1. Introduction

In recent years, there has been a growing interest in slow light planar photonic crystal (PhC) waveguides both in the context of optical delay lines and more generally for nonlinear optics [1–3]. In particular, it is expected that light-matter interaction is enhanced in the slow light regime, which should translate into a reduction of either the power or the physical length needed to observe the same (linear and nonlinear) effects as in the fast light regime [3–6]. However, the slow light regime is usually accompanied by a high group velocity dispersion (GVD), which typically distorts the optical pulses, thereby compromising the benefit of slow light [7]. Hence, it is important to realize slow-light structures with tailored nearly dispersionless properties. This has been recently achieved by engineering the geometry of the PhC waveguide [8–12], resulting in sophisticated designs that typically require nanometer-scale technological precision.

Optofluidics is a new branch within photonics which attempts to unify concepts from optics and microfluidics [13, 14]. In particular, the combination of PhC structures and microfluids has attracted some attention as a means to tune the optical properties of PhC devices [15]. Due to their intrinsic porous nature, PhCs present a large fraction of air that can be readily infiltrated with a liquid having highly tunable optical properties, such as liquid crystals [16–18]. In addition, selective liquid infiltration within individual air pores of a planar PhC lattice has been recently investigated, extending the number of opportunities associated to this optofluidic platform [19–23]. This offers the potential for realizing integrated microphotonic devices and circuits which could be (re)configured by simply changing the liquid and/ or the pattern of the infiltrated area within the PhC lattice [19, 20, 23–25].

In this paper we present an optofluidic-based method for achieving very small dispersion slow light PhC waveguides. By contrast with earlier approaches that rely on the nanometer–scale engineering of the PhC lattice geometry, this technique exploits the selective liquid infiltration within an otherwise standard PhC waveguide. We show numerically that the liquid infiltration gives an extra free parameter in the waveguide design, which allows fine tuning of its group velocity and dispersion. In particular, a large range of group indices (between 25 and 110 depending on the PhC hole radii) can be achieved with minimal dispersion from a single PhC waveguide geometry, simply by changing the index of the infused liquid. We also study the tolerance of this approach to the variations in the initial PhC waveguide geometry. We show numerically that within a 10% deviation in the hole radius, it is still possible to achieve the desirable nearly dispersionless slow light regime by freely choosing the appropriate liquid refractive index a posteriori. This relaxes the constraint on the PhC waveguide fabrication as compared to previous structural-based methods for slow light dispersion engineering, and simultaneously offers the potential to realize (re)configurable slow PhC waveguides.

2. Nearly dispersionless slow light in photonic crystal waveguides using liquid infiltration

Nearly dispersionless slow light PhC waveguides have been achieved by modifying the waveguide width [8], changing the hole size [9] or position [10, 11] around the PhC waveguide, or using ring-shaped holes [12]. Most of these approaches exploit the coupling between the modes supported by the PhC waveguide and the fact that this coupling can be adjusted by carefully changing the waveguide geometry; for instance, each mode has a distinct sensitivity to the PhC holes that directly border the center of the PhC waveguide. For appropriate designs, the existence of a flat band slow light, with both reduced group velocity and dispersion, has been demonstrated. However, all of these methods depend on the technological capability to realize a specific design with a high accuracy. In particular, it is quite difficult to control the hole size within the PhC lattice accurately and reproducibly. In addition, optimizing the position of two rows of holes to obtain a desired dispersion requires the scanning of several PhC parameters (hole size and position) during the fabrication. We show below that these disadvantages can be circumvented by adopting a technique based on the local infiltration of holes within the PhC lattice rather than careful nanometer-scale modification of the PhC geometry.

The goal of the present work is to achieve a PhC waveguide that has a large slow down factor over a large bandwidth with a reduced dispersion. We focus on a planar W09 PhC waveguide etched onto a 220nm-thick silicon membrane suspended in air, as displayed on the inset of Fig. 1(a). The W09 waveguide is formed by omitting a row of air holes in the Γ-K direction (W1), and shifting the two PhC regions closer to one another resulting in a PhC waveguide width 0.9 times that of a W1. Calculations are performed using a 3D plane-wave expansion method (PWM) for a triangular PhC lattice that consists of air holes with a periodicity of a=420 nm and a radius of 126 nm (0.3a). We restrict the calculations to polarisation states in the plane of the slab, denoted as TE and consider the refractive index of the silicon slab equal to 3.52.

To illustrate that the infiltration-based approach can produce nearly dispersionless slow light, we first compare the band structures associated to a conventional and an infiltrated W09 waveguide. Since the previous studies have highlighted the strong influence of the first two rows of holes that directly surround the PhC waveguide, we calculate the dispersion associated to a W09 having its first two rows infiltrated with a fluid of refractive index equal to 1.85 (see the inset of Fig. 1(b)). The results are displayed on Fig. 1(a-b). As expected from the increase in the effective refractive index associated to the infiltration, the even fundamental mode sustained by the infiltrated W09 is shifted towards lower frequencies as compared to the standard W09.

