1. Introduction

In an ideal medium in which both the electric permittivity ε and magnetic permeability µ are simultaneously real and negative, electromagnetic (EM) waves propagate in an unusual way. Since the electric field E, the magnetic field H and the wave vector k form a left-handed (LH) set, the phase velocity v _p, which is related to the evolution of the phase fronts of the EM wave, and the group velocity v _g, which characterizes the flow of energy, point in opposite directions [1]. Thus, it is usually said that EM waves propagate backwards in an LH medium. This property gives rise to several “reversed” phenomena, being one of them the negative refraction of EM waves when crossing from a conventional right-handed (RH) medium such as free space to an LH one or vice versa. In this case, Snell’s law is accomplished if a negative index of refraction $n=-\sqrt{\epsilon \mu }=-c\mid \mathbf{k}\mid ⁄\omega$ (c is the speed of light in free space) is assigned to the LH medium. Once the wave has entered the LH medium, it propagates with a negative phase velocity v _p=|v _p|=c/n, accumulating a total negative phase shift equal to -|k|d=nωd/c after propagated over a certain distance d. As in the case of a conventional homogeneous RH medium, the refractive index used to predict the refracted angle using the Snell’s law and the phase index that is used to characterize the evolution of the phase fronts have the same value.

Although an ideal homogeneous LH medium does not exist in nature, artificial materials can be designed and built to accomplish simultaneously ε and µ<0 at certain frequency ranges, with the inherent drawbacks of loss and dispersion. For example, a composite structure consisting of metallic split-ring resonators (behaving as an effective medium with µ<0) and thin metal wires (behaving as an effective medium with ε<0) was demonstrated to become an LH medium and refract negatively EM waves impinging in it from free space, accomplishing the Snell’s law [2–3]. This artificial LH medium is not homogeneous as the ideal LH substance addressed in [1], but a periodically structured material. However, since the effective wavelength λ at which the LH behavior is found is much larger than the period a of the structure (a ratio of 6 in [3]) the EM waves see an effective LH medium in terms of both negative refraction at the interface with an RH material and EM wave propagation along the structure (negative phase velocity) [2]. A similar effective LH behavior also occurs for artificial media based on transmission line grids loaded with series capacitors and shunt inductors, with a ratio of 35 between wavelength and period [4].

Negative refraction can also take place at the interface between free space and a properly designed photonic crystal (PhC) [5]. A PhC can behave as an “almost” isotropic refractive material in certain spectral regions for which the so-called equifrequency contours (EFCs) become rounded. The radius of the EFCs is given by the amplitude of the fundamental wave vector k _0,0 of the photonic band at a given frequency. If the EFC radius shrinks with increasing frequency, then the group velocity v _g points in a direction opposite to the direction of k _0,0. Thus, some authors have concluded that under this condition (rounded and shrinking EFCs, which hereinafter we will refer to as LH-PhC condition) a PhC displays an LH behavior [6–7]. In fact, under the LH-PhC condition and properly choosing the interface, negative refraction occurs at the free space-PhC interface and the refracted angle can be predicted with the Snell’s law if a negative effective index of refraction n _eff=-c|k _0,0|/ω is associated to the PhC [8]. Thus, in terms of refraction at the interface with a homogeneous RH medium, a PhC with negative index of refraction n _eff can resemble a LH material [6–8].

However, EM propagation inside a PhC at frequencies for which the so-called LH-PhC condition is satisfied is quite different from propagation in an ideal LH medium or even in an artificial LH composite. The LH-PhC condition is met at high frequencies, at least at the second photonic band. This means that the LH-PhC, which by nature is not a homogeneous medium as the ideal LH medium, neither works in the long-wavelength region, as is the case of the artificial LH composite. Inside the LH-PhC there is not a single EM wave with wave vector k _0,0, but a set of plane waves with wave vectors k _0,0+G (G is a reciprocal lattice vector) and different amplitudes, which is known as a Bloch wave [5]. The natural question that arises is if the PhC that satisfies the LH-PhC condition and gives rise to negative refraction at an interface with a homogeneous RH medium also behaves as a LH medium in terms of EM propagation, so that phase evolution along the LH-PhC is really opposite to the energy flow (negative phase velocity). There is a little controversy regarding this issue. Whilst some authors point out that the component of the Bloch wave with wave vector k _0,0, which gives the value of n _eff used to explain satisfactorily the negative refraction, has a negligible influence on the wave propagation in a region accomplishing the LH-PhC condition [9–11], other authors have shown a backward evolution of the phase fronts when negative refraction takes place [12] or have obtained a negative phase velocity [13] in a LH-PhC, in agreement with the expected behavior of an LH medium. It seems important to clarify this point, since if the term LH is applied to a material, backward propagation (that is, evolution of phase fronts opposed to the energy flow) associated to the negative phase velocity must take place inside such a material (the phenomenon of negative refraction is a simple consequence of the negative phase velocity inside the medium). In this work, we consider this issue by a thorough analysis of the EM propagation inside a PhC satisfying the LH-PhC condition by means of finite-difference time-domain (FDTD) simulations and Fourier decomposition of the EM field, and compare it to that expected in an ideal LH medium. We obtain that EM propagation is quite different at frequencies far from the negative-index band edge and close to it and explain it in terms of the Fourier components of the propagating Bloch wave. In addition, we find that phase evolution is positive over a period, which is in opposition to the behavior of a LH medium.

