Method for constructing self-focusing enhanced circular symmetry Airy beam in frequency space

Abstract

The invention discloses a method for constructing a self-focusing enhanced circular symmetry Airy-like beam in a frequency space, which can keep the non-diffraction property of the beam but obviously improve the self-focusing performance, and comprises the following steps of: (1) selecting a wavelength and a beam initial surface main ring radius control factor, a scale factor and an attenuation factor; (2) calculating a complex amplitude matrix of a light beam in a spectral range corresponding to an SLM pixel matrix according to the focal length of an experimental lens, the size of a spatial light modulator (SLM) and a frequency space light beam model provided by the invention; (3) calculating a normalized amplitude matrix and a phase matrix of the matrix obtained in the step (2) (the phase value is reserved between 0 and 2pi); and (4) a corrected amplitude matrix of the normalized amplitude matrix obtained in the step (3) is calculated through table look-up in the figure 1, and the corrected amplitude matrix is multiplied by the phase matrix to obtain the phase plate. And (5) loading the phase plate obtained in the step (4) onto the SLM, irradiating by using the expanded laser beam, and shielding 0-level diffraction on the rear focal plane of the lens by using a high-pass filter to obtain the self-focusing enhanced circular symmetry Airy-like beam.

Description

Technical Field

[0001] The present invention relates to the field of optical technology, and particularly relates to a construction mode and an experimental method for generating a circular Airy-like beam with strong self-focusing performance. Background Art

[0002] Airy beams have the characteristics of non-diffraction and self-acceleration, but they have infinite energy. Experimentally, by setting the attenuation coefficient, a finite-energy Airy beam can be obtained, and at the same time, the characteristics of non-diffraction and self-acceleration can be retained within a certain range. By performing a circular symmetry operation on the expression of a one-dimensional finite-energy Airy beam in the spatial domain, researchers proposed the circular Airy beam. This kind of beam has the characteristic of self-focusing, that is, during the initial transmission process, its maximum light intensity remains at a relatively low level. When approaching the self-focusing plane, its light intensity can increase by two orders of magnitude within a very short distance. Therefore, this beam has potential application value in many fields such as biomedical treatment, particle manipulation, laser ignition, and laser processing. The so-called self-focusing performance refers to the ratio of the maximum light intensity of the beam near the self-focusing plane to the maximum light intensity of the initial plane.

[0003] While conducting application exploration, researchers have also carried out a series of studies on improving the self-focusing performance. Nikolaos K. Efremidis first proposed the concept of suddenly self-focusing beams in 2010. The author analyzed the self-focusing performance of the circularly symmetric Airy beam u0(r) = Ai(r0 - r)exp(a(r0 - r)). When the attenuation coefficient a is 0.05, by optimizing the control parameter r0 of the main ring radius of the initial plane, the maximum achievable light intensity contrast is 156. The author also tried to coherently superpose circular Airy beams to form a new circularly symmetric self-focusing beam. After optimization, the self-focusing light intensity contrast can reach about 220. Generally speaking, reducing the attenuation factor a and appropriately increasing the main ring size factor r0 of the beam in the initial plane can improve the self-focusing performance.

[0004] In optimizing the self-focusing performance, in addition to parameter optimization and superposition method optimization, researchers have also made many attempts. In 2011, Ioannis Chremmos and Nikolaos K. Efremidis collaborated and proposed a new construction method for a class of self-focusing beams, that is: u0(r) = exp[a(r0 - r)]sin[C(r - r0) β , r ≥ r0; u0(r) = 0, r < r0. The author pointed out that this construction method can greatly improve the self-focusing performance, but it will reduce the non-diffraction characteristic of the beam. From this literature Figure 2 it can be seen that when the parameters are selected as C = π, β = 1.5, r0 = 4, a = 0.2, the maximum light intensity contrast of the beam can reach 300, but the diffraction effect near the self-focusing plane is relatively significant.

[0005] In 2014, Jiang Yunfeng et al. improved the self-focusing performance to around 200 by blocking the first few bright rings of the initial circularly symmetric Airy beam. In 2015, they high-pass filtered the spectrum of the circularly symmetric Airy beam, removing some low-frequency components, and improved the beam's self-focusing performance to around 250. In 2020, GENG TAO et al. applied a Gaussian modulation to the circular Airy beam spectrum, achieving an enhanced self-focusing contrast. With proper parameter selection, the theoretical self-focusing contrast can reach 694. However, the initial beam expansion is very severe, which also reduces the non-diffraction characteristics and is unfavorable for experimental generation.

