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Abstract
Countless waveguides have been designed based on four basic bends: circular bend, sine/cosine bend, Euler bend (developed in 1744) and Bezier bend (developed in 1962). This paper proposes an n-adjustable (NA) bend, which has superior properties compared to other basic bends. Simulations and experiments indicate that the NA bends can show lower losses than other basic bends by adjusting n values. The circular bend and Euler bend are special cases of the proposed NA bend as n equals 0 and 1, respectively. The proposed bend are promising candidates for low-loss compact photonic integrated circuits.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Photonic integrated circuits (PICs) are playing an increasingly important role in the information technology era [1] and can be used in a variety of applications, such as quantum communication [2–5], optical interconnection [6–10], modulation [11,12], biosensing [13], and medical diagnostics [14,15]. These applications benefit from the strongly confined light, stable operation, low power consumption, and small footprint of PICs. However, the integration density of PICs is significantly limited by the losses incurred by waveguide bends, including 90°, 180°, and S-bends used for optical signal routing. For example, several tens or hundreds of bends are required in some complex components, such as switch fabrics [16] and multichannel lidar applications [17]. Therefore, the optimization of bends is of pivotal importance when designing high integration density PICs or ultralow-loss devices.
In recent years, many studies have focused on the design of waveguide bend geometries to reduce bending losses. The main waveguide bends currently available include four basic types: circular bends, sine/cosine generated bends, Bezier bends, Euler bends, and new waveguide bends designed by various improvements to the four basic bends, as shown in Fig. 1. Circular arcs are thought to have been the first curves used to design waveguide bends. Circular curves offer mathematical simplicity, and by far, the majority of waveguide bends in PICs have been designed as circular bends. However, waveguide bends based on circular designs are not the least lossy, as the constant curvature gives rise to nonnegligible mode mismatch losses. Euler introduced the Eulerian spiral in 1744, a curve whose curvature increases linearly along the bend path length and is widely used in areas such as road design. Interestingly, researchers have found that natural conches, mouse whiskers, etc., can also be described by Euler curves [18]. In 1957, the Euler curve was first used in photonics by H.G. Unger et al., relatively low losses were achieved with the advantage of a continuous transition from straight waveguide to curves [19]. Currently, circular arcs and Euler curves are the two most common curves used in bending waveguide designs. In addition, Bezier curves have been used to design 90° and S-shaped waveguide bends since 2013. At a certain radius, a curve with the lowest loss can be found by parameter optimization [4]. Sine/cosine generated curves are often used to design S-shaped waveguide bends with lower losses than the two-arc combination type [3].
Further research has shown that new waveguide bends based on the four types of bends mentioned above can be used to further reduce losses by applying various improvements. Common improvements currently available include the use of a progressive increase in width, two bending hybrid methods, offsets between straight and bending waveguides and implementations of grooves and subwavelength grating like structures [20], as shown in Fig. 1 and Table 1. In addition, there are currently two other algorithm-based methods: the variational method and the inverse design method.
Table 1. Improvement methods based on basic waveguide bends
To reduce the losses in waveguide bends, researchers are currently focusing more on improving the basic bends using various methods. Compared to the improved waveguide bends, the study of basic bends is inadequate. We can see that the last forty years has witnessed the rapid development of PICs; for the main kinds of basic bends, circular arcs were one of the first geometric curves constructed by man, and sine/cosine curves were proposed early, as were Bezier curves in 1962 and Euler curves in 1744. In addition, a variational approach to design waveguide bend with minimal intermodal coupling was proposed in 2012 [36]. In 2020, a basic bend was proposed in which the waveguide radius follows a hyberbolic tangent variation [47]. Mathematically, there exist an infinite number of curves connecting two points. A more in-depth study of basic bend has an important influence on the design of low-loss waveguide bends.
We have attempted to investigate the basic bends for PICs, in conjunction with mathematical theory, with exciting results obtained. The new basic waveguide bend has superior properties compared to the four basic bends currently available. We call it the n-adjustable (NA) bend, where is a unitless number used to determine the nature of the bend, called the bend parameter. Moreover, circular bends and Euler bends are special cases of this class of bend for which equals 0 and 1, respectively. We have also attempted improvements based on the NA bend by using a hybrid method, and the results are also encouraging. As expected, the proposed NA bend has important implications for the design of low-loss waveguide bends.
