Method for optimizing a bifocal optical system based on a fundamental gaussian beam

Abstract

The application relates to the technical field of optical system design and optimization, and discloses a double-focus optical system optimization method based on a fundamental mode Gaussian light beam. The method comprises the following steps: constructing an optical field transmission model based on a generalized Huygens-Fresnel diffraction principle, and calculating an exit field and an on-axis light intensity distribution after passing through a double-focus system. The occurrence criterion of a quantitative focal switch phenomenon is formed by identifying a light intensity main maximum point and calculating the offset amount of the main maximum point and a geometric focal point, and then the focal point stability is analyzed. On this basis, the light beam truncation parameter and the Fresnel number are jointly iteratively optimized, the system parameters are adjusted, until the stability meets a preset threshold, and finally an optimized system configuration scheme is output. The method can effectively predict and inhibit the focal point position jump caused by physical optical effects in the design stage, and improves the reliability and stability of the double-focus system in actual work.

Description

Technical Field

[0001] This invention relates to the field of optical system design and optimization technology, specifically to a method for optimizing a bifocal optical system based on a fundamental Gaussian beam. Background Technology

[0002] In fields such as laser applications, optical communications, and precision measurement, optical systems are required to generate two or more separate and stable focal points. Existing technologies typically design bifocal optical systems based on geometrical optics theory, achieving bifocal points at predetermined locations by combining optical elements. This method is conceptually straightforward and computationally relatively simple, but it does not fully consider the physical and optical transmission characteristics of actual laser beams, especially fundamental Gaussian beams.

[0003] The design method based on geometric optics has a drawback: it assumes a fixed focal position uniquely determined by the geometric optical path, neglecting the wave nature of the beam, particularly the combined effects of diffraction and the system aperture. In actual physical optical transmission, when a Gaussian beam passes through a confined optical system, its on-axis intensity distribution undergoes complex changes, potentially exhibiting multiple maxima of similar intensity. Furthermore, the location of the principal maximum can change discontinuously and abruptly with minute variations in beam parameters, a phenomenon known as "focal switching." This causes the actual focal position to deviate from the design value, and even unpredictably switch between two predetermined positions under different conditions, severely compromising the system's output stability and reliability—a fundamental problem that conventional design methods fail to address.

[0004] The core challenge in the design of current bifocal optical systems lies in how to effectively predict and suppress focal position instability caused by physical optical effects. This requires design methods that go beyond the framework of geometric optics, enabling quantitative assessment of focal jump risks and targeted optimization at the system parameter level to ensure that the bifocal positions remain stable as designed under actual physical beam transmission conditions. Summary of the Invention

[0005] The purpose of this invention is to provide an optimization method for bifocal optical systems based on a fundamental Gaussian beam, in order to solve the problems mentioned in the background art.

[0006] To achieve the above objectives, the present invention provides an optimization method for a bifocal optical system based on a fundamental Gaussian beam, the method comprising:

[0007] Obtain the system parameter set of the dual-focal optical system, and construct an initial optical field model to describe the transmission of the fundamental mode Gaussian beam through the dual-focal optical system based on the generalized Huygens-Fresnel diffraction integral formula;

[0008] The initial light field model, the system parameter set, and the beam parameters of the fundamental Gaussian beam are input into the initial light field model for transmission calculation to obtain the output field distribution function of the beam after passing through the bifocal optical system.

[0009] Based on the emission field distribution function, the light intensity distribution of the fundamental mode Gaussian beam on the transmission axis is calculated, and on-axis light intensity distribution data is generated.

[0010] The light intensity distribution data on the axis is processed to detect the location of the principal maximum light intensity, and at least two principal maximum light intensity points and their corresponding focal positions are identified.

[0011] The occurrence criterion of focal switching phenomenon is calculated based on the offset between the focal position and the geometric focal position, and the focal stability analysis results are generated.

[0012] Based on the focal stability analysis results, the beam truncation parameter and beam Fresnel number in the system parameter set are jointly optimized and adjusted to generate an optimized system parameter set.

[0013] The optimized system parameter set is input into the generalized Huygens-Fresnel diffraction integral formula for iterative calculation until the focus stability analysis result meets the preset focus stability threshold condition, thereby generating the final optimized bifocal optical system configuration scheme.

[0014] Preferably, the acquisition of the system parameter set of the bifocal optical system, based on the generalized Huygens-Fresnel diffraction integral formula, constructs an initial light field model to describe the transmission of the fundamental mode Gaussian beam through the bifocal optical system, including:

[0015] The system parameter set includes the Fresnel half-wave zone number of the Fresnel zone plate, the focal length of the thin lens, and the beam cutoff parameters of the optical system.

[0016] Obtain the beam waist radius and beam wavelength of the fundamental Gaussian beam, and construct the initial optical field expression of the fundamental Gaussian beam;

[0017] The complex amplitude transmittance function of the Fresnel zone plate is multiplied with the phase transformation function of the thin lens to generate the composite transmittance function of the bifocal optical system.

[0018] Multiplying the initial optical field expression of the fundamental Gaussian beam with the composite transmittance function yields the modulation optical field expression at the incident plane of the bifocal optical system.

[0019] Substituting the modulation light field expression into the generalized Huygens-Fresnel diffraction integral formula, we obtain the output field integral expression of the beam at any observation plane behind the bifocal optical system, and use the output field integral expression as the initial light field model.

[0020] Preferably, the step of calculating the intensity distribution of the fundamental mode Gaussian beam along the transmission axis based on the emission field distribution function, and generating on-axis intensity distribution data, includes:

[0021] By limiting the observation plane position coordinates of the emission field distribution function to the transmission axis, a simplified emission field function at the observation point on the axis is obtained;

[0022] Calculate the product of the simplified outgoing field function and its complex conjugate function to obtain the light intensity value at each point on the axis, and generate an initial on-axis light intensity distribution sequence;

[0023] The initial on-axis light intensity distribution sequence is normalized so that the maximum light intensity value is the reference value, thereby generating normalized on-axis light intensity distribution data.

