Homework SourseA uniform plane wave propagates in the y direction in air wi
Solution
Instantaneous electric energy and electric power dissipation in dispersive media Wonseok Shin,1 Aaswath Raman,2 and Shanhui Fan1,* 1 Department of Electrical Engineering, Stanford University, Stanford, California 94305, USA 2 Department of Applied Physics, Stanford University, Stanford, California 94305, USA *Corresponding author: shanhui@stanford.edu Received November 14, 2011; revised December 26, 2011; accepted December 27, 2011; posted January 5, 2012 (Doc. ID 158103); published April 24, 2012 We derive the instantaneous densities of electric energy and electric power dissipation in lossless and lossy dispersive media for a time-harmonic electric field. The instantaneous quantities are decomposed into DC and AC components, some of which are shown to be independent of the dispersion of dielectric constants. The AC component of the instantaneous energy density can be used to visualize propagation of electromagnetic waves through complex 3D structures. © 2012 Optical Society of America OCIS codes: 260.2110, 260.2030, 260.3910. 1. INTRODUCTION Understanding the forms of the densities of energy and power dissipation in dispersive media has been a topic of interest for decades [1–4] and has found renewed interest recently due to the development of metamaterials [5–11]. When electric dispersion is concerned [12], a dispersive medium is characterized by a frequency-dependent dielectric constant 0 i 00. Assuming a time-harmonic electric field (E-field) Et E0eit , it is well-known that in a lossless dispersive medium, where 00 0, the time average of the electric energy density is [3] ue 1 4 d d jE0j 2. (1) The time average of the electric energy density is known also in a lossy dispersive medium, where 00 > 0, provided that the medium is a Lorentz medium, i.e., described by a Lorentz pole [2,6,8,13]. In addition, the time average of the electric power dissipation density in a lossy dispersive medium is [3] qe 1 2 00 jE0j 2; (2) which holds for any lossy dispersive medium including the Lorentz medium. Most of the previous works on electric energy and power dissipation have focused on the time-averaged quantities [Eq. (1) and Eq. (2)]. The present paper, on the other hand, derives the formulae for instantaneous electric energy density and electric power dissipation density. The instantaneous quantities in a dispersive medium were also studied in [7], but their analysis was limited to a dielectric constant described by a single Lorentz pole. In contrast, we derive, for a lossless dispersive medium, a formula for the instantaneous energy density for a generic dielectric constant, provided that the E-field is harmonic in time. Because we consider a time-harmonic E-field, each instantaneous quantity is expressed as the sum of the aforementioned time-averaged quantity and sinusoidal oscillation. We refer to the time average and sinusoidal oscillation as the DC and AC components of the instantaneous quantity, respectively. Examining Eq.(1), we notice that evaluating ue at a given frequency requires the values of both and d d at the frequency. In other words, measuring at the frequency alone is insufficient to determine ue; we need additional information on the dispersion of , which is characterized by d d in this case. We refer to quantities such as ue as being “dispersiondependent.” Interestingly, in contrast to ue, qe of Eq. (2) requires only the value of at the frequency and does not require the information on the dispersion of . We refer to quantities such as qe as being “dispersion-independent.” One of our objectives in this paper is to show that some DC and AC components of the instantaneous quantities in dispersive media are dispersion-independent. The paper is organized as follows. In Section 2 we introduce the notations and conventions used in this paper. In Section 3 we provide a derivation of the instantaneous electric energy density in a lossless dispersive medium, which is the main result of this paper. Since instantaneous energy density has