[0006] Existing methods for optimizing the self-focusing performance of circular Airy beams are generally carried out by blocking the main ring of the initial surface, filtering the spectrum surface, and modulating the radial chirp of the initial surface. Some of these methods can achieve a large self-focusing contrast but will reduce the non-diffraction characteristics, while others will cause the initial surface spot to expand, which is detrimental to the experiment. Summary of the Invention

[0007] In order to improve the self-focusing performance of traditional circular Airy beams as much as possible (i.e., the ratio of the maximum light intensity value during transmission to the maximum light intensity value on the initial surface; for ease of description, this patent uniformly uses "self-focusing contrast" to refer to it) without significantly expanding the initial surface beam profile, changing the beam's approximately non-diffraction properties, and affecting experimental generation and application, the present invention proposes a method for constructing a circularly symmetric Airy-like beam with enhanced self-focusing in frequency space.

[0008] The present invention solves the above problems through the following technical measures:

[0009] A method for constructing a self-focusing enhanced circularly symmetric Airy-like beam in frequency space comprises the following steps:

[0010] S1. Select the working wavelength λ, and reasonably set the beam radius control factor r′0, scale factor x0, and attenuation factor a on the initial surface;

[0011] S2. Based on the focal length of the lens used in the experiment and the size of the SLM, the frequency-space complex amplitude distribution matrix of the light beam is calculated on a computer using the frequency-space beam construction model proposed in the present invention and the frequency spectrum range corresponding to the size of the SLM pixel matrix;

[0012] S3, calculate the normalized amplitude matrix and phase matrix of the complex amplitude matrix in S2 (the phase value is retained between 0 and 2π through modulo operation);

[0013] S4. Use Figure 1 The curve lookup table method shown calculates the modified amplitude matrix of the normalized amplitude matrix in S3;

[0014] S5, multiplying the corrected amplitude matrix in S4 by the phase matrix in S3 to obtain the phase plate matrix used to generate the beam;

[0015] S6. Load the phase plate matrix in S5 onto the SLM and illuminate it with an expanded laser beam. Use a high-pass filter to block the 0th order diffraction at the focal plane (initial plane) of the Fourier lens, thus obtaining a self-focusing enhanced circularly symmetric Airy-like beam.

[0016] S7. By allowing the light beam to propagate freely, it can exhibit enhanced self-focusing effect and near-diffraction-free characteristics.

[0017] Furthermore, the spectrum expression of the self-focusing enhanced circularly symmetric Airy-like beam constructed in frequency space is: k is the distance from any point in the spectrum space to the origin, x0 is the beam scaling factor, which has the dimension of length, and a is the beam attenuation factor, which is dimensionless. 2 <<1, r′0 is the control factor of the main ring radius of the beam on the initial plane, which has the dimension of length and is used to control the main ring radius R0 of the beam on the initial plane in real space.

[0018] Furthermore, in the spectrum expression of the self-focusing enhanced circularly symmetric Airy-like beam constructed in frequency space, the sign before the imaginary number in the exponential term exp(ir′0k) is positive, and the parameter r′0 is a value greater than 0, so that the beam generated on the initial surface in real space has a ring-shaped hollow structure similar to that of the circular Airy beam, and the maximum light intensity ring radius R0 of the initial surface beam can be controlled by the parameter r′0. Within a certain range, R0≈r′0+2x0 is numerically calculated, and when the scale factor x0<<r′0, R0≈r′0.

[0019] Furthermore, the distance from the focusing position to the initial plane of the beam, i.e. the focal length, can be controlled by the parameters r′0 and x0. λ is the wavelength of the laser in vacuum.

[0020] Furthermore, increasing the parameters r′0 and can expand the radius R0 of the initial beam's maximum intensity ring, while also increasing the autofocus distance. Increasing the parameter r′0 within a certain range can also increase the autofocus contrast. Reducing the beam attenuation factor a within a certain range can also increase the autofocus contrast. Excessively small attenuation factors can cause the beam spectrum to extend beyond the range of the spatial light modulator, thus exceeding the experimental conditions. In practice, a is typically greater than or equal to 0.05. Excessively small scale factors x0 can also cause the spectrum to extend beyond the range of the spatial light modulator, thus exceeding the experimental conditions.

