VOLUME 90, N UMBER 9
Comment on ‘‘Direction of Optical Energy Flow in a Transverse Magnetic Field Employing both theoretical analysis and experimental measurement, Rikken and van Tiggelen concluded in their Letter [1] that the optical energy flux S in a homogeneous, absorbing medium is not given by the Poynting vector cE H. We do not agree with this conclusion. They arrived at this result theoretically by calculating the propagation in space of a quantity they identified as the optical energy density, Uop 12E D H2 ;
(1)
for a linearly polarized Gaussian wave packet of light, in the presence of an applied static, transverse magnetic field B0 . They found that —if the system is dissipative —the propagation direction of the maximum of Uop is different from the direction of the Poynting vector, and parallel to cE H q, with q r T perpendicular to z, the direction of light propagation in vacuum. q is uniform in the x; y plane and constant with respect to time. In other words, it remains present even after the wave packet is long gone. They suggest to include q in the energy flux. We note that (i) Uop as given above is not the total optical energy; (ii) the Poynting vector is under fairly general conditions the sole energy flux; there is no discretion for adding terms such as q r T; (iii) the above q explicitly violates general principles. Generally speaking, there is only one conserved energy for any given system —the total energy U including field and material energy, and including heat. Because it is conserved, it satisfies a continuity equation: U_ r S 0:
(2)
As shown in [2] in a model-independent, general framework, the flux S is given by the Poynting vector cE H for dispersive, dissipative, nonlinear systems, and in the presence of static fields, provided three constraints are met: (1) absence of medium motion: If the medium is nonstationary (v 0), there will be terms v in S; (2) absence of dissipative processes other than that produced by the electromagnetic waves, especially heat currents; and (3) absence of spatial dispersions. Adding a term r T to S alters neither U_ nor the fact that U is conserved; it is permissible if U is the only quantity of interest. The authors of [1], however, are focused on the direction of S, then the results of [2] clearly state that T 0. This includes especially the above q: Since it is finite in dielectrics and vanishing in vacuum, it must produce (or eliminate) energy at the lateral boundaries (say x or y const), where the Poynting vector is continuous. (A finite q in vacuum has more problems.) Let us for simplicity consider an isotropic medium with a linear constitutive relation, though we shall retain dis-
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PHYSICA L R EVIEW LET T ERS
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sipation and dispersion. Then the conserved energy density U is a function of the density , entropy density s, electric field D, polarization P, magnetic field B B0 B, and P’s canonical momentum a [2], U U0 ; s 12D2 D P 12P2 = 12B2 12!2p a2 B0 a P;
(3)
where U0 denotes the energy density in the absence of fields, and a is given as a P_ B0 P=!2p here. , !p , and are various parameters and also functions of and s. ( is a susceptibility, !p the relevant dielectric resonant frequency, and gives rise to the Faraday effect.) If the system is transparent and nondissipative, the energy in the wave packet is not being transformed into heat, then U0 ; s remains independently constant, and U U0 also satisfies Eq. (2) with S cEH, E D P, and H B. In addition, the energy density U U0 may now be expressed rather concisely using the dielectric function "ij , if one averages Eq. (3) over a cycle, hU U0 i 12 Ei E j "ij !d"ij =d! hH 2 i:
(4)
This is the famous formula by Brillouin [3]. The difference to hUop i of Eq. (1), with hE Di E i Ej "ij , is 1
2Ei Ej "ij
" ij !d"ij =d!:
(5)
Clearly, Uop of Eq. (1) is not even conserved in the absence of dissipation. Because the dielectric function was taken to be "ij n2 ij i ijk B0k in [1], the first two terms of Eq. (5) are iB0 E E , the energetic contribution of the Pitaevskii magnetization [3]. This probably larger term [4] and U0 should, given sufficient accuracy for analysis and data, account for the discrepancy. Jiang Yimin and Mario Liu* Theoretische Physik Universita¨t Tu¨bingen 72076 Tu¨bingen, Germany Received 10 April 2002; published 4 March 2003 DOI: 10.1103/PhysRevLett.90.099401 PACS numbers: 41.20.Jb, 03.50.De, 78.20.Ls *Email address:
[email protected] [1] G. L. J. A. Rikken and B. A. van Tiggelen, Phys. Rev. Lett. 78, 847 (1997). [2] Y. Jiang and M. Liu, Phys. Rev. Lett. 77, 1043 (1996); Phys. Rev. E 58, 6685 (1998). [3] L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, Oxford, 1984). [4] Although this term vanishes in the Voigt geometry, it concerns only a fraction of 10 4 of the light flux, see G. L. J. A. Rikken and B. A. van Tiggelen, Phys. Rev. Lett. 80, 1115 (1998).
2003 The American Physical Society
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