A graded-index curved waveguide and design method
Patent Information
- Application Number
- CN202610858063.3
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2026-06-15
- Publication Date
- 2026-08-11
AI Technical Summary
[0003]当等效弯曲半径超过3μm时，上述特殊弯曲波导相较于圆形弯曲，总损耗可实现有效降低；然而当等效弯曲半径小于3 μm时，其损耗反而难以改善，甚至低于圆形弯曲
[0033]本发明通过在弯曲波导段引入与局部曲率协同变化的渐变折射率分布，使得光从直波导进入弯曲段时，模场能够平缓过渡，显著减小了直-弯界面处的模式转换损耗，在微米尺寸下实现光场强束缚与超低损耗传输；此外，本发明将变换光学原理转化为可操作的设计步骤，使得设计人员无需依赖经验试错，即可针对任意给定的截面尺寸、工作波长和弯曲形状，计算获得最优的渐变折射率分布；通过将等效折射率与局部曲率作为输入，计算出的渐变折射率分布天然满足“与局部曲率动态协同变化”的关系。
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Figure CN122546376A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to a graded refractive index curved waveguide and its design method, belonging to the field of photonic integrated circuit technology. Background Technology
[0002] With the rapid development of integrated photonics, the requirements for miniaturization and low loss in photonic chips are becoming increasingly stringent. As a core passive device in optical link cabling, the minimum bending radius of the bent waveguide directly determines the physical area of the chip. To reduce bending loss while maintaining integration density and process tolerance, the mainstream design approach currently employs Bezier bends, Euler bends, and other low-loss bending structures based on special functions, aiming to reduce mode conversion loss at the junction of straight waveguides and bends while minimizing device size.
[0003] When the equivalent bending radius exceeds 3 μm, the total loss of the aforementioned special bent waveguide can be effectively reduced compared to circular bending. However, when the equivalent bending radius is less than 3 μm, the loss is difficult to improve and may even be lower than that of circular bending. Therefore, how to further reduce the loss of bent waveguides in a compact layout with an equivalent bending radius of less than 3 μm remains a pressing technical problem to be solved in the field of integrated photonics. Summary of the Invention
[0004] The purpose of this invention is to provide a graded refractive index curved waveguide and its design method. By making the refractive index distribution and local curvature of the curved waveguide segment dynamically and synergistically change, strong optical field confinement and ultra-low loss transmission can be achieved at the micrometer scale.
[0005] To achieve the above objectives, the present invention is implemented using the following technical solution.
[0006] On one hand, the present invention provides a graded refractive index curved waveguide, characterized in that it comprises:
[0007] A first straight waveguide, a second straight waveguide, and a curved waveguide segment connecting the first straight waveguide and the second straight waveguide;
[0008] The curved waveguide segment has a gradually changing refractive index distribution; the gradually changing refractive index distribution and the local curvature of the curved waveguide segment change dynamically and in tandem.
[0009] Furthermore, the graded refractive index curved waveguide also includes a substrate layer;
[0010] The first straight waveguide, the second straight waveguide, and the curved waveguide segment are disposed on the substrate layer.
[0011] Furthermore, the bending radius of the bent waveguide segment is set to 1μm~10μm, the height is set to 200nm~300nm, and the width is set to 400nm~600nm.
[0012] Furthermore, the lengths of the first and second straight waveguides are set to 1~10μm, and their heights and widths are consistent with the heights and widths of the curved waveguide segment.
[0013] Furthermore, the shape of the curved waveguide segment is a curve that can be specifically described in functional form, including circular arcs, Bézier curves, and Euler curves.
[0014] Furthermore, the curved waveguide segment is a 90° bend, a 180° bend, or an S-shaped bend, wherein the S-shaped bend is a waveguide segment comprising a first curved segment bending along a first direction and a second curved segment bending along a second direction opposite to the first direction.
[0015] Furthermore, the first straight waveguide and the second straight waveguide adopt a silicon / silicon nitride stacked structure, and the substrate layer material includes silicon dioxide; the curved waveguide section adopts a graded refractive index material.
[0016] On the other hand, the present invention also provides a method for designing a graded refractive index curved waveguide, the method comprising:
[0017] Obtain the waveguide cross-section parameters and target operating wavelength of the first and second straight waveguides;
[0018] The equivalent refractive index of the curved waveguide is obtained based on the waveguide cross-sectional parameters and the target operating wavelength.
