1. INTRODUCTION

The concepts of group, energy, and phase velocity in periodic systems were first discussed in detail in the pioneering works of Yariv and Yeh [1] and Yeh [2]. For the group and energy velocity, Yariv and Yeh showed that their definitions are equivalent in systems composed of nonabsorbing materials. This result led to the general acceptance of their definitions. The phase velocity, on the other hand, was discussed more controversially. In the photonic crystal community, there appears to be a general consensus that phase velocity has no definite meaning in the context of Bloch modes [3,4]. A careful review of the three velocities reveals that there are a few inconsistencies in their definitions. It is the purpose of this article to clarify them. Notably, the following issues will be addressed:

The article is organized as follows: first, the notation of the Bloch modes used throughout the article is introduced in Section 2. Then the definitions of group, phase, and energy velocity will be discussed in subsequent sections. For the definition of group velocity in Section 3, a group or pulse will be formed using an arbitrary spectrum of Bloch modes, and its evolution will be studied. Unlike the standard derivations, this approach truly justifies the term group velocity of the resulting definition because it does not rely on the validity of analogies between the Bloch index and the wave number of a plane wave. We will obtain a quantity that fluctuates along the propagation direction and whose average value over a unit cell will conform to the standard definition, ${(dk/d\omega )}^{-1}$. Since there are many ways to compute an average value, we will verify that the obtained average is physically meaningful. We will also point out that ignoring the spatial fluctuations of the group velocity can result in underestimations of nonlinear effects in slow-light media. For the phase velocity, we will introduce a straightforward definition in Section 4 that takes into account the full phase information contained in a Bloch mode. We will show that the average of the resulting quantity over a unit cell is in accordance with the definitions given by Yariv and Yeh using their so-called principal value of $k$ [1,2]. For completeness sake, the energy velocity is discussed in Section 5. The standard definition is $v_{\mathrm{en}}={〈{〈S〉}_T〉}_{\mathrm{\Gamma }}/{〈{〈\mathcal{U}〉}_T〉}_{\mathrm{\Gamma }}$, where $S$ is the component in propagation direction of the Poynting vector $\mathbf{S}$, $\mathcal{U}$ is the energy density, and ${〈·〉}_T$ and ${〈·〉}_{\mathrm{\Gamma }}$ denote averages over periods in time and space, respectively. Since this article puts a lot of emphasis on physically meaningful averages, we briefly want to address the seemingly unphysical procedures of averaging an energy density over time and averaging an energy flux over space along the propagation direction. Finally, all definitions are summarized and discussed in Section 6, and conclusive remarks are made in Section 7.

2. NOTATION

We will distinguish between periodic and homogeneous dielectric systems, according to their symmetry properties. For systems with a continuous translational symmetry along one particular direction, extensive literature is available [7,8]. We will refer to this type of system as homogeneous along one direction or simply homogeneous. Monochromatic electromagnetic waves propagating along this direction of translational symmetry (here the $x$ axis) are characterized by a harmonic evolution, i.e., the fields (we will use the $\mathbf{H}$ field) can be written as

In the present article, the focus shall be on systems with a periodic modulation of the permittivity along the propagation direction of the electromagnetic waves. In other words, the continuous translational symmetry of homogeneous media is replaced by a discrete translational symmetry. In particular, we shall be concerned with periods $\mathrm{\Gamma }$ on the order of the wavelength $\lambda$ of the propagating waves. This means that no effective macroscopic behavior can be assumed. For the present purposes, it is sufficient to consider systems composed of isotropic, nonabsorbing dielectric materials with negligible material dispersion. For the permittivity, we have $\epsilon (x+\mathrm{\Gamma },y,z)=\epsilon (x,y,z)$, where $\mathrm{\Gamma }$ is the period of the modulation. Monochromatic waves propagating along the direction of periodicity are referred to as Bloch modes. They can be written as