The fundamental mode dispersion of the infiltrated W09 also exhibits a nearly constant slope between 0.37<k_z<0.46. We plot on Figures 1(c-d) the group index n_g and the group velocity dispersion (GVD), β₂, of the even fundamental mode sustained by the standard and the infiltrated W09 waveguide. The values of n_g and β₂ are calculated from the first and second derivative of the band structures in Fig. 1(a-b), respectively. Unlike the standard W09 case, the dispersion of the infiltrated W09 exhibits a “flat-band slow light” window (between 1560 nm and 1570 nm) where the group index is quasi-constant and equal to 31, with an associated β₂ reduced from 2.5×10^-20s/m² (for the standard W09) to -1×10^-22s/m², i.e. by more than two orders of magnitude. Moreover, β₂ is nearly constant over the flatband region, which indicates that the associated third-order dispersion is also very low. This is important because third order dispersion can also induce pulse distortion in the slow light regime having a low β₂ value [7]. These results thus show that the infiltration approach provides an effective and alternative technique to produce nearly dispersionless slow light over a large bandwidth (10nm) without requiring nanometer-scale design engineering.

 

Fig. 1. (a–b) TE-polarization band structures of a conventional W09 PhC waveguide (a) and a W09 that has its first two rows infiltrated with a liquid having a refractive index of 1.85 (b). On the insets, the respective PhC waveguide geometries (with hole radii r/a =0.3) are shown. (c–d) Associated group index (ng, green) and GVD (β2 blue) of the fundamental W09 mode in (a–b).

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3. Dispersion engineering of infiltrated PhC waveguides: control of the group velocity and tolerances of the method

We next show that the group velocity of the very small dispersion slow light window can be controlled by tuning the liquid refractive index. For that, we calculate the dispersion of the even fundamental mode when varying the refractive index of the liquid infiltrated in the first two rows of the W09 waveguide between 1.75 and 1.95 (see Fig. 2(a) and 2(b)).

Figure 2(b) shows that by choosing the liquid refractive index, different regimes of slow light (with v_g between c/25 and c/35) can be achieved with a reasonable bandwidth. As the fluid refractive index is increased, the bandwidth associated to the flatband slow light region is enlarged, while the corresponding group index decreases. These results suggest that there is a trade-off between the achieved group index and the bandwidth over which this index is constant, with an optimum in terms of the maximum group index-bandwidth (n_g × BW) product that is obtained here for a fluid refractive index equal to 1.85.

This group index-bandwidth trade-off is general to all dispersion engineering approaches and the associated optimum generally depends on the PhC nominal parameters [11]. Similarly, here, the optimum in the fluid refractive index will most likely vary with the W09 nominal characteristics, such as the PhC hole diameter. It is therefore important to study the tolerance of the infiltration based approach for dispersion engineering by investigating the variations in the W09 design that still allow us to observe a flat band slow light regime. We calculate the n_g×BW product for a large range of fluid refractive indices and hole radii (+/-10% variation) as the hole size is the most difficult parameter to control experimentally [11].

The results are plotted on Fig. 3. The bandwidth BW is calculated by considering the group index constant within ±5% (Fig. 3(a)) and ±1% (Fig. 3(b)). These results show that a high n_g×BW product is achievable for a wide range of hole radii and fluid refractive indices, with the optimum fluid refractive index decreasing for larger hole radii.

 

Fig. 2. Even fundamental mode dispersion (bandstructure (a) and group index (b)) of the infiltrated W09 (with r/a =0.3) for various fluid refractive indices between 1.75 and 1.95. Only the first two rows are infiltrated.

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Fig. 3. Maps of the group index–bandwidth product that is achieved for the infiltrated W09 for various liquid refractive indices and hole radii. The bandwidth is calculated by considering the group index constant within ± 5% (a) and ± 1% (b). The superimposed solid lines curves correspond to constant values of the group index.

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Two main points can be derived from these results. First, they show that the infiltration based approach is very robust, because the deviations in hole radii from the PhC parameters targeted during the fabrication can be compensated for by using a different fluid index a posteriori, thereby making this specific dispersion engineering technique more tolerant upon the fabrication inaccuracies. As an example, a flatband slow light can be achieved for a nominal r/a=0.3 by using a fluid refractive index of 1.85; however if the actual hole radius of the fabricated PhC structure deviates to r/a=0.31, then, choosing a fluid refractive index of 1.8 can still provide the desirable flat band slow light regime. The second point is that for a given hole radius, the group velocity can be controlled by choosing the fluid refractive index to infiltrate, without degrading much the slow light bandwidth. For instance, a W09 with hole radii of 0.33 can be engineered to exhibit very small dispersion slow light with a group velocity ranging between c/45 and c/90. This factor of two in the group velocity can be achieved for almost all the hole radii displayed on Fig. 3, by simply adapting the fluid refractive index range for each case.

