Copyright 2006 Regents of University of California1EE 210 Applied EM Fall 2006, Neureuther Lecture #15 Ver 11/12/06EE243 Advanced Electromagnetic TheoryLec # 15 Plasmons• (Group and Phase Velocity for Guided Waves)• (Sign Reversals for Propagation Direction Reversal)• Plasmon Fields, B.C. and Dispersion Equation• Physical Interpretation• Typical Values for Silver• ω−β diagram and fields at various points• Assistance from topographical featuresReading: These PPT notes and Harrington on corrugated surface 4.8.Fixed Slides 6, 12Copyright 2006 Regents of University of California2EE 210 Applied EM Fall 2006, Neureuther Lecture #15 Ver 11/12/06Midterm 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 dispersionReview Th, Oct 19Copyright 2006 Regents of University of California3EE 210 Applied EM Fall 2006, Neureuther Lecture #15 Ver 11/12/06Overview• (First touch up – product of group and phase = c2for guided waves– Reverse direction reverse sign Ezand Ht).• Surface Plasmons on metals – are TM guided waves – are made possible by free electrons oscillating in the propagating field just below their plasma frequency – this makes the permittivity both negative and larger in magnitude than the permittivity of air or a coating dielectric.• Plasmons are described by Maxwell Equations – Their dispersion relationship is found by matching B.C.– They propagate a distance of up to 30 µm– Near the plasma frequency their wavelength along the surface can be 5X smaller that the free space wavelength– The can be used to form high resolution (λ/6) intensity probes but the probe height is roughly equal to the resolution and thusdecreases.Copyright 2006 Regents of University of California4EE 210 Applied EM Fall 2006, Neureuther Lecture #15 Ver 11/12/06Clarification on Phase and Group Velocity• The fact that the product of the phase and group velocity is equal to the square of the speed of light holds for any (propagating) mode in a lossless sytemdue to its eigenfunction and eigenvalue γ2.()22222222222211122111221cvvckkkkvckkffvkkgpgp=<+=+=∂∂=>+====+=−=λµελµεωλµεωλππλλµεωµεωγCopyright 2006 Regents of University of California5EE 210 Applied EM Fall 2006, Neureuther Lecture #15 Ver 11/12/06Sign Reversal with Direction Reversal• Take longitudinal and transverse components of curl equations with z phase variation.• With propagation direction reversal the sign of βλchanges• But if sign of Ezand Htare also reversed the original equations result• Thus the reverse guided wave solution can be nothing more than the forward guided wave solution with these changesλλλλλλλλλλλλωεβωµβωεωµttztttztzttzttEiHziiHHiEziEEiHHiE−=×+×∇=×+×∇−=×∇=×∇ˆˆCopyright 2006 Regents of University of California6EE 210 Applied EM Fall 2006, Neureuther Lecture #15 Ver 11/12/06Are There Waves on Material Surfaces?• Consider TM w/r z case with Hy given and same z phase variation• Will have Hy, Ez and Ex (but Ey = Hx = Hz = 0)zx2022210221221121ˆˆεµωεµω−=−===+−zzzikxvyzikxvykvkveeyHHeeyHHzz2 = metal1 = dielectricCopyright 2006 Regents of University of California7EE 210 Applied EM Fall 2006, Neureuther Lecture #15 Ver 11/12/06Boundary Conditions• Hy continuous (or) D normal continuous gives H10=H20.• Ez continuous gives final constraint to find kz.• This constraint is the same as setting the impedance looking upward equal to the negative of the impedance looking downward.• Impedance looking upward is capacitive (neg imy).• Impedance looking downward thus need to be inductive.xxyzyzyzyzZivivZHEHEivivHivEHivE−+=−+=−−=−=+=−−+=−−=22112211221122221111ωεωεεεωεωεCopyright 2006 Regents of University of California8EE 210 Applied EM Fall 2006, Neureuther Lecture #15 Ver 11/12/06Solving for Surface Wave Conditions• Constraint• Substitute definition of v1and v2to solve for kz.• Substitute solution for kzto find other properties –v1and v2(localization in x)– Resolution in z with large kz– Probe height in x()()12111122120222102212211εεεεεεεµωεµωεε+−=+=−=−=−=kvkkkvkvvvzzzCopyright 2006 Regents of University of California9EE 210 Applied EM Fall 2006, Neureuther Lecture #15 Ver 11/12/06Physical Interpretation•Want v1real so exponential decay away from boundary•Since ε1is real and positive this means that the denominator must go negative.• This occurs when ε2< - ε1. (plasmon condition for a single surface to support a guided wave)• In metals just below their plasma resonance frequency a negative permittivity can occur (Ag, Au, Cu, Al)• In general kzis complex and the waves die out in a few microns()12111εεε+−= kv1212εε−=vv()1221εεε+= kkzCopyright 2006 Regents of University of California10EE 210 Applied EM Fall 2006, Neureuther Lecture #15 Ver 11/12/06Refractive Index of Silver• http://www.microe.rit.edu/research/lithography/utilities.htmCopyright 2006 Regents of University of California11EE 210 Applied EM Fall 2006, Neureuther Lecture #15 Ver 11/12/06Refractive Index of Gold• http://www.microe.rit.edu/research/lithography/utilities.htmCopyright 2006 Regents of University of California12EE 210 Applied EM Fall 2006, Neureuther Lecture #15 Ver 11/12/06Physical Estimates for Air and Silver• Approximately for silver in the visible–nr=0.2, and K = 2 (410 nm) and 4 (630 nm)– the velocity is 0.87c and 0.97c, λsurface= (2π/kz) can be smaller than λairby over 3X => 3X resolution improve– the 1/e length is 4λ (1.8 µm) and 295λ (2.9 µm) – The 1/e penetration into the metal is about λsurface/5– The 1/e penetration into air is 4 to 16 times larger0.112=+=niKnnr()()⎥⎥⎦⎤⎢⎢⎣⎡−+−≈+=2/32201220121KinKKkkkrzεεε121212Kvv≈−=εε()rznKl2/3212 −=λCopyright 2006 Regents of University of California13EE 210 Applied EM Fall 2006, Neureuther Lecture #15 Ver 11/12/06ω−β Diagram for Plasmon• The plasmons start as frequency is increased– close to the speed of light line,– become slightly slower, and – turn into a very slow wave (horizontal line) at the plasma