Abstract

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

1. Introduction

Since the original papers of Yablonovitch [1] and John [2], 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 [5]. 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 [8], 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 [1116] 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 [6] 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 [14], 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 [15] and in finite one-dimensional (1D) photonic structures [16]. Specifically, the authors of the last reference derived the relation:

vE=TT+T0vg=Tvg(ω)
between vE and the group delay velocity vg for the case of normal incidence and showed that vE =vg only at the resonance frequencies of the transmission coefficient, where T0 is a frequency dependent parameter, and
vg(ω)=vgT+T0
is the group velocity defined in terms of the electromagnetic dwell time [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 〈SL 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 〈SL 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 〈SL in terms of those of the group velocity vg. The usefulness of the correlation between 〈SL 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 [19]

E(r)=yE(z)exp(ixKx)=u(r)exp[iϕ(x,z)],
H(r)=v(r)exp[iϕ(x,z)],
where
u(r)=y|E(z)|,
v(r)=icg(z,ω)[|E(z)|z+i|E(z)|φ(z)z]x+cKxg(z,ω)|E(z)|z.

 

Fig. 1 Schematic of the process of scattering by a finite photonic structure, localized between the z = 0 and z = L planes and sandwiched between two semi-infinite layers made of the same optical materials. Arrows indicate the incident, reflected and transmitted waves.

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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),

ϕ(x,z)=φ(z)+xKx
is the total phase of the electric field E(r), and g(z, ω) = ωμ(z), where μ(z) represents the magnetic permeability of the structure.

Substituting Eqs. (3) and (4) into the complex Maxwell’s Eqs., we get

×u(r)+ik×u(r)=icg(z,ω)v(r),
×v(r)+ik×v(r)=icf(z,ω)u(r),
where f (z, ω) = ωε(z) and
k=ϕ(x,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.

Taking the Kα-derivative in Eqs. (8) and (9) and combining both results, we obtain the Poynting theorem [1618] in the form

FKα+i32πckKαS=i32πcvgαU,
where
FKα=2ic|E(z)|2Kα[β(z)|E(z)|]z,
β(z)=1g(z,ω)|E(z)|z,
S=c8πRe[E×H*]
U=116π(f(z,ω)ωEE*+g(z,ω)ωHH*)

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 [22] and vgz corresponds to the group delay velocity.

Using Eqs. (3)(6) and (14), we find the expression

S=c28πg(z,ω){|E(z)|2Kxx+|E(z)|2φ(z)zz}
for the Poynting vector. This vector has normal Sz and lateral Sx components and determines the local energy flux density inside the structure. The lateral component disappears for normal propagation.

From Eqs. (7) and (10) we get

kKx=x+2φ(z)Kxzz,
kKz=2φ(z)Kzzz.

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

c216πLGz(K)+1L[φ(L)φ(0)]KzzSL=vgzUL,
c216πLGx(K)+SxL+1L[φ(L)φ(0)]KxzSL=vgxUL,
for α = z, x, respectively, where φ(L) and φ(0) are the phases of the electric field E(z) at the right (z = L) and left (z = 0) interfaces of the photonic structure, and
Gα(K)=Gα(L,K)Gα(0,K),
Gα(z,K)=|E(z)|2Kα[β(z)|E(z)|].

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

E(z)=exp(izQL)+rexp(izQL),
while for z > L it has only a transmitted component
E(z)=texp[iQL(zL)]=|t|expi[Φ+QL(zL)],
where Φ = LKz is the phase of t, QL=ω2c2εLμLKx2, and εL, μL are the permittivity and permeability for z < 0 and z > L.

In consequence,

φ(L)φ(0)=ΦθLKzθ,
where θ is the phase of E(0) = 1 + r = 1 + r1 + ir2 and satisfies the relation
tanθ=r21+r1.

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 by

SzL=zSL=c28πQLgLT,
Equations (19) and (20) can be written as
c2QL8πgL1L{12gLQLGα(K)θKαT}+SαL=vgαUL,
for α = x, z, where T = |t|2 is the transmission coefficient and gL = ωμL.

Using the continuity of Gα(z, K) at the right (z = L) and left (z = 0) interfaces of the structure and Eqs. (23) and (24), it is straightforward to show that Gα(L, K) = 0 and

Gα(K)=Gα(0,K)=i|1+r|2Kα[QLgL(rr*)|1+r|2]
for α = x, z.

It should be noted that the factor QL(rr*) = 2ir2QL in the latter Eq. arises from the interference between incident and reflected waves.

Substituting Gα(K) and ∂θ/∂Kα calculated from (26) into Eq. (28), we get

SαL=vgαULc28πr2LKα(QLgL)c28πQLgLRLΦRKα,
where R = |r|2 is the reflection coefficient and ΦR is the phase of the complex reflection amplitude r.