3. GROUP VELOCITY

A. Definition

There are several ways to motivate a definition of group velocity. A well-founded motivation must involve a group or packet of waves that are carried by the eigenmode(s) of the studied system according to the spectral content of the excitation signal [9]. The simplest example [10] of a packet is the beating of only two modes. In a homogeneous system, the two modes might be written as $A\cos (k_{1}x-{\omega }_{1}t)$ and $A\cos (k_{2}x-{\omega }_{2}t)$, with $k_1=k+\mathrm{\Delta }k$, $k_2=k-\mathrm{\Delta }k$, ${\omega }_1=\omega +\mathrm{\Delta }\omega$, ${\omega }_2=\omega -\mathrm{\Delta }\omega$, and amplitude $A$. The sum of the two waves is $2A\cos (kx-\omega t)\cos (\mathrm{\Delta }kx-\mathrm{\Delta }\omega t)$, and we can easily distinguish between the velocity of the phase fronts $v_{\mathrm{ph}}=\omega /k$ and the envelope $v_{\mathrm{gr}}=\mathrm{\Delta }\omega /\mathrm{\Delta }k\to d\omega /dk$ for $\mathrm{\Delta }k\to 0$. Another typical example of a packet, which is often used to motivate the definition of group velocity, is the Gaussian pulse, formed by a continuous spectrum of harmonic waves. Again, an analytical treatment reveals that the envelope propagates at the group velocity $v_{\mathrm{gr}}=d\omega /dk$, and it retains a Gaussian shape if third and higher order dispersion terms can be neglected [11]. Generally speaking, the concept of group velocity is only applicable to narrow-band signals.

Since Bloch modes are harmonic in time, the concept of pulse formation in the time domain is analogous to homogeneous systems. At a fixed location in space, the temporal evolution of a pulse can be constructed from a continuous spectrum of monochromatic modes by Fourier theory. Therefore, let’s consider an arbitrary group of Bloch modes whose spectrum shall be given by the complex function $G(\omega )$, which contains the amplitude of all components as well as their phase relation at an initial position $x=0$. The temporal evolution of the field at $x=0$ is described by (dependencies on $y$ and $z$ coordinates are omitted for simplicity):

B. Between the Periods

So far, we have only considered locations that are integer multiples of the spatial period. To gain further insight about what happens between the periods, we need some knowledge about the periodic function $u_{\omega }(x)$. First, we represent $u_{\omega }(x)$ in polar form, $u_{\omega }(x)=|u_{\omega }(x)|e^{i\phi (\omega ,x)}$, where $\phi (\omega ,x)$ is a real function satisfying the relation $\phi (\omega ,x+\mathrm{\Gamma })-\phi (\omega ,x)=2\pi m$, $m\in \mathbb{N}$, for all $x$. This lets us rewrite the Bloch mode of Eq. (2) as

Future numerical and experimental investigations will have to show how significant the effect of local variations in $v_{\mathrm{gr}}$ is in specific experimental situations. However, even in cases where the effect is substantial, it might be challenging to measure it experimentally. Other important effects such as optical losses, dispersion, and mode shape variations for different values of $〈v_{\mathrm{gr}}〉$ can make it difficult to quantify each contribution individually. For instance, in the context of four-wave mixing, it took quite a lot of effort to experimentally verify the trend that the conversion efficiency increases with the fourth power of $1/v_{\mathrm{gr}}$ [14].

C. Remark About Averaging of Physical Quantities

In Eq. (15), a spatial average is taken of the quantity $v_{\mathrm{gr}}^{-1}$. Why not take the average of $v_{\mathrm{gr}}$ directly? The answer is simple, but worth recapitulating. Physically meaningful values are obtained by either averaging $v_{\mathrm{gr}}$ over time or $v_{\mathrm{gr}}^{-1}$ over space. This can be seen by considering a 1D trajectory $r(t)$ of a particle. The average velocity $〈v〉$ of the particle traveling a distance $\mathrm{\Delta }r$ during the time interval $\mathrm{\Delta }t$ is given by

4. PHASE VELOCITY

From the polar representation [Eq. (10)] of a Bloch mode, we can formally define a phase velocity, i.e., the velocity at which the phase fronts [15] (lines of constant phase) propagate in space [16]:

Following the same averaging procedure as in Eq. (15), we obtain the average phase velocity

These considerations unify the different interpretations of phase velocity that exist in the context of Bloch modes using a single unambiguous definition. Again, the essential new insight is that the standard definition corresponds to an averaged quantity.