[0024] The normalized on-axis light intensity distribution data is sampled at high density along the transmission axis to generate continuous on-axis light intensity distribution data.

[0025] Preferably, the step of performing principal intensity maxima location detection processing on the on-axis light intensity distribution data to identify at least two principal intensity maxima points and their corresponding focal positions includes:

[0026] The first derivative of the light intensity distribution data on the axis is used to obtain the light intensity change rate sequence.

[0027] Identify the zero-crossing points in the light intensity change rate sequence that change from positive to negative to obtain the set of potential light intensity extreme point locations;

[0028] Calculate the light intensity value corresponding to each position in the set of potential light intensity extreme points, and filter out the light intensity extreme points whose light intensity values ​​exceed the preset principal maximum threshold as candidate principal maxima.

[0029] Local peak verification processing is performed on the candidate principal maxima to filter out the points with the highest light intensity values ​​within a preset neighborhood range and determine them as the principal maxima of light intensity.

[0030] The coordinates of the principal maximum point of light intensity on the transmission axis are read and used as the focal position.

[0031] Preferably, the step of calculating the criterion for the occurrence of focus switching phenomenon based on the offset between the focal position and the geometric focal position, and generating the focal stability analysis result, includes:

[0032] Obtain the geometric focus position coordinates from the system parameter set;

[0033] Calculate the absolute distance between each of the focal positions and the coordinates of the geometric focal position, and use it as the focal offset;

[0034] Determine whether any of the focal offsets exceeds a preset focal switch threshold distance;

[0035] If it exists, it is determined that a focus switching phenomenon has occurred in the bifocal optical system, and the focus instability state and the focus position where the focus switching occurred are marked in the focus stability analysis results;

[0036] If it does not exist, mark the focus as stable in the focus stability analysis results.

[0037] Preferably, the method further includes a correction process based on the Fresnel half-wave band number focusing switch phenomenon occurrence criterion, including:

[0038] Determine the parity of the Fresnel half-wave zone number in the system parameter set;

[0039] If the Fresnel half-wave zone number is even, then the focal switch threshold distance is adjusted to a preset even-number half-wave zone threshold distance;

[0040] If the Fresnel half-wave zone number is odd, the focal switch threshold distance is adjusted to the preset odd half-wave zone threshold distance, and the light intensity value weighting factor at the geometric focal point is set.

[0041] Based on the adjusted focus switch threshold distance and the light intensity value weighting factor, the focus stability analysis results are recalculated.

[0042] Preferably, the joint optimization and adjustment of the beam truncation parameter and beam Fresnel number in the system parameter set based on the focus stability analysis results, to generate an optimized system parameter set, includes:

[0043] When the focus stability analysis result is a focus unstable state, read the critical values ​​of the beam truncation parameter and the critical value of the beam Fresnel number that cause the focus switching phenomenon.

[0044] Based on the critical values ​​of the beam truncation parameter and the critical value of the beam Fresnel number, the parameters are adjusted stepwise in the increasing and decreasing directions in the parameter space to generate multiple sets of candidate values ​​for the beam truncation parameter and the beam Fresnel number.

[0045] Each set of candidate beam truncation parameters and candidate beam Fresnel numbers is combined to replace the corresponding parameters in the original system parameter set, forming multiple candidate system parameter sets.

[0046] Substitute each set of candidate system parameters into the generalized Huygens-Fresnel diffraction integral formula to perform focus stability verification calculations.

[0047] The set of candidate system parameters that causes the focus stability analysis results to transform into a stable state is selected as the set of optimized system parameters.

[0048] Preferably, the step of substituting each of the candidate system parameter sets into the generalized Huygens-Fresnel diffraction integral formula for focal stability verification calculation includes:

[0049] Update the system parameters in the generalized Huygens-Fresnel diffraction integral formula using the candidate system parameter set;

[0050] Based on the updated generalized Huygens-Fresnel diffraction integral formula, the output field distribution function of the fundamental mode Gaussian beam after passing through the bifocal optical system is recalculated.

[0051] New on-axis light intensity distribution data are generated based on the recalculated outgoing field distribution function;

[0052] The new on-axis light intensity distribution data is processed to detect the location of the principal maximum of light intensity, and the new focal position is obtained.

[0053] Determine whether the new focal position meets the focal stability condition, which is that the offset of all focal positions from the geometric focal position is less than a preset focal stability tolerance threshold.

[0054] Preferably, the step of inputting the optimized system parameter set into the generalized Huygens-Fresnel diffraction integral formula for iterative calculation until the focus stability analysis result meets the preset focus stability threshold condition, generating the final optimized bifocal optical system configuration scheme, includes:

[0055] The optimized system parameter set is used as the system parameter input for the current iteration to calculate the focus stability analysis result for the current iteration;

[0056] Determine whether the maximum focus offset in the focus stability analysis result of the current iteration is less than the focus stability threshold condition;

[0057] If it is less than, then the optimized system parameter set of the current iteration is determined as the final optimized bifocal optical system configuration scheme;

[0058] If it is not less than, then the beam truncation parameter and beam Fresnel number in the optimized system parameter set are fine-tuned according to the focus stability analysis results of the current iteration, and a new set of optimized system parameters for the next iteration is generated.

[0059] Repeat the iterative optimization steps until the focus stability threshold condition is met or the preset maximum number of iterations is reached.

[0060] Preferably, the method includes:

[0061] Obtain the system parameter set of the bifocal optical system and the beam parameters of the fundamental Gaussian beam;

[0062] An initial light field model is constructed based on the generalized Huygens-Fresnel diffraction integral formula, and the output field distribution function of the beam after passing through the system is calculated.

[0063] Calculate the light intensity distribution on the transmission axis based on the emission field distribution function, and identify the focal position corresponding to the main maximum point of light intensity.