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 〈SαL = vUL = 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

Kz=vgzω,
Kx=vgxω+(Kx)ω,
for α = z and x, respectively, where the latter term on the right-hand side of (32) represents the derivative with respect to Kx keeping the frequency ω constant.

If we substitute (31) and (32) into (30) for α = z and x, respectively, and use Eq. (27) and the relation 〈SαL =vUL, we obtain, after some algebraic manipulation, the expressions

vEz=TT+T0vgz=Tvg(ω),
vEx=[T+(R/τd)ΦR/ω]T+T0vgxTxT+T0,
where τd = L/vgz is the group delay, vg(ω)=vgz/(T+T0) and
T0=1τd{r2gLQLω(QLgL)+RΦRω},
Tx=1τd{r2gLQL[Kx(QLgL)]ω+RΦRKx}.

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 QL=ω2c2εLμLKx2, we obtain:

T0=Rτiτd
Tx=τilτd,
where
τi=r2QL{QLgLgLωQLω},
is the self-interference delay time[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
τil=Kxr2QL2
may be also interpreted as a self-interference time, but along the lateral direction. Noting that for lossless media T + R = 1, it is easy to see that expressions (33) and (34) become:
vEz=T1τi/τdvgz=Tvg(ω),
vEx=11τi/τdvgx+τil/τd1τi/τd.
If vgz = L/τd and vg(ω)=L/τD are substituted into (41), we obtain immediately the relation:
τd=τD+τi
between the group delay τd, the dwell time τD and the self-interference delay τi, as expected [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 vg(ω) is independent of Kx, that is to say, the effects of oblique propagation do not modify it.

Dividing (34) by (33) we obtain the formula

SxLSzL=vExvEz=[T+(R/τd)ΦR/ω]TvgxvgzτdTxLT,
which relates explicitly the components of the energy flux density to those of the group velocity. This latter formula, derived for the first time in this work, together with Eq. (27), which fully determines the properties of 〈SzL, 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 as [16]
(t0)=T^(1r).
This Eq. and the fact that is an unimodular matrix lead to the expressions
1)t=1T22,2)r=T21T22=T21t,
which will be used to derive general formulas for the quantities involved on the right-hand side of Eq. (44), where Tij are the matrix elements of .

In fact, the dispersion relation of a finite photonic structure is determined from the transcendental Eq. [20, 23]

tanΦ=tanLKz=YX=F(ω,Kx),
where X and Y are the real and imaginary parts of t, F(ω, Kx) depends only explicitly on ω and Kx, as discussed above, and Φ = LKz is the phase of t = X + iY.

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:

τd=Φω=Lvgz=X2T(Fω)Kx,
vgxvgz=X2LT(FKx)ω,
where T = X2 + Y2 is the transmission coefficient.

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 [13], periodic superlattices containing left-handed materials [19, 24], chirped periodic structures [25], 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 [23], we obtain the following formulas for T22 and T21 :

T22=cosNβisinNβsinβg,
T21=i2sinNβsinβ(η1η)sinbQ2exp(iaQ1),
where Qi=ω2c2εiμiKx2, with i = 1, 2, η = μ2Q1/μ1Q2, N is the number of unit cells, a and b are the widths of layers A and B, respectively,
g=sinaQ1cosbQ2+12(η+1η)cosaQ1sinbQ2,
and β is the Bloch phase associated with the corresponding infinite photonic crystal which satisfies the dispersion relation:
cosβ=cosaQ1cosbQ212(η+1η)sinaQ1sinbQ2=f(ω,Kx)

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 or to π + in the energy gap regions, where ψ is a real angle, the function g and both cos and sin/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:

t=1T22=T{cosNβ+isinNβsinβg},
tanΦ=tanLKz=gtanNβsinβ=F(ω,Kx),
where
T=1|T22|2=1cos2Nβ+(sin2Nβ/sin2β)g2
is the transmission coefficient and
X=TcosNβ
is the real part of t.

Moreover, using again Eqs. (46) we get the expression:

ΦR=±π/2+ΦaQ1=±π/2+KzLaQ1
for the phase ΦR of the complex reflection amplitude r, which leads immediately to ΦR/∂ω = τdτa and (ΦR/∂Kx)ω = aKx/Q1, where τa = a∂Q1/∂ω is the time the electromagnetic wave spends in layer A. If the latter relations are used in combination with Eq. (36) and the fact that T + R = 1, Eq. (44) becomes:
SxLSzL=1T{(1Rτaτd)vgxvgz+KxQ11LQ1(r2aQ1R)}

In this latter Eq., the ratio vgx/vgz should be calculated by combining Eqs. (49), (53) and (55). As a result, we obtain:

vgxvgz=TL{[(1f2)g+fgf]sin2Nβ2(1f2)3/2Ngf1f2}
where g and f are the functions shown in 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 Kx=(ω/c)μ1ε1sinθi. Using the latter relation, it is easy to see that Q1=(ω/c)μ1ε1cosθi, Q2=(ω/c)μ1ε1μ2ε2/μ1ε1sin2θi and Eq. (27) can be written down as:

SzLS0=Tcosθi,
with S0=(c/8π)ε1/μ1.

