Two-dimensional rectangular optical waveguide mode analysis method based on mode matching method

Abstract

The invention relates to the field of electromagnetic field mode matching, and discloses a two-dimensional rectangular optical waveguide mode analysis method based on a mode matching method, and the method comprises the following steps: obtaining the related structure parameters of a two-dimensional rectangular optical waveguide; slicing to obtain a corresponding one-dimensional slab waveguide, and obtaining a one-dimensional analysis mode; expressing an LSE mode and an LSM mode by using a one-dimensional analysis mode; obtaining an electromagnetic field component expression; obtaining a relational expression and a characteristic equation between mode coefficients; and solving the characteristic equation by using an optimization algorithm to obtain an analyzed complex propagation constant and a mode coefficient, and substituting the complex propagation constant and the mode coefficient into the electromagnetic field component expression to obtain the electromagnetic field distribution of any mode on the two-dimensional plane. According to the method disclosed by the invention, high-precision simulation can be carried out on two-dimensional rectangular optical waveguides of various sizes. The method is high in calculation precision, good in stability and few in occupied resources, and the problems that a traditional numerical method is low in calculation precision, long in calculation time and poor in high-order mode orthogonality are solved.

Description

Technical Field

[0001] The present invention relates to the field of electromagnetic field mode matching, and particularly to an analytical method for two-dimensional rectangular optical waveguide modes based on a mode matching method. Background Art

[0002] A two-dimensional rectangular optical waveguide is a key structure for optical communication and integrated optics. The material generally consists of a core layer and a cladding layer. The core layer in the center of the waveguide uses a high refractive index material, such as silicon (Si), silicon nitride (Si3N4), or lithium niobate (LiNbO3), to confine and guide light waves. The cladding layers above and below the core layer use low refractive index materials, such as silica (SiO2), polymer, or air, to surround the core layer to achieve refractive index contrast and ensure total internal reflection. The core layer is rectangular in the cross-sectional (x - y plane), and the waveguide extends along the z-axis direction, and light waves propagate therein. It is very necessary to study the mode characteristics supported by the rectangular optical waveguide.

[0003] The so-called mode is the electromagnetic field solution that satisfies Maxwell's equations and boundary conditions, representing a specific propagation form of light in the waveguide. Usually, the propagation constant is used to characterize a mode, and each mode has a unique spatial distribution of the electric field and magnetic field. The existence of the mode confines the light energy as much as possible in the core layer of the waveguide, avoiding energy leakage to the outside, thus achieving low-loss transmission. The modes in the optical waveguide directly determine what functions the optical waveguides with different shapes and different materials can play. Almost all functions, from high-speed communication to precision sensing, rely on the precise control and utilization of the modes. Precise mode parameters are the core basis for judging whether the waveguide can achieve the expected function. In a silicon-based photonic chip, if the mode calculation error exceeds 0.1%, the optical signal crosstalk rate will increase by 30%. Therefore, high-precision modes are the cornerstone and core of the modeling of an optical waveguide device. And if one hopes to simulate an optical waveguide structure with longitudinal non-uniformity, usually in addition to the guided modes, high-precision higher-order radiation complex modes are also required as the basis to model the waveguide structure.

[0004] The mainstream commercial method for solving modes (propagation constant and field distribution) through Maxwell's equations and boundary conditions is a numerical mode solver based on numerical methods. The core idea of the numerical mode solver is discretization approximation. First, the waveguide structure is meshed, Maxwell's equations are transformed into algebraic equations at each grid point, and a global matrix equation is formed through grid iteration. Then, its eigenvalues and eigenvectors are solved, which respectively correspond to the propagation constant and the electromagnetic field distribution of the mode that satisfies Maxwell's equations and boundary conditions.

[0005] Mainstream numerical mode solvers include the finite difference method and the finite element method. These methods are effective and general, but their accuracy mainly depends on the level of numerical discretization. Therefore, if high-precision calculation results are required, their calculation costs will be very high. Additionally, when dealing with high-order complex modes, these numerical simulation methods may lead to serious errors, especially the loss of mode orthogonality. Moreover, the mode field distributions obtained by numerical methods are discrete values given at grid points, making it difficult to perform subsequent analyses and optimizations such as sensitivity analysis and yield optimization. Summary of the Invention

[0006] To solve the above technical problems, the present invention provides an analytical method for two-dimensional rectangular optical waveguide modes based on the mode matching method, aiming to solve the disadvantages of low calculation accuracy and long calculation time of numerical simulation algorithms, and achieve the purpose of fully analytically and quickly solving the two-dimensional rectangular optical waveguide modes.

[0007] To achieve the above object, the technical solution of the present invention is as follows:

[0008] An analytical method for two-dimensional rectangular optical waveguide modes based on the mode matching method, comprising the following steps:

[0009] Step 1, obtain the relevant structural parameters of the two-dimensional rectangular optical waveguide;

[0010] Step 2, in combination with the structural parameters, slice the two-dimensional rectangular optical waveguide along the longitudinal direction according to different material refractive indices to obtain corresponding one-dimensional slab waveguides, and obtain the propagation constants and electromagnetic field distributions of the one-dimensional analytical modes according to the nonlinear control equations of the one-dimensional analytical modes of the one-dimensional slab waveguides;

[0011] Step 3, represent the LSE and LSM modes using the projection relationship of the one-dimensional analytical mode on the two-dimensional plane;

[0012] Step 4, expand each component of the electromagnetic field of the two-dimensional rectangular optical waveguide mode in the horizontal direction with the LSM and LSM modes as the basis to obtain the expression of the electromagnetic field components of the two-dimensional rectangular optical waveguide mode on the two-dimensional plane;

[0013] Step 5, use the expression of the electromagnetic field components of the two-dimensional rectangular optical waveguide on the two-dimensional plane, in combination with the field continuity condition and the boundary condition, to obtain the relationship between the mode coefficients and the characteristic equation characterizing the complex propagation constant and mode coefficients of the two-dimensional rectangular optical waveguide;

[0014] Step 6, use an optimization algorithm to solve the characteristic equation to obtain the analytical complex propagation constant and mode coefficients, and substitute the complex propagation constant and mode coefficients into the expression of the electromagnetic field components to obtain the electromagnetic field distribution of any mode on the two-dimensional plane.

[0015] In the above solution, the two-dimensional rectangular optical waveguide includes a silica lower cladding, a silicon core layer, a vacuum upper cladding, and a perfectly matched layer; the silicon core layer is located at the center above the silica lower cladding, the vacuum upper cladding is located above the silica lower cladding and the silicon core layer, and the perfectly matched layer wraps around the outside of the silica lower cladding, the silicon core layer, and the vacuum upper cladding; its related structural parameters include the material refractive index, height, and width of the silica lower cladding, the material refractive index, height, and width of the silicon core layer, the material refractive index, height, and width of the vacuum upper cladding, and the material refractive index, height, and absorption factor parameters of the perfectly matched layer.

[0016] In the above solution, in step 2, the method of slicing the two-dimensional rectangular optical waveguide is as follows:

[0017] First, establish a (x, y, z) coordinate system for the two-dimensional rectangular optical waveguide, with the center of the silicon core layer as the origin, the horizontal right direction as the positive x-axis direction; the vertical upward direction as the positive y-axis direction; and the direction perpendicular to the paper surface from the origin as the positive z-axis direction;

[0018] Then, slice the two-dimensional rectangular optical waveguide along the y direction at the boundary interface between the silicon core layer and the vacuum upper cladding, and divide it into three one-dimensional slab waveguides, including slab one, slab two, and slab three, where slab two contains the silicon core layer, and slab one and slab three do not contain the silicon core layer.