The phase velocity is mainly of conceptual interest and is hardly accessible experimentally. While for a homogeneous system, it can be linked to the refractive index of the material, this relation is not readily established for a periodic system. The phenomenon of wave refraction at interfaces between a homogeneous medium and a photonic crystal is discussed in detail in [18]. Depending on the angle of incidence, the wave can be split up into more than one component (similar to birefringence), or it can be deflected in unusual ways (e.g., negative refraction) through a mechanism that is more closely related to diffraction than to refraction (the zeroth diffraction order is suppressed by a photonic band gap).

5. ENERGY VELOCITY

The objective of this section is to produce a definition of the energy velocity without having to perform averages of quantities that are not physically motivated, such as the average of energy density over time or the average of energy flux along the propagation direction. To our knowledge, there is no similar treatment reported in the literature, although it is essential for a full understanding of the physical meaning behind the definition of energy velocity.

Consider a wave propagating along the $x$ coordinate of a nonabsorbing system. The system shall be either homogeneous or periodic along $x$. Further consider a box of length $L$ along the $x$ axis and with a cross section $A$ as shown in Fig. 1. We assume a 2D mode confinement in the $y$ and $z$ plane and that $A$ is large compared to the transverse mode size, such that energy fluxes through surfaces perpendicular to $A$ can be neglected. If the energy contained in the volume $V=L·A$ flows out through the area $A$ within time $\tau$, then we can define the energy velocity as $v_{\mathrm{en}}=L/\tau$. This definition is on solid ground if $\tau$ can be unambiguously determined, i.e., if the energy in $V$ is constant in time. We will see that for harmonic waves, this can be guaranteed by choosing the length of the box to be an integer multiple of the wavelength. For Bloch modes, this is not possible, and the issue has to be solved by choosing large values of $L$. Since an integration over the volume $V$ is involved, $v_{\mathrm{en}}$ cannot be understood as a local quantity. We will think of it as an average energy velocity and denote it $〈v_{\mathrm{en}}〉$.

 

Fig. 1. Box of length $L$ and cross-section area $A$. An electromagnetic wave propagates along $x$. The energy density in the box is $\mathcal{U}$, and the energy flux density through the area $A$ is $S=\mathbf{S}·\widehat{\mathbf{x}}$, where $\widehat{\mathbf{x}}$ is the unit vector along the $x$ coordinate.

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Let the energy density be denoted by $\mathcal{U}(x,y,z,t)$ and the energy flux density in propagation direction by $S(x,y,z,t)$. The total energy flowing through the area $A$ during a time interval $\tau$ is $E(\tau )={\int }_A{\int }_0^{\tau }S\mathrm{d}t\mathrm{d}y\mathrm{d}z$, and the total energy contained in the box is $E_{\mathrm{box}}={\int }_V\mathcal{U}\mathrm{d}x\mathrm{d}y\mathrm{d}z$, where $V=L·A$. In the case of a harmonic time dependence of the fields with period $T$, we can write ${\int }_0^{\tau }S\mathrm{d}t=\tau {〈S〉}_T$ for large $\tau$. Moreover, if $\mathcal{U}$ is the product of periodic functions along $x$, we have ${\int }_V\mathcal{U}\mathrm{d}x\mathrm{d}y\mathrm{d}z=L·{\int }_A{〈\mathcal{U}〉}_L\mathrm{d}y\mathrm{d}z$ for large $L$. The energy velocity can now be defined as $〈v_{\mathrm{en}}〉=L/\tau$, for the time $\tau$ satisfying $E(\tau )=E_{\mathrm{box}}$, which is equivalent to $\tau {\int }_A{〈S〉}_T\mathrm{d}y\mathrm{d}z=L{\int }_A{〈\mathcal{U}〉}_L\mathrm{d}y\mathrm{d}z$. This justifies the definition

The standard textbook definition [10,11,19] of energy velocity in homogeneous media is $〈v_{\mathrm{en}}〉={〈S〉}_T/{〈\mathcal{U}〉}_T$. Integrals over the surface $A$ might be added if the mode is confined. The difference to our definition is the time average of $\mathcal{U}$ instead of a space average. For harmonic waves with $\mathcal{U}\sim {\cos }^2(kx-\omega t)$, the two definitions are equivalent since both averages return the same result. To our understanding, the time average of $\mathcal{U}$ in the textbook definition can only be justified through this equivalence.

