CN114942490B - Multi-cladding step optical fiber design method based on characteristic matrix - Google Patents

Abstract

The invention discloses a multi-cladding step optical fiber design method based on a characteristic matrix. The method comprises the following steps: 1) Inputting calculation parameters of an initial structure of the optical fiber; 2) Discrete sampling is carried out in the effective propagation constant calculation range determined according to the refractive index of the fiber core and the refractive index of the outermost cladding, and an effective propagation constant sampling value is obtained; 3) Determining a segment boundary according to the number of claddings; for each effective propagation constant sampling value, calculating the normalization parameters of the corresponding refractive index layers; 4) Constructing an oscillation submatrix or an attenuation submatrix of each effective propagation constant sampling value in the optical fiber; 5) Constructing a characteristic matrix according to the oscillator sub-matrix and the attenuator sub-matrix; 6) When the determinant value of the characteristic matrix is solved to be 0, the constant value is effectively propagated; 7) Calculating equivalent refractive indexes corresponding to effective propagation constant values under different mode orders; 8) And (4) the obtained equivalent refractive index is substituted back into the characteristic matrix to obtain an equation set of the corresponding mode, and the mode field distribution in the optical fiber under each corresponding mode is obtained through calculation.

Description

A design method for multi-cladding step-index optical fiber based on characteristic matrix

Technical Field

The invention relates to the fields of simulation and optical fiber design, and in particular to a method for solving a matrix of characteristic equations of multi-cladding step-index optical fiber modes.

Background Art

Step Index Fiber (SIF) has a simple structure and is easy to draw. The preparation process and technology of fiber devices based on SIF are very mature. In terms of application, most of the near-diffraction-limited high-power fiber lasers are currently based on SIF. The SIF fiber used in high-average-power and high-beam-quality fiber lasers is generally not a strict single-mode fiber, but a few-mode fiber that supports a certain number of modes. The composition of the fundamental mode and the high-order mode in the output beam determines the beam quality, so it is necessary to accurately control the mode of the multi-cladding step-index fiber structure. The improved SIF design of the multi-cladding structure has become a research hotspot in the field of optical fiber design.

At present, in the design of optical fiber structure, for single-clad optical fiber, whether calculating scalar mode or vector mode, the solution is mostly solved in analytical form, combined with the commonly used numerical method of equation rooting. For the three-clad optical fiber with a ‘W’ structure as shown in Figure 1 or an optical fiber with more cladding structures, the derivation and differentiation process of the characteristic equation analytical form will become very complicated. In specific applications such as fiber lasers/amplifiers and optical fiber communication systems, in order to meet the requirements of larger mode field diameter, higher tolerable power, better output beam quality and low loss, the optical fiber structure needs to be precisely controlled. The existing methods for designing multi-clad optical fibers are mostly calculated for specific cladding refractive index distribution or rely on empirical design. For a specific cladding refractive index distribution, the corresponding relationship between linear polarization and vector mode is used, and the corresponding vector mode is replaced by the effective refractive index or propagation constant of the LP mode. The calculation result is very inaccurate, the design freedom is not high, and the applicability is not strong. The method of adjusting the number of optical fiber layers and performing mode calculation based on experience is too cumbersome, and it takes a lot of time and cost to optimize it, and it is difficult to obtain parameters that meet the requirements.

Summary of the invention

In order to solve the problems that the design of the multi-clad optical fiber described in the above background technology for the optimal performance optical fiber for a specific application scenario is time-consuming, has poor scalability and convenience, and is highly complex, the present invention provides a multi-clad step-index optical fiber design method based on a characteristic matrix.

The method of the present invention comprises the following steps:

1) Input the initial fiber structure calculation parameters, including the working wavelength λ, the refractive index and radius from the core to the outermost cladding, etc. If the core and each cladding are collectively referred to as the refractive index layer of the optical fiber, the total number of layers is N, and the radius of each layer is represented by r ₁ , r ₂ ,…, _ri ,…, r _N , and the refractive index value is represented by n ₁ , n ₂ ,…, n _j ,…, n _N. The radius number is represented by subscript i, and the refractive index is represented by subscript j.