2. Theory and structure of the NA bend
2.1 Construction of the NA bend
In this section, a detailed description of the construction of the NA bend is presented. The key to determining the geometry of a curve is to determine its curvature K or radius of curvature ρ (ρ=1/K). For circular curves, the curvature K is a constant. For Euler curves, the curvature K increases linearly with the path length s:
where k is scaling factor, in most studies of Euler curves, for the convenience of the later integral calculations, is considered to be a constant, and in most cases, it is set to 1.0 [24,25].Inspired by Euler curves, a new type of curve is developed in this paper by introducing an adjustable parameter n, which is called as n-adjustable (NA) bend. The relationship between the bend curvature K at one bend point and bend path length s for the NA bend is expressed as follows:
Similar to most studies of Euler curves, the scaling factor k is set as 1.0 in this paper. For the new curve, the curvature K varies exponentially with the path length s. The bend parameter n, a key parameter that determines the geometry of a curve, is adjustable. By varying the value of n, a series of curves with different geometries can be obtained. It should be noted that circular arcs and Euler curves are special cases of the new curve. When n = 0, the curve is a circular curve. When n = 1, the curve is a Eulerian spiral.
According to the definition of curvature, for a point on the curve, the corresponding differential expression of the curvature is
where θ (s, n) is the angle of the tangent direction at a bend point and s is the path length along the bend. As shown in Fig. 2(a), for one curve point P, the path length from the starting point O to point P is sp; then, the angle of the tangent direction at point P can be obtained by integration of Eq. (3)Note that when k = 1, K(s)=sn; then, it can be deduced that
Fig. 2. (a) Schematic of the composition of a 90° NA bend with an effective radius of Reff. (b) Comparison of the geometry of NA bends with different bend parameters n for the same Reff. (c) The relationship between bend curvature K and bend path length s for 90° NA bends with different bend parameters n for the same Reff.
Then, the waveguide coordinates can be expressed as integrals of the cosine and sine of the tangent angle:
Most of the research into low-loss waveguide bends has focused on 90° bends; therefore, we chose 90° NA bends for analysis. Fig. 2(a) shows a schematic diagram of the composition of a 90° NA bend with an effective radius of Reff. The curvature is gradually increased until the angle of the tangent direction reaches 45° (orange solid line) and then decreases symmetrically (orange dotted line). Fig. 2(b) shows 90° NA bends based on different n values, including -0.5, 0, 0.5, and 1. When n = 0, the bend is a circular bend (solid blue line). When n = 1, the bend is a Euler bend (green dotted line). Fig. 2(c) gives the relationship between the curvature K and path length s for different values of n. It can be seen that a change in the n value has a significant effect on the curvature K. Considering that the curvature is a critical factor for the waveguide losses, choosing a suitable n value is effective in reducing waveguide losses.
2.2 Discussion for basic bends
In this section, we try to analyze the advantages and disadvantages of the NA bend, circular bend, Euler bend and Bezier bend, as illustrated in Fig. 3(a). First, we compare the four bends in terms of the sources of loss generation. Bending losses arise from three main sources, including losses due to sidewall roughness, losses due to radiation and losses due to mode mismatch between straight and curved waveguides [37]. The fabrication process determines the losses due to sidewall roughness, and the geometry of the waveguide bends affects the mode mismatch losses and radiation losses. The latter two are the focus of this article. As illustrated in Fig. 3(b), the curvature along the path length of a circular bend is constant, which means that when light is transmitted from a straight waveguide (with zero curvature) to a circular waveguide (with non-zero constant curvature), the mode mismatch losses will be caused. Similar to the circular bend, the curvature of a Bezier bend does not go to zero at the starting and ending points (see Fig. 3(b)), which will lead to mode mismatch losses. Compared to the circular bend and Bezier bend, a Euler bend can significantly reduce mode mismatch losses, as the curvature of the Euler bend in the interface is zero (see Fig. 3(b)). However, the curvature of the Euler bend varies linearly along the path length, and the curvature in the middle of a Euler bend is higher than that in a circular bend or Bezier bend for the same Reff [24], which will introduce significant radiation losses.