[0019] In the above solution, in step 2, the one-dimensional analytical modes of the one-dimensional slab waveguide include TE mode and TM mode, specifically as follows:

[0020] Establish a (y, v, u) coordinate system for the three obtained one-dimensional slab waveguides. For slab two, take the intersection point of the core layer diagonals as the origin of the (y, v, u) coordinate system, the vertical upward direction from the origin as the positive y-axis direction, that is, the direction of material refractive index change; the horizontal right direction from the origin as the positive u-axis direction, and this direction is the one-dimensional mode propagation direction; the direction perpendicular to the paper surface from the origin as the positive v-axis direction, and the electromagnetic field distribution is uniform in this direction; for slab one and slab three, the height of the origin is the same as that of slab two, the horizontal position of the origin is at the center of the vacuum upper cladding in this area, and establish a (y, v, u) coordinate system with the vertical upward direction from the origin as the positive y-axis direction and the horizontal right direction from the origin as the positive u-axis direction;

[0021] Assume that the one-dimensional slab waveguide mode propagates along the u direction, the material refractive index changes in the y direction, and the v direction is uniformly distributed. In this coordinate system, the mode of the one-dimensional slab waveguide is decoupled into a TE mode containing H y 、H u and E v and a TM mode containing E y 、E u and H v where, H yRepresents the magnetic field component propagating in the y direction, H u Represents the magnetic field component propagating in the u direction, E v Represents the electric field component propagating in the v direction, E y Represents the electric field component propagating in the y direction, E u Represents the electric field component propagating in the u direction, H v Represents the magnetic field component propagating in the v direction; the propagation constants along the u direction for the TE mode and the TM mode are respectively and

[0022] In the above solution, in step 2, the specific method for obtaining the propagation constant and the electromagnetic field distribution of the one-dimensional analytical mode is as follows:

[0023] First, the initial value of the propagation constant of the one-dimensional slab waveguide mode is obtained by the second-order finite difference algorithm;

[0024] Then, the initial value of the propagation constant is substituted into the nonlinear control equation of the one-dimensional slab waveguide mode using the fsolve function in MATLAB for calculation to obtain the propagation constant of the one-dimensional analytical mode;

[0025] Secondly, substituting the propagation constant into the analytical formula of the field distribution of the one-dimensional slab waveguide, the electromagnetic field distribution of the one-dimensional analytical mode can be obtained.

[0026] In the above solution, the specific method for step 3 is as follows:

[0027] (1) Decompose the TE / TM mode of the one-dimensional analytical mode into the LSE / LSM mode in the two-dimensional rectangular optical waveguide:

[0028] For the TE mode, H y Remains unchanged, and H u Is projected and decomposed into H x 、H z Components on the x-z plane, where H y Represents the magnetic field component in the y direction, H x Represents the magnetic field component in the x direction, H z Represents the magnetic field component in the z direction; E v Is projected and decomposed into E x 、E z Components on the x-z plane, where E x Represents the electric field component in the x direction, E z Represents the electric field component in the z direction. The mode composed of H y 、H x 、H z 、E x 、E z These five components is the LSE mode;

[0029] The same decomposition method is adopted for the TM mode to obtain the LSM mode including E y , E x , E z , H x , H z . E x is the electric field component in the x direction, E y is the electric field component in the y direction, E z is the electric field component in the z direction, H x is the magnetic field component in the x direction, H z is the magnetic field component in the z direction;

[0030] (2) Define the propagation constants in the ±x directions and the angular relationships of the propagation constants in different directions:

[0031] According to the dispersion relation, the propagation constants of the LSE and LSM modes in the +x direction are respectively defined as and The propagation constants in the -x direction are respectively and

[0032] According to the relationship between the two coordinate systems, the ratio θ E of the propagation constant of the LSE mode in the x direction to the propagation constant in the u direction is defined as The ratio θ M of the propagation constant of the LSM mode in the x direction to the propagation constant in the u direction is defined as

[0033] In the above solution, the specific method of step 4 is as follows:

[0034] Assume that the number of LSE modes is N E , and the number of LSM modes is N M . Each LSE and LSM mode that composes the electromagnetic components of the two-dimensional rectangular optical waveguide is assigned a mode coefficient. The coefficient of the LSE mode is p represents the order of the LSE mode, ranging from the 1st to the N E th LSE mode, ± represents forward propagation or backward propagation; the coefficient of the LSM mode is q represents the order of the LSM mode, ranging from the 1st to the N M th LSM mode;

[0035] Each electromagnetic component in the x-y interface of the two-dimensional rectangular optical waveguide is composed of the superposition of LSE and LSM modes with different coefficients. The specific expressions of the electromagnetic field components are as follows:

[0036]

[0037]

[0038] Among them, E x (x, y) is the x-component of the electric field in the (x, y) plane, and E y (x, y) is the y-component of the electric field in the (x, y) plane, and E z (x, y) is the z-component of the electric field in the (x, y) plane, and H x (x, y) is the x-component of the magnetic field in the (x, y) plane, and H y (x, y) is the y-component of the magnetic field in the (x, y) plane, and H z (x, y) is the z-component of the magnetic field in the (x, y) plane; p represents the order of the LSE mode and the order of the TE mode, and q represents the order of the LSM mode and the order of the TM mode. represents the E v component of the p-th TE mode, represents the E u component of the q-th TM mode, represents the E y component of the q-th TM mode, represents the H u component of the p-th TE mode, represents the H v component of the q-th TM mode, represents the H y component of the p-th TE mode; represents the angular conversion relationship of the p-th LSE, represents the angular conversion relationship of the q-th LSM; represents the mode coefficient of the p-th LSE mode propagating in the forward direction, represents the mode coefficient of the p-th LSE mode propagating in the reverse direction, represents the mode coefficient of the q-th LSM mode propagating in the forward direction, represents the mode coefficient of the q-th LSM mode propagating in the reverse direction; in is the propagation constant of the p-th TE mode in the x direction, x is the coordinate position in the x direction, in is the propagation constant of the q-th TM mode in the x direction, x is the coordinate position in the x direction.

[0039] In the above solution, in step 5, by combining the field continuity condition and the boundary condition, the specific method for obtaining the relationship between the mode coefficients is as follows:

[0040] (1) According to the field continuity condition, the tangential electromagnetic field components E y and Ez , H y , H z Keep continuous, substitute \(x = 0\) into \(E\) in step 4 y (x, y), E z (x, y), H y (x, y) and H z In the electromagnetic field component expressions of \((x, y)\), simplify to obtain an equation in matrix form;

[0041] (2) For the outer boundary, there are: the tangential component of the electric field \(E\) y (x = P, y) = 0 and \(E\) z (x = P, y) = 0, where \(P\) is the position of the outer boundary of the two-dimensional rectangular optical waveguide in the positive \(x\)-direction. Substitute \(x = P\) into \(E\) in step 4 y (x, y) and \(E\) z In the expression of \((x, y)\), let \(E\) y (x, y) and \(E\) z The expression of \((x, y)\) is equal to 0, and the relationship between the mode coefficients is obtained through simplification and C 2+ represents the mode coefficient of the LSE mode propagating along the +x direction on the right side of the \(x = 0\) interface; \(C\) 2- represents the mode coefficient of the LSE mode propagating along the -x direction on the right side of the \(x = 0\) interface; \(D\) 2+ represents the mode coefficient of the LSM mode propagating along the +x direction on the right side of the \(x = 0\) interface; \(D\) 2- represents the mode coefficient of the LSM mode propagating along the -x direction on the right side of the \(x = 0\) interface;

[0042] Boundary condition matrix Boundary condition matrix diag represents the matrix form, and \(j\) is the imaginary unit, represents the propagation constant of the \(p\)th TE mode at \(x = P\); represents the propagation constant of the \(q\)th TM mode at \(x = P\);

[0043] (3) The center position of the two-dimensional rectangular optical waveguide is at \(x = -W / 2\). The modes of the two-dimensional rectangular optical waveguide can be divided into odd modes and even modes:

[0044] When it is an odd mode, \(E\) y (x = -W / 2, y) = \(E\) z (x = -W / 2, y) = 0, where \(x = -W / 2\) is the \(x\)-coordinate at the center of the core layer of the two-dimensional rectangular optical waveguide. Substitute \(x = -W / 2\) into \(E\) in step 4 y (x, y) and \(E\) z In the expression of \((x, y)\), let \(E\) y (x, y) and \(E\) z(x, y) is equal to 0, and by simplification, the relational expression between the mode coefficients is obtained. and

[0045] When it is the even mode, H y H(x = -W / 2, y) = z H(x = -W / 2, y) = 0. Substitute x = -W / 2 into H in step 4. y H(x, y) and z In the expression of H(x, y), let y H(x, y) and z H(x, y) is equal to 0, and by simplification, the relational expression between the mode coefficients is obtained. and

[0046] where C 1+ represents the mode coefficient of the LSE mode propagating in the +x direction on the left side of the x = 0 interface; C 1- represents the mode coefficient of the LSE mode propagating in the -x direction on the left side of the x = 0 interface; D 1+ represents the mode coefficient of the LSM mode propagating in the +x direction on the left side of the x = 0 interface; D 1- represents the mode coefficient of the LSM mode propagating in the -x direction on the left side of the x = 0 interface; The boundary condition matrix represents the propagation constant of the p-th TE mode at x = -W / 2; The boundary condition matrix represents the propagation constant of the q-th TM mode at x = -W / 2.

[0047] In the above solution, in step 5, the odd-mode nonlinear characteristic equation in the two-dimensional rectangular optical waveguide containing the complex propagation constant and the mode coefficients is as follows:

[0048]

[0049] The even-mode nonlinear characteristic equation in the two-dimensional rectangular optical waveguide containing the complex propagation constant and the mode coefficients is as follows:

[0050]

[0051] where I is the identity matrix, and T1, T2, T3, and T4 are the transfer matrices.

[0052] In the above solution, in step 6, the fsolve function in MATLAB is used to solve the nonlinear equation to obtain the required complex propagation constant, and then the complex propagation constant is substituted back into the nonlinear characteristic equation in step 5 to obtain the eigenvector D 1- ; Through D 1- the remaining mode coefficients C 1+ 、C2- , D 2- is represented; the relationship between the mode coefficients is as follows:

[0053]

[0054] In addition, at the x = 0 interface, the relationship between the mode coefficients on the left and right sides is:

[0055]

[0056]

[0057] where H v1 represents the H v component on the left side of the x = 0 interface, and its superscript represents the mode numbers from 1 to N M1 on the left side of the x = 0 interface; E y1 represents the E y component on the left side of the x = 0 interface, and its superscript represents the mode numbers from 1 to N M1 on the left side of the x = 0 interface; E y2 represents the E y component on the right side of the x = 0 interface, and its superscript represents the mode numbers from 1 to N M2 on the right side of the x = 0 interface; E v1 represents the E v component on the left side of the x = 0 interface, and its superscript represents the mode numbers from 1 to N E1 on the left side of the x = 0 interface; H y1 represents the H y component on the left side of the x = 0 interface, and its superscript represents the mode numbers from 1 to N E1 on the left side of the x = 0 interface; H y2 represents the H y component on the right side of the x = 0 interface, and its superscript represents the mode numbers from 1 to N E2 on the right side of the x = 0 interface.

[0058] Thus, the final complex propagation constant and the LSE mode coefficients and LSM mode coefficients are obtained. Substituting them into the electromagnetic component expressions at the x - y interface of the two - dimensional rectangular optical waveguide obtained in step 4, the electromagnetic field distribution of any mode in the two - dimensional rectangular optical waveguide can be obtained.

[0059] Through the above technical solution, the analytical method for the modes of a two - dimensional rectangular optical waveguide provided by the present invention based on the mode - matching method has the following beneficial effects:

[0060] 1. The present invention adopts an analytical method for two-dimensional rectangular optical waveguide modes based on the pattern matching method, which can quickly and accurately calculate and simulate the effective refractive index and field distribution of two-dimensional rectangular optical waveguides, solving the problems of low calculation accuracy and long calculation time of traditional numerical methods.

[0061] 2. The simulation effect of the present invention at the singularity of the silicon core layer is better than that of the FDM simulation with a grid accuracy of 1 nm.

[0062] 3. The present invention can obtain the field distribution of accurately stable simulated high-order complex modes.

[0063] 4. The present invention solves the problem of poor mode orthogonality of traditional numerical algorithms.

[0064] In summary, the present invention can perform high-precision simulations on various optical waveguides of two-dimensional rectangular optical waveguides. The present invention has high calculation accuracy, good stability, and low resource consumption, solving the problems of low calculation accuracy, long calculation time, and poor high-order mode orthogonality of traditional numerical methods. BRIEF DESCRIPTION OF THE DRAWINGS

[0065] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for use in the description of the embodiments or the prior art.

[0066] Figure 1 It is a schematic flow chart of an analytical method for two-dimensional rectangular optical waveguide modes based on the pattern matching method disclosed in the embodiments of the present invention.

[0067] Figure 2 It is a schematic diagram of the two-dimensional rectangular optical waveguide structure;

[0068] Figure 3 It is a schematic diagram of slicing the two-dimensional rectangular optical waveguide;

[0069] Figure 4 It is a schematic diagram of establishing a coordinate system;

[0070] Figure 5 It is an analysis diagram of the convergence of the effective refractive index;

[0071] Figure 6 It is the log 10 (FRE) value of the Ex field component under different one-dimensional mode numbers;

[0072] Figure 7 It is the comparison of the effective refractive indices between the present invention and the FDE method under different grid sizes;

[0073] Figure 8The two-dimensional guided-mode electric field distribution diagram obtained for the present invention. (a) The Ex distribution diagram of the two-dimensional guided mode obtained for the present invention; (b) The Ey distribution diagram of the two-dimensional guided mode obtained for the present invention; (c) The Ez distribution diagram of the two-dimensional guided mode obtained for the present invention;

[0074] Figure 9 The log 10 (FRE) values between the present invention and FDE under different grid sizes;

[0075] Figure 10 The simulation of the present invention at the singularity under different one-dimensional mode numbers. (a) The singularity simulation diagram obtained by the present invention when the one-dimensional mode number is 50; (b) The singularity simulation diagram obtained by the present invention when the one-dimensional mode number is 100; (c) The singularity simulation diagram obtained by the present invention when the one-dimensional mode number is 150; (d) The singularity simulation diagram obtained by the present invention when the one-dimensional mode number is 200;

[0076] Figure 11 The simulation of FDE at the singularity under different grid sizes. (a) The singularity simulation diagram obtained by FDM when the grid size is 5nm; (b) The singularity simulation diagram obtained by FDM when the grid size is 1nm;

[0077] Figure 12 The 3rd-order PML mode diagram obtained for the present invention. (a) The Ex field component diagram of the 3rd-order PML mode; (b) The Ey field component diagram of the 3rd-order PML mode; (c) The Ez field component diagram of the 3rd-order PML mode;

[0078] Figure 13 The 12th-order PML mode diagram obtained for the present invention. (a) The Ex field component diagram of the 12th-order PML mode; (b) The Ey field component diagram of the 12th-order PML mode; (c) The Ez field component diagram of the 12th-order PML mode;

[0079] Figure 14 The Ex field distribution diagram of the 12th-order PML mode under different one-dimensional mode numbers. (a) The Ex field distribution diagram of the 12th-order PML mode when the one-dimensional mode number is 50; (b) The Ex field distribution diagram of the 12th-order PML mode when the one-dimensional mode number is 100; (c) The Ex field distribution diagram of the 12th-order PML mode when the one-dimensional mode number is 200;

[0080] Figure 15 The mode orthogonality comparison diagram. (a) The two-dimensional mode orthogonality obtained by the FDE method; (b) The two-dimensional mode orthogonality obtained by the present invention when using 100 one-dimensional modes; (c) The two-dimensional mode orthogonality obtained by the present invention when using 200 one-dimensional modes.

