An exact separation of both electric and magnetic energies into stored and lost energies is shown to be possible in the special case when the wave impedance is independent of frequency. A general expression for the electromagnetic energy density in such a dispersive medium having a negative refractive index is shown to be accurate in comparison with numerical results. Using an example metamaterial response that provides a negative refractive index, it is shown that negative time-averaged stored energy can occur. The physical meaning of this negative energy is explained as the energy temporarily borrowed by the field from the material. This observation for negative index materials is of interest when approaching properties for a perfect lens. In the broader context, the observation of negative stored energy is of consequence in the study of dispersive materials.
© 2012 Optical Society of America
A negative refractive index  can be achieved when both the dielectric constant ε and the permeability μ are negative, which requires an operating frequency near a resonance in both constitutive parameters. In this frequency regime, the material properties can be highly dispersive for the candidate metamaterials, and the loss can be substantial . However, theoretically it has been shown that a perfect-lens-material condition can be obtained at some intervening frequency between absorptive and gain resonances in a material .
Poynting’s theorem, describing conservation of energy, can be written in a simple form when the material properties do not change with frequency. Brillouin suggested that in a causal system, one with frequency-dependent material properties, a different form of Poynting’s theorem was appropriate . With this view, Veselago has presented requirements for positive energy density , based on the result given by Landau and Lifshitz  and Brillouin  for transparent media. Loudon , Polevoi , Ruppin , Tretyakov , Boardman and Marinov , Cui and Kong , and Nunes et al.  have all studied energy density issues in dispersive media. This substantial body of work presents results where all energies (total, stored and dissipated) remain positive. However, Ziolkowski  questions whether the electromagnetic energy must remain positive. Previously, we have shown that the electric field energy density in dispersive materials can be exactly separated into stored and lost energies when using the electric field as the basis . Our analytical and numerical results for a narrowband modulated wave near a dielectric material resonance indicated that the stored energy can be negative, while the total energy remains positive in this high-loss regime. Ben-Aryeh  has given a correction to the transparent energy density described by Landau and Lifshitz , extending our electric field energy density representation for narrowband fields  to include both electric and magnetic responses. This work considered this analytic energy density expression for a negative refractive index material.
Here, we provide the exact energy density for dispersive negative refractive index materials and give analytic expressions for the electric as well as magnetic energy densities for fields having narrow temporal frequency bandwidth. We consider the interdependence of electric and magnetic field energy, through the wave impedance, leading to the conclusion that the total energy is generally not separable into stored and lost terms. For the special case of a frequency-independent wave impedance in the material, the exact separation of both electric and magnetic energy densities into stored (electric and magnetic) and lost (electric and magnetic) components is shown to be possible. This allows conclusions to be drawn about the signs and magnitudes of the stored and lost energies for dielectric and magnetic materials that was not evident in previous work . Using an example passive negative refractive index material response, the possibility of negative stored energy density is shown through an exact decomposition into stored and lost terms. The stored energy densities in an example active negative refractive index material (with identical electric and magnetic responses) are shown to be negative, and the total energy density is negative by virtue of the external pump energy provided.
2. Poynting’s theorem and exact energy decomposition
With electric field E and magnetic field H, the vector identity ∇ · (E × H) = H · (∇ × E) − E · (∇ × H), in conjunction with Maxwell’s curl equations, upon volume integration and application of the divergence theorem, leads toEquation (1) is known as Poynting’s theorem, a statement of conservation of energy. A unique solution of Maxwell’s equations requires a model for the material constitutive parameters, which in turn can allow Poynting’s theorem to be expressed in terms of only E and H. Here it suffices to choose the linear and isotropic relationships D = ε0εE and B = μ0μH, in the frequency domain, where ε0 and μ0 are the free space permittivity and permeability, respectively, ε is the dielectric constant, and μ is the relative permeability.
The scalar electric and magnetic flux densities can be written as an exact Fourier superposition in the formEq. (2) and Eq. (3), we assume the most general local in space and non-local in time form , and assume homogenized responses and no spatial dispersion . In the electric field case, with the electric field as a basis and using a Fourier superposition and referring to Eq. (1), the time derivative of the electric field energy density uE (J/m−3) becomes Eq. (5) leads to Eq. (8) results in Eq. (11). When the electric field is chosen as the basis, uE is exactly separable into wE (stored electric field energy density) and qE (lost electric field energy density), but uH is generally not. Similarly, when the magnetic field is chosen as the basis, uH is exactly separable. However, in the special case when both ε (ω) and μ (ω) are constant or ε (ω) = aμ (ω), with a some real constant, η (ω) is independent of frequency, giving H(t) = ηE(t), where η is the constant value of the wave impedance in the frequency domain. In these cases, the total energy density, u = uE + uH, is exactly separable into stored (which is returned to the field, w = wE + wH) and lost (converted to a non-electromagnetic form, q = qE + qH) energies. Combining Eq. (6) and Eq. (9) gives Eq. (12) is physical in the sense that the stored and lost energies at each frequency are dictated by the real and imaginary parts of the dielectric constant, respectively, and through spectral superposition, the temporal energies are thus exact. The real and imaginary parts of the dielectric constant are related by the Kramers-Kronig relations , a consequence of a causal response. Causality therefore enforces a relationship between the two terms in Eq. (6) and Eq. (9).