For a Bloch mode, it can be shown that ${\lim }_{L\to \infty }{〈\mathcal{U}〉}_L={〈{〈\mathcal{U}〉}_T〉}_{\mathrm{\Gamma }}$. This allows us to rewrite Eq. (23) as

To conclude this section, we emphasize again that the standard definitions of group and energy velocity have to be understood as averaged quantities. As pointed out in Section 3.C, it is essential that the averaging procedure is motivated by physical arguments. In this sense, the findings of Sections 3 and 5 retrospectively form an indispensable basis for considerations such as the proof of the equivalence $v_{\mathrm{gr}}=v_{\mathrm{en}}$ in lossless systems [1,2] or the decomposition of $v_{\mathrm{en}}$ into weighted contributions from each Fourier component of a Bloch mode [20]. In fact, the equivalence result, $v_{\mathrm{gr}}=v_{\mathrm{en}}$, only now unfolds its full physical meaning.

6. DISCUSSION

Table 1 summarizes the results of Sections 3–5. If we motivate the definitions of all velocities from first principles, we can obtain a fully consistent set. If we do not explicitly consider the long-wavelength regime, we have to allow the velocities to vary along the direction of propagation. For the group and phase velocity, we derived the $x$-dependent expressions. For the group velocity, we assumed a special form of the periodic function $u_{\omega }(x)=u_{\omega }{e}^{i\phi (\omega ,x)}$. If $|u_{\omega }(x)|$ contains an additional $x$ dependence, then the pulse shape is distorted between the periods, and defining a local group velocity is no longer straightforward. However, the average group velocity can still be defined, and it corresponds to the standard definition found in the literature. In fact, both the average group velocity and the average phase velocity derived here agree with the definitions given by Yariv and Yeh [1,2], except that they have not previously been recognized as being averaged quantities. We can conclude that, if the Bloch index $k$ is treated as a full analogue to the wave number of a plane wave, then the fact that $k$ does not contain the full phase information of the mode naturally leads to averaged quantities. This result is nontrivial, and we believe it has not received due attention, possibly because of its intuitive appeal.

Table 1. Definitions of the Group, Energy, and Phase Velocity of Bloch Modes, According to Sections 3–5^a

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The averaged velocities often provide an appropriate description of the physics of a waveguide, and local phase fluctuations between the periods can often be neglected. However, there can be situations where the microscopic phase properties will matter and where we must be alert to subtle effects of the periodicity. For instance, problems can arise in measurements involving interference between two Bloch modes. In particular, we believe that for the experimental measurement of the group velocity using interference fringe patterns, microscopic effects should be taken into account. However, an analytical treatment of such interference effects is rather challenging because interference phenomena typically involve structures with nonperfect periodicity (e.g., a structure terminated at one end), and a treatment in terms of pure Bloch modes is, therefore, hardly appropriate. Another example where microscopic propagation properties can be of importance is the estimation of nonlinear effects in slow-light waveguides, as addressed in Section 3.B. We have seen that local fluctuations of the group velocity can affect the strength of a Kerr nonlinearity. In cases like this, the field patterns of the excited Bloch modes have to be investigated to guarantee that local intensity maxima are accurately taken into account.

7. CONCLUSION

We have seen that the standard definitions of group and energy velocity of Bloch modes in periodic systems represent meaningful quantities in the sense of average values. Both definitions were motivated from first principles, and the averaging procedures were critically reviewed for the first time. The well-known equivalence result, $v_{\mathrm{gr}}=v_{\mathrm{en}}$, now displays its full physical meaning.

The example of nonlinear effects in a Kerr medium suggests there is a practical relevance of our findings. We demonstrated formally that using the average group velocity $〈v_{\mathrm{gr}}〉$ can lead to an underestimation of the nonlinear effect.

Using a polar representation of the Bloch mode and taking the phase evolution explicitly into account, the phase velocity can be defined in a straightforward manner. The definition is fully consistent with that of the group velocity in the sense that the average values of both quantities correspond to the definitions that result if the Bloch index $k$ is treated as a full analogue to the wave number of a plane wave. As it turns out, the fact that the Bloch index does not contain the full phase information of the mode naturally leads to averaged quantities.