Download Nanophotonics: Determining the Dielectric Function and Light-Matter Interaction - Prof. Pc and more Study notes Electrical and Electronics Engineering in PDF only on Docsity! Lecture 7 – Light-Matter Interaction Part 1 Basic excitation and coupling EECS 598-002 Winter 2006 Nanophotonics and Nano-scale Fabrication P.C.Ku 2EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku What we have learned? Nanophotonics studies the interaction of photons and matters (including electrons, nuclei, phonons, plasmons, excitons, and etc.) ε(r) Uncertainty principle and quantization How the EM wave interacts with a medium with known ε(r)? 5EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku What happens at ωp? ω R ω T ωp ωp 0 1 0 If the electron density changes slightly from its equilibrium: There will be an induced electric field: ( ) ( )The continuity equation: (- ) 0 Newton's law: If there n n n E e n n enenv t eE mv = + ∇⋅ = − − ∂ − ∇⋅ + = ∂ − = 2 21 1 12 is no field when 0, we get 0p nn n t ω∂= + = ∂ SHO for electron density fluctuation at ωp Without interaction with external EM field: 6EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku Plasmons ωp or λp: In metals: ~100nm In semiconductors: ~1µm 7EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku Interaction between light and matter In reality, however, we cannot treat microscopic physics with Newton’s Laws. But without going into details, we can make a few observations just by simple physical intuition. Despite of the need for QM, from the example in metals, we can still treat the interaction between light and matter as the coupling between two harmonic oscillators. 10EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku Fermi’s golden rule The discussions above can be described quantitatively by the Fermi’s golden rule which results from the lowest- order contribution from the time-dependent perturbation theory. 22 ( ) ( ( ))I f f iw f H i g E E E E π δ= − − Energy conservation Final state must exist Momentum conservation and selection rule 11EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku Time-dependent perturbation theory First order Second order 12EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku Framework of the calculation for ε(r) Near resonance, the first order dominates. But if far away from the resonance, we need to include the second order effect (i.e. one intermediate state). This makes the QM calculation of the entire dispersion curve rather tedious. In most of the cases, the determination of only the imaginary part of ε(r) is much simpler. It relates to only a few resonances. We will prove in the following simply by knowing Im(ε(r)) allows us to determine the entire Re(ε(r)) without going through any lengthy QM calculations. 15EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku Conservation of E & P revisited ω k photon optical phonon For resonance (excitation) to happen, two dispersion curves must intersect one another. 16EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku Coupling of two classical harmonic oscillators Please refer to S. L. Chuang, Physics of Optoelectronic Devices, section 8.2. 17EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku ε(r) for an ensemble of harmonic oscillators 2 0 2 2 0 22 0 0 0 0 2 22 2 00 2 0 2 2 0 0 2 0 0 2 0 0 ( ) / / 1 ( )( ) / 2If , Im ( ) ( / 2) Re 1 2 ( ) ( / i t i t i i i t i p p p mx m x kx qE e m x qE e qE ex m i nqP E nqx E im i ω ω ω γ ω ω ω γω ω ε ε ε ε ε ω ω γωω ω γω ω γω ω ε ω ω ω γ ω ω ωε ε ω ω ω γ − − − + = − − ≡ − − ⇒ = ⎡ ⎤− +⎣ ⎦ ⎡ ⎤ = + = − = − = −⎢ ⎥− +⎡ ⎤− + ⎢ ⎥⎣ ⎦⎣ ⎦ ≈ ≈ − + − ≈ − − + 22) ⎡ ⎤ ⎢ ⎥ ⎢ ⎥⎣ ⎦