[0081] In the figure, 1 is the silica lower cladding; 2 is the silicon core layer; 3 is the vacuum upper cladding; 4 is the perfectly matched layer. Detailed implementation mode

[0082] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention.

[0083] The present invention provides an analytical method for two-dimensional rectangular optical waveguide modes based on a pattern matching method, as Figure 1 shown, including the following steps:

[0084] Step 1, obtain relevant structural parameters of the two-dimensional rectangular optical waveguide.

[0085] As Figure 2 shown, the two-dimensional rectangular optical waveguide includes a silica lower cladding 1, a silicon core layer 2, a vacuum upper cladding 3, and a perfectly matched layer 4; the silicon core layer 2 is located at the upper center position of the silica lower cladding 1, the vacuum upper cladding 3 is located above the silica lower cladding 1 and the silicon core layer 2, and the perfectly matched layer 4 is wrapped outside the silica lower cladding 1, the silicon core layer 2, and the vacuum upper cladding 3; its relevant structural parameters include the material refractive index, height, and width of the silica lower cladding 1, the material refractive index, height, and width of the silicon core layer 2, the material refractive index, height, and width of the vacuum upper cladding 3, and the material refractive index, height, and absorption factor parameters of the perfectly matched layer 4.

[0086] Taking Figure 2 the two-dimensional rectangular optical waveguide shown as an example, the coordinate system is a Cartesian coordinate system (x, y, z), x is the horizontal direction, y is the refractive index change direction, and z is the mode propagation direction. The refractive index of the silicon core layer 2 is 3.47, the thickness is 0.22 μm, the width is 0.5 μm, the refractive index of the silica lower cladding 1 is 1.44, the thickness is 0.5 μm, the width is 1.5 μm, the refractive index of the air upper cladding is 1, the thickness is 0.5 μm, and the width is 1.5 μm. The outermost boundary adopts the boundary condition of perfect electric conductor PEC + perfectly matched layer 4PML, and the thickness D pml of the PML is 0.5 μm, and the refractive index is the refractive index of the adjacent medium. The expression of the coordinate stretching factor s in the PML region is:

[0087]

[0088] where j is the imaginary unit, n is the refractive index of the medium, λ is the wavelength of 1550 nm, R represents the reflectivity between the PML region and the PEC boundary interface, set to 1e-3, D pml is the thickness of the PML region, and ρ represents the thickness to the inner interface of the PML. The waveguide center position is at x = -W / 2, W is -0.25 μm, and the outer boundary position is at x = P, P = 1 μm.

[0089] Based on obtaining the structural parameters of the two-dimensional rectangular optical waveguide, the two-dimensional rectangular optical waveguide is segmented.

[0090] Step 2: Combining the structural parameters, slice the two-dimensional rectangular optical waveguide along the longitudinal direction according to different material refractive indices to obtain corresponding one-dimensional slab waveguides, and obtain the propagation constant and electromagnetic field distribution of the one-dimensional analytical mode according to the nonlinear control equation of the one-dimensional analytical mode of the one-dimensional slab waveguide.

[0091] The method for slicing the two-dimensional rectangular optical waveguide is as follows:

[0092] First, establish a (x, y, z) coordinate system for the two-dimensional rectangular optical waveguide, with the center of the silicon core layer 2 as the origin, the horizontal right direction as the positive x-axis direction; the vertical upward direction as the positive y-axis direction; and the direction perpendicular to the paper surface from the origin as the positive z-axis direction.

[0093] Then, slice the two-dimensional rectangular optical waveguide at the boundary interface between the silicon core layer 2 and the vacuum upper cladding layer 3 along the y direction, and divide it into three one-dimensional slab waveguides, including slab one, slab two, and slab three, where slab two contains the silicon core layer 2, and slab one and slab three do not contain the silicon core layer 2. The two-dimensional rectangular optical waveguide in this embodiment is symmetric about x = -W / 2, and only one of slab three and slab one needs to be calculated.

[0094] Using the structural parameters in step 1, it can be known that W = 0.25 μm. In slab two, the thickness d of the silicon core layer 2 Si is 0.22 μm, and the thickness d of the vacuum upper cladding layer 3 Air is 0.5 μm, and the thickness d of the silicon dioxide layer SiO2 is 0.5 μm, and D pml is 0.5 μm. In slab three, the thickness d of the vacuum upper cladding layer 3 Air is 0.72 μm, and the thickness d of the silicon dioxide layer SiO2 is 0.5 μm, and D pml is 0.5 μm. See step 1 for the refractive indices of relevant materials.

[0095] Determine the mode components of the one-dimensional slab waveguide.

[0096] The one-dimensional analytical modes of the one-dimensional slab waveguide include TE (transverse electric) mode and TM (transverse magnetic) mode, specifically as follows:

[0097] A (y, v, u) coordinate system is established for the three obtained one-dimensional planar waveguides. For planar waveguide two, the intersection point of the core layer diagonals is taken as the origin of the (y, v, u) coordinate system. The positive y-axis direction is vertically upward from the origin, which is the direction of the change in material refractive index. The positive u-axis direction is horizontally to the right from the origin, and this direction is the one-dimensional mode propagation direction. The positive v-axis direction is perpendicular to the paper surface from the origin, and the electromagnetic field distribution is uniform in this direction. For planar waveguide one and planar waveguide three, the height of the origin is the same as that of planar waveguide two, and the horizontal position of the origin is at the center of the vacuum upper cladding 3 in this region. A (y, v, u) coordinate system is established with the positive y-axis direction vertically upward from the origin and the positive u-axis direction horizontally to the right from the origin.

[0098] Assume that the one-dimensional planar waveguide mode propagates along the u direction, the material refractive index changes in the y direction, and the distribution in the v direction is uniform. In this coordinate system, the one-dimensional planar waveguide mode is decoupled into a TE mode containing H y 、H u and E v and a TM mode containing E y 、E u and H v . Among them, H y represents the magnetic field component propagating in the y direction, H u represents the magnetic field component propagating in the u direction, E v represents the electric field component propagating in the v direction, E y represents the electric field component propagating in the y direction, E u represents the electric field component propagating in the u direction, H v represents the magnetic field component propagating in the v direction; the propagation constants of the TE mode and the TM mode along the u direction are respectively and

[0099] The specific methods for obtaining the propagation constant and the electromagnetic field distribution of the one-dimensional analytical mode are as follows:

[0100] First, the initial value of the propagation constant of the one-dimensional planar waveguide mode is obtained by the second-order finite difference algorithm.

[0101] Then, the initial value of the propagation constant is substituted into the nonlinear control equation of the one-dimensional planar waveguide mode using the fsolve function in MATLAB for calculation to obtain the propagation constant of the one-dimensional analytical mode.

[0102] Secondly, substituting the propagation constant into the analytical formula for the field distribution of the one-dimensional planar waveguide, the electromagnetic field distribution of the one-dimensional analytical mode can be obtained.

[0103] Step 3, represent the LSE (longitudinal electric field mode) and LSM (longitudinal magnetic field mode) modes using the projection relationship of the one-dimensional analytical mode on the two-dimensional plane.

[0104] The specific method is as follows:

[0105] (1) Decompose the TE / TM modes in the one-dimensional analytical mode into the LSE / LSM modes in the two-dimensional rectangular optical waveguide:

[0106] As Figure 4 shown, both the u direction and the v direction are located in the x-z plane. Therefore, for the TE mode, H y remains unchanged, and H u is projected and decomposed into H x , H z components on the x-z plane. H y represents the magnetic field component in the y direction, and H x represents the magnetic field component in the x direction, and H z represents the magnetic field component in the z direction; E v is projected and decomposed into E x , E z components on the x-z plane. E x represents the electric field component in the x direction, and E z represents the electric field component in the z direction. The mode composed of the five components of H y , H x , H z , E x , and E z is the LSE mode;

[0107] For the TM mode, the same decomposition method is adopted to obtain the LSM mode including E y , E x , E z , H x , and H z . E x is the electric field component in the x direction, E y is the electric field component in the y direction, E z is the electric field component in the z direction, H x is the magnetic field component in the x direction, and H z is the magnetic field component in the z direction;

[0108] In the two-dimensional rectangular optical waveguide structure, the LSE and LSM modes propagate in the x direction. Since reflections will form at the physical boundaries, a fixed two-dimensional field distribution is finally formed. In the z propagation direction of the two-dimensional modes, these LSE and LSM modes have the same propagation constant β. This β is also the propagation constant of the two-dimensional modes existing in the two-dimensional rectangular optical waveguide.