Finally, substituting (61) into (59), one obtains immediately the formula:

SxLS0=cosθi{(1Rτaτd)vgxvgz+tanθiLQ1(r2aQ1R)},
for the x -component of the space-averaged energy flux density.

Equation (61) shows that, for a given value of the propagation angle θi, the structure of 〈SzL 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 = , with m = ±1, ±2,...,±(N − 1); bQ2 = , 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, 〈SzL 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 〈SzL should disappear. These properties of the 〈SzL-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, n1=ε1=1 and n2=ε2=1.41, 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 〈SzL-spectra to higher frequencies and to reduce the corresponding resonant-peak values.

 

Fig. 2 Normal energy flux density normalized to S0=(c/8π)ε1 as a function of the frequency ω in units of ω0, for a finite, N-period, quarter-wave-stack, with n1=ε1=1, n2=ε2=1.41, N = 5 (left-hand panel), 10 (right-hand panel) and various values of the angle of incidence θi.

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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 〈SxLand the ratio Vfin = vgx/vgz for the finite photonic crystals should exhibit resonant structures similar to those of 〈SzL. It is clearly seen in Eq. (62) that the peak values of 〈SxL 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.

 

Fig. 3 Lateral energy flux density normalized to S0=(c/8π)ε1 (black lines) and the ratio Vfin = vgx/vgz (red lines) of the finite, N-period, quarter-wave-stack as functions of ω/ω0, for the same parameters as in Fig. 2, except for θi = 0.

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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.

 

Fig. 4 Ratio of velocities vgx/vgz for the finite (red lines) and infinite (black lines) λ0/4 photonic structure as a function of ω/ω0, for the same parameters as in Fig. 3.

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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.

5. Conclusion

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 [10] and [13] 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 [24], chirped periodic structures [25], 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.

Acknowledgments

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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12. G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. M. Haus, and M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures,” Phys. Rev. E 63,036610 (2001). [CrossRef]  

13. W. Frias, A. Smolyakov, and A. Hirose, “Non-local energy transport in tunneling and plasmonic structures,” Opt. Express 19, 15281–15296 (2011). [CrossRef]   [PubMed]  

14. R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299, 309–312 (2002). [CrossRef]  

15. N. Le Thomas and R. Houdré, “Group velocity and energy transport velocity near the band edge of a disordered coupled cavity waveguide: an analytical approach,” J. Opt. Soc. Am. B 27, 2095–2101 (2010). [CrossRef]  

16. M. de Dios-Leyva and J. C. Drake-Pérez, “Group velocity and nonlocal energy transport velocity in finite photonic structures,” J. Opt. Soc. Am. B 29, 2275–2281 (2012). [CrossRef]  

17. H. G. Winful, “Group delay, stored energy, and the tunneling of evanescent electromagnetic waves,” Phys. Rev. E 68,016615 (2003). [CrossRef]  

18. R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, 1992).

19. M. de Dios-Leyva and O. E. González-Vasquez, “Band structure and associated electromagnetic fields in one-dimensional photonic crystals with left-handed materials,” Phys. Rev. B 77,125102 (2008). [CrossRef]  

20. J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996). [CrossRef]  

21. M. Centini, C. Sabilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: Applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999). [CrossRef]  

22. E. P. Wigner, “Lower Limit for the Energy Derivative of the Scattering Phase Shift,” Phys. Rev. 98, 145–147 (1955). [CrossRef]  

23. M. de Dios-Leyva and J. C. Drake-Pérez, “Properties of the dispersion relation in finite one-dimensional photonic crystals,” J. Appl. Phys. 109,103526 (2011). [CrossRef]  

24. H. Daninthe, S. Foteinopoulou, and C. M. Soukoulis, “Omni-reflectance and enhanced resonant tunneling from multilayers containing left-handed materials,” Photonics and Nanostructures-Fundamentals and Applications 4, 123–131 (2006). [CrossRef]  

25. A. R. Davoyan, A. A. Sukhorukov, I. V. Shadrivov, and Y. S. Kivshar, Beam oscillations and curling in chirped periodic structures with metamaterials,” Phys. Rev. A 79,013820 (2009). [CrossRef]  

26. G Torrese, J. Taylor, T. J. Hall, and P. Mégret, ”Effective-medium theory for energy velocity in one-dimensional finite lossless photonic crystals,” Phys. Rev. E 73,066616 (2006). [CrossRef]  