rarely been discussed in the literature, for completeness we provide the corresponding formula for a lossy Lorentz medium in Section 4 based on a recent work [13]. In Section 5 we derive the instantaneous electric power dissipation density for the same Lorentz medium and investigate the phase relationship between the oscillations of energy and power dissipation. Finally, in Section 6, we demonstrate that direct visualization of the AC component of the instantaneous electromagnetic (EM) energy density provides new insights in frequencydomain simulations of EM wave propagation compared to the conventional approaches. 2. NOTATIONS AND CONVENTIONS Throughout this paper, bold capital symbols are used for complex vector fields, whereas the corresponding curly capital 1048 J. Opt. Soc. Am. B / Vol. 29, No. 5 / May 2012 Shin et al. 0740-3224/12/051048-07$15.00/0 © 2012 Optical Society of America symbols indicate their real parts; for example, Et RefEtg RefE0 ei t g for a time-harmonic E-field. In equations, c.c. refers to the complex conjugate of the preceding quantity; for instance, Et RefEtg 1 2 Et c:c:. Finally, jEj 2 E · E and E2 E · E; note that jEj 2 E2. 3. INSTANTANEOUS ELECTRIC ENERGY DENSITY IN LOSSLESS DISPERSIVE MEDIA The time-averaged electric energy density in a lossless dispersive medium is Eq. (1). In this section, we extend the derivation of Eq. (1) to calculate the instantaneous electric energy density. Even though a dispersive medium cannot be lossless for all frequencies due to Kramers–Kronig relations [14], many materials such as dielectrics have negligible loss in some frequency bands, where the analysis in this section applies. By Poynting’s theorem and conservation of energy, the instantaneous electric energy density uet in a lossless medium satisfies ue t E · D t 1 4 E · D t E · D t c:c: (3) We integrate Eq. (3) over time to calculate uet. Here, following [3], we consider Et E0t ei t as an approximation to a purely time-harmonic E-field, where the envelope E0t varies much more slowly than ei t . Because E0t 1 2 p R E0; ei t d, we have D t 1 2 p Z i E0; ei t d. (4) Because of the slowly varying envelope assumption, E0;, and thus the integrand of Eq. (4), is nonzero only for 0. Since d d for 0, Eq. (4) is approximated to D t iE0t d d E0t t eit : (5) From now on, we write as for simplicity. Substituting Eq. (5) to Eq. (3) gives ue t 1 4 iE0t2 d d E0t · E0t t ei2t ijE0tj2 d d E0t · E0t t c:c: (6) Because for a lossless medium, Eq. (6) is further simplified to ue t 1 4 2RefiE0t2 ei2t g d d Re 2E0t· E0t t ei 2 t d d t jE0tj2 : (7) To express uet in terms of Et, we use E0t Et eit . Then, Eq. (7) reduces to ue t 1 2 2 d d Refi Et2g 1 2 d d t Et2: (8) By integrating Eq. (8) over time, we obtain the instantaneous electric energy density uet 1 2 d d Et2 1 2 2 d d ReZ i Et2 dt C; (9) where C is a constant of integration. For a purely time-harmonic field Et E0eit , Eq. (9) reduces to uet 1 2 d d Et2 1 4 d d RefEt2g C 1 4 d d jE0j 2 1 4 RefEt2g C; (10) where we have utilized an identity W2 1 4 W W2 1 2 jWj 2 1 2 RefW2g (11) that holds for any complex vector field W and its real part W. The comparison of Eq. (10) with the well-known result [Eq. (1)] proves C 0 because the time average of the second term of Eq. (10) is zero. Therefore, the instantaneous electric energy density for a time-harmonic field is uet ue u~et; (12) where the DC component ue is Eq. (1) and the AC component u~et is u~et 1 4 0 RefEt2g for 00 0: (13) By comparing Eq. (1) and Eq. (13) we notice that the AC component of uet is dispersion-independent, whereas the DC component of uet is dispersion-dependent. Therefore, in a lossless dispersive medium, if we are interested only in the amplitude and phase of the oscillation of the instantaneous electric energy density, we do not require knowledge of the dispersion of ; the only information of needed is the value of at the frequency of the E-field. We also note that the AC component [Eq. (13)] for a lossless medium is either in-phase with RefEt2g for 0 > 0 or 180° outof-phase with RefEt2g for 0 < 0. 