[0019] The shape of the curved waveguide segment in physical space is selected, and based on the inverse conformal transformation, the gradient effective refractive index distribution required for the curved waveguide segment along the propagation arc length in physical space is obtained according to the equivalent refractive index and local curvature.
[0020] Based on the gradient effective refractive index distribution, a curved waveguide structure with a corresponding gradient refractive index material is constructed to obtain the gradient refractive index curved waveguide.
[0021] Furthermore, the gradient effective refractive index distribution and the equivalent refractive index satisfy the following expression:
[0022] (1),
[0023] In the formula, This represents the gradually changing effective refractive index distribution at the arc length s along the propagation direction of the curved waveguide segment;
[0024] Indicates the equivalent refractive index;
[0025] The mapping transformation factor represents the inverse conformal transformation at the arc length s along the propagation direction of the curved waveguide segment. ; This represents the centerline trajectory of a curved waveguide segment in physical space at an arc length s along the propagation direction. , The coordinate of the curved waveguide segment in physical space along the propagation direction is represented by the x-axis, where i represents the imaginary unit. This represents the y-axis coordinate of the arc length s along the propagation direction of the curved waveguide segment in physical space.
[0026] Indicates conformal transformation The virtual space of the straight waveguide at the arc length s along the propagation direction of the curved waveguide segment afterwards. .
[0027] Furthermore, the inverse conformal transformation is the inverse mapping of the conformal transformation, and the differential form of the mapping of the conformal transformation can be expressed as:
[0028] (2),
[0029] This represents the angle between the tangent direction at the arc length s along the propagation direction of the curved waveguide segment in physical space and the vertical axis. s is the upper limit of integration, representing the arc length on the center line of the curved waveguide, measured from the starting point;
[0030] Let be the integral variable, representing any arc length position from the starting point to s;
[0031] Represents the arc length of a curved waveguide segment in physical space along the propagation direction. The radius of curvature at that point Represents the arc length of a curved waveguide segment in physical space along the propagation direction. curvature at that point .
[0032] Compared with the prior art, the beneficial effects achieved by the present invention are as follows:
[0033] This invention introduces a gradient refractive index distribution that changes in tandem with the local curvature in the curved waveguide section. This allows for a smooth transition of the mode field when light enters the curved section from the straight waveguide, significantly reducing mode conversion loss at the straight-curved interface. This achieves strong light field confinement and ultra-low loss transmission at the micrometer scale. Furthermore, this invention transforms the principles of transformation optics into operable design steps, enabling designers to calculate the optimal gradient refractive index distribution for any given cross-sectional size, operating wavelength, and curved shape without relying on trial and error. By using the equivalent refractive index and local curvature as inputs, the calculated gradient refractive index distribution naturally satisfies the relationship of "dynamically changing in tandem with the local curvature." Attached Figure Description
[0034] Figure 1 The figure shown is a three-dimensional structural schematic diagram of a graded refractive index curved waveguide provided in this embodiment;
[0035] Figure 2 The figure shown is a schematic diagram of the core cross-section of a graded refractive index curved waveguide provided in this embodiment;
[0036] Figure 3 The figure shown is a top view schematic diagram of a graded refractive index curved waveguide provided in this embodiment;
[0037] Figure 4 The figure shown is a schematic diagram of the refractive index distribution of silicon material with bending radii of 2μm, 3μm, and 4μm in a graded refractive index bent waveguide provided in this embodiment, calculated based on inverse conformal transformation.
[0038] Figure 5 The figure shown is a schematic diagram of the refractive index distribution of silicon nitride material with bending radii of 4μm, 5μm and 6μm in a graded refractive index bent waveguide provided in this embodiment, calculated based on inverse conformal transformation.
[0039] Figure 6 The diagram shown is a comparison of the normalized electric field distribution before and after the application of reverse conformal transformation to a silicon nitride material with a bending radius of 4 μm in a graded refractive index curved waveguide provided in this embodiment.
[0040] Figure 7 The diagram shown is a schematic diagram of the refractive index distribution of silicon material with Euler curves in the curved waveguide segment of a graded refractive index curved waveguide provided in this embodiment, calculated based on inverse conformal transformation.
