MODELING FLUCTUATIONS IN SCATTERED WAVES
CONTENTS 1 INTRODUCTION 1 1 1 RELEVANT LENGTH SCALES 1 1 2 DI USIVE TRANSPORT 2 1 3 COHERENT BACKSCATTERING AND THE APPROACH TO LOCALIZATION 3 1 4 SAMPLE SIZE DEPENDENCE 4 1 5 LOCALIZATION AND SCALING 6 1 6 SPATIAL DIMENSIONALITY IN LOCALIZATION AND DI USION 8 1 7 MESOSCOPIC PHENOMENA 11 1 8 LOCALIZATION VS CON NEMENT 13 1 9 TOPICS NOT COVERED 13 2 QUANTUM AND CLASSICAL WAVES 15 2 1 PRELIMINARIES 15 2 1 1 QUANTUM AND CLASSICAL WAVE EQUATIONS 16 2 2 GREEN FUNCTIONS FOR WAVES IN A UNIFORM MEDIUM 18 2 2 1 GREEN FUNCTION FOR THE UNIFORM MEDIUM 19 2 2 2 GREEN FUNCTION AND THE DENSITY OF STATES 21 2 2 3 REAL SPACE GREEN FUNCTIONS 23 2 3 WAVES ON A DISCRETE LATTICE 24 2 3 1 DISPERSION RELATION 24 2 3 2 RANDOM DISCRETE LATTICES THE ANDERSON MODEL 28 2 4 LATTICE GREEN FUNCTIONS 29 2 4 1 1D LATTICE GREEN FUNCTION 30 2 4 2 2D LATTICE GREEN FUNCTION 32 2 4 3 3D LATTICE GREEN FUNCTION 33 2 5 TREATING CONTINUUM PROBLEMS ON A LATTICE 34 2 6 PROBLEMS AND SOLUTIONS 36 3 WAVE SCATTERING AND THE COHERENT POTENTIAL APPROXIMATION 45 3 1 AN OVERVIEW OF THE APPROACH 45 3 1 1 INFORMATION FROM THE FIRST MOMENT 46 3 1 2 INFORMATION FROM THE SECOND MOMENT 46 3 1 3 INFORMATION FROM HIGHER MOMENTS 46 3 2 WAVE SCATTERING FORMALISM 47 3 2 1 THE OPERATOR NOTATION 48 3 2 2 THE T OPERATOR 49 3 2 3 CON GURATIONAL AVERAGING 50 3 2 4 THE SELF ENERGY 50 3 3 SINGLE SCATTERER THE LATTICE CASE 51 3 3 1 IMPURITY BOUND STATE 52 3 4 SINGLE SCATTERER THE CONTINUUM CASE 53 3 4 1 WAVE FUNCTION IN THE SINGLE SCATTERER CASE 55 3 4 2 A DIGRESSION ON NOTATION 55 3 4 3 SCATTERING AMPLITUDE AND THE T OPERATOR 56 3 4 4 SCATTERING AMPLITUDE AND THE OPTICAL THEOREM 3D CASE 57 3 4 5 SCATTERING AMPLITUDE AND THE OPTICAL THEOREM 2D CASE 58 3 4 6 SCATTERING AMPLITUDE AND THE OPTICAL THEOREM 1D CASE 59 3 5 IN NITE NUMBER OF SCATTERERS THE E ECTIVE MEDIUM 60 3 5 1 MEANING OF A K INDEPENDENT SELF ENERGY 60 3 5 2 THE MEAN FREE PATH 60 3 5 3 THE COHERENT POTENTIAL APPROXIMATION CPA 61 3 6 ACCURACY OF THE CPA 62 4 COHERENT WAVES AND E ECTIVE MEDIA 75 4 1 COHERENCE AND HOMOGENIZATION 75 4 2 CPA THE ANDERSON MODEL 76 4 2 1 E CIENT GREEN FUNCTION EVALUATION FROM DOS 78 4 3 CPA THE CLASSICAL WAVES 80 4 3 1 THE ROLE OF MICROSTRUCTURE 81 4 3 2 THE SYMMETRIC MICROSTRUCTURE 83 4 3 3 THE DISPERSION MICROSTRUCTURE 88 4 4 E ECTIVE MEDIUM MODELING OF INHOMOGENEOUS MATERIALS 92 4 4 1 GRANULAR METALS 92 4 4 2 SEDIMENTARY ROCKS 94 4 5 THE SPECTRAL FUNCTION APPROACH COHERENT QUASIMODES 102 4 5 1 FORMULATION 104 4 5 2 COHERENT ACOUSTIC WAVES IN THE INTERMEDIATE WAVELENGTH REGIME 106 4 5 3 NOVEL ACOUSTIC MODES IN COLLOIDAL SUSPENSIONS 108 4 5 4 GROUP VELOCITY IN STRONGLY SCATTERING MEDIA 112 4 5 5 ACCURACY OF THE SPECTRAL FUNCTION APPROACH 116 5 DI USIVE WAVES 127 5 