## Abstract

Ultimately low-loss 90° waveguide bend composed of clothoid and normal curves is proposed for dense optical interconnect photonic integrated circuits. By using clothoid curves at the input and output of 90° waveguide bend, straight and bent waveguides are smoothly connected without increasing the footprint. We found that there is an optimum ratio of clothoid curves in the bend and the bending loss can be significantly reduced compared with normal bend. 90% reduction of the bending loss for the bending radius of 4 μm is experimentally demonstrated with excellent agreement between theory and experiment. The performance is compared with the waveguide bend with offset, and the proposed bend is superior to the waveguide bend with offset in terms of fabrication tolerance.

© 2017 Optical Society of America

## 1. Introduction

Photonic integrated circuits (PICs) for optical interconnects have attracted much attention for optical computing, communication, and so on. Si photonics has been widely used for the platform of the PICs and various large-scale PICs were demonstrated [1]. In these PICs, Si-wire waveguides are used for signal transmission and there are a lot of waveguide bends and intersections to form the circuits. Typical loss of normal 90° waveguide bend of a single-mode Si-wire waveguide is on the order of 0.01 dB for the bending radius (*R*) of several microns [2]. Although the value is already very low, if there are hundreds of bends, total bending loss becomes several dB. Therefore, further reduction of the bending loss is desirable for future large-scale PICs, if possible.

Various approaches for reducing the bending loss have been proposed. For example, in [3], spline curves are connected to normal bend to smoothly connect input/output and bending waveguides. Here, normal bend means input and output straight waveguides are connected through bending waveguide with constant curvature, 1/*R*. In [3], the effect of the length of the spline curves added to the input and output of normal bend were investigated and it was demonstrated that longer spline curves lead to reduced bending loss due to the smooth connections. However, since the footprint was not fixed and always larger than that of normal bend (*R*^{2}) due to the use of input/output spline curves, the effect of spline curves for the fixed footprint was not clear.

In [4] and [5], it was theoretically proposed that TIR mirror [4] and outside trench [5] reduce the bending loss, by reducing the leakage of the field to the outside of the waveguide. However, these approaches require additional fabrication processes aside from forming Si-wire waveguides. In [6], clothoid curves were used to reduce coupling loss between straight and bent waveguide with constant curvature for silica waveguides. The clothoid curves enable smooth connections between straight waveguides and bending waveguides since the curvature of clothoid curves linearly increases from zero. Recently, the application of this curve for Si waveguide bend was demonstrated [7–9]. In [7], the S-bend waveguide using clothoid curve in an optical delay line was demonstrated. In [8], a relatively large bending radius (20 μm) Si-rib waveguide L-bend composed of 100% clothoid curve was incorporated in Si-PIC for measuring the waveguide loss (No theoretical and experimental data on bending loss was presented). In [9], a suppression of unwanted coupling at the specific *R* to higher order modes in a multimode 90° waveguide bend composed of 100% clothoid curve has been reported.

In this paper, a novel 90° waveguide bend composed of clothoid and normal curves is proposed to ultimately reduce the bending loss. From the numerical results based on three-dimensional (3-D) finite element method (FEM) [10], we found that there is an optimum ratio of clothoid curves in the bend and the bending loss can be significantly reduced compared with normal bend. In other words, the waveguide bend composed of 100% clothoid curve is not optimum, and the combined use of normal and clothoid curve is critically important to reduce the bending loss ultimately. The proposed bend is fabricated in CMOS platform and the bending loss of 0.002 dB/90° for *R* = 4 µm is achieved, which is 1/10 of normal bend. Excellent agreement between theory and experiment is presented, showing the validity of our results. Furthermore, the performance comparison between the proposed bend and waveguide bend with offset [11] is made and the proposed bend is superior in terms of fabrication tolerance. The proposed bend can be used in the PICs as conventional bend since the input/output position and footprint are the same with normal bend, and therefore, it is useful for reducing the bending loss of future large-scale PICs.

