Low-Loss Silicon Directional Couplers with Arbitrary Coupling Ratios for Broadband Wavelength Operation

Introduction

Optical power splitters are indispensable components in photonic integrated circuits (PICs), serving as fundamental building blocks with significant applications such as modulation, signal switching, and wavelength division multiplexing (WDM). The 2x2 splitter, in particular, emerges as a crucial element in these applications. It is often essential to design broadband 2x2 splitters capable of both balanced and unbalanced splitting ratios. Furthermore, for large-scale integration in PICs, these splitters must exhibit minimal excess loss, maintain a compact footprint, and not complicate the fabrication process.

The directional coupler (DC) has traditionally been the primary choice for 2x2 splitters owing to its simplicity and ease of fabrication. However, the performance of traditional DCs suffers from significant wavelength dependence due to dispersion, leading to degraded broadband performance. Various design schemes have been explored to overcome this limitation, including multimode interferometers (MMIs), adiabatic directional couplers (ADCs), and variations of the traditional DC design. However, these approaches often face challenges such as limited coupling ratio options, high excess loss, large footprints, or fabrication complexities.

This tutorial presents a design for a high-performance 2x2 splitter based on bent directional couplers (DCs) that meets the essential requirements of broadband coupling, support for arbitrary coupling ratios, ultra-low loss, high fabrication tolerance, and a compact footprint. The design leverages a rigorous coupled-mode theory (CMT) analysis and an experimental-based coupling model to achieve broadband operation at arbitrary coupling ratios.

Coupling Wavelength Dependence Analysis for Straight and Bent DCs

The coupling behavior of straight and bent DCs can be analyzed using CMT. The through (r^2) and cross (κ^2) coupling ratios of a directional coupler can be expressed as:

r^2 = 1 - m sin^2(β_c l + φ)

κ^2 = m sin^2(β_c l + φ)

Where m is the matching coefficient, β_c is the coupling strength per unit length, l is the coupling length (for straight DCs) or coupling angle (for bent DCs), and φ accounts for the coupling contributed by the input and output connection bends.

For a straight symmetric DC, the wavelength dependence arises from the term Δn_g = n_g,even - n_g,odd, which is consistently positive, as shown in Figure 2(a). This implies that achieving broadband coupling at an arbitrary ratio is not possible with a straight symmetric DC based on strip waveguides.

On the other hand, the analysis for bent DCs reveals that Δn_g and dΔn_eff/dλ can take on negative, zero, or positive values, as depicted in Figures 2(b) and 2(c), respectively. This indicates the feasibility of fine-tuning the design parameters to operate in a regime where the wavelength dependence terms cancel each other out, enabling broadband coupling (i.e., dκ^2/dλ = 0).

Fig. 2 examines the wavelength dependence for straight and bent directional couplers (DCs). (a) Features a contour plot of the group index difference (∆ng) for the straight symmetric DC, varying by gap and waveguide width. (b) and (c) Show contour plots for the bent DC, illustrating ∆ng and the rate of change of the effective index with wavelength (d∆neff/dλ), respectively, as functions of gap and radius. The waveguide widths are fixed at 0.38 µm. Both plots include lines indicating where ∆ng = 0 (blue) and d∆neff/dλ = 0 (black), crucial for achieving broadband performance.

Experimental Verification and Coupling Model

To build an experimental coupling model for straight and bent DCs, several devices were fabricated and measured using imec's iSiPP300 platform, which allows high waveguide quality and access to feature dimensions well below 100 nm. The fabricated devices are based on strip silicon waveguides with a nominal thickness of 220 nm, a width of 380 nm, and a coupling gap of 100 nm.

Figure 1 depicts two designs of directional couplers (DC). (a) Shows a traditional straight DC with a coupling length L. (b) Illustrates and (c) presents a SEM image of a modified bent DC featuring a coupling radius R and an angle θ, designed with low-loss bends, contrasted with the traditional circular bends with a 5 µm radius on the straight DC ports. (d) Provides a SEM cross-sectional view of the bent DC at the coupling region. These waveguides, made from strip waveguides on the IMEC iSiPP300 platform, have a silicon oxide top cladding with a nominal silicon thickness of 220 nm and width of 380 nm, and a coupling gap of 100 nm.

The coupling ratios were measured using the cutback method for the bent DCs, where multiple identical bent DCs were cascaded, and the power coupling ratio was extracted from the slope of the linear regression between the transmitted powers and the port numbers, as shown in Figure 3.

Figure 3: Microscope image of the cascaded identical bent DCs used for robust measurements. The power is input at the port (in) and measured at all the cascaded output stages. The labels determine whether the cascaded through (Th) or cascaded cross (Cr) coupled power is extracted, and the number indicates the number of times the measured value (through or cross) was repeated for that measurement.

