Absorption and Spatial Dispersion in Electromagnetic Susceptibility Models Robin Jeffrey Churchill School of Physics University of Exeter A thesis submitted for the degree of Doctor of Philosophy in Physics August 2017 Absorption and Spatial Dispersion in Electromagnetic Susceptibility Models Submitted by Robin Jeffrey Churchill, to the University of Exeter as a thesis for the degree of Doctor of Philosophy in Physics. August 2017 This thesis is available for Library use on the understanding that it is copyright mate- rial and that no quotation from the thesis may be published without proper acknowledge- ment. I certify that all material in this thesis which is not my own work has been identified and that no material has previously been submitted and approved for the award of a degree by this or any other University. Robin Jeffrey Churchill Date 1 Acknowledgments Firstly, I would like to thank my main supervisor Dr Tom Philbin for all the discus- sions, advice and guidance he has provided over the course of my research. Next, a big thank you to my second supervisor Dr Simon Horsley, Dr Janet Anders and the rest of the PhD students who made up our little theory group. My thanks also go out to Charles Downing and Tom Sturges of the QSN group. Finally, I would like to thank my family, both for their constant support and their willingness to listen to me ramble on about my work over the past four years. 2 Abstract This thesis focuses on two key aspects of light-matter interactions: absorption and spatial dispersion, both of which are described on a macroscopic scale by the electromag- netic susceptibility. The first part of this thesis contains an investigation into the microscopic origin of absorption in dielectric models, providing a detailed calculation for a long-held assump- tion of the Hopfield model that has formed the basis of many key works on the subject. While previous work has either focused on the quantum regime or used phenomenological methods which lack a clear relationship to the underlying physics, the microscopic model and calculations presented here are purely classical in nature, matching Hopfields initial proposal. A discrete model of a dielectric is developed, containing nonlinear interaction terms between polarizable dipoles and lattice vibrations. The lattice vibrations are found to act as a pseudo-reservoir, leading to broadband absorption of electromagnetic radia- tion that naturally emerges from the model, without the need to add damping terms to the dynamics. The effective linear susceptibility is calculated using a perturbative iteration method and is found to match the form of a model that is widely used for real dielectrics. The second half of the thesis presents a series of modifications to the Halevi-Fuchs susceptibility model, which is used to calculate the electromagnetic reflection and trans- mission coefficients of a spatially-dispersive half-infinite medium. The initial model, valid only for an idealized single-resonance scalar susceptibility with a specific wave vector dependence, is extended to include many more of the susceptibility features found in real materials, including unequal transverse and longitudinal components, multiple res- onances, anisotropy and alternate wave vector dependences. In each case, the effect of the boundary is characterized by a set of phenomenological reflection coefficients for the polarization waves in the medium, with specific values corresponding to various addi- tional boundary conditions for Maxwell’s equations. The exact expressions derived for the electromagnetic reflection and transmission coefficients can be used in the calculation of a range of physical phenomena near the boundary of the medium. This thesis consid- ers the spectral energy density of thermal and zero-point radiation outside the medium, with the key result that the inclusion of spatial dispersion naturally removes an unphysical divergence associated with the use of a spatially local susceptibility model. 3 Contents Publications 7 List of Symbols 8 List of Figures 13 List of Tables 22 List of Fourier transformations 24 1 Introduction 25 1.1 Absorption . 26 1.2 Spatial dispersion . 27 1.3 Thesis outline . 29 2 Background Theory 30 2.1 The electric susceptibility . 31 2.1.1 Temporal nonlocality . 31 2.1.2 Spatial nonlocality . 32 2.1.3 Properties of the electric susceptibility . 33 2.1.4 The electromagnetic wave equation . 34 2.1.5 The Lorentz model . 37 2.1.6 Excitons . 39 2.2 Absorption in dielectrics . 42 2.2.1 The Hopfield model . 42 2.2.2 The phenomenological reservoir . 44 2.3 Reflection and transmission at planar boundaries . 47 2.3.1 The local medium interface . 47 4 Contents 2.3.2 The nonlocal medium interface . 50 2.3.3 Additional boundary conditions . 52 2.3.4 The generalized ABC model . 54 2.3.5 Additional work . 58 2.3.6 Applications . 60 3 Absorption in Dielectrics 66 3.1 The linear model . 67 3.2 The nonlinear model . 72 3.3 The perturbative solution . 