[0041] Explanation of reference numerals in the attached figures:
[0042] 1-1: First straight waveguide; 1-2: Second straight waveguide; 2: Curved waveguide segment; 3: Substrate layer. Detailed Implementation
[0043] It should be noted that:
[0044] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the embodiments of the present invention and the specific features in the embodiments are detailed descriptions of the technical solution of the present invention, rather than limitations thereof. In the absence of conflict, the embodiments of the present invention and the technical features in the embodiments can be combined with each other.
[0045] The term "and / or" simply describes the relationship between related objects, indicating that three relationships can exist. For example, A and / or B can represent: A alone, A and B simultaneously, or B alone. Additionally, the character " / " generally indicates that the preceding and following related objects have an "or" relationship.
[0046] Example 1
[0047] See Figure 1 This embodiment provides a graded refractive index curved waveguide, including a substrate layer, a first straight waveguide, a second straight waveguide, and a curved waveguide segment connecting the first straight waveguide and the second straight waveguide;
[0048] The first straight waveguide is the input straight waveguide, and the second straight waveguide is the output waveguide; the first straight waveguide, the second straight waveguide, and the curved waveguide segment are disposed on the substrate layer;
[0049] The curved waveguide section has a gradually changing refractive index distribution; the gradually changing refractive index distribution and the local curvature of the curved waveguide section change dynamically and in tandem.
[0050] The bending radius of the curved waveguide section is set to 1μm~10μm, the height is set to 200nm~300nm, and the width is set to 400nm~600nm.
[0051] The lengths of the first and second straight waveguides are set to 1~10μm, and their heights and widths are consistent with those of the curved waveguide section.
[0052] See Figure 2 In one specific implementation, the height of the first straight waveguide and the second straight waveguide in this embodiment is set to 220nm and the width is set to 450nm; the height of the curved waveguide segment is set to 220nm and the width is set to 450nm.
[0053] The shape of the curved waveguide segment can be one or more combinations of circular arcs, Bézier curves, and Euler curves, or it can be a curve that can be specifically described in the form of a function.
[0054] The curved waveguide segment can be 90° curved, 180° curved, or S-shaped curved. The S-shaped curved segment is a waveguide segment that includes a first curved segment that bends along a first direction and a second curved segment that bends along a second direction opposite to the first direction.
[0055] See Figure 3 As a specific implementation method, the curved waveguide segment in this embodiment adopts a 90° curved arc.
[0056] In terms of material selection, this embodiment uses silicon dioxide (SiO2) as the substrate material. SiO2 is the most standard substrate material for silicon-based optoelectronics. Its deposition process is extremely mature and does not require additional development. Furthermore, SiO2 is corrosion-resistant, high-temperature resistant, and has good compatibility with subsequent processes.
[0057] The first and second straight waveguides adopt a silicon / silicon nitride stacked structure Si / SiN. The SiO2 substrate + Si / SiN stack is a classic combination of silicon-based optoelectronics, with low interface stress and less susceptibility to warping and defects.
[0058] The curved waveguide section employs a graded refractive index material. Graded refractive index materials are those whose refractive index changes continuously along a specific direction or space, allowing for optimized optical functionality through precise control of material composition or microstructure. These include silicon-based graded refractive index materials, organic polymer-based graded refractive index materials, inorganic dielectric thin films, and special crystals. In this embodiment, silicon-based graded refractive index materials are preferred. The refractive index can be continuously controlled through doping concentration, meeting the optical path design requirements of transformation optics. This effectively guides light along the curved path and suppresses light radiation loss at the bend. Furthermore, silicon-based graded refractive index materials are compatible with silicon dioxide (SiO2) substrates and silicon / silicon nitride (Si / SiN) stacked straight waveguide processes, exhibiting good interface bonding and excellent overall device integration.
[0059] Example 2
[0060] This embodiment provides a method for designing a graded refractive index bent waveguide, with the goal of designing the graded refractive index bent waveguide in Embodiment 1. The method includes:
[0061] Step S1: Obtain the waveguide cross-section parameters and target operating wavelength of the first and second straight waveguides;
[0062] The graded refractive index curved waveguide design method provided in this embodiment is applied to different platforms, and the selected target operating wavelengths correspond to the typical operating bands of each platform.
