A method and device for shaping a bessel beam with adjustable focal depth and on-axis energy uniformity

Abstract

The application relates to a focal depth adjustable and on-axis energy uniform Bessel beam shaping method and device, which are used in the field of laser processing. The method takes a standard conical lens reference phase corresponding to a preset non-diffraction distance as a physical anchor point, modifies a radial phase through B-spline parameterization, obtains a pre-optimization phase function by using an analytical gradient to drive quasi-Newton optimization under a micro-Fresnel diffraction forward model, avoids the problems of uncontrollable focal depth and slow convergence caused by random initial values in a traditional method, and thus realizes fast, flexible and adjustable focal depth; through a target compensation mechanism guided by feedback, a feedback residual is constructed by comparing an experimental axial energy response curve with a fixed flat baseline, front end drop correction and tail end lift correction are applied to a source target, and re-optimization is carried out by taking pre-optimized B-spline coefficients as a hot start initial value, axial intensity deviation under non-ideal experimental conditions is effectively inhibited, and the axial uniformity of the beam and the robustness to experimental non-ideal factors are improved.

Description

Technical Field

[0001] This invention discloses a Bessel beam shaping method and apparatus with adjustable focal depth and uniform on-axis energy, which is used in the field of laser processing technology. Background Technology

[0002] Bessel beams, due to their non-diffraction properties and long focal depth transmission capabilities, have significant application value in fields such as laser precision machining, microscopic imaging, and optical communication. In particular, applications in precision laser machining, such as high aspect ratio drilling and thick glass cutting, require the beam to maintain a flat on-axis energy distribution within the long focal depth range; fluctuations in axial intensity directly affect the consistency of machining depth and modification quality.

[0003] However, the Bessel beam directly generated by the standard conical lens has a single-peak distribution in axial intensity and oscillates violently. Overexposure at the front end can easily lead to overexposure of the workpiece surface, while attenuation at the tail end results in insufficient modification at great depths, making it difficult to meet the requirements of precision machining for axial uniformity.

[0004] To address the aforementioned issues, existing research has proposed various strategies, such as frozen wave superposition, improved GS algorithms, dual-element structures, and analytical phase design. However, these strategies generally suffer from limitations such as limited depth of focus, complex elements, or alignment sensitivity. While optimization methods based on high-order polynomial phase combined with genetic algorithms can improve axial uniformity, their polynomial form has limited local phase freedom, requiring parameters to be re-optimized according to changes in beam conditions, and the optimization process is time-consuming. More importantly, these methods typically load a fixed hologram after optimization, failing to incorporate non-ideal factors such as beam quality, alignment errors, and spatial light modulator effects in the actual optical path. This leads to deviations of the measured axial envelope from the ideal prediction, resulting in significant tail attenuation under long depth of focus conditions, making it difficult to meet the dual requirements of precision machining for axial uniformity and flexible depth of focus control. Summary of the Invention

[0005] To address the shortcomings of the aforementioned technologies, the present invention aims to provide a method and apparatus for shaping a Bessel beam with adjustable focal depth and uniform axial energy, so as to achieve the goal of rapidly obtaining a Bessel beam with a specified focal depth and high axial uniformity under non-ideal experimental conditions.

[0006] To achieve the above objectives, in a first aspect, the present invention provides a Bessel beam shaping method with adjustable depth of focus and uniform on-axis energy, comprising: Based on a preset non-diffraction distance, a standard conical lens reference phase is constructed, and the standard conical lens reference phase is projected onto the B-spline function space to obtain the initial values ​​of the B-spline coefficients. Based on the initial values ​​of the B-spline coefficients, under the differentiable Fresnel diffraction forward model, a total loss function is constructed with the complete axial target intensity distribution including the rising segment, the middle flat-top segment, and the falling segment as the target. The B-spline coefficients are solved by a quasi-Newton optimizer to obtain the pre-optimized phase function. The pre-optimized phase function is used to determine the experimental axial energy response curve. Based on the experimental axial energy response curve and the fixed flat-top baseline, the feedback residual is determined, and the front-end descent correction and the tail-end lift correction are applied to the source target according to the feedback residual. The compensated target is obtained by integral conservation scaling. Using the B-spline coefficients corresponding to the pre-optimized phase function as the initial values ​​for hot start, and replacing the original flat-top target with the compensation target, hot start re-optimization is performed under the differentiable Fresnel diffraction forward model to obtain the target phase function; Based on the target phase function, a Bessel beam with adjustable focal depth and uniform on-axis energy is generated.

[0007] In one possible implementation, the standard conical lens reference phase The expression is:

[0008] For wave number, Where is the aperture radius, For radial coordinates.

[0009] In one possible implementation, the B-spline is a cubic B-spline; the initial values ​​of the B-spline coefficients are... This is obtained by minimizing the cumulative squared deviation between the B-spline representation of the phase and the reference phase of the standard conical lens over the aperture range:

[0010] Among them, B j (r) is the basis function of the j-th B-spline, c j Here, K represents the corresponding control coefficient, and K is the number of control points, based on the approximate Fresnel function. The nodes are adaptively determined, and the B-spline node vectors are non-uniformly distributed. A denser arrangement of nodes is made within the radially lower aperture region, which primarily contributes to the axial section of the flat top. This is the reference phase for the standard conical lens.