[0064] Analyze the offset between the focal position and the geometric focal position, determine the focal switching phenomenon, and generate focal stability analysis results;

[0065] Based on the focal stability analysis results, the beam cutoff parameter and beam Fresnel number in the system parameter set are jointly optimized and adjusted, and iterative calculations are performed using the generalized Huygens-Fresnel diffraction integral formula until the focal stability meets the preset threshold, generating the final optimized bifocal optical system configuration scheme.

[0066] Compared with the prior art, the beneficial effects of the present invention are:

[0067] By establishing an optical field transmission model based on generalized Huygens-Fresnel diffraction, the principal maxima of on-axis light intensity are calculated and identified, and their offset from the geometric focal position is calculated to construct a specific quantitative criterion for focal switching phenomena. This transforms the risk of focal position jumps from an unpredictable phenomenon into a physical quantity that can be accurately assessed during the design phase. The design process can proactively identify parameter regions that may cause focal instability, thereby expanding the optimization objective from a single geometric position matching to a comprehensive index that includes physical optical stability.

[0068] Based on the aforementioned stability criteria, a joint iterative optimization of the beam cutoff parameter and the beam Fresnel number is performed. These two parameters control the degree of aperture confinement on the beam and the beam diffraction characteristics, respectively, and their coupling effect directly affects the formation and stability of the focus. This joint optimization technique systematically coordinates these two key physical quantities to find the parameter combination that stabilizes the actual focus position and minimizes its shift. This scheme directly intervenes in the physical mechanism that causes focus jumps. Through parameter matching design, it suppresses focus instability caused by the combined effects of diffraction and aperture confinement, thereby ensuring that the position of the dual focus under actual working conditions conforms to the design expectations and remains robust. Attached Figure Description

[0069] Figure 1 This is a schematic diagram illustrating the working principle of the bifocal optical system optimization method based on a fundamental Gaussian beam as described in this invention.

[0070] Figure 2 A flowchart for constructing the initial light field model;

[0071] Figure 3A flowchart for generating on-axis light intensity distribution data;

[0072] Figure 4 A diagram illustrating the influence of Fresnel half-wave zone number on focal point offset in a bifocal optical system;

[0073] Figure 5 Scatter heatmap for parameter optimization of bifocal optical system. Detailed Implementation

[0074] 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 embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0075] Please see Figure 1 This invention provides an optimization method for a bifocal optical system based on a fundamental mode Gaussian beam. The method includes: acquiring a set of system parameters for the bifocal optical system, including the half-band number of the Fresnel zone plate, the focal length of the thin lens, and beam truncation parameters of the optical system; constructing an initial optical field model to describe the transmission of the fundamental mode Gaussian beam through the bifocal optical system based on the generalized Huygens-Fresnel diffraction integral formula; inputting the initial optical field model, the set of system parameters, and beam parameters such as the beam waist radius and wavelength of the fundamental mode Gaussian beam into the model for transmission calculation to obtain the output field distribution function after the beam passes through the system; calculating the intensity distribution of the fundamental mode Gaussian beam on the transmission axis based on the output field distribution function to generate on-axis intensity distribution data; performing intensity principal maxima detection processing on the on-axis intensity distribution data to identify at least two principal intensity maxima points and their corresponding focal positions on the transmission axis; and calculating the focal switching phenomenon criterion based on the offset between the identified focal positions and the theoretical geometric focal positions to generate focal stability analysis results. Based on the focal stability analysis results, the beam cutoff parameter and beam Fresnel number in the system parameter set are jointly optimized and adjusted to generate an optimized system parameter set. The optimized system parameter set is then input into the generalized Huygens-Fresnel diffraction integral formula for iterative calculation until the output focal stability analysis results meet the preset focal stability threshold condition, ultimately generating an optimized bifocal optical system configuration scheme.

[0076] Example 1: See Figure 2The system parameters of the bifocal optical system are obtained, including the Fresnel half-band number of the Fresnel zone plate, the focal length of the thin lens, and the beam truncation parameters of the optical system. The beam waist radius and wavelength of the fundamental Gaussian beam are obtained to construct the initial optical field expression of the fundamental Gaussian beam before the incident plane. The complex amplitude transmittance function of the Fresnel zone plate is multiplied by the phase transformation function of the thin lens to generate the composite transmittance function of the bifocal optical system. The initial optical field expression of the fundamental Gaussian beam is multiplied by this composite transmittance function to obtain the modulated optical field expression of the beam at the incident plane of the bifocal optical system. Substituting this modulated optical field expression into the generalized Huygens-Fresnel diffraction integral formula, the integral expression of the exit field of the beam at any observation plane behind the bifocal optical system is obtained. This integral expression is the constructed initial optical field model.

[0077] In practical implementation, a set of system parameters for a bifocal optical system is obtained. This set of parameters explicitly includes the Fresnel half-wave zone number of the Fresnel zone plate, the focal length of the thin lens, and the beam cutoff parameters of the optical system. In a bifocal scanning system for laser processing, the Fresnel half-wave zone number can be set to 100, the focal length of the thin lens can be set to 200 mm, and the beam cutoff parameter can be set to 0.8. The beam waist radius and beam wavelength of the fundamental mode Gaussian beam are obtained. The beam waist radius can be set to 0.5 mm, and the beam wavelength can be set to 1064 nm. Based on these parameters, an initial optical field expression for the fundamental mode Gaussian beam before its incident face in the optical system is constructed. This initial optical field expression describes the transverse distribution and phase profile of the beam in free space.