[0063] As a specific implementation method, when the design method is applied to platform one made of silicon-on-insulator (SOI), the target operating wavelength is selected as 1550nm, the waveguide cross-sectional width of the first and second straight waveguides is selected as 450nm, and the height is selected as 220nm; when the design method is applied to platform two made of silicon-based silicon nitride (SiN), the target operating wavelength is selected as 850nm, the waveguide cross-sectional width of the first and second straight waveguides is selected as 600nm, and the height is selected as 300nm.
[0064] This embodiment focuses on the design of a bent waveguide for a silicon-on-insulator (SOI) platform.
[0065] Step S2: Obtain the equivalent refractive index of the bent waveguide based on the waveguide cross-sectional parameters and the target operating wavelength;
[0066] Specifically, based on the target operating wavelength and fixed waveguide cross-sectional parameters, the equivalent refractive index of the equivalent straight waveguide is selected. And set it as the desired refractive index in virtual space.
[0067] Determine that the desired refractive index in virtual space is constant. Furthermore, in subsequent steps, the equivalent refractive index of the equivalent straight waveguide in the virtual space is always kept equal to... .
[0068] As a specific implementation method, for Si, the working wavelength is 1550nm and the refractive index n0 is set to 3.455; for SiN, the working wavelength is 850nm and the refractive index n0 is set to 1.88.
[0069] An equivalent straight waveguide refers to a curved waveguide that, after being transformed and mapped to virtual space using optical coordinates, behaves identically to a straight waveguide in terms of optical transmission behavior. Therefore, the straight waveguide in this virtual space is the equivalent straight waveguide corresponding to the curved waveguide.
[0070] Step S3: Select the shape of the curved waveguide segment in the physical space, and based on the inverse conformal transformation, obtain the required gradient effective refractive index distribution of the curved waveguide segment in the physical space along the propagation arc length according to the equivalent refractive index;
[0071] Step S3A1: Define the centerline trajectory of the curved waveguide segment in physical space at the arc length s along the propagation direction as... , The coordinate of the curved waveguide segment in physical space along the propagation direction is represented by the x-axis, where i represents the imaginary unit. This represents the y-axis coordinate of the arc length s along the propagation direction of the curved waveguide segment in physical space.
[0072] Step S3A2: Introduce a conformal transformation mapping to transform the curved waveguide segment in physical space into a straight waveguide in virtual space. The differential form of the conformal transformation mapping can be expressed as:
[0073] (2),
[0074] In the formula, Indicates conformal transformation The straight waveguide in virtual space at the arc length s along the propagation direction of the curved waveguide segment. Virtual space coordinates are ,in The vertical axis corresponds to the propagation direction of the waveguide. The axis is horizontal;
[0075] This represents the angle between the tangent direction at the arc length s along the propagation direction of the curved waveguide segment in physical space and the vertical axis. s is the upper limit of integration, representing the arc length on the center line of the curved waveguide, measured from the starting point;
[0076] Let be the integral variable, representing any arc length position from the starting point to s; s and s are the same arc length coordinates. The apostrophe is added only to distinguish the upper limit of integration and does not contain any meaning of derivative or rate of change.
[0077] Represents the arc length of a curved waveguide segment in physical space along the propagation direction. The radius of curvature at that point Represents the arc length of a curved waveguide segment in physical space along the propagation direction. curvature at that point .
[0078] Conformal mapping transforms a curved trajectory in physical space into a straight line in virtual space, but this only applies to the centerline. For the entire cross-section of the waveguide, the mapping also transforms points on the cross-section into virtual space, and points at different lateral positions will deviate from the centerline in virtual space after the transformation, forming a lateral distribution.
[0079] Step S3A3: Obtain the graded effective refractive index distribution required along the propagation arc length of the curved waveguide segment in physical space based on the equivalent refractive index; the graded effective refractive index distribution and the equivalent refractive index satisfy the following expression:
[0080] (1),
[0081] In the formula, This represents the gradually changing effective refractive index distribution at the arc length s along the propagation direction of a curved waveguide segment in physical space.
[0082] The mapping transformation factor represents the inverse conformal transformation at the arc length s along the propagation direction of the curved waveguide segment. .
[0083] On the imaginary axis of virtual space, that is, when When corresponding to the waveguide centerline: there exists ,at this time For real numbers, , , The gradual effective refractive index distribution of the curved waveguide segment in physical space along the propagation direction at the waveguide centerline is the same as that of the waveguide.
