We have derived, for oblique propagation, an equation relating the averaged energy flux density to energy fluxes arising in the process of scattering by a lossless finite photonic structure. The latter fluxes include those associated with the dispersion relation of the structure, reflection, and interference between the incident and reflected waves. We have also derived an explicit relation between the energy flux density and the group velocity, which provides a simple and systematical procedure for studying theoretically and experimentally the properties of the energy transport through a wide variety of finite photonic structures. Such a relation may be regarded as a generalization of the corresponding one for infinite periodic systems to finite photonic structures. A finite, N-period, photonic crystal was used to illustrate the usefulness of our results.
© 2014 Optical Society of America
Since the original papers of Yablonovitch  and John , many theoretical and experimental works have been devoted to the study of the transport of electromagnetic radiation through photonic structures [3, 4]. This interest has been motivated by the interesting basic electromagnetic properties of these systems as well as by their potential applications in a wide range of optical devices. A quantity of fundamental importance in these studies is the energy velocity, which is defined as the ratio of time-averaged energy flux density S to time-averaged energy density U . According to this definition, this velocity is in general a local quantity that provides an appropriate measure of the energy transport velocity inside the medium. Now, when describing the global propagation properties of these structures, the quantity of interest is the averaged energy transport velocity, defined as vE = 〈S〉/〈U〉 [3, 6], where 〈S〉 and 〈U〉 are the space-averaged energy flux density and space-averaged energy density, respectively. For infinite, higher dimensional photonic crystals, the average is taken over the unit cell [3, 6, 7], whereas in one-dimensional (1D) structures, where it is always possible to define S and U at each point of the system , the average must be taken within the unit cell for infinite crystals and over the entire sample for finite ones (see Sec. 2). In the following, we will use the symbol 〈...〉L to represent the latter average.
A consequence of the above definition of the averaged energy transport velocity is that it allows to establish an explicit connection between vE and the electromagnetic properties of the structure. Specifically, it has been shown that such a connection may be achieved through the group velocity, defined as vg = ∇Kω, where K is the wavevector and ω = ω(K) represents the structure’s dispersion relation. The link between these velocities has been investigated in the presence [5, 9, 10] and absence [11–16] of losses. It has been shown that in transparent photonic structures, which will be the focus of our attention, the properties of vE and vg are closely correlated. For instance, it was demonstrated theoretically that vE = vg for unbounded homogeneous media  and infinite periodic structures [3, 11]. This equivalence is a direct consequence of the translational symmetry of these media.
When this symmetry is broken or losses are taken properly into account , the link between vE and vg and therefore between the space-averaged energy flux density and vg may be substantially modified. Recently, these modifications were investigated in periodic photonic crystals subject to a uniform residual disorder  and in finite one-dimensional (1D) photonic structures . Specifically, the authors of the last reference derived the relation:12]. We point out that these results were established for the case of normal incidence, and one of our main goals is to extend them to the case of oblique propagation. Certainly, here we will derive, for the first time to our knowledge, a formula connecting the energy transport velocity vE to the group velocity vg for finite dispersive photonic structures and any angle of incidence. An important consequence of this formula is that it allows to establish a direct correlation between the components of the energy flux density 〈S〉L and those of vg, which not only provides an analytical and systematical procedure for the study of the energy transport through the considered structures, but also it highlights the role of the dispersion relation in these studies. Accordingly, we will focus our attention on the properties of that correlation, which may be regarded as a generalization of the corresponding one for infinite periodic systems to finite photonic structures.
The paper is organized as follows. In Sec. 2, we use the Poynting theorem [17, 18] for TE-polarized waves to derive two Eqs. containing the components of 〈S〉L and those of the energy fluxes arising in the process of scattering by the photonic structure. These Eqs. are used in Sec. 3 to express the components of both vE and 〈S〉L in terms of those of the group velocity vg. The usefulness of the correlation between 〈S〉L and vg in the description and understanding of the energy transport through the considered structures is illustrated in Sec. 4 by applying it to a specific photonic structure. Finally, our conclusions are given in Sec. 5.
