## 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 **v*** _{E}* = 〈

**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 〈...〉

*to represent the latter average.*

_{L}A consequence of the above definition of the averaged energy transport velocity is that it allows to establish an explicit connection between **v*** _{E}* 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

**v**

*= ∇*

_{g}

_{K}*ω*, 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

**v**

*and*

_{E}**v**

*are closely correlated. For instance, it was demonstrated theoretically that*

_{g}**v**

*=*

_{E}**v**

*for unbounded homogeneous media [6] and infinite periodic structures [3, 11]. This equivalence is a direct consequence of the translational symmetry of these media.*

_{g}When this symmetry is broken or losses are taken properly into account [14], the link between **v*** _{E}* and

**v**

*and therefore between the space-averaged energy flux density and*

_{g}**v**

*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:*

_{g}**v**

*and the group delay velocity*

_{E}**v**

*for the case of normal incidence and showed that*

_{g}**v**

*=*

_{E}**v**

*only at the resonance frequencies of the transmission coefficient, where*

_{g}*T*

_{0}is a frequency dependent parameter, and 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

**v**

*to the group velocity*

_{E}**v**

*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 〈*

_{g}**S**〉

*and those of*

_{L}**v**

*, 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.*

_{g}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

**v**

*and 〈*

_{E}**S**〉

*in terms of those of the group velocity*

_{L}**v**

*. The usefulness of the correlation between 〈*

_{g}**S**〉

*and*

_{L}**v**

*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.*

_{g}## 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 *K _{x}* along the

*x*-axis. For TE modes, the spatial part of the electric and magnetic fields can be written as [19]

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*),

**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

*f*(

*z*,

*ω*) =

*ωε*(

*z*) and

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 *K _{x}* and the effective wavevector

*K*, which is defined in terms of the phase Φ of the complex transmission amplitude

_{z}*t*as Φ =

*LK*[20, 21]. This means that

_{z}*K*and

_{x}*K*are independent quantities and the dispersion relation of the structure is characterized by the dependence of the frequency

_{z}*ω*on the wavevector

**K**=

**x**

*K*+

_{x}**z**

*K*, i. e.

_{z}*ω*=

*ω*(

*K*,

_{x}*K*). Further, since Eqs. (8) and (9) depend explicitly on

_{z}*ω*and

*K*, the vectors

_{x}**u**and

**v**also depend on

**K**. Note, however, that the dependence of

**u**and

**v**on

*K*is only through the dispersion relation. That is, these vectors are composite functions of

_{z}*K*. This difference between

_{z}*K*and

_{x}*K*will be taken into account in our calculations.

_{z}Taking the *K _{α}*-derivative in Eqs. (8) and (9) and combining both results, we obtain the Poynting theorem [16–18] in the form

In the Eqs. above, **S** and *U* are the time-averaged Poynting vector and energy density, respectively, and *v _{g}_{α}* =

*∂ω*(

**K**)/

*∂K*is the

_{α}*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 =

*LK*, the

_{z}*z*-component of

**v**

*is given by*

_{g}*v*=

_{gz}*∂ω*(

**K**)/

*∂K*=

_{z}*L/τ*, where

_{d}*τ*=

_{d}*∂*Φ/

*∂ω*. That is,

*τ*represents the group delay or Wigner delay time [22] and v

_{d}*corresponds to the group delay velocity.*

_{gz}Using Eqs. (3)–(6) and (14), we find the expression

*S*and lateral

_{z}*S*components and determines the local energy flux density inside the structure. The lateral component disappears for normal propagation.

_{x}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

*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

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

*z*>

*L*it has only a transmitted component where Φ =

*LK*is the phase of

_{z}*t*, ${Q}_{L}=\sqrt{\frac{{\omega}^{2}}{{c}^{2}}{\epsilon}_{L}{\mu}_{L}-{K}_{x}^{2}}$, and

*ε*,

_{L}*μ*are the permittivity and permeability for

_{L}*z*< 0 and

*z*>

*L*.

In consequence,

where*θ*is the phase of

*E*(0) = 1 +

*r*= 1 +

*r*

_{1}+

*ir*

_{2}and satisfies the relation

Using (25) and taking into account the relation *∂K _{i}/∂K_{j}* =

*δ*, with

_{i,j}*i*,

*j*=

*x*,

*z*, and the fact that the normal energy flow is given by

*α*=

*x*,

*z*, where

*T*= |

*t*|

^{2}is the transmission coefficient and

*g*=

_{L}*ωμ*.

_{L}Using the continuity of *G _{α}*(

*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

*G*(

_{α}*L*,

**K**) = 0 and

*α*=

*x*,

*z*.

