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)
26CHAPTER2.MAXWELL-BLOCHEOUATIONSwhere E(t) = Eejut. By using the ansatz r(t) = ejwt, we obtain for thecomplex amplitude of the dipole moment p = eor(t) = peiwtE(2.35)P:(w-w2)+2iwForthe susceptibility,wegetNEImEo(2.36)x(w) :(wg - w2) + 2jw号and thuswg(2.37)x(w) =(%-w)+2iwwith the plasma frequency wp, determined by w, = Nea/meo.Figure 2.1shows the real part and imaginary part of the classical susceptiblity (2.37).1.00.6Q=100.4 020Z0.500.0(o),x-0.20.40.02.00.00.51.01.5el.Figure2.l:Realpartand imaginarypart of thesusceptibilityofthe classicaloscillator model for the electric polarizability.Note, there is a small resonance shift due to the loss.Off resonance,the imaginary part approaches very quickly zero. Not so the real part, itapproaches a constant value wp/w below resonance, and approaches zero forabove resonance,but slower than the real part,i.e. offresonancethere is stilla contribution to the index but practically no loss
26 CHAPTER 2. MAXWELL-BLOCH EQUATIONS where E(t) = Eeˆ jωt. By using the ansatz x (t)=ˆxejωt, we obtain for the complex amplitude of the dipole moment p = e0x(t)=ˆpejωt pˆ = e2 0 m (ω2 0 − ω2) + 2jω0 Q ω E. ˆ (2.35) For the susceptibility, we get χ(ω) = N e2 0 m 1 0 (ω2 0 − ω2) + 2jω ω0 Q (2.36) and thus χ(ω) = ω2 p (ω2 0 − ω2) + 2jω ω0 Q , (2.37) with the plasma frequency ωp, determined by ω2 p = Ne2 0/m0. Figure 2.1 shows the real part and imaginary part of the classical susceptiblity (2.37). 1.0 0.5 0.0 χ ''(ω ) *2 Q/ 0.0 0.5 1.0 1.5 2.0 ω / ω 0 0.6 0.4 0.2 0.0 -0.2 -0.4 χ '(ω ) *2 Q/ 2 Q Q=10 Figure 2.1: Real part and imaginary part of the susceptibility of the classical oscillator model for the electric polarizability. Note, there is a small resonance shift due to the loss. Off resonance, the imaginary part approaches very quickly zero. Not so the real part, it approaches a constant value ω2 p/ω2 0 below resonance, and approaches zero for above resonance, but slower than the real part, i.e. off resonance there is still a contribution to the index but practically no loss
272.3.BLOCHEQUATIONS2.3Bloch EquationsAtoms in lowconcentration show line spectra as found in gas-,dye-and somesolid-state laser media. Usually, there are infinitely many energy eigenstatesin an atomic, molecular or solid-state medium and the spectral lines areassociated with allowed transitions between two of these energy eigenstatesFor nany physical considerations it is already sufficient to take only two ofthe possible energy eigenstates into account, for example those which arerelated to the laser transition. The pumping of the laser can be describedby phenomenological relaxation processes into the upper laser level and outof the lower laser level.The resulting simple model is often called a two-level atom, which is mathematically also equivalent to a spin1/2particlein an external magnetic field, because the spin can only be parallel or anti-parallel to the field, i.e. it has two energy levels and energy eigenstates. Theinteraction of the two-level atom or the spin with the electric or magneticfield is described by the Bloch equations.2.3.1The Two-Level ModelAn atom having only two energy eigenvalues is described by a two-dimensionalstate space spanned by the two energy eigenstates le > and lg >. The twostates constitute a complete orthonormal system.The corresponding energyeigenvalues are Ee and Eg (Fig. 2.2).E+EeEgFigure 2.2: Two-level atomIn the position-, i.e. x-representation, these states correspond to the wave