[0109] (2) Define the propagation constants in the ±x directions and the angular relationships of the propagation constants in different directions:

[0110] According to the dispersion relation, the propagation constants of the LSE and LSM modes in the +x direction are respectively defined as and The propagation constants in the -x direction are respectively and

[0111] According to Figure 4 the relationship between the two coordinate systems in E the ratio θ of the propagation constant of the LSE mode in the x direction to the propagation constant in the u direction is defined as The ratio θ of the propagation constant of the LSM mode in the x direction to the propagation constant in the u direction is M defined as

[0112] Step 4: Based on the LSE and LSM modes, expand each component of the electromagnetic field of the two-dimensional rectangular optical waveguide mode in the horizontal direction to obtain the expression of the electromagnetic field components of the two-dimensional rectangular optical waveguide mode in the two-dimensional plane.

[0113] The specific method is as follows:

[0114] Assume the number of LSE modes is N E , and the number of LSM modes is N M . Each LSE and LSM mode that makes up the electromagnetic components of the two-dimensional rectangular optical waveguide is assigned a mode coefficient. The coefficient of the LSE mode is p represents the order of the LSE mode, ranging from the 1st to the N E th LSE mode, ± represents forward or backward propagation; the coefficient of the LSM mode is q represents the order of the LSM mode, ranging from the 1st to the N M th LSM mode;

[0115] Each electromagnetic component in the x-y interface of the two-dimensional rectangular optical waveguide is composed of the superposition of LSE and LSM modes with different coefficients. The expression of the electromagnetic field components is specifically as follows:

[0116]

[0117]

[0118] Among them, E x (x, y) is the x-component of the electric field in the (x, y) plane, E y (x, y) is the y-component of the electric field in the (x, y) plane, E z (x, y) is the z-component of the electric field in the (x, y) plane, H x (x, y) is the x-component of the magnetic field in the (x, y) plane, H y(x, y) is the y-component of the magnetic field in the (x, y) plane, H z (x, y) is the z-component of the magnetic field in the (x, y) plane; p represents the order of the LSE mode and the order of the TE mode, q represents the order of the LSM mode and the order of the TM mode, represents the E of the p-th TE mode v component, represents the E of the q-th TM mode u component, represents the E of the q-th TM mode y component, represents the H of the p-th TE mode u component, represents the H of the q-th TM mode v component, represents the H of the p-th TE mode y component; represents the angular conversion relationship of the p-th LSE, represents the angular conversion relationship of the q-th LSM; represents the mode coefficient of the p-th LSE mode propagating in the forward direction, represents the mode coefficient of the p-th LSE mode propagating in the reverse direction, represents the mode coefficient of the q-th LSM mode propagating in the forward direction, represents the mode coefficient of the q-th LSM mode propagating in the reverse direction; in is the propagation constant of the p-th TE mode in the x direction, x is the coordinate position in the x direction, in is the propagation constant of the q-th TM mode in the x direction, x is the coordinate position in the x direction.

[0119] The above step 4 completes the expression of the distribution of each electromagnetic component at any (x, y) coordinate within the x-y interface of the two-dimensional rectangular optical waveguide. It can be seen that the unknown parameters involved in the above formula are only the propagation constant β in addition to the mode coefficients and . Therefore, the next step will start from the continuity conditions and boundary conditions of the two-dimensional rectangular optical waveguide to obtain a nonlinear equation about the propagation constant β and the mode coefficients and solve it.

[0120] Step 5: Using the expression of the electromagnetic field components of the two-dimensional rectangular optical waveguide in the two-dimensional plane, combined with the field continuity conditions and boundary conditions, obtain the relationship between the mode coefficients and the characteristic equation characterizing the complex propagation constant and mode coefficients of the two-dimensional rectangular optical waveguide.

[0121] The specific method for obtaining the relationship between the mode coefficients by combining the field continuity conditions and boundary conditions is as follows:

[0122] (1) According to the field continuity condition, the electromagnetic field components obtained in step 4 above Figure 3 The tangential electromagnetic field components E y 、E z 、H y 、H z remain continuous on both sides of x = 0. Substitute x = 0 into E y (x, y), E z (x, y), H y (x, y) and H z (x, y) electromagnetic field component expressions, and simplify to obtain an equation in matrix form;

[0123] (2) Since the outer boundary of this two-dimensional rectangular optical waveguide is PEC, for the outer boundary, there are: the tangential component of the electric field E y (x = P, y) = 0 and E z (x = P, y) = 0, where P is the position of the outer boundary of the two-dimensional rectangular optical waveguide in the positive x direction. Substitute x = P into E y (x, y) and E z (x, y) equations, and let E y (x, y) and E z (x, y) equations be equal to 0, and obtain the relationship between the mode coefficients through simplification and represent the mode coefficients of the LSE mode propagating in the +x direction on the right side of the x = 0 interface; C 2- represents the mode coefficient of the LSE mode propagating in the -x direction on the right side of the x = 0 interface; D 2+ represents the mode coefficient of the LSM mode propagating in the +x direction on the right side of the x = 0 interface; D 2- represents the mode coefficient of the LSM mode propagating in the -x direction on the right side of the x = 0 interface;

[0124] Boundary condition matrix Boundary condition matrix diag represents the matrix form, j is the imaginary unit, represents the propagation constant of the p-th TE mode at x = P; represents the propagation constant of the q-th TM mode at x = P;

[0125] (3) Due to the symmetry of the two-dimensional rectangular optical waveguide, the center position of the two-dimensional rectangular optical waveguide is at x = -W / 2. The modes of the two-dimensional rectangular optical waveguide can be divided into odd modes and even modes, and the two show different characteristics:

[0126] When it is an odd mode, E y (x = -W / 2, y) = E z(x = -W / 2, y) = 0, where x = -W / 2 is the x-coordinate at the center of the core layer of the two-dimensional rectangular optical waveguide. Substitute x = -W / 2 into E in Step 4 y (x, y) and E z (x, y). Let E y (x, y) and E z (x, y) be equal to 0, and obtain the relationship between the mode coefficients through simplification and

[0127] When it is an even mode, H y (x = -W / 2, y) = H z (x = -W / 2, y) = 0. Substitute x = -W / 2 into H in Step 4 y (x, y) and H z (x, y). Let H y (x, y) and H z (x, y) be equal to 0, and obtain the relationship between the mode coefficients through simplification and

[0128] Among them, C 1+ represents the mode coefficient of the LSE mode propagating in the +x direction on the left side of the x = 0 interface; C 1- represents the mode coefficient of the LSE mode propagating in the -x direction on the left side of the x = 0 interface; D 1+ represents the mode coefficient of the LSM mode propagating in the +x direction on the left side of the x = 0 interface; D 1- represents the mode coefficient of the LSM mode propagating in the -x direction on the left side of the x = 0 interface; The boundary condition matrix represents the propagation constant of the p-th TE mode at x = -W / 2; The boundary condition matrix represents the propagation constant of the q-th TM mode at x = -W / 2.

