An annular flattop beam optimization method based on improved GS algorithm

Abstract

The application provides an annular flat-top beam optimization method based on an improved GS algorithm, utilizes the diffraction effect of a binary optical element to multiply an amplitude distribution and a phase function to obtain a light field complex amplitude distribution, utilizes Fourier transform to represent the diffraction effect, compares the energy conversion efficiency of an actual shaping light beam and a target annular flat-top beam with a preset threshold in a real-time manner in the algorithm iteration process, obtains the phase distribution of an optimal binary optical element, and shapes a Gaussian light beam into a hollow annular flat-top beam. The application solves the problems that the traditional algorithm is sensitive to an initial phase and a convergence path is easy to deviate from an optimal solution, and ensures that the output light beam has higher energy utilization while meeting the amplitude distribution requirement, thereby effectively improving the stability and practicality of annular flat-top beam shaping.

Description

Technical Field

[0001] This invention relates to the field of optical technology, and more specifically to a method for optimizing a ring flat-top beam based on an improved GS algorithm. Background Technology

[0002] To improve the beam quality of lasers, laser beam shaping technology has become a key research focus both domestically and internationally. Different applications have different requirements for beam shaping. Due to their advantages such as high diffraction efficiency, large design freedom, strong operability, and good dispersion, binary optical elements are widely used in beam shaping technology.

[0003] The original GS algorithm is an iterative process based on phase retrieval, repeatedly replacing amplitude and inverse transformation, with the mean square error of amplitude as the convergence condition. This algorithm belongs to the steepest descent algorithm category, with a simple and clear approach, local convergence, and sensitivity to initial phase assumptions. It is highly susceptible to Gibbs oscillations, can only obtain local extrema, and therefore cannot converge to the correct phase. Furthermore, the accuracy of light field reconstruction at the edges is relatively low.

[0004] Therefore, there is an urgent need for a GS improvement method that can improve convergence speed and stability and enhance laser beam quality. Summary of the Invention

[0005] This invention addresses the problems of slow convergence, susceptibility to local optima, and low diffraction efficiency in the original GS algorithm for annular flat-top beam shaping. It proposes an improved GS algorithm for annular flat-top beam optimization. The improved GS algorithm changes the convergence condition from the mean square error of the optical field amplitude to diffraction efficiency. During iteration, it calculates the energy conversion efficiency between the actual shaped beam and the target annular flat-top beam in real time. The iteration terminates when the efficiency reaches a preset threshold. This solves the problems of sensitivity to initial phase and easy deviation from the convergence path in traditional algorithms, while ensuring that the output beam meets the amplitude distribution requirements and has higher energy utilization.

[0006] This invention provides a method for optimizing annular flat-top beams based on an improved GS algorithm, comprising the following steps:

[0007] S1. Using the plane of the binary optical element as the input plane, according to the preset initial phase value... The complex amplitude distribution U of the incident light field is obtained from the Gaussian amplitude distribution A(x,y) of the incident light. in (x,y), where (x,y) is the free space position;

[0008] S2. Establish a diffraction model based on the distance z between the incident and exit surfaces, and use the diffraction model to perform a Fourier transform to obtain the complex amplitude U of the exit light field. out (x',y');

[0009] S3, based on the complex amplitude distribution of the incident light field U in (x,y) and the complex amplitude U of the emitted light field out (x',y') yields the diffraction efficiency. We then determine whether the diffraction efficiency meets the convergence condition. If it does, the phase parameter is the globally optimal phase distribution of the binary optical element. The optimization of the annular flat-top beam is now complete.

[0010] If not, proceed to step S4;

[0011] S4, n = n + 1, where n is the number of updates. The phase is updated to obtain the updated phase. According to the updated phase The diffraction model affects the complex amplitude U of the emitted light field. out The inverse Fourier transform of (x', y') yields the new light field amplitude U. in '(x,y), return to step S2, and use the diffraction model to measure the amplitude U of the new light field. in Perform a Fourier transform on (x,y); where n = 1, 2, ...

[0012] The present invention provides a method for optimizing a ring-shaped flat-top beam based on an improved GS algorithm. In a preferred embodiment, step S2, when… The diffraction model is the Fresnel diffraction model; when When the diffraction model is Fraunhofer diffraction model, D is the size of the diffraction aperture and λ is the wavelength of the laser in vacuum, in meters.