4. INSTANTANEOUS ELECTRIC ENERGY DENSITY IN LOSSY DISPERSIVE MEDIA In general, Poynting’s theorem I S E × H · da Z V E · D t H · B t dv (14) holds for both lossy and lossless media, with H SE × H · da being the power influx through the surface S enclosing a volume V. On the other hand, by considering energy conservation alone, in a lossy medium we expect that Shin et al. Vol. 29, No. 5 / May 2012 / J. Opt. Soc. Am. B 1049 I S E × H · da Z V ue t um t dv Z V qe qmdv; (15) where umt is the instantaneous magnetic energy density; qet and qmt are the instantaneous electric and magnetic power dissipation densities, respectively. Comparing Eq. (14) and Eq. (15), we see that for lossy media it is no longer possible to identify the integrand of the right-hand side of Eq. (14) as the time derivative of the energy density, like we did in Section 3 for lossless media. As a result, Eq. (1) is in general not correct for a lossy dispersive medium [2,6,7,9]. In this and the next sections, we provide derivations of uet and qet in a lossy dispersive medium. For this purpose, we follow [2,6–8,13] and consider a medium whose dielectric function is described by multiple Lorentz poles: 1 XN i1 2 p;i 2 0;i 2 ii : (16) The ith pole in the dielectric function can be understood as resulting from electrons moving in a harmonic potential characterized by the resonance frequency 0;i while experiencing a damping force with the damping coefficient i. When an Efield is applied, the equation of motion for such an electron is m d2ri dt2 mi dri dt m2 0;iri eE; (17) where m, e, ri are the mass, charge, and the displacement of the electron, respectively. We further assume that the density of such electrons is ni. The polarization field Pi due to these electrons is then Pi nieri, and its dynamics is described by 2Pi t 2 i Pi t 2 0;iPi 2 p;iE; (18) where p;i nie2m p . For time-harmonic fields, Eq. (18) dictates 2Pi iiPi 2 0;iPi 2 p;iE; (19) from which Eq. (16) recovered. Now, we define Vi Pit that corresponds to the polarization velocity field [13]. Then Eq. (18) can be written as Vi t iVi 2 0;iPi 2 p;iE: (20) Using D E P and Eq. (20), we obtain E · D t XN i1 1 2 p;i iV2 i t 1 2 E2 XN i1 2 0;i 22 p;i P2 i XN i1 1 22 p;i V2 i : (21) Substituting Eq. (21) in Eq. (14) and equating Eq. (14) to Eq (15), we have qet XN i1 1 2 p;i iV2 i ; (22) which is the density of the power dissipated by electric damping, and uet 1 2 E2 XN i1 2 0;i 22 p;i P2 i XN i1 1 22 p;i V2 i ; (23) where the three terms in the right-hand side are the densities of the energy of the E-field, potential energy of the electrons, and kinetic energy of the electrons. Equations (22) and (23) are consistent with [2,6,7,13]. We note that Eq. (22) and Eq. (23) are valid for any time-varying Et. For a time-harmonic Et, we now reduce Eq. (23) into a formula that is independent of the fields other than Et. From Eq. (19) we have Pit 2 p;i 2 0;i 2 ii Et; (24) and therefore Vit Pit t i2 p;i 2 0;i 2 ii Et: (25) We decompose E2, P2 i , and V2 i in Eq. (23) into DC and AC components by Eq. (11) and use Eq. (24) and Eq. (25) to obtain uet ue u~et (26) with the DC component ue 1 4 1 XN i1 2 p;i2 0;i 2 2 0;i 22 22 i jE0j 2 (27) and the AC component u~et 1 4 Re1 XN i1 2 p;i2 0;i 2 2 0;i 2 ii2 Et2 : (28) Equations (27) and (28) agree with Eq. (1) and Eq. (13), respectively, when we use the dielectric function of Eq. (16) with i 0. We note that Eq. (28) is considerably simplified in cases where the Lorentz medium consists of a single pole, as shown in Appendix A. For uet, both its DC component [Eq. (27)] and AC component [Eq. (28)] depend on the parameters 0;i, p;i, i, and . These parameters cannot be determined from the value of the of Eq. (16) at single . Therefore, both ue and u~et for a lossy dispersive medium are dispersion-dependent. We also note that the AC component [Eq. (28)] for a lossy medium is in general out-of-phase with RefEt2g. This is due to the phase delay in Pit with respect to Et; in a damped harmonic oscillator, the displacement typically lags behind the driving force [15,16]. Because Pit, and thus Vit, is