Electromagnetic energy transport in finite photonic structures

M. de Dios-Leyva; C. A. Duque; J. C. Drake-Pérez

doi:10.1364/OE.22.012760

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 〈...〉_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 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 = ∇_Kω, where K is the wavevector and ω = ω(K) represents the structure’s dispersion relation. The link between these velocities has been investigated in the presence [5, 9, 10] and absence [11–16] of losses. It has been shown that in transparent photonic structures, which will be the focus of our attention, the properties of v_E and v_g are closely correlated. For instance, it was demonstrated theoretically that v_E = v_g 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 v_E and v_g and therefore between the space-averaged energy flux density and v_g 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:

v_{E} = \frac{T}{T + T_{0}} v_{g} = T v_{g}^{(ω)}

between v_E and the group delay velocity v_g for the case of normal incidence and showed that v_E =v_g only at the resonance frequencies of the transmission coefficient, where T₀ is a frequency dependent parameter, and

v_{g}^{(ω)} = \frac{v_{g}}{T + T_{0}}

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_E to the group velocity v_g for finite dispersive photonic structures and any angle of incidence. An important consequence of this formula is that it allows to establish a direct correlation between the components of the energy flux density 〈S〉_L and those of v_g, which not only provides an analytical and systematical procedure for the study of the energy transport through the considered structures, but also it highlights the role of the dispersion relation in these studies. Accordingly, we will focus our attention on the properties of that correlation, which may be regarded as a generalization of the corresponding one for infinite periodic systems to finite photonic structures.

The paper is organized as follows. In Sec. 2, we use the Poynting theorem [17, 18] for TE-polarized waves to derive two Eqs. containing the components of 〈S〉_L and those of the energy fluxes arising in the process of scattering by the photonic structure. These Eqs. are used in Sec. 3 to express the components of both v_E and 〈S〉_L in terms of those of the group velocity v_g. The usefulness of the correlation between 〈S〉_L and v_g 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 K_x along the x -axis. For TE modes, the spatial part of the electric and magnetic fields can be written as [19]

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)],

where

u (r) = y | E (z) |,

v (r) = \frac{ic}{g (z, ω)} [\frac{\partial | E (z) |}{\partial z} + i | E (z) | \frac{\partial φ (z)}{\partial z}] x + \frac{c K_{x}}{g (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) + x K_{x}

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

\nabla \times u (r) + i k \times u (r) = \frac{i}{c} g (z, ω) v (r),

\nabla \times v (r) + i k \times v (r) = - \frac{i}{c} f (z, ω) u (r),

where f (z, ω) = ωε(z) and

k = \nabla 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 K_x and the effective wavevector K_z, which is defined in terms of the phase Φ of the complex transmission amplitude t as Φ = LK_z [20, 21]. This means that K_x and K_z are independent quantities and the dispersion relation of the structure is characterized by the dependence of the frequency ω on the wavevector K = xK_x + zK_z, i. e. ω = ω(K_x, K_z). Further, since Eqs. (8) and (9) depend explicitly on ω and K_x, the vectors u and v also depend on K. Note, however, that the dependence of u and v on K_z is only through the dispersion relation. That is, these vectors are composite functions of K_z. This difference between K_x and K_z will be taken into account in our calculations.

Taking the K_α-derivative in Eqs. (8) and (9) and combining both results, we obtain the Poynting theorem [16–18] in the form

\nabla \cdot \frac{\partial F}{\partial K_{α}} + i \frac{32 π}{c} \frac{\partial k}{\partial K_{α}} \cdot S = i \frac{32 π}{c} v_{g α} U,

where

\frac{\partial F}{\partial K_{α}} = - 2 i c {| E (z) |}^{2} \frac{\partial}{\partial K_{α}} [\frac{β (z)}{| E (z) |}] z,

β (z) = \frac{1}{g (z, ω)} \frac{\partial | E (z) |}{\partial z},

S = \frac{c}{8 π} Re [E \times H^{*}]

U = \frac{1}{16 π} (\frac{\partial f (z, ω)}{\partial ω} E \cdot E^{*} + \frac{\partial g (z, ω)}{\partial ω} H \cdot H^{*})

In the Eqs. above, S and U are the time-averaged Poynting vector and energy density, respectively, and v_g_α = ∂ω(K)/∂K_α 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 = LK_z, the z -component of v_g is given by v_gz = ∂ω(K)/∂K_z = L/τ_d, where τ_d = ∂Φ/∂ω. That is, τ_d represents the group delay or Wigner delay time [22] and v_gz corresponds to the group delay velocity.

