z + l (l + 1) − w = 0. 1−z dz dz − z2 (C.1) Solution is the associated Legendre function as ) 12 m dm Pl (z) , (z) = z − dz m m )1 (m) m( 2 m d Pl (z) Pl (z) = (−1) − z , dz m (m) Pl ( (C.2a) (C.2b) l dl (z − 1) . where ≤ m < l and Rodrigue’s formula states Pl (z) = n l! dz l In some books such as Rade [37], Born [34], Bohren [66], the Legendre function is defined in the form of Eq. (C.2b) with the omission of (−1)m term. However in this research, we follow the convention in Kong [63] and Abramowitz [49] which contains the (−1)m term as Eq. (C.2b). 116 Recurrence relations as defined in Abramowitz [49] are given by ( ) dPl(m) (z) (m) (m) = lzPl (z) − (m + l) Pl−1 (z) , z −1 dz (m) (m) (l − m + 1) Pl+1 (z) = (2l + 1) zPl (C.3a) (m) (z) − (l + m) Pl−1 (z) . (C.3b) The first-order derivative of Legendre function is related to associated Legendre function with order as ′ dPl (cos θ) dPl (cos θ) d cos θ = dθ d cos θ dθ (1) P (cos θ) (− sin θ) = − l sin θ Pl (cos θ) = (1) = Pl (cos θ). (C.4) [...]... of inelastic scattering include Brillioun scattering, Raman scattering, inelastic X-ray scattering and Compton scattering In this work, we are concerned with elastic scattering, to be more specific, Mie scattering extended to the complex media, i.e., radially uniaxial anisotropic media The Mie scattering model is an analytical tool for solving Maxwell’s equations while dealing with electromagnetic scattering. .. its ease of implementation In our case, we simplify the structure to a single-layer sphere and add one more degree of freedom to the material parameters by making use of the radially uniaxial anisotropic materials We attempt to design a single sphere which is transparent with the introduction of radial anisotropy In addition, we will examine the contributions of tangential and radial components of permittivity... encountered forms of scattering, can be broadly divided into two types, namely, elastic and inelastic scattering Elastic scattering includes Mie scattering [2] and Rayleigh scattering [3] where the kinetic energy of the incident wave is conserved 1 Chapter 1 Introduction 2 in the absence of dissipation On the other hand, inelastic scattering is common in molecular collisions where the kinetic energy of the incident... represented by Rayleigh scattering which is a first-order approximation of the Mie theory Rayleigh scattering is characterized by a scattering intensity inversely proportional to the fourth power of the incident wavelength It gives an explanation to the blueness of the sky where the blue component, having a shorter wavelength than the red one, is scattered the most In Rayleigh scattering, the weak dipolar scattering. .. formulation of electromagnetic fields in all regions in the presence of a stratified spherical scatterer with radial anisotropy, which lays the theoretical foundation for solving transparency and resonance problems to be discussed in later chapters Based on full-wave Mie scattering theory, field solutions for scattering from uniaxial anisotropic spheres can be analytically derived by a myriad of techniques,... section, we highlight two of the techniques, namely, coordinate transformation and scattering cancellation technique The advantages and drawbacks of the transformation cloak are illustrated as follows Theoretically speaking, the transformation cloak is closest to the criteria of an ideal cloak The basic idea is to control the path of electromagnetic waves by the careful engineering of permittivity and permeability... 2010 (b) A P Vinogradov, A V Dorofeenko, E O Liznev, H.-Z Liu, and S Zouhdi, “Employing Epsilon-near-zero Material in Cloaking”, PIERS 2009, Progress in Electromagnetics Research Symposium, Moscow, Russia, 18-21 August 2009 xiii Chapter 1 Introduction In this thesis the interaction between plane electromagnetic (EM) waves and stratified radially uniaxial anisotropic spheres is analytically studied based... maximizing scattering are considered, i.e., transparency and resonance Special attention is paid to the role of radial anisotropy in controlling the scattering intensity This introductory chapter covers the background and the structure of the thesis 1.1 Background Scattering takes place when propagating waves or particles encounter obstacles along their paths Electromagnetic scattering, being one of the... components of permittivity is investigated By making comparisons with pertinent Rayleigh results, an expression of equivalent permittivity for sphere with radial anisotropy is derived Numerical analysis is performed to validate the derived transparency Chapter 1 Introduction 10 condition by examining the scattering cross section and the first few orders of scattering coefficient The size tolerance of the... anisotropy on the size of permissible regions in a contour map Furthermore, the applicability of the resonance condition is validated by inspecting the density plots of the magnitudes of scattering coefficients in the material-structural space for various configurations, particle sizes, orders of resonance and losses Last but not least, a summary of the thesis is provided in chapter 5 Limitations of this work and . Analysis of Scattering of Plane Electromagnetic Waves by Stratified Spheres of Radially Uniaxial Anisotropic Materials Liu Huizhe (B. Eng., National University of Singapore) A. plane wave by a stratified radially uniaxial anisotropic spherical scatterer. . . . . . . . . . . . 12 3.1 Configuration of scattering of an incident plane wave by a radially uniaxial anisotropic. . . 59 4.1 Configuration of scattering of an incident plane wave by a coated radially uniaxial anisotropic spherical scatterer. . . . . . . . . . . . 65 4.2 Contour plot of resonance condition