CN112904555A - Gaussian beam shaping lens design method based on partial differential equation solution - Google Patents

Abstract

The invention discloses a free-form surface lens design method for laser shaping, which comprises the following steps: based on the snell's law, the relation between incident Gaussian light, the curved surface of the lens and the illumination of the target surface is found, the partial differential equation of the curved surface of the lens is constructed, programming is carried out in MATLAB, and the curve discrete points of the lens are solved by using a Runge-Kutta method. And (3) importing the obtained data into solidworks software, constructing a lens 3D model, importing the model into optical software ZEMAX software for verification, wherein simulation analysis results show that the design successfully shapes the Gaussian beam into a flat-top beam, and the uniformity of the illumination of the target surface is more than 95%.

Description

Gaussian beam shaping lens design method based on partial differential equation solution

Technical Field

The invention belongs to the field of optical system design, and particularly relates to a Gaussian beam shaping lens design method based on partial differential equation solution.

Background

Laser processing technology is in the present day industry where energy control of a laser beam is one of the most important factors. Various optical systems have been developed for beam shaping of laser light, but most designs have aspheric surfaces set in optical software, and the lens is obtained by optimizing an optimization function step by step. The method cannot effectively ensure that the optimal solution is obtained, and a large amount of time is spent in the process of repeated optimization. It is therefore far reaching how to quickly design an effective shaping lens.

Disclosure of Invention

A Matlab program written by the design method can quickly construct a lens meeting the requirements, and the shaping of Gaussian light is realized.

In order to realize the purpose, the invention adopts the technical scheme that: a Gaussian beam shaping lens design method based on partial differential equation solution comprises the following steps:

step 1, confirming basic parameters of a structure according to design requirements, wherein the basic parameters comprise an initial position of a central point of a first lens; the distance between the first lens and the second lens; target surface illumination values, etc.

And 2, performing energy integration on the parallel incident Gaussian beams, and establishing an energy mapping relation with the target surface.

And 3, establishing a partial differential equation between the light ray, the lens curved surface and the target surface according to the Snell's law.

And 4, using Matlab programming, and solving by using a Runge-Kutta method in numerical analysis to obtain a partial differential equation solution so as to obtain the discrete points of the free-form surface of the first shaping lens.

And 5, calculating the discrete points of the free-form surface of the second lens by the light rays emitted by the first lens, wherein the lens is used for collimating and emitting the shaped light rays.

And 6, importing the discrete points of the free curved surfaces of the first lens and the second lens obtained by Matlab calculation into Solidworks mechanical software, and establishing a 3D model.

And 7, importing the established 3D model into optical software Zemax, and verifying the effect of the shaping system.

Further, in step 2, the gaussian function is divided by a Newton-Cotes formula, and the steps are as follows:

step 2-1, setting incident light as Gaussian function distribution with beam waist of omega₀。

And 2-2, setting n nodes.

And 2-3, performing numerical integration on the Gaussian functions in the nodes from n-1 to n by using a Newton-Cotes formula when the Cotes coefficient is 4.

Further, in step 3, the detailed steps of constructing the partial differential equation are as follows:

step 3-1, setting the refractive index value n of the lens₁Let the lens curve expression be y (x), where y is the light incidence direction.

Step 3-2, establishing an energy mapping relation f (x) ═ g (r), wherein f (x) is the energy within the radius x of the lens, and g (r) is the energy within the radius r of the m target surface.

And 3-3, deriving a partial differential equation related to the coordinates of the discrete points of the lens according to the energy mapping relation and the Snell equation. The expression is as follows:

further, in step 4, a partial differential equation is solved by using a Runge-Kutta method, wherein the formula is as follows:

k₁＝f(x_n，y_n)

further, in step 6, a 3D model is established according to the calculated data, which specifically comprises the steps of:

step 6-1, store the x, y, z coordinates of the lens curve into a three column matrix.

And 6-2, deriving a matrix as a txt document by using the save command in Matlab.

And 6-3, importing the txt document by using a characteristic-curve passing through an xyz point operation in Solidworks to generate a curve.