It should be noted that although the bending loss of Si-wire waveguides are investigated in this paper, the bending loss reduction of rib waveguides [12] is also important (and may be more challenging) because the rib waveguide is essential for composing the active components, such as switches and modulators.

## 2. Bending waveguide structure and theory

#### 2.1 90° waveguide bend composed of clothoid and normal curves

Here, we consider 90° waveguide bend composed of clothoid (red lines) and normal (blue line) curves as shown in Fig. 1 [13]. We call this proposed bend as “clothoid bend” hereafter. Green dashed line in Fig. 1 shows normal bend for comparison. The clothoid curve (also called Euler curve) is one of the easement curves and its curvature linearly increases from zero to 1/*R*_{min}. If a starting point of the clothoid curve is set as origin as shown in Fig. 1, a point with the coordinate of (*x*, *z*) on the clothoid curve is expressed as Eqs. (1) and (2). At the same time, the curvature radius *R* and the curve length *L* at the point satisfy Eq. (3).

*A*is a constant number called as a clothoid parameter. The ending point of the clothoid curve has a minimum curvature radius

*R*

_{min}and the length of the curve is

*L*

_{max}. The clothoid curve at the input is connected to a symmetric clothoid curve at the output of a 90° waveguide bend via a normal bend with the curvature of 1/

*R*

_{min}. Therefore, starting and ending points of the proposed bend are (0, 0) and (

*R*

_{eff},

*R*

_{eff}) as shown in Fig. 1. The ratio of clothoid curves can be varied from 0 to 100% by changing

*A*. We fairly compare the bending loss of the clothoid bend and normal bend with the bending radius of

*R*

_{eff}, since both bends have the same footprint as shown in Fig. 1. Note that

*R*

_{eff}for normal bend is just a usual definition of bending radius. The insets show outlines of the 90° waveguide bend with clothoid curves for

*R*

_{eff}= 4 µm and

*A*= 1.3, 2.4, and 2.68. Black dashed lines are the outline of the waveguide with normal curves for comparison.

For practical application, the proposed bend can be written as follows. From Eq. (3), two of three parameters (*R*_{min}, *L*, and *A*) are necessary to define the clothoid curve. For example, if *R*_{min} and *A* are given for fixed *R*_{eff}, *L*_{max} and the coordinate of ending point of the clothoid curve are determined, and therefore, two symmetric clothoid curves can be drawn. From these data, the coordinate of the center of the circle with the bending radius of *R*_{min} can be obtained by using, for example, bisection method, and the normal bend between two clothoid curves can be drawn.

The reason for choosing the clothoid curve for the easement curve is as follows. The shift of the peak of the transverse electric field distributions in the bending waveguide is proportional to 1/*R*. In the clothoid curve, since the curvature is linearly increased from 0 to 1/*R*_{min}, the peak of the transverse electric field distributions is smoothly shifted to that of the bending waveguide with the constant curvature of 1/*R*_{min}. Of course, although there is a possibility that other easement curves, such as spline curves [3], can reduce the bending loss as well, we employ the clothoid curve here due to above reason.

#### 2.2 Waveguide structure and theory

Figure 2 shows the top-view of bending waveguide and the cross section of the Si-wire waveguide. The waveguide width is *w* = 400 nm, and the height is *h* = 210 nm. The Si core is surrounded by silica and the refractive indices of Si and silica are 3.476 and 1.444. Since the curvature in the clothoid curve is not constant, conventional estimation of bending loss based on equivalent effective index method [14] cannot be used and full 3-D simulation is necessary. We use home-made 3-D FEM [10] for the analysis. In 3-D FEM, following vector wave equation is considered.

**is the electric field,**

*E**k*

_{0}is the free-space wavenumber, [μ

_{r}] and [ε

_{r}] are permeability and permittivity tensors.

