Scientific Report|Ultra-compact and Efficient Photonic Waveguide Bends Designed Using Topology Optimization

Introduction

Photonic integrated circuits (PICs) have seen tremendous growth over the past decades, with advancements in devices and circuits enabling a wide range of applications such as data communication, sensing, quantum photonics, and biomedical applications. The silicon-on-insulator (SOI) platform has been particularly instrumental to this progress, providing high refractive index contrast that allows for the creation of compact photonic devices and circuits.

A fundamental component within these PICs are photonic waveguide bends, which guide optical signals along desired pathways. However, these waveguide bends are still prone to significant propagation losses, especially as smaller bend radii are employed. The loss in optical mode in a bend waveguide can be attributed to three main factors: (i) loss due to the non-zero curvature of the waveguide, (ii) mode mismatch between the input and output waveguides, and (iii) loss due to scattering and absorption in the waveguide.

To address these challenges, extensive research has explored various improvements over the years. These include transitioning from sharp corners to circular bends, using adiabatic curvature matching to smooth the transition between straight and curved waveguide sections, introducing lateral offsets between input and output waveguides, and exploring non-circular bend geometries such as Euler bends, hybrid bends, and photonic crystal-based bends. While these approaches have shown promise, there are still limitations in achieving ultra-compact bends with minimized losses.

In this tutorial, we discuss the design of ultra-compact and efficient photonic waveguide bends using topology optimization, a powerful inverse design technique that can generate creative and unexpected nanophotonic structures. We demonstrate the effectiveness of this approach in designing compact L-bend and U-bend structures on a 220 nm silicon-on-insulator (SOI) platform, achieving significant reductions in insertion loss compared to conventional bend designs.

Topology Optimization for Photonic Device Design

Topology optimization originates from the field of solid mechanics, but it has now attracted considerable attention in many areas of physics, including the field of photonics. This powerful technique reshapes the material layout within a device's design space to achieve optimal performance, often leading to creative and unexpected topologies.

The key advantage of topology optimization lies in its ability to explore the design space more freely compared to other inverse design methods. Unlike direct binary search or particle swarm optimization, which are limited by pixel-by-pixel alterations or block constraints, topology optimization applies filters concurrently during the optimization process. This not only saves computational time but also allows for the design of broadband devices, overcoming the single-wavelength operation limitation of objective-first algorithms.

The topology optimization process can be mathematically represented as a continuous constrained optimization problem, where a design field ξ(r) ∈ [0, 1] is used to control the material distribution within the design domain Ω. The objective is to maximize a defined figure-of-merit (FOM), Φ, subject to various equality and inequality constraints, such as feature size limitations.

In the context of photonic device design, the FOM is typically defined using electromagnetic field monitors to assess the coupling and overlap between the desired mode and the actual fields within the optimized structure. By maximizing this FOM, the optimization process aims to generate a material distribution that facilitates efficient light propagation and minimizes losses.

The optimization is carried out using an adjoint-based sensitivity analysis, which requires solving only a single adjoint equation for the FOM and one for each constraint. This approach is computationally efficient compared to finite difference-based methods, where the system equations must be solved for perturbations of each design variable.

To ensure the fabricability of the optimized structures, various constraints can be incorporated into the optimization problem, such as minimum feature size requirements. Additionally, filter functions can be applied to smooth sharp edges and corners, making the designs more compatible with practical fabrication processes.

L-Bend and U-Bend Design using Topology Optimization

In this tutorial, we focus on the design of ultra-compact L-bend and U-bend structures on a 220 nm silicon-on-insulator (SOI) platform using topology optimization. The goal is to minimize the propagation loss along the bending regions, including mode mismatch, radiation, and sidewall roughness effects.

The design model for the waveguide bends is shown in Figure 1. The input and output waveguides have a width of 500 nm, and the design region lengths Lx and Ly are varied to realize different bend radii. The device is excited with a transverse electric (TE)-polarized source with a bandwidth of 200 nm and a center wavelength of 1550 nm.

