On the possibility of superluminal energy propagation in a hyperbolic metamaterial of metal-dielectric layers

The energy propagation of electromagnetic fields in the effective medium of a one-dimensional photonic crystal consisting of dielectric and metallic layers is investigated. We show that the medium behaves like Drude and Lorentz medium, respectively, when the electric field is parallel and perpendicular to the layers. For arbitrary time-varying electromagnetic fields in this medium, the energy density formula is derived. We prove rigorously that the group velocity of any propagating mode obeying the hyperbolic dispersion must be slower than the speed of light in vacuum, taking into account the frequency dependence of the permittivity tensor. That is, it is not possible to have superluminal propagation in this dispersive hyperbolic medium consisting of real dielectric and metallic material layers. The propagation velocity of a wave packet is also studied numerically. This packet velocity is very close to the velocity of the propagating mode having the central frequency and central wave vector of the wave packet. When the frequency spread of the wave packet is not narrow enough, small discrepancy between these two velocities manifests, which is caused by the non-penetration effect of the evanescent modes. This work reveals that no superluminal phenomenon can happen in a dispersive anisotropic metamaterial medium made of real materials.

2 It is well known that any kind of wave propagation process must satisfy causality [14]. That is, the cause must happen before the appearance of the result. Special relativity tells us that if superluminal signals can be sent as messages connecting event A and event B in an inertial frame, then it is always possible to find an inertial frame in which the observed orders of time for them are exchanged, violating the constraint of causality. Another restriction deduced from relativity is that no massive particle can move faster than the speed of light in a vacuum [15]. These considerations imply that waves can never have superluminal propagation velocity for the energy or information carried by them. However, recent studies concerning the propagation of electromagnetic [16], acoustic [17], and quantum waves [18] in various kinds of dispersive and dissipative media revealed that, under suitable conditions and through various mechanisms, superluminal phenomena do exist and can be observed, though in some subtle ways they never break causality and do not violate the restriction of speed limitation from relativity [19].
Up to now, most studies about the superluminal phenomena considered isotropic media or one-dimensional propagation [20][21][22][23][24][25][26]. About a decade ago, a study claimed that the group velocity of hyperbolic propagating mode can become superluminal under suitable conditions [27]. However, a recent theoretical work concerning passive linear media with negligible loss proved that the group/energy velocity of the electromagnetic propagating modes can never exceed the speed of light in a vacuum [28]. These contradictive results attracted our attention and inspired us to further explore the propagation behaviors of wave energy in a dispersive-anisotropic medium.
In this paper, we provide a rigorous proof showing that the group velocity of any propagating mode inside the effective medium that derived from the periodic structure of dielectric and metallic layers, no matter whether it obeys elliptical or hyperbolic dispersion, is always slower than the speed of light in a vacuum. This is the most important result of this paper.
We also numerically simulate the wave packet propagating behavior based on the Fourier transform method and the energy density formula derived in this paper. Although our result on the mode propagating velocity confirmed the conclusion given in [28], our proof is concrete in details and directly based on the material properties, different from that of the more abstract approach in [28]. Besides, though in the lossless limit our time dependent energy density formula applying to a propagating mode is the same as that derived from the corresponding formula in [28], for the case containing nonzero loss our energy density formula follows our previous work [29], excluding the "dissipation part" in the formula of [28] that represents the time integration of the power loss. In addition, very detailed results about the signatures and dispersions of the principal values of permittivity tensor are given, which is rare in existing publications.

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The metamaterials discussed in this paper can be realized effectively in the long-wavelength limit by the one-dimensional photonic crystals consisting of alternatively arranged dielectric and metallic layers. In Sec.2.1, we start with a one-dimensional photonic crystal (1DPC) of dielectric-metal layers and show that the principal values of the effective permittivity tensor along the two directions parallel to the material layers is of the same form as in the Drude model, implying a plasma behavior, whereas in the direction perpendicular to the layers, the effective medium mimics a Lorentz medium. We then derive the formula of group velocity for propagating mode in Sec. II.B and discuss in Sec. II.C the restrictions on its propagation directions and prove rigorously in Sec. II.D that they must be slower than the light speed in vacuum if the dielectric materials used in this 1DPC is a real material having relative permittivity larger than one. We also derive the formula of the dynamical wave energy density in Sec.II.E for arbitrary time varying electromagnetic fields in the effective medium. The energy formula is then used in the numerical study of the energy propagation velocity of a wave packet in Sec. III.A and III.B The wave packet is formed by a supposition of many plane waves with properly chosen coefficients for different propagation directions and frequencies. We then compare the numerical results for the wave packet and the theoretical results for the propagating mode and discuss the reasons of their differences. The conclusion is presented in Sec.IV.

