1. Introduction

Slow light propagation in structured periodic media has received a lot of attention in recent years [1,2]. After introduction of the effect in line-defect photonic crystal (PhC) waveguides by Notomi et al. [3] many publications appeared concerning different aspects of this effect. In particular, the small group velocity of light leads to: large time delays [4,5], increased phase shift [6–8], large [9,10] or vanishing [11,12] dispersion, enhanced nonlinear interaction [13]. Slow light becomes especially attractive combined with silicon photonics approach [14].

A critical problem with PhC waveguides is their propagation losses which are caused, to a large extent, by small inaccuracies in the manufactured structures. Such disorder-induced scattering losses in the slow light waveguides were investigated experimentally [15,16] and theoretically [17,18]. The scattering loss consists of two parts: vertical scattering (away from the waveguide) and backscattering into the guided Bloch mode propagating in the opposite direction. Vertical scattering is considered to be dominant for the “non-slow” modes of line-defect waveguides and was estimated to grow proportionally to the inverse group velocity [15,18]. In active media, vertical scattering could be compensated by in-line amplification.

On the other hand, the disorder-induced backscattering loss has been predicted to increase with inverse group velocity squared [15,19], which makes it a dominant loss factor at small group velocities [18]. At the same time, the backscattering can lead to strong coupling between forward and backward propagating modes in disordered slow light structures. Even small manufacturing inaccuracies can lead to localization effects at light frequencies employing small group velocities [20–23]. An in-line amplification of the disordered structure will also amplify the backscattered light and will enhance localization phenomena [24]. Thus it is important to fathom out the potential how to decrease the backscattering intensity by a proper design of the slow light waveguide.

The backscattering phenomena are usually investigated with perturbation methods [17, 25,19] or considered as band diagram deterioration [26, 27]. For slow light coupled cavity waveguides models are developed to simulate disorder as a variation of resonance frequencies and coupling constants [20,28]. In this paper we use a more direct method to investigate the effects of backscattering in PhC slow-light waveguides: we numerically calculate the backscattering at a single disorder defect and present a comparison of two different dispersion-engineered slow-light waveguides. From the calculations for a single disorder defect we then determine the mean free path in the slow light waveguides as a function of group index.

2. Approach

We have concentrated on the direct calculation of the Bloch mode reflection at a single disorder defect, i.e., we assume the waveguide is perfectly periodic except for a single hole whose boundaries are slightly perturbed. The reflections at all possible hole-boundary shifts can be calculated separately. The boundary perturbations in the photonic crystal waveguides from hole to hole are considered to be uncorrelated and thus all reflection amplitudes add up with random phase shifts. When the single-defect reflections are known, due to the random phase condition, the overall reflection intensity as a first approximation can be considered as an incoherent sum of intensities from all the defects. The mean free path of the disordered slow light structure is defined as the length where the initial intensity would reduce to 1/e if second order backscattering is not considered, which is the backscattering of the already scattered light back into the forward propagating mode.

To investigate small reflections at the defects we have used the eigenmode expansion method. This method discretizes the simulation volume into slices of constant lateral refractive-index distribution (see Fig. 1). Any field distribution in the slice can be presented as a sum of the eigenmodes of this slice. The eigenmodes propagate inside each slice according to their propagation constants. By requiring continuity of the expanded fields between adjacent slices, overlap integrals are obtained that describe the transition between the various slices. From the scattering matrix of a unit cell of the periodic line-defect waveguide the Bloch modes may then be calculated. The advantage of this method is the possibility to excite Bloch modes directly and investigate the reflection into the same Bloch mode as well. Thus we avoid any reflection artifacts at the input and output of the PhC structure and concentrate only on the reflection at the single defect. We have used a two-dimensional (2D) method to investigate reflections and thus disregard vertical disorder induced losses. The structures were simulated with two implementations of the method: a self-written implementation [29] and CAMFR [30,31] to verify the results.

 

Fig. 1. Schematic representation of the disorder defects in a line-defect waveguide. In our calculations, a single disorder defect is introduced by shifting the holes’ boundary. In this figure several boundaries of the holes in the 1st, 2nd, or 3rd rows are named for later reference. The assumed refractive indices are 3.5 (corresponding to silicon) for the grey area, and 1.0 (corresponding to Air) for the white areas.

