## Abstract

We report new results about the improvement of delay-bandwidth product in photonic crystal slow light waveguides. Previous studies have obtained large delay-bandwidth product at the price of small average group index. It is pointed out here that the radius and the distance between the two boundary rows of holes have a key contribution for delay-bandwidth product. We show the possibility of improving this factor of merit meanwhile maintaining the same group index. We succeed in improving normal delay-bandwidth product from 0.15 to 0.35, keeping at the same time the group index unchanged at high value of 90. This optimization approach may be applicable for previous flat band slow light devices.

© 2010 OSA

## 1. Introduction

Slow light in nanophotonics has raised for some years a strong interest because it is seen as a key technology for the realization of future on-chip optical networks mixing electronics and optics [1]. Photonic crystal waveguide (PCW) is an effective way to generate slow light in micro-space at room temperature. Slow light in PCW has been observed in several experiments [2,3]. However, the bandwidth is extremely narrow and huge group velocity dispersion (GVD) effects accompany when light is slowed down [3,4].

The concept of delay-bandwidth Product (DBP) is a good indication for the highest slow light capacity that the device potentially provides [4]. DBP is defined as the product of the time group delay Δt and the bandwidth δω in slow light:

It can be seen from the above equation that due to the extremely narrow bandwidth, conventional PCWs have low slow light capacity (low DBP value). To overcome this problem, various solutions about “flat band” slow light (linear dispersion curve) have been proposed in previous works. These approaches rely on the fact that slow wave modes tend to spread perpendicularly to the direction of light propagation, and are thus sensitive to structural changes of the waveguide geometry. Some of these solutions rely on special photonic crystal (PC) lattices like annular photonic crystals (PCs) [5,6]. The other approaches rely on the geometrical adjustment of conventional PCW, i.e. by shifting the bordering of waveguide which produce unconventional U-like slow light [7], changing the radius size [8] or position [16,17] of the central two rows of holes which will cause flat band slow light.

These approaches [5–12,16,17] have greatly widened the bandwidth of slow light compared with conventional PCWs. However, if under the same group index (n_{G}) tolerance ± 10% (n_{G} variation with respect to its mean n_{G} value), most of normalized delay-bandwidth product (NDBP) values in these approaches are below 0.3 [5,6,8,11,12]. NDBP values larger than 0.3 are only obtained in chirped photonic crystal (PC) coupled waveguides [9,10] characterized by a complicated design and huge losses. Moreover, these large NDBP values (NDBP>0.3) are obtained at the price of small group index (n_{G}<40) [7,9,10]. For example, in Ref. 7, a new type of U-like slow light has been introduced, and the maximum NDBP value was 0.31. However, the corresponding n_{G} was only 11. At large n_{G} value of 211, the corresponding NDBP was only 0.14.

To effectively enlarge NDBP value, it is important to improve the bandwidth, but more importantly, this improvement must maintain large n_{G} at the same time. Aiming at this objective, this article proposes a step-by-step optimization procedure for PCW. All possible geometry parameters are optimized, including the slab thickness, the hole radius within the full PC lattice, the lattice distortion, the waveguide width, and the shifts of the two boundary rows of holes. The NDBP value of previous U-like slow light waveguide has been successfully improved from 0.15 to 0.35, while keeping large value n_{G} unchanged at 90.

Moreover, our results reveal the relationship between geometrical parameters and slow light performance DBP, pointed out that the radius of holes R and the lateral interval distance between two successive rows of holes are vital factors which influence DBP. By properly arranging R and dy, it is possible to improve the DBP value of PC waveguides.

## 2. Previous U-dispersion PCW

The proposed optimization approach is presented here, but it is important first to emphasize the concept of NDBP which characterizes the compromise between the light slowing down factor and the bandwidth [4]. NDBP is given by:

where <n_{G}> is the obtained average group index calculated in bandwidth Δω, and ω

_{0}the central normalized frequency (ω = a/λ, with a the PC lattice parameter). The critical point is then to clarify the definition of bandwidth, i.e. explicating what is considered as flat n

_{G}. This is done by fixing a given tolerable flat ratio µ defined by [7]:Another critical issue before the optimization is at which n

_{G}we should work at. Of course, the larger n

_{G}, the better slow light performances. However, optical losses are known to scale dramatically with n

_{G}. Some experimental results suggest losses have <n

_{G}>

^{2}dependence in the back-scattering propagation regime [18]. Very large group indices (n

_{G}>100) raise serious problems in terms of losses [13]. To make balance of both sides, large group index of 90 is fixed here for the following sections.

