1. Introduction

Waveguide bends are key components in integrated optics devices and circuits that have a strong impact on the integration density. In order to avoid power loss due to radiation [1], the radii of conventional waveguides are much bigger than the wavelength which leads to huge size of conventional integrated optics structures. Additionally, substantial mode conversion is observed in the case of multi-mode waveguides. Several propositions were made for reducing the radiation and enhance the transmission [2–6]. Recently, Photonic Crystal (PhC) waveguides attracted much attention because sharp bends with zero reflection and radiation are theoretically possible for certain frequencies or even over almost the entire band gap of the PhC [5]. The main problems of integrating such PhC waveguide bends in conventional integrated optics waveguides are the coupling of the shallow-etched rib or deeply-etched trench waveguides with PhC waveguides and the high impact of fabrication imperfections.

For sharp 90° bends in high-index contrast waveguides, several approaches were described [7], namely corner mirrors and waveguide resonators [6] as well as PhCs acting as 45 degree corner mirrors [8]. To our knowledge, single mode operation is assumed throughout and this leads to very narrow, deeply etched trench waveguides (typically 200nm width) that are very hard to fabricate. For wider waveguides, multimode operation poses the additional problem of mode conversion that makes the numerical analysis substantially more difficult. The waveguide bend must be then designed in such a way that not only radiation but also mode conversion is avoided. However, by combining an accurate field solver with an efficient numerical optimizer one still may find ultra-compact bend configurations with almost 100% transmission of the fundamental mode, as demonstrated below, where we consider a 400nm wide trench waveguide that exhibits two propagating modes. In addition to the optimization of mirror- and resonator-based sharp bend structures, we introduce an ultra-small local photonic crystal in the bend area. Our simulations show that this local PhC suppresses radiation and reduces problems caused by fabrication imperfections at the same time.

2. Simulation and optimization issues

Figure 1 shows the 3D model of the trench waveguide that will be fabricated on an InP substrate. Note that the wave is weakly confined in vertical direction but strongly confined in the horizontal direction.

 

Fig. 1. Schematic of the 3D structure (left-hand side) and the corresponding 2D effective index structure (right-hand side, inset: Poynting vector at 1.55 µm).

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Since we intend to optimize the bend geometry, we need a highly efficient field solver that should be robust and accurate at the same time. Since brute-force 3D time-domain computations are time consuming and often exhibit discretization noise that disturbs the optimization process, a 2D effective index model [9] is used and solved with the 2D MMP solver contained in the MaX-1 platform. The simplified 2D structure (width 400nm) is shown in Fig. 1 (right-hand side) with effective index n_eff=3.236558 (ε_eff=10.4753), surrounded by air. The waveguide exhibits two TM modes (E field perpendicular to the 2D plane) at the telecommunication wavelength 1.55µm. This wavelength choice is based on the constraints of our research. The propagation constants and field distributions of both modes are easily obtained from the MMP eigenvalue solver [4] of MaX-1.

Pioneering works for ultra-sharp bends were presented by Melloni et al. [11, 12], Manolotou et al. [6], Espinola et al. [7]. These configurations - based on complex physical rules and direct design methods - are difficult to master and limited in many ways. First of all, new structures with enhanced functionality can often not be obtained by physical intuition [15]. Like in the Inverse Design procedure (ID), we optimize the design parameters within some constrains in such a way that the transmission coefficient of the fundamental TM mode is maximized.

For complex optimization problems with many local optima, conventional optimization methods are not favorable. Genetic Algorithms (GA) [12] and Evolutionary Strategies (ES) are therefore widely used for solving such problems [10]. An (n+m) ES with n=15, m=100 and adaptive deviation values was used for obtaining the results shown in the following. Global optima were usually found within less than 25 generations. The parameter search space was limited to compact bend structures because structures with size much larger than the width of the waveguide are not desirable. The fitness function was specified as the transmission coefficient of the fundamental mode. Since high transmission may only be obtained when radiation, reflection, and mode conversion are simultaneously low, this simple fitness definition implicitly minimizes radiation, reflection, and mode conversion.

For the numeric fitness evaluation, a Maxwell solver is required, for example, BPM [2] or FDTD [6]. In this paper, the semi-analytic Multiple Multipole Program (MMP) contained in the MaX-1 platform [4] is used. This solver provides high accuracy with low discretization noise (that would disturb the optimizer). Furthermore, MaX-1 provides all routine required for post-processing, fitness definition, and linkage with the numeric optimizer.