[0011] In one possible implementation, the differentiable Fresnel diffraction forward model is based on axisymmetric Fresnel diffraction integrals, with a propagation distance of... The diffraction field at that point is represented as:

[0012] in, Let be the waist radius of the incident Gaussian beam. It is the radial phase function. It is a zero-order Bessel function. For wavelength, The radial coordinates of the observation surface Aperture radius, It is the imaginary unit.

[0013] In one possible implementation, the analytical gradient of the total loss function is calculated using the chain rule. Adding the analytical gradient eliminates the need for extensive forward simulation estimation, thus improving optimization efficiency. This applies to the ... The gradient of each B-spline coefficient includes the partial derivative contribution of the deviation between the calculated intensity and the target intensity at each discrete sampling point within the axial flat-top interval to the complex field.

[0014] In one possible implementation, the target The construction method is as follows:

[0015] Among them, flat-top strength The on-axis intensity integral of a standard conical lens in the flat-top region The upper limit is determined by multiplying the average value within the range by a residual factor less than 1, typically between 0.8 and 0.95. For the front-end window function, take the smaller value between the normalized response of the standard conical lens and 1; This is a smooth cutoff window function applied to the back end of the flat top.

[0016]

[0017] in, It is the original axial light intensity distribution of the standard axis pyramid, which is directly calculated using a differentiable Fresnel diffraction forward model; It is the strength of the flat top; It is the axial propagation coordinate; It is the attenuation cutoff center position, and the specific position is set according to the required tail attenuation length, controlling the axial position of the falling edge; It is the truncation transition width, which controls the length of the interval from 1 to 0 light intensity smoothly. The specific value is adjusted according to the smoothness requirements of the target curve and the optimization of convergence.

[0018] In one possible implementation, the total loss function consists of five parts: a bias term, a total variation term, a curvature term, a shape term, and a coefficient smoothing regularization term.

[0019] Where: L dev The deviation term is used to calculate the difference between the axial strength and the target axial strength; L tvThe total variational term to suppress the first-order axial oscillation; L curv The curvature term is used to suppress second-order axial oscillations; L shape The shape term used to align the local axial trend with the target profile; L smooth A coefficient smoothing regularization term is used to suppress excessive oscillations in B-spline coefficients; , , , , These are the weighting coefficients for each part. The specific values ​​should be determined based on the corresponding adjustment functions and the actual experimental calibration.

[0020] The deviation term measures the deviation between the calculated axial strength and the target axial strength. The total variation term suppresses the first-order axial oscillation, the curvature term suppresses the second-order axial oscillation, the shape term aligns the local axial trend with the target profile, and the coefficient smoothing regularization term suppresses excessive oscillation of the B-spline coefficients.

[0021] In one possible implementation, the experimental axial energy response curve is determined by generating a phase hologram from the pre-optimized phase function and loading it onto a spatial light modulator. Lateral light intensity frames are acquired along the axial direction of the phase-modulated beam. The beam center of each frame is determined by a gray-scale weighted centroid positioning method. The average gray value within a fixed pixel window at this center is taken as the on-axis intensity at each axial position to obtain the experimental axial energy response curve.

[0022] In one possible implementation, the feedback residual is a normalized and smoothed fixed flat-top baseline. Subtract the normalized smoothed experimental axial energy response Difference:

[0023] The fixed flat-top baseline is the target defined in the pre-optimization stage. The experimental axial energy response is obtained by acquiring transverse light intensity frames along the axial direction, locating the beam center, and averaging them via a window. Both are aligned to the same simulation grid via coordinate mapping and then normalized and smoothed in a consistent manner to obtain the final result. and .

[0024] In one possible implementation, the front-end descent correction and tail-end lift correction are applied to the source target, which uses the fixed flat-top baseline during the first feedback and the compensated target from the previous feedback during subsequent feedbacks. The front-end descent correction is to subtract a front-end descent correction amount defined by the residual amplitude, i.e., the maximum value of the feedback residual in the front-end window, from the source target within the front-end terminal window of the flat-top region. The tail-end lift correction is to add a tail-end lift correction amount derived from the measured tail-end deficit to the source target within the tail-end terminal window of the flat-top region, where the tail-end deficit is the attenuation amount by which the on-axis energy is lower than the target flat-top intensity within the tail-end terminal window.

[0025] In one possible implementation, the integral conservation scaling factor is the ratio of the on-axis intensity integral of the source target within the simulation window to the on-axis intensity integral of the compensation target within the simulation window before scaling, so that the total on-axis intensity integrals of the compensation target and the source target are equal.