Figure 1. (a) Schematic design of L-bend and (b) U-bend

L-Bend Designs

For the L-bend structures, three different footprints are considered, as shown in Figure 2. These include:

• L-bend 1: Lx = 2.5 μm, Ly = 2.5 μm (radii = 1.76 μm)

• L-bend 2: Lx = 1.5 μm, Ly = 1.5 μm (radii = 1.06 μm)

• L-bend 3: Lx = 1 μm, Ly = 1 μm (radii = 0.71 μm)

The initial structures for optimization are arc-type bends, as shown in Figures 2(a-c). The topology optimization process then modifies these initial structures to maximize the coupling and minimize the mode mismatch between the input and output waveguides over the broad wavelength range of 1450-1650 nm.

Figure 2. Initial structures of (a) L-bend 1 (radii: 1.76 µm), (b) L-bend 2 (radii: 1.06 µm), (c) L-bend 3 (radii: 0.71 µm). Topology-optimized structures of (d) L-bend 1 (radii: 1.76 µm), (e) L-bend 2 (radii: 1.06 µm), and (f) L-bend 3 (radii: 0.71 µm).

Figure 3. Simulated electric field profles of (a) initial L-bend 1 (radii: 1.75 µm), (b) initial L-bend 2 (radii: 1.06 µm), (c) initial L-bend 3 (radii: 0.71 µm). (d) topology-optimized L-bend 1 (radii: 1.75 µm), (e) topologyoptimized L-bend 2 (radii: 1.06 µm), and (f) topology-optimized L-bend 3 (radii: 0.71 µm). (g) Comparison of simulated insertion losses of optimized L-bends and un-optimized L-bends with varied radii.

Figure 3 shows the simulated electric field profiles for the conventional L-bends and the topology-optimized L-bends. As the bending radii decrease, the un-optimized arc-type bends exhibit increased loss and mode distortion. In contrast, the topology-optimized structures are able to preserve the mode profile and maintain high transmission.

The performance comparison between the optimized and un-optimized L-bends is summarized in Figure 3(g). The trend clearly demonstrates the significant improvement in insertion loss achieved through topology optimization. For the L-bend 1 (radii = 1.76 μm), the optimized design exhibits a maximum insertion loss of only 0.07 dB, compared to 0.37 dB for the un-optimized arc-type bend - a reduction of over 81%. Similarly, for the L-bend 2 (radii = 1.06 μm) and L-bend 3 (radii = 0.71 μm), the optimized designs show maximum insertion losses of 0.26 dB and 0.78 dB, respectively, which are 64.8% and 51.55% lower than their un-optimized counterparts.

U-Bend Designs

In addition to the L-bend structures, we also designed U-bend configurations to guide light through 180-degree turns, as shown in Figure 4. Three different design regions were explored:

• U-bend 1: 3.0 μm × 3.6 μm (radii = 2.5 μm)

• U-bend 2: 2.5 μm × 2.5 μm (radii = 2.0 μm)

• U-bend 3: 1.5 μm × 1.5 μm (radii = 1.0 μm)

Similar to the L-bends, the initial arc-type U-bends were used as the starting point for topology optimization, as shown in Figures 4(a-c). The optimization process then modified these structures to improve the coupling and mode matching between the input and output waveguides.

Figure 4. Initial structures of (a) U-bend 1 (radii: 2.5 µm), (b) U-bend 2 (radii: 2.0 µm), and (c) U-bend 3 (radii: 1.0 µm). Topology-optimized structures of (d) U-bend 1 (radii: 2.5 µm), (e) U-bend 2 (radii: 2.0 µm), and (f) U-bend 3 (radii: 1.0 µm).

The simulated electric field profiles in Figure 5 demonstrate the effectiveness of the topology optimization approach. The un-optimized arc-type U-bends exhibit significant mode distortion and high transmission losses, particularly as the bend radii are reduced. In contrast, the topology-optimized U-bends are able to maintain the mode profile and achieve much higher transmission.

Figure 5. Simulated electric feld profles for (a) initial U-bend 1 (radii: 2.5 µm), (b) initial U-bend 2 (radii: 2.0 µm), (c) initial U-bend 3 (radii: 1.0 µm), (d) topology-optimized U-bend 1 (radii: 2.5 µm), (e) topologyoptimized U-bend 2 (radii: 2.0 µm), and (f) topology-optimized U-bend 3 (radii: 1.0 µm). (g) Comparison of simulated insertion losses of optimized and un-optimized U-bends with varied radii

The performance comparison between the optimized and un-optimized U-bends is presented in Figure 5(g). The results clearly show the advantages of the topology-optimized designs. For the U-bend 1 (radii = 2.5 μm), the optimized structure achieves a maximum insertion loss of only 0.07 dB, compared to 0.46 dB for the un-optimized bend - a reduction of over 84%. Similarly, for the U-bend 2 (radii = 2.0 μm) and U-bend 3 (radii = 1.0 μm), the optimized designs exhibit maximum insertion losses of 0.21 dB and 3.16 dB, respectively, which are 73.8% and 29.9% lower than their un-optimized counterparts.

