Bent Waveguide

Abstract

In an optical waveguide according to the present disclosure, a configuration of a bent waveguide having a novel structure is proposed. Regarding four types of curves A, B, C, and D that are proposed, values obtained by differentiating curvatures thereof along the waveguide become 0 at points at which one end l of the bent waveguide is 0 and the other end l thereof is L. It is possible to suppress optical loss and intermode crosstalk that occur in a bent waveguide part. Even with a bent waveguide having a multi-mode waveguide width, superior intermode crosstalk properties and low loss are realized, as compared with a clothoid curve in the prior art.

Description

[Technical Field]

[0001] The present invention relates to an optical waveguide for an optical circuit, and more particularly to a bent waveguide. [Background technology]

[0002] In order to increase the communication capacity per optical communication device, research and development of smaller and more functional optical modules is actively underway. One promising technology for this is silicon photonics (SiP).

[0003] Silicon photonics is an optical circuit technology that uses waveguides formed on an SOI (Silicon On Insulator) wafer, with a core material of silicon (Si) and a clad material of silica glass (SiO2). Silicon waveguides have a high relative refractive index difference between the core and clad, making it possible to confine light in a tiny area, thereby enabling the realization of extremely small optical circuits.

[0004] In bending waveguides, which change the direction of light in optical circuits, a small bending radius causes optical loss, which can be a problem. If the bending radius is made large enough, this can be reduced to a negligible level, but the circuit size increases.

[0005] To solve the above problems, curved waveguides with different shapes are sometimes used instead of the usual circular arc shape. One example is a clothoid curve, where the product of the length along the propagation direction and the radius of curvature is constant. Bent waveguides using clothoid curves can achieve lower loss than conventional circular arc-shaped waveguides, but research is also being conducted on curved shapes with even lower loss. [Prior art documents] [Non-patent literature]

[0006] [Non-Patent Document 1] Xiaohui Jiang, Hao Wu, Daoxin Dai, “Low-loss and low-crosstalk multimode waveguide bend on silicon,” Optics Express, Vol. 26, No. 13 (2018) Summary of the Invention [Problem to be solved by the invention]

[0007] In particular, for waveguides with relatively large propagation losses, such as silicon waveguides, another characteristic is sometimes required for bent waveguides: low inter-mode crosstalk. Inter-mode crosstalk can be reduced by designing a wide waveguide under multimode conditions, for example, by increasing the waveguide width. However, single-mode conditions must be satisfied for bent waveguides. If a bent waveguide with low inter-mode crosstalk can be realized, even in multimode applications, the aforementioned benefits of a wide waveguide, such as reduced loss and reflection, can be achieved in the bent waveguide as well, improving overall device performance. [Means for solving the problem]

[0008] One aspect of the present invention is a curved waveguide in which the radius of curvature r gradually changes from a first value (R1) at a first end to a second value (R2) at a second end, the curvature being represented by the reciprocal 1 / r of the radius of curvature, and the derivative of the curvature with respect to the length l along the waveguide being zero at the first end and the second end. [Effects of the Invention]

[0009] A bent waveguide is provided that is miniaturized and has low optical loss and intermodal crosstalk. [Brief explanation of the drawings]

[0010] [Figure 1] 1A and 1B are diagrams illustrating the configuration of a clothoid curved waveguide including a diminishing curve. [Figure 2]FIG. 1 is a diagram showing the relationship between curve length and curvature for four proposed curves A to D. [Figure 3] FIG. 10 is a diagram illustrating the derivation of x and y values ​​for drawing four curves in Cartesian coordinates. [Figure 4] FIG. 10 is a diagram showing the inter-modal crosstalk characteristics of the four proposed curves A to D. [Figure 5] 10 is a table comparing the zeroth-order mode loss of a curved waveguide according to four types of curves A to D. [Figure 6] 10 is a table showing a comparison of zero-order mode loss under single-mode conditions. DETAILED DESCRIPTION OF THE INVENTION

[0011] The optical waveguide of this disclosure provides a novel bent waveguide configuration. The loss and inter-modal crosstalk characteristics of four types of bent waveguide configurations are described, in comparison with a conventional clothoid bent waveguide. The bent waveguide can be used to realize, for example, a direction-changing circuit.

[0012] First, we explain the characteristics of the clothoid curve, which is widely used in bent waveguides, and the design conditions required for bent waveguides. Then, we explain the configurations of the four proposed bent waveguides.