[0078] In some embodiments, the complex amplitude transmittance function of the Fresnel zone plate is multiplied by the phase transformation function of the thin lens. The complex amplitude transmittance function of the Fresnel zone plate is determined by its annular structure and the designed wavelength, while the phase transformation function of the thin lens is determined by its focal length. The product of the two generates the composite transmittance function of the bifocal optical system, which spatially integrates the diffraction modulation effect of the zone plate and the converging phase effect of the lens. The initial optical field expression of the fundamental Gaussian beam is multiplied by the composite transmittance function of the bifocal optical system. The multiplication operation is performed point-by-point in the spatial domain to obtain the optical field expression of the fundamental Gaussian beam modulated at the incident plane of the bifocal optical system. The modulated optical field expression simultaneously contains the amplitude information of the original Gaussian beam and the phase modulation information introduced by the Fresnel zone plate and the lens. The modulated light field expression is substituted into the generalized Huygens-Fresnel diffraction integral formula for integration. This formula describes the physical process of light propagating from the incident plane to the observation plane. The integration yields the integral expression of the exit field at any observation plane behind the bifocal optical system. This exit field integral expression is used as the initial light field model describing the light field transmission behavior of the entire system. It can be understood that constructing the initial light field model strictly depends on the precise input of the system parameter set and beam parameters. For example, when the beam truncation parameter is adjusted from 0.8 to 1.2, the modulated light field distribution at the incident plane will change significantly, directly altering the input conditions for subsequent integration operations and the final form of the initial light field model.

[0079] Example 2: See Figure 3 The observation plane position coordinates in the outgoing field distribution function are constrained to the transmission axis, and the lateral coordinate is set to zero, resulting in a simplified outgoing field function at the observation point on the axis. The product of this simplified outgoing field function and its complex conjugate function is calculated to obtain the light intensity value at each point on the transmission axis, generating an initial on-axis light intensity distribution sequence. This initial on-axis light intensity distribution sequence is normalized so that the maximum light intensity value is the baseline value, generating normalized on-axis light intensity distribution data. High-density sampling is performed on this normalized on-axis light intensity distribution data along the transmission axis to generate continuous on-axis light intensity distribution data. The first derivative of the on-axis light intensity distribution data is calculated to obtain a sequence of light intensity change rates along the transmission axis. Zero-crossing points in this light intensity change rate sequence where values ​​change from positive to negative are identified, resulting in a set of potential light intensity extrema locations. The light intensity value corresponding to each point in this location set is calculated, and light intensity extrema points whose light intensity values ​​exceed a preset principal maximum threshold are selected as candidate principal maxima. For each candidate principal maximum, local peak verification is performed to select the point with the highest light intensity value within a preset neighborhood, which is then determined as the principal maximum point. The coordinates of these principal maximum points on the transmission axis are read as the actual focal positions of the system.

[0080] In practical implementation, the observation plane position coordinates in the emission field distribution function are limited to the transmission axis. Specifically, the lateral coordinate value is set to zero, resulting in the simplified emission field function at the observation point on the axis. The simplified emission field function is a complex function that only relates to the axial coordinate variables. The product of the simplified emission field function and its own complex conjugate function is calculated using the following formula:

[0081]

[0082] in: Indicates the on-axis light intensity. This represents the simplified emission field function. Indicates axial coordinates, superscript This represents taking the complex conjugate, through which the light intensity values ​​at each discrete point on the transmission axis are obtained, thereby generating the initial axis light intensity distribution sequence.

[0083] In some embodiments, the initial on-axis light intensity distribution sequence is normalized. This involves finding the maximum light intensity value in the sequence and dividing all light intensity values ​​by this maximum value, making the normalized maximum light intensity a reference unit of 1, thus generating normalized on-axis light intensity distribution data. High-density sampling is performed on the normalized on-axis light intensity distribution data along the transmission axis, with sampling intervals much smaller than the system's characteristic diffraction length, thereby generating a continuous on-axis light intensity distribution data curve for subsequent precise peak analysis. The first derivative of the on-axis light intensity distribution data is calculated using a numerical difference method, yielding a sequence of light intensity change rates along the transmission axis. This sequence characterizes the trend of light intensity increasing or decreasing with increasing position. Zero-crossing points in the light intensity change rate sequence where values ​​change from positive to negative are identified. This process is achieved by detecting changes in the sign of adjacent data points, thereby obtaining a set of potential light intensity extreme point locations.

[0084] Optionally, the normalized light intensity value corresponding to each position in the potential set of light intensity extrema is calculated. A master maxima threshold is preset, for example, set to 0.3. All light intensity extrema points with light intensity values ​​exceeding the master maxima threshold of 0.3 are selected as candidate master maxima. Local peak verification processing is performed on each candidate master maxima. This local peak verification process compares the light intensity values ​​of all points within a preset neighborhood centered on that point, for example, within a range of 5 sampling points to the left and right. The point with the highest light intensity value within the preset neighborhood is selected, and the point that meets the condition is finally determined as the light intensity master maxima. This can be understood as reading the axial coordinates corresponding to all light intensity master maxima points on the transmission axis. These coordinate values ​​are used as the actual focal positions formed by the bifocal optical system. For example, in a single calculation, two master maxima points with axial coordinates of 150 mm and 210 mm may be identified. It is understandable that the setting of the principal maximum threshold and the neighborhood range will affect the recognition result of the focal position. For example, when the beam truncation parameter is 0.5, setting the principal maximum threshold to 0.3 can effectively filter out noise peaks, while when the beam truncation parameter becomes 1.5, it may be necessary to adjust the principal maximum threshold to 0.25 to accurately capture a slightly weaker focal signal.

