PROPAGATION OF ELECTROMAGNETIC ENERGY AND 1077 netic and material contributions, as the polarization P, ex A. Dielectric function pressed in the form(2. 12), is a material variable Consider a plane wave of frequency o and wave vector k The final results(3.23)-(3. 25)can also be obtained di- that is propagated parallel to the axis with its electric and ctly from the simple theory of Sec. I. Thus multiplication magnetic vectors oriented in the directions of the x and y of Eq(2.)by as /ari gives axes, respectively. The real electric field is written conven- tionally as a sum of positive-and negative-frequency contri- ds E(=,t)=E+(x,t)+E-(=,t) where Eq.(2.12)has been used. Subtraction of Eq. (3.26) Et(aexp(-iot+ik=)+E(exp(iot-ikz) from Eg.(2.21)gives (4.1) Here Et(o) is the complex amplitude at ==0, t=0 o{(Tm)n+m(s2-0s2)6+E E-(o)=[E+(a)]*, (Gcm)-ms}+pE+(j×B)=ms可 and a similar notation is used for the other fields The am- plitude of the field at ==0 is assumed to be time indepen- (3.27) dent, and the model therefore provides for a constant supply It is not difficult to show with the use of Eqs. ( 2.3),(2.5), dielectric material is still assumed to fill all of space, and the boundary condition at ==0 does not imply the existence of (2.6)and standard vector operator identities that It follows from Eq. (2.9)that m2+pE,+(xB),=2 (P×B)+r-(E SE(o/ (E,P) (3. 28) The electric displacement D(o) and the dielectric function (o) are defined by Thus Eq.(3.27)can be written in the form of the continuity equation(3. 23)with the same definitions (3.24)and(3.25) DT(o)=EoE(oET(O)=EoE()+Pt(o),(4.4) the wave momentum densities. However, in contrast to this and use of Eqs. (2. 12)and(4.3)leads to the explicit expres direct derivation, the lagrangian formulation establishes the sion nature of the wave momentum. Its two distinct contributions arising from the conserved momentum and the dissipating pseudomomentum, are unambiguously identified. Their sepa- ()=1+ (4.5) rate conservation and continuity properties are expressed by Eqs. (3. 14)and(3.19)respectively The refractive index no) and extinction coefficient K(a) are defined in the usual way by IV MONOCHROMATIC WAVE e(u)=[m(o)+ik(a)]2 (46) No assumptions have so far been made about the time dependences of the fields. We now evaluate the various den- and it follows from Eq. (4.5)that sities that have been derived in the previous two sections for the simple example of a monochromatic plane wave. The (4.7) cycle averages of the various energy and momentum densi- ties associated with the electromagnetic fields and the inter- and time in this example, and they have exponentially decaying spatial dependences. The electromagnetic and internal parts 2n(o)k(a)= (48) of the system are thus subjected to a steady-state excitation Eom(0r-02))2+62r2 and the monochromatic case usefully displays the full fre- fluency dependences of the energy and momentum densities. The wave vector is given by the usual expression, k=ln(o)+ix(oJo/ which maintains the steady state, results in a center-of-mass momentum density that grows linearly with the time, pro- and the complex magnetic and electric field amplitudes are vided that the velocity remains nonrelativistic. The results related by derived in the present section are valid for arbitrarily strong B(o=Ln()+iK(o)JE(o/c (410)netic and material contributions, as the polarization P, ex￾pressed in the form ~2.12!, is a material variable. The final results ~3.23!