2.3. BLOCH EQUATIONS 27 2.3 Bloch Equations Atoms in low concentration show line spectra as found in gas-, dye- and some solid-state laser media. Usually, there are infinitely many energy eigenstates in an atomic, molecular or solid-state medium and the spectral lines are associated with allowed transitions between two of these energy eigenstates. For many physical considerations it is already sufficient to take only two of the possible energy eigenstates into account, for example those which are related to the laser transition. The pumping of the laser can be described by phenomenological relaxation processes into the upper laser level and out of the lower laser level. The resulting simple model is often called a twolevel atom, which is mathematically also equivalent to a spin 1/2 particle in an external magnetic field, because the spin can only be parallel or antiparallel to the field, i.e. it has two energy levels and energy eigenstates. The interaction of the two-level atom or the spin with the electric or magnetic field is described by the Bloch equations. 2.3.1 The Two-Level Model An atom having only two energy eigenvalues is described by a two-dimensional state space spanned by the two energy eigenstates |e > and |g >. The two states constitute a complete orthonormal system. The corresponding energy eigenvalues are Ee and Eg (Fig. 2.2). Figure 2.2: Two-level atom In the position-, i.e. x-representation, these states correspond to the wave
28CHAPTER2.MAXWELL-BLOCHEOUATIONSfunctions(2.38)e() =,and Φg(r) = :The Hamiltonian of the atom is given by(2.39)HA=Eele>= cglg>+cele>Weobtain(2.47)a+[ > =cgle>,(2.48)o-b > =celg>,(2.49)ob > =cele>-cglg>:The operator o+generates a transition from the ground to the excited state.and o- does the opposite. In contrast to α+ and o-, is a Hermitianoperator, and its expectation value is an observable physical quantity withexpectation value(2.50)=|ce[2-Jca/2=w,the inversion w of the atom, since |ce/? and Ic/? are the probabilities forfinding the atom in state Je > or |g > upon a corresponding measurement
28 CHAPTER 2. MAXWELL-BLOCH EQUATIONS functions ψe(x) =, and ψg(x) =. (2.38) The Hamiltonian of the atom is given by HA = Ee|e >= cg|g > + ce|e>. (2.46) We obtain σ+|ψ > = cg|e >, (2.47) σ−|ψ > = ce|g >, (2.48) σz|ψ > = ce|e > −cg|g>. (2.49) The operator σ+ generates a transition from the ground to the excited state, and σ− does the opposite. In contrast to σ+ and σ−, σz is a Hermitian operator, and its expectation value is an observable physical quantity with expectation value = |ce| 2 − |cg| 2 = w, (2.50) the inversion w of the atom, since |ce| 2 and |cg| 2 are the probabilities for finding the atom in state |e > or |g > upon a corresponding measurement
292.3.BLOCHEQUATIONSIf we consider an ensemble of N atoms the total inversion would be =N . If we separate from the Hamiltonian (2.38) the term (Ee +Eg)/2 -1, where 1 denotes the unity matrix, we rescale the energy valuescorrespondingly and obtainfortheHamiltonian of the two-level system1(2.51)HA=hwegozSwith the transition frequency(2.52)- Eg),(EWeg方This form of the Hamiltonian is favorable. There are the following commu-tatorrelationsbetweenoperators(2.41)to(2.43)(2.53)[ot,o] = az][g+,a] = -2g+,(2.54)(2.55)[g-,0] = 2g-,and anti-commutator relations, respectively[ot,a-]+ = 1,(2.56)[o+,a]+ = 0,(2.57)[-,α]+ = 0,(2.58)[α-,a-]+ = [α+,a+]+ =0.(2.59)The operators r,u,,fulfill the angular momentum commutator relations(2.60)[0a,Qy] = 2jo2,(2.61)[oy,o:] = 2jor,(2.62)[o,Q] = 2joy.The two-dimensional state space can be represented as vectors in C2 accord-ing to the rule:Ce[b >=cele>+cglg > -→(2.63)