[0013] The annular flat-top beam optimization method based on the improved GS algorithm described in this invention, as a preferred embodiment, in step S2, when performing Fourier transform using the Fresnel diffraction model:

[0014] Where i is the imaginary part unit, and k is the wave number in meters. -1 , k = 2π / λ.

[0015] The annular flat-top beam optimization method based on the improved GS algorithm described in this invention, as a preferred embodiment, in step S3, when using the Fresnel diffraction model inverse Fourier transform,

[0016]

[0017] Where ds is a two-dimensional integral area element, ds=dx′dy′.

[0018] The annular flat-top beam optimization method based on the improved GS algorithm described in this invention, as a preferred embodiment, in step S2, when performing Fourier transform using the Fraunhofer diffraction model,

[0019]

[0020] Where i is the imaginary part unit, and k is the wave number in meters. -1 , k = 2π / λ.

[0021] The annular flat-top beam optimization method based on the improved GS algorithm described in this invention, as a preferred embodiment, in step S3, when performing the inverse Fourier transform using the Fraunhofer diffraction model,

[0022]

[0023] Where ds is a two-dimensional integral area element, ds=dx′dy′.

[0024] The annular flat-top beam optimization method based on the improved GS algorithm described in this invention, as a preferred embodiment, in step S1, the initial phase value... Multiplying the incident light by the Gaussian amplitude distribution A(x,y) yields the complex amplitude distribution U of the incident light field. in (x,y);

[0025]

[0026] The present invention provides a method for optimizing a ring flat-top beam based on an improved GS algorithm. In a preferred embodiment, step S1 involves normalizing the amplitude of the incident Gaussian beam to obtain the Gaussian amplitude distribution A(x,y) of the incident light.

[0027] The annular flat-top beam optimization method based on the improved GS algorithm described in this invention, as a preferred embodiment, in step S1, the initial phase value... It is a random value within the range of 0 to 2π;

[0028] In step S4, the phase is updated. The update objective is to increase the diffraction efficiency.

[0029] In practical applications, various improved algorithms based on the GS algorithm are often used.

[0030] This invention is based on scalar diffraction theory combined with an improved GS algorithm. Using diffraction efficiency as the convergence condition, it improves the consistency between the design accuracy of diffraction optical elements and the actual diffraction effect by increasing the preset diffraction efficiency and the number of iterations. It enhances phase retrieval accuracy through iterative optimization algorithms, solving problems such as slow convergence speed and local optima in traditional GS algorithms, ultimately obtaining a ring-shaped flat-top beam with uniform energy distribution and steep edges. It has wide applications in fields such as lidar, laser communication, materials processing, medical aesthetics, and photolithography.

[0031] This invention provides a method for optimizing a ring-shaped flat-top beam based on an improved Gaussian beam (GS) algorithm, belonging to the field of beam shaping in laser technology and optical engineering. This invention utilizes the diffraction effect of binary optical elements, multiplying the amplitude distribution by a phase function to obtain the complex amplitude distribution of the optical field. Fourier transform is then used to characterize the diffraction effect. By comparing the energy conversion efficiency of the actual shaped beam and the target ring-shaped flat-top beam with a preset threshold during real-time algorithm iteration, the optimal phase distribution of the binary optical elements is obtained, shaping the Gaussian beam into a hollow ring-shaped flat-top beam. This invention solves the problems of traditional algorithms being sensitive to initial phase and prone to deviation from the optimal solution during convergence, while ensuring that the output beam meets the amplitude distribution requirements and has higher energy utilization, effectively improving the stability and practicality of ring-shaped flat-top beam shaping. Applying the improved GS algorithm of this invention to related engineering fields provides key technical ideas for beam shaping and precise design parameters for optical elements, further expanding the application scope of beam shaping technology.

[0032] The present invention has the following advantages:

[0033] (1) This invention obtains the precise phase of the binary optical element through software calculation, which solves the problems of low accuracy, low operability and difficult processing of existing traditional methods; by using Gaussian light through scalar diffraction theory combined with the improved GS algorithm, the phase variable is iteratively optimized with diffraction efficiency as the convergence criterion, which greatly improves the shaping accuracy and makes the output beam closer to the ideal shape.

[0034] (2) The present invention takes into account the distance between the incident surface and the exit surface to create an accurate diffraction model and improve measurement accuracy.