References

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  1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [CrossRef] [PubMed]
  2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
    [CrossRef] [PubMed]
  3. K. Sakoda, Optical Properties of Photonic Crystals (Springer, Berlin, 2001).
    [CrossRef]
  4. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, NJ, 2008).
  5. L. Brillouin, Wave Propagation and Group Velocity (Academic Press, 1960).
  6. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, 1984).
  7. S. Foteinopoulou, C. M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects,” Phys. Rev. B 72,165112 (2005).
    [CrossRef]
  8. G. Torrese, J. Taylor, H. P. Schriemer, M. Cada, “Energy transport through structures with finite electromagnetic stop gaps,” J. Opt. A: Pure Appl. Opt. 8, 973–980 (2006).
    [CrossRef]
  9. R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970).
    [CrossRef]
  10. P. Y. Chen, R. C. Mc Phedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82,053825 (2010).
    [CrossRef]
  11. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742–756 (1979).
    [CrossRef]
  12. G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. M. Haus, M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures,” Phys. Rev. E 63,036610 (2001).
    [CrossRef]
  13. W. Frias, A. Smolyakov, A. Hirose, “Non-local energy transport in tunneling and plasmonic structures,” Opt. Express 19, 15281–15296 (2011).
    [CrossRef] [PubMed]
  14. R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299, 309–312 (2002).
    [CrossRef]
  15. N. Le Thomas, R. Houdré, “Group velocity and energy transport velocity near the band edge of a disordered coupled cavity waveguide: an analytical approach,” J. Opt. Soc. Am. B 27, 2095–2101 (2010).
    [CrossRef]
  16. M. de Dios-Leyva, J. C. Drake-Pérez, “Group velocity and nonlocal energy transport velocity in finite photonic structures,” J. Opt. Soc. Am. B 29, 2275–2281 (2012).
    [CrossRef]
  17. H. G. Winful, “Group delay, stored energy, and the tunneling of evanescent electromagnetic waves,” Phys. Rev. E 68,016615 (2003).
    [CrossRef]
  18. R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, 1992).
  19. M. de Dios-Leyva, O. E. González-Vasquez, “Band structure and associated electromagnetic fields in one-dimensional photonic crystals with left-handed materials,” Phys. Rev. B 77,125102 (2008).
    [CrossRef]
  20. J. M. Bendickson, J. P. Dowling, M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
    [CrossRef]
  21. M. Centini, C. Sabilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: Applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
    [CrossRef]
  22. E. P. Wigner, “Lower Limit for the Energy Derivative of the Scattering Phase Shift,” Phys. Rev. 98, 145–147 (1955).
    [CrossRef]
  23. M. de Dios-Leyva, J. C. Drake-Pérez, “Properties of the dispersion relation in finite one-dimensional photonic crystals,” J. Appl. Phys. 109,103526 (2011).
    [CrossRef]
  24. H. Daninthe, S. Foteinopoulou, C. M. Soukoulis, “Omni-reflectance and enhanced resonant tunneling from multilayers containing left-handed materials,” Photonics and Nanostructures-Fundamentals and Applications 4, 123–131 (2006).
    [CrossRef]
  25. A. R. Davoyan, A. A. Sukhorukov, I. V. Shadrivov, Y. S. Kivshar, Beam oscillations and curling in chirped periodic structures with metamaterials,” Phys. Rev. A 79,013820 (2009).
    [CrossRef]
  26. G Torrese, J. Taylor, T. J. Hall, P. Mégret, ”Effective-medium theory for energy velocity in one-dimensional finite lossless photonic crystals,” Phys. Rev. E 73,066616 (2006).
    [CrossRef]

2012 (1)

2011 (2)

W. Frias, A. Smolyakov, A. Hirose, “Non-local energy transport in tunneling and plasmonic structures,” Opt. Express 19, 15281–15296 (2011).
[CrossRef] [PubMed]

M. de Dios-Leyva, J. C. Drake-Pérez, “Properties of the dispersion relation in finite one-dimensional photonic crystals,” J. Appl. Phys. 109,103526 (2011).
[CrossRef]

2010 (2)

P. Y. Chen, R. C. Mc Phedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82,053825 (2010).
[CrossRef]

N. Le Thomas, R. Houdré, “Group velocity and energy transport velocity near the band edge of a disordered coupled cavity waveguide: an analytical approach,” J. Opt. Soc. Am. B 27, 2095–2101 (2010).
[CrossRef]

2009 (1)

A. R. Davoyan, A. A. Sukhorukov, I. V. Shadrivov, Y. S. Kivshar, Beam oscillations and curling in chirped periodic structures with metamaterials,” Phys. Rev. A 79,013820 (2009).
[CrossRef]

2008 (1)

