Chapter 2Maxwell-Bloch Equations2.1Maxwell's EquationsMaxwell's equations aregivenbyaDRVxH=j+(2.1a)OtOBVxE(2.1b)ot.D = p,(2.1c).B = 0.(2.1d)The material equations accompanying Maxwell's equations are:D = E+P,(2.2a)B = μoH+M.(2.2b)Here, E and H are the electric and magnetic field, D the dielectric flux, Bthe magnetic flux,j the current density of free carriers, p is the free chargedensity, P is the polarization, and M the magnetization. By taking the curlof Eq. (2.1b) and considering × (×E)=(E) -△E, we obtainOE0ap×M+(.E)E-oa(2.3)j+0atotot21
Chapter 2 Maxwell-Bloch Equations 2.1 Maxwell’s Equations Maxwell’s equations are given by ∇ ×H = j + ∂D ∂t , (2.1a) ∇ ×E = −∂B ∂t , (2.1b) ∇ · D = ρ, (2.1c) ∇ · B = 0. (2.1d) The material equations accompanying Maxwell’s equations are: D = 0E + P, (2.2a) B = µ0H + M. (2.2b) Here, E and H are the electric and magnetic field, D the dielectric flux, B the magnetic flux, j the current density of free carriers, ρ is the free charge density, P is the polarization, and M the magnetization. By taking the curl of Eq. (2.1b) and considering ∇ × ³ ∇ ×E ´ = ∇ ³ ∇ E ´ − ∆E , we obtain ∆E − µ0 ∂ ∂t à j + 0 ∂E ∂t + ∂P ∂t ! = ∂ ∂t∇ ×M +∇ ³ ∇ · E ´ (2.3) 21
22CHAPTER2.MAXWELL-BLOCHEOUATIONSand henceai0202M+(.(2.4)cot2tOt2OtThevacuumvelocity of light is1(2.5)ofO2.2Linear Pulse Propagation in Isotropic MediaFor dielectric non magnetic media, with no free charges and currents dueto free charges, there is M = 0, j = 0, p = 0. We obtain with D =E()E=EOEr()E. (e(r)E) = 0(2.6)In addition for homogeneous media, we obtain . E = 0 and the waveequation (2.4) greatly simplifies1 202元(2.7)poat2coot2This is the wave equation driven by the polarization in the medium. Ifthe medium is linear and has only an induced polarization described by thesusceptibility x(w)= er(w)-1, we obtain in the frequencydomainP(w) = Eox(w)E(w).(2.8)Substituted in (2.7)(+)()=-w/o:(a);(),(2.9)where D = oe,(w)E(w), and thus(μ+(+x)w) E(w) = 0,(2.10)
22 CHAPTER 2. MAXWELL-BLOCH EQUATIONS and hence µ ∆ − 1 c2 0 ∂2 ∂t2 ¶ E = µ0 Ã ∂j ∂t + ∂2 ∂t2P ! + ∂ ∂t∇ ×M +∇ ³ ∇ · E ´ . (2.4) The vacuum velocity of light is c0 = s 1 µ00 . (2.5) 2.2 Linear Pulse Propagation in Isotropic Media For dielectric non magnetic media, with no free charges and currents due to free charges, there is M = 0, j = 0, ρ = 0. We obtain with D = (r) E=0r (r) E ∇ · ((r) E )=0. (2.6) In addition for homogeneous media, we obtain ∇ · E = 0 and the wave equation (2.4) greatly simplifies µ ∆ − 1 c2 0 ∂2 ∂t2 ¶ E = µ0 ∂2 ∂t2P. (2.7) This is the wave equation driven by the polarization in the medium. If the medium is linear and has only an induced polarization described by the susceptibility χ(ω) = r(ω) − 1, we obtain in the frequency domain b P (ω) = 0χ(ω) ˆ E (ω). (2.8) Substituted in (2.7) µ ∆ + ω2 c2 0 ¶ ˆ E (ω) = −ω2 µ00χ(ω) ˆ E (ω), (2.9) where b D = 0r(ω) ˆ E (ω), and thus µ ∆ + ω2 c2 0 (1 + χ(ω) ¶ ˆ E (ω)=0, (2.10)