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The slow light structures were considered in the photonic crystal line-defect waveguide based on the triangular lattice of square air holes in silicon (see Fig. 1). Transverse electric modes were excited with electric field vector lying in the xz plane. The side of the square hole is defined exactly as half the lattice constant a . In practice, PhC waveguides are fabricated using round holes. However, here we use square holes which are better adapted to the eigenmode expansion method and thus allow to decrease substantially the number of slices necessary for the device simulation which reduces the simulation time and increases method accuracy. The basic PhC functionality is not changed by using square instead of round holes, and as shown later it allows slow light mode engineering similar to that known for round holes. To justify the approximation a single simulation was conducted with discretized round holes whose areas are equal to those of the presented square holes. A difference in the field distribution at the boundaries of square and round holes leads to only 30% difference in reflection coefficients which is an acceptable deviation. The defects were implemented as hole boundary shifts normal to the boundary. The boundaries of the holes in the first, second and third rows around line-defect were shifted. Examples of the shifted boundaries are presented in Fig. 1.

The discussed 2D model provides an approximation for the three dimensional (3D) slab photonic crystals. The vertical scattering of the slab structure can be integrated in the 2D calculation as an additional loss mechanism. The reflections due to boundary shifts in the slab structures will be a little bit less than in 2D simulations. Part of the field in the 3D structures propagates in the air cladding and thus has no overlap with shifted boundaries of the holes. Therefore, the disorder effects discussed here are to be interpreted as upper estimates to real PhC slabs.

3. Results

3.1 Slow light waveguides

In this article we want to compare two slow light modes with different concentration of the fields on the hole boundaries as we expect these to react distinctly different to disorder. The first mode is presented in Fig. 2(a). The dispersionless slow light is achieved in the line-defect waveguide where the PhC side walls are shifted together thus that the waveguide width is W0.8, where waveguide width is defined as a distance between centers of the holes adjacent to the waveguide channel and W equals a√3 . The reason for the dispersionless behavior was presented elsewhere [11]. The simulated group velocity at the point of vanishing dispersion (at the normalized frequency 0.2357) is 0.007c. Due to the reduced waveguide width the mode has strong field amplitudes on the boundary of the first hole.

Another possibility to achieve dispersionless behavior was described in Ref. [12]. This approach involves position optimization of the first, second and third rows of holes. As presented in Fig. 2(b). in the waveguide W1 it was possible to achieve a similar dispersion curve with group velocity around 0.007c. The distance between first and second rows is 0.41W and between second and third row 0.53W. As can be seen in the electric field distribution in Fig. 2(b)., there is much less field amplitude on the boundary of the first row of holes in the W1 waveguide. The channel is wider and thus there is more space to guide power without touching boundaries. A similar effect is observed in normal dielectric waveguides with increasing width of the guiding channel. In addition, in the W1 waveguide the field is more distributed in the channel and between the first and second row of holes.

 

Fig. 2. Group velocity as a function of normalized frequency and electric field amplitude distribution at the point of vanishing dispersion for the waveguides (a) W0.8 and (b) W1. Strong field concentration at the boundary of the first row of holes is observed in W0.8 waveguide. On the other hand electric field is evenly distributed among the first and second row of holes in W1 waveguide. Both false color plots refer to the same amplitude scale. Field distribution was calculated with finite integration technique of CST, Darmstadt.

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3.2 Reflection at a single defect

The reflection amplitude of the fundamental Bloch mode into its oppositely propagating counterpart caused by a single disorder defect should be proportional to the inverse group velocity and to the extent of the boundary shift [18]. Therefore we here consider the normalized reflectivity

where Δd is the absolute shift of the boundary, r is the (numerically calculated) value of the Bloch-mode amplitude reflectivity, and n_gr = c / v_gr is the group index. The normalization over the group index is justified by the perturbation theory where the modes should be normalized by the group velocity to carry a fixed power [19]. Following the perturbation theory the reflectivity is also linearly proportional to the boundary shift, thus normalization over boundary shifts is justified for small shifts. The simulations confirm the linear dependency of the reflectivity on such small boundary shifts. Thus the normalized reflectivity α defined by Eq. (1), i.e., the amplitude reflectivity per normalized boundary shift per unit group index, is generally a function of the field amplitude and polarization at the considered boundary [19].