2) According to the refractive index of the core and the refractive index of the outermost cladding, determine the effective propagation constant range. The effective propagation constant range is between the propagation constant of the outermost cladding and the propagation constant of the core. According to linear discrete sampling, Z is the total number of sampling points, and each point is recorded as β = β ₁ , β ₂ , β ₃ , ... β _t ... β _Z , where t represents the ordinal number of the sampling point.

为第j层折射率层所对应的归一化横向相位参数或归一化衰减参数，其中，k₀为真空中的波矢，n_j为该折射率层的折射率，β_t为步骤2中有效常数的取样值。当(k₀n_j)²-β_t ²＞0，光纤模式在该层有振荡解，f_j为该层的归一化横向相位参数；当(k₀n_j)²-β_t ²＜0，光纤模式在该层有衰减解，f_j为该层的归一化横向衰减参数。对于每个折射率层，根据设定光纤的各折射率层的折射率，其归一化横向参数分别为f₁,f₂,…,f_j,…,f_N。3) For β _t , find the normalized lateral phase parameter or normalized attenuation parameter of each refractive index layer. Definition

is the normalized transverse phase parameter or normalized attenuation parameter corresponding to the j-th refractive index layer, wherein k ₀ is the wave vector in vacuum, n _j is the refractive index of the refractive index layer, and β _t is the sampled value of the effective constant in step 2. When (k ₀ n _j ) ² -β _t ² > 0, the fiber mode has an oscillation solution in this layer, and f _j is the normalized transverse phase parameter of this layer; when (k ₀ n _j ) ² -β _t ² < 0, the fiber mode has an attenuation solution in this layer, and f _j is the normalized transverse attenuation parameter of this layer. For each refractive index layer, according to the refractive index of each refractive index layer of the set optical fiber, its normalized transverse parameters are f ₁ , f ₂ ,…, f _j ,…, f _N , respectively.

4) According to the mode calculation method (vector mode/scalar mode), construct each oscillator matrix and attenuation submatrix for β _t . The scalar mode is represented by LP _mn (m＝0,1,2…), and the vector mode is represented by TE _0n , TE _0n (m＝0), HE _mn , EH _mn (m＝1,2…). The mode order m represents the mode number in the circumferential direction of the fiber mode, and the root number n represents the mode number in the radial direction of the fiber. The two are used as subscripts to name the fiber mode. Among them, the mode order m is numerically equal to the order of the Bessel function below.

Each submatrix of β _t includes an oscillation solution submatrix or an attenuation submatrix of each fiber refractive index layer. In each fiber refractive index layer, the size of the β _t value determines whether to construct an oscillator submatrix or an attenuation submatrix in the layer. When (k ₀ n _j ) ² -β _t ² > 0, the oscillator submatrix of the layer is constructed; when (k ₀ n _j ) ² -β _t ² < 0, the attenuation submatrix of the layer is constructed. The oscillator matrix is a matrix about the order m of the Bessel function (m = 0, 1, 2...), the radius of each fiber refractive index layer _ri , the normalized transverse parameter f _j , the first kind Bessel function J _m and the second kind modified Bessel function N _m . The attenuation submatrix is a matrix about the order m of the Bessel function, the radius of each fiber refractive index layer _ri , the normalized transverse phase parameter or the normalized transverse attenuation parameter f _j , the first kind modified Bessel function K _m and the second kind Bessel function I _m . When considering the scalar mode, the core oscillator matrix and the outermost cladding attenuation matrix are 2×1 matrices, and their characteristic forms are:

In the non-outermost cladding, the oscillator matrix or attenuation matrix formed by the cladding in the middle is a 2×2 matrix, and its characteristic forms are:

When considering the vector mode, the treatment is the same as that of the scalar mode core, the outermost cladding, and the non-outermost cladding. The core oscillator matrix and the outermost cladding attenuation matrix are 4×2 matrices, and their characteristic forms are:

Where ω is the circular frequency, ω = 2πc/λ, c is the speed of light propagation in a vacuum, ε ₀ and μ ₀ are the dielectric constant and magnetic permeability in a vacuum, ε _j is the dielectric constant related to the refractive index, ε _j = ε ₀ n _j ² .