1 BEYOND THE COHERENT REGIME 127 5 2 PULSE INTENSITY EVOLUTION IN A RANDOM MEDIUM 128 5 3 THE BETHE SALPETER EQUATION AND ITS SOLUTION BY MOMENTS 131 5 3 1 NOTATIONS 132 5 3 2 IRREDUCIBLE VERTEX FUNCTION AND THE BETHE SALPETER EQUATION 134 5 3 3 DI USIVE BEHAVIOR FROM A SIMPLI ED MODEL 136 5 3 4 THE WARD IDENTITY IN THE SIMPLI ED MODEL 137 5 3 5 FORMAL DERIVATION OF THE DI USIVE POLE AND THE DI USION CONSTANT 139 5 3 6 HIGHER MOMENT OF THE TRANSPORT EQUATION 140 5 3 7 EXPANSION OF K 141 5 3 8 SOLUTION FOR THE DI USIVE POLE AND THE DI USION CONSTANT 143 5 3 9 CON GURATIONAL AVERAGING AND DISSIPATION 145 5 4 THE VERTEX FUNCTION 146 5 4 1 THE REDUCIBLE VERTEX AND DIAGRAMMATIC REPRESENTATIONS 146 5 4 2 CPA FOR THE LADDER DIAGRAMS AND THE IRREDUCIBLE VERTEX 147 5 4 3 IRREDUCIBLE VERTEX THE ANDERSON MODEL 150 5 4 4 IRREDUCIBLE VERTEX CLASSICAL WAVES 152 5 5 THE WARD IDENTITY 155 5 5 1 CANCELLATION OF THE IMPURITY SCATTERING TERMS 157 5 5 2 REDUCTION TO THE SIMPLI ED MODEL 159 5 5 3 MEANING OF THE WARD IDENTITY 160 5 6 DI USION CONSTANT MODI CATION FOR CLASSICAL WAVES 160 5 7 EVALUATION OF THE WAVE DI USION CONSTANT 162 5 7 1 QUANTUM CASE THE ANDERSON MODEL 162 5 7 2 CLASSICAL WAVE DI USION CONSTANT 165 5 7 3 ANISOTROPIC SCATTERING AND THE TRANSPORT MEAN FREE PATH 167 5 8 APPLICATION DI USIVE WAVE SPECTROSCOPY 168 5 8 1 THE SIEGERT RELATION 168 5 8 2 AVERAGING OVER THE DI USIVE MOTION OF SCATTERERS 170 5 8 3 AVERAGING OVER THE SCATTERING ANGLES 170 5 8 4 DI USIVE TRANSPORT OF WAVES 171 5 8 5 BOUNDARY CONDITIONS FOR DI USIVE LIGHT FLUX 173 5 8 6 SOLUTIONS FOR THE SLAB GEOMETRY 174 6 THE COHERENT BACKSCATTERING E ECT 183 6 1 WAVE DI USION VERSUS CLASSICAL DI USION 183 6 2 COHERENCE IN THE BACKSCATTERING DIRECTION 184 6 2 1 TIME REVERSAL INVARIANCE 185 6 3 ANGULAR PRO LE OF THE COHERENT BACKSCATTERING 186 6 3 1 SINGLE CON GURATION AND CON GURATIONAL AVERAGING 187 6 3 2 COUNTING THE SCATTERING PATHS 188 6 3 3 BACKSCATTERING FROM A RANDOM HALF SPACE 188 6 3 4 FEATURES OF THE COHERENT BACKSCATTERING PEAK 189 6 3 5 BACKSCATTERING FROM A FINITE THICKNESS SLAB 190 6 4 SAMPLE SIZE PATH LENGTH DEPENDENCE 191 6 4 1 E ECT OF ABSORPTION 194 7 RENORMALIZED DI USION 199 7 1 COHERENT BACKSCATTERING E ECT IN THE DIAGRAMMATIC REPRESENTATION 199 7 2 EVALUATION OF THE MAXIMALLY CROSSED DIAGRAMS 201 7 2 1 MOMENTA TRANSFORMATIONS 201 7 2 2 SUMMATION OF MAXIMALLY CROSSED DIAGRAMS 203 7 3 RENORMALIZED DI USION CONSTANT 204 7 4 SAMPLE SIZE AND SPATIAL DIMENSIONALITY DEPENDENCIES OF WAVE DI USION 206 7 4 1 CONDUCTIVITY CORRECTION 206 7 4 2 DI USION CONSTANT CORRECTION 206 7 4 3 THE EMERGENCE OF LOCALIZATION LENGTH 207 7 4 4 LOCALIZATION IN 3D THE IO E REGEL CRITERION 207 7 5 LOCALIZATION IN ONE DIMENSION THE HERBERT JONES THOULESS FORMULA 208 7 5 1 LOCALIZATION