232.2.LINEARPULSEPROPAGATIONINISOTROPICMEDIAwith the refractive index n and 1 + x(w) = n2) E(w) = 0,△+(2.11)-where c= co/n is the velocity of light in the medium.2.2.1Plane-WaveSolutions(TEM-Waves)The complex plane-wave solution of Eq: (2.11) is given byE(+)(w,r) = E(+)(w)e-ik-r = Eoe-k-r . (2.12)with廊= k2.(2.13)2Thus, the dispersion relation is given by(a) = n(a).(2.14)CoFrom .E = O, we see that Kk I e. In time domain, we obtainE(+)(r,t) = Eoe.elut-jk.r(2.15)with(2.16)k = 2元/入,where is the wavelength, w the angular frequency, k the wave vector, e thepolarization vector, and f = w/2 the frequency. From Eq. (2.1b), we getfor the magnetic field-jk × Eoce(ut-Kr) = -jow Hi(+),(2.17)orH(+) = Eo el(ut-Er) × é = Hohei(ut-kn)(2.18)Howwithkh(2.19)xe[因;]
2.2. LINEAR PULSE PROPAGATION IN ISOTROPIC MEDIA 23 with the refractive index n and 1 + χ(ω) = n2 µ ∆ + ω2 c2 ¶ ˆ E (ω)=0, (2.11) where c = c0/n is the velocity of light in the medium. 2.2.1 Plane-Wave Solutions (TEM-Waves) The complex plane-wave solution of Eq. (2.11) is given by ˆ E (+)(ω,r) = ˆ E (+)(ω)e−j k·r = E0e−j k·r · e (2.12) with | k| 2 = ω2 c2 = k2 . (2.13) Thus, the dispersion relation is given by k(ω) = ω c0 n(ω). (2.14) From ∇ · E = 0, we see that k ⊥ e. In time domain, we obtain E (+)(r, t) = E0e · ejωt−j k·r (2.15) with k = 2π/λ, (2.16) where λ is the wavelength, ω the angular frequency, k the wave vector, e the polarization vector, and f = ω/2π the frequency. From Eq. (2.1b), we get for the magnetic field −j k × E0eej(ωt− kr) = −jµ0ωH (+), (2.17) or H (+) = E0 µ0ω ej(ωt− kr) k × e = H0 hej(ωt− kr) (2.18) with h = k |k| × e (2.19)
24CHAPTER2.MAXWELL-BLOCHEOUATIONSand1[A]Eo(2.20)HoZFHowThenatural impedanceis1Ho(2.21)ZFoZF=HoCnEOErwith the free space impedancePoZFo= 377 2.(2.22)EoFor a backward propagating wave with E(+)(r,t) = Eoe. ejut+ik-r there isH(+) = Hohei(ut-kn) withELEo.(2.23)Ho =HowNote that the vectors e, h and k form an orthogonal trihedral,elh,kle,klh(2.24)2.2.2Complex NotationsPhysical E, H fields are real:((+)(F,t) +E(一)(,t))E(r,t) =(2.25)with E(-)(r,t) = E(+)(r,t)*. A general temporal shape can be obtained byaddingdifferent spectralcomponents,dw(+)E"(w)e(ut-R-r),E(+)(F,t) =(2.26)2元1Correspondingly, the magnetic field is given byH(,t)=(H(+)(r,t) +H(-(,t)(2.27)with H(-)(r, t) = H(+)(r,t)*. The general solution is given by(w)ej(ut-K-r)H(+)(r,t) =(2.28)2元with(+)EonH(2.29) (w) =ZF