[0013] As mentioned above, instead of a circular arc shape, a clothoid curve (Euler spiral) is used as a curved waveguide. In a clothoid curve, the radius of curvature changes continuously from infinity to a specified value, so the change in propagation mode also becomes continuous, making it possible to reduce the loss that occurs in a curved waveguide. A curve with a continuously decreasing curvature, such as a clothoid curve, is also called a diminishing curve. A 90° change in direction in a clothoid curve is realized using a diminishing curve as follows:

[0014] FIG. 1 illustrates the configuration of a clothoid-shaped curved waveguide including a decay curve. The waveguide 100 in FIG. 1 consists of two straight waveguides 101-1 and 101-2 arranged at a 90° angle relative to each other, connected by two decay curves 102-1 and 102-2. Two decay curves 102-1 and 102-2 are created such that the radius of curvature is ∞ when the angle θ is 0° (point P) and r is R when the angle θ is 45° (point Q). These two decay curves are then joined symmetrically at the point θ=45°. This allows the radius of curvature to be ∞ at θ=0° and 90°, enabling the decay curves 102-1 and 102-2 to be coupled with the adjacent straight waveguides 101-1 and 101-2 with low loss. The waveguide 100 in FIG. 1 constitutes a direction change circuit.

[0015] Although silicon waveguides can realize extremely small optical circuits, they have relatively large propagation losses. As mentioned above, in silicon waveguides, low inter-mode crosstalk is required for bent waveguides.

[0016] Silicon waveguides are sometimes designed with a width wider than that which satisfies the single-mode condition, i.e., multimode conditions. By widening the waveguide to meet the multimode condition, it is possible to reduce the overlap between the waveguide sidewall and the optical mode, thereby reducing radiation loss and reflection due to sidewall roughness. It also reduces phase errors caused by manufacturing variations in the waveguide width and suppresses nonlinear effects by lowering the peak intensity of the light within the waveguide.

[0017] However, such wide waveguides are generally limited to straight waveguides because widening a curved waveguide to make it multimode results in inter-mode coupling from the zeroth mode to higher modes, resulting in inter-mode crosstalk. Inter-mode crosstalk from the zeroth mode to higher modes is undesirable because it ultimately causes optical loss and group delay ripple. Therefore, curved waveguides have been designed to satisfy the single-mode condition.

[0018] The inventors came up with the idea that if a bent waveguide with low inter-modal crosstalk could be realized even in a multimode design, the effects of a wide waveguide, such as suppression of loss and reflection, could be obtained, and the performance of devices using silicon waveguides could be improved. Non-Patent Document 1 reports that by using the above-mentioned clothoid curve, inter-modal crosstalk can be kept low even when the waveguide is bent while maintaining a wide width that results in multimode operation. However, the inter-modal crosstalk reduction effect shown in Non-Patent Document 1 was not sufficient.

[0019] Below, we will explain the configuration of four novel curved waveguides disclosed herein and their excellent inter-modal crosstalk and loss characteristics. Unless otherwise specified, the term "curvature" refers to the reciprocal 1 / r of the radius of curvature r. Furthermore, since sine and cosine only differ in phase, the shape of the waveguide may sometimes be referred to as sine, even though it is mathematically a cosine.

[0020] Figure 2 shows the relationship between the curve length and curvature for the four proposed curves, A to D. Figure 2(a) is a graph showing the length l (μm) along the curve from the connection point with the linear waveguide on the horizontal axis and the curvature 1 / r on the vertical axis for the four curves, A, B, C, and D. Figure 2(b) is a graph showing the length l (μm) along the curve on the horizontal axis and the derivative of the curvature with respect to l on the vertical axis. For comparison, the two graphs also show a clothoid curve. In the two graphs in Figure 2, l = 0 corresponds to point P in Figure 1, where the curvature is 0. The maximum value of l, i.e., the total length of the curve, is L = 7.85 μm, which corresponds to point Q in Figure 1 where θ = 45°. Here, r = R = 5 μm. As will be described later, the relationship l = 2rθ exists between r, l, and θ for all curves in the two graphs shown in Figure 2.