2. Analysis of EM propagation inside the LH-PhC

In our analysis, we consider a two-dimensional (2D) hexagonal array of air cylinders (r=0.4a, a being the lattice constant) on a high-index substrate (ε=12.96). Only TM modes (electric field parallel to the cylinders) are considered. The analysis here also holds for a three-dimensional PhC provided that the so-called LH-PhC condition is met, but we choose the 2D PhC for the sake of simplicity. Figure 1(a) shows the EFCs in the first Brillouin zone of the proposed PhC for normalized frequencies from 0.29 to 0.34 (in units of ωa/2πc) in the second photonic band [14]. The LH-PhC condition previously described is met since for frequencies higher than 0.29 and up to the band edge that occurs at 0.3495 the EFCs are rounded and their radii diminish as the frequency increases. We also consider in Fig. 1(a) that an EM wave propagates in free space along the y-axis and impinges on the PhC to excite a propagating Bloch mode. The group velocity v _g=dω/dk _0,0 of the second band for wave vectors along ΓM is shown in Fig. 1(b). Since conventionally the direction of propagation of energy (in our case, towards positive values of y) is taken as “positive”, it can be considered that the fundamental wave vector k _0,0 for the second band is “negative”, which gives the LH-PhC behavior. In fact, the refraction at the free space-PhC interface can be well explained by modeling the PhC as an effective medium with a negative effective index [8]. However, to understand in depth EM propagation inside the LH-PhC, we have to consider other wave vectors that also contribute to the whole Bloch wave.

 

Fig. 1. (a) Schematic EFCs of the PhC under study for frequencies between 0.29 and 0.34. The direction of the input external wave, as well as the direction of k _0,0 and v _g inside the PhC, are depicted; (b) v_g (normalized to c) vs. k _0,0 (normalized to G_y ) for the frequency range under study: results obtained by derivation of the photonic band (solid line) and by weighted summation of the group velocity of the different Fourier components (open circles). Inset: detailed view of the group velocity near the band edge.

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For TM polarization and assuming propagation along the y axis, the electric field of the Bloch wave can be expressed as follows:

where e _m,n is the normalized field amplitude (∑_m,n $e_{m,n}^2$=1) of the (m,n) Fourier component, G _x=2π/a x, G _y=2π/(√3 a)y, G _x=|G _x |, G _y=|G _y|, and m+n must be zero or an even integer. This notation is a bit different to that conventionally used to express a Bloch wave in a hexagonal PhC as a summation of plane waves, but is useful to analyze propagation as described below.

To calculate the values of e _m,n we obtain the field distribution inside the PhC with the FDTD method (as in [9] and [13]) at frequencies of the second photonic band for which the LH-PhC condition is met and then apply a 2D Fourier transformation. It has to be mentioned that the Fourier decomposition of the Bloch wave can provide useful information on the behavior of the studied structure [15]. The FDTD domain is a rectangle that extends from - 7.5a to 7.5a in the x direction and from 0 to 100a in the y direction with a grid size of a/50 in each direction. This permits that the PhC can be considered semi-infinite in the sense that it is sufficiently large in the y direction (100 periods) so that there is not reflected wave when the field is captured. Perfectly matched layer conditions are applied at the boundaries of the computational domain. A Gaussian monochromatic beam (initial width of 10a to ensure no divergence) is launched at y=0 in free-space and propagates along y. At y=√3 a the beam impinges the PhC (see details of the interface in the left-hand side of Fig. 3(e)). The PhC is placed so that the ΓM (ΓK) main symmetry direction is along the y (x) axis. Under these conditions, the incident wave impinges the interface and creates a Bloch wave inside the PhC that propagates along y, so k _0,0=k _0,0 y, as assumed in Eq. (1). Since the Bloch mode has even symmetry with respect to the y-axis for the second photonic band and propagation along ΓM, it results that e _m,n=e _-m,n. The field distribution (real part of E _z) used to obtain the Fourier coefficients e _m,n is captured after a simulation time long enough so that the EM wave has propagated a large distance along the y axis. Then, the 2D Fourier transformation is carried out on the field distribution.

