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ENERGY AND MOMENTUM OF LIGHT IN DIELECTRIC PHYSICAL REVIEW E 73. 026606(2006) F(o, z,= (2, o)exp(- iot) (71) Using these expressions, the density and flow of energy and 0o-(o+iy)- w0-(o'-iy linear me nsity of the rate of work and force can be calculated. The attention is restricted to energy [E(o)-1o-[e(o)-1]ja and linear momentum, as angular momentum does not play a a’+iy role for the wave packets under discussion E(o)o-E()o G(ω)+G(o In the following, the shorthand D(o,)')f(o,o')≡ (22f(o,)E(, a) where it has been used that y is infinitesimally small and where the expression(20)for the dielectric function is used The total energy density may now be divided into two parts XE(, w)exp[-i(o-w)r] (72) gating part, the nonpropagating part being due to dissipation alone will be used, which is convenient as most relevant quantities are bilinear in field components. The integrand may be split u=uPr +I P into parts f(o,')=fs(o, )+fA(o,w), where fs(o, o') =fs(o, o) and fA(o, o')=-fA(o,o). As D( D(o', o)it follows that only the part fs(o, o')contributes to the integral. This often helps to simplify equations. L cont o o) where the propagating and nonpropagating parts are 2E0 D(o, w EAo)o-E(o")/ +vE(oE(o') (77) A Energy For the field part of the energy density it is found that D(o, w ') (78) 0)+iy The nonpropagating contribution may be further rewritten 2p0 using the Fourier representation of the step function D(au)[1+v(o)E(a’)].(73) I dre(r)exp[i(o-o)I'].(79) The material part is more involved to the time integral of the rate of work T drw"(r) (80) Eo D(o, o')X(o, a'). (74) where the rate of work is given by W=Eo D(o,o'[G(o)+G(G) (81) X(o, c') This may be written as the product of the dissipative part of he current density and the electric field [an2-(a+iy)2∞2-( W"=j(x,)E2(z,1), (82) where the dissipative current density is given by (a+iy)2-(a-iy)2 woo(oo) j=8J27 GoE(z, ) exp(- iot).(83) This dissipative current density is the time derivative of the dissipative part of the dielectric polarization, i.e., the part of the dielectric polarization that involves only the imaginary u′+2iy)丌Jo part of the susceptibility. The remaining, conservative part of (a+iy)2+oo′(a-iy)2+oo′ the dielectric polarization contributes to the energy of the propagating wave. It follows that the nonpropagating part of the total energy may be identified as the local energy of the 026606-7Fx,z,t = − d 2 1 2 − + i 2E ˆz,exp− it. 71 Using these expressions, the density and flow of energy and linear momentum, and the density of the rate of work and force can be calculated. The attention is restricted to energy and linear momentum, as angular momentum does not play a role for the wave packets under discussion. In the following, the shorthand D,f, − − dd 2 2 f,E ˆz, E ˆz, * exp− i − t, 72 will be used, which is convenient as most relevant quantities are bilinear in field components. The integrand may be split into parts f,= fS,+ fA,, where fS, = fS, * and fA,=−fA, * . As D, =D, * it follows that only the part fS, contributes to the integral. This often helps to simplify equations. A. Energy For the field part of the energy density it is found that uEM = 1 2 0Ex 2 + 1 2 0 By 2 = 1 2 0 D,1 + ˆˆ * . 73 The material part is more involved uMT = 0 0 d0ˆ0t Fx 2 + 0 2 Fx 2 = 1 2 0 D,X,, 74 with X, = 2 0 d0ˆ0 0 2 + 0 2 − + i 2 0 2 − − i 2 = 1 + i 2 − − i 2 2 0 d0ˆ0 0 2 + 0 2 − + i 2 − 0 2 + 0 2 − − i 2 = 1 + − + 2i 2 0 d0ˆ0 + i 2 + 0 2 − + i 2 − − i 2 + 0 2 − − i 2 = 1 − + i 2 − d0ˆ0 0 2 − + i 2 − 0 2 − − i 2 = ˆ − 1 − ˆ * − 1 − + i = ˆr − ˆr − − 1 + iˆ + ˆ − + i , 75 where it has been used that is infinitesimally small and where the expression 20 for the dielectric function is used. The total energy density may now be divided into two parts in yet a third way, namely into a propagating and nonpropagating part, the nonpropagating part being due to dissipation alone u = uPR + uNP, 76 where the propagating and nonpropagating parts are uPR = 1 2 0 D, ˆr − ˆr − + ˆˆ * , 77 uNP = 1 2 0 D, iˆ + ˆ − + i . 78 The nonpropagating contribution may be further rewritten using the Fourier representation of the step function i − + i = − dttexpi − t, 79 to the time integral of the rate of work uNP = − t dtWt, 80 where the rate of work is given by W = 1 2 0 D,ˆ + ˆ. 81 This may be written as the product of the dissipative part of the current density and the electric field W = jxz,tExz,t, 82 where the dissipative current density is given by jxz,t = 0 − d 2 ˆE ˆz,exp− it. 83 This dissipative current density is the time derivative of the dissipative part of the dielectric polarization, i.e., the part of the dielectric polarization that involves only the imaginary part of the susceptibility. The remaining, conservative part of the dielectric polarization contributes to the energy of the propagating wave. It follows that the nonpropagating part of the total energy may be identified as the local energy of the ENERGY AND MOMENTUM OF LIGHT IN DIELECTRIC... PHYSICAL REVIEW E 73, 026606 2006 026606-7