M. de Dios-Leyva, O. E. González-Vasquez, “Band structure and associated electromagnetic fields in one-dimensional photonic crystals with left-handed materials,” Phys. Rev. B 77,125102 (2008).
[CrossRef]

2006 (3)

G Torrese, J. Taylor, T. J. Hall, P. Mégret, ”Effective-medium theory for energy velocity in one-dimensional finite lossless photonic crystals,” Phys. Rev. E 73,066616 (2006).
[CrossRef]

H. Daninthe, S. Foteinopoulou, C. M. Soukoulis, “Omni-reflectance and enhanced resonant tunneling from multilayers containing left-handed materials,” Photonics and Nanostructures-Fundamentals and Applications 4, 123–131 (2006).
[CrossRef]

G. Torrese, J. Taylor, H. P. Schriemer, M. Cada, “Energy transport through structures with finite electromagnetic stop gaps,” J. Opt. A: Pure Appl. Opt. 8, 973–980 (2006).
[CrossRef]

2005 (1)

S. Foteinopoulou, C. M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects,” Phys. Rev. B 72,165112 (2005).
[CrossRef]

2003 (1)

H. G. Winful, “Group delay, stored energy, and the tunneling of evanescent electromagnetic waves,” Phys. Rev. E 68,016615 (2003).
[CrossRef]

2002 (1)

R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299, 309–312 (2002).
[CrossRef]

2001 (1)

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. M. Haus, M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures,” Phys. Rev. E 63,036610 (2001).
[CrossRef]

1999 (1)

M. Centini, C. Sabilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: Applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

1996 (1)

J. M. Bendickson, J. P. Dowling, M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

1987 (2)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

1979 (1)

1970 (1)

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970).
[CrossRef]

1955 (1)

E. P. Wigner, “Lower Limit for the Energy Derivative of the Scattering Phase Shift,” Phys. Rev. 98, 145–147 (1955).
[CrossRef]

Asatryan, A. A.

P. Y. Chen, R. C. Mc Phedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82,053825 (2010).
[CrossRef]

Bendickson, J. M.

J. M. Bendickson, J. P. Dowling, M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

Bertolotti, M.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. M. Haus, M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures,” Phys. Rev. E 63,036610 (2001).
[CrossRef]

M. Centini, C. Sabilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: Applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

Bloemer, M. J.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. M. Haus, M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures,” Phys. Rev. E 63,036610 (2001).
[CrossRef]

M. Centini, C. Sabilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: Applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

Botten, L. C.

P. Y. Chen, R. C. Mc Phedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82,053825 (2010).
[CrossRef]

Bowden, C. M.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. M. Haus, M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures,” Phys. Rev. E 63,036610 (2001).
[CrossRef]

M. Centini, C. Sabilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: Applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

Brillouin, L.

L. Brillouin, Wave Propagation and Group Velocity (Academic Press, 1960).

Cada, M.

G. Torrese, J. Taylor, H. P. Schriemer, M. Cada, “Energy transport through structures with finite electromagnetic stop gaps,” J. Opt. A: Pure Appl. Opt. 8, 973–980 (2006).
[CrossRef]

Centini, M.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. M. Haus, M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures,” Phys. Rev. E 63,036610 (2001).
[CrossRef]

M. Centini, C. Sabilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: Applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

Chen, P. Y.

P. Y. Chen, R. C. Mc Phedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82,053825 (2010).
[CrossRef]

Collin, R. E.

R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, 1992).

D’Aguanno, G.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. M. Haus, M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures,” Phys. Rev. E 63,036610 (2001).
[CrossRef]

M. Centini, C. Sabilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: Applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

Daninthe, H.

H. Daninthe, S. Foteinopoulou, C. M. Soukoulis, “Omni-reflectance and enhanced resonant tunneling from multilayers containing left-handed materials,” Photonics and Nanostructures-Fundamentals and Applications 4, 123–131 (2006).
[CrossRef]

Davoyan, A. R.

A. R. Davoyan, A. A. Sukhorukov, I. V. Shadrivov, Y. S. Kivshar, Beam oscillations and curling in chirped periodic structures with metamaterials,” Phys. Rev. A 79,013820 (2009).
[CrossRef]

de Dios-Leyva, M.

M. de Dios-Leyva, J. C. Drake-Pérez, “Group velocity and nonlocal energy transport velocity in finite photonic structures,” J. Opt. Soc. Am. B 29, 2275–2281 (2012).
[CrossRef]

M. de Dios-Leyva, J. C. Drake-Pérez, “Properties of the dispersion relation in finite one-dimensional photonic crystals,” J. Appl. Phys. 109,103526 (2011).
[CrossRef]

M. de Dios-Leyva, O. E. González-Vasquez, “Band structure and associated electromagnetic fields in one-dimensional photonic crystals with left-handed materials,” Phys. Rev. B 77,125102 (2008).
[CrossRef]

de Sterke, C. M.

