An optimization design method and system for a self-focusing annular Airy beam array

Abstract

The present invention relates to a method and system for optimizing the design of a self-focusing annular Airy beam array, belonging to the field of beam array design. The method uses maximum light intensity to characterize the self-focusing capability of the annular Airy beam array. Based on the correlation between the self-focusing capability of the annular Airy beam array and the number of Airy beams, the optimal number of Airy beams corresponding to saturation of the annular Airy beam array is determined. When the number of Airy beams reaches the optimal value, the annular Airy beam array has the maximum self-focusing capability under given beam parameter conditions. Therefore, the present invention overcomes the blind selection of the number of Airy beams and achieves a scientific selection of the number of Airy beams, ensuring that the annular Airy beam array has the maximum self-focusing capability while avoiding the blind increase of the number of beams.

Description

Technical Field

[0001] The present invention relates to the field of beam array design, and in particular to an optimization design method and system for a self-focusing annular Airy beam array. Background Art

[0002] Optics researchers first discovered the phenomenon of sudden self-focusing during theoretical studies of the properties of radially symmetric Airy beams. Sudden self-focusing refers to a beam that can be focused without the aid of optical components. It maintains a low intensity before reaching the focal point, but upon reaching it, the intensity suddenly increases by tens or even hundreds of times, before rapidly decreasing again. A year later, optics researchers successfully observed sudden self-focusing experimentally. By properly selecting two apodization parameters, the sudden self-focusing capability of improved circular Airy beams can be significantly enhanced. The focusing intensity of partially coherent circular Airy beams decreases as the spatial coherence length decreases. Under the same conditions, the focusing intensity of radially polarized symmetric Airy beams is nearly three times that of symmetric Airy beams. While the sudden self-focusing effect of Airy beams has a wide range of applications, the limited intensity of individual Airy beams limits their effectiveness. Consequently, Airy beam arrays with self-focusing capabilities have emerged. Annular Airy beam arrays exhibit both sudden and triangular self-focusing properties. Radial Airy beam arrays maintain their self-focusing capability even when propagating through atmospheric turbulence. Airy beam arrays can even self-focus twice when propagating through a self-focusing medium with a strong Kerr effect. Airy beam arrays can be used for high-quality image transmission, stable optical capture, food safety testing, and other applications. In these Airy beam arrays, the number of beams sometimes reaches 128. Currently, it remains unknown whether the self-focusing performance of an Airy beam array improves with a larger number of beams, or whether the self-focusing capability of such an Airy beam array can be increased indefinitely by increasing the number of beams. Therefore, the correlation between the self-focusing capability of an annular Airy beam array and the number of beams has become a pressing issue. Summary of the Invention

[0003] The purpose of the present invention is to provide a method and system for optimizing the design of a self-focusing annular Airy beam array, thereby optimizing the number of selected beams and ensuring that the self-focusing capability of the annular Airy beam array is maximized.

[0004] To achieve the above object, the present invention provides the following solutions:

[0005] A method for optimizing a self-focusing annular Airy beam array, the method comprising:

[0006] Determining an electric field distribution expression of an annular Airy beam array on an initial plane; the annular Airy beam array is formed by rotating the Airy beam multiple times around a rotation axis on the initial plane;

[0007] According to the electric field distribution expression, an analytical expression of the electric field of the annular Airy beam array during free space transmission is determined;

[0008] Determine a simplified expression for the electric field of a point on the beam transmission direction axis of the annular Airy beam array based on the electric field analytical expression;

[0009] Using the simplified expression for the on-axis point electric field, the maximum light intensity on the focal plane of annular Airy beam arrays corresponding to different numbers of Airy beams during free-space transmission is calculated under given beam parameter conditions, and a curve showing the relationship between the maximum light intensity and the number of Airy beams is plotted; the maximum light intensity is used to characterize the self-focusing capability of the annular Airy beam array;

[0010] The number of Airy beams when the maximum light intensity first reaches a maximum value is selected on the variation relationship curve, and is determined as the optimal number of Airy beams when the self-focusing ability of the annular Airy beam array is saturated under given beam parameter conditions.

[0011] Optionally, the electric field distribution expression is:

[0012]

[0013] Where E(x,y,0) is the electric field distribution of the annular Airy beam array on the initial plane z = 0; x is the coordinate value on the x-axis in the rectangular coordinate system; y is the coordinate value on the y-axis in the rectangular coordinate system; X = xcosθ n -ysinθ n +c, X represents the position of the nth Airy beam in the x-axis direction; Y = xsinθ n +ycosθ n +c,Y represents the position of the nth Airy beam in the y-axis direction; θ n =2(n-1)π / N,θ n represents the angle of rotation of the nth Airy beam around the z-axis; N is the number of Airy beams, which is a positive integer greater than 1; c is the eccentricity distance; A is the intensity control parameter, which makes the maximum intensity of the annular Airy beam array on the initial plane 1; a is the exponential decay factor; w0 is the scale factor; Ai(·) is the Airy function.