Figure 3 presents the normalized reflectivity α as a function of frequency, rescaled to group index, for different boundary shifts defined in Fig. 1. Rescaling of the frequency axis to group indices is performed for better data presentation, otherwise the slow light region is very narrow in frequency range. The region of small group indices corresponds to the frequencies above the vanishing dispersion point, 0.007c or 140 group index, and the region of large group indices corresponds to frequencies close to the band edge (see Fig. 2). There is a clear α step at group index 140, which is coursed by strong variation of field distribution at the vanishing dispersion point. This can be explained by the fact that the point of vanishing dispersion is built by the anticrossing of two different modes [11]. At the same time the normalized reflectivity is changing slowly for the group indices away from the anticrossing point, where mode profile also changes slowly with frequency. Unfortunately, it is not possible to vary the group index without changing the field distribution at the same time, thus α is not a constant but an implicit function of group index. At small group indices the mode is strongly concentrated at the channel and less field on holes boundaries and thus lower α. On the other hand at large group indices or small group velocity the mode has wider field profile with more field at hole boundaries what corresponds to the increasing α. At large group indices the shift of the lateral boundary of the second hole 2.x leads to stronger reflections as boundary 1.x in both waveguides, which stresses the importance of the second row in PhC slow light waveguides. The reflection from the boundary shift of the third hole is vanishing as can be seen in Fig. 3(b).

 

Fig. 3. The normalized reflectivity α as a function of the group index in the (a) W0.8 and (b) W1 waveguides. The boundaries parallel to the direction of propagation are shown in red color and boundaries orthogonal to direction of propagation with blue color. The labels of the curves refer to the boundary labels in Fig. 1.

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The anticrossing point is important to take into account for the experimental verifications of group velocity dependence. Close to the anticrossing point the scattered intensity may strongly differ from inverse group velocity for vertical scattering or inverse group velocity squared for backscattering. At the same time line-defect waveguide modes are intrinsically built by anticrossing of two modes [3]. Thus the mode profile variation should be investigated carefully for the correct experimental demonstration.

There is strong reflection from the boundary 1.z, closest to the channel waveguide of W0.8 waveguide. In W1 waveguide the α factor is comparable for first and second row of holes as a comparable fraction of the total field amplitude is located there. The concentration of the field in W0.8 waveguide, which is unnecessary to achieve slow light properties, leads to significant increase in backscattering intensity. The reflection amplitude from the boundary 1.z is approximately 4 times stronger than from the other boundaries. If the boundaries are arbitrarily shifted then overall reflection intensity can be presented as intensity sum from all boundaries, what makes the contribution from boundary 1.z 16 times stronger as from other boundaries. From the picture the field at the boundary 1.x looks high for the W0.8 waveguide. Therefore, at first glance, it may be confusing that a shift of this boundary causes a much smaller effect on reflection as compared to a shift of the 1.z boundary. This fact can be explained when polarization is analyzed in detail. In the W0.8 waveguide the electric field turns out to be polarized normal to the 1.x boundary, on the other hand in the W1 waveguide it is mostly polarized parallel to that boundary. Due to the normal polarization in the W0.8 case and continuity of the dielectric displacement, the electric field is high inside the hole region but low inside in the material of the waveguide. Thus the shift of the 1.x boundary in W0.8 and W1 waveguide can lead to comparable effect.

4. Discussion

4.1 Effect of field concentration

The presented results demonstrate that it is possible to decrease the backscattering intensity per disorder (i.e., the alpha factor) without losing functionality such as slow light and vanishing dispersion. The influence of light concentration was already presented for the vertical scattering of non slow light modes in PhCs by Gerace and Andreani [32]. Here we demonstrate that the same approach is possible for slow light modes. The concentration of the field on the boundaries of the holes should be avoided. Following this concept it is interesting to compare photonic crystal coupled cavities waveguides [33–35] with line-defect waveguides on the basis of equal group velocity and disorder. The group velocity can be calculated as [36]:

where power flow through waveguide cross section is taken, and W_E and W_M are the energies of the electric and magnetic field correspondingly in the unit cell of length Λ. Both types of waveguides are taken with the same power flows and group velocities. Thus the average energy per unit length should be equal for coupled cavity and line-defect waveguides. We will consider a coupled cavity waveguide with periodicity Λ = ma , where m is an integer. In the extreme case the field energy is almost entirely concentrated in the center of the cavity, then the intensity of the field is m times larger there than in the line-defect waveguide. That leads to m times larger reflection amplitude at the defect due to the fact that reflection is proportional to electric field squared r~E ² Ref. [19]. Taking into account that a coupled cavity waveguide has m times less scattering points, it follows from the intensity sum for the same device length Na:

where R_CC and R_LD are average reflection intensities of coupled cavity waveguide and line-defect waveguide correspondingly. In the extreme case the coupled cavities waveguides will have approximately m time stronger reflection for the same disorder. The concentration of the field in the coupled cavity waveguide can be reduced by choosing higher order modes of the cavities or increasing coupling strength between the cavities. But any field concentration exceeding that in a line-defect waveguide will lead to larger scattering losses. Thus, to decrease the reflection, the concentration of the field along the waveguide should be avoided. Another possibility for reducing backscattering is the dynamic photonics [37] where light is slowed down without spatial pulse compression and intensity increase. Field concentration is also avoided in this case and thus less backscattering intensity is expected.