2. Energy flux density
In this work, we study the process of scattering by a one-dimensional (1D) photonic structure localized between the z = 0 and z = L planes, as shown schematically in Fig. 1. For simplicity, we assume the structure is sandwiched between two semi-infinite layers made of the same optical materials. We focus our attention on a monochromatic electromagnetic field propagating in the (x, z) plane with wave vector component Kx along the x -axis. For TE modes, the spatial part of the electric and magnetic fields can be written as 
In these Eqs., x, y and z are the unit vectors along the x, y and z axes, respectively, φ(z) is the phase of E(z),
The magnetic permeability μ(z) and the dielectric permittivity ε(z) are real quantities and may be frequency dependent.
In order to characterize the electromagnetic modes in a finite photonic structure, one can use both Kx and the effective wavevector Kz, which is defined in terms of the phase Φ of the complex transmission amplitude t as Φ = LKz [20, 21]. This means that Kx and Kz are independent quantities and the dispersion relation of the structure is characterized by the dependence of the frequency ω on the wavevector K = xKx + zKz, i. e. ω = ω(Kx, Kz). Further, since Eqs. (8) and (9) depend explicitly on ω and Kx, the vectors u and v also depend on K. Note, however, that the dependence of u and v on Kz is only through the dispersion relation. That is, these vectors are composite functions of Kz. This difference between Kx and Kz will be taken into account in our calculations.
In the Eqs. above, S and U are the time-averaged Poynting vector and energy density, respectively, and vgα = ∂ω(K)/∂Kα is the α-component of the group velocity. To obtain these expressions we followed the procedure used in [3, 11] for the case of periodic photonic crystals and the fact that u is a real quantity. It is important to note that since Φ = LKz, the z -component of vg is given by vgz = ∂ω(K)/∂Kz = L/τd, where τd = ∂Φ/∂ω. That is, τd represents the group delay or Wigner delay time  and vgz corresponds to the group delay velocity.
Using Eqs. (12), (17) and (18) and taking into account that the z -component of the Poynting vector z · S is conserved throughout the structure in the absence of losses, the spatial average of Eq. (11) over the entire sample along the z -direction leads to the formulas
As is clearly seen in Eqs. (19)–(22), the phase and modulus of the electric field E(z) at z = 0 and z = L play an important role in the study of the average behavior of the energy flow through the structure. Since these quantities are directly related to the complex reflection r and transmission t amplitudes, let us express the above Eqs. in terms of them. This is achieved by noting that for z < 0 the electric field is a linear combination of incoming (incident) and outgoing (reflected) waves
Using (25) and taking into account the relation ∂Ki/∂Kj = δi,j, with i, j = x, z, and the fact that the normal energy flow is given byEquations (19) and (20) can be written as
It should be noted that the factor QL(r − r*) = 2ir2QL in the latter Eq. arises from the interference between incident and reflected waves.
The first term on the right-hand side of Eq. (30) represents an energy flow Sgα whose velocity is determined by the dispersion relation of the structure. The second one is the energy flow along the α-axis arising from the interference between the incident and reflected waves, as noted above. In order to interpret the third term we consider, for simplicity, a symmetric photonic structure. In this case, ΦR = Φ ±π/2 = LKz ± π/2 and (1/L)∂ΦR/∂Kα = 0 and 1 for α = x and z, respectively. Thus, the energy flow associated with the third term vanishes along the lateral direction, whereas it is exactly equal to the reflected energy flow along the normal direction.
It follows immediately from Eq. (30) that at transmission resonances 〈Sα〉L = vEα〈U〉L = Sgα and therefore vEα = vgα for α = x, z. As one sees, Eq. (30) connects the interference and reflection energy fluxes to two quantities of special interest: the group velocity vg, which may be superluminal away from resonance, and the energy velocity vE, which in general remains causal [13, 16, 17]. It is then clear that, in the superluminal regime, the subluminal behavior of vE is closely related to effects of interference and reflection on the energy transport through the structure. In other words, these effects avoid the violation of causality.