[0014] Optionally, the electric field analytical expression is

[0015]

[0016] Where E(x, y, z) represents the electric field distribution of the annular Airy beam array during free space transmission; z is the coordinate value on the z-axis in the rectangular coordinate system; the z-axis is both the direction of beam transmission and the rotation axis of the Airy beam; z0 represents the Rayleigh distance, k represents the wave number; i represents the imaginary unit.

[0017] Optionally, the simplified expression of the electric field at the axis point is:

[0018]

[0019] Where E(0,0,z) represents the electric field distribution of a point on the beam propagation axis when the annular Airy beam array propagates in free space.

[0020] Optionally, using the simplified expression of the on-axis point electric field, the maximum light intensity on the focal plane of annular Airy beam arrays corresponding to different numbers of Airy beams during free-space transmission is calculated under given beam parameter conditions, specifically including:

[0021] Using the formula z f =2kw0(w0x p ) 1 / 2 Determine the position of the focal plane; where z f represents the position of the focal plane, x p represents the position of the first peak intensity of the annular Airy beam array in the positive direction of the x-axis of the initial plane;

[0022] According to the position of the focal plane and the simplified expression of the electric field at the axis point, the maximum light intensity expression on the focal plane of the annular Airy beam array during free space transmission is determined as I(0, 0, z f )=|E(0,0,z f )| 2 Where, I(0,0,z f ) represents the maximum light intensity on the focal plane of the annular Airy beam array during free space transmission, E(0,0,z f ) represents the electric field at a point on the focal plane axis when the annular Airy beam array propagates in free space;

[0023] The number of Airy beams N is changed, and each number of Airy beams is substituted into the maximum light intensity expression under given beam parameter conditions to obtain the maximum light intensity corresponding to each number of Airy beams.

[0024] Optionally, the relationship curve of the maximum light intensity versus the number of Airy beams follows the rule that the maximum light intensity increases with the increase in the number of Airy beams until the number of Airy beams increases to the optimal number of Airy beams, at which point the maximum light intensity reaches a maximum value. Thereafter, the number of Airy beams gradually increases from the optimal number of Airy beams, but the maximum light intensity remains at the maximum value and no longer changes, and the self-focusing ability of the annular Airy beam array reaches saturation.

[0025] Optionally, the annular Airy beam array is formed by:

[0026] On the initial spatial plane, the Airy beam is rotated N times around the rotation axis at a rotation angle of 2π / N to obtain a ring-shaped Airy beam array.

[0027] An optimization design system for a self-focusing annular Airy beam array, the system comprising:

[0028] An electric field distribution expression determination module is used to determine the electric field distribution expression of an annular Airy beam array on an initial plane; the annular Airy beam array is formed by rotating the Airy beam multiple times around the rotation axis on the initial plane;

[0029] An electric field analytical expression determination module is used to determine the electric field analytical expression of the annular Airy beam array during free space transmission based on the electric field distribution expression;

[0030] A simplified expression determination module for the on-axis point electric field, configured to determine a simplified expression for the on-axis point electric field of the annular Airy beam array in the beam transmission direction based on the electric field analytical expression;

[0031] a variation relationship curve drawing module, for calculating the maximum light intensity on the focal plane of annular Airy beam arrays corresponding to different numbers of Airy beams during free-space transmission under given beam parameter conditions using the simplified expression of the on-axis point electric field, and drawing a variation relationship curve of the maximum light intensity as a function of the number of Airy beams; the maximum light intensity is used to characterize the self-focusing capability of the annular Airy beam array;

[0032] The optimal Airy beam number determination module is used to select the Airy beam number when the maximum light intensity first reaches the maximum value on the change relationship curve, and determine it as the optimal Airy beam number when the self-focusing ability of the annular Airy beam array is saturated under given beam parameter conditions.

[0033] Optionally, the change relationship curve drawing module specifically includes:

[0034] The focal plane determination submodule is used to use the formula z f =2kw0(w0x p ) 1 / 2 Determine the position of the focal plane; where z f represents the position of the focal plane, x p represents the position of the first peak intensity of the annular Airy beam array in the positive direction of the x-axis of the initial plane, k represents the wave number, and w0 is the scale factor;

[0035] The maximum light intensity expression determination submodule is used to determine the maximum light intensity expression on the focal plane of the annular Airy beam array during free space transmission as I(0, 0, z f )=|E(0,0,z f )|2 Where, I(0,0,z f ) represents the maximum light intensity on the focal plane of the annular Airy beam array during free space transmission, E(0,0,z f ) represents the electric field at a point on the focal plane axis when the annular Airy beam array propagates in free space;

[0036] The maximum light intensity calculation submodule is used to change the number N of Airy beams and substitute each number of Airy beams into the maximum light intensity expression under given beam parameter conditions to obtain the maximum light intensity corresponding to each number of Airy beams.

[0037] Optionally, the relationship curve of the maximum light intensity versus the number of Airy beams follows the rule that the maximum light intensity increases with the increase in the number of Airy beams until the number of Airy beams increases to the optimal number of Airy beams, at which point the maximum light intensity reaches a maximum value. Thereafter, the number of Airy beams gradually increases from the optimal number of Airy beams, but the maximum light intensity remains at the maximum value and no longer changes, and the self-focusing ability of the annular Airy beam array reaches saturation.

