Poynting’s theorem and luminal total energy transport in passive dielectric media

Abstract

Without approximation the energy density in Poynting's theorem for the generally dispersive and passive dielectric medium is demonstrated to be a system total dynamical energy density. Thus the density in Poynting's theorem is a conserved form that by virtue of its positive definiteness prescribes important qualitative and quantitative features of the medium-field dynamics by rendering the system dynamically closed. This fully three-dimensional result, applicable to anisotropic and inhomogeneous media, is model independent, relying solely on the complex-analytic consequences of causality and passivity. As direct applications of this result, we show (1). that a causal medium responds to a virtual, "instantaneous" field spectrum, (2). that a causal, passive medium supports only a luminal front velocity, (3). that the spatial "center-of-mass" motion of the total dynamical energy is also always luminal and (4). that contrary to (3). the spatial center-of-mass speed of subsets of the total dynamical energy can be arbitrarily large. Thus we show that in passive media superluminal estimations of energy transport velocity for spatially extended pulses is inextricably associated with incomplete energy accounting.

Speed-of-light limitations in passive linear media

“…Furthermore, our results apply to arbitrary anisotropic and bianisotropic (chiral) dispersive media, both homogeneous (Sec. II, similar to Yaghjian [9,10] and Glasgow et al [8]) and periodic (Sec. III) media, include certain classes of spatially nonlocal media and only require that the medium be passive "on average" in the unit cell of periodicity.…”

“…In lossy media, Poynting's theorem can be generalized to define a "dynamical" energy density [8,9,12,[33][34][35], from which one can define a "dynamical" energy velocity via the ratio of the Poynting flux to the energy density [8,12]. Recently, Glasgow et al [8] showed that this dynamical energy velocity is c in passive media, showed that it bounds the front velocity [2], [23,Sec. 5.18] (the speed at which a region of nonzero fields expands) by generalizing a result of Sommerfeld and Brillouin [2,[58][59][60] 3], and also bounded the velocity of the mean energyweighted position.…”

“…This was first pointed out by Loudon [3,4] who, after correcting this error, obtained a subluminal expression for the energy velocity in a Lorentz medium, and Loudon's formulation was subsequently extended by other authors [5,6]. However, some later works argued that one should include an additional term in the energy density (which appears in the denominator of the energy velocity), equal to a total dissipated energy [8,9,34,35], and this also results in a subluminal energy velocity [8]; our bounds apply regardless of whether this term is included. Whereas the early work was limited to a specific (usually Lorentz "oscillator") model of dispersion, the newer "dynamical" energy density has the advantage of being model independent.…”

“…84], for homogeneous passive isotropic or anisotropic dispersive media [8][9][10], or for periodic nondispersive media [11, p. 41], usually assuming negligible loss (i.e., in a "transparency window"). For lossy media various authors have proved similar bounds, but with some disagreement over the definition of energy velocity [2,[4][5][6]8,12]. In this paper, we extend those results by both greatly simplifying the underlying assumptions and by generalizing the reach of the results.…”

“…Nevertheless, the results of Yaghjian can be viewed as a precursor of our results. For lossy media, Glasgow et al [8] employed similar passivity assumptions in order to obtain positivity of the dissipated energy and hence subluminal energy velocity. All of these previous results, however, were for energy velocity defined pointwise in space, or equivalently for light propagation in a homogeneous medium (or possibly a homogenized approximation to an inhomogeneous "metamaterial"), but a more general setting is that of periodic media in which Bloch waves propagate with a well-defined group and energy velocity [11,[26][27][28][29][30][31][32]36].…”