By discretizing the computational domain by FEM, vector wave Eq. (4) is reduced to simultaneous equation as

where {*E*} is the electric field vector to be solved, {

*E*

_{in}} is the input electric field vector, and [

*P*] is the finite-element matrix. The detailed formulation can be found in [10]. By solving Eq. (5) with the arbitrary input field (in this paper, the eigenmode of Si-wire waveguide), the electric field of the system is obtained. The number of unknowns for discretizing the 3-D computational volume is on the order of 10

^{7}.

## 3. Bending loss of clothoid bends

Figure 3 and Fig. 4 show the bending loss per 90° calculated by 3-D FEM as a function of the ratio of clothoid curves for *R*_{eff} = 3, 4, and 5 µm, and *R*_{eff} = 2 μm. The wavelength is λ = 1.55 μm and quasi-TE mode is considered. The horizontal line is the bending loss of normal bend. For *R*_{eff} = 3, 4, and 5 µm, the bending loss is reduced by introducing clothoid curves and there are optimum ratios for reducing the loss. The bending loss can be divided into two losses: the connection loss between input/output straight waveguide and bending waveguide and the radiation loss in the bending waveguide. The clothoid bend is effective to reduce the connection loss since the curvature of the clothoid curve varies continuously. However, the bending radius of normal curve contained in the proposed bend, *R*_{min}, is smaller than that of *R*_{eff}, leading to the increased radiation loss. Therefore, if the ratio of clothoid curve is too small, the field transition is not enough and the loss is generated at the boundary of clothoid and normal curves. On the other hand, if the ratio of clothoid curve is too large, *R*_{min} becomes small and the radiation loss is generated around the normal bend. Therefore, there is an optimum ratio for reducing the bending loss in the clothoid bend and they are 68, 60, and 50% for *R*_{eff} = 3, 4, and 5 µm, respectively. For *R*_{eff} = 2 μm, the bending loss is increased by adding the clothoid curves. From the result, the clothoid bend is not effective for too small *R*_{eff} since the radiation loss is dominant rather than the connection loss. Figure 5 shows *xz*-plane electric field distributions (|** E**|

^{2}) of normal and clothoid bends (

*A*= 2.4) for

*R*

_{eff}= 4 μm. The vertical position is at the center of the waveguide. For the normal bend, the field is more radiated into cladding region, especially around the interface between input straight and bending waveguides.

The clothoid bend with *R*_{eff} = 4 µm is fabricated in CMOS platform to verify its validity and effectiveness. Four different bends with clothoid parameters *A* = 0, 1.3, 2.4, and 2.68 are fabricated. For each bend, four waveguides with the number of bends, *N _{b}* = 98, 198, 298, and 398 turns, are fabricated. The

*A*= 0, 1.3, 2.4, and 2.68 correspond to the ratio of 0 (normal bend), 14, 58, and 98%, respectively, and are named as normal bend, clothoid-A, clothoid-B, and clothoid-C for convenience. The clothoid-B has the optimum ratio for

*R*

_{eff}= 4 µm as shown in Fig. 3. Figure 6 shows the micrographs of the winding layout of the normal bend and clothoid-A, -B, and -C. The clothoid-B has more roundabout profiles than normal bend. We measured 10 chips containing the same patterns of 16 waveguides (4 types of

*N*for 4 types of bend shape). TE-polarized light is coupled to the chip through inverse taper spot size converter fabricated at the both edges of the chip [15]. Transmitted light is received by optical spectrum analyzer. The bending loss is measured by subtracting the transmitted power through a reference straight waveguide fabricated in the same chip from the transmitted power through the bending waveguide. Figure 7 shows the bending loss of 4 types of 90° waveguide bend as a function of

_{b}*N*, where points represent the averages of 10 chips and top and bottom of bars represent the maximum and minimum values in the 10 chips. The bending loss linearly increases in proportion to

_{b}*N*. The dashed lines are regression line for each bend and the slopes correspond to the bending loss per 90° and they are 0.019, 0.016, 0.002, and 0.006 dB/90° for normal bend, clothoid-A, clothoid-B, and clothoid-C, respectively. The 90% reduction in the bending loss compared with the normal bend is obtained for clothoid-B.