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

S = \frac{c^{2}}{8 π g (z, ω)} {{| E (z) |}^{2} K_{x} x + {| E (z) |}^{2} \frac{\partial φ (z)}{\partial z} z}

for the Poynting vector. This vector has normal S_z and lateral S_x 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

\frac{\partial k}{\partial K_{x}} = x + \frac{\partial^{2} φ (z)}{\partial K_{x} \partial z} z,

\frac{\partial k}{\partial K_{z}} = \frac{\partial^{2} φ (z)}{\partial K_{z} \partial z} z .

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

\frac{c^{2}}{16 π L} G_{z} (K) + \frac{1}{L} \frac{\partial [φ (L) - φ (0)]}{\partial K_{z}} z \cdot {〈 S 〉}_{L} = v_{g z} {〈 U 〉}_{L},

\frac{c^{2}}{16 π L} G_{x} (K) + {〈 S_{x} 〉}_{L} + \frac{1}{L} \frac{\partial [φ (L) - φ (0)]}{\partial K_{x}} z \cdot {〈 S 〉}_{L} = v_{g x} {〈 U 〉}_{L},

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

G_{α} (K) = G_{α} (L, K) - G_{α} (0, K),

G_{α} (z, K) = - {| E (z) |}^{2} \frac{\partial}{\partial K_{α}} [\frac{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 (i z Q_{L}) + r exp (- i z Q_{L}),

while for z > L it has only a transmitted component

E (z) = t exp [i Q_{L} (z - L)] = | t | exp i [Φ + Q_{L} (z - L)],

where Φ = LK_z is the phase of t,

Q_{L} = \sqrt{\frac{ω^{2}}{c^{2}} ε_{L} μ_{L} - K_{x}^{2}}

, and ε_L, μ_L are the permittivity and permeability for z < 0 and z > L.

In consequence,

φ (L) - φ (0) = Φ - θ - {LK}_{z} - θ,

where θ is the phase of E(0) = 1 + r = 1 + r₁ + ir₂ and satisfies the relation

tan θ = \frac{r_{2}}{1 + r_{1}} .

Using (25) and taking into account the relation ∂K_i/∂K_j = δ_i,j, with i, j = x, z, and the fact that the normal energy flow is given by

{〈 S_{z} 〉}_{L} = z \cdot {〈 S 〉}_{L} = \frac{c^{2}}{8 π} \frac{Q_{L}}{g_{L}} T,

Equations (19) and (20) can be written as

\frac{c^{2} Q_{L}}{8 π g_{L}} \frac{1}{L} {\frac{1}{2} \frac{g_{L}}{Q_{L}} G_{α} (K) - \frac{\partial θ}{\partial K_{α}} T} + {〈 S_{α} 〉}_{L} = v_{g α} {〈 U 〉}_{L},

for 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

G_{α} (K) = - G_{α} (0, K) = - i {| 1 + r |}^{2} \frac{\partial}{\partial K_{α}} [\frac{Q_{L}}{g_{L}} \frac{(r - r^{*})}{{| 1 + r |}^{2}}]

for α = x, z.

It should be noted that the factor Q_L(r − r^*) = 2ir₂Q_L in the latter Eq. arises from the interference between incident and reflected waves.