[0026] In one possible implementation, the loss function for hot-start reoptimization is superimposed with a feedback-specific regularization term on the total loss function; the initial B-spline coefficients for reoptimization are the B-spline coefficients corresponding to the pre-optimized phase function during the first feedback, and the optimization result of the previous feedback is used as the initial value for hot-start in subsequent feedbacks.

[0027] In one possible implementation, the quasi-Newton optimizer is the L-BFGS-B algorithm.

[0028] One possible implementation also includes: using a convergence index to evaluate the feedback compensation effect, wherein the convergence index is the ratio of the maximum absolute value of the feedback residual to the maximum absolute value of the fixed flat-top baseline; when the convergence index is higher than a preset tolerance, the next round of feedback compensation and re-optimization continues until the convergence condition is met.

[0029] In one possible implementation, the axial acquisition of the beam is achieved through a 4-f relay imaging system, with the camera acquisition position converted to the object-side equivalent axial coordinates: in, This is the axial magnification. and These are the focal lengths of the two positive lenses in the 4-f system.

[0030] In one possible implementation, the axial uniformity of the Bessel beam... Defined in the flat-top interval superior: in, and These represent the maximum and minimum values ​​of the intensity on the axis within that interval.

[0031] Secondly, the present invention provides a Bessel beam shaping device with adjustable focal depth and uniform on-axis energy, comprising: An initialization module is used to construct a standard conical lens reference phase based on a preset diffraction-free distance, and project the standard conical lens reference phase onto the B-spline function space to obtain the initial values ​​of the B-spline coefficients; The pre-optimization module is used to construct a total loss function based on the initial values ​​of the B-spline coefficients under a differentiable Fresnel diffraction forward model, with the complete axial target intensity distribution including the rising segment, the middle flat-top segment, and the falling segment as the target. The B-spline coefficients are solved by a quasi-Newton optimizer to obtain the pre-optimized phase function. The pre-optimized phase function is used to determine the experimental axial energy response curve. The compensation module is used to determine the feedback residual based on the experimental axial energy response curve and the fixed flat-top baseline, and to apply front-end descent correction and tail-end elevation correction to the source target according to the feedback residual, and obtain the compensation target through integral conservation scaling. The re-optimization module is used to take the B-spline coefficients corresponding to the pre-optimized phase function as the initial value for hot start, replace the original flat-top target with the compensation target, and perform hot start re-optimization under the differentiable Fresnel diffraction forward model to obtain the target phase function. The generation module is used to generate a Bessel beam with adjustable focal depth and uniform on-axis energy based on the target phase function.

[0032] Compared with existing technologies, the present invention has at least the following beneficial effects: The method provided by the present invention initializes the optimization process to a standard conical lens reference phase corresponding to a preset non-diffraction distance, and uses this as a physical anchor point for B-spline parameterized radial phase correction, avoiding the slow convergence and uncontrollable depth of focus problems caused by random or zero initial values ​​in traditional methods, thereby achieving rapid and flexible adjustment of the depth of focus; Through a differentiable Fresnel diffraction forward model and analytical gradient-driven quasi-Newton optimization, the axial flat-top energy distribution can be obtained in the feedback-free pre-optimization stage; Through a feedback-guided target compensation mechanism and hot-start re-optimization, the experimental axial energy response is used as the feedback input, and the phase function is iteratively updated after comparison, correction and compensation, effectively suppressing axial intensity deviation and tail energy attenuation under non-ideal experimental conditions, improving the axial uniformity of the beam and robustness to non-ideal experimental factors, making this method have outstanding practical value in scenarios with extremely high requirements for beam uniformity and depth of focus, such as laser precision processing, long depth of focus microscopic imaging and optical communication. Attached Figure Description

[0033] To more clearly illustrate the technical solutions in the embodiments of this application, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0034] Figure 1 The optical path structure design diagram provided for this invention; Figure 2 A flowchart of the overall design method provided for this invention; Figure 3 The differentiable physics optimization flowchart provided by this invention; Figure 4 The experimental feedback compensation flowchart provided by this invention; Figure 5 The pre-optimized phase and axial intensity distribution diagram provided by this invention, wherein, Figure 5 (a), 5(b), and 5(c) are the radial phase delay curves corresponding to the target's non-diffraction distances of 600 mm, 1000 mm, and 1400 mm, respectively; Figure 5 (d), 5(e), and 5(f) are the axial intensity distribution curves corresponding to the target's non-diffraction distances of 600 mm, 1000 mm, and 1400 mm, respectively; Figure 6 This is a comparison diagram of experimental axial strength before and after feedback compensation provided by the present invention, wherein, Figure 6 (a), 6(b), and 6(c) correspond to the three target non-diffraction distances of 600 mm, 1000 mm, and 1400 mm, respectively; Figure 7 The experimental comparison diagram of the transverse light spot provided by the present invention, wherein, Figure 7 (a), 7(b), and 7(c) correspond to three sets of target non-diffraction distances of 600 mm, 1000 mm, and 1400 mm, respectively. In each figure, the top row (Standard) is the CCD measured transverse spot sequence of the Bessel beam generated by the standard conical lens at different axial positions z, and the bottom row (Optimized) is the CCD measured transverse spot sequence of the Bessel beam at the corresponding axial positions after complete optimization by this method. Figure 8 This is a schematic diagram of an embodiment of the depth-of-focus controllable Gaussian-Bessel beam generation device provided by the present invention. Detailed Implementation