[0021] The beneficial effects of the present invention include at least:

[0022] The present invention adopts a new spectral analytical expression of a circular Airy-like beam. The initial surface beam size can be flexibly adjusted by the parameter r′0. The self-focusing distance of the beam can be flexibly controlled in combination with the scale factor x0. At the same time, the self-focusing contrast can be changed by the parameter r′0 and the attenuation factor a. Without significantly expanding the initial surface beam profile, without changing the approximately non-diffraction property of the beam, and without affecting the experimental generation and application, a higher self-focusing contrast can be obtained than that of a circular Airy beam with the same parameters. BRIEF DESCRIPTION OF THE DRAWINGS

[0023] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the following briefly introduces the drawings required for describing the embodiments.

[0024] Figure 1 It is the lookup curve when calculating the corrected amplitude in the present invention.

[0025] Figure 2 Figure 1 shows matrix visualizations of the spatial light modulator phase plate calculation process in Example 1 of the present invention. Figure (a) shows the normalized amplitude matrix visualization distribution; Figure (b) shows the modified normalized amplitude matrix visualization distribution; Figure (c) shows the phase matrix visualization distribution; and Figure (d) shows the final spatial light modulator phase plate matrix visualization distribution.

[0026] Figure 3 This is a simulation diagram comparing the intensity distribution of the self-focusing enhanced circularly symmetric Airy beam produced by Example 1 of the present invention and the classical circular Airy beam with the same parameters. Figures (a1), (b1), (c1), and (d1) correspond to the self-focusing enhanced circularly symmetric Airy beam produced by Example 1, and Figures (a2), (b2), (c2), and (d2) correspond to the classical circular Airy beam with the same initial surface main ring radius control parameters, scale factor, and attenuation factor as the self-focusing enhanced circularly symmetric Airy beam produced by Example 1. Figures (a1) and (a2) are the intensity diagrams of the beam within the optical axis section; Figures (b1) and (b2) are the initial plane intensity distribution diagrams of the beam, and the superimposed curves represent the radial intensity distribution profile; Figures (c1) and (c2) are the intensity distribution diagrams of the beam's self-focusing plane, that is, the maximum intensity plane, and the superimposed curves represent the radial intensity distribution profile; Figures (d1) and (d2) are the ratios of the maximum light intensity during transmission to the initial surface maximum light intensity as a function of the transmission distance z.

[0027] Figure 4 Figure 2 shows the matrix diagrams used in calculating the spatial light modulator phase plate in Example 2 of the present invention. Figure (a) shows the visualization of the normalized amplitude matrix; Figure (b) shows the visualization of the modified normalized amplitude matrix; Figure (c) shows the phase matrix; and Figure (d) shows the final spatial light modulator phase plate.

[0028] Figure 5This is a simulation diagram comparing the intensity distribution of the self-focusing enhanced circularly symmetric Airy beam produced by Example 2 of the present invention and the classical circular Airy beam with the same parameters. Figures (a1), (b1), (c1), and (d1) correspond to the self-focusing enhanced circularly symmetric Airy beam produced by Example 2, and Figures (a2), (b2), (c2), and (d2) correspond to the classical circular Airy beam with the same initial surface main ring radius control parameters, scale factor, and attenuation factor as the self-focusing enhanced circularly symmetric Airy beam produced by Example 2. Figures (a1) and (a2) are the intensity diagrams of the beam within the optical axis section; Figures (b1) and (b2) are the initial plane intensity distribution diagrams of the beam, and the superimposed curves represent the radial intensity distribution profile; Figures (c1) and (c2) are the intensity distribution diagrams of the beam's self-focusing plane, that is, the maximum intensity plane, and the superimposed curves represent the radial intensity distribution profile; Figures (d1) and (d2) are the ratios of the maximum light intensity during transmission to the initial surface maximum light intensity as a function of the transmission distance z.

[0029] Figure 6 This is a flow chart of the method for constructing a self-focusing enhanced circularly symmetric Airy-like beam in frequency space according to the present invention. DETAILED DESCRIPTION

[0030] The present invention is further described below with reference to the accompanying drawings. In particular, for those skilled in the art, without inventive effort, by changing the relevant parameters, it is possible to obtain a light beam with a higher self-focusing contrast than that of the embodiment. Without departing from the spatial expression of the light beam spectrum proposed by the present invention, the light beam self-focusing contrast can also be further improved by methods other than but not limited to superimposing lens phase or radial secondary chirp phase, superimposing positive and negative vortices, etc. Therefore, the scope of protection of the patent of the present invention shall be based on the attached claims, and the scope of protection of the present invention shall not be limited by the embodiments.

