Lecture 2: Dispersion in Materials 5 nm Course Info Course webpage •Is now up and running www.ecepurdue.edu/~shalaev Let me know what you think! • Direct questions • Topics ? • Format ? • Big comments on the nanocourse are most welcome! • Ask questions any time………. Previous Lecture: Maxwell + Wave Equation Speed of the EM wave: 2 2 2 ∂ E 2 1 ∂ U (rt, ) Compare ∇=E µεε00r and general wave Equation: ∇=U (rt, ) ∂t 2 v22∂t 11 c2 v2 = = 0 µε00 εrr ε 2 -12 2 3 -7 2 8 2 Where c0 = 1/(ε0 μ0) = 1/((8.85x10 C /m kg) (4π x 10 m kg/C )) = ( 3.0 x 10 m/s) Optical refractive index c Refractive index is defined by: n =0 =εχ =1 + v r Note: Including polarization results in same wave equation with a different εr c0 becomes v Refractive Index Various Materials 3.4 3.0 2.0 Refractive index: n index: Refractive 1.0 0.1 1.0 10 λ (µm) Dispersion Relations Dispersion relation: ω = ω(k) n2 ∂2 Er( ,t) Derived from wave equation ∇=2 Er( ,t) c22∂t Substitute: EE( z, t) = Re{ ( z ,ωω) exp(−+ ikz i t)} n2 Result: k 22= ω ω c2 Check this! c2 ω 22= k v n2 g k dωω cc c Group velocity: vg ≡==== dk k n εχr 1+ ω Phase velocity: v ≡ ph k A z Absorption and Dispersion of EM Waves Transparent materials can be described by a purely real refractive index n EM wave: EE( z, t) = Re{ ( z ,ωω) exp(−+ ikz i t)} c2 ω Dispersion relation ω 22= k kn= ± n2 c Absorbing materials can be described by a complex n: n= n' + in '' ω ωω α It follows that: k=±( nin' + '') =± ni ' + n '' ≡±β − i c cc 2 α Investigate + sign: EE( zt,) = Re( z ,ωβ) exp−− i z z + i ω t 2 Traveling wave Decay ω Note: β =n'' = kn n’ acts as a regular refractive index c ω α =−=−2n '' 2 kn '' α is the absorption coefficient c Summary and Future Directions Maxwell’s Equations ∂B ∂D ∇⋅D = ρ f ∇⋅B =0 ∇×E =− ∇×HJ = + ∂t ∂t Curl Equations lead to 22 2 ∂∂EP ∇=E µε + µ (under certain conditions) 00∂∂tt220 Linear, Homogeneous, and Isotropic Media PE= εχ0 Wave Equation with v = c/n n2 ∂2 Er( ,t) ∇=2 Er( ,t) Solutions: c22∂t Today: In real life the relation between P and E is dynamic ω= εχω ω Pk( ,,) 0 ( ) Ek( ) Materials have a response time Lorentz model: Helps to understand the optical response of real materials χ(ω), ε (ω), n(ω) n’ and n’’ vs χ’ and χ’’ vs εr’ and εr’’ All pairs (n’ and n’’, χ’ and χ’’, εr’ and εr’’) describe the same physics For some problems one set is preferable for others another n’ and n’’ used when discussing wave propagation α EE( zt,) = Re( z ,ωβ) exp−− i z z + i ω t where β = kn0 ' and α = −2kn '' 2 0 Phase propagation absorption χ’ and χ’’ used when discussing microscopic origin of optical effects εr’ and εr’’ As we will see today… Inter relationships 22 Example: n and ε (εr') ++( εε rr '') ' r n' = 22 2 ε r '=(nn ') − ( '') From n = ε r and ε = 22 r '' 2nn ' '' (ε') +−( εε '') ' n'' = r rr n'+= in ''εεrr ' + i '' 2 Light Propagation Dispersive Media Relation between P and E is dynamic The relation : Pr ( ,, tt ) = εχ 0 Er ( ) assumes an instantaneous response +∞ =ε − In real life: Pr( ,t) 0 ∫ dt ' x( t t ') Er( ,' t ) −∞ P results from response to E over some characteristic time τ : Function x(t) is a scalar function lasting a characteristic time τ : E(t’) x(t-t’) x(t-t’) = 0 for t’ > t (causality) t’ t’ = t - τ t’ = t EM waves in Dispersive Media Relation between P and E is dynamic +∞ =ε − Pr( ,t) 0 ∫ dt ' x( t t ') Er( ,' t ) −∞ EM wave: Er( ,t) = Re{ Ek( ,ωω) exp(−i kr ⋅+ it)} Pr( ,t) = Re{ Pk( ,ωω) exp(−i kr ⋅+ it)} Relation between complex amplitudes Pk( ,,ω) = εχω0 ( ) Ek( ω) (Slow response of matter ω-dependent behavior) This follows by equation of the coefficients of exp(iωt) ..check this! It also follows that: εr ( ω) = ε0 1 + χω( ) Next: We will model the frequency dependence of χ, ε, and n Linear Dielectric Response of Matter Behavior of bound electrons in an electromagnetic field • Optical properties of insulators are determined by bound electrons EL - Lorentz model e ,m • Charges in a material are treated as harmonic oscillators C, γ r maFel=++ E, Local F Damping F Spring (one oscillator) p = −er dd2rr m+ mγω +=− CrE eexp( − it) 2 L + dt dt Nucleus • The electric dipole moment of this system is: pr= −e dd2pp m+ mγω += CpE e2 exp( − it) dt2 dt L • Guess a solution of the form: dp d 2p pp=exp( −itω ) ;=−−iωωp exp( it) ; =−−ωω2p exp( it) 0 dt 0 dt 2 0 22 −mωp0 − im γω pp 00 += C e EL Solve for p0(EL ) Atomic Polarizability Determination of atomic polarizability 22 • Last slide: −mωp0 − im γω pp 00 += C e EL Ce2 −−ω2pi γω pp + = E (Divide by m) 0 00mmL 2 Define as ω0 (turns out to be the resonance ω) e2 1 = pE0 22 L miω0 −− ω γω Atomic polarizability (in SI units) p e2 1 αω≡=0 • Define atomic polarizability: ( ) 22 EL miω0 −− ω γω Resonance frequency Damping term Characteristics of the Atomic Polarizability Response of matter (P) is not instantaneous ω-dependent response p e2 1 αω=0 = = θω • Atomic polarizability: ( ) 22 Aiexp ( ) EL miω0 −− ω γω • Amplitude e2 1 A = smaller γ 2 1/2 m ω2−+ ω 2 γω 22 ( 0 ) Amplitude • Phase lag of α with E: 180 γω γ θ = −1 smaller tan 22 ωω0 − 90 Phase lag 0 ω0 ω Relation Atomic Polarizability (α) and χ: 2 cases Case 1: Rarified media (.. gasses) E-field photon • Dipole moment one atom, j : pEjj= αω( ) L Density 11 • Polarization vector: P=∑∑ pj =ααjL EE = N L VVjj Occurs in Maxwell’s equation.. sum over all atoms e2 1 αω= ( ) 22 miω0 −− ω γω Ne2 1 = =εχ = εχ P 22 ELL00 EE miω0 −− ω γω Ne2 1 χω= Microscopic origin susceptibility: ( ) 22 ε00mi ω−− ω γω 2 ω 2 2 Ne p • Plasma frequency defined as: ω = χω( ) = p ω22−− ω γω ε 0m 0 i Remember: ε and n follow directly from χ Frequency dependence ε ω 2 • Relation of ε to χ : εχ=+=+ p r 1122 ω0 −− ωi γω ω 2 εε'+ '' =++=+ χχ'" p rrii1122 ω0 −− ωi γω εωr ( ) ε r '' 22 2 ωωp ( 0 − ω) εr '1=+=+ χω '( ) 1 2 ω2−+ ω 2 γω 22 ( 0 ) ω 2 ε ' ω2 γω + p r χ '0= p 1 2 εr ''= χω ''( ) = ω 2 22 22 0 (ω0 −+ ω) γω 1 2 ω p ω χ = ω0 2 ≈ ω p ω0 Propagation of EM-waves: Need n’ and n’’ Relation between n and εr εωr ( ) ε n = ε r r '' 22 ε r ' ε r '=(nn ') − ( '') χ '0= ε r ''= 2nn ' '' 1 ω ω0 n(ω) n' • ω << ω0 : High n’ low vph =c/n’ n'' n'1= • ω ≈ ω0 : Strong ω dependence vph Large absorption (~ n’’) • ω >> ω : 0 n’ = 1 vph =c ω ω0 Realistic Rarefied Media Realistic atoms have many resonances • Resonances occur due to motion of the atoms (low ω) and electrons (high ω) Ne2 1 χ = k ∑ 22 k ε0mi ωk −− ω γω Where Nk is the density of the electrons/atoms with a resonance at ωk Example of a realistic dependence of n’ and n’’ Molecule Atomic Electron rotation vibration excitation α= 2k0n’’ n’ n’ > 1 indicates presence high ω oscillators ω1 ω2 ω3 Log (ω) Back to Relation Atomic Polarizability (α) and χ: Case 2: Solids Solid pi E-field? • Atom “feels” field from: 1) Incident light beam 2) Induced dipolar field from other atoms, pi • Local field: ELI= EE0 + Local field Induced dipolar field from all the other atoms Field without matter Electric Susceptibility of a Solid Local field • Local field: ELI= EE0 + Local field Field without matter All the other atoms Induced dipolar field • Example: For cubic symmetry: (Solid state Phys. Books, e.g. Kittel p381-390) P EEL = + (Similar relations can be derived for any solid) 3ε 0 Polarization of a solid • Solid consists of atom type j at a concentration Nj PP PE=∑εα00NjjL = ∑ εα Njj E += ε 0 ∑∑ NNjj α E + jj α j j33ε 0 jj p j ∑ N jjα 1 P j PE1−=NNαε α χ = = ∑∑jj 0 jj ε 1 3 jj 0 E 1− ∑ N jjα 3 j Clausius-Mossotti Relation Polarization of a solid • Susceptibility: ∑ N jjα P χ = = j ε 1 0 E I 1− ∑ N jjα 3 j • Limit of low atomic concentration: ….or weak polarizability: χα≈ ∑ N jj II pretty good for gasses and glasses j Clausius-Mossotti • By definition: εχ = 1 + ε −11 = ∑ N jjα III • Rearranging I gives ε + 23j • Conclusion: Dielectric properties solids related to atomic polarizability • This is very general!!.