[0129] The obtained odd-mode nonlinear characteristic equation in the two-dimensional rectangular optical waveguide containing the complex propagation constant and mode coefficients is as follows:

[0130]

[0131] The obtained even-mode nonlinear characteristic equation in the two-dimensional rectangular optical waveguide containing the complex propagation constant and mode coefficients is as follows:

[0132]

[0133] In the formula, I is the identity matrix, and T1, T2, T3, and T4 are transmission matrices.

[0134] The transmission matrices T1, T2, T3, T4 and the mode coefficient matrix D1- , the boundary condition matrix The specific expression is:

[0135]

[0136] In the formula, E1, E2, E3, E4, E5, E6, The specific expression is:

[0137]

[0138] In the formula, the detailed expressions of the matrices L1, L2, L3, L4, L5, L6, R1, R2, R3, R4, R5, R6 are respectively:

[0139]

[0140]

[0141]

[0142] Step 6, use the optimization algorithm to solve the characteristic equation to obtain the analytical complex propagation constant and mode coefficients. Substitute the complex propagation constant and mode coefficients into the electromagnetic field component expressions to obtain the electromagnetic field distribution of any mode on the two-dimensional plane.

[0143] In this embodiment, use the fsolve function in MATLAB to solve the nonlinear equation to obtain the required complex propagation constant, and then substitute the complex propagation constant back into the nonlinear characteristic equation in Step 5 to obtain the eigenvector D 1- ; through D 1- express the remaining mode coefficients C 1+ , C 2- , D 2- ; the superscript numbers represent the left or right side of the interface, and ± represents forward or backward propagation. The relationship between the mode coefficients is as follows:

[0144]

[0145] If the odd mode:

[0146]

[0147] If the even mode:

[0148]

[0149] In addition, at the x = 0 interface, the relationship between the mode coefficients on the left and right sides is:

[0150]

[0151] Among them, H v1 represents the H component on the left side of the x = 0 interface, and its superscript represents the mode numbers from 1 to N on the left side of the x = 0 interface v ; E M1 represents the E component on the left side of the x = 0 interface, and its superscript represents the mode numbers from 1 to N on the left side of the x = 0 interface y1 ; E y represents the E component on the right side of the x = 0 interface, and its superscript represents the mode numbers from 1 to N on the right side of the x = 0 interface M1 ; E y2 represents the E component on the left side of the x = 0 interface, and its superscript represents the mode numbers from 1 to N on the left side of the x = 0 interface y ; H M2 represents the H component on the left side of the x = 0 interface, and its superscript represents the mode numbers from 1 to N on the left side of the x = 0 interface v1 ; H v represents the H component on the right side of the x = 0 interface, and its superscript represents the mode numbers from 1 to N on the right side of the x = 0 interface E1 ; y1 ; y ; E1 ; y2 represents the H component on the right side of the x = 0 interface, and its superscript represents the mode numbers from 1 to N on the right side of the x = 0 interface y ; E2 ;

[0152] Thus, the final complex propagation constant, the LSE mode coefficients, and the LSM mode coefficients are obtained. Substituting them into the electromagnetic component expressions at the x - y interface of the two - dimensional rectangular optical waveguide obtained in step 4, the electromagnetic field distribution of any mode in the two - dimensional rectangular optical waveguide can be obtained.

[0153] The present invention will elaborate on the beneficial effects of the present invention compared with the FDE (finite - difference method) module in Lumerical software from seven aspects, including the convergence of the algorithm, the comparison of the effective refractive index obtained by solving with the FDE module of Lumerical software, the comparison of the field - distribution - related errors under different grid sizes, the simulation effect at the singularity of the silicon core layer, the simulation effect on the two - dimensional high - order PML mode, the orthogonality effect of the high - order modes of the two - dimensional rectangular optical waveguide, and the comparison of the operation time.

[0154] First, introduce the normalized field - component - related error formula FRE, which is used to describe the gap between the test component and the reference component. As follows:

[0155]

[0156] Among them, H represents the field component, the subscript t represents the test component, and t, b represents the reference component. For example Figure 5As shown, the convergence of the present invention is only related to the number of one-dimensional modes applied. As the number of one-dimensional modes increases, the effective refractive index of the two-dimensional guided mode converges rapidly. As Figure 6 shown, in order to better describe the convergence of the Ex field component, the log 10 (FRE) form is adopted. For the reference mode, the present invention selects the two-dimensional guided mode field distribution calculated by the present invention when the number of one-dimensional modes is 200. It can be seen that as the number of one-dimensional modes increases, log 10 (FRE) tends to converge. For Figure 2 this embodiment, the number of one-dimensional modes used to achieve convergence is approximately 100 or 110. Therefore, in the following verification, it is appropriate to use 100 one-dimensional modes.

[0157] Secondly, the FDE module of Lumerical software is used to evaluate the calculation accuracy of the present invention. The comparison method is the log 10 form of the relative error of the effective refractive index between the present invention and the FDE method. The number of one-dimensional modes applied in the present invention is 100. It can be seen from Figure 7 that as the grid size decreases, log 10 (the relevant error) continues to decrease. This proves that as the grid size decreases, the FDE result converges to the result of the present invention. At the grid size dx = dy = 0.5 nm, the relevant error between the two methods reaches -5 dB. However, at this time, the FDE module of Lumerical software will consume a large amount of computing resources and computing time, while the present invention is not affected by the grid size, providing higher computing efficiency and computing accuracy, and solving the problems of long computing time and low computing accuracy of traditional numerical simulation methods.

[0158] In Figure 8 , figures (a)-(c) show the simulation effects of the three components of the two-dimensional guided mode electric field of the present invention, and compare the log 10 (FRE) values of the Ex results obtained by the present invention and the FDE module under different grid sizes. The number of one-dimensional modes used in the present invention is 100. As shown in Figure 9 , log 10 (FRE) continuously decreases, indicating that as the grid size decreases, the accuracy of the FDE method continuously improves, while the high-precision result of the present invention is not affected by the grid size.

[0159] At the singularity at the interface of different materials, the field strength should approach infinity. Then, for a simulation algorithm, the simulation effect at the singularity of the core layer is also an important criterion for judging the performance of a simulation method. In Figure 7 , the amplitude verification of the Ex field component is carried out at the singularity in the upper right corner of the core layer. First, in Figure 10(a)-(d) verify the convergence of the present invention at the singularity. As the number of one-dimensional modes increases, the normalized amplitude at the singularity begins to converge. Figure 11 (a) and (b) show the simulation of the singularity by FDE with different grid sizes. By comparison, it is proved that the present invention has better simulation performance than FDE at the singularity.

[0160] Additionally, the present invention verifies the field distribution of the high-order PML mode. The electric field distribution of the third-order PML mode is shown in Figure 12 . It can be seen from Figure 12 (a) to Figure 12 (c) that the present invention can accurately simulate the field distribution of the third-order PML mode; the electric field distribution of the twelfth-order PML mode is shown in Figure 13 . It can be seen from Figure 13 (a) to Figure 13 (c) that the present invention can accurately simulate the field distribution of the higher-order twelfth-order PML mode; and it is proved in Figure 14 (a)-(c) by using the Ex field distribution of the twelfth-order PML mode that the present invention can achieve consistent simulation accuracy under different numbers of one-dimensional modes.

[0161] The orthogonality of two-dimensional modes is very important for advanced high-precision simulation methods and will directly affect the accurate modeling of optical devices. The orthogonality of the high-order modes of two-dimensional rectangular optical waveguides obtained by the present invention and FDE is compared in Figure 15 . Figure 15 In (a), the mode orthogonality obtained by FDE is shown. Figure 15 (b) is the two-dimensional mode orthogonality diagram obtained by the present invention when the number of one-dimensional modes is 100. Figure 15 In (c), the two-dimensional mode orthogonality diagram obtained by the present invention when the number of one-dimensional modes is 200 is shown. It can be clearly seen that the mode orthogonality obtained by FDE is relatively poor, while since the present invention is a fully analytical method for solving the modes of two-dimensional rectangular optical waveguides, the orthogonality of two-dimensional high-order complex modes remains stable under different numbers of one-dimensional modes. It solves the problems of low accuracy and poor mode orthogonality in solving the modes of two-dimensional rectangular optical waveguides by traditional numerical algorithms.