The oscillator matrix or attenuation matrix formed by the cladding in the middle is a 4×4 matrix, and its characteristic forms are:

5) Construct a feature matrix, which is composed of an oscillator matrix and an attenuation matrix.

的组合形式为：The characteristic matrix of the optical fiber is composed of the oscillation submatrix or attenuation submatrix obtained in step (4) as a block matrix. The number of oscillation/attenuation submatrices is determined by the total number of optical fiber refractive index layers. In the core and the outermost cladding, the number of submatrices is one each, and in the non-outermost cladding, the number of submatrices is two. Combining all these matrices can obtain the characteristic matrix. Characteristic matrix

The combination form is:

The “/” symbol in the determinant represents a logical OR relationship. The specific selection of matrix A _ij or matrix B _ij depends on the size of its refractive index. When k ₀ n _j ≥ β _t, the oscillator matrix A _i,j is taken, and when k ₀ n _j ＜β _t, the attenuation submatrix B _i,j is taken.

6) Repeat steps (3) to (5) to find the number of fiber modes and the corresponding propagation constants for each Bessel function order m (m = 0, 1, 2, ...).

能够使得特征矩阵行列式为零，则该传播常数

所对应的光纤模式可以在光纤中传输。因此，求解传播常数

的步骤为：①依次求出各β_t所对应的特征矩阵行列式的值，利用零点存在性定理，找出其中能使所有满足行列式的值存在零点的区间，例如，若存在G个满足零点存在的区间，则分别记为(β_g,β_g+1)，g为解区间的序数，g＝1,2,3,…G；②采用牛顿法等离散数值方法在所有满足条件的区间(β_g,β_g+1)求出能令特征矩阵行列式为0的数值解。假设能令光纤模式存在的数值解个数为x，则各数值解对应的传播常数记为

T表示数值解的序数。When the fiber mode exists, the determinant corresponding to the characteristic matrix must be zero. It can be deduced that if there are certain propagation constants

Can make the characteristic matrix determinant zero, then the propagation constant

The corresponding fiber mode can be transmitted in the fiber. Therefore, solve the propagation constant

The steps are as follows: ① find the value of the characteristic matrix determinant corresponding to each β _t in turn, and use the zero point existence theorem to find the interval in which all values satisfying the determinant have zero points. For example, if there are G intervals that satisfy the existence of zero points, they are recorded as (β _g , β _g+1 ), g is the ordinal number of the solution interval, g = 1, 2, 3, ... G; ② Use discrete numerical methods such as Newton's method to find the numerical solution that can make the characteristic matrix determinant equal to 0 in all satisfying intervals (β _g , β _g+1 ). Assuming that the number of numerical solutions that can make the fiber mode exist is x, the propagation constant corresponding to each numerical solution is recorded as

T represents the ordinal number of the numerical solution.

值由大到小的顺序进行排序并命名，并进一步计算得到标量模式或矢量模式等效折射率。7) According to the mode order m, the fiber mode

The values are sorted and named in descending order, and the scalar mode or vector mode equivalent refractive index is further calculated.

值，随n增大

值增大，且TE_0n的β值略大于TM_0n的

值。当m≠0时，可以计算获得EH_ln和HE_ln模式的值，随n增大

值增大，HE_ln的

值略大于EH_ln的

值；等效折射率n_eff传播常数

和的关系为：

As n increases, β _T decreases; the vector mode is represented by TE mode, TM mode, EH mode, and HE mode. When m = 0, the TE _0n and TM _0n modes can be calculated.

As n increases

The value increases, and the β value of TE _0n is slightly larger than that of TM _0n.

When m≠0, the values of EH _ln and HE _ln modes can be calculated. As n increases

The value increases, HE _ln

The value is slightly larger than that of EH _ln

Value; equivalent refractive index n _eff propagation constant

The relationship between and is:

8) Find the mode field distribution of each fiber mode in the optical fiber.