3. Relation between group velocity and energy flux density
Let us first use Eq. (30) to derive an explicit relation between the group and energy transport velocities. This may be achieved by noting that, as discussed above, we can substitute the operator ∂/∂Kα in (30) by(32) represents the derivative with respect to Kx keeping the frequency ω constant.
Equations (33) and (34) relate the components of the energy transport velocity vE to those of the group velocity vg for the case of oblique propagation in 1D lossless photonic structures of finite length. In general, these vectors do point in different directions and have different magnitudes, except at transmission resonances where vE = vg. Since these Eqs. provide a simple correlation between the energy transport velocity and the dispersion relation, the latter notion is of great importance in describing the properties of the energy flux through the photonic structure. A similar role is played by the frequency-dependent parameters T0 and Tx. Due to this, it is very important to understand the meaning of these parameters and their possible connection with measurable quantities. In order to simply the analysis, this issue will be treated for symmetric photonic structures, that are materials of great practical interest. In this case, ΦR = Φ ±π/2 = LKz ± π/2 and ∂ΦR/∂Kx = 0, ∂ΦR/∂Kz = L and ∂ΦR/∂ω = τd. Using these relations in Eq. (35) and Eq. (36) and taking into account that , we obtain:17] arising from the overlap between incident and reflected waves in the region before the scattering medium z < 0 and along the normal direction, and (33) and (34) become: (41), we obtain immediately the relation: 17]. In conclusion, the energy transport velocity vE and the group velocity vg are related through quantities having specific physical meaning.
Note, finally, that the relation between vEz, vgz and is independent of Kx, that is to say, the effects of oblique propagation do not modify it.Eq. (27), which fully determines the properties of 〈Sz〉L, form the basic Eqs. for studying the energy flux densities in finite photonic structures. To carry out such a study, it is necessary to know the complex reflection r and transmission t amplitudes, which are related through the total transfer matrix T̃ as  Eq. (44), where Tij are the matrix elements of T̃.
Using Eq. (47), it is straightforward to obtain the following formulas for the group delay τd and the ratio between the components of the group velocity:
4. Application to finite, N-period, photonic crystals
Formulas (27) and (44) are general enough and may be used to investigate the properties of the space-averaged energy flux density in a wide variety of finite photonic structures, such as plasma slab , periodic superlattices containing left-handed materials [19, 24], chirped periodic structures , etc. Here, in order to illustrate the usefulness of these formulas, we choose a finite, N-period, photonic structure A[BABA...BA]A sandwiched between two semi-infinite layers made of the same optical materials A, characterized by positive and frequency independent optical parameters ε1 and μ1. For now we will leave the nature of layer B unspecified, beyond requiring that their optical parameters ε2(ω) and μ2(ω) be real quantities, as assumed above.
Taking into account that the electric field E(z) for z < 0 and z > L are giving by Eqs. (23) and (24), respectively, and using the transfer-matrix technique , we obtain the following formulas for T22 and T21 :
Noting that Q1 is real, Q2 may be real or purely imaginary, and the Bloch phase β is real inside the allowed bands and equal to iψ or to π + iψ in the energy gap regions, where ψ is a real angle, the function g and both cosNβ and sinNβ/sinβ are always real quantities. These properties and the first relation in Eq. (46) lead to the following formulas for the transmission amplitude and the dispersion relation of the finite photonic crystal:
Moreover, using again Eqs. (46) we get the expression:Eq. (36) and the fact that T + R = 1, Eq. (44) becomes: Eqs. (52)–(53) and g′ = (∂g/∂Kx)ω and f′ = (∂f/∂Kx)ω.
