1. Introduction

The use of multiple spatial modes is gaining interest for increasing the throughput of optical information processing systems [1]. A basic component for integrated systems is a waveguide bend that can simultaneously redirect multiple spatial modes with near total transmission for each mode and low crosstalk between the modes. It is possible to use the self-collimation effect to achieve bending that preserves modal content [2, 3]. However, integrated optics is typically based on waveguide structures. Recent work on transformation optics approaches has achieved a waveguide bend structure with high transmission and low crosstalk, but the device has a spatial extent of many wavelengths [4].

Photonic crystal structures have been widely used for creating very compact integrated optical components. In integrated 2D photonic crystal systems, the design of bends for single-mode waveguides has been extensively studied both theoretically [5–9] and experimentally [10–15]. In this paper we present the design of a bend for a photonic crystal waveguide supporting three modes with nearly total transmission and almost no crosstalk. The physical size of the bend is less than 3 wavelengths long on a side.

2. The optimized bend structure

Figure 1 shows the optimized device structure, consisting of an input waveguide on the left and an output waveguide on the bottom, highlighted in blue. The structure is embedded within a background photonic crystal of silicon rods (refractive index n = 3.4) of radius 0.2a in air (n = 1) on a square lattice of lattice constant a. The waveguides are formed by removing three rows of rods and support three modes. The dispersion relations of the three modes are shown in Fig. 2. Also shown in Fig. 2 are the out-of-plane electric field profiles for the three modes at the operating frequency.

 

Fig. 1 Schematic of the optimized bend structure. The blue highlighted regions show the waveguides, and the orange highlighted region shows the bend region where search and optimization were performed. Solid gray circles indicate dielectric rods (refractive index 3.4) surrounded by air. Unlabeled rods have radius 0.2a where a is the lattice constant. The table on the right lists the radii of the labeled rods. The dashed green diagonal line shows the plane of mirror symmetry.

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Fig. 2 Band structure of the waveguide modes within the bandgap of the background photonic crystal. The out-of-plane electric field profiles of the three modes at the operating frequency are shown on the right. The profiles are taken along the edge of the waveguide unit cell and are on the same horizontal scale as the cell shown at the bottom right.

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The bend region is highlighted in orange in Fig. 1 and consists of rods centered on lattice sites but with varying radii listed in the table on the right of Fig. 1. Although the presented design is an idealized 2D structure, there have been recent efforts to implement such designs using effective index models in 3D for planar fabrication [16].

The structure in Fig. 1 has a mirror symmetry plane along the diagonal dashed line. In general, in our design we considered only structures with such a mirror symmetry. We enforced such a constraint since single-mode photonic crystal waveguide bends with 100% transmission coefficients all have such a symmetry [5]. In [5], it was argued that a single-mode photonic crystal waveguide bend can be described by a resonator model, with a resonance located at the bend region coupled to the two waveguides. Such a resonator model supports 100% transmission only when the resonance couples to the two waveguides with the same strength. The most straightforward way to implement such a symmetric resonator model is to have a structure with mirror symmetry. For multi-mode waveguide bends, there is no strict theoretical basis for enforcing such a symmetry; we simply carried over the intuition that was developed from the case of a single-mode waveguide bend. The success in obtaining a functional structure in such a restricted structural space provides some support of this intuition.

We further restricted the design space by considering only structures with rods centered on the 7×7 possible lattice sites within the bend region. It was assumed that all rods have the same refractive index as the rods in the background photonic crystal, but the radius of the rods in the bend region was allowed to vary. Combined with the mirror symmetry constraint, only 28 of the 49 rods in the bend region were free parameters for design.

3. The design process

Within the design space, we searched for structures with the desired performance by using an aperiodic design methodology involving combinatorial search followed by continuous parameter adjustment. The process was discussed in more detail in Refs. [17–19]. We will briefly review the process here, focusing primarily on those aspects that are specific to our current design.

All bend structures were evaluated using the following error metric:

The design process proceeded in two phases. In the first phase, a combinatorial search was performed where every possible combination of rods (presence or absence) on the lattice sites in the bend region was considered, requiring evaluation of 2²⁸ possible structures. In this phase, the rods considered were identical to those in the background photonic crystal (with radius 0.2a). Any structures encountered in the combinatorial search that showed promising behavior (J < 0.05) were considered initial candidates for further optimization.

In the second phase, the initial candidates were fine tuned by adjusting the rod radii in the bend region using a simple gradient descent method to minimize the error metric J in Eq. (1). The gradient of J with respect to rod radius is straightforward to calculate by applying the adjoint variable method throughout the computational process [20,21]. For the structure shown, the second phase reduced the initial value of J = 0.0155 to a final value of J = 0.000798.