[0085] Example 3: Obtain the theoretically calculated geometric focal point coordinates from the system parameter set. Calculate the absolute distance between each identified focal point position and its coordinates, using this distance as the focal offset for each focal point. Determine if any focal offset exceeds a preset focal switching threshold distance. If such a focal offset exists, a focal switching phenomenon is determined to have occurred in the bifocal optical system, and it is marked as an unstable state in the focal stability analysis results, with the focal point position where the focal switching occurred recorded. If none of the focal offsets exceed the focal switching threshold distance, it is marked as a stable state in the focal stability analysis results. Further, the criterion for the occurrence of the focal switching phenomenon is modified based on the parity of the Fresnel half-wave zone number, determining the parity of the Fresnel half-wave zone number in the system parameter set. If the Fresnel half-wave zone number is even, the focal switching threshold distance is adjusted to a preset even-numbered half-wave zone threshold distance. If the Fresnel half-wave zone number is odd, the focal switching threshold distance is adjusted to a preset odd-numbered half-wave zone threshold distance, and a weighting factor is set for the light intensity value at the geometric focal point. Based on the adjusted focus switching threshold distance and light intensity value weighting factor, the focus stability analysis results are recalculated and generated.

[0086] In practical implementation, the theoretically calculated geometric focal point coordinates are obtained from the system parameter set of the bifocal optical system. For example, for a system consisting of a Fresnel zone plate and a thin lens with a focal length of 200 mm, its geometric focal point coordinates might be calculated to be 205.0 mm from the design parameters. The absolute distance between each identified focal point and its geometric focal point coordinates is calculated using the following formula:

[0087]

[0088] in: This represents the focus offset of the nth focus. This represents the coordinates of the nth focal point identified through light intensity analysis. This represents the coordinates of the theoretical geometric focal point, and the focal offset of each focal point is calculated.

[0089] In some embodiments, it is determined whether any focal offset exceeds a preset focal switching threshold distance, for example, the preset focal switching threshold distance is 2.0 mm. If a focal offset exceeds the focal switching threshold distance of 2.0 mm, it is determined that a focal switching phenomenon has occurred in the bifocal optical system. The generated focal stability analysis result explicitly marks the system as being in a focal unstable state, and specifically records the focal position where the focal switching occurred. For example, the offset of the focal position at 203.5 mm is recorded as 1.5 mm, which does not exceed the threshold, and the offset of the focal position at 206.8 mm is recorded as 1.8 mm, which also does not exceed the threshold. If another focal position at 208.5 mm has an offset of 3.5 mm, then it exceeds the threshold. If all calculated focal offsets do not exceed the focal switching threshold distance of 2.0 mm, then the focal stability analysis result marks the system as being in a focal stable state.

[0090] Optionally, the criterion for the occurrence of focus switching phenomena is modified based on the parity of the Fresnel half-wave zone number. The parity of the Fresnel half-wave zone number in the system parameter set is determined; for example, a Fresnel half-wave zone number of 100 is even. If the Fresnel half-wave zone number is even, the focus switching threshold distance is adjusted to a preset even-numbered half-wave zone threshold distance, for example, adjusting the focus switching threshold distance from the common 2.0 mm to 2.2 mm designed for even-numbered rings. If the Fresnel half-wave zone number is odd, for example, 99, the focus switching threshold distance is adjusted to a preset odd-numbered half-wave zone threshold distance, for example, 1.8 mm, and a weighting factor is set for the light intensity value at the geometric focus, for example, setting the weight of the light intensity at the geometric focus in the judgment to 1.1 times. It can be understood that, based on the adjusted focus switching threshold distance and the light intensity value weighting factor, the modified criterion is used to recalculate and generate the focus stability analysis results. It is understandable that when the Fresnel half-wave zone number changes from 100 to 99, not only does the focal switching threshold distance change from 2.2 mm to 1.8 mm, but the introduction of the geometric focal intensity weight will also change the final judgment of focal stability. This reflects the dependence of the criterion correction process on the details of system parameters.

[0091] See Figure 4This is a graph analyzing the impact of the Fresnel half-wave zone number on focus shift in a bifocal optical system. As the Fresnel half-wave zone number increases, the maximum focus shift generally decreases, reaching its minimum value when the half-wave zone number approaches 110. When the half-wave zone number is small (80-95), the shift at some data points exceeds the general threshold (2.0mm), posing a risk of focus switching. When the half-wave zone number is large (above 100), the shift stabilizes below the threshold. There is no significant difference in shift between even and odd half-wave zone numbers; both tend to stabilize as the half-wave zone number increases. This type of graph is used in the focus switching determination stage of bifocal optical system optimization. Analyzing the relationship between the half-wave zone number and focus shift clarifies this. Choosing a larger Fresnel half-wave zone number (e.g., above 100) can effectively reduce focus shift and avoid focus switching. Adjusting the focus switching threshold based on the parity of the half-wave zone number (e.g., 2.2mm for even numbers and 1.8mm for odd numbers) allows for more accurate determination of system stability.

[0092] Example 4: When the focus stability analysis result indicates a focus instability state, the critical values ​​of the beam truncation parameter and the critical value of the beam Fresnel number that cause the focus switching phenomenon are read. Based on these critical values, step adjustments are made in the parameter space along the increasing and decreasing directions, generating multiple sets of candidate values ​​for the beam truncation parameter and the beam Fresnel number. Each set of candidate values ​​for the beam truncation parameter and the beam Fresnel number is combined to replace the corresponding parameters in the original system parameter set, forming multiple candidate system parameter sets. Each candidate system parameter set is substituted into the generalized Huygens-Fresnel diffraction integral formula for focus stability verification calculation. The relevant system parameters in the generalized Huygens-Fresnel diffraction integral formula are updated using the candidate system parameter sets. Based on the updated formula, the exit field distribution function of the fundamental mode Gaussian beam after passing through the bifocal optical system is recalculated. Based on the recalculated exit field distribution function, new on-axis light intensity distribution data is generated. The new on-axis light intensity distribution data is processed to detect the principal maximum position of the light intensity, obtaining a new set of focus positions. Determine whether the new focal positions satisfy the focal stability condition, which is defined as the offset of all new focal positions from the geometric focal positions being less than a preset focal stability tolerance threshold. Select a set of candidate system parameters that can transform the focal stability analysis results into a stable state, and use this set as the optimized system parameter set.