 

Fig. 2. (a), (b) Amplitudes of the coefficients e _m,n and Δ_m,n obtained from the 2D Fourier transformation as a function of frequency. The inset in (b) shows in detail the region close to the band edge.

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The obtained values of e _m,n for the frequency range consideration are shown in Fig. 2(a). Only those components with a normalized power above 0.05 at some frequency are shown. It can be seen that the amplitudes of the different (m,n) Fourier components change with frequency, and the composition of the Bloch wave is quite different for frequencies far from the photonic band edge and close to it. To better understand the influence of the different Fourier components on the Bloch wave propagation at different frequencies, we can express Eq. (1) as follows:

where Δ_m,n=e _m,n-e _m,-n, and only most significant terms are written. The amplitudes of the components e _m,n and Δ_m,n in Eq. (2) are shown in Fig. 2(b). The inset shows in detail the components in the frequency region near the photonic band edge. The terms in each row of the right-hand side in Eq. (2) have different interpretations and their contribution is different depending of the frequency range under consideration. We explain the influence of each of these terms in what follows.

The terms in the first row have an amplitude proportional to e _m,0 and correspond to EM waves whose net wave vector in the y direction is k _0,0 and gives the n _eff used to explain negative refraction and the negative phase velocity as defined in [13]. It is interesting to note from Fig. 2(b) (green curves) that the amplitude of these e _m,0 components is very small and tends to zero when approaching the band edge so they have negligible influence on the propagating Bloch wave. This is in clear contrast with the case of an ideal or artificial LH medium in which propagation can be characterized in terms of a unique wave with negative wave vector k, which is equivalent to the k _0,0 of the studied PhC in the sense that it is used to describe the phenomenon of refraction.

The terms in the second row in Eq. (2) correspond to EM waves propagating forwards with positive wave vectors n G _y +k _0,0 and amplitudes proportional to Δ_m,ncos(mG _x x), which provides a spatial modulation of the wave along x. As shown in Fig. 2(b) (blue curves), these forward-propagating EM waves dominate the composition of the Bloch wave at frequencies of the regime under study which are far from the photonic band edge. As an example, we consider the normalized frequency of 0.3, at which k _0,0=-0.55G _y and the value of n _eff used to explain refraction is close to -1. We try to identify if there is backward propagation with phase fronts spaced 2π/|k _0,0|=3.1a at this frequency inside the PhC, as it would occur for an LH medium with a negative index of -1 [16]. To this end, we obtain the electric field inside the PhC at different time steps using FDTD, as shown in Figs. 3(a)–(d). The time spacing between adjacent snapshots is much lower than the time period of the expected backward wave with wave vector k _0,0. We can observe in the field patterns of Fig. 3 that the propagated electric field is spatially modulated along the x direction due to the periodicity of the PhC, as expected from the considerations about the second term in the right-hand side of Eq. (2), but phase fronts spaced 3.1a and moving backwards are not observed. Taking a closer look into the dashed rectangle in Fig. 3(d), which is shown in Fig. 3(e), we can identify some strongly-modulated phase-like fronts [13] spaced 1.2a and moving forwards, as highlighted by the vertical dashed lines and the horizontal arrows in Figs. 3(a) to (d). These spatially modulated phase-like fronts can be associated to the wave component with amplitude Δ_2,0 and wave vector 2G _y+k _0,0, of Eq. (2), which, from Fig. 2(b), is by far the main contribution of the Bloch wave at the frequency of 0.3. Thus, it can be expected that the phase evolution of the whole Bloch wave will be characterized by that of the Δ_2,0 term, and not by the phase velocity of the component with negative wave vector k _0,0.