2.3. BLOCH EQUATIONS 29 If we consider an ensemble of N atoms the total inversion would be σ = N. If we separate from the Hamiltonian (2.38) the term (Ee + Eg)/2 ·1, where 1 denotes the unity matrix, we rescale the energy values correspondingly and obtain for the Hamiltonian of the two-level system HA = 1 2 ~ωegσz, (2.51) with the transition frequency ωeg = 1 ~ (Ee − Eg). (2.52) This form of the Hamiltonian is favorable. There are the following commutator relations between operators (2.41) to (2.43) [σ+,σ−] = σz, (2.53) [σ+,σz] = −2σ+, (2.54) [σ−,σz]=2σ−, (2.55) and anti-commutator relations, respectively [σ+,σ−]+ = 1, (2.56) [σ+,σz]+ = 0, (2.57) [σ−,σz]+ = 0, (2.58) [σ−,σ−]+ = [σ+, σ+]+ = 0. (2.59) The operators σx, σy, σz fulfill the angular momentum commutator relations [σx,σy] = 2jσz, (2.60) [σy,σz] = 2jσx, (2.61) [σz,σx] = 2jσy. (2.62) The two-dimensional state space can be represented as vectors in C2 according to the rule: |ψ >= ce|e > + cg|g > → µ ce cg ¶ . (2.63)
30CHAPTER2.MAXWELL-BLOCHEQUATIONSThe operators are then represented by matrices(2.64)(2.65)(2.66)dz1(2.67)2.3.2The Atom-Field Interaction In Dipole Approxi-mationThe dipole moment of an atom p is essentially determined by the positionoperatorxvia(2.68)p= -eo x.Then the expectation value for the dipole moment of an atom in state (2.46)is(2.69)<>=-eo(cej2+cec,+ Cgc+|cgl).For simplicity, we may assume that that the medium is an atomic gas. Theatoms posses inversion symmetry, therefore, energy eigenstates must be sym-metric or anti-symmetric, i.e. == 0. We obtain(2.70)=-eo (cec+cgc*). (Note, this means, there is no permanent dipole moment in an atom, whichis in an energy eigenstate. Note, this might not be the case in a solid. Theatoms consituting the solid are oriented in a lattice, which may break thesymmetry.If so,there arepermanent dipole moments and consequently thematrix elements and would not vanish. If so, thereare also crystal fields, which then imply level shifts, via the linear Starkeffect.) Thus an atom does only exhibit a dipole moment in the average, ifthe product cecg + O, i.e. the state of the atom is in a superposition of statesle > and Ig >
30 CHAPTER 2. MAXWELL-BLOCH EQUATIONS The operators are then represented by matrices σ+ → µ 0 1 0 0 ¶ , (2.64) σ− → µ 0 0 1 0 ¶ , (2.65) σz → µ 1 0 0 −1 ¶ , (2.66) 1 → µ 1 0 0 1 ¶ . (2.67) 2.3.2 The Atom-Field Interaction In Dipole Approximation The dipole moment of an atom p˜ is essentially determined by the position operator x via p = −e0 x. (2.68) Then the expectation value for the dipole moment of an atom in state (2.46) is = −e0(|ce| 2 +cec∗ g (2.69) + cgc∗ e +|cg| 2 ). For simplicity, we may assume that that the medium is an atomic gas. The atoms posses inversion symmetry, therefore, energy eigenstates must be symmetric or anti-symmetric, i.e. == 0. We obtain = −e0 (cec∗ g +cgc∗ e ∗ ). (2.70) (Note, this means, there is no permanent dipole moment in an atom, which is in an energy eigenstate. Note, this might not be the case in a solid. The atoms consituting the solid are oriented in a lattice, which may break the symmetry. If so, there are permanent dipole moments and consequently the matrix elements and would not vanish. If so, there are also crystal fields, which then imply level shifts, via the linear Stark effect.) Thus an atom does only exhibit a dipole moment in the average, if the product cec∗ g 6= 0, i.e. the state of the atom is in a superposition of states |e > and |g >