As a main example, we come back to our previous study of “U-dispersion” slow light [7]. A U-dispersion PC waveguide with δx = 0.272a at n_{G} = 90 is chosen here as a starting point, defined as “waveguide A” and depicted in Fig. 1
.

For all the waveguides proposed in this paper, a central silicon layer of 0.65a, top cladding of air and SiO_{2} as bottom cladding, based on the widely used silicon on insulator technology, are introduced. Owing to the indices of Si (3.45) and SiO_{2} (1.45), the slab PCW geometry is reduced using the effective index method, with an effective index of 2.98 around the wavelength of 1550nm in TE-like polarization. Plane-wave expansion method is used with MIT Photonic Bands package [19].

Waveguide A is a special slow light device which has a large n_{G} (n_{G} = 90), but a moderately large NDBP (NDBP = 0.15) under the flat ratio of 0.2.

## 3. First waveguide optimization

One problem in waveguide A is that its bands have been curved up due to the shift of bordering holes [7]. This tends to cause the increase of central frequency, and thus the decrease of the NDBP factor of merit.

To remove this drawback, we propose here a new PCW geometry arrangement consisting in shifting the second rows of holes simultaneously when the first rows of holes are shifted (see the inset picture of Fig. 2 b
)). With respect to Ref. 7, it turns out that shifting the second rows of holes brings an additional degree of freedom to engineer the PC waveguide dispersion diagram. We have applied this approach and systematically scanned the δx_{1} and δx_{2} parameters. It was found that with δx_{1} = 0.4a and δx_{2} = 0.2a, the guided band is now curved down, as shown in Fig. 2 a), while keeping the average group index unchanged to n_{G} = 90. We define this new structure as “waveguide B”.

Figure 2 a) shows a comparison between the band curves of waveguide A and waveguide B, respectively. There are two sections of linear band observed but the origins are different: waveguide A introduces an inflection point which makes its band curved up, thus forming the upwards linear band. However, in waveguide B, the simultaneous shift of δx_{2} decreases the degree of the inflection point, thus leading to the downwards linear band. It is obvious that the central frequency of waveguide B is smaller than the one of waveguide A. According to Eq [2], it thus can be inferred that waveguide B has a larger NDBP than waveguide A.

Figure 2 b) shows the corresponding group index curve for waveguide B. The group index curve has been changed from U-like to step-like. Under the same flat ratio µ = 0.2 [7], the NDBP value for waveguide B is now 0.26. If compared with waveguide A, the value of NDBP has been improved by 72%.

## 4. Overall optimization

Results of section 3 have shown the possibility to increase the flat band slow light bandwidth of PCWs by adjusting the two first pairs of rows of holes. To further optimize NDBP, other degrees of freedom are needed.

To serve as a guideline to a broader optimization approach, we develop here a simple model. Suppose a slow light device is working over a certain bandwidth Δω, and all frequencies inside this bandwidth propagate with the same group index (linear band curve). Then with the constant group index, we have:

within the linear band range, we have:where K is the wavenumber and ω is the normalized frequency.Substitute Eqs [4]& [5] into Eq [2], we have:Eq [6] shows two factors that strongly influence NDBP: the wave number range ΔK of the linear band and the central frequency ω_{0}of the linear band. The optimization of geometry parameter can thus follow two main guidelines: a) decrease ω

_{0}, and b) increase ΔK.

#### 4.1 Guideline a): optimization of ω_{0}

The improvement of NDBP from waveguide A to waveguide B partly relies on the decrease of ω_{0}. However, there is still space for a further decrease of ω_{0}. Decreasing the central frequency means increasing the slab effective index. Thus, any parameter changes that cause index increase could be used, including the increase of the slab thickness, increase of the waveguide width, and decrease of the hole radii. As the increase of the slab waveguide thickness implies multi-mode operation, we fix the slab thickness to its maximum single mode values [14] and focus here on the optimization of the air hole radius.