outof-phase with Et in general, we see from Eq. (23) that uet and Et2, or their AC components u~et and 1 2 RefEt2g, do not oscillate in-phase for a time-harmonic Et. 1050 J. Opt. Soc. Am. B / Vol. 29, No. 5 / May 2012 Shin et al. We end this section by noting that the instantaneous energy density uet is always positive, as can be seen from Eq. (23). From Eq. (26) we therefore have u~et ue. Since the AC component u~et oscillates sinusoidally, this further implies that ue u~et ue: (29) In other words, the amplitude of the AC component of the instantaneous energy density cannot exceed the time-averaged energy density. Also, 0 uet 2 ue; (30) so the instantaneous energy density, which is always positive, does not exceed twice the time-averaged energy density. 5. INSTANTANEOUS ELECTRIC POWER DISSIPATION DENSITY IN LOSSY DISPERSIVE MEDIA For a time-harmonic E-field, we decompose V2 i in Eq. (22) into DC and AC components by Eq. (11) and use Eq. (25) to obtain qet qe ~qet (31) with the DC component qe 1 2 XN i1 2 p;i2i 2 0;i 22 22 i jE0j 2 (32) and the AC component ~qet 1 2 ReXN i1 2 p;i2i 2 0;i 2 ii2 Et2 : (33) Because Eqs. (32) and (33) depend on the parameters 0;i, p;i, i, and that cannot be determined by the value of the of Eq. (16) at single , it would seem that both the DC and AC components of qet are dispersion-dependent. However, it turns out that Eq. (33) reduces to Eq. (2), so the DC component of qet is dispersion-independent. On the other hand, the AC component of qet is dispersion-dependent, because it satisfies ~qet 4 2 u~et 1 4 Ref Et2g; (34) which can be proved by direct substitution of Eqs. (16), (28), and (33). The first term of the right-hand side of Eq. (34) is dispersion-dependent as shown at the end of Section 4, whereas the second term is not. Hence, the left-hand side of Eq. (34) should be dispersion-dependent. Table 1 summarizes the dispersion dependencies of the DC and AC components of the instantaneous quantities examined so far. We now investigate the relationship between the oscillation phases of the instantaneous electric energy density, power dissipation density, and electric field. We consider silver as an example of a lossy material. The parameters 0;i, p;i, i, and describing the of silver are taken from [17]. Using these parameters, we evaluate u~et and ~qet in silver for two vacuum wavelengths 0 620 nm and 0 1550 nm, for both of which 00 Ag0 Ag 0.08. Figure 1 displays u~et and ~qet for the two vacuum wavelengths; RefEt2g is also plotted for comparison. We first note that u~et is nearly 180° out-of-phase with RefEt2g. This is because j00 Ag0 Agj 1 and thus silver can be treated as nearly lossless for both the wavelengths; see the comment at the end of Section 3. We also compare the phases of u~et and ~qet. Because u~et Equation 13 1 4 RefEt2g for j000 j 1; (35) the magnitude of u~et 1 4 Ref Et2g is close to zero. However, Eq. (35) does not impose any restriction on the phase of u~et 1 4 RefEt2g. As a result, the phase of ~qet in Eq. (34) can have a range of values, as demonstrated in Figs. 1(a) and 1(b). In particular, Fig. 1(a) implies that the instantaneous densities of electric energy and electric power dissipation are not necessarily maximized at the same time, even if the medium is nearly lossless. We also note that the phases of ~qet in Figs. 1(a) and 1(b) are quite different regardless of the nearly identical loss tangents 00 Ag0 Ag 0.08 for the two wavelengths used. 6. VISUALIZATION OF THE AC COMPONENT OF THE INSTANTANEOUS EM ENERGY DENSITY CARRIED BY EM WAVES In this section, we demonstrate that visualizing the AC component of the instantaneous EM energy density in numerical simulations provides new insights into photonic device physics. In frequency-domain EM simulations, it is conventional to visualize simulation results by plotting a single coordinate component of the E- or H-field (e.g., Ex in the Cartesian coordinate system, H in the spherical coordinate system). However, this method describes the propagation of EM waves correctly only when the field is polarized dominantly in one