[0162] Finally, the operation time of the present invention is described. The simulation environment is as follows: 12th Gen Intel(R) Core(TM) i5-12400F (12 CPUs), ~2.5 GHz. The RAM is 16384 MB. Build in MATLAB Figure 2For the optical waveguide model shown, the running time of the present invention is 14 s. When the grid size is set to dx = dy = 1 nm in the FDE module of the Lumerical software, the running time is 710 s. Therefore, compared with the existing numerical simulation technology, the present invention has the beneficial effects of high calculation efficiency and high calculation accuracy.

[0163] The above description of the disclosed embodiments enables those skilled in the art to implement or use the present invention. Various modifications to these embodiments will be apparent to those skilled in the art, and the general principles defined herein can be implemented in other embodiments without departing from the spirit or scope of the present invention. Therefore, the present invention will not be limited to the embodiments shown herein, but rather to the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. An analytical method for two-dimensional rectangular optical waveguide modes based on a pattern matching method, characterized in that, It includes the following steps: Step 1: Obtain the relevant structural parameters of the two-dimensional rectangular optical waveguide; Step 2: Combine the structural parameters, and slice the two-dimensional rectangular optical waveguide along the longitudinal direction according to different material refractive indices to obtain corresponding one-dimensional slab waveguides, and obtain the propagation constant and electromagnetic field distribution of the one-dimensional analytical mode according to the nonlinear control equation of the one-dimensional analytical mode of the one-dimensional slab waveguide; Step 3: Use the projection relationship of the one-dimensional analytical mode on the two-dimensional plane to represent the LSE and LSM modes; Step 4: Expand each component of the electromagnetic field of the two-dimensional rectangular optical waveguide mode in the horizontal direction based on the LSM and LSM modes to obtain the expression of the electromagnetic field components of the two-dimensional rectangular optical waveguide mode on the two-dimensional plane; Step 5: Use the expression of the electromagnetic field components of the two-dimensional rectangular optical waveguide on the two-dimensional plane, combine the field continuity conditions and boundary conditions to obtain the relationship between the mode coefficients and the characteristic equation characterizing the complex propagation constant and mode coefficients of the two-dimensional rectangular optical waveguide; Step 6: Use the optimization algorithm to solve the characteristic equation to obtain the analytical complex propagation constant and mode coefficients, and substitute the complex propagation constant and mode coefficients into the electromagnetic field component expression to obtain the electromagnetic field distribution of any mode on the two-dimensional plane.

2. The analytical method for two-dimensional rectangular optical waveguide modes based on the pattern matching method according to claim 1, wherein The two-dimensional rectangular optical waveguide includes a silica cladding, a silicon core layer, a vacuum upper cladding, and a perfectly matched layer; the silicon core layer is located at the upper center position of the silica cladding, the vacuum upper cladding is located above the silica cladding and the silicon core layer, and the perfectly matched layer is wrapped outside the silica cladding, the silicon core layer, and the vacuum upper cladding; Its relevant structural parameters include the material refractive index, height, width of the silica cladding, the material refractive index, height, width of the silicon core layer, the material refractive index, height, width of the vacuum upper cladding, and the material refractive index, height, and absorption factor parameters of the perfectly matched layer.

3. The analytical method for two-dimensional rectangular optical waveguide modes based on the pattern matching method according to claim 1, wherein In Step 2, the method for slicing the two-dimensional rectangular optical waveguide is as follows: First, establish a (x, y, z) coordinate system for the two-dimensional rectangular optical waveguide, with the center of the silicon core layer as the origin, the horizontal right direction as the positive x-axis direction; the vertical upward direction as the positive y-axis direction; and the direction perpendicular to the paper surface from the origin as the positive z-axis direction; Then, slice the two-dimensional rectangular optical waveguide along the y direction at the boundary interface between the silicon core layer and the vacuum upper cladding, and divide it into three one-dimensional slab waveguides, including Slab One, Slab Two, and Slab Three, where Slab Two contains the silicon core layer, and Slab One and Slab Three do not contain the silicon core layer.

4. The analytical method for two-dimensional rectangular optical waveguide modes based on the pattern matching method according to claim 3, characterized in that, In Step 2, the one-dimensional analytical modes of the one-dimensional slab waveguide include TE mode and TM mode, specifically as follows: Establish a (y, v, u) coordinate system for the three obtained one-dimensional planar waveguides. For planar waveguide two, take the intersection point of the core layer diagonals as the origin of the (y, v, u) coordinate system. The positive direction of the y-axis is vertically upward from the origin, which is the direction of the change in material refractive index. The positive direction of the u-axis is horizontally to the right from the origin, and this direction is the propagation direction of the one-dimensional mode. The positive direction of the v-axis is perpendicular to the paper surface from the origin, and the electromagnetic field distribution is uniform in this direction. For planar waveguide one and planar waveguide three, the height of the origin is the same as that of planar waveguide two, and the horizontal position of the origin is at the center of the vacuum upper cladding in this region. Establish a (y, v, u) coordinate system with the positive direction of the y-axis vertically upward from the origin and the positive direction of the u-axis horizontally to the right from the origin. Assume that the one-dimensional planar waveguide mode propagates along the u direction, the refractive index of the material varies in the y direction, and is uniform in the v direction. In this coordinate system, the modes of the one-dimensional planar waveguide are decoupled into TE modes containing H y , H u and E v , and TM modes containing E y , E u and H v . Among them, H y represents the magnetic field component propagating in the y direction, H u represents the magnetic field component propagating in the u direction, E v represents the electric field component propagating in the v direction, E y represents the electric field component propagating in the y direction, E u represents the electric field component propagating in the u direction, H v represents the magnetic field component propagating in the v direction; the propagation constants of the TE mode and the TM mode along the u direction are respectively and 5. The analytical method for two-dimensional rectangular optical waveguide modes based on the pattern matching method according to claim 4, characterized in that, In step 2, the specific method for obtaining the propagation constant and electromagnetic field distribution of the one-dimensional analytical mode is as follows: First, obtain the initial value of the propagation constant of the one-dimensional planar waveguide mode by the second-order finite difference algorithm. Then, use the fsolve function in MATLAB to substitute the initial value of the propagation constant into the nonlinear control equation of the one-dimensional planar waveguide mode for calculation to obtain the propagation constant of the one-dimensional analytical mode. Secondly, substitute the propagation constant into the analytical formula for the field distribution of the one-dimensional planar waveguide to obtain the electromagnetic field distribution of the one-dimensional analytical mode.

