Topology optimization has been used to design a 60° bend in a single-mode planar photonic crystal waveguide. The design has been realized in a silicon-on-insulator material and we demonstrate a record-breaking 200nm transmission bandwidth with an average bend loss of 0.43±0.27 dB for the TE polarization. The experimental results agree well with 3D finite-difference-time-domain simulations.
©2004 Optical Society of America
Modulating the refractive index profile periodically in an optical material can create photonic bandgaps (PBGs) wherein no optical modes can exist [1–3]. Such structures are often referred to as photonic crystals (PhCs) and have attracted a lot of attention as they potentially allow ultra-compact photonic integrated circuits (PICs) to be realized [4–6]. Planar PhC structures are often defined as triangular arrangements of low dielectric pillars in a high dielectric material. This configuration gives rise to a large PBG for the transverse-electric (TE) polarization [7, 8]. Defects in the PhC can introduce modes in the PBG. In this way, photonic crystal waveguides (PhCWs) can be formed by locally breaking the periodicity along a specific direction of the PhC lattice. Due to the triangular lattice configuration, such PhCWs are naturally bent in steps of 60°, thus, making the 60° PhCW bend a key component in PhC-based PICs.
In the field of planar photonic crystals, research has within the last decade mostly relied on an Edisonian design approach combining physical arguments and experimental/numerical verifications [9, 10]. Further optimizations have typically been done in an iterative trial-and-error procedure to improve a chosen performance measure of the PhC component. Such an approach is very time-consuming and does not guarantee optimal solutions. Recently, Smajic et al.  have shown that sensitivity analyses can be assistive in choosing the critical PhC rods/holes to be altered. A different approach was suggested in our previous work , in which we used an inverse design strategy called topology optimization to optimize the performance of a PhCW containing two consecutive 120° bends. This design method offers an effective and robust optimization of the photonic crystal structure irrespectively of the device under consideration. Here, we apply the topology optimization method to the much more important and commonly used PhCW 60° bend and demonstrate an experimental 1-dB transmission bandwidth of more than 200nm. Experimental transmission spectra are compared to spectra obtained from 3D finite-difference-time-domain (FDTD) calculations  and good agreement is found.
2. Design and fabrication
Silicon-on-insulator (SOI) is an excellent choice of material for a monolithic integration of PhC-based PICs and electronic devices. We define the PhC structures in the top silicon layer of a SOI material by utilizing e-beam lithography and standard anisotropic reactive-ion etch . The regular PhCs are defined as air holes arranged in a triangular lattice and the PhCWs are carved out as W1 waveguides by removing one row of holes in the nearest-neighbor direction of the crystal lattice. The lattice pitch is Λ≈400nm and the diameter of the holes D≈275nm. This configuration gives rise to a broad PBG below the silica-line from Λ/0.3463-Λ/0.2592 and allows TE-like single-mode propagation in the PhCW.
The optimization of the 60° PhCW bend has typically relied on attempts to smooth out the bend by altering, displacing, and/or removing holes in the bend region. However, the useful bandwidth (~30nm) of practical waveguide bends has usually been one order of magnitude smaller than the bandgap [9, 10, 15, 16]. Thus, a careful design, tolerant to fabrication deviations, is very important to utilize the full bandgap of the PhC.
Figure 1 shows schematics of PhCWs defined in a triangular lattice and containing two consecutive 60° bends. The distance between the bends has been chosen arbitrarily, but sufficiently long to achieve steady-state behavior of the PBG mode in the waveguide section separating the bends so that these bends can be treated as separate components. The left image illustrates a PhCW with two simple 60° bends. Such generic bends form severe discontinuities in the PhCW and introduce large reflections and excite higher order modes, which are not necessarily guided in the PhCW.
We use the method of topology optimization (see e.g. ) to optimize the performance of the component. This is done by changing the material distribution in the designated design domains indicated by the two red areas in Fig. 1 (left). No geometrical restrictions are enforced so the resulting design in these domains may consist of an arbitrary number of holes of arbitrary sizes and shapes.
The optimization algorithm is based on a 2D finite-element frequency-domain solver, which produces an accurate 2D field solution. The solver is used repeatedly in an iterative scheme, in which the material distribution is updated every iteration based on analytical sensitivity analysis and use of a mathematical programming tool . On a standard 2.66GHz PC with 1 GB RAM each iteration takes about 10s. More details about the general method can be found in  and its application in optimization of photonic/phononic crystal structures in [12, 19, 20].