4.2 Length limitations

The start of the localization phenomena can be estimated by the mean free path calculation:

where R_a is the reflection intensity at a single disordered unit cell. If the shifts of all boundaries are uncorrelated, α_a is the square root of the sum of the squared factors α ² from all boundaries of a periodical unit cell and thus the normalized amplitude-reflectivity coefficient for the random shift of all boundaries in the unit cell. The uncorrelated boundary shift is similar in our case to hole boundary roughness approach described in [17] with correlation length in the order of a hole side length 0.5a. Disorder can be also described as a statistical variation of the hole radius or hole position [32,38], which will lead in our case to a correlated shift of the hole boundaries. This approach is not considered in this paper but would lead to comparable reflection amplitudes.

We have added up reflection intensities from all boundaries of first two rows of holes on both sides of the channel, assuming the third and other rows of holes have vanishing effect, for the reason of low field amplitude. For the W1 waveguide the coefficient α_a can be estimated as 0.3 at group index 100. We assume that this level of normalized reflectivity can be obtained even for larger indices, for example by shifting the anti-crossing point for different group velocities. For an approximation we have set then the α_a as a constant and calculated mean free path lengths presented in Fig. 4 for different standard deviation of boundary shifts and different group indices. For the length calculation lattice constant of 0.4 μm is assumed. Three main standard deviations are presented. Disorder levels of 5 nm standard deviation presents the conventionally available precision of the current technology [38,39]. The 1nm precision is the state of the art manufacturing technology [16,40]. The next technology step in E-beam and deep UV lithography may allow better precision, though the silicon lattice constant will be reached already at 0.5 nm. For comparison reasons the disorder level of 0.3 nm is also presented. An important characteristic of the slow light component is its time delay. We present in the Fig. 4 with dashed lines the curves for time delay 10 ps and 1 ns. Time delay of 10 ps is the approximate time delay needed to switch a Mach-Zehnder interferometer with refractive index variation in the order of Δn = 0.001, which is the level achieved with current tuning approaches [14,41,42]. A time delay of 1ns opens new application of slow light as optical buffers [4,5], however, will only be feasible at very sophisticated manufacturing precisions.

It should be mentioned that the presented estimation takes into account no attenuation or gain. This estimation can be considered as referring to a slow light waveguide where losses are compensated by an additional amplification mechanism. At the same time in real slab structures there is strong attenuation already due to the vertical scattering. The attenuation makes the second order backscattering weaker in comparison to the original wave due to the longer propagation distance. Thus the phase distortion will be smaller in real structures, what will allow longer slow light waveguides though at the cost of increasing signal attenuation.

 

Fig. 4. Mean free path of the slow light waveguides as a function of group index for the normalized reflectivity α_a = 0.3 and different levels of disorder. The lattice constant is set to a typical value of a = 0.4 μm. The dashed lines represent the lengths where constant time delay 10 ps and 1 ns will be obtained.

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5. Conclusion

We presented an investigation of backscattering in slow light line-defect waveguides. It is demonstrated that by a reduction of the field intensity at the hole's boundary the backscattering can be decreased at the same group velocity. Thus designs are recommended which avoid field concentration on boundaries of slow light structures. The same consideration favors slow light line-defect waveguides over photonic crystal coupled cavity waveguides. The reflection intensity is strongly influenced by the modes anticrossing in line-defect waveguides. Thus the experimental demonstration of the one over group velocity squared dependency will fail close to the anticrossing point which is generally present in the considered line-defect waveguides.

The length limitations of the slow light devices are discussed. The backscattering in slow light waveguides does not impose an additional loss mechanism but sets length limitations due to localization effects. The application of slow light to modulators and switches is possible with current manufacturing technology. On the other hand it is argued that the current technological stage can not allow slow light structures in 1mm length for 1ns time delay without significant localization phenomena.