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Resonances and optical constants of dielectrics: basic light-matter interaction Understanding the Rainbow Dielectric materials: All charges are attached to specific atoms or molecules Response to an electric field E: Microscopic displacement of charges P 0 E 0 1E Relative dielectric permittivity describes how a material is polarized in response to an electric field depends on frequency: () If we know the relation between P and E we can solve Maxwell’s equations E 1 0 P H 0 E P H 0 J t t H E 0 t leading to the wave equation: 2 2 1 E P J 2 E 0 0 2 2 2 c t t t In vacuum (P = J = 0): E E 0 e i (k r t ) k 2 c c 1 0 0 Deriving the relation between P and E in a dielectric Equation of motion of the electron: P Nex d 2x dx m 2 m kx eE dt dt : damping coefficient for given material k: restoring-force constant resonance frequency: 0 k / m assume E is varying harmonically, and also Ne2 / m P Nex 2 E 2 0 i x x 0 e it Inserting P(E) in wave equation 1 2E 2P J E 2 2 0 2 0 c t t t 2 gives: 2 1 Ne 2 E 2 1 c m 0 2E 1 2 2 i t 2 0 solution: E E0ei ( k z z t ) with complex propagation constant kz = + iα : 2 kz c 2 E E0e Ne 2 1 m 0 1 2 2 0 i z i ( z t ) e kz 2 So that… n c So that we find the refractive index of the dielectric: Ne 2 n 1 m 0 2 1 2 2 0 i For a dielectric with multiple resonances: 2 Ne n2 1 m 0 j fj 2 j 2 i j Rainbow: why red outside, blue inside ? Rainbow: why red outside, blue inside ? Red (small frequency): smaller n Blue (high frequency): larger n Light scattering from small resonant particles Metal nanoparticle plasmons What is a plasmon? “plasma-oscillation”: density fluctuation of free electrons + + + Ne2 Plasmons in the bulk oscillate at p m 0 determined by the free electron density and effective mass drude - - - k + - + Plasmons confined to surfaces that can interact with light to form propagating “surface plasmon polaritons (SPP)” Confinement effects result in resonant SPP modes in nanoparticles Sphere in a uniform static electric field particle can be considered as a dipole: in a metal cluster placed in an electric field, the negative charges are displaced from the positive ones 3 m p 40 R m E0 2 m electric polarizability of a sphere α resonant enhancement of p if ( ) 2 m minimum negative real dielectric constant ε1(ω) Bohren and Huffman (1983), p.136 4 0 R 3 m 2 m ε = ε1(ω)+i ε2(ω) = dielectric constant of the metal particle εm = dielectric constant of the embedding medium usually real and taken independent of frequency Derivation using quasi-static approximation r r r ik rr t r 2 V r ,t f r e , k r r r r r it r k r 2 V r ,t f r e Einc k i kr t E inc r ,t E 0 e y x Einc k it Einc r ,t E 0 e y x Derivation using quasi-static approximation Equations: 1 0 2 0 E0 r a r a m r q z a Boundary conditions: 1 2 r a , 1 m 2 r r r a , lim 2 E 0 z r Jackson (1998), p.157 Bohren and Huffman (1983), p.136 Derivation using quasi-static approximation Equations: 1 0 2 0 E0 r a r a m r q z a Solution: m 3 m 1 E 0 r cos q E 0 r cosq E 0 r cos q 2 m 2 m cos q pcos q 3 m 2 E 0 r cos q a E 0 2 E 0 r cos q 2 2 r 4 r m m r r p 0 m E 0 with: Sphere in electromagnetic field (a << ): r it m r 4 a 3 p 0 m E 0e 2 m Jackson (1998), p.157 Bohren and Huffman (1983), p.136 Metal nanoparticles: • Extinction = scattering + absorption • Large field enhancement near particle n=1.5 Ienh 20 Au 550nm At resonance, both scattering and absorption are large albedo = scattering / extinction = ssca/(sabs+ssca) Reosnance spectra Groupings of 35nm Au NPs are obtained after surface ligand exchange (thio-PEG instead of BSPP) Optical density 0.8 35nm Dimer Trimer 0.6 0.4 0.2 0.0 400 450 500 550 600 650 700 Wavelength (nm) Extinction spectra in water Resonance tunable by dielectric environment Ag, D=100 nm Si3N4 (n=2.00) Q D O Si (n=3.5) Q D H Optics Express (2008), in press Resonance spectra for particles on surface σscat normalized to particle area 14 12 Q Qscat, Qsubs 10 30 nm 30nm tot 8 sub 6 D 4 2 0 500 10 nm10nm 600 700 800 wavelength (nm) 900 1000 Appl. Phys. Lett. 93, 191113 (2008) Other applications of nanoparticles Old: New: (but the same principle) Different materials/shapes: distinct colors All particles are driven by the external field and by each other Au colloids in water (M. Faraday ~1856) Focusing and guidance of light at nanometer length scales (image: CALTECH) Interaction between particles An isolated sphere is symmetric, so the polarization direction does not matter. LONGITUDINAL: restoring force reduced by coupling to neighbor Resonance shifts to lower frequency TRANSVERSE: restoring force increased by coupling to neighbor Resonance shifts to higher frequency Near field enhancement in gaps between particles: nanoscale antenna