direction, which is in general not true for waves propagating through 3D structures. For example, consider EM wave propagation through a metallic slot waveguide bend [18] illustrated in Fig. 2. Such a slot waveguide structure is important in integrated nanophotonics since it provides broadband nanoscale guidance of EM waves [19]. Our numerical simulation of a metallic slot waveguide Table 1. Dispersion Dependencies of the DC and AC Components of the Instantaneous Electric Energy Density uet and Instantaneous Electric Power Dissipation Density qet in Lossless and Lossy Dispersive Mediaa Lossless Lossy ue ue qe DC D DI AC I DD a D indicates dispersion-dependent, and I indicates dispersionindependent. Shin et al. Vol. 29, No. 5 / May 2012 / J. Opt. Soc. Am. B 1051 bend described in Fig. 2 reveals that the transmission is about 80% at a vacuum wavelength 0 1550 nm. The rest of the incident power divides into the two leakage channels indicated in Fig. 2: about 10% of the incident power couples into a surface plasmon mode bound to the metal film, and another 10% radiates into the background dielectric. Now, we consider the visualization of the EM wave that best illustrates the underlying physics. One could plot a single Cartesian component of the EM field. In this case, because Hy is the only EM field component that is dominant inside the slot waveguide both before and after the bend, it is natural to plot Hy over all space. However, as shown in Fig. 3(a), while the plot of Hy indeed describes light going around the bend, it displays neither the energy leaking into the surface mode nor the spherical wavefronts that are characteristic of the radiation into the background dielectric. Alternatively, one may plot the time-averaged EM energy density u as in Fig. 3(b). The time-averaged EM energy density is a scalar quantity that contains contributions from all components of the E- and H-fields, so it can show the distribution of energy in space properly without being sensitive to variation in the dominant polarization direction. However, since the time-averaged EM energy density does not contain any phase information, it cannot describe wave propagation. To overcome the limitations of plotting a single Cartesian component of the EM field or the time-averaged EM energy density, we can plot the AC component u~ u~e u~m of the instantaneous EM energy density. The AC component of the instantaneous EM energy density has the merits of both quantities discussed above: it retains phase information like a single Cartesian component of the EM field, and it is insensitive to variation in the dominant polarization direction of the EM field like the time-averaged EM energy density. Therefore, u~ is indeed an appropriate quantity to plot to visualize EM wave propagation through arbitrary 3D routes. Figure 3(c) visualizes u~ for the same solution of the frequency-domain Maxwell’s equations used in Fig. 3(a) and 3(b). The resulting plot successfully shows high transmission through the bend, and it also correctly describes the energy coupled to the surface plasmon mode and the spherical wavefronts of the radiation. Moreover, the plot clearly shows the phase variation of the field. Note, however, that the spatial frequency of u~ in Fig. 3(c) is twice as fast as that of Hy in Fig. 3(a) because u~ contains E2 and H2. One could visualize RefE2g to achieve the same advantages that the plot of u~ has. However, we note that u~ has a direct physical significance, and it also includes the contributions from both the electric and magnetic fields. In calculating u~ u~e u~m for Fig. 3(c), we use Eq. (13) for u~e and 1 4 0 RefH2g for u~m at each spatial point. Even though silver is lossy, the loss is very small at the wavelength used; linear interpolation of the experimentally measured data tabulated in [20] gives Ag 129 i 3.28 0 as the dielectric constant of silver at 0 1550 nm, so j00 Ag0 Agj 0.0254 1. Therefore, silver is nearly lossless to the infrared wave used in the numerical simulation, so the use of Eq. (13) is valid as ue t qe t Re E t 2 0 2 4 6 8 10 12 4 2 0 2 4 