It should be noted that the factor *Q _{L}*(

*r*−

*r*

^{*}) = 2

*ir*

_{2}

*Q*in the latter Eq. arises from the interference between incident and reflected waves.

_{L}Substituting *G _{α}*(

**K**) and

*∂θ/∂K*calculated from (26) into Eq. (28), we get

_{α}*R*= |

*r*|

^{2}is the reflection coefficient and Φ

*is the phase of the complex reflection amplitude*

_{R}*r*.

The first term on the right-hand side of Eq. (30) represents an energy flow *S _{g}_{α}* 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, Φ

*= Φ ±*

_{R}*π*/2 =

*LK*±

_{z}*π*/2 and (1/

*L*)

*∂*Φ

*= 0 and 1 for*

_{R}/∂K_{α}*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 〈*S _{α}*〉

*= v*

_{L}*〈*

_{Eα}*U*〉

*=*

_{L}*S*and therefore v

_{g}_{α}*= v*

_{E}_{α}*for*

_{g}_{α}*α*=

*x*,

*z*. As one sees, Eq. (30) connects the interference and reflection energy fluxes to two quantities of special interest: the group velocity

**v**

*, which may be superluminal away from resonance, and the energy velocity*

_{g}**v**

*, which in general remains causal [13, 16, 17]. It is then clear that, in the superluminal regime, the subluminal behavior of*

_{E}**v**

*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.*

_{E}## 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 *∂/∂K _{α}* in (30) by

*α*=

*z*and

*x*, respectively, where the latter term on the right-hand side of (32) represents the derivative with respect to

*K*keeping the frequency

_{x}*ω*constant.

If we substitute (31) and (32) into (30) for *α* = *z* and *x*, respectively, and use Eq. (27) and the relation 〈*S _{α}*〉

*=*

_{L}*v*〈

_{Eα}*U*〉

*, we obtain, after some algebraic manipulation, the expressions*

_{L}*τ*=

_{d}*L/v*is the group delay, ${v}_{g}^{(\omega )}={v}_{gz}/(T+{T}_{0})$ and

_{gz}Equations (33) and (34) relate the components of the energy transport velocity **v*** _{E}* to those of the group velocity

**v**

*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*

_{g}**v**

*=*

_{E}**v**

*. 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*

_{g}*T*

_{0}and

*T*. 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, Φ

_{x}*= Φ ±*

_{R}*π*/2 =

*LK*±

_{z}*π*/2 and

*∂*Φ

*= 0,*

_{R}/∂K_{x}*∂*Φ

*=*

_{R}/∂K_{z}*L*and

*∂*Φ

*=*

_{R}/∂ω*τ*. Using these relations in Eq. (35) and Eq. (36) and taking into account that ${Q}_{L}=\sqrt{\frac{{\omega}^{2}}{{c}^{2}}{\epsilon}_{L}{\mu}_{L}-{K}_{x}^{2}}$, we obtain:

_{d}*z*< 0 and along the normal direction, and 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:

*v*=

_{gz}*L/τ*and ${v}_{g}^{(\omega )}=L/{\tau}_{D}$ are substituted into (41), we obtain immediately the relation: between the group delay

_{d}*τ*, the dwell time

_{d}*τ*and the self-interference delay

_{D}*τ*, as expected [17]. In conclusion, the energy transport velocity

_{i}**v**

*and the group velocity*

_{E}**v**

*are related through quantities having specific physical meaning.*

_{g}Note, finally, that the relation between *v _{Ez}*,

*v*and ${v}_{g}^{(\omega )}$ is independent of

_{gz}*K*, that is to say, the effects of oblique propagation do not modify it.

_{x}Dividing (34) by (33) we obtain the formula

*S*〉

_{z}*, 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*

_{L}*r*and transmission

*t*amplitudes, which are related through the total transfer matrix

*T̃*as [16]

*T̃*is an unimodular matrix lead to the expressions

*T*are the matrix elements of

_{ij}*T̃*.

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

where*X*and

*Y*are the real and imaginary parts of

*t*,

*F*(

*ω*,

*K*) depends only explicitly on

_{x}*ω*and

*K*, as discussed above, and Φ =

_{x}*LK*is the phase of

_{z}*t*=

*X*+

*iY*.