[0021] Next, we will explain what each of the four curves A, B, C, and D is. - Curve A: Sine half wavelength type - The differential curvature (1 / r)' of curve A is expressed by the following formula (1). Since the range of l = 0 to L is half the wavelength of a sine wave (sine) function, it is called a sine half-wave type. TIFF0007787466000001.tif9150

[0022] If we integrate while paying attention to the boundary conditions of r=∞ at l=0 and r=R at l=L, the curvature becomes as shown in equation (2), and although the phase is shifted, it still has the shape of half the wavelength of the sine function. TIFF0007787466000002.tif9150

[0023] - Curve B: Sine type - The differential curvature (1 / r)' of curve B is expressed by equation (3). Since the range of l=0 to L is one wavelength of a sine wave (sine function), it is called a sine wave. TIFF0007787466000003.tif9150

[0024] If we integrate while keeping in mind the boundary conditions mentioned above, the curvature becomes as shown in equation (4), and the shape is expressed in the form of the sum of a linear function and a sine function. TIFF0007787466000004.tif9150

[0025] - Curve C: Quadratic function type - The differential curvature (1 / r)' of curve C is expressed by equation (5). Since it is a quadratic function of l, it is called a quadratic function type. TIFF0007787466000005.tif9150

[0026] By integrating while keeping in mind the boundary conditions mentioned above, the curvature becomes as shown in equation (6), which is a cubic function. TIFF0007787466000006.tif11150

[0027] - Curve D: Linear function type - The curve D is called a linear function type because the differential (1 / r)' of the curvature is expressed by two different linear functions at the boundary of l=L / 2, as shown in equations (7) and (8). TIFF0007787466000007.tif9150 TIFF0007787466000008.tif9150

[0028] If we integrate while keeping in mind the boundary conditions mentioned above, the curvature becomes as shown in equations (9) and (10), which are expressed by two different quadratic functions with l = L / 2 as the boundary, and are called quadratic function types. TIFF0007787466000009.tif10150 TIFF0007787466000010.tif10150

[0029] For all four curves A, B, C, and D described above, it can be easily confirmed that 1 / r=0 and (1 / r)'=0 when l=0, and 1 / r=1 / R and (1 / r)'=0 when l=L.

[0030] Referring to Figure 2, for each of these four curved waveguides, A, B, C, and D, the curvature changes smoothly when it is connected to a straight waveguide where 1 / r = 0 at l = 0 (point P). Furthermore, when it is connected to its own inverted version where r = R at l = L (point Q), for example, when curve A and its inverted version A are connected, the curvature changes smoothly. In contrast, for a clothoid curve, the product of the radius of curvature r and the length l is constant by definition, so the curvature 1 / r and the length l are proportional, as shown in Figure 2(a). The differential of the curvature, (1 / r)', does not become 0 at any length l of the curved waveguide, as shown in Figure 2(b). In other words, for a clothoid curve, the change in curvature at the connection points with the preceding and following straight waveguides is continuous but not smooth. The four curves A, B, C, and D differ from the clothoid curve in that the value obtained by differentiating the curvature along the waveguide is 0 at one end l=0 of the curved waveguide and at the other end l=L.

[0031] The ends (end points) of the four curves A, B, C, and D shown in Figure 2 refer to the starting and ending points when each curve is expressed mathematically, and correspond to points P and Q of the clothoid curved waveguide in Figure 1. Therefore, if the curves of curvature and their derivatives in Figure 2 are folded symmetrically along the line l = L, a 90° bent waveguide is represented. Both ends of this bent waveguide are continuously connected to straight waveguides, forming part of an optical circuit.

[0032] Therefore, the curved waveguide of the present disclosure can be implemented as a curved waveguide in which the radius of curvature r gradually changes from a first value (R1, point P) at a first end to a second value (R2, point Q) at a second end, the curvature being represented by the reciprocal of the radius of curvature, 1 / r, and the differential coefficient of the curvature differentiated with respect to the length l along the waveguide being 0 at the first end and the second end.

[0033] Next, we will explain how to derive the x and y coordinates in the Cartesian coordinate system when actually drawing these four curves A, B, C, and D as optical circuit patterns. In the above equations (2), (4), (6), (9), and (10), the curved waveguide is expressed by the radius of curvature r, which is a function of the length l. Here, the four curves A, B, C, and D are expressed in Cartesian coordinates x(l) and y(l) via the angle θ, which will be described later.