t fs ue t qe t Re E t 2 0 5 10 15 20 25 30 30 20 10 0 10 20 30 t fs Fig. 1. (Color online) Plots of u~et, ~qet and RefEt2g for time-harmonic electric fields Et x E^ 0 eit in silver (Ag). The quantities are plotted for two vacuum wavelengths: (a) 0 620 nm and (b) 0 1550 nm. Note that u~et is nearly 180° out-phase with RefEt2g for both cases. Also notice that ~qet is out-of-phase with u~et in (a) whereas it is nearly in-phase with u~et in (b). The units of u~et, ~qet, and RefEt2g in the vertical axes are 0 E2 0, 0 E2 0, and E2 0, respectively. Fig 2. (Color online) Metallic slot waveguide bend composed of two silver (Ag) films immersed in silica (SiO2). The two red arrows indicate the directions of the energy flow inside the slot region before and after the bend. The dashed blue arrow indicates the energy leakage into a mode bound to the metal film. The blue arcs indicate the radiation into the background silica. The dominant H-field component is shown for each energy flow channel. Note that the leakage channels have the dominant H-field polarized in different directions than the slot waveguide channel. The vacuum wavelength and relevant dimensions of the structure are indicated in the figure. At the specified vacuum wavelength, the dielectric constants of Ag [20] and SiO2 [21] are Ag 129 i 3.280 and SiO2 2.0850 , respectively. The magnetic permeabilities of both materials are 0. 1052 J. Opt. Soc. Am. B / Vol. 29, No. 5 / May 2012 Shin et al. shown in Eq. (35). In cases where substantially lossy materials are used, we can still calculate u~e using Eq. (28) with fitting parameters tabulated in the literature such as [17] or using a simple method introduced in Appendix A that requires a single fitting parameter when is close to a resonance frequency of . 7. CONCLUSION We have derived an expression for the instantaneous electric energy density in lossless dispersive media for a timeharmonic electric field. For lossy dispersive media described by Lorentz poles, we have derived an expression for the instantaneous electric power dissipation density as well as the instantaneous electric energy density. Each instantaneous quantity is decomposed into DC and AC components. Interestingly, some of the DC and AC components do not depend on the dispersion of dielectric constants. The AC component of the instantaneous EM energy density exhibits wavelike oscillation and contains contributions from all polarization components of the EM field. Plotting such a quantity therefore provides a convenient and informative way to visualize energy transport in complex EM structures and future integrated nanophotonic devices. APPENDIX A: SIMPLE METHOD TO CALCULATE THE AC COMPONENTS OF uet AND qet FOR LOSSY MEDIA DESCRIBED BY A SINGLE LORENTZ POLE In this section, we provide a simple method to calculate the AC components of the electric energy density and electric power dissipation density in a lossy medium described by a dielectric function with a single Lorentz pole: 1 2 p 2 0 2 i : (A1) The method involves only one fitting parameter . For a medium described by the dielectric function [Eq. (A1)], Eq. (28) reduces to u~et 1 4 Re 1 2 p2 0 2 2 0 2 i2 Et2 : (A2) The four parameters 0, p, , and needed to calculate Eq. (A2) are typically obtained by fitting experimentally measured around 0 to Eq. (A1). We seek to simplify Eq. (A2) into a form that involves only the value of at a given and . Using the electric susceptibility e 2 p2 0 2 i, we rewrite the factor multiplied to Et2 within the curly braces in Eq. (A2) as 1 2 p2 0 2 2 0 2 i2 1 e2 Re 1 e : (A3) Because e 1, Eq. (A3) is also written as 1 2 p2 0 2 2 0 2 i2 2 Re 1 : (A4) By substituting Eq. (A4) in Eq. (A2), we reach u~et 1 4 RefEt2g 1 4 Re 1 Ref 2 Et2g: (A5) Equation (A5) allows us to calculate u~et using a single fitting parameter ; the other parameter in Eq. (A5) is an experimentally measured quantity found in references such as [20,21]. We also obtain a simplified formula for ~qet by substituting Eq. (A5) into Eq. (34): ~qet 1 2 Imf Et2g 1 2 Re 1 Imf 2 Et2g: (A6) Fig 3. (Color online) Visualization