[0035] (3) This invention can be used in fields such as laser processing and optical communication, further expanding the application scope of the ring flat-top beam in precision manufacturing, medical equipment, remote sensing and other scenarios. Attached Figure Description

[0036] Figure 1 A flowchart of an annular flat-top beam optimization method based on an improved GS algorithm;

[0037] Figure 2 This is a diagram showing the output light field distribution of an ideal ring-shaped flat-top beam.

[0038] Figure 3 A phase distribution diagram of a binary optical element for an annular flat-top beam optimization method based on an improved GS algorithm;

[0039] Figure 4 The final output light field distribution diagram of a ring flat-top beam optimization method based on an improved GS algorithm at 99% diffraction efficiency;

[0040] Figure 5This is a graph showing the relationship between the number of iterations and diffraction efficiency of an annular flat-top beam optimization method based on an improved GS algorithm. Detailed Implementation

[0041] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments.

[0042] Example 1

[0043] like Figure 1 As shown, an optimization method for annular flat-top beams based on an improved GS algorithm is proposed. Based on scalar diffraction theory and combined with the improved Gerchberg-Saxton (GS) algorithm, the method uses diffraction efficiency as the convergence condition. By increasing the preset diffraction efficiency and the number of iterations, the design accuracy of diffractive optical elements and the consistency with the actual diffraction effect are improved, thus accurately shaping the incident Gaussian beam into a hollow annular flat-top beam.

[0044] A laser beam propagates along the Z-axis in free space, and its electric field amplitude can be expressed as:

[0045]

[0046] Where E(x,y,z) is the electric field amplitude at the free space position (x,y,z), with units of V·m. -1 E0 represents the peak electric field amplitude at the beam waist position z = 0 and the optical axis r = 0, in V·m. -1 ω(z) is the beam cross-sectional dimension, i.e., the beam waist radius, expressed as: z_R is the Rayleigh length, in meters (m); when z = 0, ω(z) decreases to a minimum of ω0, which is the beam waist radius in the diffraction-limited state; r is the radial distance to the optical axis, in meters (m), expressed as: k is the wave number, in meters (m). -1 The expression is k = 2π / λ; λ is the wavelength of the laser in vacuum, in meters; ρ(z) is the normalized radial coordinate; η(z) is the normalized amplitude factor.

[0047] Scalar diffraction theory treats the electromagnetic field of a vector as a scalar, ignoring the coupling relationship between the components of the electromagnetic field vector and only considering the complex amplitude of a transverse component. Kirchhoff diffraction theory is the most widely used among these theories.

[0048] The essence of beam shaping using binary optical elements (BOEs) can be understood as using discretized phase modulation degrees of freedom to inversely solve for the phase distribution that satisfies the target optical field constraint within a diffraction framework. An improved GS algorithm is used to design a binary optical element that optimizes the wavefront of a Gaussian beam by updating the phase distribution on the element surface, thereby accurately achieving energy distribution conversion.

[0049] Applying Kirchhoff's diffraction theory to the design of binary optical elements, treating the plane of the binary optical element as the input plane, and pre-setting an initial phase value. Input the amplitude of the incident Gaussian beam; the product of the two is the complex amplitude distribution of the incident light field, expressed as:

[0050]

[0051] Where A(x,y) is the Gaussian amplitude distribution of the incident light. This represents the initial phase distribution of the plane of the binary optical element. After propagating a distance *r* in space, diffraction occurs. Mathematically, the diffraction effect is represented by a Fourier transform, U out (x′,y′) is the complex amplitude of any point P(x′,y′) within the exit surface; θ is the diffraction angle. The general expression for the complex amplitude distribution of the output surface light field is:

[0052]

[0053] Based on the distance z between the incident and exit planes, diffraction can be specifically divided into Fresnel diffraction and Fraunhofer diffraction. When the distance z between the incident and exit planes... When (where D is the size of the diffraction aperture), the application conditions of Fresnel diffraction are satisfied, and the expression for the complex amplitude distribution of the outgoing light field can be further expressed as:

[0054]

[0055] When the distance between the incident surface and the imaging surface When the condition for Fraunhofer diffraction is met, the expression for the complex amplitude distribution of the emitted light field is:

[0056]

[0057] The output surface is constrained by replacing the amplitude distribution after the forward transform with the target output beam amplitude B(x,y) while keeping the phase unchanged. Then, an inverse Fourier transform is performed to obtain a new optical field amplitude and phase distribution. For n = 0, 1, 2, ..., where n is the update number, the expressions for the complex amplitude distribution of the new Fresnel and Fraunhofer diffraction incident light fields are as follows:

[0058]

[0059] Where ds represents a two-dimensional integral area element, and the expression is ds = dx′dy′.