[0034] Figure 3 is a diagram explaining how to derive coordinate values ​​when drawing four types of curves in a Cartesian coordinate system. At a certain point on a curve, the radius of curvature is r, the length from the starting point is l, and the angle between the normal to the curve and the y-axis is θ, with counterclockwise angles being positive. Assume that (r, θ, l)=(0, 0, 0) at the starting point and the following equation at the end point. TIFF0007787466000011.tif6150 For an infinitesimal portion of a curve, the angle, length, x component of the length, and y component of the length are dθ, dl, dx, and dy, respectively.

[0035] From FIG. 3, the following relationship holds between the parameters in the polar coordinate system of FIG. 1 and the Cartesian coordinate system. TIFF0007787466000012.tif6150 TIFF0007787466000013.tif6150 TIFF0007787466000014.tif6150 From equation (11), the following equation is obtained for θ: TIFF0007787466000015.tif11150

[0036] Furthermore, by applying θ in equation (14) to equations (12) and (13), the following equations are obtained for the x component of length and the y component of length. TIFF0007787466000016.tif11150 TIFF0007787466000017.tif11150

[0037] Since 1 / r is known, for example as in equation (2) for curve A, the x and y coordinates can be found by substituting it into equations (15) and (16). The integrals of equations (15) and (16) generally cannot be solved analytically, but can be solved by numerical integration.

[0038] Furthermore, by applying equation (2) etc. to equation (14) and substituting (l, θ) = (L, Θ), the following relationship can be obtained for the length L of the curved waveguide for all of curves A, B, C, and D. TIFF0007787466000018.tif6150

[0039] Although the derivation will be omitted, equation (17) also holds for clothoid curves, and in the case of circular arcs, L = RΘ. The total length of the clothoid curve and the four curves A, B, C, and D is equal to the total length of a circular arc with a radius (2R) twice the minimum bending radius R of these curves. It can be seen that the clothoid curve and the four curves A, B, C, and D are roughly the same size.

[0040] Next, we will explain that the bent waveguides using the four types of curves A, B, C, and D have lower loss and inter-modal crosstalk than the bent waveguides using the conventional circular arcs and clothoid curves.

[0041] Figure 4 shows the intermodal crosstalk characteristics of the four proposed curves, A to D. For each curve, a 90° direction-changing circuit similar to that shown in Figure 1 was fabricated, and the vertical axis shows the first-order mode conversion ratio (dB), calculated as the coupling rate of light input in the zeroth-order mode to the first-order mode. The horizontal axis shows wavelength (μm). In addition to the four curves, A to D, the graph also shows the cases of a circular arc and a clothoid curve. The core material is Si, the cladding material is SiO2, the waveguide thickness is 0.22 μm, and the waveguide width is 0.8 μm. The polarization is TE polarization. To ensure that the overall curve lengths are consistent, the bending radius is set to 10 μm for the circular arc and half that, 5 μm, for the other curves, in accordance with the relationship between the length L of the bent waveguide mentioned above.

[0042] Referring to Figure 4, the coupling coefficient to the first-order mode at a wavelength of 1.55 μm was -18.3 dB for the circular arc, -33.2 dB for the clothoid, -55.6 dB for (A) the sine half-wavelength type, -40.1 dB for (B) the sine wave type, -50.8 dB for (C) the quadratic function type, and -41.3 dB for (D) the linear function type. While the clothoid also significantly reduced the first-order mode coupling coefficient compared to the circular arc, the curved waveguides shown in curves A, B, C, and D of this disclosure all exhibited even lower first-order mode coupling coefficients than the clothoid. In particular, the direction-changing circuit using the sine half-wavelength type shown in curve A exhibited inter-mode crosstalk characteristics that were more than 22 dB better than the clothoid curve. This superiority was confirmed across almost the entire C-band and L-band, wavelength bands commonly used in communications.

[0043] Figure 5 is a table comparing the zeroth-order mode loss of the bent waveguides based on the four curves A to D. This table shows the same intermodal crosstalk characteristics as those compared in Figure 4, but plots the zeroth-order mode loss (dB) at a wavelength of 1.57 μm. The bent waveguides based on curves A to D all exhibit lower transmission loss, i.e., higher transmittance, than the arc and clothoid curves. Table 1 also shows the footprint, which is the area occupied by the bent waveguide. It can be seen that, assuming the same curve length, the footprints of the four curves A to D are smaller than those of the arc and clothoid curves. As is clear from the table in Figure 5, the bent waveguides based on the four curves A to D are superior to the arc and clothoid curves in terms of lower zeroth-order mode loss, lower coupling to the first mode, and smaller footprint.