3. Mirror type and resonator type bends

The original sharp 90 degree waveguide bend structure is shown in Fig.1. The wave is incident from the left side while the output port is at the top of the figure (right hand side of Fig.1). In this configuration, the transmission and reflection coefficients of the first and the second mode are T₁=2.34e-3, R₁=0.22, T₂=5.99e-2, R₂=5.99e-2. Obviously, most of the energy is radiated away from the bend, resulting in extremely poor transmission.

When a 45 degree mirror or a 45 degree waveguide piece is introduced at the corner [see Fig. 2(a)], the transmission may be improved [6, 7]. Actually, many known structures [6, 7, 11, 12] may be obtained by changing the positions of the corner points 1, 2, 3, 4 shown in Fig. 2(a) and Fig. 2(b). When the designed structure are symmetric with respect to the line y=-x [dashed line in Fig. 2(a)], only two parameters need to be optimized, i.e., the x coordinates of points 1 and 3. Optimization of two real-valued parameters is relatively easy and quick, but with more degrees of freedom one has the chance to find better solutions. Since it is reasonable to keep the symmetry with respect to y=-x, on may, for example, insert two points between the points 1 and 2 and another point (7) between the points 3 and 4 (see Fig. 2(b)). One then has 3 additional real parameters for the optimization, namely the x and y coordinates of the new point 5 and the x coordinate of the new point 7.

Several selected results of the optimization procedure are shown in the animated Fig. 2(c) and in Fig. 2(d). Note that the optimizer may find various local optima. As special cases, structures similar to those presented in other papers [6, 7, 11, 12] were found with the optimization procedure shown here. Note that the optimizer does not always find a structure proposed by other authors. This is either because it finds better solutions or because of optimization constraints that do not allow finding certain solutions. Since we search for ultra-compact solutions, our constraints do not allow the optimizer to find big ones. Figure 3 lists the geometry of four typical configurations with high transmission. The global optima of all structures exhibit transmission coefficients above 98%. Each of these solutions has different properties concerning radiation, reflection, mode conversion, as well as sensitivity with respect to fabrication imperfections such as accuracy of the locations of the corner points and surface roughness. In order to overcome fabrication imperfection problems, one can add small PhCs as demonstrated in the next section. Note that PhCs will be added to a sub-optimal solution that might be obtained from an optimal one when fabrication tolerances of around 20nm affect both the locations of the corners and the radii of curvature in the corners.

 

Fig. 2. (a) “Mirror” structure for the first optimization; (b) “Resonator” structure for the second optimization; (c) (1.95 MB)Animated results of the first optimization; (d) (2.17 MB)Animated results of the second optimization.

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Fig. 3. Some selected results from animated Fig.2(c) and Fig.2(d) with transmission above 90%. The global optima exhibit transmission above 98%. The coordinates of all points for all of the structures are shown in inset of figures, corner radius r=1e-7 is used in MMP calculation.

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4. Local PhCs in the bend region

Because of its photonic band gap, a PhC can act similar to a ‘perfectly reflecting wall’ within some frequency range. By properly selecting the geometry and material properties, one can design a PhC that provides total reflection within the desired frequency range. Instead of introducing a PhC acting as a 45 degree mirror [6, 7] near the waveguide corner, we arrange two very small PhCs on both sides of a not very well optimized 45° waveguide bend, for example, the one shown in Fig. 3(a). When the coordinates of four corner points of the 45° bend (see Fig.2(a)) are set at 1: (-3.71e-7, -2e-7), 2: (2e-7, 3.71e-7), 3: (-4.73e-7, 2e-7), and 4: (-2e-7, 4.73e-7) - which corresponds to a sub-optimal solution that might be obtained due to fabrication imperfections - the transmission and reflection coefficients for the first and the second mode are T₁=0.933, R₁=2.3e-3, T₂=2.07e-4, R₂=1.73e-3, respectively. The resulting structure after adding two PhC parts is shown in Fig. 4(a). The positions of the rods A and B are (1e-7,-1e-7), (-7e-7, 7e-7), respectively. The PhCs consists of circular rods with radii r=120 nm on a square lattice with lattice constant a=600 nm, the circular rods have the same material properties as the waveguide. Therefore, the PhCs can be fabricated in one step together with the trench waveguide, only the mask for the etching needs to be defined using electron-beam lithography accordingly. Furthermore, the normalized band diagram (not show here) shows that the operation wavelength λ=1.55 µm is in the center of the first band gap (Note that a/λ=0.3–0.42 corresponds to the wavelength range λ=1.43–2.0µm.).