[0093] In practical implementation, when the focus stability analysis result indicates a focus instability state, the critical values ​​of the beam truncation parameter and the critical value of the beam Fresnel number that cause the focus switching phenomenon are read. For example, the critical value of the beam truncation parameter is 0.8, and the critical value of the beam Fresnel number is 3.0. Based on the critical values ​​of 0.8 for the beam truncation parameter and 3.0 for the beam Fresnel number, adjustments are made in the parameter space along the increasing and decreasing directions with fixed step sizes, for example, a step size of 0.05, generating multiple sets of candidate values ​​for the beam truncation parameter and the beam Fresnel number. The candidate values ​​for the beam truncation parameter may include 0.75 and 0.85, and the candidate values ​​for the beam Fresnel number may include 2.8 and 3.2. Each set of candidate beam truncation parameters is combined with candidate beam Fresnel number values ​​to form parameter pairs such as (0.75,2.8), (0.75,3.2), (0.85,2.8), and (0.85,3.2). These parameter pairs are used to replace the corresponding parameters in the original system parameter set to form multiple candidate system parameter sets.

[0094] In some embodiments, each candidate system parameter set is substituted into the generalized Huygens-Fresnel diffraction integral formula for focus stability verification calculation. The system parameters in the generalized Huygens-Fresnel diffraction integral formula are updated using the candidate system parameter set, for example, the beam truncation parameter is updated from 0.8 to 0.85, and the beam Fresnel number is updated from 3.0 to 2.8. Based on the updated generalized Huygens-Fresnel diffraction integral formula, the exit field distribution function of the fundamental mode Gaussian beam after passing through the bifocal optical system is recalculated. New on-axis light intensity distribution data is generated based on the recalculated exit field distribution function. The new on-axis light intensity distribution data is processed to detect the principal maximum position of light intensity, resulting in a new set of focus positions. It is determined whether the new focus positions meet the focus stability condition, which is defined as the offset of all new focus positions from the geometric focus position being less than a preset focus stability tolerance threshold, for example, the focus stability tolerance threshold is set to 1.5 mm, and the determination formula is as follows: ,in This represents the set of offsets between the new focal position and the geometric focal position. This represents the focal stability tolerance threshold. A set of candidate system parameters that can transform the focal stability analysis results into a stable state is selected and used as the optimized system parameter set.

[0095] Optionally, the validation results of candidate parameter combinations can be recorded in a table for comparative analysis, see Table 1:

[0096] Table 1: Verification Results of Parameter Combinations

[0097] Parameter combination number Beam cutoff parameters Beam Fresnel Number Calculated maximum focal offset (mm) Does it meet the stability condition (threshold δ=1.5mm)? 1 0.75 2.8 1.8 no 2 0.75 3.2 2.1 no 3 0.85 2.8 1.2 yes 4 0.85 3.2 1.7 no

[0098] It is understandable that different parameter combinations will lead to different focus offsets. Only combinations that meet the stability conditions can be selected as the optimal system parameter set, such as combination 3 in the table above. It is also understandable that the adjustment direction and step size of the beam truncation parameter and the beam Fresnel number directly affect the optimization results. For example, when the beam truncation parameter increases from 0.8 and the beam Fresnel number decreases from 3.0, it may be easier to achieve the stability condition, which reflects the necessity of joint optimization adjustment.

[0099] Example 5: The optimized system parameter set is used as the system parameter input for the current iteration cycle to calculate the focus stability analysis result of the current iteration. It is then determined whether the maximum focus offset in the current iteration's focus stability analysis result is less than a preset focus stability threshold condition. If the maximum focus offset is less than the threshold condition, the optimized system parameter set used in the current iteration is determined as the final optimized bifocal optical system configuration scheme. If the maximum focus offset is not less than the threshold condition, the beam truncation parameter and beam Fresnel number in the optimized system parameter set are fine-tuned according to the offset trend reflected in the current iteration's focus stability analysis result, generating a new set of optimized system parameters for the next iteration. This iterative optimization step is repeated until the maximum focus offset meets the focus stability threshold condition or reaches the preset maximum number of iterations. This method integrates the entire process of acquiring system and beam parameters, constructing and calculating the optical field model, analyzing on-axis light intensity and focus position, determining focus switching phenomena, and performing joint optimization iterations of key parameters based on stability results, ultimately generating a bifocal optical system configuration scheme that meets the focus stability requirements.

[0100] In practice, the optimized system parameter set is used as the system parameter input for the current iteration cycle. For example, the input beam cutoff parameter is 0.85 and the beam Fresnel number is 2.8. The focus stability analysis result for the current iteration is calculated, and the maximum focus offset is recorded in the analysis result. It is then determined whether the maximum focus offset in the focus stability analysis result of the current iteration is less than a preset focus stability threshold condition. For example, the focus stability threshold condition is set to a maximum offset not exceeding 1.0 mm. If the calculated maximum focus offset is 0.9 mm, then the condition of less than 1.0 mm is met. If the maximum focus offset is less than the focus stability threshold condition, then the optimized system parameter set used in the current iteration is determined as the final optimized bifocal optical system configuration scheme. The configuration scheme explicitly includes the beam cutoff parameter of 0.85 and the beam Fresnel number of 2.8.

[0101] In some embodiments, if the maximum focus offset calculated in the current iteration is not less than the focus stability threshold condition, for example, the maximum focus offset is 1.3 mm, which is greater than the threshold of 1.0 mm, then the beam truncation parameter and beam Fresnel number in the optimized system parameter set are fine-tuned according to the focus stability analysis results of the current iteration. The fine-tuning rule is that if the focus position is generally ahead of the geometric focus, the beam Fresnel number is increased by a small step, for example, increasing the beam Fresnel number from 2.8 to 2.82 by 0.02; if the focus position is generally lagging, the beam truncation parameter is decreased by a small step, for example, decreasing the beam truncation parameter from 0.85 to 0.83, generating a new set of optimized system parameters for the next iteration.