后，将之依次回代入步骤5)中式(9)所示的特征矩阵。若用标量法，则特征矩阵为2(N-1)×2(N-1)阶矩阵；若用矢量法，则特征矩阵为4(N-1)×4(N-1)阶矩阵。对于标量模，各模式的特征方程组

如式(10)所示：Solve the propagation constants of each mode as in step 7

Then, substitute them back into the characteristic matrix shown in equation (9) in step 5). If the scalar method is used, the characteristic matrix is a 2(N-1)×2(N-1) matrix; if the vector method is used, the characteristic matrix is a 4(N-1)×4(N-1) matrix. For the scalar mode, the characteristic equations of each mode are

As shown in formula (10):

可依次求出不同传播常数

所对应的电场分量C_k。至此本发明已获得光纤模式所对应

的电场分量表达式各项系数，将每一光纤模式所对应的各项系数回代入电场分量表达式即能获得该光纤模式在光纤中的模场分布。Where C _k is the Bessel function coefficient of each electric field component (k = 1, 2, 3...2N-1). C ₁ is the Bessel function coefficient of the electric field component of the core, C _2N-1 is the Bessel function coefficient of the electric field component of the outermost cladding, and the middle C _k is the Bessel function coefficient of the non-outermost cladding. Each group of 2 is the Bessel function coefficient of the electric field component of the cladding from the inside to the outside. Let

Different propagation constants can be obtained in turn

The corresponding electric field component C _k . So far, the present invention has obtained the corresponding electric field component C k of the optical fiber mode.

The coefficients of the electric field component expression are obtained. Substituting the coefficients corresponding to each fiber mode into the electric field component expression can obtain the mode field distribution of the fiber mode in the fiber.

后，回代入步骤(5)中式(9)所示的特征矩阵。对于矢量模，各模式的特征方程组

如式(11)所示：The solution of the vector solution is similar to that of the propagation constant of each mode.

Then, substitute back into the characteristic matrix shown in equation (9) in step (5). For the vector mode, the characteristic equation group of each mode is

As shown in formula (11):

可依次求出不同传播常数

所对应的电场分量和磁场分量C_k。至此本发明已获得光纤模式所对应

的电场分量表达式各项系数，将每一光纤模式所对应的各项系数回代入电场分量表达式即能获得该光纤模式在光纤中的矢量模场分布。At this time, C _k is the Bessel function coefficient of each electric field component and magnetic field component (k＝1,2,3...4N-1), C ₁ is the Bessel function coefficient of the electric field component of the core, C ₂ is the Bessel function coefficient of the magnetic field component of the core, C _4(N-1)-1 is the Bessel function coefficient of the electric field component of the outermost cladding, and C _4(N-1) is the Bessel function coefficient of the magnetic field component of the outermost cladding. The middle C _k is the Bessel function coefficient of the non-outermost cladding, and each group of 4 is composed of the first 2 Bessel function coefficients of the electric field components of each inner cladding in sequence from the core to the outer cladding, and the last 2 are the Bessel function coefficients of the magnetic field components of each inner cladding. Let

Different propagation constants can be obtained in turn

The corresponding electric field component and magnetic field component C _k . So far, the present invention has obtained the corresponding optical fiber mode

Substituting the coefficients corresponding to each fiber mode into the electric field component expression can obtain the vector mode field distribution of the fiber mode in the optical fiber.

After calculating the equivalent refractive index, propagation constant and transmission modes of the optical fiber, it can be used as a theoretical basis for the next step of calculating the mode field diameter, insertion loss, cutoff wavelength and fiber dispersion of the optical fiber. It provides rich and accurate multi-cladding optical fiber design solutions for various single-mode/few-mode fiber laser/amplifier systems.

Compared with the prior art, the multi-clad step-index optical fiber design method based on the characteristic matrix of the present invention is as follows:

①Fast calculation speed, high accuracy and high reliability.

② The number of layers of multi-clad optical fiber can be set arbitrarily, and the design of the core and cladding is flexible, which can provide customized design solutions for various application fields.

③ The present invention can provide theoretical support for the optical fiber design scheme of the current high-power optical fiber laser/amplifier project, and can simulate and calculate the characteristics of the optical fiber laser/amplifier before the experiment, saving precious R&D time and R&D funds.