P. Y. Chen, R. C. Mc Phedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82,053825 (2010).
[CrossRef]

Dowling, J. P.

J. M. Bendickson, J. P. Dowling, M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

Drake-Pérez, J. C.

M. de Dios-Leyva, J. C. Drake-Pérez, “Group velocity and nonlocal energy transport velocity in finite photonic structures,” J. Opt. Soc. Am. B 29, 2275–2281 (2012).
[CrossRef]

M. de Dios-Leyva, J. C. Drake-Pérez, “Properties of the dispersion relation in finite one-dimensional photonic crystals,” J. Appl. Phys. 109,103526 (2011).
[CrossRef]

Foteinopoulou, S.

H. Daninthe, S. Foteinopoulou, C. M. Soukoulis, “Omni-reflectance and enhanced resonant tunneling from multilayers containing left-handed materials,” Photonics and Nanostructures-Fundamentals and Applications 4, 123–131 (2006).
[CrossRef]

S. Foteinopoulou, C. M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects,” Phys. Rev. B 72,165112 (2005).
[CrossRef]

Frias, W.

González-Vasquez, O. E.

M. de Dios-Leyva, O. E. González-Vasquez, “Band structure and associated electromagnetic fields in one-dimensional photonic crystals with left-handed materials,” Phys. Rev. B 77,125102 (2008).
[CrossRef]

Hall, T. J.

G Torrese, J. Taylor, T. J. Hall, P. Mégret, ”Effective-medium theory for energy velocity in one-dimensional finite lossless photonic crystals,” Phys. Rev. E 73,066616 (2006).
[CrossRef]

Haus, J. M.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. M. Haus, M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures,” Phys. Rev. E 63,036610 (2001).
[CrossRef]

Hirose, A.

Houdré, R.

Joannopoulos, J. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, NJ, 2008).

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

Johnson, S. G.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, NJ, 2008).

Kivshar, Y. S.

A. R. Davoyan, A. A. Sukhorukov, I. V. Shadrivov, Y. S. Kivshar, Beam oscillations and curling in chirped periodic structures with metamaterials,” Phys. Rev. A 79,013820 (2009).
[CrossRef]

Le Thomas, N.

Loudon, R.

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970).
[CrossRef]

Mc Phedran, R. C.

P. Y. Chen, R. C. Mc Phedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82,053825 (2010).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, NJ, 2008).

Mégret, P.

G Torrese, J. Taylor, T. J. Hall, P. Mégret, ”Effective-medium theory for energy velocity in one-dimensional finite lossless photonic crystals,” Phys. Rev. E 73,066616 (2006).
[CrossRef]

Nefedov, I.

M. Centini, C. Sabilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: Applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

Poulton, C. G.

P. Y. Chen, R. C. Mc Phedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82,053825 (2010).
[CrossRef]

Ruppin, R.

R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299, 309–312 (2002).
[CrossRef]

Sabilia, C.

M. Centini, C. Sabilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: Applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals (Springer, Berlin, 2001).
[CrossRef]

Scalora, M.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. M. Haus, M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures,” Phys. Rev. E 63,036610 (2001).
[CrossRef]

M. Centini, C. Sabilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: Applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

J. M. Bendickson, J. P. Dowling, M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

Schriemer, H. P.

G. Torrese, J. Taylor, H. P. Schriemer, M. Cada, “Energy transport through structures with finite electromagnetic stop gaps,” J. Opt. A: Pure Appl. Opt. 8, 973–980 (2006).
[CrossRef]

Shadrivov, I. V.

A. R. Davoyan, A. A. Sukhorukov, I. V. Shadrivov, Y. S. Kivshar, Beam oscillations and curling in chirped periodic structures with metamaterials,” Phys. Rev. A 79,013820 (2009).
[CrossRef]

Sibilia, C.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. M. Haus, M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures,” Phys. Rev. E 63,036610 (2001).
[CrossRef]

Smolyakov, A.

Soukoulis, C. M.

H. Daninthe, S. Foteinopoulou, C. M. Soukoulis, “Omni-reflectance and enhanced resonant tunneling from multilayers containing left-handed materials,” Photonics and Nanostructures-Fundamentals and Applications 4, 123–131 (2006).
[CrossRef]

S. Foteinopoulou, C. M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects,” Phys. Rev. B 72,165112 (2005).
[CrossRef]

Steel, M. J.

P. Y. Chen, R. C. Mc Phedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82,053825 (2010).
[CrossRef]

Sukhorukov, A. A.

A. R. Davoyan, A. A. Sukhorukov, I. V. Shadrivov, Y. S. Kivshar, Beam oscillations and curling in chirped periodic structures with metamaterials,” Phys. Rev. A 79,013820 (2009).
[CrossRef]

Taylor, J.