–~3.25! can also be obtained di￾rectly from the simple theory of Sec. II. Thus multiplication of Eq. ~2.9! by ]si/]rj gives ms¨i ]si ]rj 1mGs˙i ]si ]rj 1mvT 2 si ]si ]rj 5E• ]P ]rj , ~3.26! where Eq. ~2.12! has been used. Subtraction of Eq. ~3.26! from Eq. ~2.21! gives ] ]ri $~Tem!ji1 1 2 m~s˙ 22vT 2 s2!d ji%1E• ]P ]rj 1 ] ]t H ~Gem!j2ms˙• ]s ]rj J 1rEj1~j3B!j5mGs˙• ]s ]rj . ~3.27! It is not difficult to show with the use of Eqs. ~2.3!, ~2.5!, ~2.6! and standard vector operator identities that E• ]P ]rj 1rEj1~j3B!j5 ] ]t ~P3B!j1 ] ]rj ~E•P! 2 ] ]ri ~EjPi!. ~3.28! Thus Eq. ~3.27! can be written in the form of the continuity equation ~3.23! with the same definitions ~3.24! and ~3.25! of the wave momentum densities. However, in contrast to this direct derivation, the Lagrangian formulation establishes the nature of the wave momentum. Its two distinct contributions, arising from the conserved momentum and the dissipating pseudomomentum, are unambiguously identified. Their sepa￾rate conservation and continuity properties are expressed by Eqs. ~3.14! and ~3.19! respectively. IV. MONOCHROMATIC WAVE No assumptions have so far been made about the time dependences of the fields. We now evaluate the various den￾sities that have been derived in the previous two sections for the simple example of a monochromatic plane wave. The cycle averages of the various energy and momentum densi￾ties associated with the electromagnetic fields and the inter￾nal motion of the crystal lattice are all independent of the time in this example, and they have exponentially decaying spatial dependences. The electromagnetic and internal parts of the system are thus subjected to a steady-state excitation, and the monochromatic case usefully displays the full fre￾quency dependences of the energy and momentum densities. However, the constant supply of electromagnetic energy, which maintains the steady state, results in a center-of-mass momentum density that grows linearly with the time, pro￾vided that the velocity remains nonrelativistic. The results derived in the present section are valid for arbitrarily strong damping. A. Dielectric function Consider a plane wave of frequency v and wave vector k that is propagated parallel to the z axis with its electric and magnetic vectors oriented in the directions of the x and y axes, respectively. The real electric field is written conven￾tionally as a sum of positive- and negative-frequency contri￾butions, E~z,t!5E1~z,t!1E2~z,t! 5E1~v!exp~2ivt1ikz!1E2~v!exp~ivt2ikz!. ~4.1! Here E1~v! is the complex amplitude at z50, t50, E2~v!5@E1~v!#*, ~4.2! and a similar notation is used for the other fields. The am￾plitude of the field at z50 is assumed to be time indepen￾dent, and the model therefore provides for a constant supply of energy at the coordinate origin. It is emphasized that the dielectric material is still assumed to fill all of space, and the boundary condition at z50 does not imply the existence of any real boundary. It follows from Eq. ~2.9! that si 1~v!5 §Ei 1~v!/m vT 22v22ivG . ~4.3! The electric displacement D~v! and the dielectric function «~v! are defined by Di 1~v!5«0«~v!Ei 1~v!5«0Ei 1~v!1Pi 1~v!, ~4.4! and use of Eqs. ~2.12! and ~4.3! leads to the explicit expres￾sion «~v!511 §2 «0m 1 vT 22v22ivG . ~4.5! The refractive index h~v! and extinction coefficient k~v! are defined in the usual way by «~v!5@h~v!1ik~v!# 2, ~4.6! and it follows from Eq. ~4.5! that h~v! 22k~v! 2511 §2 «0m vT 22v2 ~vT 22v2! 21v2G2 ~4.7! and 2h~v!k~v!5 §2 «0m vG ~vT 22v2! 21v2G2 . ~4.8! The wave vector is given by the usual expression, k5@h~v!1ik~v!#v/c, ~4.9! and the complex magnetic and electric field amplitudes are related by B1~v!5@h~v!1ik~v!#E1~v!/c. ~4.10! 55 PROPAGATION OF ELECTROMAGNETIC ENERGY AND... 1077