Fig. 3. Comparison of the four basic bends. (a) Schematic of 90° bends composed of a circular bend, a Euler bend, two Bezier bends and two NA bends at the same effective radius Reff. (b) The relationship between bend curvature K and bend path length s.
Unlike the circular bend and Euler bend, a series of different NA bends or Bezier bends can be generated between two fixed points by adjusting the bending parameters. 90° Bezier bends are typically defined by 4 points, with (0, 0) and (Reff, Reff) as the start and end points, respectively. The other two middle points characterize how the bend travels from the start to the end and are defined as ((1-B)Reff, 0) and (Reff, B). B is a unitless number used to define the gradual change in the radius at the interface [4]. As shown in Fig. 3(a), the two orange dashed lines show the 90° Bezier bends when B = 0.12 and B = 0.36. B can be optimized to reduce the waveguide losses. Similarly, the waveguide losses can be reduced by optimizing n values of NA bends.
As shown in Table 2, the NA bends proposed in this paper has the advantages of Euler and Bezier bends and circumvents their shortcomings. Similar to Euler bends, when n > 0, the curvature of the NA bend at the interface between the straight and curved waveguides is zero, and the curvature varies gradually along the path length, and the mode mismatch losses will be reduced. Similar to Bezier bends, a series of bends can be obtained by adjusting n, and an optimal NA bend with the lowest losses can be found for a defined Reff. Therefore, it is expected that the NA bend proposed in this paper will have lower losses than the circular bend, Euler bend and Bezier bend, the detailed analysis for which is developed in Section 3.
Table 2. Comparison of the characteristics of basic bends
3. Numerical experiments and analysis
In this section, we varied the n value and the B value to find the optimal 90° NA bend and optimal 90° Bezier bend for small Reff (≤5µm), respectively. The wavelength used was λ=1550 nm, and the TE mode was considered. Comparing the losses for 90° bends based on circular (n = 0), Euler (n = 1), optimal Bezier and optimal NA bends, we found that the optimal NA bend shows the best performance in terms of bending losses. We also performed wavelength scans for the four bends in the C-band.
Silicon photonics holds great promise for a variety of applications due to its wide transparency window and ultrahigh index contrast between the silicon core (∼3.5 at 1550 nm) and silica cladding (∼1.45 at 1550 nm), which makes it possible to realize high confinement of the optical mode, leading to ultradense PICs [37,48,49]. Therefore, we demonstrate the performance of basic bends on the silicon-on-insulator (SOI) platform. The cross section of the embedded waveguide was set to 400 nm × 220 nm to ensure single-mode operation.
The 3-dimensional finite-difference time-domain (3D-FDTD) method was adopted in the simulation through commercial software Lumerical FDTD. To find the optimal NA bend at a certain radius, the n values were swept in steps of 1/32, and the results are shown in Fig. 4.
Fig. 4. Bending losses for 90° bends as a function of bend parameter n for Reff =1.0, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, and 5.0 µm.
Based on the simulation results shown in Fig. 4, the optimum n values for different Reff are summarized in Table 3. Obviously, the optimum n value varies with Reff. When Reff < 3.0 µm, the optimal n value increases with Reff. When 3.0 µm ≤ Reff ≤ 4.5 µm, the optimal n value is 17/32. When Reff = 5.0 µm, the optimum n value is 21/32. It is worth mentioning that when Reff =1.0 µm, the optimum n value is less than zero. In general, the optimal values of n gradually increase as the effective radius Reff increases. This is related to mode mismatch losses and radiation losses. As illustrated in Fig. 2(b), as the values of n increases, the bending degree in the middle of NA bend gradually increases. When Reff is small (e.g., Reff =1 µm), the radiation losses due to the sharp turns in the middle of the bend have a dominant impact on the overall losses. As Reff increases, the mode mismatch loss becomes a nonnegligible source of loss. Therefore, as Reff increases, the optimal values of bend parameter n gradually increase.