[0035] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0036] like Figure 1 As shown, the optical path system provided in this embodiment of the invention includes: a fiber laser, a polarization beam splitter (PBS), an attenuator, a phase-type spatial light modulator (SLM), a first lens Lens1 (focal length f1), a second lens Lens2 (focal length f2), and a CCD camera. The beam emitted by the fiber laser passes sequentially through the polarization beam splitter and the attenuator before being incident on the spatial light modulator for phase modulation. The phase-modulated beam is then relayed by a 4-f relay imaging system composed of the first and second lenses, and finally received and recorded by the CCD camera for the lateral light intensity distribution. The CCD camera is mounted on an electrically driven stage and can scan and acquire lateral light intensity frames at different propagation distances point by point along the optical axis (z-direction).

[0037] For example, the focal length f1 of the first lens is 200 mm, and the focal length f2 of the second lens is 50 mm. The system's horizontal scaling factor is The axial scaling factor is By using a 4-f relay imaging system, the beam propagation characteristics are maintained while compressing the axial working distance. The camera coordinates at each axial position acquired by the CCD camera... Transform to the equivalent axial coordinates of the object using the axial scaling factor: .

[0038] Based on the above optical path system, this invention provides a Bessel beam shaping method with adjustable focal depth and uniform on-axis energy. For example... Figure 2 As shown, the method includes the following steps: S101: Input design parameters, including the target non-diffraction distance, the target axial light intensity distribution, and constraint conditions.

[0039] S102: Construct a standard conical lens reference phase and perform B-spline projection, initialize the design variables, and obtain the initial values ​​of the B-spline coefficients.

[0040] S103: Execute the differentiable physical optimization model. Under the differentiable Fresnel diffraction forward model, construct the total loss function with the complete axial target intensity distribution including the rising segment, the middle flat-top segment and the falling segment as the target. Use the analytical gradient to solve the B-spline coefficients through a quasi-Newton optimizer to obtain the pre-optimized phase function.

[0041] S104: Generate a phase mask from the pre-optimized phase function and load it onto the spatial light modulator (SLM) to build the experimental optical path system. Scan the phase-modulated beam along the axial direction to acquire transverse light intensity frames, extract the on-axis intensity at each axial position, and obtain the experimental axial energy response curve.

[0042] S105: Execute the experimental feedback compensation model. Compare the experimental axial energy response curve with the fixed flat-top baseline to determine the feedback residual. Based on the feedback residual, apply front-end descent correction and tail-end elevation correction to the source target, and obtain the compensation target through integral conservation scaling. Use the pre-optimized B-spline coefficients as the initial values ​​for hot start, replace the original flat-top target with the compensation target, and perform hot start re-optimization under the differentiable Fresnel diffraction forward model to obtain the target phase function.

[0043] S106: Determine whether the design and experimental indicators meet the preset requirements. If not, return to S103 and use the B-spline coefficients corresponding to the target phase function as the new initial values ​​for hot start for the next iteration; if they meet the requirements, proceed to S107.

[0044] S107: Outputs the final phase mask design results, generating a Bessel beam with adjustable focal depth and uniform on-axis energy based on the target phase function.

[0045] like Figure 3 As shown, the differentiable physics optimization module in S103 further includes the following steps: S201: Input design parameters and current target axial strength. Design parameters include a preset non-diffraction distance. The incident beam waist radius ω and wavelength λ, etc. The target axial intensity is target I. tar (z).

[0046] target The construction method is as follows: Among them, flat-top strength The on-axis intensity integral of a standard conical lens in the flat-top region The upper limit is determined by multiplying the average value within the range by a residual factor less than 1, typically between 0.8 and 0.95. For the front-end window function, take the smaller value between the normalized response of the standard conical lens and 1; This is a smooth cutoff window function applied to the back end of the flat top.

[0047]

[0048] in, It is the original axial light intensity distribution of the standard axis pyramid, which is directly calculated using a differentiable Fresnel diffraction forward model; It is the strength of the flat top; It is the axial propagation coordinate; It is the attenuation cutoff center position, and the specific position is set according to the required tail attenuation length, controlling the axial position of the falling edge; It is the truncation transition width, which controls the length of the interval from 1 to 0 light intensity smoothly. The specific value is adjusted according to the smoothness requirements of the target curve and the optimization of convergence.

[0049] S202: Initialize the phase function. First, based on the preset diffraction-free distance z... nd Constructing a standard conical lens reference phase Where k is the wavenumber, R is the aperture radius, and r is the radial coordinate. Then, the reference phase of the standard conical lens is projected onto a cubic B-spline function space. The initial value c of the B-spline coefficient is obtained by minimizing the cumulative squared deviation between the B-spline represented phase and the reference phase of the standard conical lens over the aperture interval. (0) .