G. Torrese, J. Taylor, H. P. Schriemer, M. Cada, “Energy transport through structures with finite electromagnetic stop gaps,” J. Opt. A: Pure Appl. Opt. 8, 973–980 (2006).
[CrossRef]

G Torrese, J. Taylor, T. J. Hall, P. Mégret, ”Effective-medium theory for energy velocity in one-dimensional finite lossless photonic crystals,” Phys. Rev. E 73,066616 (2006).
[CrossRef]

Torrese, G

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[CrossRef]

Torrese, G.

G. Torrese, J. Taylor, H. P. Schriemer, M. Cada, “Energy transport through structures with finite electromagnetic stop gaps,” J. Opt. A: Pure Appl. Opt. 8, 973–980 (2006).
[CrossRef]

Wigner, E. P.

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[CrossRef]

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H. G. Winful, “Group delay, stored energy, and the tunneling of evanescent electromagnetic waves,” Phys. Rev. E 68,016615 (2003).
[CrossRef]

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M. de Dios-Leyva, J. C. Drake-Pérez, “Properties of the dispersion relation in finite one-dimensional photonic crystals,” J. Appl. Phys. 109,103526 (2011).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

G. Torrese, J. Taylor, H. P. Schriemer, M. Cada, “Energy transport through structures with finite electromagnetic stop gaps,” J. Opt. A: Pure Appl. Opt. 8, 973–980 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

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Opt. Express (1)

Photonics and Nanostructures-Fundamentals and Applications (1)

H. Daninthe, S. Foteinopoulou, C. M. Soukoulis, “Omni-reflectance and enhanced resonant tunneling from multilayers containing left-handed materials,” Photonics and Nanostructures-Fundamentals and Applications 4, 123–131 (2006).
[CrossRef]

Phys. Lett. A (1)

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[CrossRef]

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[CrossRef]

Phys. Rev. A (2)

A. R. Davoyan, A. A. Sukhorukov, I. V. Shadrivov, Y. S. Kivshar, Beam oscillations and curling in chirped periodic structures with metamaterials,” Phys. Rev. A 79,013820 (2009).
[CrossRef]

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[CrossRef]

Phys. Rev. B (2)

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[CrossRef]

M. de Dios-Leyva, O. E. González-Vasquez, “Band structure and associated electromagnetic fields in one-dimensional photonic crystals with left-handed materials,” Phys. Rev. B 77,125102 (2008).
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[CrossRef]

G Torrese, J. Taylor, T. J. Hall, P. Mégret, ”Effective-medium theory for energy velocity in one-dimensional finite lossless photonic crystals,” Phys. Rev. E 73,066616 (2006).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. M. Haus, M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures,” Phys. Rev. E 63,036610 (2001).
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[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Schematic of the process of scattering by a finite photonic structure, localized between the z = 0 and z = L planes and sandwiched between two semi-infinite layers made of the same optical materials. Arrows indicate the incident, reflected and transmitted waves.

Fig. 2
Fig. 2

Normal energy flux density normalized to S 0 = ( c / 8 π ) ε 1 as a function of the frequency ω in units of ω0, for a finite, N-period, quarter-wave-stack, with n 1 = ε 1 = 1, n 2 = ε 2 = 1.41, N = 5 (left-hand panel), 10 (right-hand panel) and various values of the angle of incidence θi.

Fig. 3
Fig. 3

Lateral energy flux density normalized to S 0 = ( c / 8 π ) ε 1 (black lines) and the ratio Vfin = vgx/vgz (red lines) of the finite, N-period, quarter-wave-stack as functions of ω/ω0, for the same parameters as in Fig. 2, except for θi = 0.

Fig. 4
Fig. 4

Ratio of velocities vgx/vgz for the finite (red lines) and infinite (black lines) λ0/4 photonic structure as a function of ω/ω0, for the same parameters as in Fig. 3.

Equations (62)