Figure 3 a ) b) c) show the band gaps of the two-dimensional PCs vary under three radii: R = 0.30a, R = 0.23a, and R = 0.12a, respectively. As shown in Fig. 3, decrease of R is responsible for the decrease of the operating frequency, but also on the shrinkage of the PC bandgap size. When R decreases to certain value, the operating frequency is not significantly reduced. However, the optical band gap of PCs has completely closed, e. g, from R = 0.23a to R = 0.12a in Fig. 3. Another feature is that the reduction of radius causes the increase of the available K range, e. g, from R = 0.3a to R = 0.23a in Fig. 3.

The decrease of operating frequency will lead to the decrease of central frequency and the increase of available K range, which is desirable for our design. However, the decrease of band gap size will lead to the band gap closing, which is not welcomed in our design. As a conclusion, the hole radius choice is the result of a compromise between reduction of ω_{0} and enough space for band engineering. R = 0.23a is considered for this in the following, because it provides low central frequency (around 0.24) and at the same time a sufficiently large band gap space for the further engineering of a linear band.

#### 4.2 Guideline b): optimization of ΔK

ΔK has been previously defined as the wavevector range of linear dispersion curve. As it is yet not straightforward to obtain a linear dispersion curve in all conditions, we extend here the definition of ΔK to the available wavevector range δK for dispersion curve optimization. In other words, we try first enlarging the available wavevector range, and then we will optimize lateral rows of holes to obtain a linear dispersion curve hereafter.

According to Ref [15], the band curve of PCWs is made up of two parts: the index-guided modes part and the gap-guided modes part. For low wavenumber, the waveguide modes are dominated by the index guiding mechanism with a low group index (fast light). For large wavenumber, the waveguide modes are dominated by the gap-guided mechanism (slow light). More importantly, large n_{G} is known only obtained in the gap-guided modes part. The strategy targeted here is to enlarge the gap-dominated mode area as much as possible. In this purpose, it is necessary to modify the interaction between the index-dominated guided modes and the gap-dominated modes.

One possible way to achieve this is to squeeze the waveguide modes energy into the lateral lattice, which is indication of enhancing the slow light modes [5]. Figure 4 a ) shows the new geometry PC lattice and possible adjustment of each parameter. Attention should be paid on two of them: dy is the interval distance between two adjacent rows in the lateral direction, i.e, for a conventional W1 waveguide, dy = (√3/2)a, and 2 × dw is the additional distance added to the central waveguide width with respect to 2dy. By increasing the hole density of the new PC lattice, i.e. shrink dy, the optical energy can be pushed out of the waveguide core, thus spreading in the lateral direction. The interaction between the index-dominated modes and the gap-dominated modes is then influenced.

Figure 4 b) shows typical band curves in three adjusted PC lattices. Points P, Q and M in the three curves are marked as the connection points between the gap-dominated modes part and the index-dominated modes part, respectively. The black dash line is the band curve for W1 waveguide, dy = (√3/2)a and dw = 0. Starting from the W1 waveguide, the hole density is increased by reducing dy, i.e. the blue dash dot line for dy = 0.65a and dw = 0. In this way, gap-guided modes are pulled up and index-guided modes are slightly pulled down. The connection point Q is thus pushed to lower wavenumber if compared with point P of W1 waveguide. And it is clearly seen that the available wavenumber range δK has been enlarged in comparison with the result of W1 waveguide. This confirms our inference that reducing dy will cause the increase of gap-dominated mode area. The blue dotted line in Fig. 4 b) has a rather large δK range which would be helpful for increasing DBP value. Yet, a significant fraction of the dispersion curve is now located above the light line. Moreover, the central frequency has been also increased, thus counteracting the DBP increase from previous radius optimization. To fix this, the waveguide width is increased by choosing positive dw values. The red solid line in Fig. 3 b) shows the optimized band curve obtained with dy = 0.78a and dw = 0.18a, respectively. Compared with the W1 waveguide (black dash line), the red line has a slightly lower central frequency and meanwhile has a rather larger δK range, both of which would lead to DBP increase.

As a summary of the above analysis, by increasing the PC lattice density and appropriately adjusting the radius and waveguide width, we succeeded in enlarging the available K range while at the same time maintaining a low central frequency.

The final optimized parameters for the new PC lattice are R = 0.23a, dy = 0.78a, dw = 0.18a, respectively.