3. Energy density for modulated light
Information transfer (or a causal system, where the field is zero at times prior to the propagation delay l/c, with l the source-detector distance and c the speed of light in vacuum) implies modulation of a carrier signal. For the current purpose, this modulation signal can have arbitrarily small bandwidth relative to the carrier frequency, ω0, or we assume the field has been applied for a sufficiently long time.
We assume E = e(t)cos(ω0t), with slowly varying modulation signal e(t) relative to t0 = 2π/ω0, producing an effective bandwidth that is small relative to the features of ε (ω). Therefore, in the spectral domain, the electric field becomes , where e(ω) is the Fourier transform of e(t). D(t) for the slowly varying field is obtained by substituting the complex E(ω) and ε (ω) into Eq. (2). Following our earlier work , the approximate form of ∂D/∂t is then obtained by using the first two terms in the Taylor series expansion of (ω + ω0)ε (ω + ω0) about ω = 0. The product of E(t) and the approximate form of ∂D(t)/∂t gives the approximate form of ∂uE/∂t. After some additional steps (see ), we obtain the average (over t0 and as a function of time) electric field energiesEquations (13) – (16) apply regardless of the degree of loss, provided that the two-term Taylor series expansions for ∂D(t)/∂t and ∂B(t)/∂t are sufficiently accurate.
Taking the electric field as the basis and using Eq. (11), we writeEq. (17) and Eq. (18) respectively into the scalar form of Eq. (7), and taking the time average, we obtain Equation (19) cannot be exactly separated into stored and lost energy terms when there is material dispersion, because u and v cannot be separated into terms involving inly real or imaginary components of the constitutive parameters. In the special case where η (ω) is independent of frequency, u(ω) becomes the real constant u′ (ω0) (because η is a real constant) and Eq. (19) reduces to
Conservation of energy for a passive medium, with frequency-independent η (ω), dictates that
4. Numerical example
For the simple case when η is independent of frequency and the energy density is exactly separable, we choose identical electric and magnetic susceptibilities (χE = χM) to arrive at a negative index material given byEq. (22). Example dispersive dielectric constant and relative permeability responses having the form of Eq. (22) are shown in Fig. 1. The material parameters are chosen so that the real part of the refractive index is n′ (ω0) = −1 for ω0 = 8/7.
We assume a Gaussian modulation so that the electric field becomesEq. (4) was evaluated numerically, with D(t) formed as in Eq. (2) by Fourier transforming (using an FFT) the temporal electric field, multiplying by ε (ω), and then inverse Fourier transforming the result. The quantities ∂wE/∂t and ∂qE/∂t were formed using the exact decompositions. Numerical integration over time followed by time-averaging over the period t0 resulted in 〈wE〉, 〈∂qE/∂t〉 and 〈uE〉. Using the magnetic field instead of the electric field and following a similar procedure, the average quantities 〈wH〉, 〈∂qH/∂t〉 and 〈uH〉 were obtained. Subject to numerical precision, we consider these as exact results.
Figure 2 shows the numerical results for 〈wE〉 and 〈wH〉 obtained for the three different carrier frequencies (ω0 = 4/9, 8/7 and 4/3) and the approximate results of Eq. (13) and Eq. (15), the model. When the carrier frequency is sufficiently far from the resonant frequency, 〈wE〉 as well as 〈wH〉 are positive, just as they would have been in free space. However, when the carrier frequency is just above the resonant frequency (ω0 = 8/7), and in this case where the refractive index is negative, 〈wE〉 as well as 〈wH〉 become negative. Figure 3 shows the numerical results for 〈∂qE/∂t〉 and 〈∂qH/∂t〉 obtained for the same three carrier frequencies, along with the results from the approximate analytical expressions in Eq. (14) and Eq. (16). Throughout, the total average energy (〈uE〉 + 〈uH〉) remains positive, as Fig. 4 shows. At times long after the pulse has gone, 〈qE〉 = 〈uE〉 and 〈qH〉 = 〈uH〉, indicating that no additional energy beyond that delivered by the electromagnetic field is involved, despite 〈wE〉 and 〈wH〉 being negative for some time. Notice the excellent agreement between the exact results and the approximate analytical expressions in Figs. 2 and 3.
Consider now the important case of an active dispersive material having3]. Small deviation from α = 1 gives rise to a material with small gain or loss, depending on whether α < 1 (gain) or α > 1 (loss). The dielectric constant having the form of Eq. (24) is shown in Fig. 5 and the carrier frequency ω0 = 1 is marked by a vertical solid line. Note that in a hypothetical material where χM = χE, with χE defined as in Eq. (24) and with α = 1, we obtain a lossless negative refractive index material at ω0. This realizes the perfect lens material condition , which requires a lossless material response at that frequency [3, 19]. In such a material, the total time-averaged electric 〈uE〉 and magnetic 〈uH〉 energy densities would be equal, giving the total electromagnetic energy density 〈u〉 = 2〈uE〉. The stored and lost energy densities in this material are 〈w〉 = 2〈wE〉 and 〈q〉 = 2〈qE〉, respectively.