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v E = T T + T 0 v g = T v g ( ω )
v g ( ω ) = v g T + T 0
E ( r ) = y E ( z ) exp ( i x K x ) = u ( r ) exp [ i ϕ ( x , z ) ] ,
H ( r ) = v ( r ) exp [ i ϕ ( x , z ) ] ,
u ( r ) = y | E ( z ) | ,
v ( r ) = ic g ( z , ω ) [ | E ( z ) | z + i | E ( z ) | φ ( z ) z ] x + c K x g ( z , ω ) | E ( z ) | z .
ϕ ( x , z ) = φ ( z ) + x K x
× u ( r ) + i k × u ( r ) = i c g ( z , ω ) v ( r ) ,
× v ( r ) + i k × v ( r ) = i c f ( z , ω ) u ( r ) ,
k = ϕ ( x , z ) .
F K α + i 32 π c k K α S = i 32 π c v g α U ,
F K α = 2 i c | E ( z ) | 2 K α [ β ( z ) | E ( z ) | ] z ,
β ( z ) = 1 g ( z , ω ) | E ( z ) | z ,
S = c 8 π Re [ E × H * ]
U = 1 16 π ( f ( z , ω ) ω E E * + g ( z , ω ) ω H H * )
S = c 2 8 π g ( z , ω ) { | E ( z ) | 2 K x x + | E ( z ) | 2 φ ( z ) z z }
k K x = x + 2 φ ( z ) K x z z ,
k K z = 2 φ ( z ) K z z z .
c 2 16 π L G z ( K ) + 1 L [ φ ( L ) φ ( 0 ) ] K z z S L = v g z U L ,
c 2 16 π L G x ( K ) + S x L + 1 L [ φ ( L ) φ ( 0 ) ] K x z S L = v g x U L ,
G α ( K ) = G α ( L , K ) G α ( 0 , K ) ,
G α ( z , K ) = | E ( z ) | 2 K α [ β ( z ) | E ( z ) | ] .
E ( z ) = exp ( i z Q L ) + r exp ( i z Q L ) ,
E ( z ) = t exp [ i Q L ( z L ) ] = | t | exp i [ Φ + Q L ( z L ) ] ,
φ ( L ) φ ( 0 ) = Φ θ LK z θ ,
tan θ = r 2 1 + r 1 .
S z L = z S L = c 2 8 π Q L g L T ,
c 2 Q L 8 π g L 1 L { 1 2 g L Q L G α ( K ) θ K α T } + S α L = v g α U L ,
G α ( K ) = G α ( 0 , K ) = i | 1 + r | 2 K α [ Q L g L ( r r * ) | 1 + r | 2 ]
S α L = v g α U L c 2 8 π r 2 L K α ( Q L g L ) c 2 8 π Q L g L R L Φ R K α ,
K z = v g z ω ,
K x = v g x ω + ( K x ) ω ,
v E z = T T + T 0 v g z = T v g ( ω ) ,
v E x = [ T + ( R / τ d ) Φ R / ω ] T + T 0 v g x T x T + T 0 ,
T 0 = 1 τ d { r 2 g L Q L ω ( Q L g L ) + R Φ R ω } ,
T x = 1 τ d { r 2 g L Q L [ K x ( Q L g L ) ] ω + R Φ R K x } .
T 0 = R τ i τ d
T x = τ i l τ d ,
τ i = r 2 Q L { Q L g L g L ω Q L ω } ,
τ i l = K x r 2 Q L 2
v E z = T 1 τ i / τ d v g z = T v g ( ω ) ,
v E x = 1 1 τ i / τ d v g x + τ i l / τ d 1 τ i / τ d .
τ d = τ D + τ i
S x L S z L = v E x v E z = [ T + ( R / τ d ) Φ R / ω ] T v g x v g z τ d T x L T ,
( t 0 ) = T ^ ( 1 r ) .
1 ) t = 1 T 22 , 2 ) r = T 21 T 22 = T 21 t ,
tan Φ = tan LK z = Y X = F ( ω , K x ) ,
τ d = Φ ω = L v g z = X 2 T ( F ω ) K x ,
v g x v g z = X 2 L T ( F K x ) ω ,
T 22 = cos N β i sin N β sin β g ,
T 21 = i 2 sin N β sin β ( η 1 η ) sin b Q 2 exp ( i a Q 1 ) ,
g = sin a Q 1 cos b Q 2 + 1 2 ( η + 1 η ) cos a Q 1 sin b Q 2 ,
cos β = cos a Q 1 cos b Q 2 1 2 ( η + 1 η ) sin a Q 1 sin b Q 2 = f ( ω , K x )
t = 1 T 22 = T { cos N β + i sin N β sin β g } ,
tan Φ = tan LK z = g tan N β sin β = F ( ω , K x ) ,
T = 1 | T 22 | 2 = 1 cos 2 N β + ( sin 2 N β / sin 2 β ) g 2
X = T cos N β
Φ R = ± π / 2 + Φ a Q 1 = ± π / 2 + K z L a Q 1
S x L S z L = 1 T { ( 1 R τ a τ d ) v g x v g z + K x Q 1 1 L Q 1 ( r 2 a Q 1 R ) }
v g x v g z = T L { [ ( 1 f 2 ) g + f g f ] sin 2 N β 2 ( 1 f 2 ) 3 / 2 N g f 1 f 2 }
S z L S 0 = T cos θ i ,
S x L S 0 = cos θ i { ( 1 R τ a τ d ) v g x v g z + tan θ i L Q 1 ( r 2 a Q 1 R ) } ,

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