1078 R LOUDON. L. ALLEN. AND D. F. NELSON It is convenient to simplify the expressions that occur in the remainder of the section by removal of explicit o depen dence from the notation for the dielectric properties and field amplitudes B Energy propagation The total-energy current density(2. 18)has only a nonzero z component for the geometry assumed here, and its cycle average is Eoc FIG. 1. Frequency dependence of the energy velocity (4. 16)for the dielectric function (4.5)with seom=0.136 this gives the L=c/20K 4.12) ratio of longitudinal to transverse optic mode frequencies WL/o=1066 found in GaAs. The curves are labeled with the ap- is the attenuation length, the distance after which the inten- propriate values of T/or sity of an electromagnetic wave in the dielectric decays to l/e of its initial value. The total-energy density(2. 19)has a cycle average Up LN and we note that it is not possible to express this quantity entirely in terms of macroscopic electromagnetic functions, is the phase velocity. Figure 1 shows the frequency depen- independent of the parameters of the optic mode. These ex- dence of the energy velocity in the vicinity of the transverse pressions for the total energy densities agree with a previous resonance for several values of the dampin derivation [5]. The cycle average of the energy dissipation Although derived for a specific model, relation (4. 17)be rate on the right of Eq.(2.17)is tween energy and phase velocities, decay length and damp- ing rate is found to apply to a wide range of systems, includ- -(mT5)=-4E0onKEPe-il=- 2EoCn1E+Pe. dielectrics in regions of resonant absorption [ 15, 16] and ir ing the propagation of pulsed optical signals through (4.14) regions of resonant amplification [17], self-induced transpar ency in two-level atoms [18], and energy transport in media 2. 17 cycle average of the energy continuity equation containing randomly distributed scatterers [19]. A similar re- takes the form lation is also valid for the propagation of ultrasonic signals [20]. Each of these systems has a detailed theoretical treat- 0(S2) ment for the relevant attenuation or amplification process, =-(m32) (4.15) but the derived and measured propagation velocities gene ally agree with the common form of energy velocity given and it is readily verified from Eqs. (4.11), (4. 12)and (4. 14) by Eq. (4. I7) that this relation is indeed satisfied. The energy that is con stantly supplied at ==0 in the example considered here tion,until none is left for propagation distances =>L. The (3. 18)and(2.20) has three nonzero components for the 9s steadily drains into the reservoir associated with the dissipa- Momentum. The momentum current density given by Eqs kinetic energy delivered to the dielectric material also grows ometry assumed here. The two transverse components have steadily in this example, but the material velocity is assumed cycle-averaged values to be always sufficiently small that the accumulated kinetic energy is negligible (Tm)x)=E0(1-2+3k2)E+Pe-m(419) Just as the ratio of the values of(S-) and(w)in a lossless dielectric gives the ray or energy velocity [10], so the ratio of and the energy densities is taken to define the velocity ve of energy transport through the absorbing dielectric as (Tm)y)=80(1-2-k2)E+P2e-.(4.20) 〈S2) W)刀+(2ok/T) (4.16) The cycle-averaged longitudinal component Is ((Tm)==((Tem) This can be rearranged in the form (421)It is convenient to simplify the expressions that occur in the remainder of the section by removal of explicit v depen￾dence from the notation for the dielectric properties and field amplitudes. B. Energy propagation The total-energy current density ~2.18! has only a nonzero z component for the geometry assumed here, and its cycle average is ^Sz&52«0chuE1u 2e22vkz/c 52«0chuE1u 2e2z/L, ~4.11! where L5c/2vk ~4.12! is the attenuation length, the distance after which the inten￾sity of an electromagnetic wave in the dielectric decays to 1/e of its initial value. The total-energy density ~2.19! has a cycle average ^W&52«0S h21 2vhk G DuE1u 2e2z/L, ~4.13! and we note that it is not possible to express this quantity entirely in terms of macroscopic electromagnetic functions, independent of the parameters of the optic mode. These ex￾pressions for the total energy densities agree with a previous derivation @5#. The cycle average of the energy dissipation rate on the right of Eq. ~2.17! is 2^mGs˙ 2 &524«0vhkuE1u 2e2z/L52 2«0ch L uE1u 2e2z/L. ~4.14! The cycle average of the energy continuity equation ~2.17! takes the form ]^Sz& ]z 52^mGs˙ 2 &, ~4.15! and it is readily verified from Eqs. ~4.11!, ~4.12! and ~4.14! that this relation is indeed satisfied. The energy that is con￾stantly supplied at z50 in the example considered here steadily drains into the reservoir associated with the dissipa￾tion, until none is left for propagation distances z@L. The kinetic energy delivered to the dielectric material also grows steadily in this example, but the material velocity is assumed to be always sufficiently small that the accumulated kinetic energy is negligible. Just as the ratio of the values of ^Sz& and ^W& in a lossless dielectric gives the ray or energy velocity @10#, so the ratio of the energy densities is taken to define the velocity ve of energy transport through the absorbing dielectric as ve5^Sz& ^W& 5 c h1~2vk/G! . ~4.16! This can be rearranged in the form 1 ve 5 1 vp 1 1 LG , ~4.17! where vp5c/h, ~4.18! is the phase velocity. Figure 1 shows the frequency depen￾dence of the energy velocity in the vicinity of the transverse resonance for several values of the damping. Although derived for a specific model, relation ~4.17! be￾tween energy and phase velocities, decay length and damp￾ing rate is found to apply to a wide range of systems, includ￾ing the propagation of pulsed optical signals through dielectrics in regions of resonant absorption @15,16# and in regions of resonant amplification @17#, self-induced transpar￾ency in two-level atoms @18#, and energy transport in media containing randomly distributed scatterers @19#. A similar re￾lation is also valid for the propagation of ultrasonic signals @20#. Each of these systems has a detailed theoretical treat￾ment for the relevant attenuation or amplification process, but the derived and measured propagation velocities gener￾ally agree with the common form of energy velocity given by Eq. ~4.17!. C. Momentum propagation Momentum. The momentum current density given by Eqs. ~3.18! and ~2.20! has three nonzero components for the ge￾ometry assumed here. The two transverse components have cycle-averaged values ^~Tm!xx&5«0~12h213k2!uE1u 2e2z/L ~4.19! and ^~Tm!yy&5«0~12h22k2!uE1u 2e2z/L. ~4.20! The cycle-averaged longitudinal component is ^~Tm!zz&5^~Tem!zz&5«0~11h21k2!uE1u 2e2z/L. ~4.21! FIG. 1. Frequency dependence of the energy velocity ~4.16! for the dielectric function ~4.5! with §2 /«0mv T 250.136; this gives the ratio of longitudinal to transverse optic mode frequencies vL/vT51.066 found in GaAs. The curves are labeled with the ap￾propriate values of G/vT . 1078 R. LOUDON, L. ALLEN, AND D. F. NELSON 55