[0102] The method iterates and optimizes the process, recalculating the focus stability analysis results with a new set of parameters and determining the maximum focus offset until the maximum focus offset meets the focus stability threshold or the number of iterations reaches a preset maximum limit, such as 5 iterations. Optionally, this method integrates the following steps: obtaining the system parameter set and beam parameters of the fundamental Gaussian beam; constructing an initial light field model based on the generalized Huygens-Fresnel diffraction integral formula and calculating the output field distribution function after the beam passes through the system; calculating the light intensity distribution on the transmission axis based on the output field distribution function and identifying the focus position corresponding to the principal maximum point of light intensity; analyzing the offset between the focus position and the geometric focus position and determining the focus switching phenomenon to generate focus stability analysis results; and jointly optimizing and adjusting the beam truncation parameters and beam Fresnel number in the system parameter set based on the focus stability analysis results, iteratively calculating using the generalized Huygens-Fresnel diffraction integral formula until the focus stability meets the preset threshold to generate the final optimized bifocal optical system configuration. The convergence of the iterative process is determined by the adjustment step size and the threshold condition. The step size formula is as follows:

[0103]

[0104] in: This indicates the parameter value for the new iteration. Indicates the current parameter value. This indicates a fixed fine-tuning step size. This indicates the maximum focus offset. For a sign function, a step size that is too large will cause oscillations, while a step size that is too small will result in slow convergence.

[0105] See Figure 5This is a scatter plot heatmap of bifocal optical system parameter optimization, used to analyze the impact of "beam truncation parameters" and "beam Fresnel number" on "maximum focus shift". Near the green dashed line (optimal truncation parameters), the scatter plots are lighter in color and have smaller shifts, indicating that this region represents the parameter combination with the best focus stability. In regions far from the optimal parameters (such as around truncation parameters of 0.80 or 0.90), the scatter plots are darker in color and have larger shifts, reflecting the negative impact of parameter deviations on focus stability. Visually identifying the parameter combination with the "minimum focus shift" (optimal truncation parameters + optimal Fresnel number) provides a precise target for parameter tuning of the bifocal optical system. The more significant impact of "truncation parameters on focus shift" is clearly shown along the horizontal axis, indicating that subsequent optimization should prioritize controlling the accuracy of the truncation parameters.

[0106] It should be noted that, in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such process, method, article, or apparatus.

[0107] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A method for optimizing a bifocal optical system based on a fundamental Gaussian beam, characterized in that, The method includes: Obtain the system parameter set of the dual-focal optical system, and construct an initial optical field model to describe the transmission of the fundamental mode Gaussian beam through the dual-focal optical system based on the generalized Huygens-Fresnel diffraction integral formula; The system parameter set and the beam parameters of the fundamental Gaussian beam are input into the initial optical field model for transmission calculation to obtain the output field distribution function of the beam after passing through the bifocal optical system; Based on the emission field distribution function, the light intensity distribution of the fundamental mode Gaussian beam on the transmission axis is calculated, and on-axis light intensity distribution data is generated. The light intensity distribution data on the axis is processed to detect the location of the principal maximum light intensity, and at least two principal maximum light intensity points and their corresponding focal positions are identified. The occurrence criterion of focal switching phenomenon is calculated based on the offset between the focal position and the geometric focal position, and the focal stability analysis results are generated. Based on the focal stability analysis results, the beam truncation parameter and beam Fresnel number in the system parameter set are jointly optimized and adjusted to generate an optimized system parameter set. The optimized system parameter set is input into the generalized Huygens-Fresnel diffraction integral formula for iterative calculation until the focus stability analysis result meets the preset focus stability threshold condition, thereby generating the final optimized bifocal optical system configuration scheme. The beam truncation parameter and beam Fresnel number in the system parameter set are jointly optimized and adjusted based on the focus stability analysis results to generate an optimized system parameter set, including: When the focus stability analysis result is a focus unstable state, read the critical values ​​of the beam truncation parameter and the critical value of the beam Fresnel number that cause the focus switching phenomenon. Based on the critical values ​​of the beam truncation parameter and the critical value of the beam Fresnel number, the parameters are adjusted stepwise in the increasing and decreasing directions in the parameter space to generate multiple sets of candidate values ​​for the beam truncation parameter and the beam Fresnel number. Each set of candidate beam truncation parameters and candidate beam Fresnel numbers is combined to replace the corresponding parameters in the original system parameter set, forming multiple candidate system parameter sets. Substitute each set of candidate system parameters into the generalized Huygens-Fresnel diffraction integral formula to perform focus stability verification calculations. The set of candidate system parameters that causes the focus stability analysis results to transform into a stable state is selected as the set of optimized system parameters.

2. The optimization method for a bifocal optical system based on a fundamental Gaussian beam according to claim 1, characterized in that, The acquisition of the system parameter set of the bifocal optical system involves constructing an initial optical field model based on the generalized Huygens-Fresnel diffraction integral formula to describe the transmission of the fundamental mode Gaussian beam through the bifocal optical system, including: The system parameter set includes the Fresnel half-wave zone number of the Fresnel zone plate, the focal length of the thin lens, and the beam cutoff parameters of the optical system. Obtain the beam waist radius and beam wavelength of the fundamental Gaussian beam, and construct the initial optical field expression of the fundamental Gaussian beam; The complex amplitude transmittance function of the Fresnel zone plate is multiplied with the phase transformation function of the thin lens to generate the composite transmittance function of the bifocal optical system. Multiplying the initial optical field expression of the fundamental Gaussian beam with the composite transmittance function yields the modulation optical field expression at the incident plane of the bifocal optical system. Substituting the modulation light field expression into the generalized Huygens-Fresnel diffraction integral formula, we obtain the output field integral expression of the beam at any observation plane behind the bifocal optical system, and use the output field integral expression as the initial light field model.