[0038] According to the specific embodiments provided by the present invention, the present invention discloses the following technical effects:

[0039] The present invention discloses a method and system for optimizing the design of a self-focusing annular Airy beam array. This method uses maximum light intensity to characterize the self-focusing capability of the annular Airy beam array. Based on the correlation between the self-focusing capability of the annular Airy beam array and the number of Airy beams, the optimal number of Airy beams corresponding to saturation of the annular Airy beam array is determined. When the number of Airy beams reaches the optimal value, the annular Airy beam array has the maximum self-focusing capability under given beam parameter conditions. Therefore, the present invention overcomes the blind selection of the number of Airy beams and achieves a scientific selection of the number of Airy beams, ensuring that the annular Airy beam array has the maximum self-focusing capability while avoiding the blind increase of the number of beams. BRIEF DESCRIPTION OF THE DRAWINGS

[0040] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0041] Figure 1 A flow chart of a method for optimizing the design of a self-focusing annular Airy beam array according to an embodiment of the present invention;

[0042] Figure 2Schematic diagram of the arrangement of an annular Airy beam array with a=0.1, w0=0.1 mm, c=0.25 mm and N=6 on the initial plane z=0 provided in an embodiment of the present invention;

[0043] Figure 3 A graph showing the relationship between the self-focusing capability of an annular Airy beam array with a=0.1, w0=0.1 mm, and c=0.25 mm and the number of Airy beams N provided in an embodiment of the present invention;

[0044] Figure 4 Light intensity distribution diagram of an annular Airy beam array with a=0.1, w0=0.1mm, c=0.25mm and N=10 on different observation planes in free space provided by an embodiment of the present invention; Figure 4 (a) is the light intensity distribution diagram of the initial plane z=0, Figure 4 (b) is the light intensity distribution diagram at the observation plane z = 0.25m, Figure 4 (c) is the light intensity distribution diagram at the observation plane z = 0.3m, Figure 4 (d) is the light intensity distribution diagram at the observation plane z = 0.33m, Figure 4 (e) is the focal plane z=z f =Light intensity distribution diagram of 0.434m, Figure 4 (f) is the light intensity distribution diagram at the observation plane z = 0.5m;

[0045] Figure 5 Light intensity distribution diagram of an annular Airy beam array with a=0.1, w0=0.1mm, c=0.25mm and N=20 on different observation planes in free space provided by an embodiment of the present invention; Figure 5 (a) is the light intensity distribution diagram of the initial plane z=0, Figure 5 (b) is the light intensity distribution diagram at the observation plane z = 0.25m, Figure 5 (c) is the light intensity distribution diagram at the observation plane z = 0.3m, Figure 5 (d) is the light intensity distribution diagram at the observation plane z = 0.33m, Figure 5 (e) is the focal plane z=z f =Light intensity distribution diagram of 0.434m, Figure 5 (f) is the light intensity distribution diagram at the observation plane z = 0.5m;

[0046] Figure 6 Light intensity distribution diagram of an annular Airy beam array with a=0.1, w0=0.1mm, c=0.25mm and N=54 on different observation planes in free space provided by an embodiment of the present invention; Figure 6 (a) is the light intensity distribution diagram of the initial plane z=0, Figure 6(b) is the light intensity distribution diagram at the observation plane z = 0.25m, Figure 6 (c) is the light intensity distribution diagram at the observation plane z = 0.3m, Figure 6 (d) is the light intensity distribution diagram at the observation plane z = 0.33m, Figure 6 (e) is the focal plane z=z f =Light intensity distribution diagram of 0.434m, Figure 6 (f) is the light intensity distribution diagram at the observation plane z = 0.5m;

[0047] Figure 7 This is a graph showing the relationship between the power ratio of the central bright spot of the focal spot of an annular Airy beam array with a=0.1, w0=0.1mm, and c=0.25mm and the number of Airy beams N provided in an embodiment of the present invention. DETAILED DESCRIPTION

[0048] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0049] The purpose of the present invention is to provide a method and system for optimizing the design of a self-focusing annular Airy beam array, thereby optimizing the number of selected beams and ensuring that the self-focusing capability of the annular Airy beam array is maximized.

[0050] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0051] An embodiment of the present invention provides an optimization design method for a self-focusing annular Airy beam array. The method finds the optimal value of the number of beams N required to form the annular Airy beam array when the self-focusing ability is maximized under given beam parameter conditions. This method scientifically determines the value of the number of beams N, ensuring that the annular Airy beam array has the maximum self-focusing ability while avoiding a blind increase in the number of beams.

[0052] See also Figure 1 , the optimization design method of the self-focusing annular Airy beam array includes the following steps:

[0053] Step S1, determining the electric field distribution expression of the annular Airy beam array on the initial plane.

[0054] On the initial plane, the Airy beam is rotated N times around the z axis (rotation axis) at a rotation angle of 2π / N each time, to obtain an Airy beam array arranged in a ring, which is a ring Airy beam array.