6. The analytical method for two-dimensional rectangular optical waveguide modes based on the pattern matching method according to claim 4, characterized in that, The specific method for step 3 is as follows: (1) Decompose the TE / TM mode of the one-dimensional analytical mode into the LSE / LSM mode in the two-dimensional rectangular optical waveguide. For the TE mode, H y remains unchanged, and H u is decomposed by projection on the x-z plane into H x , H z components. H y represents the magnetic field component in the y direction, and H x represents the magnetic field component in the x direction. H z represents the magnetic field component in the z direction; E v is decomposed by projection on the x-z plane into E x , E z components. E x represents the electric field component in the x direction, and E z represents the electric field component in the z direction. The mode composed of the five components H y , H x , H z , E x , and E z is the LSE mode; The same decomposition method is adopted for the TM mode, resulting in the LSM mode including E y , E x , E z , H x , H z . Among them, E x is the electric field component in the x direction, E y is the electric field component in the y direction, E z is the electric field component in the z direction, H x is the magnetic field component in the x direction, H z is the magnetic field component in the z direction; (2) Define the propagation constants in the ±x directions and the angular relationship of the propagation constants in different directions. According to the dispersion relation, the propagation constants of the LSE and LSM modes in the +x direction are respectively defined as and The propagation constants in the -x direction are respectively and According to the relationship between the two coordinate systems, the ratio θ of the propagation constant in the x direction to the propagation constant in the u direction in the LSE mode E is defined as The ratio θ of the propagation constant in the x direction to the propagation constant in the u direction in the LSM mode M is defined as 7. An analytical method for two-dimensional rectangular optical waveguide modes based on the pattern matching method according to claim 6, characterized in that, The specific method for step 4 is as follows: Assume that the number of LSE modes is N E , and the number of LSM modes is N M . A mode coefficient is assigned to each LSE and LSM mode that constitutes the electromagnetic components of the two-dimensional rectangular optical waveguide. The coefficient of the LSE mode is where p represents the order of the LSE mode, ranging from the 1st to the N E LSE modes, and ± represents forward or backward propagation; the coefficient of the LSM mode is where q represents the order of the LSM mode, ranging from the 1st to the N M LSM modes; Each electromagnetic component in the x - y interface of the two-dimensional rectangular optical waveguide is composed of the superposition of the LSE and LSM modes with different coefficients. The specific expressions of the electromagnetic field components are as follows: Among them, E x (x, y) is the x-component of the electric field in the (x, y) plane, and E y (x, y) is the y-component of the electric field in the (x, y) plane, and E z (x, y) is the z-component of the electric field in the (x, y) plane, and H x (x, y) is the x-component of the magnetic field in the (x, y) plane, and H y (x, y) is the y-component of the magnetic field in the (x, y) plane, and H z (x, y) is the z-component of the magnetic field in the (x, y) plane; p represents the order of the LSE mode and the order of the TE mode, and q represents the order of the LSM mode and the order of the TM mode. represents the E v component of the p-th TE mode, represents the E u component of the q-th TM mode, represents the E y component of the q-th TM mode, represents the H u component of the p-th TE mode, represents the H v component of the q-th TM mode, represents the H y component of the p-th TE mode; represents the angular conversion relationship of the p-th LSE, represents the angular conversion relationship of the q-th LSM; represents the mode coefficient of the p-th LSE mode propagating in the forward direction, represents the mode coefficient of the p-th LSE mode propagating in the reverse direction, represents the mode coefficient of the q-th LSM mode propagating in the forward direction, represents the mode coefficient of the q-th LSM mode propagating in the reverse direction; in is the propagation constant of the p-th TE mode in the x direction, and x is the coordinate position in the x direction. in is the propagation constant of the q-th TM mode in the x direction, and x is the coordinate position in the x direction.

8. An analytical method for two-dimensional rectangular optical waveguide modes based on the pattern matching method according to claim 7, characterized in that, In step 5, the specific method for obtaining the relationship between the mode coefficients by combining the field continuity condition and the boundary condition is as follows: (1) According to the field continuity condition, the tangential electromagnetic field components E y , E z , H y , and H z on both sides of x = 0 are continuous. Substitute x = 0 into the expressions of the electromagnetic field components E y (x, y), E z (x, y), H y (x, y), and H z (x, y) obtained in step 4 above, and simplify to obtain an equation in matrix form; (2) At the outer boundary, there is: the tangential component of the electric field E y (x = P, y) = 0 and E z (x = P, y) = 0, where P is the position of the outer boundary of the two-dimensional rectangular optical waveguide in the positive x direction. Substitute x = P into E y (x, y) and E z (x, y). Let E y (x, y) and E z (x, y) be equal to 0, and obtain the relationship between the mode coefficients through simplification and C 2+ represents the mode coefficient of the LSE mode propagating along the +x direction on the right side of the x = 0 interface; C 2- represents the mode coefficient of the LSE mode propagating along the -x direction on the right side of the x = 0 interface; D 2+ represents the mode coefficient of the LSM mode propagating along the +x direction on the right side of the x = 0 interface; D 2- represents the mode coefficient of the LSM mode propagating along the -x direction on the right side of the x = 0 interface; Boundary condition matrix Boundary condition matrix diag represents the matrix form, and j is the imaginary unit, represents the propagation constant of the p-th TE mode at x = P; represents the propagation constant of the q-th TM mode at x = P; (3) The center position of the two-dimensional rectangular optical waveguide is at x = -W / 2. The two-dimensional rectangular optical waveguide mode can be divided into odd mode and even mode. When it is the odd mode, E y (x = -W / 2, y) = E z (x = -W / 2, y) = 0, where x = -W / 2 is the x - coordinate at the center of the core layer of the two - dimensional rectangular optical waveguide. Substitute x = -W / 2 into E y (x, y) and E z (x, y). Let E y (x, y) and E z (x, y) be equal to 0. Through simplification, the relationship between the mode coefficients is obtained and When it is the even mode, H y (x = -W / 2, y) = H z (x = -W / 2, y) = 0. Substitute x = -W / 2 into H in step 4 y (x, y) and H z In the expression of (x, y) and H y (x, y) and H z Let (x, y) and H be equal to 0, and obtain the relationship between the mode coefficients through simplification and where, C 1+ represents the mode coefficient of the LSE mode propagating in the +x direction on the left side of the x = 0 interface; C 1- represents the mode coefficient of the LSE mode propagating in the -x direction on the left side of the x = 0 interface; D 1+ represents the mode coefficient of the LSM mode propagating in the +x direction on the left side of the x = 0 interface; D 1- represents the mode coefficient of the LSM mode propagating in the -x direction on the left side of the x = 0 interface; the boundary condition matrix represents the propagation constant of the p-th TE mode at x = -W / 2; the boundary condition matrix represents the propagation constant of the q-th TM mode at x = -W / 2.

9. The analytical method for two-dimensional rectangular optical waveguide modes based on the pattern matching method according to claim 8, characterized in that, In step 5, the odd-mode nonlinear characteristic equation in the two-dimensional rectangular optical waveguide containing the complex propagation constant and mode coefficients is as follows: The even-mode nonlinear characteristic equation in the two-dimensional rectangular optical waveguide containing the complex propagation constant and mode coefficients is as follows: In the formula, I is the identity matrix, and T1, T2, T3, and T4 are transmission matrices.

10. An analytical method for two-dimensional rectangular optical waveguide modes based on the pattern matching method according to claim 9, characterized in that, In step 6, the fsolve function in MATLAB is used to solve the nonlinear equation to obtain the required complex propagation constant, and then the complex propagation constant is substituted back into the nonlinear characteristic equation in step 5 to obtain the eigenvector D 1- ; Through D 1- The remaining mode coefficients C 1+ 、C 2- 、D 2- are expressed; The relationship between the mode coefficients is as follows: In addition, at the x = 0 interface, the relationship between the mode coefficients on the left and right sides is: Among them, H v1 represents the H v component on the left side of the x = 0 interface, and its superscript represents the mode numbers from 1 to N M1 on the left side of the x = 0 interface; E y1 represents the E y component on the left side of the x = 0 interface, and its superscript represents the mode numbers from 1 to N M1 on the left side of the x = 0 interface; E y2 represents the E y component on the right side of the x = 0 interface, and its superscript represents the mode numbers from 1 to N M2 on the right side of the x = 0 interface; E v1 represents the E v component on the left side of the x = 0 interface, and its superscript represents the mode numbers from 1 to N E1 on the left side of the x = 0 interface; H y1 represents the H y component on the left side of the x = 0 interface, and its superscript represents the mode numbers from 1 to N E1 on the left side of the x = 0 interface; H y2 represents the H y component on the right side of the x = 0 interface, and its superscript represents the mode numbers from 1 to N E2 on the right side of the x = 0 interface; Thus, the final complex propagation constant and the coefficients of each LSE mode and LSM mode are obtained. Substitute them into the expressions of the electromagnetic components in the x - y interface of the two-dimensional rectangular optical waveguide obtained in step 4 to obtain the electromagnetic field distribution of any mode in the two-dimensional rectangular optical waveguide.

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