76 3.4 Nonlinear equations and diagrams . 79 3.4.1 Applications to the linear model . 81 3.5 The effective susceptibility . 83 3.5.1 Leading-order F terms . 86 3.5.2 Higher-order F terms . 92 3.6 Numerical calculations . 93 3.6.1 Frequency dependence . 95 3.6.2 Wave vector dependence . 98 3.7 Chapter summary . 100 4 Assumptions in ABC models 102 4.1 The Pekar ABC . 103 4.1.1 Nearest-neighbour coupling . 104 4.1.2 jth-nearest-neighbour coupling . 108 4.1.3 Conclusions . 111 4.2 The Halevi-Fuchs ansatz . 111 4.2.1 Calculating the electric field . 114 4.2.2 Comparisons to the Halevi-Fuchs ansatz . 118 4.2.3 Conclusions . 120 4.3 Chapter summary . 121 5 Extending the Halevi-Fuchs model I. The single-resonance susceptibility 122 5.1 The infinite medium . 124 5.2 The half-infinite medium . 126 5 Contents 5.3 The p-polarization . 130 5.3.1 The surface impedance . 131 5.3.2 Field amplitude ratios . 131 5.3.3 Electromagnetic reflection and transmission coefficients . 133 5.4 Results . 136 5.4.1 The p-polarization reflection coefficient . 137 5.4.2 The p-polarization transmission coefficients . 140 5.5 Spectral energy density . 142 5.5.1 The p-polarization . 143 5.5.2 The s-polarization . 145 5.5.3 Results . 146 5.6 Chapter summary . 151 6 Extending the Halevi-Fuchs model II. Multiple parabolic exciton bands 152 6.1 The infinite medium . 153 6.2 The half-infinite medium . 154 6.3 The p-polarization . 157 6.3.1 The surface impedance . 157 6.3.2 Field amplitude ratios . 157 6.3.3 Electromagnetic reflection and transmission coefficients . 158 6.4 The s-polarization . 159 6.4.1 The surface impedance . 159 6.4.2 Field amplitude ratios . 159 6.4.3 Electromagnetic reflection and transmission coefficients . 160 6.5 Results . 161 6.5.1 Non-interacting exciton bands . 162 6.5.2 Heavy/light exciton bands . 164 6.6 Spectral energy density . 166 6.6.1 Non-interacting exciton bands . 166 6.6.2 Heavy/light exciton bands . 169 6.7 Chapter summary . 172 6 Contents 7 Extending the Halevi-Fuchs model III. Uniaxial crystals and linear k terms 173 7.1 The infinite medium . 174 7.1.1 The c k y^ orientation . 177 7.1.2 The c k x^ and c k z^ orientations . 177 7.1.3 Other c orientations . 178 7.2 The half-infinite medium . 179 7.3 Electromagnetic reflection coefficients . 180 7.3.1 The c k y^ orientation . 180 7.3.2 The c k x^ and c k z^ orientations . 181 7.4 Results . 182 7.4.1 Electromagnetic reflection coefficients . 183 7.4.2 Spectral energy density . 186 7.5 Chapter summary . 188 8 Conclusions 190 8.1 Thesis summary . 190 8.2 Future work . 195 Publications Absorption in dipole-lattice models of dielectrics R. J. Churchill and T. G. Philbin, Phys. Rev. A, 93, 053809 (2016). Electromagnetic reflection, transmission, and energy density at boundaries of nonlocal media R. J. Churchill and T. G. Philbin, Phys. Rev. B, 94, 235422 (2016). Reflection and transmission in nonlocal susceptibility models with multiple resonances R. J. Churchill and T. G. Philbin, Phys. Rev. B, 95, 205406 (2017). 7 Contents List of Symbols Chapter 3: Absorption in Dielectrics Notation Definition A(z; t) Scalar field acting as projection of electromagnetic vector potential Ah(q; !) Homogeneous solution of the A field in the absence of coupling a Lattice spacing c Speed of light in vacuum F (q; !) Sum over all possible intermediate steps for iteration processes that start and end with p(q; !), under certain restrictions n Fn(q; !) Sum of all terms in F (q; !) with a factor of σ F2p, F2u, F4p Terms of Fn corresponding to specific iteration processes 0 f1=2(q; q ) Lppu nonlinear coupling function GA(k; !) Retarded Green function of A field in the absence of coupling Gp(q; !) Retarded Green function of p field in the absence of coupling d Gp(q; !) Retarded Green function of p field dressed with A by linear interactions in the ML model 0 Gp(q; !) Effective Green function of p field dressed with u by nonlinear interactions in the MN model Gu(q; !) Retarded Green function of u field in the absence of coupling H Collection of homogeneous solutions k Wave vector LA, Lp, Lu “Free” Lagrangians LpA, Lppu Interaction Lagrangians M Particle mass P Principal-valued term ph(q; !) Homogeneous solution of the p field in the absence of coupling th pn(t) Electric dipole moment of n particle in 1D model Q(!) Inverse of ν(q) q Wave vector (limited to first Brillouin zone) t Time 8 Contents Notation Definition U(q) Choice of function for uh U0 Amplitude of U(q) uh(q; !) Homogeneous solution of the u field in the absence of coupling th un(t) Displacement of n particle in 1D model V0 Dipole-dipole interaction strength ZL[A] Collection of linear A terms resulting from iteration process ZNL[A] Collection of nonlinear A terms resulting from iteration process z Position th zn Lattice position of n particle in 1D model α(z) Function describing finite size of particles in 1D model β LpA linear coupling parameter γ Damping coefficient of the Lorentz model δ(z), δ(q) Dirac delta function th κn n -nearest-neighbor displacement.