Meander Line with Optimized U-Bends and Experimental Verification

To further validate the effectiveness of the topology-optimized bends, we fabricated and tested a meander line device consisting of 16 U-bend 1 structures (radii = 2.5 μm) connecting strip waveguides. This meander line design allows us to assess the scalability and performance of the optimized bends in a more realistic PIC configuration.

Figure 6 shows the structural difference between the meander line with optimized U-bends and the one with un-optimized U-bends of the same radius. The simulated electric field profiles clearly demonstrate the superior light guidance and reduced radiative losses in the meander line with the topology-optimized bends.

Figure 6. Structures of (a) meander line having normal U-bends with radii of 2.5 µm, and (b) meander line having optimized U-bends with radii of 2.5 µm. Simulated electric feld profle of (c) meander line having normal U-bends with radii of 2.5 µm, and (b) meander line having optimized U-bends with radii of 2.5 µm

The fabrication of the devices was carried out using electron beam lithography and etching processes on a 220 nm SOI platform. Figure 7(b) shows the layout of the optimized bends-based meander line, as well as a scanning electron microscope (SEM) image of the fabricated U-bend 1 device.

The experimental setup for measuring the transmission of the fabricated devices is illustrated in Figure 7(a). A tunable laser diode with a wavelength range of 1520-1580 nm was used to couple light into the input grating coupler, and the output power was measured using a power meter.

Figure 7(c) shows the comparison of the measured transmission for the optimized and un-optimized U-bend 1 structures. The optimized U-bend 1 demonstrates an average insertion loss of only 0.1 dB near the 1550 nm wavelength, while the un-optimized counterpart exhibits an insertion loss of 1.1 dB. These results are consistent with the simulated performance, where the optimized U-bend 1 showed a maximum insertion loss of 0.05 dB and the un-optimized U-bend 1 had a maximum loss of 0.23 dB at 1550 nm.

Furthermore, the transmission measurement of the meander line structures is shown in Figure 7(d). The meander line with the optimized U-bends exhibits an averaged insertion loss of 1.23 dB in the wavelength range of 1520-1580 nm, while the meander line with un-optimized U-bends has a significantly higher insertion loss of 11.12 dB. The simulated values also show a consistent trend, with the optimized meander line having a maximum insertion loss of 1.23 dB and the un-optimized meander line reaching 10.17 dB in the broader wavelength range of 1450-1650 nm.

Figure 7. (a) Experimental setup for optical transmission measurement. (b) Layout for optimized bends based meander line structure with SEM image of U-bend 1. (c) Simulated and experimental results for the transmission of optimized U-bend 1 and un-optimized U-bend 1. (d) Simulated and experimental results for the transmission of optimized U-bends-based meander lines and un-optimized U-bends-based meander lines

The experimental results are in good agreement with the simulated data, validating the effectiveness of the topology-optimized bends in reducing insertion loss and maintaining signal integrity, even in a complex meander line configuration.

Benchmark and Comparison with Other Approaches

The performance of the topology-optimized bends designed in this work is benchmarked against various analytical and inverse design methods reported in the literature, as summarized in Table 1.

Table 1. Comparison of our designed bends with simulated results of bends made with diferent analytical approaches and inverse design methods.

