Refractive index dispersion and Drude model Optics, Eugene Hecht, Chpt. 3

Dielectrics Electric field is reduced inside dielectric –Space charge partly cancels –E / E v =  /  0 Also possible for magnetic fields –but usually B = B v and  =  0 Result: light speed reduced v = c  (    ) = c/n < c Wavelength also reduced  = 0 /n E-field Dielectric Index of refraction: n

Conventions Polarization of materials Separate into material and vacuum parts –  E =  0 E + P –linear material: P =  0  E Material part is due to small charge displacement Similar equation for magnetic polarization – B /  = B /  0 + M Most optical materials have  =  0 Refractive index n 2 = (  /  0 ) (  /  0 ) = [1 + P / (  0 E)] / [1 +  0 M/B] Drop magnetic part n 2 = [1 + P / (  0 E)]

Material part of polarization Polarization due to small displacements Examples: –Polar molecules align in field –Non-polar molecules – electron cloud distorts Optical frequencies –Nucleus cannot follow fast enough Too heavy –Consider mainly electron cloud Distorted electron cloud

Model of atom Lowest order – everything is harmonic oscillator Model atom as nucleus and electron connected by spring Newton’s law: F = m a Spring restoring force: F R = - k x = - m   2 x –Resonant freq of mass-spring:   =  k/m Driving force: F D = q e E Damping force: F  = - m  v Resultant equation: q e E - m  dx/dt - m   2 x = m d 2 x/dt 2 Free oscillation: (E=0,  =0) –d 2 x/dt 2 +   2 x = 0 Use complex representation for E –E = E 0 e i  t Forced oscillation: –motion matched drive frequency –x = x 0 e i  t Result: x 0 = (q/m) E 0 / [   2 + i  ]

Refractive index & dispersion Drude model Polarization of atom –Define as charge times separation –P A = q e x Material has many atoms: N Material polarization: P = q e x N Recall previous results n 2 = [1 + P / (  0 E)] x 0 = (q/m) E 0 / [   2 + i  ] Result is dispersion equation: Correction for real world complications: Sum over all resonances in material f is oscillator strength of each transition ~ 1 for allowed transition

Sample materials Polar materials Refractive index approx. follows formula Resonances in UV Polar materials also have IR resonances –Nuclear motion – orientation

Anomalous dispersion Above all resonance frequencies Dispersion negative Refractive index < 1 v > c X-ray region

Metals and plasma frequency “Free” conduction electrons – resonance at zero  0 = 0 Metals become transparent at very high frequency – X-ray Neglect damping At low frequency n 2 < 0 –refractive index complex –absorption At high frequency –n becomes real –like dielectric –transparency Plasma freq

Skin depth in metals Electrons not bound Current can flow Conductance  ~ 1/R causes loss Maxwell’s equations modified Wave solution also modified –Express as complex refractive index –n complex = n R – i  c / (2  ) –E = E 0 e -  z/2 e i(kz-  t) Result for propagation in metal: I = I 0 e -  z, 1/  = skin depth Metals: 1/  << Example copper: – = 100 nm, 1/  = 0.6 nm = / 170 – = 10  m, 1/  = 6 nm = / 1700 – = 10 mm, 1/  = 0.2  m = / 50,000 – 1/  ~  Similar to n >> 1 Strong reflection – not much absorption Metal Density Ro f skin depth (microOhm cm) (GHz) (microns) Aluminum 2.70 g/cc2.824; Copper 8.89 g/cc1.7241; Gold 19.3 g/cc2.44; Mercury g/cc ;10, Silver 10.5 g/cc1.59; Drude -- low frequency limit   0

Reflectivity of metals Assume perfect conductor No electric field parallel to interface Reflectivity at normal incidence (assume n i = 1) Power reflected R = r r*  1 for large absorption E field incident reflected metal Standing wave -- zero at surface Normal incidence reflection from metal

Plasmons Assume  0 = 0 for conduction electrons -- keep damping Transition occurs when optical frequency exceeds collision frequency –depends on dc resistivity –lower resistivity = higher frequency transition Above collision frequency -- Plasmons Plasmons quenched at plasma frequency Example -- silver –  = 6.17 x 10 7 /  -m,  plasma = 9.65 x Hz (311 nm, 4 eV) –  e = 1/(13 fs) = 7.7 x Hz –plasmons beyond ~ 23.5 microns wavelength

Plasmons and nano optics Small metal particles can act like inductors, capacitors Maxwell’s equation for current density: –Separate into vacuum and metal parts Vacuum (or dielectric) part is capacitor Metal part is inductor plus series resistor RLC circuit parameters –Resonance frequency  0 =1/sqrt(LC) =  plasma –Resonance width  = R/L = collision Structure geometry can increase L and C –Strong local field enhancement possible in capacitor conductivity Displacement current Vacuummetal metal dielectric LC Nano optic RLC circuit

“Left hand” materials: (E in plane of incidence) Sign of  and  both negative Strange properties Refraction backward Example -- E parallel, P-polarization Two components of E Parallel to surface –E i cos  i + - E r cos  r = E t cos  t Perpendicular to surface –1. Space charge attenuates E t –  i E i sin  i +  r E r sin  r =  t E t sin  t –Sign of  t is negative –2. Use Snell’s law –n i E i + n r E r = n t E t B is parallel to surface –same as perpendicular E r parallel = (n t cos  i - n i cos  t ) / (n t cos  i + n i cos  t ) t parallel = (2n i cos  i ) / (n t cos  i + n i cos  t ) Interface ii rr tt EiEi ErEr EtEt E’ t ’t’t nini ntnt Propagation direction E x B Momentum

Left handed materials - fabrication Need sign of  and  both negative Problem: magnetic part usually ~1 Solution: Fool the EM field –LC circuit – material in capacitor gap indirectly modifies magnetic material Loops are inductors Gap is capacitor LC circuit E B k Artificial “left-hand” material