 

Fig. 3. (a)–(d) Distribution of the real part of E _z in the computational domain at a normalized frequency of 0.3 at different time steps (see the figure) of the FDTD simulations. (e) Left-hand side: detail of the computation domain at the free space-PhC interface; Right-hand side: detail of E _z inside the dashed rectangle in (d).

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We can obtain the phase evolution along the y axis by inserting the calculated values of e _m,n in Eq. (1) so that the complex electric field E_z (x,y) is obtained. The plane y=0 is considered as the phase reference and phase continuity is imposed along the y axis in the calculations. For the case of a plane wave in a conventional RH or LH medium, the phase evolution does not vary with the x coordinate. However, in the LH-PhC under study it will also depend on the x coordinate, so spatial paths along y with different values of x will experience different phase shifts. This can be observed in Fig. 4(a) which shows the phase accumulated by the Bloch wave with normalized frequency 0.3 when it propagates over a period od √3a along the y direction and for a whole period a in the x direction. The phase evolution closely resembles that of a single plane wave with positive wave vector 2G_y +k _0,0 , despite of the fact that there is a slight modulation of the phase due to the presence of other weaker Fourier components. A phase velocity v_p =0.239c y can then be associated to the Bloch wave, taking into account its resemblance with the single plane wave in terms of phase evolution (the spatial modulation in x does not occur in the single plane wave). This phase velocity is very close to the group velocity v_g =0.25c y (see Fig. 1(b)) of the whole Bloch wave at the frequency under consideration.

This positive phase evolution is also observed in Fig. 4(b), which represents the different phase shifts experienced by the Bloch wave in a period along the y axis for all the x-paths between -a/2 and a/2 (shown in green). The paths for x=+a/4 and x=-a/4 are shown by a red straight line corresponding to (2G _y+k _0,0)y, whilst all possible phase shifts are included between the paths for x=0 and x=±a/2 (in blue). From the representation of the phase evolution it becomes clear that it is totally different from that expected for an ideal LH medium with negative wave vector k _0,0 (black line with negative slope in Fig. 4(b)). If we consider only the phase shift accumulated after one period, it becomes equal to 4π+k _0,0 √3a, which, if the phase is folded within [-π,π], results in a negative phase shift of k _0,0 √3a. This explains the misleading interpretation of a negative phase velocity reported in [13]. These results confirm that for the frequency 0.3 (and also for the frequencies far from the band edge of the negative-index photonic band), in which negative refraction can be predicted by using the effective index n _eff=c k _0,0/ω, phase velocity is positive and can be quite accurately expressed as v _p≅ω/(2G _y+k _0,0). In other words, the refractive index n _eff used to explain refraction at interfaces with RH media is different from the phase index n _p=c/v _p used to characterize the evolution of the EM wave phase fronts along the PhC.

 

Fig. 4. Phase evolution of the Bloch wave with normalized frequency of 0.3 along a period in the y direction for different values of the x coordinate in a period. (b) is a the projection of (a) on a plane perpendicular to the x-axis. Values in (b) stand for the values of the x coordinate for which the phase evolution is obtained. The phase shifts that would occur in an ideal LH medium with negative wave vector k _0,0 are also shown as a black line in (b).

