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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 11 — Jun. 2, 2014
  • pp: 12760–12772
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Electromagnetic energy transport in finite photonic structures

M. de Dios-Leyva, C. A. Duque, and J. C. Drake-Pérez  »View Author Affiliations


Optics Express, Vol. 22, Issue 11, pp. 12760-12772 (2014)
http://dx.doi.org/10.1364/OE.22.012760


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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

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

] and John [2

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

], many theoretical and experimental works have been devoted to the study of the transport of electromagnetic radiation through photonic structures [3

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

, 4

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

]. 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

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

]. 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

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

, 6

6. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

], 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

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

, 6

6. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

, 7

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

], whereas in one-dimensional (1D) structures, where it is always possible to define S and U at each point of the system [8

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

], 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.

The paper is organized as follows. In Sec. 2, we use the Poynting theorem [17

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

, 18

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

] 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

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]

]
E(r)=yE(z)exp(ixKx)=u(r)exp[iϕ(x,z)],
(3)
H(r)=v(r)exp[iϕ(x,z)],
(4)
where
u(r)=y|E(z)|,
(5)
v(r)=icg(z,ω)[|E(z)|z+i|E(z)|φ(z)z]x+cKxg(z,ω)|E(z)|z.
(6)

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.

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
(7)
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),
(8)
×v(r)+ik×v(r)=icf(z,ω)u(r),
(9)
where f (z, ω) = ωε(z) and
k=ϕ(x,z).
(10)

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

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

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]

]. 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 [16

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]

18

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

] in the form
FKα+i32πckKαS=i32πcvgαU,
(11)
where
FKα=2ic|E(z)|2Kα[β(z)|E(z)|]z,
(12)
β(z)=1g(z,ω)|E(z)|z,
(13)
S=c8πRe[E×H*]
(14)
U=116π(f(z,ω)ωEE*+g(z,ω)ωHH*)
(15)

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

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

, 11

11. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742–756 (1979). [CrossRef]

] 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

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

] 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}
(16)
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,
(17)
kKz=2φ(z)Kzzz.
(18)

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,
(19)
c216πLGx(K)+SxL+1L[φ(L)φ(0)]KxzSL=vgxUL,
(20)
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),
(21)
Gα(z,K)=|E(z)|2Kα[β(z)|E(z)|].
(22)

In consequence,
φ(L)φ(0)=ΦθLKzθ,
(25)
where θ is the phase of E(0) = 1 + r = 1 + r1 + ir2 and satisfies the relation
tanθ=r21+r1.
(26)

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,
(27)
Equations (19) and (20) can be written as
c2QL8πgL1L{12gLQLGα(K)θKαT}+SαL=vgαUL,
(28)
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]
(29)
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,
(30)
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.

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,
(31)
Kx=vgxω+(Kx)ω,
(32)
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),
(33)
vEx=[T+(R/τd)ΦR/ω]T+T0vgxTxT+T0,
(34)
where τd = L/vgz is the group delay, vg(ω)=vgz/(T+T0) and
T0=1τd{r2gLQLω(QLgL)+RΦRω},
(35)
Tx=1τd{r2gLQL[Kx(QLgL)]ω+RΦRKx}.
(36)

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.

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

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]

, 23

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]

]
tanΦ=tanLKz=YX=F(ω,Kx),
(47)
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,
(48)
vgxvgz=X2LT(FKx)ω,
(49)
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

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]

], periodic superlattices containing left-handed materials [19

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]

, 24

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]

], chirped periodic structures [25

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]

], 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

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]

], we obtain the following formulas for T22 and T21 :
T22=cosNβisinNβsinβg,
(50)
T21=i2sinNβsinβ(η1η)sinbQ2exp(iaQ1),
(51)
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,
(52)
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)
(53)

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},
(54)
tanΦ=tanLKz=gtanNβsinβ=F(ω,Kx),
(55)
where
T=1|T22|2=1cos2Nβ+(sin2Nβ/sin2β)g2
(56)
is the transmission coefficient and
X=TcosNβ
(57)
is the real part of t.

Moreover, using again Eqs. (46) we get the expression:
ΦR=±π/2+ΦaQ1=±π/2+KzLaQ1
(58)
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)}
(59)

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}
(60)
where g and f are the functions shown in Eqs. (52)(53) and g′ = (∂g/∂Kx)ω and f′ = (∂f/∂Kx)ω.

Finally, substituting (61) into (59), one obtains immediately the formula:
SxLS0=cosθi{(1Rτaτd)vgxvgz+tanθiLQ1(r2aQ1R)},
(62)
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

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]

, 26

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]

], 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.

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.

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.

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

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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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]

OCIS Codes
(260.2160) Physical optics : Energy transfer
(160.5293) Materials : Photonic bandgap materials
(050.5298) Diffraction and gratings : Photonic crystals

ToC Category:
Photonic Crystals

History
Original Manuscript: February 24, 2014
Revised Manuscript: April 21, 2014
Manuscript Accepted: April 22, 2014
Published: May 19, 2014

Citation
M. de Dios-Leyva, C. A. Duque, and J. C. Drake-Pérez, "Electromagnetic energy transport in finite photonic structures," Opt. Express 22, 12760-12772 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-11-12760


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References

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  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]

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