[0060] By constraining the input surface, i.e. keeping the phase unchanged, the amplitude of the new optical field is replaced by the amplitude of the input beam A(x,y). The diffraction effect is simulated by Fourier transform and iterated continuously until the preset diffraction efficiency is met. At this time, the phase parameter is the globally optimal phase distribution of the binary optical element, which can realize the optimization of the ring flat-top beam.

[0061] The expression for calculating diffraction efficiency is:

[0062]

[0063] Among them, ∫|U out (x′,y′)| 2 dx′dy′ represents the output light energy; ∫|U in (x,y)| 2 dxdy represents the incident light energy.

[0064] The technical solution adopted in this invention is: given the initial phase of the binary optical element The selected diffraction principle is determined based on the distance between the incident and exit surfaces, and a diffraction model is constructed. The incident light field distribution is coupled through the transmittance function to transform it into the output light field distribution. The complex amplitude transmittance function expression is as follows:

[0065]

[0066] in The phase function is represented by . A real-time diffraction efficiency criterion is introduced in the GS iteration, and the phase distribution on the element surface is repeatedly adjusted until an ideal annular flat-top beam is obtained.

[0067] Improved GS algorithm process as follows Figure 1 As shown, it includes the following steps:

[0068] S1. Obtain the complex amplitude distribution of the incident light field, and preset the initial phase of a binary optical element. Multiplying the amplitude of the incident Gaussian beam by the amplitude of the incident beam gives the complex amplitude distribution of the incident light field. i is the imaginary unit.

[0069] Before step one, the method also includes normalizing the amplitude of the incident Gaussian beam to ensure the accuracy of subsequent calculations.

[0070] S2. Obtain the diffraction distribution of the Gaussian beam after passing through a binary optical element, and input the complex amplitude distribution U of the Gaussian beam. inPerform a Fourier transform on (x,y). Select the corresponding diffraction model based on the diffraction distance z (i.e., the distance between the incident and exit planes). If the near-field diffraction conditions are met, use the Fresnel diffraction model; if the far-field diffraction conditions are met, use the Fraunhofer diffraction model.

[0071] In step S2, the corresponding diffraction model is selected based on the distance z between the incident and exit surfaces, specifically including: if Choose the Fresnel diffraction model; if The Fraunhofer diffraction model was selected.

[0072] The complex amplitude distribution of the outgoing Gaussian beam is obtained. Taking the phase part of the expression after the Fourier transform in step two, and combining it with the amplitude B(x′,y′) of the outgoing Gaussian beam, the product of the two is the complex amplitude distribution of the outgoing Gaussian beam in step S2.

[0073] S3. Calculate the diffraction efficiency according to formula (8), and determine whether the iteration condition is met in this cycle. When the iteration condition is met, the target parameters are obtained, and the optimization of the Gaussian beam with a ring flat top is realized, and the phase function is obtained. The phase of the binary optical element obtained after the cycle is taken, i.e., the complex amplitude distribution U of the Gaussian beam in the output field. out The phase part of (x′,y′) is used to complete the shaping of the Gaussian beam into a ring-shaped flat-top beam.

[0074] If the conditions are not met, a newer complex amplitude distribution of the incident light field is obtained. Based on the reversibility of the optical path, the complex amplitude distribution U of the newly generated Gaussian beam in the output field is then calculated. out Performing an inverse Fourier transform on (x′,y′) yields U in ′(x,y), return to step S2 for a new round of iteration until the iteration condition is met.

[0075] In this embodiment, the annular flat-top beam has the advantages of uniform energy and a central void, which can effectively solve the problem of overheating at the center of the Gaussian beam, suppress thermal distortion, and improve processing consistency. It is widely used in laser additive manufacturing, precision drilling, free-space optical communication, and optical tweezers particle manipulation. The 1550nm wavelength is within the communication window, with low atmospheric attenuation and low transmission loss, making it suitable for long-distance transmission. At the same time, 1550nm also has good eye safety characteristics, balancing performance and safety, making it the preferred wavelength for beam shaping in many scenarios. Based on this, a method for optimizing the annular flat-top beam based on an improved GS algorithm is proposed. This method can accurately obtain annular flat-top beams with different ring diameters and has extremely high diffraction efficiency. In the example, the Gaussian beam has a wavelength of 1550nm, a waist radius of 1.8mm, an optical element aperture of 4mm, an inner ring diameter of 0.7mm, an outer ring diameter of 1.8mm, a preset diffraction efficiency threshold of 99%, and a transmission distance of 100mm. In this case, the Fresnel diffraction model is used for algorithm calculation.