The group index of the slow light flatband can be even more widely controlled by scanning the liquid refractive index and the hole radius. Figure 4 shows four operating points corresponding to different (hole radius, fluid refractive index) combinations taken from Fig. 3(b) that can provide very small dispersion slow light over a bandwidth between 3nm and 7nm and with a group velocity ranging between c/40 and c/110, i.e. almost a factor 3.

 

Fig. 4. Group index of the even fundamental mode of the infiltrated W09 PhC with (a) r/a=0.31, and nf=1.8 (b) r/a=0.32, and nf=1.8 (c) r/a=0.33, and nf=1.75 (d) r/a=0.34, and nf=1.7. These operating points are highlighted with white dots on Fig. 3.

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4. Discussion

The previous dispersion engineering works relied on the optimization of the PhC lattice geometry, where high accuracy is essential as even nanometre-scale deviations typically lead to a significant degradation in the dispersion properties. By contrast, the infiltration-based approach can achieve a wide range of slow light regimes with different group velocities and reduced dispersion from a single PhC geometry, namely a W09 waveguide. In addition, this approach remains valid for various PhC hole radii, which is the most difficult parameter to target experimentally with a high accuracy. The liquid infiltration therefore offers an additional and free parameter (which is independent on the fabrication) for achieving a desired modal dispersion. In addition, the post-process nature of this approach -the infiltration step is performed after the waveguide fabrication–allows it to be adapted depending on the actual PhC structure produced by the fabrication, while the microfluidic aspect offers the potential for (re)configuring the slow light waveguide at will.

An important question is the accuracy of the infiltration method itself. In that respect, various selective infiltration techniques of planar PhC structures have been experimentally demonstrated [19–23,26] using for instance an integrated microfluidic circuit [19], an actuated microtip [20,22,23] or micropipette [21] whose size is comparable to the holes to infiltrate. The local filling of very small PhC regions could be achieved with a high precision and a good reproducibility. In addition, the mobile nature of liquids has been shown to provide a way to reconfigure the infiltrated structure, simply by removing the liquid out of the PhC lattice [19,20,23], which adds another degree of flexibility in the engineering of the PhC waveguides. The range of liquids that are available for infiltration have a refractive index that spans most of the refractive index window investigated in Fig. 3, namely between 1.3 for water based solution to 1.5 for silica oil matching fluids and to above 1.8 for Cargille liquids. Liquid crystals could also be used, providing the basis for externally tuning the group velocity of the waveguide, while maintaining the low dispersion characteristic.

Another relevant problem to slow light waveguides is the capability to couple light efficiently into the slow light regime. One method to improve the light insertion from conventional ridge waveguides into slow light PhC waveguides is to add short intermediate “fast” PhC waveguides (coupler sections) [27]. Using 3D finite difference Time domain calculation (FDTD), we investigate the validity of this approach in the case of the slow light PhC waveguide engineered through infiltration. Figure 5 shows the calculated transmission through a 28 period long infiltrated W09 (n_liquid= 1.85) that is surrounded by two coupler sections consisting of an infiltrated W09 with a stretched period (a_c=480nm, for a 7 period length) and two silicon nanowires on both sides see Fig. 5 (a). The group indices calculated by PWM and inferred from the FDTD simulations are also displayed on Fig. 5 (b). The FDTD n_g-curve is 6nm red-shifted but has a similar shape, which confirms the nearly dispersionless behaviour of the infiltrated PhC waveguide. A high transmission is obtained for a wide bandwidth that spans across the slow light flat band window when the coupler sections are added (blue curve) as compared to the poor coupling obtained when inserting light directly from the nanowires to the slow light W09 (green curve). This attests that infiltrated dipersionless PhC waveguides behave like their geometrical counterparts [10] and do not suffer from unexpected high losses. In addition, it shows that an efficient coupling into the slow light regime of the infiltrated PhC waveguides can be achieved using the coupler based approach demonstrated in Ref 27.

 

Fig. 5. (a) Schematic of the waveguide with a_c=480nm and a=420nm (b) Transmission spectra obtained through FDTD simulation for a 28 period infiltrated W09 PhC waveguide surrounded by two ridge waveguides (green) and two coupler sections and ridge waveguides (blue). The group index dispersion calculated through PWM (red) and inferred from the FDTD calculations (blue circles) is also plotted.

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5. Conclusion

A novel technique based on the selective liquid infiltration of PhC waveguides to produce nearly dispersionless slow light over a substantial bandwidth has been discussed. We numerically demonstrated that this approach allows one to control the group velocity (between c/20 to c/110 depending on the PhC hole radius) from a single PhC waveguide design, simply by choosing the index of the liquid to infiltrate. In addition, we showed that this post-process microfluidic method has the potential for realizing (re)configurable slow light waveguides, and is tolerant to deviations in the PhC parameters such as the hole size, which relaxes the constraint on the PhC fabrication accuracy as compared to previous structural-based methods for slow light dispersion engineering.