Lecture 9 - Metal Optics An Introduction

# We are looking for solutions that look like 0 0 exp m

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• We are looking for solutions that look like: ( ) ( ) 0, ,0 exp m ym xm zm H i k x k z t ω = + H ( ) ( ) ,0, exp m xm zm xm zm E E i k x k z t ω = + E ( ) ( ) 0, ,0 exp d yd xd zd H i k x k z t ω = + H ( ) ( ) ,0, exp d xd zd xd zd E E i k x k z t ω = + E • Mathematically: z<0 z>0

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i i i ε ∇× = E H t ( Dispersion Relation Surface-Plasmon Polaritons • Start with curl equation for H in medium i (as we did for EM waves in vacuum) ) ( ) 0, ,0 exp i yi xi zi H i k x k z t ω = + H where ( ) ( ) ,0, exp i xi zi xi zi E E i k x k z t ω = + E ( ) ( ) , , ,0, ,0, yi yi zi xi zi xi zi yi xi yi i xi i zi H H H H H H ik H ik H i E i E y z z x x y ωε ωε = = − zi yi i xi k H E ωε = − • We will use that: zm yi m xm k H E ωε = − zd yd d xd k H E ωε = − • E // across boundary is continuous: , , x m x d E E = zm zd ym yd m d k k H H ε ε = • H // across boundary is continuous: ym yd H H = Combine with: zm zd ym yd m d k k H H ε ε = zm zd m d k k ε ε =
Dispersion Relation Surface-Plasmon Polaritons Relations between k vectors • Condition for SP’s to exist: zm zd m d k k ε ε = • Relation for k x ( Continuity E // , H // ) : xm xd k k = • For any EM wave: zd k zm k Example 1 m ε = − 1 d ε = z z 2 2 2 x zi i k k c ω ε + = 2 2 sp x i zi k k k c ω ε = = • Both in the metal and dielectric: zm zd m d k k ε ε = 1/ 2 m d x m d k c ε ε ω ε ε = + Dispersion relation homework Example Air SiO 2 true at any boundary

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Dispersion Relation Surface-Plasmon Polaritons 1/ 2 m d x m d k c ε ε ω ε ε = + Plot of the dispersion relation • Last page: • Plot dielectric constants • Low ω : 1/ 2 lim m m d x d m d k c c ε ε ε ω ω ε ε ε →−∞ = + • At ω = ω sp (when ε m = - ε d ): ω r ε p ω sp ω d ε d ε dielectric metal x k → ∞ ω k sp ω d k c ω ε = • Note: Solution lies below the light line
Dispersion Relation Surface-Plasmon Polaritons Dispersion relation plasma modes and SPP • Note: Higher index medium on metal results in lower ω sp 2 2 1 p m d ω ε ε ω = = − 2 2 2 p d ω ω ε ω = − 2 2 1 p d ω ω ε = + ω = ω sp when: Metal/air Metal/dielectric with ε d 1 p d ω ω ε = + , SP Air ω , d SP ε ω

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Excitation Surface-Plasmon Polaritons with Electrons Excitation with electrons • First experiments with high energy electrons • Whole dispersion relation can be investigated • Measurement: Energy loss Direction of e’s: • Low k’s are hard!
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