We consider the cases of lossless (α = 1), small loss (α = 1.0001), and small gain (α = 0.9999) materials. Figure 6 shows the stored energy density obtained for these different materials, and indicates that 〈wE〉 is approximately the same for all three cases because of the very small change in ε′ associated with the small change in α. However, this small change in α causes a noticeable change in the lost power 〈∂q/∂t〉, as shown in Fig. 7. The time averaged total energy densities 〈uE〉 for these three cases are shown in Fig. 8.
No power is lost in the lossless case because ε″ = 0. However, when there is a small loss (ε″ > 0), the power is positive. In the case of small gain (ε″ < 0) the power is negative. The total energy density 〈uE〉 becomes negative in all three cases because of the large magnitude of 〈wE〉. Note that 〈wE〉 is negative for the three active material cases, even when the carrier frequency is away from resonance. Referring to Eq. (13), this happens because we achieve ε′ < −ω(∂ε′/∂ω) at the carrier frequency. In the optical regime, the carrier frequency is very large compared to ε′, so negative stored energy would generally be obtained when (∂ε′/∂ω) < 0, just as in the cases studied for the passive material given by Eq. (22). A negative stored energy is physical and does not violate energy conservation because the additional energy comes from the gain in the material, provided by an external source. For the passive medium described by Eq. (22), 〈uE〉 remained positive. In the active medium, 〈uE〉 < 0, as in the case of a stimulated emission.
Poynting’s theorem describes the change in energy density per unit time, and this is exhibited in Maxwell’s equations. We have considered the energies associated with a plane electromagnetic wave in infinite homogeneous material. Considering the microstructure, this implies that the homogenized material responses can be described by complex μ and ε. The spatial variation was suppressed for notational convenience, but use of the wave impedance to relate the electric and magnetic fields implies the spatial dependence of a plane wave. We have evaluated the kernel of the integral on the right hand side of Eq. (1). Volume-integrated results simply require the product of the density with the differential volume. Integration of the Poynting vector (the left-hand side of Eq. (1)) requires the spatial representation of the Poynting vector with the differential volume and surface area limit applied, and although we do not verify that result here, it is implied through the equality in Poynting’s theorem, from Maxwell’s equations.
The sign of 〈∂w/∂t〉 gives the transfer direction of the transient energy. When 〈∂w/∂t〉 is positive during the first half of the Gaussian pulse and negative during the second half, 〈w〉 in Fig. 2 is positive throughout and zero for large time, after the pulse has left that point in space. In the case where we find negative stored energy, 〈∂w/∂t〉 is first negative, and then positive. This opposite phase, relative to the cases where 〈w〉 is positive, results in negative stored energy (that in the field in vacuum plus that in the dipole moments in the material). Looking at this from the point of view of the material, the means that the material first delivers some energy to the field, and then the field returns this energy to the material. The negative field energy is energy borrowed from the material, and during the second half of the pulse, the field repays this energy. In the case of passive material, this means that during the first half of the pulse more energy was dissipated in the material than the field delivered. However, in the second half, less (than what the field delivered) was dissipated in the material. The net energy delivered to this point in the material after the pulse has passed is equal to that dissipated. The net accumulation or declination is due to a small contribution over each period at the carrier frequency. However, in the case of the lossless material considered here, there is no dissipation of energy in the material. Hence, 〈uE〉 = 0 after the pulse is gone. In the case of material with small gain, 〈uE〉 < 0, showing that the material delivered some energy to the incident field, thereby amplifying the outgoing pulse.
Note that the cases treated with a frequency-independent wave impedance, the total energy is exactly separable into stored and lost energies. For a material having a real part of the refractive index equal to −1, we obtain a negative total stored energy and the total electromagnetic energy remains positive. This provides insight regarding the energy density in an engineered negative refractive index metamaterial.
Based on Maxwell’s equations and more specifically Poynting’s theorem, we separated the total energy density into stored and lost components for a plane wave in homogeneous media described by both an electric (dielectric constant) and magnetic (permeability) response. Negative stored energy can occur for both passive and active material. In the passive case, negative stored energy occurs with the carrier frequency close to a material resonance. For active materials, both the total and stored energies can be negative because of the external pump energy provided. Importantly for sub-wavelength imaging applications requiring a negative refractive index, the stored field energy can be (but is not necessarily) negative in a regime where the refractive index is negative and real, which is the desirable condition to amplify evanescent fields.
The analytical model we provide for a narrowband pulse was validated by comparison with exact numerical results for the energies densities. This model provides a basis to evaluate the material response conditions that lead to negative stored energy.
This work was supported in part by the National Science Foundation (Grants 0824185, 0901383, 1028610), the Department of Energy (Grant DE-FG52-06NA27505), and the Army Research Office (Grant W911NF-10-1-0492).
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