The invention has the following beneficial effects: the energy of the laser beam can be arbitrarily shaped and collimated out. The programming program obtained by the method can be operated and completed within a few seconds, and the free-form surface lens structure is obtained. Compared with the traditional design method, the lens provided by the invention has the advantages of higher precision, more excellent shaping effect and shorter design time. For different shaping requirements, the setting of the target surface in the program can be modified, so that the program can be recycled. The shaping of a gaussian beam into a flat-topped beam is described herein as an example.

Drawings

Fig. 1 is an optical schematic diagram of the present invention, and the process of shaping a gaussian beam into a flat-top beam can be seen from the diagram.

FIG. 2 is a schematic diagram of the equation construction of the present invention, in which various parameters such as the lens curved surface, the light incident angle, the emergent angle, etc. are given, and the equation is established according to the parameters shown in the diagram.

FIG. 3 is a cross-sectional illumination diagram of a Gaussian beam.

Fig. 4 is a planar illuminance diagram of a gaussian beam.

FIG. 5 is a line section illuminance diagram of a flat-topped beam at the target surface.

FIG. 6 is a planar illuminance diagram of a flat-topped beam at the target surface.

Detailed Description

The technical scheme of the invention is further explained in detail by combining the drawings in the specification.

Referring to fig. 1, the 1 st lens is a plano-convex lens, the gaussian light is incident from the plane 1 and then refracted to the convex surface 3 from the convex surface 2, and the collimated light is emergent after refraction.

Referring to fig. 2, the direction of incidence of light is taken as the y-axis and the perpendicular direction is taken as the x-axis in the figure. The specific partial differential equation construction process is as follows:

the snell's law can be applied on the surface 2:

n₁sinθ₁＝n₂sinθ₂

n₁，n₂refractive index of the lens and refractive index of air, respectively, and theta₁，θ₂It can be expressed by the normal slope k1 at the refraction of the surface 2 and the slope k2 of the outgoing ray, thus obtaining the system of equations:

the relation between the lens curved surface derivative and the emergent light can be deduced by the following four formulas:

next, an energy mapping relation is established, and since the system is a structure which is circularly symmetrical around the y axis, the x-y axis section can be taken for analysis.

Let the Gaussian beam initial position illumination expression be

Integral to this function is represented by f (x) ═ f (x) dx, f (x) can be represented as enclosed by a gaussian beam within the section of radius xThe energy contained in the water. Let the illumination of the target surface be expressed as g (r), since the illumination distribution of the target surface is uniform, the energy enclosed in the section of radius r can be expressed as

Wherein T is the total energy and R is the radius of the light spot of the target surface 3. From f (x) ═ g (r), then:

since R and T are both known numbers, R is a simple function of x. According to

The final partial differential equation is obtained, where y₀Is the y coordinate when x is 0 on the surface 2 and d is the distance from the center of the surface 2 to the target surface.

The solution to the partial differential equation can be solved by the Runge-Kutta method:

k₁＝f(x_n，y_n)

the center (0, y0) of the surface 2 is used as an initial point, the step size is set to h, and the discrete points of the surface 2 can be found. The solving process of the surface 4 is similar to that of the surface 2, and because the distance between the surface 2 and the surface 4 is long, the curvature of the surface 4 is large and close to a plane, and the influence caused by the difference between the surface 4 and the target surface 3 can be ignored. And (3) calculating the curvature of each node by using the partial differential equation in the step (3), taking the curvature as the intersection point of the tangent and the next ray as the next point of the curved surface, and circulating the steps until the last ray iteration is finished. The parameters set herein are as follows:

d＝100mm，R＝3mm，y₀＝3mm，n₁＝1.5，ω₀＝3mm

after the calculation is finished, the obtained data is stored in the matrix and is exported to be a txt document, and the txt document is imported by using the operation of 'feature-curve passing through xyz points' to generate a curve. And (3) constructing an entity file of the lens by using the rotation characteristic, importing the entity file into Zemax, and establishing a non-sequence model to analyze the lens effect. Simulated light rays are set to be 1000000 ten thousand, the half width of the Gaussian beam is 1.5, a light ray detector with the length of 5mm x 5mm is arranged at the position 300mm on the axis, and the simulation result is observed.