[0050] Initial value of B spline coefficient This is obtained by minimizing the cumulative squared deviation between the B-spline representation of the phase and the reference phase of the standard conical lens over the aperture range:

[0051] Among them, B j (r) is the basis function of the j-th B-spline, c j Here, K represents the corresponding control coefficient, and K represents the number of control points. This is the reference phase for the standard conical lens.

[0052] For example, the number of control points K of the B-spline is based on the approximate Fresnel number. The nodes are adaptively determined, and the B-spline node vectors are non-uniformly distributed, with denser nodes arranged in the radial lower aperture that mainly contributes to the axial section of the flat top.

[0053] S203: Calculate the light field propagation. Based on the axisymmetric Fresnel diffraction integral, the two-dimensional Fresnel diffraction integral is degenerated into a radial one-dimensional integral in the form of a zero-order Hankel transform. The diffraction field E(r1,z) and the on-axis intensity I(z) = |E(0,z)| at each axial position are calculated. 2 .

[0054] Differentiable Fresnel diffraction forward model is based on axisymmetric Fresnel diffraction integral, propagation distance The diffraction field at that point is represented as:

[0055] in, Let be the waist radius of the incident Gaussian beam. It is the radial phase function. It is a zero-order Bessel function. For wavelength, The radial coordinates of the observation surface Aperture radius, It is the imaginary unit.

[0056] S204: Calculate the objective difference function and constraints. Construct a total loss function containing five terms. Among them, L dev The deviation term is used to calculate the difference between the axial strength and the target axial strength; L tv The total variational term to suppress the first-order axial oscillation; L curv The curvature term is used to suppress second-order axial oscillations; L shape The shape term used to align the local axial trend with the target profile; L smooth A coefficient smoothing regularization term is used to suppress excessive oscillations in B-spline coefficients; , , , , These correspond sequentially to the weight coefficients of each loss function. In this embodiment, the weight coefficients are first determined through empirical initial values, combined with experimental calibration iterations: using the bias term weights... Assign initial benchmark values ​​to the core benchmark terms, prioritizing the accuracy of the axial strength fit; the remaining weighting coefficients... , , Initial values ​​are assigned based on their respective adjustment functions and constraint targets. During the actual calibration process, based on the first-order oscillation degree, second-order curvature fluctuation, local contour alignment effect, B-spline coefficient oscillation state, and overall fitting accuracy requirements of the axial curve obtained from the experiment, the weight coefficients are calibrated and iteratively adjusted using the control variable method or orthogonal experimental method. Through multiple sets of comparative experiments, the axial strength fitting error, curve smoothness, and target contour conformity under different weight combinations are verified. Finally, the weight coefficient combination that meets the preset engineering accuracy requirements and achieves the expected curve shape and smoothness is selected.

[0057] S205: Calculate the gradient and update the phase function. Since the forward model of differentiable Fresnel diffraction is differentiable with respect to the B-spline coefficients, the partial derivative of the total loss function with respect to each B-spline coefficient can be analytically obtained using the chain rule, without relying on numerical difference approximation. First, the current B-spline coefficients are substituted into the forward model to obtain the computational intensity at each axial position and its deviation contribution from the target. Then, backpropagation along the computation graph yields the gradient vector of the total loss with respect to each coefficient. Finally, the gradient vector is fed into the L-BFGS-B optimizer, which uses historical gradient information to construct a curvature approximation matrix and updates the B-spline coefficients along the quasi-Newton direction to reduce the total loss. Because analytical gradients are used instead of numerical differences, the entire gradient calculation can be completed in one forward and one backpropagation, ensuring the solution efficiency of the pre-optimization stage.

[0058] S206: Determine if the optimization has converged. Convergence is determined by the decrease in the total loss function being lower than a preset threshold, the gradient norm being lower than a preset threshold, or the number of iterations reaching a preset upper limit. If convergence has not occurred, return to S203 to continue iteration. If convergence has occurred, proceed to S207 and output the pre-optimized phase function.

[0059] S207: Outputs the pre-optimized phase function and passes it to the experimental calibration in S104 and the feedback compensation model in S105.

[0060] like Figure 4 As shown, the experimental feedback compensation model in S105 further includes the following steps: S301: Load optimized phase onto SLM mask. Generate a phase hologram from the phase function obtained in the pre-optimization stage and load it onto the spatial light modulator.

[0061] S302: Acquiring the experimental axial energy response. A CCD camera scans point-by-point along the optical axis, acquiring transverse light intensity frames at each axial position. For each frame, the beam center coordinates are determined using a gray-scale weighted centroid localization method. Then, the average grayscale value within a fixed pixel window (e.g., 3×3 pixels) centered on these coordinates is taken as the on-axis intensity at that axial position, obtaining the experimental axial energy response curve I. exp (z).

[0062] S303: Experimental Axial Energy Response Coordinate Mapping. The axial position coordinates acquired by the CCD are transformed to the object-side equivalent coordinates using the axial scaling factor of the 4-f system, aligning the experimental data with the simulation grid of the differentiable Fresnel diffraction forward model.