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Finally, the terms in the third row of the right-hand side of Eq. (2) are EM waves with amplitude proportional to 2e _m,ncos(m G _x x)cos(n G _y y) (n being negative) –which is the amplitude of a standing wave along the x and y axes- and propagating backwards, since they are affected by a plane wave with negative wave vector k _0,0. From Fig. 2(b), we observe that these terms dominate the wave propagation near the photonic band edge. Just at the band edge, k _0,0=0 and the Bloch wave is a summation of standing waves with zero group velocity so there is not power flux along y. This situation is quite different from that observed at frequencies far from the band edge. We repeat the simulations shown in Fig. 3 but at a frequency of 0.346, which is close to the band edge, and obtain the field distributions shown in Figs. 5(a)–(d). Again, the time between consecutive snapshots is much lower than the period associated to a wave with wave vector k _0,0 . Now, it can be clearly observed that there are strongly-modulated lobes that move backwards. Figure 5(e) shows a detail of two consecutive modulated lobes. The observed periodicity in x and y corresponds exactly to the field distribution expected for the mentioned standing waves. It can also be observed a 180° phase shift between them, which would correspond to half a wavelength if the observed strongly-modulated lobes are considered to be phase-like fronts [13]. Then, the distance between adjacent strongly-modulated phase-like fronts is 13a, which closely corresponds to 2π/k _0,0 at the studied frequency, the wavelength associated to the a wave with wave vector k _0,0. Then, close to the band edge, EM propagation of the Bloch wave resembles backward propagation of strongly-modulated lobes with wave vector k _0,0 , which explains the apparent backward propagation reported in [12]. However, this is not a real backward propagation as in an LH medium, since the phase evolution along y is also positive as shown in Figs. 6(a) and 6(b), which are similar to Figs. 4(a) and 4(b) but for the normalized frequency of 0.346. In this case, the strong mixing of Fourier components near the band edge results in discontinuities of the phase in the x direction so the total phase accumulated after a period in the y direction depends on the x coordinate. It has been observed that the discontinuities in x appear at normalized frequencies higher than 0.315. The phase paths along y also present strong curvatures although the general trend is positive, so the total phase shift along a period can be 4π+k _0,0√3a or 2π+k _0,0√3a depending on the x coordinate. Since there is not a single predominating component (see Fig. 2), it is not possible to associate a single phase velocity to describe the evolution of the phase fronts along y, as it occurred at frequencies far from the band edge. Anyway, folding the accumulated phase shifts after a period within [-π,π] results again in the negative phase shift of k _0,0√3a regardless of the x coordinate, which occurs for all the frequencies under study. However, it has to be highlighted that this is a result of the periodicity of the PhC and the folding of the phase, and is not caused by a real negative evolution of the phase and a negative phase velocity as it occurs in an ideal or artificial LH medium.

3. On the group velocity of the Bloch wave

In this section, we make a brief consideration about the group velocity of the Bloch wave as a result of the group velocity of each individual Fourier component. The group velocity can be calculated from the photonic band calculated for an infinite PhC, as shown by the solid curve in Fig. 1(b). However, this parameter can also be obtained as the weighted sum of the group velocities of the individual Fourier components as [13]:

where, as in Eq. (1), m+n must be zero or an even integer. The group velocity calculated from the Fourier components obtained from the FDTD electric field distribution is shown in Fig. 1(b) as open circles for different frequencies. A good agreement between the values of the group velocity obtained by means of two different methods is observed. As stated before, far from the band edge a plane wave with amplitude Δ_0,2 and wave vector 2G_y +k _0,0 clearly dominates the Bloch wave propagation. Then, in this frequency range the group velocity of the Bloch wave can be well approximated by the phase velocity of this dominating plane wave, taking into account that v_g =v_p for this wave. This has been confirmed for the particular case of a frequency of 0.3. However, when increasing the frequency the trend of the Fourier components from Fig. 2 is as follows: e _0,m→0 and e _m,n→e _m,-n (for m≠0). Since in the case under study e _m,n=e _-m,n, just at the band edge the (m,n) and (-m,-n) Fourier components have the same amplitude and opposite group velocity, which results in a zero group velocity for the whole Bloch wave and the formation of the standing waves as expressed in Eq. (2).

 

Fig. 5. (a)–(d) As in Fig. 3, but for a frequency of 0.346. (e) Detail of E_z inside the dashed rectangle in (d).

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Fig. 6. As Fig. 4, but for a normalized frequency of 0.346. In (b) the values indicate the interval of the x axis for which the phase path are obtained.

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4. Conclusion

In summary, although negative refraction following Snell’s law in a PhC under the so-called LH-PhC condition can be satisfactorily explained using the negative effective index obtained from the fundamental wave vector k _0,0 of the corresponding Bloch wave, and the folding of the phase produces that the total phase shift accumulated by a wave propagating along a period of the PhC could be interpreted as negative, a deep analysis on the EM propagation reveals that the phase evolution (phase velocity) is clearly positive over a period, so care has to be taken when applying the term LH to a PhC. In addition, two different behaviors of EM propagation in the studied PhC in a photonic band with v_g and k _0,0 being antiparallel have been found.

Far from the band edge, EM propagation resembles closely that of a plane wave with wave vector 2G_y +k _0,0 and the group velocity of the Bloch wave can be approximated by the phase velocity of this single plane wave. At these frequencies, a positive phase index that characterized the phase evolution can be associated to the Bloch wave, and it is different from the negative refractive index used to model the negative refraction.

When approaching the band edge, the propagation of the wave is more complex and, in the spatial domain, it resembles strongly-modulated lobes traveling backwards with negative wave vector k _0,0 . However, in this case, a single phase velocity cannot be associated to the PhC, since the phase evolution can be different (although positive) depending on the position in the axis transversal to the propagation.