Lecture 7 Light-Matter Interaction Part 1 Basic excitation and coupling. EECS Winter 2006 Nanophotonics and Nano-scale Fabrication P.C.

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1 Lecture 7 Light-Matter Interaction Part 1 Basic excitation and coupling EECS Winter 006 Nanophotonics and Nano-scale Fabrication P.C.Ku

2 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)? EECS Nanophotonics and Nanoscale Fabrication by P.C.Ku

3 Schedule for the rest of the semester Introduction to light-matter interaction (today): How to determine ε(r)? The relationship to basic excitations. Basic excitations and measurement of ε(r). (1/31) Structure dependence of ε(r) overview (/) Surface effects (/7 & /9): Surface EM wave Surface polaritons Size dependence Case studies (/14 /1): Quantum wells, wires, and dots Nanophotonics in microscopy Nanophotonics in plasmonics Dispersion engineering (/3 3/9): Material dispersion Waveguide dispersion (photonic crystals) EECS Nanophotonics and Nanoscale Fabrication by P.C.Ku 3

4 How to determine ε(r)? As a simplest example for metals, if we could treat the electron and the ion classically using Newton s Laws and the movement of electrons and ions follows the incoming electric field linearly: mx + mγ x = qe e x = m qe e i iωt ( ω + iγω) nq ( ω iγ) ωp ( ω iγ) ε = ε0 + P / E = ε0 nqx / E = ε0 ε 0 1 mω( ω + γ ) ω( ω + γ ) ωp If ω γ, ε i Highly absorptive ωγ ω If ω γ, ε ε 1 i iωt ( + γ ) ω ω p 0 EECS Nanophotonics and Nanoscale Fabrication by P.C.Ku Real & positive if ω > ω p 4

5 What happens at ωp? R T ωp ω If the electron density changes slightly from its equilibrium: n = n + n There will be an induced electric field: E = e( n n0 ) ( en) The continuity equation: (- env) + = 0 t Newton's law: ee = mv n1 If there is no field when n1 = 0, we get + ω pn1 = 0 t ωp Without interaction with external EM field: ω 0 1 SHO for electron density fluctuation at ωp EECS Nanophotonics and Nanoscale Fabrication by P.C.Ku 5

6 Plasmons ωp or λp: In metals: ~100nm In semiconductors: ~1µm EECS Nanophotonics and Nanoscale Fabrication by P.C.Ku 6

7 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. EECS Nanophotonics and Nanoscale Fabrication by P.C.Ku 7

8 Conservation of energy and momentum E1,P1 Ef,Pf E1+ E = E P + P = P 1 f f E,P k E = for massive particles (electrons, etc.) m or ω for massless particles (photons, phonons, etc.) P = k EECS Nanophotonics and Nanoscale Fabrication by P.C.Ku 8

9 Conservation conditions Conservation of energy and momentum b/w the initial and final states. The final state must exist! (The interaction, however, can go through one or more intermediate states which can be virtual.) In addition, if the interaction has rotational symmetry (i.e. it depends only on r e.g. the Coulomb interaction), the angular momentum must also be conserved selection rule. We have also learned the electric charge is a conserved quantity. EECS Nanophotonics and Nanoscale Fabrication by P.C.Ku 9

10 Fermi s golden rule The discussions above can be described quantitatively by the Fermi s golden rule which results from the lowestorder contribution from the time-dependent perturbation theory. w= π f HI i g( Ef ) δ ( E ( Ef Ei)) Energy conservation Final state must exist Momentum conservation and selection rule EECS Nanophotonics and Nanoscale Fabrication by P.C.Ku 10

11 Time-dependent perturbation theory First order Second order EECS Nanophotonics and Nanoscale Fabrication by P.C.Ku 11

12 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. EECS Nanophotonics and Nanoscale Fabrication by P.C.Ku 1

13 Kramers-Kronig relation Because the dielectric function represents a response of the matter to the incoming EM field and any physical response should be causal (nothing happens before the cause), ε(ω) satisfies the Karmers-Kronig relation as follows: [ ] [ εω] [ εω ] ω [ εω ] 1 Im ( ') ' Im ( ') Re ε ( ω) = ε + Pr dω' = ε + Pr dω' 0 0 π ω' ω π ω' ω 0 ω Re [ εω ( ')] ε0 dω π ω' ω 0 Im ( ) = Pr ' For the derivation of the Kramers-Kronig relation, please refer to J. D. Jackson, Classical Electrodynamics nd ed., section 7.10 (c). EECS Nanophotonics and Nanoscale Fabrication by P.C.Ku 13

14 Example of a sharp resonance [ ] Im εω ( ) = κδω ( ω) [ ] 0 [ ε ω ] ω'im ( ') Re εω ( ) = ε + Pr dω' 0 π ω' ω 0 κ ωδω ' ( ' ω0 ) Pr ' 0 dω π ω' ω = ε + = ε + κ ω 0 0 π ω0 ω Re(ε) EECS Nanophotonics and Nanoscale Fabrication by P.C.Ku 14

15 Conservation of E & P revisited ω photon optical phonon k For resonance (excitation) to happen, two dispersion curves must intersect one another. EECS Nanophotonics and Nanoscale Fabrication by P.C.Ku 15

16 Coupling of two classical harmonic oscillators Please refer to S. L. Chuang, Physics of Optoelectronic Devices, section 8.. EECS Nanophotonics and Nanoscale Fabrication by P.C.Ku 16

17 ε(r) for an ensemble of harmonic oscillators iωt iωt mx + mγx = kx qeie mω0 x qeie iωt qeie x = m ( ω ω0 ) + iγω nq ω p ε = ε0 + P / E = ε0 nqx / E = ε0 = ε 0 1 m ( ω ω0 ) + iγω ( ω ω0 ) + iγω ω If ω ω ε p γ / 0, Im ω0 ( ω ω0) + ( γ /) ω ω ω p 0 Reε ε0 1 ω0 ( ω ω0 ) + ( γ / ) EECS Nanophotonics and Nanoscale Fabrication by P.C.Ku 17

18 Polariton dispersion relation Polaritons are mixtures (coupling) of photons and harmonic oscillators in the media. ω p k ε = ε0 1 = ( ω ω0 ) + iγω ω µ 0 when γ = 0 ω ω ( ω ω 0 + ωp ) k = c ω ω0 ω + ω 0 p Photons In free space ω 0 k EECS Nanophotonics and Nanoscale Fabrication by P.C.Ku 18