[0055] The electric field distribution expression is:

[0056]

[0057] Where E(x,y,0) is the electric field distribution of the annular Airy beam array on the initial plane z = 0; x is the coordinate value on the x-axis in the rectangular coordinate system; y is the coordinate value on the y-axis in the rectangular coordinate system; X = xcosθ n -ysinθ n +c, X represents the position of the nth Airy beam in the x-axis direction; Y = xsinθ n +ycosθ n +c, Y represents the position of the nth Airy beam in the y-axis direction; n represents the nth Airy beam; θ n =2(n-1)π / N,θ n represents the angle of rotation of the nth Airy beam around the z-axis; A is the intensity control parameter, which makes the maximum intensity on the initial plane 1; a is the exponential decay factor; w0 is the scale factor; c is the eccentricity distance; Ai(·) is the Airy function; N is the number of beams, which is a positive integer greater than 1.

[0058] Step S2: determining an analytical expression for the electric field of the annular Airy beam array during free space transmission based on the electric field distribution expression.

[0059] The analytical expression of the electric field is derived from the initial electric field distribution. The electric field of the annular Airy beam array when propagating in free space is characterized by the Collins integral formula:

[0060]

[0061] Where E(x,y,z) represents the electric field distribution of the annular Airy beam array during free-space propagation; k is the wave number, k = 2π / λ, where λ is the wavelength of the beam; x′ is the integral variable on the x-axis; y′ is the integral variable on the y-axis; z is the coordinate value on the z-axis in the rectangular coordinate system; the z-axis is both the direction of beam propagation and the rotation axis of the Airy beam; and i represents the imaginary unit. First, consider the propagation of the first Airy beam in free space. The electric field distribution of the first Airy beam on the initial plane is:

[0062]

[0063] Where E1(x,y,0) is the electric field distribution of the first Airy beam on the initial plane z = 0, and the subscript 1 represents the first Airy beam. The electric field of the first Airy beam on the observation plane z when propagating in free space is:

[0064]

[0065] Where E1(x,y,z) is the electric field distribution of the first Airy beam on the observation plane z when it propagates in free space. E1(x,z) and E1(y,z) are

[0066]

[0067]

[0068] Where E1(x,z) is the electric field distribution of the first Airy beam on the cross section xz when it propagates in free space, and E1(y,z) is the electric field distribution of the first Airy beam on the cross section yz when it propagates in free space. First, derive the analytical expression of E1(x,z). First, E1(x,z) can be rewritten as

[0069]

[0070] Among them, V'=x'+c, V=x+c. Next, use the integral definition of Airy function:

[0071]

[0072] Where u is the integral variable. E1(x,z) can be expressed as

[0073]

[0074] Then use the following mathematical integration formula

[0075]

[0076] Where p and q are constants, and t is the integral variable. E1(x,z) can be simplified to

[0077]

[0078] Where z0 represents the Rayleigh distance, Finally, use the following mathematical integral formula

[0079]

[0080] Where b and c are constants, and u is the integral variable. E1(x,z) can be expressed analytically as

[0081]

[0082] Using the same integration operation as above, E1(y,z) can be expressed analytically as

[0083]

[0084] Therefore, the electric field of the first Airy beam on the observation plane z when propagating in free space can be expressed analytically as:

[0085]

[0086] The second Airy beam up to the Nth Airy beam can be obtained by rotating the first Airy beam. The electric field of the nth Airy beam on the observation plane z when it propagates in free space can be obtained by replacing the coordinate position of the first Airy beam with the coordinate position of the nth Airy beam in the above formula, that is, using X = xcosθ n -ysinθ n +c and Y = xsinθ n +ycosθ n +c replaces x+c and y+c respectively. Finally, summing over n, we get the analytical expression of the electric field on the observation plane z when the annular Airy beam array is transmitted in free space:

[0087]

[0088] Step S3: Determine a simplified expression for the electric field of a point on the beam transmission direction axis of the annular Airy beam array based on the analytical expression for the electric field.

[0089] Since the annular Airy beam array is symmetrically distributed on the initial plane, its focus must be on the z-axis during free space transmission. The simplified expression of the electric field at the point on the axis is:

[0090]

[0091] Where E(0,0,z) is the simplified expression of the electric field of a point on the axis of the beam propagation direction when the annular Airy beam array propagates in free space.

[0092] Step S4, using the simplified expression of the electric field at an axis point, calculate the maximum light intensity on the focal plane of the annular Airy beam array corresponding to different numbers of Airy beams during free space transmission under given beam parameter conditions, and draw a curve showing the relationship between the maximum light intensity and the number of Airy beams.

[0093] The position of the focal plane can be expressed by the formula z f =2kw0(w0x p ) 1 / 2 Estimate, where x p is the position of the first peak intensity of the annular Airy beam array in the positive x-axis direction of the initial plane.