For the 60° bends in Fig. 1 (left) we optimize the component by modifying the material distribution so that the transmission through the waveguide is maximized. The transmission is evaluated as the power flow at the output port. In order to create a broadband component the power flow is evaluated for several frequencies (up to 6) in the chosen frequency range. During the optimization procedure we maximize the output for all frequencies, and update these target frequencies every 10th or 20th iteration in order to eliminate transmission dips in the frequency range .
The optimized design is shown in Fig. 1 (right) where the green areas highlight the optimized design domains. This design was obtained after approximately 1000 iterations of the optimization algorithm, however, with the qualitative structure of the design appearing after about 200 iterations. Clearly, the bends have been smoothened by applying a soft curvature in the bend region and one hole has been removed on the inner side of the bend. However, the smoothening is not trivial as the design domain still contains complex structures. Note that the optimized 60° bend mostly resembles an etched mirror , whereas the topology-optimized 120° bend  retained its original crystal structure with deformed holes. The major strength of the topology optimization method is that the superior type of structure does not need to be known in advance; it will appear from the optimization procedure.
Figure 2 shows scanning electron micrographs of the fabricated PhCWs containing two un-optimized (left) and two topology-optimized (right) 60° bends. The PhC structures have been fabricated without applying any special proximity corrections to the irregular shaped holes during the e-beam patterning. Nonetheless, the fabricated topology-optimized structures nicely resemble those shown in Fig. 1 (right).
3. Simulation and experimental results
Figure 3 shows the steady-state magnetic field distribution for the fundamental PBG mode of the fabricated PhCWs simulated using 2D FDTD. The left image shows the mode behavior for light incident from the bottom-left through the PhCW with the generic bends. It is clearly seen that the generic bend forms a severe discontinuity in the straight PhCW and excite an odd mode, which is not well guided in the PhCW. Moreover, the lower bend introduces large reflections and scattering of light to the PhC structure. In contrast, the right image shows that the topology-optimized bend regions guide the fundamental PBG mode nicely through the two bends without disturbing the mode profile.
The fabricated PhCWs shown in Fig. 2 have been optically characterized using broadband light emitting diodes (LEDs) as sources. To cover the full bandwidth of the fabricated components we used three different LEDs centered around 1310nm, 1414nm, and 1538nm. Tapered lensed fibers were used to couple light in and out of the ridge waveguides connected to the PhCWs. Two polarization controllers and a polarizer with an extinction ratio better than 35 dB were used to control the polarization of the light sent into the device under test. The optical spectra for the transmitted light were recorded with a spectral resolution of 10nm using an optical spectrum analyzer. To extract the bend loss the transmission spectra have been normalized to the transmission spectrum for a straight PhCW of the same length. Figure 4 shows the measured bend loss of TE polarized light for the un-optimized generic (red curve) and the topology-optimized (green curve) 60° bend.
The topology-optimized bend displays more than 200nm bandwidth with less than 1dB loss and an average bend loss of 0.43±0.27 dB. In the same wavelength range the un-optimized bend clearly shows a large bend loss, which only reduces in a narrow range near the cut-off of the fundamental mode at longer wavelengths. Thus, applying the two-dimensional topology optimization method has dramatically boosted the performance of the 60° bend and opened for a practical implementation of the bend without the need for delicately tuning a narrow operational bandwidth of the bend to the bandwidth of the rest of the PhC component. The high transmission bandwidth of the bend is obtained for the fundamental even mode in the PBG of the PhCW. Due to the fundamental properties of the SOI PhCW [7, 8] this bandwidth is above the silica-line but in the PBG and cannot be attributed to the optimization method, as this is a purely two-dimensional algorithm.
Figure 5 shows a detailed comparison between the experimental and calculated loss of the optimized bend and a good agreement is found. The negative theoretical propagation losses are due to numerical artifacts when calculating near zero losses. The 3D FDTD spectrum has been shifted 1.2% in absolute wavelength and slightly undershoots the experimental values. These deviations are partly due to uncertainties in the experimental hole diameters, but more importantly due to the limited grid resolution in the calculations .
We have optimized the performance of a 60° planar photonic crystal waveguide bend using a two-dimensional inverse design strategy called topology optimization. The design was fabricated in silicon-on-insulator material and we experimentally obtained a record-high 1-dB transmission bandwidth exceeding 200nm for the TE polarization. The experimental results agree well with 3D finite-difference-time-domain simulations. The broadband topology-optimized 60° waveguide bend solves an important issue in designing planar photonic crystal components and opens for the realization of a wide range of ultra-compact, low-loss, and broadband optical devices.
This work was supported in part by the Danish Technical Research Council through the research programs ‘Planar Integrated PBG Elements’ (PIPE) and ‘Designing bandgap materials and structures with optimized dynamic properties’.
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