[0031] It should be noted that the spectrum expression of the self-focusing enhanced circularly symmetric Airy-like beam described in the present invention is: k is the distance from any point in the spectrum space to the origin, x0 is the beam scaling factor, which has the dimension of length, and a is the beam attenuation factor, which is dimensionless. 2 << 1, r′0 is the control factor for the primary ring radius of the beam on the initial plane. It has a length dimension and is used to control the primary ring radius R0 of the beam on the initial plane in real space. By changing the parameters x0, a, and r′0, the beam size, autofocus distance, and autofocus contrast can be changed.

[0032] Please refer to Figure 1 , Figure 1This is the lookup table curve used in calculating the corrected amplitude of the present invention. The steps for calculating the corrected amplitude for a normalized amplitude value are as follows: S1. Find a point with the same ordinate value on the curve; S2. Find the abscissa value of the point, which is the corrected amplitude value.

[0033] In all the following embodiments, the spatial light modulator (SLM) is a reflective, pure phase type with a pixel size of 12.5um*12.5um, an effective SLM pixel matrix size of 1272*1024, and a Fourier lens focal length f=0.25m. As the contrast beam for achieving self-focus contrast enhancement in the present invention, the circular Airy beam is expressed in real space as: E(r)=Ai[(r0-r) / x0]exp[a(r0-r) / x0], where x0 is the scale factor and a is the attenuation factor. The scale factor x0 and the attenuation factor a are exactly the same as those in the frequency-space beam construction model proposed in the present invention. r0 is the control factor for the radius of the main ring of the circular Airy beam on the initial surface. The main ring radius R0≈r0+x0. When x0<<r0, R0≈r0. Therefore, the r0 factor of the circular Airy beam and the r′0 factor of the frequency-space beam construction model proposed in the present invention have no essential difference and can be considered to have the same meaning. Therefore, no distinction is made in the following description of the embodiments.

[0034] It should be noted that in the following embodiments of the present invention, when compared with circular Airy light with the same parameters, the scale factor x0, the attenuation factor a, and the initial surface main ring radius control factor r0 have the same values.

[0035] Example 1

[0036] Example 1 and Figure 2 、 Figure 3 Match.

[0037] S1. Select the working wavelength λ = 632.8nm, set the beam radius control factor r0 = 1mm, scale factor x0 = 100um, and attenuation factor a = 0.08 at the initial surface;

[0038] S2. Based on the focal length of the lens used in the experiment, f = 0.25m, and the SLM dimensions of 12.5um*12.5um, 1272*1024, the frequency-space beam construction model proposed in the present invention is substituted into the parameters in S1 on a computer to calculate the frequency-space complex amplitude distribution matrix of the beam;

[0039] S3. Calculate the normalized amplitude matrix and phase matrix of the complex amplitude matrix in S2 (the phase value is retained between 0 and 2π). The normalized amplitude matrix is ​​as follows: Figure 2 As shown in (a), the phase matrix is Figure 2 (c);

[0040] S4. Use Figure 1The curve lookup table method shown in FIG. 4 is used to calculate the modified amplitude matrix of the normalized amplitude matrix in S3, as shown in FIG. Figure 2 (b)

[0041] S5. Multiply the modified amplitude matrix in S4 by the phase matrix in S3 to obtain the phase plate matrix for generating the self-focusing enhanced circularly symmetric Airy-like beam, such as Figure 2 (d);

[0042] S6. Load the phase plate matrix in S5 onto the SLM and illuminate it with an expanded laser beam. Use a high-pass filter to block the 0th order diffraction at the focal plane (initial plane) of the Fourier lens, thus obtaining a self-focusing enhanced circularly symmetric Airy-like beam.

[0043] S7, allowing the light beam to propagate freely, can show enhanced self-focusing effect and near-diffraction-free characteristics, such as Figure 3 shown.