As can be seen from fig. 3-1 to 3-4, a beam of flat-top collimated light is programmed after incident light is shaped by a lens, a circular light spot with a radius of 3mm is formed on a target surface, the uniformity is over 97%, and the design requirement is met.

The partial differential equation in the shaping process is constructed, the free-form surface type is solved in Matlab by programming, when 1000 nodes are set, the solution is completed for about 5s, the design efficiency is greatly improved, and the free-form surface type can be obtained only by modifying corresponding required parameters in subsequent similar designs, which has important significance for laser shaping realized by the lens.

The above description of the disclosed embodiments is intended to enable those skilled in the art to make modifications and alterations to the described embodiments. Therefore, the present invention is not limited to the specific embodiments disclosed and described above, and some modifications and variations of the present invention should fall within the scope of the claims of the present invention. Furthermore, although specific terms are employed herein, they are used in a generic and descriptive sense only and not for purposes of limitation.

Claims (5)

1. A Gaussian beam shaping lens design method based on partial differential equation solution is characterized in that: the method comprises the following steps:

step 1, confirming basic parameters of a structure according to design requirements, wherein the basic parameters comprise an initial position of a central point of a first lens; the distance between the first lens and the second lens; a target surface illumination value;

step 2, performing energy integration on the parallel incident Gaussian beams, and establishing an energy mapping relation with a target surface;

step 3, establishing a partial differential equation between the light ray, the lens curved surface and the target surface according to the Snell's law;

step 4, Matlab programming is used, and a partial differential equation solution is obtained by solving with a Runge-Kutta method in numerical analysis, so that discrete points of the free-form surface of the first shaping lens are obtained;

step 5, calculating the discrete points of the free-form surface of the second lens by the light rays emitted by the first lens, wherein the lens is used for collimating and emitting the shaped light rays;

step 6, importing the discrete points of the free curved surfaces of the first lens and the second lens obtained by Matlab calculation into Solidworks mechanical software, and establishing a 3D model;

and 7, importing the established 3D model into optical software Zemax, and verifying the effect of the shaping system.

2. The method of claim 1, wherein the partial differential equation solution-based Gaussian beam-shaping lens design method comprises: in step 2, the Gaussian function is divided through a Newton-Cotes formula, and the steps are as follows:

step 2-1, incident light is set to beam waist omega₀(ii) a gaussian function distribution;

step 2-2, setting n nodes;

and 2-3, performing numerical integration on the Gaussian functions in the nodes from n-1 to n by using a Newton-Cotes formula when the Cotes coefficient is 4.

3. The method of claim 1, wherein the partial differential equation solution-based Gaussian beam-shaping lens design method comprises: in step 3, the detailed steps for constructing the partial differential equation are as follows:

step 3-1, setting the refractive index value n of the lens₁Setting a lens curve expression as y (x), wherein y is a light incidence direction;

step 3-2, establishing an energy mapping relation F (x) ═ G (r), wherein F (x) is the energy within the radius of the lens x, and G (r) is the energy within the radius of the m target surface r;

and 3-3, deriving a partial differential equation related to the coordinates of the discrete points of the lens according to the energy mapping relation and the Snell equation. The expression is as follows:

4. the method of claim 1, wherein the partial differential equation solution-based Gaussian beam-shaping lens design method comprises: in step 4, a Runge-Kutta method is used for solving a partial differential equation, wherein the formula is as follows:

k₁＝f(x_n，y_n)

5. the method of claim 1, wherein the partial differential equation solution-based Gaussian beam-shaping lens design method comprises: in step 6, a 3D model is established according to the data obtained by calculation, and the specific steps are as follows:

step 6-1, storing the x, y and z coordinates of the lens curve into a three-column matrix;

step 6-2, deriving a matrix as a txt document by using a save command in Matlab;

and 6-3, importing the txt document by using a characteristic-curve passing through an xyz point operation in Solidworks to generate a curve.

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