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Transcript
Light and Matter Classical electrodynamics Tim Freegarde School of Physics & Astronomy University of Southampton Electromagnetic waves • electrostatic force acts through vacuum • retardation due to finite speed of light, enhanced by inertia of any charged particles Q1 a F Q1Q2 40 r 2 Q2 • net force due to oscillating dipole F t Q2 Q1 40 r 3 at r c 2 Maxwell’s equations • Gauss • no monopoles • Faraday • Ampère dv E.dDS 0 B.dS 0 E.ds t B.dS DE B . d s J 0 H J 0 .dS t t D E 0E P 0 B 0 0 H Μ Β E E J t E B J0 J 00 t t constitutive equations 3 Constitutive equations D DE, B D 0 r E D 0E P H HΕ, B B 0 r H B 0 H Μ J JE, B • conservation of charge J E J 0 t constitutive equations 4 Maxwell’s equations D D 0 r E D 0E P B 0 B 0 r H B 0 H Μ J E constitutive equations Β E t D H J t conservation of charge J 0 t 5 Electromagnetic wave equation use constitutive equations to reduce electric & magnetic 0 fields to single functions D D 0 r E B 0 B 0 r H B 0 H Μ J E differentiate equations to allow electric or magnetic constitutive equations field to be eliminated Β E t D H J t conservation of charge D EP Jvector relations 0 to apply t equation produce wave 6 Sinusoidal plane wave solutions D B 0 Β E t D H J t Er, t E0 cost kz E y Bx E z k y t z B E H y H z Dx x , y y , x y t z 7 Maxwell’s equations D D D0Er ,EB D 0E P B 0 BH 0Εr,H H B B 0 H Μ J JE, B constitutive equations Β E t D H J t conservation of charge J 0 t 8 Electromagnetic wave equations use constitutive equations to reduce electric & magnetic 0 fields to single functions D D 0 r E B 0 B 0 r H B 0 H Μ J E differentiate equations to allow electric or magnetic constitutive equations field to be eliminated Β E t D H J t conservation of charge D EP Jvector relations 0 to apply t equation produce wave 9 Constitutive equations D D 0 r E B 0 B 0 r H Β E t D H J t J E J 0 t use constitutive equations to reduce electric & magnetic fields to single functions differentiate equations to allow electric or magnetic field to be eliminated apply vector relations to produce wave equation 10 Electromagnetic waves in isotropic media • atoms and molecules are polarized by applied fields • induced polarization • alignment of permanent dipole moment • polarization modifies field propagation: refractive index; absorption 11 Constitutive equations • define polarization P and magnetization M • governed by properties of the optical medium • vapours, dielectrics, plasmas, metals D 0 r E apply Newtonian mechanics to determine response of medium to applied field B 0 r H J E • magnetization usually too slow to have effect at optical frequencies • assume (for now) D[E] to be linear and scalar use result to write (complex) conductivity, dielectric constant etc. insert into constitutive equations and hence derive wave equation as usual 12 Vapours and dielectrics m02 2 x 2 • bound or massive nuclei • electrons confined in harmonic potential • restoring force proportional to displacement • Newtonian dynamics • frequency dependence (dispersion) Ne2 m P 2 E 2 0 13 Metals and conductors • free charges • diffusion in response to applied field • equilibrium velocity characterized by conductivity v • frequency dependence (dispersion) E • damped solutions (absorption) • dissipation through resistive heating 14 Plasmas and the ionosphere • independent, free charges • inertia in response to applied field • Newtonian dynamics • frequency dependence (dispersion) Ne2 P E 2 m 15 Electromagnetic energy density & flow D D 0 r E D 0E P B 0 B 0 r H B 0 H Μ J E constitutive equations Β E t D H J t conservation of charge J 0 t 16 Electromagnetic energy density & flow • BBC Radio 4 long wave transmitter, Droitwich frequency: 198 kHz l = 1515 m power: 400 kW • MSF clock transmitter, Rugby frequency: 60 kHz power: 60 kW l = 5000 m 17 Constitutive equations • define polarization P and magnetization M • governed by properties of the optical medium • vapours, dielectrics, plasmas, metals D 0 r E apply Newtonian mechanics to determine response of medium to applied field B 0 r H J E • magnetization usually too slow to have effect at optical frequencies • assume (for now) D[E] to be linear and scalar use result to write (complex) conductivity, dielectric constant etc. insert into constitutive equations and hence derive wave equation as usual 18 Continuity conditions • transverse waves on a guitar string T T 2 y 2 y 2 T 2 t x x • continuity of y … finite extension • conservation of energy • continuity of y x … finite acceleration • conservation of momentum 19 Continuity conditions • electromagnetic fields • parallel components E//,1 E//,2 • conservation of energy H //,1 H //,2 • perpendicular components D,1 D, 2 • conservation of momentum B,1 B, 2 E// 1 2 D 1 2 20 Reflection at metal and dielectric interfaces • electromagnetic fields • parallel components • conservation of energy E//,1 E//,2 E1 H t k 1 z HE2 t k1 2 z //,1 //,2 D,1 D, 2 x • conservation of momentum y B ,1 A B, 2 2 apply continuity conditions for separate components E3 t k1 z • perpendicular components combine forward and reflected waves to give total E// fields for each region D B 1 z hence derive fractional transmission and reflection 2 21 Reflection at metal and dielectric interfaces E//,1 E//,2 H //,1 H //,2 E1 t k1 z E2 t k2 z D,1 D, 2 E3 t k1 z B,1 B, 2 x y A combine forward and reflected waves to give total fields for each region apply continuity conditions for separate components B z hence derive fractional transmission and reflection 22 Reflection at multiple dielectric interfaces combine forward and reflected waves to give total fields for each region E//,1 E//,2 H //,1 H //,2 D,1 D, 2 B,1 B, 2 E1 t k1 z E23 t k2 z E5 t k3 z E23 t k1 z E4 t k2 z x y A B z apply continuity conditions for separate components hence derive fractional C transmission and reflection 23 Reflection at multiple dielectric interfaces E//,1 E//,2 H //,1 H //,2 D,1 D, 2 B,1 B, 2 i t k3 z i t k 2 z i t k1 z t k3 z t k2 z E5e E1e t k1 z E3e i t k1 z i t k 2 z E2e t k1 z E4e t k2 z A 0 x y B l C z E1ei t k1 0 E2eEi 1 t kE1 02 E3ei Et 4 k2 0 E4ei t k2 0 k1 1 E e i t k1 0 1 1 i t k1 0 eE E2 E Z12 1 k12 Z2 E3ei Et 4k 0 E4ei t k 0 2 2 iik 3tl k3l i t kik22l l 2l E3ei t Ek23l e ikE e E e E e 4 4 5 k2 2 E e E e E e E e i 1 t k 2l ik2l i t kik22l l 3 4 3 Z2 4 k13 Z3 E5eiik3tlk3l 24 Reflection at multiple dielectric interfaces E//,1 E//,2 H //,1 H //,2 D,1 D, 2 B,1 B, 2 E1ei t k1z E3ei t k2 z E5ei t k3 z E2ei t k1z E4ei t k2 z x y A 0 B l C z 25