4. Numerical methods

In order to efficiently simulate all 2²⁸ possible rod configurations, we used an efficient simulation method tailored to the particular class of structures being analyzed [19]. The Dirichlet-to-Neumann (DtN) map was used to model each unique type of unit cell [22]. Because the cells in the structures we considered contain only circular inclusions, the DtN maps were computed using cylindrical wave expansions, which provided sufficient accuracy with only 5 field samples per unit cell edge. By enforcing continuity constraints on the fields of adjacent cells, a relatively small linear system Ax = b is formed. Here x represents the fields on the boundaries of the unit cells, b represents the incident wave source, and A is the system matrix formed from the DtN maps. Once a single structure is solved by computing A⁻¹, subsequent modifications such as addition or removal of a single rod or changing of a rod radius, can be accounted for straightforwardly by computing the modified A⁻¹ using a low rank update [18]. We performed the entire combinatorial search over 2²⁸ structures within 12 hours using 1024 cores on the SDSC Trestles cluster.

5. Discussion

The transmission spectra of the bend structure shown in Fig. 1 for each of the three modes is shown in Fig. 3. At the design frequency of 0.4 × 2πc/a, the transmission coefficient for all three modes exceeds 98%. The transmission peaks have a sizable relative bandwidth of 1% at a 95% transmission threshold. For operation at λ = 1.55 μm, the bend region has dimensions of 4.34 μm on a side, and a bandwidth of 15 nm. Because the bend structure is lossless, the high transmission coefficients necessarily imply that very little power is reflected or converted into other waveguide modes. For the device shown, the cross-talk parameters (transmission coefficients into other modes) are all less than 1%.

 

Fig. 3 Transmission spectra for each of the three waveguide modes shown in Fig. 2 for the structure shown in Fig. 1. The inset shows details of the transmission peaks.

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Representative field profiles of the device in operation at the center frequency are shown in Fig. 4. The incident mode enters from the left and exits from the bottom. We indeed see that the bend achieves very high transmission while preserving the modal content. Due to the mirror symmetry of the device, the transmission behavior in the reverse direction is identical.

 

Fig. 4 Representative field patterns (out-of-plane electric field) for each waveguide mode of the optimized structure shown in Fig. 1 at the operating frequency of ω = 0.4 × 2πc/a.

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In our first combinatorial search phase where we evaluated 2²⁸ structures, with all rods in the 7 × 7 bend region having the same radius, only 10 structures with J < 0.05 were found. Using a smaller 6 × 6 bend region did not produce any promising candidates. It is clear that for the relatively small search space shown in Fig. 1, a randomized search procedure would probably not encounter any working device, since the probability of encountering such a device is very low (∼ 4×10⁻⁸ for the 7×7 bend region). The use of heuristic search procedures such as simulated annealing is unlikely to significantly improve the likelihood of encountering a working device due to the very low correlation in transmission properties between devices differing by adding or removing even a single rod. Thus, an exhaustive search is actually practical for device design for this class of structures.

The six digits of accuracy presented in Fig. 1 is not actually required for a functioning device. Practical fabrication tolerances limit the precision of the rod radii to two or three significant digits. In Fig. 5 we show the transmission spectra for devices with hole radii rounded to three (dashed lines) and two (solid lines) significant digits. In both cases, the transmission coefficient at the target frequency remains above 98%, suggesting that the design is robust to perturbations in radius.

 

Fig. 5 Transmission spectra as in Fig. 3, except for the structure in Fig. 1 with rod radii rounded to three (dashed lines) and two (solid lines) significant digits.

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The design method is not limited to structures of dielectric rods in air. We have also applied the same method to design an analogous structure in a system with air holes surrounded by dielectric. We consider the TE polarization with an out-of-plane magnetic field. In Fig. 6(a), an optimized structure is shown. The structure is embedded in a background photonic crystal of air holes (n = 1, r = 0.4a) in a dielectric (n = 3.481) background. The holes in the bend region after optimization have the radii listed in the table. The corresponding transmission coefficient for each of the three modes is shown in Fig. 6(b). All three modes achieve greater than 98% transmission at the design frequency of 0.268 × 2πc/a.

 

Fig. 6 (a) Schematic of an alternate optimized bend structure. Solid white circles indicate air holes (refractive index 1) surrounded by dielectric (refractive index (3.481). Unlabeled rods have radius 0.4a where a is the lattice constant. The table on the right lists the radii of the labeled rods. (b) Transmission spectra of the bend for each of the three modes.

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Finally, we comment briefly on the size of the device with respect to fundamental limits. The resonator model of [5] may be used to estimate a lower limit on the size of the bend region required. The ability to pass three spatial modes with total transmission at a single frequency implies the presence of at least three distinct resonances in the bend region. Assuming a typical estimate of one square wavelength per resonant mode, the bend region would have to be at least 3λ² in area. The bend region we have presented is approximately 8λ² in area, making it near the theoretical optimum. Note that simply having three resonances is not sufficient to guarantee total transmission of three modes since the field patterns of the resonances must also be compatible with the waveguide spatial modes, likely increasing the minimum device area required.

6. Conclusion

We have described the design of a multi-mode photonic crystal waveguide bend structure exhibiting over 98% transmission, and less than 1% reflection and crosstalk. The structure was shown to be robust to fabrication error with respect to deviations in the rod radii. In addition to a structure of dielectric rods in air, we also demonstrated a second device implemented with air holes in a dielectric background, which may be more experimentally relevant. Finally, we note that the size of these structures is very near the fundamental size limit derived from heuristic arguments.