[0063] S304: Calculate the experimental feedback residuals and construct a compensation target. Compare the experimental axial energy response curve with the fixed flat-top baseline. Compare and construct feedback residuals The source target is subjected to front-end descent correction and tail-end rise correction based on the feedback residual: within the front terminal window of the flat-top interval, a front-end descent correction amount defined by the residual amplitude is subtracted from the source target; within the tail terminal window of the flat-top interval, a tail-end rise correction amount derived from the measured tail-end deficit is added to the source target. A fixed flat-top baseline is used for the source target during the first feedback, and the compensation target from the previous feedback is used for subsequent feedbacks.

[0064] S305: Stabilize the compensation target. Multiply the compensation target after front-end descent correction and tail-end lift correction by an integral conservation scaling factor. This factor is the ratio of the on-axis intensity integral of the source target within the simulation window to the on-axis intensity integral of the compensation target within the simulation window before scaling, so that the total on-axis intensity integral of the compensation target is equal to that of the source target.

[0065] S306: Use the differentiable physics re-optimization module to perform hot-start re-optimization of the compensation target. The re-optimized loss function is based on the original total loss function with the addition of a feedback-specific regularization term; the initial B-spline coefficients of the re-optimization use the optimization results from the pre-optimization stage during the first feedback, and the optimization results from the previous feedback are used as the initial values ​​for hot start in subsequent feedbacks.

[0066] S307: Determine whether the experimental indicators meet the preset requirements. Use convergence indicators. Evaluate the effectiveness of the feedback and compensation. Among these, For axial sampling index, Represents the first coordinate after coordinate mapping alignment. Each axial sampling position The feedback residual obtained from the above calculation, | indicates taking the maximum value at all axial sampling positions. When the convergence index is higher than the preset tolerance, return to S304 to continue the next iteration; when the convergence index meets the tolerance, proceed to S308.

[0067] S308: Outputs the final phase mask as the target phase function for generating a Bessel beam with adjustable depth of focus and uniform on-axis energy.

[0068] Figure 5 This is a pre-optimized phase and axial intensity distribution diagram provided for an embodiment of the present invention. Figure 5 (a), 5(b), and 5(c) are the radial phase delay curves corresponding to the target non-diffraction distances of 600 mm, 1000 mm, and 1400 mm, respectively. The horizontal axis is the radial coordinate r (in mm), and the vertical axis is the phase delay (in rad). The dashed line represents the linear phase of the standard conical lens, and the solid line represents the phase curve after optimization by B-spline differentiability physics. It can be seen that the optimized phase has been smoothly corrected for each radial position while maintaining the geometric relationship of the reference cone angle. Figure 5(d), 5(e), and 5(f) are the corresponding axial intensity distribution curves. The horizontal axis represents the axial propagation distance z (in mm), and the vertical axis represents the axial intensity (in any unit). The dashed line represents the axial response of the standard conical lens, which has a single-peak distribution. The solid line represents the axial response after B-spline optimization, which forms a flat intensity distribution within the preset flat-top interval.

[0069] For example, for a target distance of 600 mm without diffraction, the axial uniformity of the standard conical lens is only about 17.6%, which is improved to more than 95% after pre-optimization; similar flat-top shaping effects were also achieved for target distances of 1000 mm and 1400 mm, verifying the effectiveness of the differentiable physical optimization model at different focal depths and the ability to adjust the focal depth.

[0070] Figure 6 This is a comparison diagram of experimental axial strength before and after feedback compensation, provided in an embodiment of the present invention. Figure 6 Figures (a), (b), and (c) correspond to the three target non-diffraction distances of 600 mm, 1000 mm, and 1400 mm, respectively. In each figure, the gray dashed line represents the experimental axial intensity curve during the pre-optimization stage, and the blue solid line represents the experimental axial intensity curve after feedback target compensation and hot-start re-optimization. The green filled area represents the intensity increase range after feedback compensation relative to the pre-optimization stage, and the orange filled area represents the intensity suppression range after feedback compensation relative to the pre-optimization stage.

[0071] by Figure 6 Taking the 600 mm case in (a) as an example, the pre-optimized experimental curve showed overshoot above the flat-top baseline at the front end of the flat-top region and attenuation below the flat-top baseline at the rear end. After feedback compensation, the intensity of the overshoot region at the front end was suppressed (orange area), and the intensity of the attenuation region at the rear end was increased (green area), resulting in a flatter axial distribution in the flat-top region. The 1000 mm and 1400 mm cases also showed the compensation trend of front-end suppression and rear-end enhancement, indicating that the feedback-guided target compensation and hot-start re-optimization mechanism can effectively correct the axial intensity deviation caused by non-ideal experimental conditions and improve the axial uniformity of the beam in the actual optical path.