AND INTENSITY TRANSMISSION 208 7 5 2 FLUCTUATIONS IN INTENSITY TRANSMISSION 209 7 5 3 RANDOMNESS IS NOT ALWAYS A SU CIENT CONDITION FOR 1D LOCALIZATION 210 7 5 4 LOCALIZATION AND EIGENFUNCTIONS IN 1D THE CHARACTERISTIC FUNCTION 210 7 5 5 RELATION BETWEEN THE 1D LOCALIZATION LENGTH AND DENSITY OF STATES 211 7 5 6 HJT FORMULA FOR THE 1D ANDERSON MODEL 212 7 5 7 HJT FORMULA FOR 1D CLASSICAL SCALAR WAVES 212 7 5 8 HIGH FREQUENCY AND LOW FREQUENCY LIMITING BEHAVIORS 214 8 THE SCALING THEORY OF LOCALIZATION 219 8 1 DISTINGUISHING A LOCALIZED STATE FROM AN EXTENDED STATE 219 8 1 1 THE SCALING ARGUMENT AND SPATIAL DIMENSIONALITY 220 8 1 2 DIMENSIONLESS CONDUCTANCE 221 8 2 THE SCALING HYPOTHESIS AND ITS CONSEQUENCES 222 8 2 1 DE NING THE SCALING FUNCTION 223 8 2 2 SCALING AND THE E ECT OF RANDOMNESS 223 8 2 3 BEHAVIOR OF THE SCALING FUNCTION 224 8 2 4 WHEN SYSTEM SIZE IS LESS THAN THE LOCALIZATION LENGTH 225 8 2 5 WAVE TRANSPORT NEAR THE 3D MOBILITY EDGE 226 8 2 6 ANOMALOUS DI USION AND THE MINIMUM METALLIC CONDUCTANCE IN 3D 227 8 2 7 SCALING BEHAVIOR IN THE 3D LOCALIZED REGIME 228 8 3 NUMERICAL EVALUATION OF THE SCALING FUNCTION 229 8 3 1 THE FINITE SIZE SCALING FUNCTION 230 8 3 2 LIMITING BEHAVIORS 230 8 3 3 RELATION TO THE SCALING FUNCTIONS IN 1D 2D AND 3D 231 8 4 UNIVERSALITY AND LIMITATIONS OF THE SCALING THEORY RESULTS 235 8 4 1 LOCALIZATION IN 3D LATTICE VS CONTINUUM 236 8 4 2 IMPLICIT ASSUMPTIONS OF THE SCALING THEORY 236 9 LOCALIZED STATES AND THE APPROACH TO LOCALIZATION 243 9 1 THE SELF CONSISTENT THEORY OF LOCALIZATION 243 9 1 1 TRANSITION TO THE STRONG SCATTERING REGIME 243 9 1 2 RESPONSE OF THE LOCALIZED STATE AT FINITE MODULATION FREQUENCY 244 9 1 3 QUALITATIVE EVALUATION OF THE CORRECTION TERM 245 9 2 LOCALIZATION BEHAVIOR OF THE ANDERSON MODEL 246 9 2 1 EVALUATION OF THE MAXIMALLY CROSSED DIAGRAMS CONTRIBUTION 247 9 2 2 MOBILITY EDGE IN 3D ANDERSON MODEL 249 9 2 3 LOCALIZATION LENGTH AND CORRELATION LENGTH IN THE 3D ANDERSON MODEL 251 9 2 4 2D ANDERSON MODEL 252 9 2 5 LOCALIZATION LENGTH IN THE 2D ANDERSON MODEL 254 9 2 6 1D ANDERSON MODEL 256 9 2 7 LOCALIZATION LENGTH IN THE 1D ANDERSON MODEL 257 9 2 8 INPUT PARAMETERS TO LOCALIZATION CALCULATIONS 257 9 3 CLASSICAL SCALAR WAVE LOCALIZATION 257 9 3 1 ONE PARAMETER CRITERION FOR THE CLASSICAL WAVE 3D MOBILITY EDGE 259 9 3 2 INPUT PARAMETER VALUES AND WAVE SCATTERING REGIMES 261 9 3 3 SCATTERING AND LOCALIZATION OF CLASSICAL WAVES IN 3D 262 9 3 4 PERIODICITY IN SCATTERING STRUCTURES AND 3D LOCALIZATION 263 9 3 5 LOCALIZATION OF 2D CLASSICAL SCALAR WAVE 265 9 3 6 LOCALIZATION OF 1D CLASSICAL SCALAR WAVE 266 9 4 TRANSPORT VELOCITY OF CLASSICAL SCALAR WAVES 267 9 4 1 TRANSPORT VELOCITY AND SCATTERING RESONANCES 268 9 4 2 E ECTIVE MEDIUM BASED ON ENERGY HOMOGENIZATION 268