The Poynting vector of the structure in Fig. 4(a) is shown in Fig. 4(b). The transmission of the first mode has improved to T₁=0.9896 but this structure is still not optimal. The additional 21 rods provide 63 additional optimization parameters (coordinates and radii). Using symmetry constraints, with respect to y=-x, the number of additional parameters is 31. For current solvers, it would be extremely time consuming to optimize this parameter space. From the optimization of the 90 degree PhC waveguide bend [5] it is known that the transmission is most sensitive to rod A. When moved a little bit, i.e., from its original location (1e-7,-1e-7) to (2e-7,-2e-7), the transmission of the first mode is improved to T₁=0.9969 (see Fig. 4(c)). Since this value is very close to the ideal value 1, further optimization is not required.

 

Fig. 4. (a) Schematic of Bend structure shown in Fig. 3(a) after PhC rods are put on the bend corners; (b) Poynting vector field for the structure shown in Fig. 4(a). At λ=1.55 µm one has T₁=0.9896, R₁=4.06e-4, T₂=1.937e-4, R₂=1.15e-3. (c) Rod A moved from (-1e-7, 1e-7) to (2e-7,-2e-7). At λ=1.55µm one now has T₁=0.9969, R₁=2.48e-4, T₂=6.355e-6, R₂=1.509e-4.

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5. Fabrication imperfections and perturbations

Because of fabrication imperfections, one has to face several problems caused by the etching process. Many of them have to do with 3D aspects that cannot be taken into account in a 2D model but some of them can at least be estimated with 2D models. Two of them are considered in the following: 1) inaccurate locations of the bend corners and inaccurate radii of curvature in the corners and of the PhC rods and 2) surface roughness. Scattering loss due to surface roughness was investigated decades ago [13] based on perturbation theory. According to this theory, the roughness can be interpreted as a superposition of gratings with different periods, where each grating period causes coupling to specific radiation modes. For the sake of simplicity, only a periodic, sinusoidal perturbation along the sidewalls of the trench waveguide is introduced to characterize the roughness. Table 1 shows that the local PhC reduces the influence of the roughness on the transmission of the fundamental mode. As one can see, the transmission reduction of the first mode of the structure without photonic crystal (case 1) is larger than when the PhCs are present (case 2).

To check the influence of displacement of the corner points of the bend [shown in Fig. 3(a), as Fig. 4(a)], two schemes are used: (1) corner 1, 3 are shifted along the x direction and corner 4, 2 are shifted along the y direction; (2) Corner 1 and 2 are shifted along the x and y direction respectively while the other corners are fixed. Figure 5 shows the change in the transmission as a function of the displacement error due to fabrication. From this one can see that also the influence of inaccurate corner locations is reduced when the small local PhCs are present.

Table 1. Comparison of the transmission of the fundamental mode for a bend structure (shown in Fig.4(a)) without and with local PhCs after perturbation was introduced.

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Fig. 5. Corner displacement .vs. transmission at 1.55µm. (a) for scheme 1; (b) for scheme 2 with (black curve) and without (red curves) the local PhCs.

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4. Summary

Various sharp 90 degree bends in a dielectric trench waveguide were analyzed with the highly accurate MMP solver contained in the MaX-1 package. In order to keep the computation time sufficiently small for extensive optimizations with evolutionary strategies, a 2D effective index model was considered. Excellent properties (high transmission coefficients for the fundamental mode, low radiation, low reflection, and low mode conversion) were obtained for two types of bends that were essentially based on a 45 degree mirror concept and on a local resonator structure in the bend area. A careful analysis of the optimized structures shows strong dependence on fabrication tolerances. In order to reduce the radiation and the dependence on fabrication imperfections, a small local photonic crystal consisting of only 21 dielectric rods was introduced in the bend area. This bend structure is more complicated, but less sensitive to the displacement of bend corner positions as well as to surface roughness.