At this point, it is convenient to express the Eqs. obtained above in terms of the angle of incidence θi, which is related to the Kx -component of the wavevector K as . Using the latter relation, it is easy to see that , and Eq. (27) can be written down as:
Equation (61) shows that, for a given value of the propagation angle θi, the structure of 〈Sz〉L as a function of ω is the same as that of the transmission coefficient. In consequence, the z-component of the energy flux density exhibits maxima at transmission resonances which, according to Eq. (56), correspond to the conditions Nβ = mπ, with m = ±1, ±2,...,±(N − 1); bQ2 = nπ, with n = 1, 2, 3,...; and η = ±1. Note that only the former condition depends on the number N of unit cells. Thus, when the frequency ω varies within an allowed miniband of the corresponding infinite photonic crystal, 〈Sz〉L is an oscillating function of ω and exhibits a resonant structure. For frequencies inside the bandgaps of the infinite crystal, the Bloch phase β is a complex quantity and, according to Eq. (56), the resonant structure of 〈Sz〉L should disappear. These properties of the 〈Sz〉L-spectra are illustrated in Fig. 2 for a finite, N -period, quarter-wave-stack (λ0/4 = πc/2ω0 structure)[20, 26], with μ1 = μ2 = 1, and , for N = 5, 10 and various values of the angle of incidence θi. One sees in Fig. 2 that the main effects of increasing θi, for a given value of N, are to shift the 〈Sz〉L-spectra to higher frequencies and to reduce the corresponding resonant-peak values.
Since R = r2 = 0 at transmission resonances, it follows from Eqs.(49), (57) and (62) that, for a fixed value of θi ≠ 0 and ignoring quantitative differences, the lateral energy flux density 〈Sx〉Land the ratio Vfin = vgx/vgz for the finite photonic crystals should exhibit resonant structures similar to those of 〈Sz〉L. It is clearly seen in Eq. (62) that the peak values of 〈Sx〉L and Vfinare the same at each transmission resonance if the factor cosθi is ignored. These theoretical results are illustrated in Fig. 3 for the λ0/4 photonic structure with the same parameters used in Fig. 2.
Let us now briefly compare the properties of Vfin with those of the ratio Vinf = vgx/vgz for the corresponding infinite crystals. This will provide an appropriate understanding of the behavior of Vfin. Such a comparison is shown in Fig. 4 for the λ0/4 structure with the same parameters used in Fig. 2. One clearly sees that, for a fixed value of N, the oscillations of Vfin in a pass band always occur around the curve associated with Vinf. This means that the pass bands of the infinite crystals are similar to the corresponding ones of the finite crystal, specially for large values of N. The difference inside the bandgaps of the infinite crystals, which tends to disappear for large values of N, is due to the fact that the effect of finite crystal size is to create photon states inside these gaps.
Finally, it should be pointed out that, for a structure with given optical and geometrical parameters, formulas (61) and (62) and the fact that the transmission coefficient is a measurable quantity allow the experimental study of the normal and lateral electromagnetic energy transport in finite photonic structures.
We have derived, for oblique propagation of TE-polarized modes, a formula relating the energy flux density to energy fluxes arising in the process of scattering by a lossless, finite, 1D photonic structure. The latter fluxes include those associated with the dispersion relation of the structure, reflection, and interference between the incident and reflected waves. A simple analysis of that formula indicated that the causal behavior of the energy velocity is closely related to effects of interference and reflection on the energy transport through the structure. We have also derived an explicit relation between the energy flux velocity and the group velocity, which represents an extension of the corresponding results obtained in  and  for normal incidence to the case of oblique propagation. That relation allows us to find a simple correlation between the energy flux density and the group velocity. This correlation provides a simple and systematical procedure for studying theoretically and experimentally the energy transport through a wide variety of finite photonic structures, such as periodic superlattices containing left-handed materials , chirped periodic structures , etc. It also highlights the role of the dispersion relation in these studies. Finally, a finite, N -period, photonic crystal was used to illustrate the usefulness of the presented results.
We are grateful for the financial support provided by the Alma Mater Project of the University of Havana. MDL is grateful to Universidad de Antioquia where part of this work was done. CAD is grateful to the Colombian Agencies CODI-Universidad de Antioquia (Estrategia de Sostenibilidad 2013–2014 de la Universidad de Antioquia), Facultad de Ciencias Exactas y Naturales-Universidad de Antioquia (CAD-exclusive dedication project 2013–2014), and El Patrimonio Autónomo Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación, Francisco José de Caldas.
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