[0044] The comparative evaluation of the inter-mode crosstalk characteristics in Figure 4 and the zeroth-order mode loss in Figure 5 was performed with a waveguide width of 0.8 μm (800 nm). A waveguide width of 0.8 μm is wider than the single-mode condition. Next, we evaluated the bending loss at waveguide widths that satisfy the single-mode condition for the bent waveguides using the four curves A to D.

[0045] Figure 6 is a table comparing the zeroth-order mode loss under single-mode conditions for four curved waveguides. The comparison in the table in Figure 6 shows the zeroth-order mode coupling rate when the waveguide width is set to 500 nm to satisfy the single-mode conditions and the minimum bending radius is set to 2.5 μm (5 μm for the arc only). Because the waveguide is more resistant to bending under single-mode conditions, the minimum bending radius was set smaller for evaluation than when the waveguide width was 800 nm in Figures 4 and 5.

[0046] 6, the zeroth-order mode loss is lower for all of the curved waveguides according to the curves A, B, C, and D of the present disclosure compared to the arc and clothoid curves. Similarly, the footprints of all of the curved waveguides according to the curves A, B, C, and D are smaller than those of the arc and clothoid curves.

[0047] As explained above, the curved waveguides according to the curves A, B, C, and D of the present disclosure can suppress coupling to higher-order modes to a lower level than conventional techniques, even at widths that satisfy multimode conditions. Therefore, the benefits of a wide waveguide, such as suppression of loss, reflection, and nonlinear effects, can be achieved in the curved waveguide section, improving overall device performance. Furthermore, the optical loss of the zeroth mode is also lower than that of conventional techniques. This low loss is not limited to waveguide widths that satisfy multimode conditions, but can also be achieved when applied to waveguides that satisfy single-mode conditions. The footprint is also smaller than that of curved waveguides according to conventional techniques, contributing to device miniaturization.

[0048] In the above explanation, the structure of the bent waveguide section has been described using mathematical expressions. However, these bent waveguides can be used to form a direction change circuit in which two attenuation curves 102-1 and 102-2 are coupled with the front and rear straight waveguides 101-1 and 101-2 with low loss, as shown in Figure 1. That is, a direction change circuit can be realized using two straight waveguides and two bent waveguides that follow the curves A, B, C, and D between them. The two bent waveguides are inverted relative to each other at the connection point. The direction change circuit is formed by a first straight waveguide, a bent waveguide connected to the first straight waveguide at a first end, another bent waveguide connected to the second end of the bent waveguide and having a shape that is the inverse of the bent waveguide, and a second straight waveguide connected to the first end of the other bent waveguide. The direction change circuit shown in Figure 1 is a 90° direction change circuit, but it can also be applied to direction changes at angles gentler than 90° if the differential value of the curvature at the two ends of the curved waveguide is configured to be zero.

[0049] The curved waveguides of curves A, B, C, and D can suppress optical loss and inter-mode crosstalk by continuously changing the curvature and by making the differential value of the curvature (1 / r) zero at the two ends of the curved waveguide. Therefore, there are other curved waveguides that can achieve similar effects, not just these four curves. [Industrial Applicability]

[0050] The present invention can be used in optical circuits.

Claims

1. 1. A curved waveguide having a radius of curvature r that gradually changes from a first value (R1) at a first end to a second value (R2) at a second end, The curvature is expressed by the reciprocal of the radius of curvature, 1 / r, a differential coefficient obtained by differentiating the curvature with respect to a length l along the waveguide is 0 at the first end and the second end; When the total length of the curve is L and the minimum bending radius is R, the differential coefficient is It is expressed as either Bent waveguide.

2. 2. The curved waveguide according to claim 1, wherein the width of the waveguide satisfies the multimode condition.

3. a first straight waveguide; a curved waveguide according to claim 1 connected at the first end to the first straight waveguide; another bent waveguide connected to the second end of the bent waveguide and having an inverted shape of the bent waveguide; a second straight waveguide connected at a first end of the other curved waveguide; A direction change circuit comprising:

4. 4. The direction changing circuit according to claim 3, wherein the bent waveguide and the other bent waveguide each correspond to a 45° change in direction, achieving a 90° bend overall.

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