Using Eq. (47), it is straightforward to obtain the following formulas for the group delay *τ _{d}* and the ratio between the components of the group velocity:

*T*=

*X*

^{2}+

*Y*

^{2}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 *ε*_{1} and *μ*_{1}. For now we will leave the nature of layer *B* unspecified, beyond requiring that their optical parameters *ε*_{2}(*ω*) and *μ*_{2}(*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 *T*_{22} and *T*_{21} :

*i*= 1, 2,

*η*=

*μ*

_{2}

*Q*

_{1}/

*μ*

_{1}

*Q*

_{2},

*N*is the number of unit cells,

*a*and

*b*are the widths of layers

*A*and

*B*, respectively,

*β*is the Bloch phase associated with the corresponding infinite photonic crystal which satisfies the dispersion relation:

Noting that *Q*_{1} is real, *Q*_{2} 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 cos*Nβ* and sin*Nβ*/sin*β* 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*.

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

for the phase Φ*of the complex reflection amplitude*

_{R}*r*, which leads immediately to

*∂*Φ

*=*

_{R}/∂ω*τ*−

_{d}*τ*and (

_{a}*∂*Φ

*)*

_{R}/∂K_{x}*=*

_{ω}*aK*

_{x}/Q_{1}, where

*τ*=

_{a}*a∂Q*

_{1}

*/∂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:

In this latter Eq., the ratio *v _{gx}/v_{gz}* should be calculated by combining Eqs. (49), (53) and (55). As a result, we obtain:

*g*and

*f*are the functions shown in Eqs. (52)–(53) and

*g′*= (

*∂g/∂K*)

_{x}*and*

_{ω}*f′*= (

*∂f/∂K*)

_{x}*.*

_{ω}At this point, it is convenient to express the Eqs. obtained above in terms of the angle of incidence *θ _{i}*, which is related to the

*K*-component of the wavevector

_{x}**K**as ${K}_{x}=(\omega /c)\sqrt{{\mu}_{1}{\epsilon}_{1}}\text{sin}{\theta}_{i}$. Using the latter relation, it is easy to see that ${Q}_{1}=(\omega /c)\sqrt{{\mu}_{1}{\epsilon}_{1}}\text{cos}{\theta}_{i}$, ${Q}_{2}=(\omega /c)\sqrt{{\mu}_{1}{\epsilon}_{1}}\sqrt{{\mu}_{2}{\epsilon}_{2}/{\mu}_{1}{\epsilon}_{1}-{\text{sin}}^{2}{\theta}_{i}}$ and Eq. (27) can be written down as:

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

*x*-component of the space-averaged energy flux density.

Equation (61) shows that, for a given value of the propagation angle *θ _{i}*, the structure of 〈

*S*〉

_{z}*as a function of*

_{L}*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);

*bQ*

_{2}=

*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, 〈

*S*〉

_{z}*is an oscillating function of*

_{L}*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 〈

*S*〉

_{z}*should disappear. These properties of the 〈*

_{L}*S*〉

_{z}*-spectra are illustrated in Fig. 2 for a finite,*

_{L}*N*-period, quarter-wave-stack (

*λ*

_{0}/4 =

*πc*/2

*ω*

_{0}structure)[20, 26], with

*μ*

_{1}=

*μ*

_{2}= 1, ${n}_{1}=\sqrt{{\epsilon}_{1}}=1$ and ${n}_{2}=\sqrt{{\epsilon}_{2}}=1.41$, for

*N*= 5, 10 and various values of the angle of incidence

*θ*. One sees in Fig. 2 that the main effects of increasing

_{i}*θ*, for a given value of

_{i}*N*, are to shift the 〈

*S*〉

_{z}*-spectra to higher frequencies and to reduce the corresponding resonant-peak values.*

_{L}Since *R* = *r*_{2} = 0 at transmission resonances, it follows from Eqs.(49), (57) and (62) that, for a fixed value of *θ _{i}* ≠ 0 and ignoring quantitative differences, the lateral energy flux density 〈

*S*〉

_{x}*and the ratio*

_{L}*V*=

_{fin}*v*for the finite photonic crystals should exhibit resonant structures similar to those of 〈

_{gx}/v_{gz}*S*〉

_{z}*. It is clearly seen in Eq. (62) that the peak values of 〈*

_{L}*S*〉

_{x}*and*

_{L}*V*are the same at each transmission resonance if the factor cos

_{fin}*θ*is ignored. These theoretical results are illustrated in Fig. 3 for the

_{i}*λ*

_{0}/4 photonic structure with the same parameters used in Fig. 2.

Let us now briefly compare the properties of *V _{fin}* with those of the ratio

*V*=

_{inf}*v*for the corresponding infinite crystals. This will provide an appropriate understanding of the behavior of

_{gx}/v_{gz}*V*. Such a comparison is shown in Fig. 4 for the

_{fin}*λ*

_{0}/4 structure with the same parameters used in Fig. 2. One clearly sees that, for a fixed value of N, the oscillations of

*V*in a pass band always occur around the curve associated with

_{fin}*V*. 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

_{inf}*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.

## 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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