[0094] Using the formula I(0,0,z)=|E(0,0,z)| 2Calculate the light intensity of the annular Airy beam array at a point on the beam transmission direction axis when the annular Airy beam array is transmitted in free space. Wherein, E(0,0,z) represents the electric field distribution of the annular Airy beam array at a point on the beam transmission direction axis when the annular Airy beam array is transmitted in free space; I(0,0,z) represents the light intensity of the annular Airy beam array at a point on the beam transmission direction axis when the annular Airy beam array is transmitted in free space. Therefore, I(0,0,z f ) is the maximum light intensity on the focal plane of the annular Airy beam array when it is transmitted in free space. The maximum light intensity on the focal plane of the annular Airy beam array when it is transmitted in free space is expressed as I(0, 0, z f )=|E(0,0,z f )| 2 ; Where, E(0,0,z f ) represents the electric field at a point on the focal plane axis when the annular Airy beam array propagates in free space.

[0095] Under given beam parameter conditions, the value of the number of Airy beams N is changed, the maximum light intensity of the annular Airy beam array on the focal plane is calculated, and the maximum light intensity I(0,0,z f ) with the change of the number of Airy beams N.

[0096] The principle of maximum light intensity used to characterize the self-focusing ability of annular Airy beam array is as follows: the self-focusing ability of annular Airy beam array in free space transmission is given by I(0,0,z f ) / I 0m To describe. Among them, I(0,0,z f ) represents the maximum light intensity of the annular Airy beam array on the focal plane during free space transmission, I 0m represents the maximum light intensity of the annular Airy beam array on the initial plane, and I 0m is always 1. Therefore, the self-focusing ability of the annular Airy beam array during free space transmission is given by I(0,0,z f ) characterization, I(0,0,z f ) is larger, the stronger the self-focusing ability of the annular Airy beam array is.

[0097] Step S5, selecting the number of Airy beams when the maximum light intensity first reaches the maximum value on the variation relationship curve, and determining it as the optimal number of Airy beams when the self-focusing ability of the annular Airy beam array is saturated under given beam parameter conditions.

[0098] By plotting the maximum light intensity as a function of the number of Airy beams, N, the relationship between the self-focusing ability of the annular Airy beam array and the number of Airy beams, N, was determined. The relationship is as follows: the self-focusing ability of the annular Airy beam array quickly approaches saturation relative to the number of Airy beams, N. When the number of Airy beams, N, increases from 2 to a certain value, the self-focusing ability of the annular Airy beam array continues to increase. However, when the number of Airy beams, N, is further increased from a certain value, the self-focusing ability of the annular Airy beam array reaches saturation and no longer changes. Therefore, the value here represents the number of Airy beams at which the maximum light intensity first reaches its maximum value. This value serves as the optimal value of the number of Airy beams, N, for saturation of the self-focusing ability of the annular Airy beam array. When the number of Airy beams, N, reaches its optimal value, the annular Airy beam array exhibits the maximum self-focusing ability under given beam parameter conditions.

[0099] According to the optimal value of the number of beams, the evolution process of self-focusing of annular Airy beam arrays with three different numbers of beams during free space transmission is drawn to provide evidence.

[0100] The three different numbers of beams include a number of beams less than the optimal value, a number of beams equal to the optimal value, and a number of beams much greater than the optimal value. The light intensity distribution of the annular Airy beam array during free space transmission specifically includes:

[0101] Using the formula I(x,y,z)=|E(x,y,z)| 2 Calculate the light intensity of an annular Airy beam array. Where E(x,y,z) is the analytical expression for the electric field of the annular Airy beam array during free-space propagation, and I(x,y,z) is the light intensity distribution of the annular Airy beam array on any observation plane z during free-space propagation.

[0102] By comparing the two-dimensional light intensity images of annular Airy beam arrays with three different numbers of beams on the observation plane, the following conclusions can be drawn: when the number of beams is less than the optimal value, the self-focusing ability of the annular Airy beam array is weak; when the number of beams takes the optimal value, the self-focusing ability of the annular Airy beam array is the highest; when the number of beams is much larger than the optimal value, the self-focusing ability of the annular Airy beam array no longer improves and remains the same as when the number of beams takes the optimal value.

[0103] Figure 2 A schematic diagram of the arrangement of an annular Airy beam array with a=0.1, w0=0.1 mm, c=0.25 mm and N=6 on an initial plane z=0 provided by an embodiment of the present invention is given. Figure 2 There are 6 Airy beams in total, and the next Airy beam can be obtained by rotating the previous Airy beam by π / 3. Figure 3The relationship between the self-focusing ability of the annular Airy beam array provided by the embodiment of the present invention and the number of Airy beams N is given. Without loss of generality, in the following calculations, the beam parameters of the annular Airy beam array are fixed as follows: a = 0.1, w0 = 0.1mm, c = 0.25mm and λ = 532nm. Figure 3 The following conclusion can be drawn: The self-focusing capability of an annular Airy beam array quickly approaches saturation relative to the number of Airy beams, N. As the number of Airy beams, N, increases from 2 to 20, the self-focusing capability of the annular Airy beam array continues to increase. However, when the number of Airy beams, N, increases further from 20, the self-focusing capability of the annular Airy beam array reaches a saturation value of 35.11 and remains unchanged. Therefore, the number of Airy beams, N = 20, is considered the optimal value for the number of Airy beams in an annular Airy beam array under the given beam parameters.