A. Effective permittivity for the layered structure
To begin with, we consider a one-dimensional periodic structure which has one metal layer and one dielectric layer per unit cell, as is shown in Fig.1. In the long wavelength limit this photonic crystal behaves like an effective anisotropic medium which in some frequency ranges has indefinite eigenvalues of its permittivity tensor. We arrange these layers to be parallel to the xy plane and periodic along the z axis, so if we define xz plane to be the plane of incidence then the electromagnetic (EM) fields in this structure can be set to be independent of the y coordinate. Since the anisotropic feature appears only for the TM ( x z y E E H ) polarized waves, hereafter we consider only waves of this polarization.
Denoting the filling fraction of the dielectric layer in one unit cell as F, and d  and m  the permittivity of the dielectric and metallic layer respectively, the principal values of the effective permittivity tensor of the metamaterial along the x and z directions under the long wavelength limit are given by: Here both eff p  and 0  are functions of F , and they become the same if

B. Group velocity of a propagating mode in a dispersive anisotropic medium
In this section we derive the formula for the group velocity of a propagating mode in a general anisotropic dispersive medium. This velocity can be further shown to be equal to the energy velocity defined by the ratio between the time averaged Poynting vector and the time averaged energy density. We will not derive this result here but only recognize it as a fact. The group velocity = k   g v for this mode can be deduced by taking the k-gradient of the dispersion relation: Since the principal values of the permittivity of the medium are frequency dependent, so there are the relations Multiplying Eq.(7) by  and using the dispersion relation (5), we have thus the group velocity in the dispersive anisotropic medium is: (9) 6 C. The restrictions on the propagating directions of the modes In this section we discuss the existence condition and the restrictions on the propagation direction of a propagating mode.
To be more specific, we define 22 sin , cos , For a propagating mode all of the three quantities must be real, and conventionally k takes positive value. Note that according to Eq.

No restriction
These tables summarize the existence conditions for propagating modes and the restrictions on their propagating directions.
The 0 F  and 1 F  cases respectively correspond to pure metallic and pure dielectric materials. In the next section we will show that if 0 /1 dd     were assumed, any propagating mode can only have group velocity slower than the speed of light in vacuum.

D. The subluminality of the propagating modes
To discuss the speed of a propagating mode, we define a factor =| | / g c  g v , which represents the ratio between the group speed and the universal speed of light in vacuum. If  g can be proved to be always smaller than 1, then the subluminality is established. According to Eq.(9) and Eq. (10) If both  and  can be proved to be smaller than 1, than subluminality must be true. Using Eq.(3), we find:  Fig.2. Here the "k-angle" denotes the angle (in degrees) between the wave vector k of the propagating mode and the z-axis (the direction of the periodicity), and the color in 9 color bar represents the group velocity to light speed ratio g  defined in Eq. (12). The blank regions represent the forbidden combinations of   ,  k for propagating modes. As one can see, all the results confirm the conclusion of subluminality.
Based on this discussion, now it's easy to understand that the superluminality of the mode or wave packet velocity in [27] is caused by the fact that the frequency dependence of the permittivity tensor were not taken into account in the derivation of the group velocity formula, equation (7).
FIG. 2. The group velocity to light speed ratio for 4 filling fractions. The vertical axis represents the angle of the wave vector k for the mode. The unit frequency is the effective plasma frequency in Eq.(4). The permittivity of the dielectric in these simulation is 2. Every result shown here corresponds to subluminal group velocity.