Bending type	Design	Waveguide dimension (nm2)	Bandwidth (nm)	Device footprint (µm)
L-bend (900)	Bend with elliptical reflector (polymer)	2300 x 2300	800-900	20 x 20	< 0.33 (TE, exp.) < 0.33 (TM, exp.)
Optical bends	400 x 220	N/A	5.0 x 5.0	< 0.002 (sim.)
Sharp adiabatic bends	2400 x 220	1530-1570	2.6 x 2.6	< 1 (exp.)
Transformation optics-based multi-mode bends	4000 x 500	N/A	79 x 79	< 0.24 (exp.)
This work	500 x 220	1450-1650	2.5 x 2.5	< 0.06 (sim.)
1.5 x 1.5	< 0.21 (sim.)
1.0 x 1.0	< 3.16 (sim.)
U-bend (1800 )	Bend with parabolic reflector(polymer)	2300 x 2300	800-900	20 x 40	< 0.31 (TE, exp.) < 0.35 (TM, exp.)
Optical bends	400 x 220	N/A	5.0 x 10.0	< 0.013 (sim.)
Meta-material waveguide bends	500 x 220	1450-1650	3.0 x 3.0	< 1.87 (TE, exp.) < 1.37 (TM, exp.)
This work	500 x 220	1450-1650	3.0 x 3.6	< 0.07 (sim.)
2.5 x 2.5	< 0.60 (sim.)
1.5 x 1.5	< 0.78 (sim.)
1520-1580	3.0 x 3.6	< 0.077 (exp.)

While some prior studies have achieved lower insertion losses, it is essential to consider the critical distinction in the footprint of these structures. Our designed bends stand out for their remarkable compactness and sub-wavelength bending radii, setting a new standard for miniaturization in the realm of optical bends.

For example, the L-bends designed in this work with radii as low as 0.71 μm exhibit maximum insertion losses of only 0.78 dB, outperforming bends made with direct binary search (radii = 1 μm, loss < 1 dB) and direct range search (radii = 9.35 μm, loss < 0.04 dB) methods. Similarly, our optimized U-bends with radii of 1.0 μm achieve a maximum insertion loss of 3.16 dB, which is significantly better than the un-optimized counterparts.

Furthermore, the fabrication complexity of some reported bends, such as those involving quadratic reflectors or metamaterials, poses substantial challenges for practical implementation. By leveraging a CMOS-compatible SOI platform, our topology-optimized designs can be readily integrated into existing photonic technologies.

It is important to note that the performance of the bends can be further optimized by exploring different waveguide widths. While our analysis focused on 500 nm waveguides, the topology optimization approach can be applied to other common waveguide dimensions, potentially leading to even more compact and efficient bending structures.

Additionally, the versatility of this method extends beyond TE-polarized mode optimization, as it can be readily applied to TM-mode bends as well. Preliminary 2D simulations have already revealed distinct optimized structures for TM-mode propagation, highlighting the adaptability of the topology optimization approach to different modal requirements.

Conclusion

In this tutorial, we have presented the design of ultra-compact and efficient photonic waveguide bends using topology optimization. By leveraging the powerful capabilities of this inverse design technique, we have demonstrated the realization of L-bend and U-bend structures on a 220 nm silicon-on-insulator (SOI) platform with significantly reduced insertion losses compared to conventional bend designs.

The key highlights of our work include:

1. Topology-optimized L-bends with footprints as small as 1 μm × 1 μm (radii = 0.71 μm) achieving maximum insertion losses of only 0.78 dB.

2. Topology-optimized U-bends with footprints as small as 1.5 μm × 1.5 μm (radii = 1.0 μm) achieving maximum insertion losses of 3.16 dB.

3. Experimental verification of a meander line structure with 16 optimized U-bends (radii = 2.5 μm), demonstrating an averaged insertion loss of 1.23 dB in the wavelength range of 1520-1580 nm, consistent with the simulated results.

4. Comparison with existing analytical and inverse design approaches, showcasing the superior performance of the topology-optimized bends in terms of both insertion loss and footprint reduction.

The ability to design ultra-compact and efficient photonic bends using topology optimization holds great promise for advancing the field of photonic integrated circuits. By overcoming the limitations of conventional bend designs, this approach can enable higher integration densities, improved signal integrity, and more compact and versatile photonic systems. The insights and techniques presented in this tutorial pave the way for further exploration and innovation in the realm of inverse-designed nanophotonic devices.

Reference

[2] S. Irfan, J.-Y. Kim, and H. Kurt, "Ultra-compact and efficient photonic waveguide bends with different configurations designed by topology optimization," Scientific Reports, vol. 14, no. 6453, 2024. [Online]. Available: https://doi.org/10.1038/s41598-024-53881-9. [Accessed: 13-Apr-2024].