Table 3. The optimum bend parameter n with different Reff for NA bend
Each simulation of the waveguide bend by 3D-FDTD takes ≈1 min with relatively low accuracy. High accuracy simulations are only used to verify the final result after the relatively low accuracy simulations. It takes ≈45 min to obtain 32 sets of optimized NA bends on a computer server with an 8-core CPU for a bend with effective radius of 3 µm. The bend with smaller effective radius requires less simulation time and the bend with larger effective radius requires more simulation time. The difference in simulation time is not significant overall. The total simulation time required ranges from 6 to 7 h for the simulation cases of Fig. 4.
To verify the superiority of the NA bend, as a comparison, we also conducted simulations based on Bezier bends. As mentioned above, a series of Bezier bends can be generated by adjusting the value of the bend parameter B. We swept the B value in steps of 0.03 to determine the optimal B values for different Reff based on the simulation results (as shown in Table 4).
Table 4. The optimum bend parameter B with different Reff for Bezier bend
Based on simulation results, waveguide losses of basic bends at different Reff are shown in Table 5. By comparing the losses of the circular bend (n = 0), Euler bend (n = 1), optimal Bezier bend and optimal NA bend for Reff =1, 2, 3, 4, and 5 µm, it can be found that the optimal NA bend shows the lowest losses among the four basic bends, which is consistent with our analysis in Section 2.2. For Reff = 3 and 4 µm, the difference between the losses of the optimal Bezier bend and the optimal NA bend is not obvious, while at other Reff, the optimal NA bend shows significantly lower losses. For both circular and Euler bends, the difference between their losses and those of the optimal NA bend is more pronounced for Reff = 1, 2, 3, 4, and 5 µm. We also launch a light source in a lossless straight waveguide at an angle of 45° to the mesh and monitor the transmission at the output. Then we launch a light source in a lossless straight waveguide at an angle of 0° to the mesh and monitor the transmission at the output. The difference between the two transmission losses is 0.0008 dB, which is caused by the numerical error.
Table 5. Simulated waveguide losses of basic bends at different Reff [dB/90°]
For a more detailed analysis, the electric field distributions of the four bends when Reff =1 and 4 µm are given in Fig. 5. It can be seen from Fig. 5(a) that when the effective radius is small (Reff =1 µm), the radiation losses due to the sharp turns in the middle of the bend have a dominant impact on the overall losses. As Reff increases (Reff =4 µm), the radiation losses are not obvious, and the mode mismatch loss becomes a nonnegligible source of loss, as shown in Fig. 5(b). It can be deduced that when Reff is 1 µm, there is not enough time for the optical mode to sufficiently evolve along the bend due to the short path length of the bend [37], and the losses occur mainly in the turning section, which is not the case when Reff is 4 µm.
Fig. 5. Electric field distributions for the circular bend, Euler bend, optimal Bezier bend, and optimal NA bend. (a) when Reff = 1 µm. (b) when Reff = 4 µm.
The wavelength scans for optimal NA bends when Reff = 1, 2, 3, 4, and 5 µm are also performed, as shown in Fig. 6. The results show that when Reff =1, 2 µm, the bending loss increases monotonically with wavelength, and when Reff =3, 4, and 5 µm, the lowest bending loss is around the wavelength of 1550 nm. The undulation in Fig. 6 may be due to the interference of high-order evanescent modes, which has a small energy and has no significant overall effect when the bending loss is large, and when the bending loss is small the effect is nonnegligible. Figure 7 shows comparison of simulation results of circular, Euler, optimal Bezier, and optimal NA bends for Reff =2 µm. The results show that optimal NA bends have better loss characteristics than other three bends. The Euler bend has significantly higher losses than the other three bends over the entire range of wavelengths. Losses for the circular, optimal Bezier and optimal NA bends follow a similar trend with wavelength, with the optimal NA having the lowest losses at different wavelengths.