Based on the optical measurements, the model parameters were extracted by fitting the coupling ratios into sinusoids with respect to the coupling length or angle at each measured wavelength, as shown in Figure 4.

Figure 4: Sinusoidal fitting of the cross coupling (κ^2) and through coupling (r^2) with the coupling length for the straight DC (a) and with the coupling angle for the bent DC (b) at λ = 1.31 μm for the proposed devices. Dots represent measured values, and lines depict the fitting.

The extracted model parameters were then linearly fitted with respect to the working wavelength, as shown in Figure 5 for the straight and bent DCs.

Figure 5: Extracted model parameters wavelength response with linear fitting based on experimental data. A_t, A_c, m, β_c, and φ for the straight DC (a-d) and for the bent DC (e-h).

The analysis confirms that a broadband response for the bent DC (i.e., dκ^2/dλ = 0) is feasible at the intersection of the positive and negative parts of the sinusoid terms, as illustrated in Figure 6.

Figure 6: Cross coupling wavelength derivative (dκ^2/dλ, Eq. 2) of the bent DC, along with its constituent terms, demonstrating that broadband coupling (i.e., dκ^2/dλ = 0) can take place at the intersection of the positive and negative parts of the sinusoid terms at a specific design regime.

Broadband, Ultra Low-Loss, and Tolerant 0.5:0.5 Coupler with 300 mm Wafer Mapping

To showcase the broadband coupling and investigate the impact of asymmetry on the coupling wavelength dependence for the bent DC, a 0.5:0.5 splitter was designed using straight and bent DCs with varying lengths or coupling angles, respectively. As depicted in Figure 7, the straight DC exhibits high wavelength dependence, recording a 0.391 coupling variation over an 80 nm wavelength range. In contrast, the bent DC exhibits broadband coupling with a minimal cross coupling variation of 0.051, achieving a remarkable 7.67 times reduction in coupling variation compared to the straight DC.

Figure 7: The cross coupling (κ^2) and the cross coupling derivative (dκ^2/dλ) as a function of the coupling length for the straight DC (a) and as a function of the coupling angle for the bent DC (b).

The measured coupling ratios of the traditional straight DC and the proposed bent DC at the 0.5:0.5 coupling ratio are shown in Figure 8. The bent DC demonstrates broadband coupling with a minimal variation of 0.051 over an 80 nm wavelength range, marking a 7.67 times reduction in coupling variation compared to the straight DC.

Figure 8: The measured coupling ratios of the traditional straight DC (a) and the proposed bent DC (b) at 0.5:0.5 coupling. The bent DC shows broadband coupling with a minimal variation of 0.051, achieving a 7.67 times reduction in coupling variation compared to the straight DC.

Fabrication Tolerance

To assess the fabrication tolerance of the proposed 0.5:0.5 bent DC, the coupling behavior was simulated using 3D-FDTD models with waveguide width deviations (δw) of -20, -10, 0, 10, and 20 nm, as shown in Figure 9. The device exhibits robust performance even with δw = ±20 nm, showcasing its tolerance. Notably, the cross coupling variation over the 80 nm wavelength range is 0.058 for δw = ±10 nm and 0.061 for δw = ±20 nm.

Figure 9: Fabrication tolerance of the proposed 0.5:0.5 bent DC. Coupling spectrum simulation results of the nominal design (δw = 0) along with waveguide width deviations of -20, -10, 10, 20 nm. The inset figure depicts the waveguide width (W) deviation and the corresponding gap deviation for the proposed SOI-based bent DC.

Furthermore, the robustness of the proposed design across the 300 mm wafer was investigated through complete wafer measurements for two metrics: the cross coupling deviation averaged over the 80 nm wavelength range with respect to the central die (Figure 10a) and the cross coupling variation over the 80 nm wavelength range (Figure 10b), covering all 63 dies.

Figure 10: (a) Average cross coupling deviation with respect to the central die and (b) cross coupling variation over 80 nm bandwidth for the proposed 0.5:0.5 bent DC splitter over the 300 mm wafer, covering all 63 dies.

Around the nine central dies, the average cross coupling deviation from the central die ranged between 0.003 and 0.023, with a cross coupling variation between 0.055 and 0.081. The die with the largest coupling variation was (5, 0) at the extreme edge of the wafer, where process variations are typically high. Die (5,0) exhibited a maximum cross coupling variation of 0.112 over the 80 nm wavelength range, with an average cross coupling deviation of 0.007. In general, the results indicate low variations across most dies, with relatively higher values observed only at the extreme edges of the wafer, illustrating the potential for large-scale use of the proposed device.

Broadband Coupling with Arbitrary Coupling Ratios: Model and Coupling Examples

The asymmetry of the bent DC is inversely proportional to the bending radius (R), with higher asymmetry leading to a lower maximum coupling ratio. A comprehensive fitting was conducted for broadband coupling ratios, considering both bending radius and coupling angle while maintaining a fixed gap of 0.1 μm and waveguide widths of 0.38 μm, as shown in Figure 11.