[0104] Figures 4 to 6 The light intensity distribution diagrams of annular Airy beam arrays provided by embodiments of the present invention at different observation planes in free space are shown, with the number of Airy beams N being 10, 20, and 54, respectively. N values ​​of 10, 20, and 54 correspond to the number of beams less than the optimal value, the number of beams equal to the optimal value, and the number of beams far greater than the optimal value, respectively. Figures 4 to 6 (a)-(f) correspond to the initial plane z = 0, the observation plane z = 0.25m, the observation plane z = 0.3m, the observation plane z = 0.33m, and the focal plane z = z f = 0.434 m and observation plane z = 0.5 m. Figures 4 to 6 The color bars on each sub-image represent the light intensity. Figure 4 The curves in (e)-6(e) correspond to I(x,0,z f ). As the number of beams N increases, the number of rings in the initial plane intensity pattern of the annular Airy beam array first increases and then tends to saturation, which leads to the saturation of the self-focusing ability. Figure 4 (a) and Figure 5 (a), we can find that the light intensity pattern on the initial plane when the number of beams N = 20 has more rings than when the number of beams N = 10. When the number of beams N = 10, the bright ring outside the focal spot has not yet formed, as shown in the figure below. Figure 4 (e) shows the comparison. Figure 5 and Figure 6 , we can find that: except for Figure (e), Figure 6 Other sub-images of Figure 5The energy distribution of the corresponding sub-image is more concentrated; however, the intensity amplitude of all sub-images is the same. The focal spot when the number of beams N = 20 is exactly the same as when the number of beams N = 50, both consisting of a central bright spot and a bright outer ring, with a relatively large distance between the bright outer ring and the central bright spot. When the number of beams N = 10, the maximum light intensity of the annular Airy beam array is equal to 25.44; when the number of beams N = 20 and 54, the maximum light intensity of the annular Airy beam array is equal to 35.11. When the number of beams N = 20, the self-focusing ability of the annular Airy beam array has reached saturation. Therefore, the number of beams N = 20 is considered the optimal value for the number of beams in the annular Airy beam array.

[0105] Next, the power ratio of the central bright spot of the annular Airy beam array focal spot to the entire focal spot is analyzed. Figure 7 A graph showing the relationship between the central bright spot power ratio of the focal spot of an annular Airy beam array with a=0.1, w0=0.1mm, and c=0.25mm and the number of beams N. c The power of the central bright spot is expressed as P, and the power of the entire focal spot is expressed as P. c The power ratio of the central bright spot in the annular Airy beam array increases as the number of Airy beams N increases from 2 to 15. When the number of Airy beams N increases further from 15, the power ratio of the central bright spot in the annular Airy beam array reaches a saturation value of 0.78 and no longer changes. Therefore, when the number of Airy beams N reaches the optimal value of 20, the focal spot of the annular Airy beam array has solidified and no longer changes with further increases in the number of Airy beams N.

[0106] In summary, the present invention proposes a method for optimizing the design of a self-focusing annular Airy beam array. Under given beam parameter conditions, the optimal value for the number of Airy beams N required to form the annular Airy beam array is determined, maximizing the self-focusing capability. When the number of Airy beams N reaches the optimal value, the self-focusing capability of the annular Airy beam array has reached saturation, and the focal spot of the annular Airy beam array has already solidified. Further increasing the number of Airy beams N beyond the optimal value simply wastes the number of Airy beams and has no benefit in improving the self-focusing capability or changing the focal spot shape. Therefore, the present invention overcomes the blind practice of selecting the number of Airy beams N and achieves a scientifically determined value for the number of Airy beams N, ensuring that the annular Airy beam array has the maximum self-focusing capability while avoiding the blind practice of increasing the number of Airy beams.

[0107] An embodiment of the present invention further provides an optimization design system for a self-focusing annular Airy beam array, the system comprising:

[0108] An electric field distribution expression determination module is used to determine the electric field distribution expression of an annular Airy beam array on an initial plane; the annular Airy beam array is formed by rotating the Airy beam multiple times around the rotation axis on the initial plane;

[0109] An electric field analytical expression determination module is used to determine the electric field analytical expression of the annular Airy beam array during free space transmission based on the electric field distribution expression;

[0110] A simplified expression determination module for the on-axis point electric field, configured to determine a simplified expression for the on-axis point electric field of the annular Airy beam array in the beam transmission direction based on the electric field analytical expression;

[0111] a variation relationship curve drawing module, for calculating the maximum light intensity on the focal plane of annular Airy beam arrays corresponding to different numbers of Airy beams during free-space transmission under given beam parameter conditions using the simplified expression of the on-axis point electric field, and drawing a variation relationship curve of the maximum light intensity as a function of the number of Airy beams; the maximum light intensity is used to characterize the self-focusing capability of the annular Airy beam array;

[0112] The optimal Airy beam number determination module is used to select the Airy beam number when the maximum light intensity first reaches the maximum value on the change relationship curve, and determine it as the optimal Airy beam number when the self-focusing ability of the annular Airy beam array is saturated under given beam parameter conditions.