Fig. 6. Simulated bending loss spectrum for the optimal NA bend for silicon waveguides when Reff = 1, 2, 3, 4, and 5 µm.
Fig. 7. Simulation results of the circular, Euler, optimal Bezier, and optimal NA bends for silicon waveguides when Reff = 2 µm.
4. Experimental analysis
4.1 Fabrication and optical characterization
The optimal NA bend, circular bend, optimal Bezier bend and Euler bend were simultaneously fabricated on a SOI photonic platform to verify the superiority of the NA bend proposed in this paper. The thickness of the silicon device layer was 220 nm with a 2 µm thick bottom buried oxide and 1 µm thick cladding of silicon oxide. During the fabrication process, a layer of 250 nm ARN7520.17 resist was spin coated on top of the SOI. Next, the device was defined by high-resolution electron-beam lithography at 100 kV. Inductively coupled plasma reactive ion etching was performed using SF6/C4F8 gases after developing and rinsing. Then, a 1 µm thick layer of silicon oxide was deposited as an upper cladding by using a plasma-enhanced chemical vapor deposition process. Finally, we ground and polished the ends of the chip using chemical mechanical polishing in preparation for measuring the waveguide losses.
To verify the conclusions of Section 3, a series of bends were fabricated. First, to validate the optimal values for the bend parameter n proposed in this paper, NA bends with different bend parameters n (n = 0, 4/32, 8/32, 12/32, 16/32, 17/32, 20/32, 24/32, 28/32, 1) at Reff =3 µm were fabricated. This part of the experiment was used to verify the idea that losses can be effectively reduced by adjusting the n value for the bend, as described in Sections 2 and 3. Then, when Reff =1, 2, 3, 4, and 5 µm respectively, the comparison between the four basic bends was conducted. To facilitate the calculation of the losses, for each type of basic bend for a certain Reff, four waveguides with different numbers of bends were fabricated [24]. Fig. 8 shows the scanning electron microscopic (SEM) images of the winding layout of the circular, Euler, optimal Bezier and optimal NA bend for Reff = 4 µm.
TE-polarized light was end-face coupled via a polarization-maintaining fiber to waveguides. The output light of the PIC was collected with a single mode fiber connected to an optical power meter. The positioning of the PIC input and output fibers was realized by a piezoelectric manual alignment system (Suruga Seiki).
4.2 Results and discussion of experiments
Figure 9 shows the measurement results obtained for NA bends with different bend parameters n (n = 0, 4/32, 8/32, 12/32, 16/32, 17/32, 20/32, 24/32, 28/32, 1) for Reff = 3 µm. The wavelength used was λ=1550 nm. The experimental results are shown by the red dots, and the black squares show the simulation results obtained in Section 3. The experimental results show that the bending losses can be reduced by adjusting n value. As shown in Fig. 9, the optimal n value for Reff =3 µm is 17/32 with a lowest bending loss of 0.0046 dB/90°. The trend in the variation of simulation and experimental results is consistent, showing the validity of the optimal bend parameter n. It should be noted that based on the simulation and experimental results, the optimal n values for Reff =3 µm are the same, 17/32. This confirms the conclusions in Section 3. There are deviations between the measured losses and the simulation varies at the same design geometry. It may result from the sidewall roughness of the fabricated waveguide, which mainly determined by fabrication conditions.
The measurement results for the bending loss of the circular bend, Euler bend, optimal Bezier bend and optimal NA bend are shown in Table 6. For Reff =1, 2, 3, 4, and 5 µm, the loss of the optimal NA bend is the lowest among the four bends.
Table 6. Measured waveguide losses of basic bends at different Reff [dB/90°]
Through the optimization of the bend geometry, the proposed NA bend can reduce the losses due to radiation and mode mismatch, and the losses due to sidewall roughness are mainly determined by fabrication conditions. The experiments conducted in this research are based on high-resolution electron-beam lithography at 100 kV, as for the performances of the NA bends fabricated with commercial foundry processes, the relevant experiments need to be conducted. It is worth pointing out that the superior properties of NA bend have been demonstrated from three aspects: theoretical analysis (Section 2.2), simulation (Section 3) and experiment (Section 4.2). The circular bend and Euler bend, as the most two widely used basic bends, have been manufactured in various fabrication conditions. They are special cases of NA bends. We believe that the NA bends are promising candidates for use in low-loss compact PICs, even under restricted fabrication conditions.