[0084] For a curved waveguide with a fixed cross-section, the gradually changing effective refractive index distribution at the arc length s along the propagation direction of the curved waveguide segment in physical space... The local radius of curvature at that location and the rate of change of its arc length Decide.
[0085] See Figure 4 This embodiment presents the gradual effective refractive index distribution along the arc length calculated by the above-mentioned inverse conformal transformation when the equivalent bending radii on the SOI platform are 2μm, 3μm, and 4μm.
[0086] See Figure 5 This embodiment presents the gradual effective refractive index distribution along the arc length calculated by the above-mentioned inverse conformal transformation when the equivalent bending radii on the SiN platform are 4μm, 5μm, and 6μm.
[0087] See Figure 4 and Figure 5 It can be seen that, in order to achieve the homogenization of the virtual space, the gradient effective refractive index distribution has a smooth gradient characteristic along the arc length. This smooth gradient can be physically achieved by the continuous change of the curvature of the curved waveguide.
[0088] See Figure 6 This paper compares the normalized electric field distribution of a silicon nitride (SiN) with a bending radius of 4 μm before and after inverse conformal mapping. It shows that the contour distribution at the bottom layer provides stronger constraint on mode leakage. The mode field of the conventional waveguide is significantly shifted outwards, with a large amount of optical power diffused into the SiO2 cladding, exhibiting typical radiation mode characteristics. In contrast, the mode field of the graded-index waveguide is forcefully pulled back, tightly bound to the core center, with only a certain amount of wake. Furthermore, the contour line with the maximum value indicates the energy transfer efficiency. A comparison of the two figures shows that the energy in the left figure has significantly attenuated at the end, while the right figure maintains the energy during transmission, ensuring that the energy does not decrease significantly at the end.
[0089] Step S4: Based on the gradient effective refractive index distribution, construct a curved waveguide structure with a corresponding gradient refractive index material to obtain a gradient refractive index curved waveguide.
[0090] Construct waveguide geometry and perform simulation verification, compare data from different waveguides, summarize and draw conclusions.
[0091] The curvature function is obtained after step S3. Then, the centerline coordinates of the curved waveguide are reconstructed through numerical integration, as shown in the following expression:
[0092] (3),
[0093] This constructs the geometry of a curved waveguide, enabling three-dimensional finite-difference time-domain calculations. Transmission simulations are performed at wavelengths of 1550 nm and 850 nm. A mode light source is placed at the waveguide input, and the normalized transmission power is extracted after simulation to calculate the bending loss.
[0094] Table 1 shows the simulated transmission loss of silicon and graded-index bent optical waveguides under different bending radii:
[0095] Table 1. Simulation of transmission loss in silicon and graded-index bent optical waveguides
[0096]
[0097] Table 2 shows the simulated transmission loss of silicon nitride and graded refractive index bent optical waveguides under different bending radii:
[0098] Table 2 Simulation of transmission loss in silicon nitride and graded-index curved optical waveguides
[0099]
[0100] As shown in Tables 1 and 2, under extremely small equivalent bending radii (such as 2 μm for SOI and 4 μm for SiN), the transmission loss of the constant-width bent waveguide designed in this invention is significantly lower than that of the two traditional designs. The fundamental reason is that the curvature gradient path derived through inverse conformal transformation essentially corrects the equivalent refractive index tilt caused by bending, making the mode field distribution of light in the bent waveguide as close as possible to the modes in the straight waveguide, thereby significantly suppressing radiation loss.
[0101] It should be noted that although the above embodiment uses a 90° bend as an example, the present invention is also applicable to other angled bending structures such as Bézier bends and Euler bends. Figure 7 This refers to the refractive index distribution applied to Euler curves. Figure 7 The study demonstrates the refractive index distribution applied to Euler curves. Unlike traditional circular arcs, its curvature is constantly changing. The smooth transition at the junction of the curved and straight sections reduces mode mismatch, thereby achieving lower transmission loss.
[0102] Waveguides whose curved curves can be accurately described by function can all be optimized for refractive index distribution by reverse conformal transformation using the design concept of this invention, so as to obtain extremely low transmission loss while maintaining a compact size.