_{b}Points and bars in Fig. 8 show the measured bending loss per 90° as a function of the ratio of clothoid curves. The circle points represent the slope value of the regression lines of Fig. 7 and top and bottom of bars represent the loss calculated by the slope value of regression lines of the maximum and minimum values in Fig. 7. Dashed line shows the calculated results (the same data in Fig. 3). Calculated and measured results are in excellent agreement and the optimum point for low bending loss is well reproduced in the measured results, showing the validity and usefulness of proposed bend. Figure 9 shows the bending loss spectra for each bend with *N _{b}* = 398 for one of the 10 chips. The clothoid-B has the lowest bending loss with wide wavelength range from 1520 nm to 1620 nm. It should be noted that the spectrum oscillation is observed in clothoid-A. The oscillation probably comes from multi-path interference [16] in the multiple bends originating from undesired excitation of leaky higher-order mode at the interface of clothoid and normal curves due to abrupt field discontinuity.

## 4. Comparison with the bending waveguide with offset

Bending waveguides with an offset structure [11] (we call it offset bend, hereafter) have been commonly used for reducing the bending loss. Here, we compare the bending loss of the clothoid bend and offset bend and show the superiority of the proposed bend in terms of fabrication tolerance. Figure 10 shows the top-view of the offset bend. By shifting the center of bending waveguide, the connection loss with the input/output waveguide can be reduced. Here, the shift is Δ*x*, and the *R*_{eff} for the offset bend is defined for the center of the input and output waveguides as shown in Fig. 10 to make fair comparison in terms of the footprint. Based on 3-D FEM calculation, the optimum Δ*x* is 15 nm for 1.55 μm. Figure 11 shows the transmission spectra of normal, clothoid and offset bends for *R*_{eff} = 4 µm. From the figure, the proposed bend is superior to other bends for the wavelength shorter than 1.625 μm. Especially, for λ < 1.45 μm, although the Si-wire waveguide is multimode, the clothoid bend has almost no loss in the fundamental mode due to the smooth connection. For longer wavelength region (λ > 1.625 μm), since the radiation loss is dominant in the bending loss, the clothoid bend is inferior to the offset bend. However, the structure of the clothoid bend is optimized for the waveguide with the cross section of 210 × 400 nm^{2} at 1.55 μm (single mode at the wavelength). For longer wavelength, since the single mode condition is relaxed, optimum cross section of the waveguide will be different.

Figure 12 shows the bending loss per 90° as a function of the waveguide width for normal, clothoid and offset bends at 1.55 μm. For all bends, the loss is reduced (increased) for wider (narrower) waveguide width. And the loss of the clothoid bend is always smaller than those of other bends. Figure 13 shows the bending loss per 90° as a function of the offset width for the offset bend at 1.55 μm. The waveguide width is kept constant (400 nm). For Δ*x* = 0 nm, the bend is normal bend. If Δ*x* is changed from the optimized value (15 nm), the loss is increased rapidly. Therefore, the clothoid bend is superior to the offset bend in terms of fabrication tolerance since two parameters have to be considered in the offset bend.

## 5. Conclusion

Ultimately low-loss 90° waveguide bend composed of clothoid and normal curves was proposed for dense optical interconnect PICs. From the numerical simulation, we found that the bending loss can be significantly reduced by using the combination of clothoid and normal curves and there is optimum ratio of clothoid curves in the bend. The proposed bend for *R*_{eff} = 4 µm was fabricated in CMOS platform and the bending loss of 0.002 dB/90° was achieved, which is 1/10 of normal bend. Excellent agreement between theory and experiment was presented, showing the validity of our results. The performance is compared with that of the offset bend, and it was found that the proposed bend is superior to the offset bend in terms of fabrication tolerance. The proposed bend has the lower bending loss and does not increase the footprint compared with normal bend, and does not need any additional fabrication processes. Therefore, total loss from waveguide bends in large-scale PICs containing large number of bends can be significantly reduced by just replacing the normal bend with the proposed bend.

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