EE243 Advanced Electromagnetic Theory Lec # 15 Plasmons. Reading: These PPT notes and Harrington on corrugated surface 4.8.

Transcription

1 EE Applied EM Fall 6, Neureuther Lecture #5 Ver //6 EE43 Adanced Electromagnetic Theory Lec # 5 Plasmons Fixed Slides 6, (Group and Phase Velocity for Guided Waes) (Sign Reersals for Propagation Direction Reersal) Plasmon Fields, B.C. and Dispersion Equation Physical Interpretation Typical Values for Siler ωβ diagram and fields at arious points Assistance from topographical features Reading: These PPT notes and arrington on corrugated surface 4.8. Copyright 6 Regents of Uniersity of California

2 EE Applied EM Fall 6, Neureuther Lecture #5 Ver //6 Midterm Exam In Class Tuesday October 4 th Reiew Th, Oct 9 Coers material through today (Chapter 7) Open Boo, Open Notes, Bring Calculator, Paper Proided Topics Green s functions free space and use in Theorems and concepts with emphasis on statics Separation of ariables in rectangular coordinates using N- and N method Time-armonic ME, planewaes, boundary conditions, and dispersion Copyright 6 Regents of Uniersity of California

3 EE Applied EM Fall 6, Neureuther Lecture #5 Ver //6 Oeriew (First touch up product of group and phase c for guided waes Reerse direction reerse sign E and t ). Surface Plasmons on metals are TM guided waes are made possible by free electrons oscillating in the propagating field just below their plasma frequency this maes the permittiity both negatie and larger in magnitude than the permittiity 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 3 µm Near the plasma frequency their waelength along the surface can be 5X smaller that the free space waelength The can be used to form high resolution (λ/6) intensity probes but the probe height is roughly equal to the resolution and thus decreases. Copyright 6 Regents of Uniersity of California 3

4 EE Applied EM Fall 6, Neureuther Lecture #5 Ver //6 Clarification on Phase and Group Velocity γ ω p g p µω g µ fλ ω c λ + πf π λ µ ω ( ) λ + µ λ + > c µ λ + < c The fact that the product of the phase and group elocity is equal to the square of the speed of light holds for any (propagating) mode in a lossless sytemdue to its eigenfunction and eigenalue γ. Copyright 6 Regents of Uniersity of California 4

5 EE Applied EM Fall 6, Neureuther Lecture #5 Ver //6 Sign Reersal with Direction Reersal t t t t E E tλ tλ λ λ iωµ λ λ λ iωe λ + iβ ˆ E tλ i + iβ ˆ iωµ tλ tλ iωe tλ Tae longitudinal and transerse components of curl equations with phase ariation. With propagation direction reersal the sign of β λ changes But if sign of E and t are also reersed the original equations result Thus the reerse guided wae solution can be nothing more than the forward guided wae solution with these changes Copyright 6 Regents of Uniersity of California 5

6 Copyright 6 Regents of Uniersity of California 6 EE Applied EM Fall 6, Neureuther Lecture #5 Ver //6 Are There Waes on Material Surfaces? Consider TM w/r case with y gien and same phase ariation Will hae y, E and Ex (but Ey x ) x ˆ ˆ µ ω µ ω + i x y i x y e ye e ye metal dielectric

7 EE Applied EM Fall 6, Neureuther Lecture #5 Ver //6 Boundary Conditions E E i E y Z iω + iω + i E y y y + + x iω iω y continuous (or) D normal continuous gies. E continuous gies final constraint to find. This constraint is the same as setting the impedance looing upward equal to the negatie of the impedance looing downward. Impedance looing upward is capacitie (neg imy). Impedance looing downward thus need to be inductie. x Copyright 6 Regents of Uniersity of California Z 7

8 Copyright 6 Regents of Uniersity of California 8 EE Applied EM Fall 6, Neureuther Lecture #5 Ver //6 Soling for Surface Wae Conditions Constraint Substitute definition of and to sole for. Substitute solution for to find other properties and (localiation in x) Resolution in with large Probe height in x ( ) ( ) µ ω µ ω + +

9 EE Applied EM Fall 6, Neureuther Lecture #5 Ver //6 Physical Interpretation ( + ) ( + ) Want real so exponential decay away from boundary Since is real and positie this means that the denominator must go negatie. This occurs when < -. (plasmon condition for a single surface to support a guided wae) In metals just below their plasma resonance frequency a negatie permittiity can occur (Ag, Au, Cu, Al) In general is complex and the waes die out in a few microns Copyright 6 Regents of Uniersity of California 9

10 EE Applied EM Fall 6, Neureuther Lecture #5 Ver //6 Refractie Index of Siler Copyright 6 Regents of Uniersity of California

11 EE Applied EM Fall 6, Neureuther Lecture #5 Ver //6 Refractie Index of Gold Copyright 6 Regents of Uniersity of California

12 EE Applied EM Fall 6, Neureuther Lecture #5 Ver //6 n n Physical Estimates for Air and Siler n r. + ik K K inr + ( + ) ( ) 3/ K K K l λ nr Approximately for siler in the isible n r., and K (4 nm) and 4 (63 nm) the elocity is.87c and.97c, λ surface (π/ ) can be smaller than λ air by oer 3X > 3X resolution improe the /e length is 4λ (.8 µm) and 95λ (.9 µm) The /e penetration into the metal is about λsurface/5 The /e penetration into air is 4 to 6 times larger Copyright 6 Regents of Uniersity of California ( ) 3/

13 EE Applied EM Fall 6, Neureuther Lecture #5 Ver //6 ωβ Diagram for Plasmon f f ω 5.9 x 5 7 π ( 3 nm / s) / λnm 8 5 (.89 nm / s) / The plasmons start as frequency is increased close to the speed of light line, become slightly slower, and turn into a ery slow wae (horiontal line) at the plasma frequency. Copyright 6 Regents of Uniersity of California nm 63 nm 4 nm ω/ c 3 nm β 3

14 EE Applied EM Fall 6, Neureuther Lecture #5 Ver //6 Plasmon Characteristics ω ω/ c 5.9 x 5 63 nm 4 nm 3 nm β 63 nm:.97c, /e 46.3λ (9 µm), 6X air 4 nm:.86c, /e 4.λ (.8 µm), 4X air 3 nm: > X resolution But eanescent near fields are X faster decay Copyright 6 Regents of Uniersity of California 4

15 EE Applied EM Fall 6, Neureuther Lecture #5 Ver //6 Near Field Probe Limits R h h λ π / e / e surface ( + ) ( + ) Wor near the plasmon frequency to mae large and resolution smaller than normal by 3 to X. But the /e height reduces by the same amount that the resolution is reduced Copyright 6 Regents of Uniersity of California R R 5

16 EE Applied EM Fall 6, Neureuther Lecture #5 Ver //6 Surface Topography Can Aid Guided Waes arrington 4.8 Corrugated Surface x Z down Impedance looing down into the corrugations is inductie. Impedance looing into slot is that of a parallel plate waeguide terminated in a short. Slsot must be narrow compared to a waelength Depth must be >5% of waelength to contribute. For plasmons effects might add which waelength should be used Copyright 6 Regents of Uniersity of California Z Z up down i iω µ tan + tan d d i377tan 6 d