④ The parameters of each layer of the multi-clad step-index optical fiber designed by the present invention are flexible and controllable, with high design freedom and wide application scenarios. The design method described in the present invention has fast computing speed, high computing speed, good robustness, no need to repeatedly write code, strong adaptability, good scalability and convenience, and different types of multi-clad optical fibers can be customized for different application scenarios.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG1 shows the geometric dimensions and refractive index profile of a “W” type triple-clad optical fiber;

(a) Transverse cross-section of the optical fiber; (b) Refractive index distribution of the optical fiber along the radial direction; (c) Cross-section of the optical fiber along the radial direction.

FIG2 shows the mode field distribution image of some modes of the triple-clad optical fiber obtained after the solution.

FIG3 shows a schematic diagram of the principle steps of a method for designing a multi-cladding step-index optical fiber based on a characteristic matrix.

DETAILED DESCRIPTION

In order to clearly demonstrate the purpose, technical solutions and advantages of the present invention, the specific implementation modes of the present invention are further described in detail below in conjunction with the accompanying drawings and examples.

In one embodiment, taking the "W" type three-cladding fiber design as an example, the multi-cladding structure design method is similarly scalable. The present invention calculates the LP mode of the fiber according to the fiber parameters given in Table 1. The geometric dimensions and refractive index distribution of the fiber are shown in FIG1 .

Table 1 Triple-clad fiber parameters

The method flow of the present invention is shown in FIG3 , and the steps include:

(1) Input the initial fiber structure calculation parameters, including 1) operating wavelength λ; 2) the radius of the fiber core and each cladding; 3) the number of refractive index layers N of the fiber; 4) the refractive index of the fiber core and each cladding.

(2) According to the refractive index of the core and the refractive index of the outermost cladding, the calculation range of the effective propagation constant is determined. The effective propagation constant range is between the propagation constant of the outermost cladding and the propagation constant of the core. The effective propagation constant β is sampled linearly and discretely, first densely sampled and then gradually sparsely sampled to balance the number of stable solutions and the influence of solution efficiency.

(3) Determine the segment boundary according to the number of cladding layers, take effective propagation constant sampling values in ascending order and calculate the frequency of each segment corresponding to the sampling value, and take the absolute value of the frequency f ₁ , f ₂ ,…, f _j ,…, f _N .

(4) Choose to use the scalar method or the vector method to calculate the mode field distribution. If the scalar method is used, the characteristic matrix is a 2(N-1)×2(N-1) matrix; if the vector method is used, the characteristic matrix is a 4(N-1)×4(N-1) matrix.

(5) Construct a characteristic matrix, which is composed of an oscillator matrix and an attenuation submatrix. Take effective propagation constant sampling values in ascending order. Compare the sampling values with the propagation constants of the core and each cladding layer one by one. When the sampling value is greater than the propagation constant of the core/cladding, construct an oscillator matrix; when the sampling value is less than the propagation constant of the core/cladding, construct an attenuation submatrix. The oscillation/attenuation submatrices of each layer of the obtained optical fiber are formed into a double diagonal matrix, which is the characteristic matrix of the optical fiber.

(6) Find the value of the variable effective propagation constant when the determinant of the characteristic matrix is 0. The characteristic matrix is a square matrix. If the homogeneous equation system is to have a non-zero solution, its determinant must be zero. The discrete solution to the root of the equation can be a discrete numerical method for solving the root of the equation, such as the bisection method, Newton's method, and the chord-intercept method.

(7) By sorting the effective propagation constant values according to different mode orders and converting them through formulas, the equivalent refractive index of the scalar mode or vector mode can be obtained.

(8) Substitute the equivalent refractive index back into the characteristic matrix to obtain the equation group of the corresponding mode, and calculate the mode field distribution in the optical fiber under this mode. The results of the optical fiber mode are shown in Table 2. After further calculation, the amplitude, light intensity, phase distribution and other information of each mode can be obtained, as shown in Figure 2.

Table 2 Partial results of fiber mode

Although the specific embodiments of the present invention are disclosed for the purpose of illustration, the purpose is to help understand the content of the present invention and implement it accordingly, those skilled in the art will understand that various substitutions, changes and modifications are possible without departing from the spirit and scope of the present invention and the appended claims. Therefore, the present invention should not be limited to the content disclosed in the best embodiment, and the scope of the present invention is subject to the scope defined in the claims.

