Copyright 2006 Regents of University of California1EE 210 Applied EM Fall 2006, Neureuther Lecture #20 Ver 11/12/06EE243 Advanced Electromagnetic TheoryLec # 20 Coupled Mode Theory (Cont. 2)• Coupled Mode Eigenvalues and Eigenfunctions• Modes Lock Together as Super Modes• Leaky Waves on Structures with Radiation Loss• Mode Crossings in ω−β diagrams: two types• Bloch Waves (Propagation and Stop Bands)• Floquet Theorem for Periodic Structures• ω−β Diagrams for Periodic StructuresReading: Haus 7.6, 8.1, (9,1-9.2 lite), Tamir 3.1.4, Collin 4.8-4.9, 8.1, 8.2, 8.6, 8.8, 5.7-5.8Fixed Slides 5, 9Copyright 2006 Regents of University of California2EE 210 Applied EM Fall 2006, Neureuther Lecture #20 Ver 11/12/06OverviewImplications and generalizations of coupled modes.• Super modes are locked sets of modes with new k-vectors• Leaky modes have radiation loss, hence complex k-vectors• Relative direction of the group and phase velocity determines phase or attenuation type interactionGeneralizations for Periodic Structures• Bloch Waves (Allowed Crystal Super Modes)• Floquet Most General Representation of a periodic field• ω−b Diagrams for periodic structures with k-vector combs and intersectionsCopyright 2006 Regents of University of California3EE 210 Applied EM Fall 2006, Neureuther Lecture #20 Ver 11/12/06Coupled Modes as Eigenfunction Problem• Construct a vector of mode amplitudes• Rate equation can be written as derivative of mode vector equal to a coupling matrix M times mode vector• Look for source free solutions (eigenvalues) by substituting an arbitrary exponential variation• Determinant constrains arbitrary exponential (eigenvalues)Use to check Kogelnik Solution in Eq. 2.6.30-31.⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧=+−11nnnAaaaX[][]00=−⋅−=′−=′⋅=′IjMDetXIjMXjXXMXieiieiieiAAλλλCopyright 2006 Regents of University of California4EE 210 Applied EM Fall 2006, Neureuther Lecture #20 Ver 11/12/06Coupled Modes as Eigenfunction Problem (Cont.)• Eigenfunctions are found by back substituting eigenvalues• These eigenfunctions are the Super Modes and show the field behavior in the cross section. (It changes as ω changes)• Homogeneous solution is sum over eigenfunctions with their eigenvalue z dependence plus boundary condition at z locations• A solution driven by an imposed (forced) z variation will take on that z-variation with eigenfunctions added to match z transition conditions. (Like a circuit - forced time variation and transient.)()()zjieiAeeeeiiieXXXjMXMXMXjMkkλλλδλδλ−∑==++=+++±==−−0002,22221,2212,1121,112122222Copyright 2006 Regents of University of California5EE 210 Applied EM Fall 2006, Neureuther Lecture #20 Ver 11/12/06Leaky Waves: Uniform Structure• When ever modes can transfer part of their energy to radiation they are termed Leaky Waves and their k-vectors parallel to the propagation direction are complex.• This can occur by tunneling into the high n region (Prism Coupler)• This complex z variation can occur by giving or receiving energySubstrateGuidenGUIDE> NSUB> NAIRPrismk1kxdownwardnAIRnSUBnGkzkMODEupwardnPRISMRADIATIONnPRISM> nGUIDECopyright 2006 Regents of University of California6EE 210 Applied EM Fall 2006, Neureuther Lecture #20 Ver 11/12/06Leaky Waves: Periodic Structure• When ever modes can transfer part of their energy to radiation they are termed Leaky Waves and their k-vectors parallel to the propagation direction are complex.• A periodic geometrical variation can produce a k-vector that radiates.• The radiation loss requires an associated attenuation hnfncnskm0kairkm-1kx2π/Periodkm12π/PeriodkairForward CouplerReverse Couplerkm-1Copyright 2006 Regents of University of California7EE 210 Applied EM Fall 2006, Neureuther Lecture #20 Ver 11/12/06Wave Media Interaction• Standing waves can produce periodic media modulation– Acousto-Optical Modulator (slow moving periodic structure)• Flowing carriers can add and remove energy – Traveling-wave tubes with electron beams• Nonlinear media effects can couple modes– ModelockingFlowing or standing wave medianfncnskm0kairkm-1kx2π/Periodkm12π/PeriodkairForward CouplerReverse Couplerkm-1Copyright 2006 Regents of University of California8EE 210 Applied EM Fall 2006, Neureuther Lecture #20 Ver 11/12/06Coupled Mode: vgand vpSame DirectionWhen the group and phase velocities are in the same direction• The eigenvalues (β’s) move away from each other• The displacement is proportional to the coupling coefficient• The eigenfunctions (Super Modes) associated with eigenvalue (β) continuously change identity in passing through the crossing pointωβHaus 7.6Copyright 2006 Regents of University of California9EE 210 Applied EM Fall 2006, Neureuther Lecture #20 Ver 11/12/06Coupled Mode: vgand vpOpposite DirectionWhen the group and phase velocities are in the opposite direction• The eigenvalues (β’s) move toward each other and merge• After they merge an attenuation region appears• The region over which they merge and the level of attenuation isproportional to the coupling coefficientωβαattenuationHaus 7.6Copyright 2006 Regents of University of California10EE 210 Applied EM Fall 2006, Neureuther Lecture #20 Ver 11/12/06Periodic Media Complications and Fixes• Coupled Mode Theory – Describes the change with distance when coupling is introduced – Each eigenvalue and eigenvector gives a Super Mode distribution and its β• For a periodic structure – The coupling is not uniform with distance– The fields are not uniform with distance– A Super Mode becomes a sum over an infinite number of periodic k-vectors– But it is possible to compare fields at z values that differ by the period P = dzxExponential decayExponential decayOscillatoryµ0ε0µ1ε1µ2ε2P = dCopyright 2006 Regents of University of California11EE 210 Applied EM Fall 2006, Neureuther Lecture #20 Ver 11/12/06Bloch Wave Concept and Constraint• At each cut plane n and n+1– Super Mode to right cn+ and cn+1+ – Super Mode to the left cn-and cn+1-• Require periodic behavior e-γdbetween cut planes–cn+1+ = cn+ e-γd–cn+1-= cn-e-γd• Integrate coupling coefficients over period (A matrix)– Aij = integrate (Super Mode)i∆ε (Super Mode)jzxExponential decayExponential decayOscillatoryµ0ε0µ1ε1µ2ε2P = dnn+1Copyright 2006 Regents of University of California12EE 210 Applied EM Fall 2006, Neureuther Lecture #20 Ver 11/12/06Bloch Wave Constraint