[0113] The change relationship curve drawing module specifically includes:

[0114] The focal plane determination submodule is used to use the formula z f =2kw0(w0x p ) 1 / 2 Determine the position of the focal plane; where z f represents the position of the focal plane, x p represents the position of the first peak intensity of the annular Airy beam array in the positive direction of the x-axis of the initial plane, k represents the wave number, and w0 is the scale factor;

[0115] The maximum light intensity expression determination submodule is used to determine the maximum light intensity expression on the focal plane of the annular Airy beam array during free space transmission as I(0, 0, z f )=|E(0,0,z f )| 2 ; Where, I(0,0,z f ) represents the maximum light intensity on the focal plane of the annular Airy beam array during free space transmission, E(0,0,z f ) represents the electric field at a point on the focal plane axis when the annular Airy beam array propagates in free space;

[0116] The maximum light intensity calculation submodule is used to change the number N of Airy beams and substitute each number of Airy beams into the maximum light intensity expression under given beam parameter conditions to obtain the maximum light intensity corresponding to each number of Airy beams.

[0117] The relationship curve of the maximum light intensity versus the number of Airy beams follows the rule that the maximum light intensity increases with the increase in the number of Airy beams until the number of Airy beams increases to the optimal number of Airy beams, at which point the maximum light intensity reaches its maximum value. After that, the number of Airy beams gradually increases from the optimal number of Airy beams, but the maximum light intensity remains at its maximum value and no longer changes, and the self-focusing ability of the annular Airy beam array reaches saturation.

[0118] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on the differences from other embodiments. Reference can be made to the common and similar parts between the various embodiments. For the systems disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the description is relatively simple, and the relevant parts can be referred to the method description.

[0119] This document uses specific examples to illustrate the principles and implementation methods of the present invention. The above examples are only intended to help understand the method and core concept of the present invention. At the same time, those skilled in the art will find that the specific implementation methods and application scopes may vary based on the concept of the present invention. In summary, the contents of this specification should not be construed as limiting the present invention.

Claims

1. A method for optimizing the design of a self-focusing annular Airy beam array, characterized in that: The method comprises: Determining an electric field distribution expression of an annular Airy beam array on an initial plane; the annular Airy beam array is formed by rotating the Airy beam multiple times around a rotation axis on the initial plane; According to the electric field distribution expression, an analytical expression of the electric field of the annular Airy beam array during free space transmission is determined; Determine a simplified expression for the electric field of a point on the beam transmission direction axis of the annular Airy beam array based on the electric field analytical expression; Using the simplified expression for the on-axis point electric field, the maximum light intensity on the focal plane of annular Airy beam arrays corresponding to different numbers of Airy beams during free-space transmission is calculated under given beam parameter conditions, and a curve showing the relationship between the maximum light intensity and the number of Airy beams is plotted; the maximum light intensity is used to characterize the self-focusing capability of the annular Airy beam array; Selecting the number of Airy beams when the maximum light intensity first reaches a maximum value on the variation relationship curve, and determining it as the optimal number of Airy beams when the self-focusing ability of the annular Airy beam array is saturated under given beam parameter conditions; The simplified expression of the electric field at the on-axis point is: Where E(0,0,z) represents the electric field distribution of a point on the beam transmission direction axis when the annular Airy beam array is transmitted in free space; N is the number of Airy beams, which is a positive integer greater than 1; n represents the nth Airy beam; A is the intensity control parameter, which makes the maximum intensity of the annular Airy beam array on the initial plane 1; a is the exponential decay factor; c is the eccentricity distance; w0 is the scale factor; z is the coordinate value on the z-axis in the rectangular coordinate system; the z-axis is both the direction of beam transmission and the rotation axis of the Airy beam; z0 represents the Rayleigh distance, k represents the wave number; i represents the imaginary unit; Ai(·) is the Airy function; Using the simplified expression of the on-axis point electric field, the maximum light intensity on the focal plane of annular Airy beam arrays corresponding to different numbers of Airy beams during free-space transmission is calculated under given beam parameter conditions. Specifically, the following is done: Using the formula z f =2kw0(w0x p ) 1 / 2 Determine the position of the focal plane; where z f represents the position of the focal plane, x p represents the position of the first peak intensity of the annular Airy beam array in the positive direction of the x-axis of the initial plane; According to the position of the focal plane and the simplified expression of the electric field at the axis point, the maximum light intensity expression on the focal plane of the annular Airy beam array during free space transmission is determined as I(0,0,z f )=|E(0,0,z f )| 2 ; Where, I(0,0,z f ) represents the maximum light intensity on the focal plane of the annular Airy beam array during free space transmission, E(0,0,z f ) represents the electric field at a point on the focal plane axis when the annular Airy beam array propagates in free space; The number of Airy beams N is changed, and each number of Airy beams is substituted into the maximum light intensity expression under given beam parameter conditions to obtain the maximum light intensity corresponding to each number of Airy beams.