Claims (9)

1. A method for designing a multi-clad step-index optical fiber based on a characteristic matrix, the steps comprising: 1) Input the calculation parameters of the initial structure of the optical fiber, including the working wavelength λ, the radius from the core to each cladding, the refractive index of the core and the refractive index of each cladding; the core and each cladding are collectively referred to as the refractive index layer of the optical fiber, then the total number of refractive index layers is N, the radius from the core to the outermost cladding is represented by r ₁ , r ₂ ,…, _ri ,…, r _N , and the refractive index value is represented by n ₁ , n ₂ ,…, n _j ,…, n _N , where _ri is the radius of the i-th refractive index layer and n _j is the refractive index of the j-th refractive index layer; 2) determining the effective propagation constant calculation range according to the core refractive index and the outermost cladding refractive index; performing discrete sampling in the effective propagation constant calculation range to obtain effective propagation constant β sampling values, recorded as β=β ₁ ,β ₂ ,β ₃ ,…β _t …β _Z ; wherein β _t is the t-th effective propagation constant sampling value, and Z is the total number of sampling points; 3) Determine the segment boundary according to the number of cladding layers; for each effective propagation constant sample value, calculate the normalized lateral phase parameter or normalized lateral attenuation parameter of each refractive index layer corresponding to the effective propagation constant sample value, denoted as f ₁ , f ₂ ,…, f _j ,…, f _N ; 4) constructing an oscillator matrix or an attenuation matrix of each effective propagation constant sampling value in the designed optical fiber; 5) constructing a characteristic matrix of the designed optical fiber according to the oscillator matrix and the attenuation matrix; 6) Solve the effective propagation constant value when the determinant value of the characteristic matrix is 0

7) Calculate the effective propagation constant value under different mode orders

The corresponding equivalent refractive index n _eff ; 8) Substituting the obtained equivalent refractive index n _eff back into the characteristic matrix to obtain the equation group of the corresponding mode, and calculating the mode field distribution in the optical fiber under each corresponding mode. 2. The method according to claim 1, characterized in that when the effective propagation constant sampling value is β _t , the normalized lateral phase parameter or the normalized lateral attenuation parameter of the j-th refractive index layer is

Wherein, k ₀ is the wave vector in vacuum, when (k ₀ n _j ) ² -β _t ² > 0, f _j is the normalized transverse phase parameter, and when (k ₀ n _j ) ² -β _t ² < 0, f _j is the normalized transverse attenuation parameter. 3 . The method according to claim 2 , characterized in that when (k ₀ n _j ) ² - β _t ² > 0, an oscillator matrix of the j-th refractive index layer is constructed; when (k ₀ n _j ) ² - β _t ² < 0, an attenuation submatrix of the j-th refractive index layer is constructed. 4. The method according to claim 3, characterized in that the feature matrix is

The “/” symbol in the determinant represents a logical OR relationship. When k ₀ n _j ≥ β _t, the oscillator matrix A _i,j is taken; when k ₀ n _j ＜ β _t, the attenuation submatrix B _i,j is taken. 5. The method according to claim 4, characterized in that in step 6), the effective propagation constant value is obtained by solving

The steps are: ① find the value of the characteristic matrix determinant corresponding to each β _t in turn, and use the zero point existence theorem to find the interval in which all values satisfying the determinant have zero points; ② use discrete numerical methods to find the numerical solution where the determinant is 0 in all satisfying intervals

Where x is the number of numerical solutions for the fiber mode,

is the effective propagation constant value corresponding to the Tth numerical solution. 6. The method according to claim 1 is characterized in that the mode field diameter, insertion loss, cutoff wavelength, and fiber dispersion of the designed optical fiber are determined according to the mode field distribution, thereby determining the application scenario of the designed optical fiber. 7. The method according to claim 1 is characterized in that discrete sampling is performed within the effective propagation constant calculation range. 8. A server, characterized in that it comprises a memory and a processor, the memory stores a computer program, the computer program is configured to be executed by the processor, and the computer program comprises instructions for executing each step in any one of the methods of claims 1 to 7. 9. A computer-readable storage medium having a computer program stored thereon, wherein when the computer program is executed by a processor, the steps of the method according to any one of claims 1 to 7 are implemented.

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