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

{〈 S_{α} 〉}_{L} = v_{g α} {〈 U 〉}_{L} - \frac{c^{2}}{8 π} \frac{r_{2}}{L} \frac{\partial}{\partial K_{α}} (\frac{Q_{L}}{g_{L}}) - \frac{c^{2}}{8 π} \frac{Q_{L}}{g_{L}} \frac{R}{L} \frac{\partial Φ_{R}}{\partial K_{α}},

where R = |r|² is the reflection coefficient and Φ_R is the phase of the complex reflection amplitude 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_R = Φ ±π/2 = LK_z ± π/2 and (1/L)∂Φ_R/∂K_α = 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 〈S_α〉_L = v_Eα〈U〉_L = S_g_α and therefore v_E_α = v_g_α 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 v_g, which may be superluminal away from resonance, and the energy velocity v_E, which in general remains causal [13, 16, 17]. It is then clear that, in the superluminal regime, the subluminal behavior of v_E 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 ∂/∂K_α in (30) by

\frac{\partial}{\partial K_{z}} = v_{g z} \frac{\partial}{\partial ω},

\frac{\partial}{\partial K_{x}} = v_{g x} \frac{\partial}{\partial ω} + {(\frac{\partial}{\partial K_{x}})}_{ω},

for α = z and x, respectively, where the latter term on the right-hand side of (32) represents the derivative with respect to K_x 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 〈S_α〉_L =v_Eα〈U〉_L, we obtain, after some algebraic manipulation, the expressions

v_{E z} = \frac{T}{T + T_{0}} v_{g z} = T v_{g}^{(ω)},

v_{E x} = \frac{[T + (R / τ_{d}) \partial Φ_{R} / \partial ω]}{T + T_{0}} v_{g x} - \frac{T_{x}}{T + T_{0}},

where τ_d = L/v_gz is the group delay,

v_{g}^{(ω)} = v_{g z} / (T + T_{0})

and

T_{0} = \frac{1}{τ_{d}} {r_{2} \frac{g_{L}}{Q_{L}} \frac{\partial}{\partial ω} (\frac{Q_{L}}{g_{L}}) + R \frac{\partial Φ_{R}}{\partial ω}},

T_{x} = \frac{1}{τ_{d}} {r_{2} \frac{g_{L}}{Q_{L}} {[\frac{\partial}{\partial K_{x}} (\frac{Q_{L}}{g_{L}})]}_{ω} + R \frac{\partial Φ_{R}}{\partial K_{x}}} .

Equations (33) and (34) relate the components of the energy transport velocity v_E to those of the group velocity v_g 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 v_E = v_g. 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 T_x. 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_R = Φ ±π/2 = LK_z ± π/2 and ∂Φ_R/∂K_x = 0, ∂Φ_R/∂K_z = L and ∂Φ_R/∂ω = τ_d. Using these relations in Eq. (35) and Eq. (36) and taking into account that $Q_{L} = \sqrt{\frac{ω^{2}}{c^{2}} ε_{L} μ_{L} - K_{x}^{2}}$ , we obtain:

T_{0} = R - \frac{τ_{i}}{τ_{d}}

T_{x} = - \frac{τ_{i l}}{τ_{d}},

where

τ_{i} = \frac{r_{2}}{Q_{L}} {\frac{Q_{L}}{g_{L}} \frac{\partial g_{L}}{\partial ω} - \frac{\partial Q_{L}}{\partial ω}},

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

τ_{i l} = - \frac{K_{x} r_{2}}{Q_{L}^{2}}

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_{E z} = \frac{T}{1 - τ_{i} / τ_{d}} v_{g z} = T v_{g}^{(ω)},

v_{E x} = \frac{1}{1 - τ_{i} / τ_{d}} v_{g x} + \frac{τ_{i l} / τ_{d}}{1 - τ_{i} / τ_{d}} .

If v_gz = L/τ_d and

v_{g}^{(ω)} = L / τ_{D}

are substituted into (41), we obtain immediately the relation:

τ_{d} = τ_{D} + τ_{i}

between the group delay τ_d, the dwell time τ_D and the self-interference delay τ_i, as expected [17]. In conclusion, the energy transport velocity v_E and the group velocity v_g are related through quantities having specific physical meaning.

Note, finally, that the relation between v_Ez, v_gz and $v_{g}^{(ω)}$ is independent of K_x, that is to say, the effects of oblique propagation do not modify it.