2. The method according to claim 1, characterized in that The electric field distribution expression is: Where E(x,y,0) is the electric field distribution of the annular Airy beam array on the initial plane z = 0; x is the coordinate value on the x-axis in the rectangular coordinate system; y is the coordinate value on the y-axis in the rectangular coordinate system; X = xcosθ n -ysinθ n +c, X represents the position of the nth Airy beam in the x-axis direction; Y = xsinθ n +ycosθ n +c,Y represents the position of the nth Airy beam in the y-axis direction; θ n =2(n-1)π / N,θ n represents the rotation angle of the nth Airy beam around the z-axis.

3. The method according to claim 2, characterized in that The analytical expression of the electric field is Where E(x,y,z) represents the electric field distribution of the annular Airy beam array during free space transmission.

4. The method according to claim 1, wherein The relationship curve of the maximum light intensity versus the number of Airy beams follows the rule that the maximum light intensity increases with the increase in the number of Airy beams until the number of Airy beams increases to the optimal number of Airy beams, at which point the maximum light intensity reaches its maximum value. After that, the number of Airy beams gradually increases from the optimal number of Airy beams, but the maximum light intensity remains at its maximum value and no longer changes, and the self-focusing ability of the annular Airy beam array reaches saturation.

5. The method according to claim 1, wherein The method for forming the annular Airy beam array is as follows: On the initial spatial plane, the Airy beam is rotated N times around the rotation axis at a rotation angle of 2π / N to obtain a ring-shaped Airy beam array.

6. An optimization design system for a self-focusing annular Airy beam array, characterized in that: The system comprises: An electric field distribution expression determination module is used to determine the electric field distribution expression of an annular Airy beam array on an initial plane; the annular Airy beam array is formed by rotating the Airy beam multiple times around the rotation axis on the initial plane; An electric field analytical expression determination module is used to determine the electric field analytical expression of the annular Airy beam array during free space transmission based on the electric field distribution expression; A simplified expression determination module for the on-axis point electric field, configured to determine a simplified expression for the on-axis point electric field of the annular Airy beam array in the beam transmission direction based on the electric field analytical expression; a variation relationship curve drawing module, for calculating the maximum light intensity on the focal plane of annular Airy beam arrays corresponding to different numbers of Airy beams during free-space transmission under given beam parameter conditions using the simplified expression of the on-axis point electric field, and drawing a variation relationship curve of the maximum light intensity as a function of the number of Airy beams; the maximum light intensity is used to characterize the self-focusing capability of the annular Airy beam array; an optimal Airy beam number determination module, configured to select the number of Airy beams when the maximum light intensity first reaches a maximum value on the variation relationship curve, and determine the optimal Airy beam number when the self-focusing ability of the annular Airy beam array is saturated under given beam parameter conditions; The simplified expression of the electric field at the on-axis point is: Where E(0,0,z) represents the electric field distribution of a point on the beam transmission direction axis when the annular Airy beam array is transmitted in free space; N is the number of Airy beams, which is a positive integer greater than 1; n represents the nth Airy beam; A is the intensity control parameter, which makes the maximum intensity of the annular Airy beam array on the initial plane 1; a is the exponential decay factor; c is the eccentricity distance; w0 is the scale factor; z is the coordinate value on the z-axis in the rectangular coordinate system; the z-axis is both the direction of beam transmission and the rotation axis of the Airy beam; z0 represents the Rayleigh distance, k represents the wave number; i represents the imaginary unit; Ai(·) is the Airy function; The change relationship curve drawing module specifically includes: The focal plane determination submodule is used to use the formula z f =2kw0(w0x p ) 1 / 2 Determine the position of the focal plane; where z f represents the position of the focal plane, x p represents the position of the first peak intensity of the annular Airy beam array in the positive direction of the x-axis of the initial plane, k represents the wave number, and w0 is the scale factor; The maximum light intensity expression determination submodule is used to determine the maximum light intensity expression on the focal plane of the annular Airy beam array during free space transmission as I(0,0,z f )=|E(0,0,z f )| 2 ; Where, I(0,0,z f ) represents the maximum light intensity on the focal plane of the annular Airy beam array during free space transmission, E(0,0,z f ) represents the electric field at a point on the focal plane axis when the annular Airy beam array propagates in free space; The maximum light intensity calculation submodule is used to change the number N of Airy beams and substitute each number of Airy beams into the maximum light intensity expression under given beam parameter conditions to obtain the maximum light intensity corresponding to each number of Airy beams.

7. The system according to claim 6, characterized in that The relationship curve of the maximum light intensity versus the number of Airy beams follows the rule that the maximum light intensity increases with the increase in the number of Airy beams until the number of Airy beams increases to the optimal number of Airy beams, at which point the maximum light intensity reaches its maximum value. After that, the number of Airy beams gradually increases from the optimal number of Airy beams, but the maximum light intensity remains at its maximum value and no longer changes, and the self-focusing ability of the annular Airy beam array reaches saturation.

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