Figure 11: The proposed model for extracting broadband bent DCs with arbitrary coupling ratios. (a) Broadband cross coupling values as a function of the bending radius. (b) The corresponding coupling angle as a function of bending radius.

The fitting depicted in Figure 11 provides a valuable tool for achieving broadband coupling for arbitrary coupling ratios. This model was employed to extract multiple examples of devices with broadband cross coupling ratios of 0.4, 0.5, 0.6, and 0.7, as illustrated in Figure 12.

Figure 12: Examples of broadband bent DCs with arbitrary coupling ratios in accordance with the proposed model, where the through and cross coupling are shown as a function of wavelength. Devices with broadband (a) 0.4, (b) 0.5, (c) 0.6, and (d) 0.7 cross coupling values are presented.

The corresponding cross coupling variations over the 50 nm wavelength range for these ratios are minimally equal to 0.023, 0.023, 0.038, and 0.034, respectively. This affirms the potential of the proposed methodology for designing broadband coupling ratios with flexibility.

Performance Comparison

Compared to existing results in the literature (Table I), the proposed splitter stands out as a high-performance design that simultaneously meets all essential criteria of low wavelength dependence, ultra-low loss coupling, compact footprint, support for arbitrary coupling ratios, and high fabrication tolerance.

Specifically, this work introduces the first experimental-based model to achieve broadband coupling with arbitrary coupling ratios using the bent DC. Moreover, a compact 0.5:0.5 bent DC splitter with a length of 27.5 μm and the least coupling variation of 0.051 over an 80 nm wavelength range is demonstrated, outperforming previously reported bent DCs.

Additionally, the proposed 0.5:0.5 splitter exhibits the lowest excess loss in the literature to date (0.003 ± 0.013 dB), enabled by the introduction of low-loss bends that ensure continuous curvature and curvature derivative at all connections. This ultra-low loss feature, coupled with the capability of achieving arbitrary coupling ratios, compactness, and high fabrication tolerance, makes the proposed device an attractive component for practical applications with mass production.

Conclusion

In this tutorial, we have presented a comprehensive analysis and design methodology for low-loss silicon directional couplers with arbitrary coupling ratios for broadband wavelength operation based on bent waveguides. The key points are summarized below:

1. A rigorous coupled mode theory (CMT) analysis was performed to derive the broadband behavior of bent directional couplers (DCs), revealing the feasibility of fine-tuning the design parameters to achieve broadband coupling at arbitrary ratios.

2. An experimental-based coupling model was developed by fabricating and measuring straight and bent DCs on imec's advanced iSiPP300 platform, enabling the extraction of model parameters and their wavelength dependence.

3. The proposed model was leveraged to design and demonstrate a 0.5:0.5 bent DC splitter with a compact length of 27.5 μm, exhibiting the least coupling variation of 0.051 over an 80 nm wavelength range compared to previously reported bent DCs.

4. The introduction of low-loss bends in the proposed design resulted in an ultra-low excess loss of 0.003 ± 0.013 dB, marking the lowest reported excess loss for a silicon 2x2 splitter to date.

5. Fabrication tolerance studies confirmed the robustness of the proposed 0.5:0.5 bent DC, with a cross coupling variation of 0.061 over an 80 nm wavelength range for a waveguide width deviation of ±20 nm.

6. Wafer-scale measurements on imec's 300 mm platform showcased consistently low variations across most dies, with a maximum cross coupling variation of 0.112 observed at the extreme edge of the wafer, highlighting the potential for large-scale integration and mass production.

7. A comprehensive fitting model was developed to achieve broadband coupling at arbitrary ratios, enabling the design of devices with broadband cross coupling ratios of 0.4, 0.5, 0.6, and 0.7, with minimal coupling variations over a 50 nm wavelength range.

Overall, the proposed bent DC splitters meet all the essential requirements of low wavelength dependence, ultra-low loss, compact footprint, support for arbitrary coupling ratios, and high fabrication tolerance, making them attractive components for practical applications with mass production in highly dense photonic integrated circuits.

Reference

[1] H. El-Saeed, A. Elshazly, H. Kobbi, R. Magdziak, G. Lepage, C. Marchese, J. Rahimi Vaskasi, S. Bipul, D. Bode, M. E. Filipcic, D. Velenis, M. Chakrabarti, P. De Heyn, P. Verheyen, P. Absil, F. Ferraro, Y. Ban, J. Van Campenhout, W. Bogaerts, and Q. Deng, "Low-Loss Silicon Directional Coupler with Arbitrary Coupling Ratios for Broadband Wavelength Operation Based on Bent Waveguides," Apr. 2024.