3. The optimization method for a bifocal optical system based on a fundamental Gaussian beam according to claim 1, characterized in that, The step of calculating the intensity distribution of the fundamental mode Gaussian beam along the transmission axis based on the emission field distribution function, and generating on-axis intensity distribution data, includes: By limiting the observation plane position coordinates of the emission field distribution function to the transmission axis, a simplified emission field function at the observation point on the axis is obtained; Calculate the product of the simplified outgoing field function and its complex conjugate function to obtain the light intensity value at each point on the axis, and generate an initial on-axis light intensity distribution sequence; The initial on-axis light intensity distribution sequence is normalized so that the maximum light intensity value is the reference value, thereby generating normalized on-axis light intensity distribution data. The normalized on-axis light intensity distribution data is sampled at high density along the transmission axis to generate continuous on-axis light intensity distribution data.

4. The optimization method for a bifocal optical system based on a fundamental Gaussian beam according to claim 1, characterized in that, The process of detecting the principal maxima of light intensity in the on-axis light intensity distribution data, identifying at least two principal maxima and their corresponding focal positions, includes: The first derivative of the light intensity distribution data on the axis is used to obtain the light intensity change rate sequence. Identify the zero-crossing points in the light intensity change rate sequence that change from positive to negative to obtain the set of potential light intensity extreme point locations; Calculate the light intensity value corresponding to each position in the set of potential light intensity extreme points, and filter out the light intensity extreme points whose light intensity values ​​exceed the preset principal maximum threshold as candidate principal maxima. Local peak verification processing is performed on the candidate principal maxima to filter out the points with the highest light intensity values ​​within a preset neighborhood range and determine them as the principal maxima of light intensity. The coordinates of the principal maximum point of light intensity on the transmission axis are read and used as the focal position.

5. The optimization method for a bifocal optical system based on a fundamental Gaussian beam according to claim 1, characterized in that, The step of calculating the criterion for the occurrence of focus switching phenomenon based on the offset between the focal position and the geometric focal position, and generating focal stability analysis results, includes: Obtain the geometric focus position coordinates from the system parameter set; Calculate the absolute distance between each of the focal positions and the coordinates of the geometric focal position, and use it as the focal offset; Determine whether any of the focal offsets exceeds a preset focal switch threshold distance; If it exists, it is determined that a focus switching phenomenon has occurred in the bifocal optical system, and the focus instability state and the focus position where the focus switching occurred are marked in the focus stability analysis results; If it does not exist, mark the focus as stable in the focus stability analysis results.

6. The optimization method for a bifocal optical system based on a fundamental Gaussian beam according to claim 5, characterized in that, The method also includes a correction process based on the Fresnel half-wave band number focusing switch phenomenon occurrence criterion, including: Determine the parity of the Fresnel half-wave zone number in the system parameter set; If the Fresnel half-wave zone number is even, then the focal switch threshold distance is adjusted to a preset even-number half-wave zone threshold distance; If the Fresnel half-wave zone number is odd, the focal switch threshold distance is adjusted to the preset odd half-wave zone threshold distance, and the light intensity value weighting factor at the geometric focal point is set. Based on the adjusted focus switch threshold distance and the light intensity value weighting factor, the focus stability analysis results are recalculated.

7. The optimization method for a bifocal optical system based on a fundamental Gaussian beam according to claim 6, characterized in that, The step of substituting each candidate system parameter set into the generalized Huygens-Fresnel diffraction integral formula for focal stability verification calculation includes: Update the system parameters in the generalized Huygens-Fresnel diffraction integral formula using the candidate system parameter set; Based on the updated generalized Huygens-Fresnel diffraction integral formula, the output field distribution function of the fundamental mode Gaussian beam after passing through the bifocal optical system is recalculated. New on-axis light intensity distribution data are generated based on the recalculated outgoing field distribution function; The new on-axis light intensity distribution data is processed to detect the location of the principal maximum of light intensity, and the new focal position is obtained. Determine whether the new focal position meets the focal stability condition, which is that the offset of all focal positions from the geometric focal position is less than a preset focal stability tolerance threshold.

8. The optimization method for a bifocal optical system based on a fundamental Gaussian beam according to claim 1, characterized in that, The step of inputting the optimized system parameter set into the generalized Huygens-Fresnel diffraction integral formula for iterative calculation until the focus stability analysis result meets the preset focus stability threshold condition, and generating the final optimized bifocal optical system configuration scheme, includes: The optimized system parameter set is used as the system parameter input for the current iteration to calculate the focus stability analysis result for the current iteration; Determine whether the maximum focus offset in the focus stability analysis result of the current iteration is less than the focus stability threshold condition; If it is less than, then the optimized system parameter set of the current iteration is determined as the final optimized bifocal optical system configuration scheme; If it is not less than, then the beam truncation parameter and beam Fresnel number in the optimized system parameter set are fine-tuned according to the focus stability analysis results of the current iteration, and a new set of optimized system parameters for the next iteration is generated. Repeat the iterative optimization steps until the focus stability threshold condition is met or the preset maximum number of iterations is reached.

9. An optimization method for a bifocal optical system based on a fundamental Gaussian beam, characterized in that, The method includes: Obtain the system parameter set of the bifocal optical system and the beam parameters of the fundamental Gaussian beam; An initial light field model is constructed based on the generalized Huygens-Fresnel diffraction integral formula, and the output field distribution function of the beam after passing through the system is calculated. Calculate the light intensity distribution on the transmission axis based on the emission field distribution function, and identify the focal position corresponding to the main maximum point of light intensity. Analyze the offset between the focal position and the geometric focal position, determine the focal switching phenomenon, and generate focal stability analysis results; Based on the focal stability analysis results, the beam cutoff parameter and beam Fresnel number in the system parameter set are jointly optimized and adjusted, and iterative calculations are performed using the generalized Huygens-Fresnel diffraction integral formula until the focal stability meets the preset threshold, generating the final optimized bifocal optical system configuration scheme.

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