E. The energy density in the dispersive anisotropic metamaterial
In the previous subsection, we have rigorously proved that the group velocity of a propagating mode in the effective hyperbolic metamaterial we defined in II.A must be subluminal. It is well known that for a propagating mode of frequency  one can define its energy velocity as e U  S v and this velocity can be proved to be equal to the group velocity, i.e, eg  vv if the absorption effect can be neglected. Here A stands for the time average of the time-dependent quantity () At ,  S E H is the Poynting vector, and U is the energy density of the electromagnetic wave in the medium. However, instead of proving this relation, in the following sections we will compare g v with the traveling speed of a wave packet which has central frequency  and formed by a weighted sum of many propagating modes. We thus derive the instantaneous energy density formula here and apply it in the following sections for studying the energy propagation problem.
In classical electrodynamics the displacement field   Similar to the previous works [29][30], the right hand side of the equation can be identified as the time rate of change of the field energy plus the power loss. Since in this paper we assume the power loss of the medium is small enough to be negligible, the right-hand side can be written as is the rate of change of the electric energy density including both the contributions from the electric field itself and the material response of the medium. Similarly, is the rate of change of the magnetic field energy density. Finally, U represents the total energy density to be identified. It is obvious that the magnetic energy density is simply given by  (19)) of the energy density are from the kinetic and potential energies of the vibrating dipoles in the medium. The dynamical vibration of the dipoles is induced by the electric field in the medium.
In the next section we will numerically study the propagation of a wave packet in the hyperbolic medium. Two numerical velocities will be compared: the wave-amplitude velocity and the wave-energy velocity. The former is defined by tracking the motion of the amplitude center of the wave packet, whereas the latter is defined by tracking the motion of the energy-center of the wave packet, based on the formula derived in this subsection.

A. Building a wave packet based on the superposition principle
We can build a wave packet of arbitrary shape by adding the plane waves of propagating modes with appropriate weight function of frequency and wave vector. This is relying on the fact that the plane waves H H k (21) In practice, the number of wave bases for constructing the wave packet should be large enough to prevent the possibility of more than one packets appear in our interest spatial region of observation.

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As mentioned before, the signs of x  and z  are determined by the wave frequency  and three characteristic frequencies

IV. Conclusion
In this paper, the effective electromagnetic properties in the long wavelength limit of the 1DPC of dielectric-metal layers is studied. This metamaterial behaves like a plasma or Lorentz medium with respect to the applied electric fields parallel or perpendicular to the layers, respectively. Based on the derivation of the Poynting theorem, the formula for the dynamical energy density is obtained. This dynamical energy includes two parts: the energies from the electric and magnetic fields themselves, and the kinetic plus the potential energies from the charge carriers in the medium. For the propagating modes in the medium, we derived the restrictions on the propagating directions of them. According to our theoretical analysis and numerical results, the group velocity of a propagating mode in the effective medium is always subluminal (slower than the speed of light) if the Drude and Lorentz dispersions already lead to subluminal propagation separately, that is, they are real materials. Thus our results confirm the conclusion of the theoretical work in [28] and reveal the reason of why superluminal propagation were derived in [27]. Besides, for the wave-packet velocity, no matter how we evaluate it, based on tracking the packet amplitude or energy distribution, the result is very close to the theoretical value of the group velocity for the propagating mode having the frequency close to the central frequency of the packet if the bandwidth of the packet is not too wide. If the bandwidth of the wave packet is a little wider, however, the two possible wave packet velocities (amplitude and energy) become deviated from the theoretical group velocity of the mode. We believe this is because some Fourier components of the wave packet become evanescent and cannot penetrate through the media, and this leads to the shift of the central frequency of the wave packet in the propagation process.
Our conclusion that no superluminal propagation of wave energy can happen in a dispersive-hyperbolic metamaterial is based on two main assumptions. The first is that the dielectric layers must have permittivity greater than 1, and the second is that the metal layers have permittivity of Drude-model type dispersion. Both of which are typical dielectric properties for real dielectric and metallic materials, respectively. It is in this sense we call them "real materials" in our paper. In addition, we assume through this paper that the medium is lossless. This assumption may seem unphysical because the Kramers-Kronig relations require a nonzero positive damping coefficient in the permittivity formula [31]. However, this requirement does not give a lower bound to such damping coefficient but only restricts the sign of it. Thus a lossless medium is not unphysical because it can be treated as an absorptive medium with a negligible damping coefficient. Finally, we want to stress that our conclusion does not exclude some superluminal phenomena happen in finite thickness gain medium or evanescent regions [32].
Our results are about wave propagation in an infinite or semi-infinite medium, which is very different from these superluminal phenomena caused by reshaping [16] or tunneling [18] effects in a finite region. Our results confirm that the wave energy propagation velocity in a real system cannot exceeds the speed of light in vacuum, which is a restriction derived from special relativity.