[0103] Those skilled in the art should understand that the embodiments of the present invention have been described above in conjunction with the accompanying drawings. However, the present invention is not limited to the specific embodiments described above. The specific embodiments described above are merely illustrative and not restrictive. Those skilled in the art can make many other forms under the guidance of the present invention without departing from the spirit and scope of the claims, and these forms are all within the protection scope of the present invention.
Claims
1. A graded refractive index curved waveguide, characterized in that, include: A first straight waveguide, a second straight waveguide, and a curved waveguide segment connecting the first straight waveguide and the second straight waveguide; The curved waveguide segment has a gradually changing refractive index distribution; the gradually changing refractive index distribution and the local curvature of the curved waveguide segment change dynamically and in tandem.
2. The graded refractive index curved waveguide according to claim 1, characterized in that, The graded refractive index curved waveguide also includes a substrate layer; The first straight waveguide, the second straight waveguide, and the curved waveguide segment are disposed on the substrate layer.
3. The graded-index curved waveguide of claim 1, wherein, The bending radius of the bent waveguide section is set to 1μm~10μm, the height is set to 200nm~300nm, and the width is set to 400nm~600nm.
4. The graded-index curved waveguide of claim 1, wherein, The lengths of the first and second straight waveguides are set to 1~10μm, and their heights and widths are consistent with those of the curved waveguide segment.
5. The graded-index curved waveguide of claim 1, wherein, The shape of the curved waveguide segment is a curve that can be specifically described in the form of a function, including circular arcs, Bézier curves, and Euler curves.
6. The graded-index curved waveguide of claim 1, wherein, The curved waveguide segment is a 90° bend, a 180° bend, or an S-shaped bend. The S-shaped bend is a waveguide segment that includes a first curved segment bending along a first direction and a second curved segment bending along a second direction opposite to the first direction.
7. The graded-index curved waveguide of claim 2, wherein, The first and second straight waveguides adopt a silicon / silicon nitride stacked structure, and the substrate layer is made of silicon dioxide; the curved waveguide section adopts a graded refractive index material.
8. A method of designing a graded-index curved waveguide of the graded-index curved waveguide of claim 1, characterized by, The method includes: Obtain the waveguide cross-section parameters and target operating wavelength of the first and second straight waveguides; The equivalent refractive index of the curved waveguide is obtained based on the waveguide cross-sectional parameters and the target operating wavelength. The shape of the curved waveguide segment in physical space is selected, and based on the inverse conformal transformation, the gradient effective refractive index distribution required for the curved waveguide segment along the propagation arc length in physical space is obtained according to the equivalent refractive index and local curvature. Based on the gradient effective refractive index distribution, a curved waveguide structure with a corresponding gradient refractive index material is constructed to obtain the gradient refractive index curved waveguide.
9. The graded-index curved waveguide design method of claim 8, wherein, The gradient effective refractive index distribution and the equivalent refractive index satisfy the following expression: （1）， In the formula, This represents the gradually changing effective refractive index distribution at the arc length s along the propagation direction of the curved waveguide segment; n represents the equivalent refractive index; The mapping transformation factor represents the inverse conformal transformation at the arc length s along the propagation direction of the curved waveguide segment. ; This represents the centerline trajectory of a curved waveguide segment in physical space at an arc length s along the propagation direction. , The coordinate of the curved waveguide segment in physical space along the propagation direction is represented by the x-axis, where i represents the imaginary unit. This represents the y-axis coordinate of the arc length s along the propagation direction of the curved waveguide segment in physical space. represents a straight waveguide after a conformal transformation of the curved waveguide segment along the propagation direction at an arc length s, .
10. The graded-index curved waveguide design method of claim 9, wherein, The inverse conformal transformation is the inverse mapping of the conformal transformation, and the differential form of the mapping of the conformal transformation can be expressed as: （2）， an angle between a tangent direction at an arc length s along a propagation direction of a curved waveguide segment representing a physical space and a vertical axis, s is an upper limit of integration, representing an arc length from a starting point on a center line of the curved waveguide; is the integral variable, representing an arbitrary arc length position between the starting point and s; Represents the arc length of a curved waveguide segment in physical space along the propagation direction. The radius of curvature at that point Represents the arc length of a curved waveguide segment in physical space along the propagation direction. curvature at that point .