[0076] A beam model is constructed based on the given Gaussian beam parameters. The beam is filled by a ring model with defined inner and outer ring diameters. Under ideal conditions, the output light field distribution of the ring-shaped flat-top beam is as follows: Figure 2 As shown, its light field distribution clearly resembles a standard hollow cylinder, with the cylinder height representing the light intensity. The diffraction effect of the binary optical element is characterized using Fourier transform. Through continuous algorithm updates and iterations, the phase distribution satisfying the convergence condition is shown below. Figure 3 As shown, this is an irregular relief structure. By performing Fourier transforms and inverse transforms on the complex amplitudes of the incident and output light fields, and targeting diffraction efficiency, the phase distribution parameters of the binary optical element are updated to achieve the actual output light field distribution that satisfies the convergence condition, as shown below. Figure 4 As shown in the figure. During the update iteration process, the relationship between the number of iterations and the diffraction efficiency is as follows: Figure 5 As shown, the diffraction efficiency increases sharply in the early stages of iteration, indicating that the method quickly finds a relatively ideal phase value. Then, as the number of iterations increases significantly, the diffraction efficiency slowly increases until the convergence condition is met, ultimately outputting a solution that is closer to the ideal phase.

[0077] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be covered within the scope of protection of the present invention.

Claims

1. A method for optimizing annular flat-top beams based on an improved GS algorithm, characterized in that: Includes the following steps: S1. Using the plane of the binary optical element as the input plane, according to the preset initial phase value... Gaussian amplitude distribution of incident light The complex amplitude distribution of the incident light field was obtained. , For free space location; S2. Based on the distance between the incident surface and the exit surface z A diffraction model is established, and a Fourier transform is performed using the diffraction model to obtain the complex amplitude of the outgoing light field. ; When the diffraction distance Z satisfies The diffraction model is a Fresnel diffraction model; when When the diffraction model is the Fraunhofer diffraction model, wherein, D The size of the diffraction aperture. The wavelength of a laser in a vacuum is given by the unit λ. m ; When performing a Fourier transform using the Fresnel diffraction model: ； in, i For imaginary units, k Wave number, unit: , ; When performing Fourier transform using the Fraunhofer diffraction model: ； in, i For imaginary units, k Wave number, unit: ; S3, Based on the complex amplitude distribution of the incident light field and the complex amplitude of the emitted light field Once the diffraction efficiency is obtained, it is determined whether the diffraction efficiency meets the convergence condition. If it does, then the phase parameter is the globally optimal phase distribution of the binary optical element, and the optimization of the ring flat-top beam is completed. If not, proceed to step S4; When using the Fresnel diffraction model for inverse Fourier transform ； in, ds It is a two-dimensional integral area element. ; When performing inverse Fourier transform using the Fraunhofer diffraction model ； in, ds It is a two-dimensional integral area element. ; S4, n=n+1, n The phase is updated to the number of updates to obtain the updated phase. According to the updated phase The diffraction model affects the complex amplitude of the emitted light field. The new light field amplitude is obtained by performing an inverse Fourier transform. Return to step S2, and use the diffraction model to measure the amplitude of the new light field. Perform Fourier transform; among which n =1, 2, ...

2. The method for optimizing annular flat-top beams based on an improved GS algorithm according to claim 1, characterized in that: In step S1, the initial phase value Gaussian amplitude distribution of incident light Multiplication yields the complex amplitude distribution of the incident light field. ; 。 3. The method for optimizing annular flat-top beams based on an improved GS algorithm according to claim 1, characterized in that: In step S1, the amplitude of the incident Gaussian beam is normalized to obtain the Gaussian amplitude distribution of the incident light. .

4. The method for optimizing annular flat-top beams based on an improved GS algorithm according to claim 1, characterized in that: In step S1, the initial phase value It is a random value within the range of 0 to 2π; In step S4, the phase update The update objective is to increase the diffraction efficiency.

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