of wave propagation through the metallic slot waveguide bend. The fields here are obtained by solving the frequency-domain Maxwell’s equations using the finite-difference frequency-domain method. The quantities (a) Hy, (b) time-averaged energy density u ue um, and (c) AC component of instantaneous energy density u~ u~e u~m of the solution of the frequency-domain Maxwell’s equations are plotted on two planes: a horizontal y const: plane on top of the metal film and a vertical x const: plane containing the central axis of the input port. Red, green, and blue indicate positive, zero, and negative, respectively. The area enclosed by the white dashed line in each figure is where the coupling of the EM energy into the surface plasmon mode is supposed to be observed. Note that (a) fails to capture such a coupling; (b) captures the coupling but loses the phase information; (c) displays both the coupling and phase information properly. Also notice the weak but discernible pattern of spherical wavefronts in the vertical plane in (c). The thin orange lines around the y 0 plane outline the two metal pieces. Shin et al. Vol. 29, No. 5 / May 2012 / J. Opt. Soc. Am. B 1053 ACKNOWLEDGMENTS We thank Sung Hee Park and Dr. Hyungjin Ma for helpful discussions and comments. This work was supported by the National Science Foundation (Grant No. DMS 09- 68809), the MARCO Interconnect Focus Center, and the AFOSR MURI program (Grant No. FA9550-09-1-0704). Wonseok Shin also acknowledges the support of Samsung Scholarship Foundation. REFERENCES AND NOTE 1. L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960). 2. R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A: Gen. Phys. 3, 233–245 (1970). 3. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Butterworth–Heinemann, 1984), 2nd ed. Section 80. 4. V. Polevoi, “Maximum energy extractable from an electromanetic field,” Radiophys. Quantum Electron. 33, 603–609 (1990). 5. R. W. Ziolkowski, “Superluminal transmission of information through an electromagnetic metamaterial,” Phys. Rev. E 63, 046604 (2001). 6. R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299, 309–312 (2002). 7. T. Cui and J. Kong, “Time-domain electromagnetic energy in a frequency-dispersive left-handed medium,” Phys. Rev. B 70, 205106 (2004). 8. S. Tretyakov, “Electromagnetic field energy density in artificial microwave materials with strong dispersion and loss,” Phys. Lett. A 343, 231–237 (2005). 9. P.-G. Luan, “Power loss and electromagnetic energy density in a dispersive metamaterial medium,” Phys. Rev. E 80, 046601 (2009). 10. K. Webb and Shivanand, “Electromagnetic field energy in dispersive materials,” J. Opt. Soc. Am. B 27, 1215–1220 (2010). 11. F. D. Nunes, T. C. Vasconcelos, M. Bezerra, and J. Weiner, “Electromagnetic energy density in dispersive and dissipative media,” J. Opt. Soc. Am. B 28, 1544–1552 (2011). 12. The derivation given in this paper for electrically dispersive media can be easily extended to magnetically dispersive media, which is described by a frequency-dependent magnetic permeability 0 i 00. 13. A. Raman and S. Fan, “Photonic band structure of dispersive metamaterials formulated as a Hermitian eigenvalue problem,” Phys. Rev. Lett. 087401 (2010). 14. J. D. Jackson, Classical Electrodynamics (Wiley, 1999), 3rd ed. Section 7.10. 15. A. Sommerfeld, Mechanics (Academic, 1956). Section III.19. 16. J. B. Marion and S. T. Thornton, Classical Dynamics of Particles and Systems (Saunders College, 1995), 4th ed. Section 3.6. 17. A. D. Raki, A. B. Djuriši, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices.” Appl. Opt. 37, 5271–5283 (1998). 18. W. Cai, W. Shin, S. Fan, and M. L. Brongersma, “Elements for plasmonic nanocircuits with three-dimensional slot waveguides,” Adv. Mat. 22, 5120–5124 (2010). 19. G. Veronis and S. Fan, “Modes of subwavelength plasmonic slot waveguides,” J. Lightwave Technol. 25, 2511–2521 (2007). 20. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). 21. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985). 1054 J. Opt. Soc. Am. B / Vol. 29, No. 5 / May 2012 Shin et al.