[0072] Figure 7 The experimental comparison diagram of the transverse light spot provided in the embodiment of the present invention. Figure 7Figures (a), 7(b), and 7(c) correspond to three target non-diffraction distances of 600 mm, 1000 mm, and 1400 mm, respectively. In each figure, the top row (Standard) shows the CCD measured lateral spot sequence of the Bessel beam generated by the standard conical lens at different axial positions z, and the bottom row (Optimized) shows the CCD measured lateral spot sequence of the Bessel beam at the corresponding axial positions after complete optimization using this method. The lateral coordinates of each spot are in the x and y directions of the camera plane (unit: μm), and the grayscale represents the normalized light intensity.

[0073] from Figure 7 As can be seen, the optimized beam maintains the Bessel transverse structural features of the central bright spot and concentric side rings at multiple axial positions. Compared with the standard conical lens, the optimized spot sequence maintains higher central brightness over a longer axial range, and the transverse Bessel feature profile is not destroyed by axial energy redistribution. This indicates that the proposed method effectively preserves the transverse non-diffraction characteristics of the Bessel beam while achieving axial flat-top shaping and adjustable depth of focus.

[0074] like Figure 8 As shown, this embodiment of the invention also provides a Bessel beam shaping device 800 with adjustable focal depth and uniform on-axis energy, comprising an initialization module 801, a pre-optimization module 802, a compensation module 803, a re-optimization module 804, and a generation module 805 connected sequentially along the data flow direction. Wherein: The initialization module 801 is used to construct a standard conical lens reference phase based on a preset non-diffraction distance, and project the standard conical lens reference phase onto the B-spline function space to obtain the initial values ​​of the B-spline coefficients; The pre-optimization module 802 is used to construct the total loss function based on the initial values ​​of the B-spline coefficients under the differentiable Fresnel diffraction forward model, with the complete axial target intensity distribution including the rising segment, the middle flat-top segment and the falling segment as the target. The B-spline coefficients are solved by a quasi-Newton optimizer to obtain the pre-optimized phase function. The pre-optimized phase function is used to determine the experimental axial energy response curve. The compensation module 803 is used to determine the feedback residual based on the experimental axial energy response curve and the fixed flat-top baseline, and to apply front-end descent correction and tail-end lift correction to the source target according to the feedback residual, and obtain the compensation target through integral conservation scaling. The re-optimization module 804 is used to take the B-spline coefficients corresponding to the pre-optimized phase function as the initial value for hot start, replace the original flat-top target with the compensation target, and perform hot start re-optimization under the differentiable Fresnel diffraction forward model to obtain the target phase function. The generation module 805 is used to generate a Bessel beam with adjustable depth of focus and uniform on-axis energy based on the target phase function. The arrows between the modules indicate the data transmission direction, and the whole structure forms a pipeline processing structure from initial phase design to final beam generation.

[0075] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for shaping a Bessel beam with adjustable depth of focus and uniform on-axis energy, characterized in that, include: Based on a preset non-diffraction distance, a standard conical lens reference phase is constructed, and the standard conical lens reference phase is projected onto the B-spline function space to obtain the initial values ​​of the B-spline coefficients. Based on the initial values ​​of the B-spline coefficients, under the differentiable Fresnel diffraction forward model, a total loss function is constructed with the complete axial target intensity distribution including the rising segment, the middle flat-top segment, and the falling segment as the target. The B-spline coefficients are solved by a quasi-Newton optimizer to obtain the pre-optimized phase function. The pre-optimized phase function is used to determine the experimental axial energy response curve. Based on the experimental axial energy response curve and the fixed flat-top baseline, the feedback residual is determined, and the front-end descent correction and the tail-end lift correction are applied to the source target according to the feedback residual. The compensated target is obtained by integral conservation scaling. Using the B-spline coefficients corresponding to the pre-optimized phase function as the initial values ​​for hot start, and replacing the original flat-top target with the compensation target, hot start re-optimization is performed under the differentiable Fresnel diffraction forward model to obtain the target phase function; Based on the target phase function, a Bessel beam with adjustable focal depth and uniform on-axis energy is generated.

2. The method according to claim 1, characterized in that, The standard conical lens reference phase The expression is: ， Where k is the wave number, R is the aperture radius, and z nd The preset non-diffraction distance is r, where r is the radial coordinate. The B-spline is a cubic B-spline; the initial value of the B-spline coefficients. This is obtained by minimizing the cumulative squared deviation between the B-spline representation of the phase and the reference phase of the standard conical lens over the aperture range: ， Among them, B j (r) is the basis function of the j-th B-spline, c j Here, K represents the corresponding control coefficient, and K represents the number of control points. This is the reference phase for the standard conical lens.