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

\frac{{〈 S_{x} 〉}_{L}}{{〈 S_{z} 〉}_{L}} = \frac{v_{E x}}{v_{E z}} = \frac{[T + (R / τ_{d}) \partial Φ_{R} / \partial ω]}{T} \frac{v_{g x}}{v_{g z}} - \frac{τ_{d} T_{x}}{L T},

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 〈S_z〉_L, form the basic Eqs. for studying the energy flux densities in finite photonic structures. To carry out such a study, it is necessary to know the complex reflection r and transmission t amplitudes, which are related through the total transfer matrix T̃ as [16]

(\begin{matrix} t \\ 0 \end{matrix}) = \hat{T} (\begin{matrix} 1 \\ r \end{matrix}) .

This Eq. and the fact that T̃ is an unimodular matrix lead to the expressions

1) t = \frac{1}{T_{22}}, 2) r = - \frac{T_{21}}{T_{22}} = - T_{21} t,

which will be used to derive general formulas for the quantities involved on the right-hand side of Eq. (44), where T_ij are the matrix elements of T̃.

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

tan Φ = tan {LK}_{z} = \frac{Y}{X} = F (ω, K_{x}),

where X and Y are the real and imaginary parts of t, F(ω, K_x) depends only explicitly on ω and K_x, as discussed above, and Φ = LK_z is the phase of 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:

τ_{d} = \frac{\partial Φ}{\partial ω} = \frac{L}{v_{g z}} = \frac{X^{2}}{T} {(\frac{\partial F}{\partial ω})}_{K_{x}},

\frac{v_{g x}}{v_{g z}} = - \frac{X^{2}}{L T} {(\frac{\partial F}{\partial K_{x}})}_{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 :

T_{22} = cos N β - i \frac{sin N β}{sin β} g,

T_{21} = \frac{i}{2} \frac{sin N β}{sin β} (η - \frac{1}{η}) sin b Q_{2} exp (- i a Q_{1}),

where

Q_{i} = \sqrt{\frac{ω^{2}}{c^{2}} ε_{i} μ_{i} - K_{x}^{2}}

, 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 = sin a Q_{1} cos b Q_{2} + \frac{1}{2} (η + \frac{1}{η}) cos a Q_{1} sin b Q_{2},

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

cos β = cos a Q_{1} cos b Q_{2} - \frac{1}{2} (η + \frac{1}{η}) sin a Q_{1} sin b Q_{2} = f (ω, K_{x})

Noting that Q₁ is real, Q₂ may be real or purely imaginary, and the Bloch phase β is real inside the allowed bands and equal to iψ or to π + iψ in the energy gap regions, where ψ is a real angle, the function g and both cosNβ and sinNβ/sinβ are always real quantities. These properties and the first relation in Eq. (46) lead to the following formulas for the transmission amplitude and the dispersion relation of the finite photonic crystal:

t = \frac{1}{T_{22}} = T {cos N β + i \frac{sin N β}{sin β} g},

tan Φ = tan {LK}_{z} = \frac{g tan N β}{sin β} = F (ω, K_{x}),

where

T = \frac{1}{{| T_{22} |}^{2}} = \frac{1}{{cos}^{2} N β + ({sin}^{2} N β / {sin}^{2} β) g^{2}}

is the transmission coefficient and

X = T cos N β

is the real part of t.

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

Φ_{R} = \pm π / 2 + Φ - a Q_{1} = \pm π / 2 + K_{z} L - a Q_{1}

for the phase Φ_R of the complex reflection amplitude r, which leads immediately to ∂Φ_R/∂ω = τ_d − τ_a and (∂Φ_R/∂K_x)_ω = aK_x/Q₁, where τ_a = 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:

\frac{{〈 S_{x} 〉}_{L}}{{〈 S_{z} 〉}_{L}} = \frac{1}{T} {(1 - R \frac{τ_{a}}{τ_{d}}) \frac{v_{g x}}{v_{g z}} + \frac{K_{x}}{Q_{1}} \frac{1}{L Q_{1}} (r_{2} - a Q_{1} R)}

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:

\frac{v_{g x}}{v_{g z}} = - \frac{T}{L} {\frac{[(1 - f^{2}) g^{'} + f g f^{'}] sin 2 N β}{2 {(1 - f^{2})}^{3 / 2}} - \frac{N g f^{'}}{1 - f^{2}}}

where g and f are the functions shown in Eqs. (52)–(53) and g′ = (∂g/∂K_x)_ω and 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_x -component of the wavevector K as $K_{x} = (ω / c) \sqrt{μ_{1} ε_{1}} sin θ_{i}$ . Using the latter relation, it is easy to see that $Q_{1} = (ω / c) \sqrt{μ_{1} ε_{1}} cos θ_{i}$ , $Q_{2} = (ω / c) \sqrt{μ_{1} ε_{1}} \sqrt{μ_{2} ε_{2} / μ_{1} ε_{1} - {sin}^{2} θ_{i}}$ and Eq. (27) can be written down as:

\frac{{〈 S_{z} 〉}_{L}}{S_{0}} = T cos θ_{i},

with

S_{0} = (c / 8 π) \sqrt{ε_{1} / μ_{1}}

.

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

\frac{{〈 S_{x} 〉}_{L}}{S_{0}} = cos θ_{i} {(1 - R \frac{τ_{a}}{τ_{d}}) \frac{v_{g x}}{v_{g z}} + \frac{tan θ_{i}}{L Q_{1}} (r_{2} - a Q_{1} R)},

for the 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〉_L as a function of ω is the same as that of the transmission coefficient. In consequence, the z-component of the energy flux density exhibits maxima at transmission resonances which, according to Eq. (56), correspond to the conditions Nβ = mπ, with m = ±1, ±2,...,±(N − 1); bQ₂ = nπ, with n = 1, 2, 3,...; and η = ±1. Note that only the former condition depends on the number N of unit cells. Thus, when the frequency ω varies within an allowed miniband of the corresponding infinite photonic crystal, 〈S_z〉_L is an oscillating function of ω and exhibits a resonant structure. For frequencies inside the bandgaps of the infinite crystal, the Bloch phase β is a complex quantity and, according to Eq. (56), the resonant structure of 〈S_z〉_L should disappear. These properties of the 〈S_z〉_L-spectra are illustrated in Fig. 2 for a finite, N -period, quarter-wave-stack (λ₀/4 = πc/2ω₀ structure)[20, 26], with μ₁ = μ₂ = 1, $n_{1} = \sqrt{ε_{1}} = 1$ and $n_{2} = \sqrt{ε_{2}} = 1.41$ , for N = 5, 10 and various values of the angle of incidence θ_i. One sees in Fig. 2 that the main effects of increasing θ_i, for a given value of N, are to shift the 〈S_z〉_L-spectra to higher frequencies and to reduce the corresponding resonant-peak values.

Fig. 2 Normal energy flux density normalized to $S_{0} = (c / 8 π) \sqrt{ε_{1}}$ as a function of the frequency ω in units of ω₀, for a finite, N-period, quarter-wave-stack, with $n_{1} = \sqrt{ε_{1}} = 1$ , $n_{2} = \sqrt{ε_{2}} = 1.41$ , N = 5 (left-hand panel), 10 (right-hand panel) and various values of the angle of incidence θ_i.

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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 θ_i ≠ 0 and ignoring quantitative differences, the lateral energy flux density 〈S_x〉_Land the ratio V_fin = v_gx/v_gz for the finite photonic crystals should exhibit resonant structures similar to those of 〈S_z〉_L. It is clearly seen in Eq. (62) that the peak values of 〈S_x〉_L and V_finare the same at each transmission resonance if the factor cosθ_i 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 $S_{0} = (c / 8 π) \sqrt{ε_{1}}$ (black lines) and the ratio V_fin = v_gx/v_gz (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 θ_i = 0.

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Let us now briefly compare the properties of V_fin with those of the ratio V_inf = v_gx/v_gz for the corresponding infinite crystals. This will provide an appropriate understanding of the behavior of V_fin. 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 V_fin in a pass band always occur around the curve associated with V_inf. 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 v_gx/v_gz 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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Electromagnetic energy transport in finite photonic structures

Abstract

1. Introduction

2. Energy flux density

3. Relation between group velocity and energy flux density

4. Application to finite, N-period, photonic crystals

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (62)

Optics Express