Copyright 2006 Regents of University of California1EE 210 Applied EM Fall 2006, Neureuther Lecture #12 Ver 10/04/06EE243 Advanced Electromagnetic TheoryLec # 12: Waves in Dispersive Media• Models for Medial (harmonic oscillator)• Behavior of permittivity and refractive index vs ω• Superposition of waves and group velocity• Pulse width increase with propagation• Implications of Causality and initial/final condition • Kramers Kronig Relations• ω−β diagramsReading: Jackson Ch 7.8-7.10 (skip 7.6 and 7.7)Copyright 2006 Regents of University of California2EE 210 Applied EM Fall 2006, Neureuther Lecture #12 Ver 10/04/06Overview• A harmonic oscillator model of electrons circulating about the nucleus is the basic phenomena by which materials affect EM waves.• These ponderable media and also boundary (eigenvalue) constraints produce wave phase velocities that depend on frequency.• This so called dispersion generally makes – the energy velocity less than the speed of light and (even undefined)– And causes the pulse length to increase with distance.• ε(ω)/ε0is an analytical function and the real part can be found from the imaginary part and visa versa • The tool for characterizing wave dispersion is the ω-βdiagram that plots the radian frequency versus the propagation k-vector.Copyright 2006 Regents of University of California3EE 210 Applied EM Fall 2006, Neureuther Lecture #12 Ver 10/04/06Harmonic Oscillator Model for Material[]())(11)()(,22020220220ωωγωεεωεωωγωωγ−−+=−−=−=−=++∑iijjifmNeiEmexeptxEexxm&&&• 2nd order differential equation in time•eiωtconverts to 2ndorder algebraic equation• Polarization is charge times displacement• Dielectric constant is 1 + polarization effects• Add contributions of each oscillator typeCopyright 2006 Regents of University of California4EE 210 Applied EM Fall 2006, Neureuther Lecture #12 Ver 10/04/06Permittivity Frequency Behavior• ε(ω) has frequency dependence•Re ε(ω) generally decreases with increasing frequency•Imyε(ω) shows resonate peaks • With low damping see resonate absorption and anomolous dispersion [negative slope of ε(ω)]• Represent ε(ω) by sum of poles in complex ω plane• Free electrons give pole at zero (Drude Model)• At high frequency converges to 1 as constant/ω2Copyright 2006 Regents of University of California5EE 210 Applied EM Fall 2006, Neureuther Lecture #12 Ver 10/04/06Refractive Index Water Versus Frequency• Real part drops from 9 to 1.5 and then 1.0• Imy Part rises to peak, drops to valley (visible), rises to peak, and drops to valley.• First peak is due to vibrationalmodes in molecules and interaction among molecules.• Second valley is due to electronic states of outer and then core electrons.Plasmons on metals come into this picture when ε(ω) is negative near the real axis (free electrons responding to E).eVnmeVnmeVnmeVnmhvnm112401.25904.6193925.131240→→→→=λWavelength versus energyJackson 315Copyright 2006 Regents of University of California6EE 210 Applied EM Fall 2006, Neureuther Lecture #12 Ver 10/04/06Dispersion Single Wave Propagation• Wave equation constraint on propagation vector• (kx-ωt)=0 => vP=w/k• Length of k-vector determines phase velocity• Not all signal components remain in phase()())()(0)()()()(k0)(0000002ωωωωωωωωεωµωεωεωµωωεωµωnccnktxvtxkcnkkkkkpr=====−=====+⋅−Copyright 2006 Regents of University of California7EE 210 Applied EM Fall 2006, Neureuther Lecture #12 Ver 10/04/06Group Velocity Derivation• Useful information or energy has more than one frequency to have a finite duration• Consider Fourier Transform representation and use ω(k) = ω(-k)• Choose a finite envelope and find frequency spread ∆x∆k>1/2 •Assume ω(k) approximated as linear– Constant => phase shift– ωdω/dk gives delay => vg= dω/dk• Curvature gives pulse broadening() ( )ωωωddnncvg/+=Copyright 2006 Regents of University of California8EE 210 Applied EM Fall 2006, Neureuther Lecture #12 Ver 10/04/06Causality• D is the response to E and D(ω)=e(ω)E(ω)• Write as summation over time• Kernal is the polarizability and has duration of γ-1.• Must be careful when time is large and mean free path involves multiple neighbors (anomalous skin effects)• ε(ω)/ε0is an analytic function in the upper half-plane(){}() ()[]() ()ττεωεωεωεπττττεωτωdeGdeGdtxEGtxEtxDiti∫∫∫∞−∞+∞−+∞∞−+=−=−+=00001/1/21),(),(),(Copyright 2006 Regents of University of California9EE 210 Applied EM Fall 2006, Neureuther Lecture #12 Ver 10/04/06Kramers-Kronig Relations• Analytic => Cauchy contour intergral• Part at infinity gives zero• Integrate out singularity at pole• P is principle value at pole()()()()[]ωωωεωεπωεωεωωωεωεωπεωε′−′−′−=′−′′+=∫∫∞∞dPdP02200022001/Re2/Im/Im21/ReCopyright 2006 Regents of University of California10EE 210 Applied EM Fall 2006, Neureuther Lecture #12 Ver 10/04/06Boundary Conditions Create Dispersion• Boundary conditions contribute eigenvalues• Wave equation forces constraint that gives the dispersion relationship• Could interpret as relative permittivity() ()() ()() ()()]/221[)(0]221[022022222222222222222cbabakkbakkkkrzzzzωππωεµεωππµεωµεωππµεω+−==+−+−=+−−−=+⋅−Copyright 2006 Regents of University of California11EE 210 Applied EM Fall 2006, Neureuther Lecture #12 Ver 10/04/06ω-β or k-β Diagram (1-D)k0=ω/cβ = k• plot ω versus k dispersion relationship• Speed of light reference k0= ω/c• Phase velocity = global slope• Group velocity = local slopek-vector in propagation directionω normalized to k-vector unitsCopyright 2006 Regents of University of California12EE 210 Applied EM Fall 2006, Neureuther Lecture #12 Ver 10/04/06Midterm Exam• In Class Tuesday October 24th• Covers material through today (Chapter 7)• Open Book, Open Notes, Bring Calculator, Paper Provided•Topics– Green’s functions free space and use in Theorems and concepts with emphasis on statics– Separation of variables in rectangular coordinates using N-1 and N method– Time-Harmonic ME, planewaves, boundary conditions, and