3. The method according to claim 1, characterized in that, The differentiable Fresnel diffraction forward model is based on axisymmetric Fresnel diffraction integral, propagation distance The diffraction field at that point is represented as: ， in, Let be the waist radius of the incident Gaussian beam. It is the radial phase function. It is a zero-order Bessel function. For wavelength, The radial coordinates of the observation surface Where is the aperture radius; The analytical gradient of the total loss function is calculated using the chain rule, and its gradient with respect to the first... The gradient of each B-spline coefficient includes the contribution of the partial derivative of the deviation between the calculated intensity and the target intensity at each discrete sampling point within the axial flat-top interval to the complex field. The target The construction method is as follows: Among them, flat-top strength The on-axis intensity integral of a standard conical lens in the flat-top region The upper limit is determined by multiplying the average value within the range by a residual factor less than 1; For the front-end window function, take the smaller value between the normalized response of the standard conical lens and 1; This is a smooth cutoff window function applied to the back end of the flat top.

4. The method according to claim 1, characterized in that, The number of control points of the B spline Based on the approximate Fresnel number The nodes are adaptively determined, and the B-spline node vectors are non-uniformly distributed, with denser nodes arranged in the radial lower aperture that mainly contributes to the axial section of the flat top.

5. The method according to claim 1, characterized in that, The total loss function is expressed as: Among them, L dev The deviation term is used to calculate the difference between the axial strength and the target axial strength; L tv The total variational term to suppress the first-order axial oscillation; L curv To suppress the curvature term of the second-order axial oscillation; L shape The shape term used to align the local axial trend with the target profile; L smooth A coefficient smoothing regularization term is used to suppress excessive oscillations in B-spline coefficients; The experimental axial energy response curve is determined by generating a phase hologram from the pre-optimized phase function and loading it into a spatial light modulator. Lateral light intensity frames are collected along the axial direction of the phase-modulated beam. The beam center of each frame is determined by the gray-scale weighted centroid positioning method. The average gray value within a fixed pixel window at this center is taken as the on-axis intensity at each axial position to obtain the experimental axial energy response curve.

6. The method according to claim 1, characterized in that, The feedback residual is expressed as: in, The fixed flat-top baseline is the result of normalization and smoothing. The normalized and smoothed experimental axial response; The source target uses the fixed flat-top baseline during the first feedback, and the compensation target of the previous feedback is used during subsequent feedbacks; The front-end drop correction is to subtract a front-end drop correction amount defined by the residual amplitude from the source target within the front terminal window of the flat-top interval; the tail-end rise correction is to add a tail-end rise correction amount derived from the measured tail-end loss to the source target within the tail terminal window of the flat-top interval. The integral conservation scaling factor is the ratio of the on-axis intensity integral of the source target within the simulation window to the on-axis intensity integral of the compensation target within the simulation window before scaling, so that the total on-axis intensity integrals of the compensation target and the source target are equal.

7. The method according to claim 1, characterized in that, The loss function for hot-start reoptimization is based on the total loss function with a feedback-specific regularization term superimposed; the initial B-spline coefficients for reoptimization are the B-spline coefficients corresponding to the pre-optimized phase function during the first feedback, and the optimization result of the previous feedback is used as the initial value for hot-start in subsequent feedbacks; the quasi-Newton optimizer is the L-BFGS-B algorithm.

8. The method according to claim 1, characterized in that, Also includes: The feedback compensation effect is evaluated using a convergence index, which is the ratio of the maximum absolute value of the feedback residual to the maximum absolute value of the fixed flat-top baseline. When the convergence index is higher than the preset tolerance, the next round of feedback compensation and re-optimization continues until the convergence condition is met.

9. The method according to claim 1, characterized in that, The axial acquisition of the beam is achieved through a 4-f relay imaging system, and the camera acquisition position is converted to the object-side equivalent axial coordinates: in, This is the axial magnification. and These are the focal lengths of the two positive lenses in the 4-f system; Axial uniformity of the Bessel beam Defined in the flat-top interval superior: in, and These represent the maximum and minimum values ​​of the intensity on the axis within that interval.

10. A Bessel beam shaping device with adjustable focal depth and uniform on-axis energy used in the method according to any one of claims 1-9, characterized in that, include: An initialization module is used to construct a standard conical lens reference phase based on a preset diffraction-free distance, and project the standard conical lens reference phase onto the B-spline function space to obtain the initial values ​​of the B-spline coefficients; The pre-optimization module is used to construct a total loss function based on the initial values ​​of the B-spline coefficients under a differentiable Fresnel diffraction forward model, with the complete axial target intensity distribution including the rising segment, the middle flat-top segment, and the falling segment as the target. The B-spline coefficients are solved by a quasi-Newton optimizer to obtain the pre-optimized phase function. The pre-optimized phase function is used to determine the experimental axial energy response curve. The compensation module is used to determine the feedback residual based on the experimental axial energy response curve and the fixed flat-top baseline, and to apply front-end descent correction and tail-end elevation correction to the source target according to the feedback residual, and obtain the compensation target through integral conservation scaling. The re-optimization module is used to take the B-spline coefficients corresponding to the pre-optimized phase function as the initial value for hot start, replace the original flat-top target with the compensation target, and perform hot start re-optimization under the differentiable Fresnel diffraction forward model to obtain the target phase function. The generation module is used to generate a Bessel beam with adjustable focal depth and uniform on-axis energy based on the target phase function.

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