24 CHAPTER 2. MAXWELL-BLOCH EQUATIONS and H0 = |k| µ0ω E0 = 1 ZF E0. (2.20) The natural impedance is ZF = µ0c = r µ0 0r = 1 n ZF0 (2.21) with the free space impedance ZF0 = rµ0 0 = 377 Ω. (2.22) For a backward propagating wave with E (+)(r, t) = E0e · ejωt+j k·r there is H (+) = H0 hej(ωt− kr) with H0 = − |k| µ0ω E0. (2.23) Note that the vectors e, h and k form an orthogonal trihedral, e ⊥ h, k ⊥ e, k ⊥ h. (2.24) 2.2.2 Complex Notations Physical E , H fields are real: E (r, t) = 1 2 ³ E (+)(r, t) + E (−) (r, t) ´ (2.25) with E (−) (r, t) = E (+)(r, t)∗. A general temporal shape can be obtained by adding different spectral components, E (+)(r, t) = Z ∞ 0 dω 2π b E (+) (ω)ej(ωt− k·r) . (2.26) Correspondingly, the magnetic field is given by H (r, t) = 1 2 ³ H (+)(r, t) + H (−) (r, t) ´ (2.27) with H (−) (r, t) = H (+)(r, t)∗. The general solution is given by H (+)(r, t) = Z ∞ 0 dω 2π b H (+) (ω)ej(ωt− k·r) (2.28) with b H (+) (ω) = E0 ZF h. (2.29)
2.2.LINEARPULSEPROPAGATIONINISOTROPICMEDIA252.2.3Poynting Vectors, Energy Density and Intensityfor Plane Wave FieldsReal fieldsQuantityComplexfieldsEOEW=(o,E+Po,2)Energy densityW=H(+)+popS-ExHT=E(+)x((+)Poynting vectorI=s= cwIntensityI=T=cwS0+-0Energy Cons.For E(+)(r,t) = Eoerei(ut-kz) we obtain the energy density12r0|E0/2,(2.30)w=thepoyntingvector1T 1E012e.(2.31)2ZFand the intensityZeHo/2.(2.32)[E0/212ZF22.2.4Dielectric SusceptibilityThe polarization is given by(+)(a)=dipolemoment=N (+(w) = ox(u)(+)(a),(2.33)volumewhere N is density of elementary units and (p) is the average dipole momentof unit (atom, molecule,..).Classical harmonic oscillator modelThe damped harmonic oscillator driven by an electric force in one dimension,r, is described by the differential equation+2mk(2.34)+mwgr = eoE(t),mdt2Qndt
2.2. LINEAR PULSE PROPAGATION IN ISOTROPIC MEDIA 25 2.2.3 Poynting Vectors, Energy Density and Intensity for Plane Wave Fields Quantity Real fields Complex fields hit Energy density w = 1 2 ³ 0rE 2 + µ0µrH 2 ´ w = 1 4 ⎛ ⎝ 0r ¯ ¯ ¯ E (+) ¯ ¯ ¯ 2 +µ0µr ¯ ¯ ¯ H (+) ¯ ¯ ¯ 2 ⎞ ⎠ Poynting vector S = E×H T = 1 2E (+)× ³ H (+)´∗ Intensity I = ¯ ¯ ¯ S ¯ ¯ ¯ = cw I = ¯ ¯ ¯ T ¯ ¯ ¯ = cw Energy Cons. ∂w ∂t + ∇ S = 0 ∂w ∂t + ∇ T = 0 For E (+)(r, t) = E0exej(ωt−kz) we obtain the energy density w = 1 2 r0|E0| 2 , (2.30) the poynting vector T = 1 2ZF |E0| 2 ez (2.31) and the intensity I = 1 2ZF |E0| 2 = 1 2 ZF |H0| 2 . (2.32) 2.2.4 Dielectric Susceptibility The polarization is given by P (+)(ω) = dipole moment volume = N · hp(+)(ω)i = 0χ(ω)E (+)(ω), (2.33) where N is density of elementary units and hpi is the average dipole moment of unit (atom, molecule, .). Classical harmonic oscillator model The damped harmonic oscillator driven by an electric force in one dimension, x, is described by the differential equation m d2x dt2 + 2ω0 Q m dx dt + mω2 0x = e0E(t), (2.34)