5. Further improvement of the NA bend
In this section, by using the optimal NA bend as a basic bend, an improved method was applied to the basic bend to obtain a bend with lower bending losses. In 1957, H.G. Unger et al. first proposed the use of hybrid Euler and circular bends, and found that the bending loss can be significantly reduced [19]. In this section, an extension of the hybrid method is applied, and hybrid NA bends with lower bending losses are achieved compared with a pure optimal NA bend.
As shown in Fig. 10, the red line shows the optimal NA bend for a certain Reff, and the blue line shows a general NA bend. The red bend has a significantly lower curvature at the starting and ending points than the blue bend, while in the middle part of the bend, the blue bend shows a smoother turn. Therefore, if the advantages of the red and blue bends are combined, a lower loss bend is expected to be obtained. As shown in the dashed line in Fig. 10, by using a red bend for the starting and ending parts and a blue bend for the middle part, a hybrid NA bend can be constructed. It should be noted that the geometry of a hybrid NA bend is determined by n1, n2 and p for a certain Reff. n1 is the bend parameter of the optimal NA bend for a certain Reff, and n2 is the bend parameter of general NA bend. The parameter p represents the proportion of the part among the whole bend that has a bend parameter of n1, in this section p is set as 0.4.
Fig. 10. Geometries of theoptimal NA bend with bend parameter n1, a general NA bend with bend parameter n2, and a hybrid NA bend.
Table 7 shows a summary of the simulated bending loss of pure optimal NA bends and hybrid NA bends for Reff =1, 2, 3 and 4 µm. This table indicates that the hybrid method applied to the NA bend can result in significant loss reduction in most cases. Therefore, the proposed optimal NA bend not only has the lowest losses among the basic bends but also has even lower losses when the improved method is applied to it.
Table 7. Simulated waveguide losses of pure optimal NA bends and hybrid NA bends at different Reff [dB/90°]
We chose a hybrid NA bend with an effective radius of 3 µm for detailed analysis, as it shows the most significant reduction in losses compared to a pure optimal NA bend. As shown in Fig. 11(a), the bend parameters at both the starting and ending parts of the bend are n1= 17/32, and the bend parameter in the middle part of the bend is n2 = -4/32. As shown in Fig. 11(b), the curvature of the bend increases from 0, and there is a steep increase in the curvature at the interface of the two bends. After the curvature gradually decreases to the midpoint of the bend, it changes symmetrically until the end of the bend. It can be concluded that by using the optimal NA bend as the basic bend, the waveguide losses can be further reduced by hybrid method. It is foreseen that by combining other improved methods such as widening width method with NA bend, it may be possible to further reduce the waveguide losses. The NA bend proposed in this paper will be potential for wide application in PICs.
Fig. 11. Optimal NA bend, general NA bend, and hybrid NA bend for Reff =3 µm. (a) Geometries of bends. (b) Curvatures of bends.
6. Conclusion
In this paper, we propose and experimentally demonstrate an n-adjustable (NA) bend based on a SOI platform. The new bend has superior properties compared to the four basic bends currently available. Simulations and experiments indicate that the NA bends can show lower losses than several other basic bends via adjustment of n values when the effective radius is 1, 2, 3, 4 and 5 µm. Further research shows that the bending losses can be reduced by applying the hybrid method to NA bends. The proposed NA bend is important for PICs with hundreds or thousands of bends, and it is promising candidates for use in low-loss ultracompact PICs.
Funding
National Key Research and Development Program of China (2020YFC2004500, 2021YFB3200600); National Natural Science Foundation of China (61675203, 62073307); Chinese Academy of Medical Sciences Innovation Fund for Medical Sciences (2019-I2M-5-019); Chinese Academy of Sciences Joint Fund for Equipment Preresearch (8091A140106).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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