#### 4.3 Final step of the optimization

With above results gathered, it is now possible to re-optimize the PCW geometry to produce the desired constant n_{G}, taking full advantage of the new large available K range (δK) and new low central frequency. In accordance with section 3, the shift of the first row and second row (parameters δx_{1} and δx_{2}) have been also added here.

We have achieved a two-dimensional scan of these two parameters (0<δx_{1}<0.5a, −0.5a<δx_{2}<0.5a). Figure 5 a
) shows the optimized results of constant n_{G}. When δx_{1} is fixed at the value of 0.4a and δx_{2} varies from −0.236a to −0.244a, the group index shows step-like curves with constant n_{G} ranging from 90 to 140. As pointed out in Ref [7], the shift of δx_{1} causes inflection point and linear band in the dispersion curve. However, the shift of δx_{2} brings additional controls. The variations of δx_{2} have smoothed the inflection point and slightly adjust the slop of the linear band, which feedback in Fig. 5a) as the variation of group index value.

To compare the waveguide properties with waveguide A and waveguide B, the new waveguide with parameters δx_{1} = 0.4a and δx_{2} = −0.236a is chosen here as “waveguide C”. These three waveguides highlight a constant group index of 90. Its dispersion curve is shown in Fig. 5 b). This structure produces NDBP around 0.35 (under strict flat ratio 0.2). The value of NDBP has been improved 33% if compared with waveguide B.

## 5. Comparison between the three waveguides

To have a better depiction of this improvement, Fig. 6
summaries the group index curves of the three proposed waveguides. The bandwidths over n_{G} = 90 are 2.7nm, 4.6nm, and 6.4nm for waveguide A, waveguide B and waveguide C, respectively. Waveguide A has the minimum flat band because it originates from band-up slow light. Waveguide B presents a moderate flat band because it originates from band-down slow light in conventional PC lattice. Finally, waveguide C produces the maximum flat band due to band-down slow light in the well optimized PC lattice design.

The group velocity dispersion (GVD) parameter is another important issue which controls the signal distorsion during slow wave propagation. It is obtained by:

Figure 7
compares the GVD parameter between the waveguides A, B, and C. It can be seen that waveguide C provides the largest bandwidth range with low GVD, and waveguide A is has the worst GVD performance. If we consider D_{λ} = 10^{5}(λ^{2}/2πc) as ultralow dispersion, which is one order of magnitude smaller than Ref [12], waveguide B and C are still qualified to provide a large bandwidth in such a low GVD constraint. We should note that GVD increases with group index, and all the three waveguide work under large group index (<n_{G}> = 90).

Table 1
shows a comparison of the detailed information of the three proposed waveguides. As a reference, the properties of the simple W1 waveguide are also given. From waveguide A to waveguide B, the NDBP value has increased by 72%. From waveguide A to waveguide C, the NDBP value has increased by 130%. All NDBP values of the three waveguides are operated at <n_{G}> = 90 under strict flat ratio of 0.2, which means the increase of NDBP keeps slow light velocity unchanged, not in the price of reducing <n_{G}>.

Table 2
shows the slow light performances between our best structure and previous published structures. To make this comparison, all bandwidths of constant n_{G} in Table 2 are considered with a tolerance of ± 10% n_{G} variation with respect to its mean n_{G} value (i.e. under the same flat ratio of 0.2). We should notice that the comparison of NDBP in different devices should be performed at the same n_{G} value.

## 6. Conclusion

This paper proposes an overall approach to improve the delay-bandwidth product of slow light photonic crystal waveguides based on an opto-geometrical optimization approach including the adjustment of radius size, slab thickness, waveguide width, PC hole lattice density, and shift of bordering rows of holes. It has been proved that the radius R and the lattice density have important contribution for delay-bandwidth product. By carefully optimizing the shift parameters δx1 and δx2, constant n_{G} over certain bandwidth can be obtained. By appropriately selecting radius R and lateral distance between two successive rows, the delay-bandwidth product can be greatly improved. As a typical example, we succeeded in this paper in improving delay-bandwidth product from 0.15 to 0.35 at a constant group index of 90. Our results show the possibility of improving the delay-bandwidth product meanwhile maintaining a high group index. We believe that previous flat band slow light devices based on conventional PC lattices could be optimized in the same way, and that the proposed method is applicable to other waveguide and material configurations.

## Acknowledgement

Ran Hao is very grateful for the scholarship from the French government.

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