[0044] Figure 3 This is a simulation diagram comparing the intensity distribution of the self-focusing enhanced circularly symmetric Airy beam generated by Example 1 of the present invention and the circular Airy beam with the same parameters. Under the parameter setting conditions of Example 1, the self-focusing enhanced circularly symmetric Airy beam generated by Example 1 has a maximum light intensity ring radius of 1.2 mm on the initial surface, a self-focusing focal length of 0.6553 m, a self-focusing contrast of 489.5, and a self-focusing plane light intensity distribution profile full width at half maximum (FWHM) of about 70 um. For the circular Airy beam with the same parameters, the maximum light intensity ring radius is 1.1 mm, the self-focusing focal length is 0.6572 m, the self-focusing contrast is 73.8, and the self-focusing plane light intensity distribution profile FWHM is about 70 um. Analysis shows that the self-focusing contrast enhancement of the beam produced by the present invention is 6.6 times that of the circular Airy beam with the same parameters.

[0045] Example 2

[0046] Example 2 and Figure 4 、 Figure 5 Match.

[0047] S1. Select the working wavelength λ = 632.8nm, set the beam radius control factor r0 = 1mm, scale factor x0 = 50um, and attenuation factor a = 0.07 at the initial surface;

[0048] S2. Based on the focal length of the lens used in the experiment, f = 0.25m, and the SLM dimensions of 12.5um*12.5um, 1272*1024, the frequency-space beam construction model proposed in the present invention is substituted into the parameters in S1 on a computer to calculate the frequency-space complex amplitude distribution matrix of the beam;

[0049] S3. Calculate the normalized amplitude matrix and phase matrix of the complex amplitude matrix in S2 (the phase value is retained between 0 and 2π). The normalized amplitude matrix is ​​as follows: Figure 4 As shown in (a), the phase matrix is Figure 4 (c);

[0050] S4. Use Figure 1 The curve lookup table method shown in FIG. 4 is used to calculate the modified amplitude matrix of the normalized amplitude matrix in S3, as shown in FIG. Figure 4 (b)

[0051] S5. Multiply the corrected amplitude matrix in S4 by the phase matrix in S3 to obtain the phase plate matrix used to generate the beam, such as Figure 4 (d);

[0052] S6. Load the phase plate matrix in S5 onto the SLM and illuminate it with an expanded laser beam. Use a high-pass filter to block the 0th order diffraction at the focal plane (initial plane) of the Fourier lens, thus obtaining a self-focusing enhanced circularly symmetric Airy-like beam.

[0053] S7, allowing the light beam to propagate freely, can show enhanced self-focusing effect and near-diffraction-free characteristics, such as Figure 5 shown.

[0054] Figure 5 This is a simulation diagram comparing the intensity distribution of the self-focusing enhanced circularly symmetric Airy-like beam generated by Example 2 of the present invention and the circular Airy beam with the same parameters. Under the parameter setting conditions of Example 2, the maximum light intensity ring radius of the beam generated by Example 2 on the initial surface is 1.1mm, the self-focusing focal length is 0.2273m, the self-focusing contrast is 586, and the self-focusing plane light intensity distribution profile FWHM is about 30um. For the circular Airy beam with the same parameters, the maximum light intensity ring radius is 1.05mm, the self-focusing focal length is 0.2274m, the self-focusing contrast is 68.8, and the self-focusing plane light intensity distribution profile FWHM is about 30um. Analysis shows that the self-focusing contrast enhancement of the light speed produced by the present invention is 8.5 times that of the circular Airy beam with the same parameters.

[0055] The generation methods of Example 1 and Example 2 are similar, so they can be used Figure 6 The flowchart shown summarizes the implementation process of the method for constructing circularly symmetric Airy-like beams in frequency space of the present invention.

[0056] In summary, the present invention provides a method for generating a self-focusing beam in frequency space with significantly enhanced self-focusing performance compared to a circular Airy beam with the same parameters, improving the self-focusing performance by approximately an order of magnitude. This patent provides an analytical expression for the frequency space of the beam, as well as a process for fabricating a spatial light modulator phase plate. The intensity distribution during free propagation is also compared and analyzed with that of a circular Airy beam with the same parameters. The present invention utilizes a single pure phase spatial light modulator and a single Fourier lens to generate the beam, which requires minimal experimental instrumentation. By setting parameters, the phase plate can be easily calculated, allowing for convenient control of beam characteristics such as the self-focusing contrast. While the descriptions of Examples 1 and 2 above are relatively specific and detailed, they should not be construed as limiting the scope of protection of the present invention. It should be noted that, without departing from the spatial expression of the beam spectrum proposed by the present invention, methods other than superimposing lens phases, or radial quadratic chirping phases, or superimposing positive and negative vortices, can also be employed to further enhance the self-focusing performance of the beam, without departing from the spatial expression of the beam spectrum proposed by the present invention. Therefore, the scope of protection of the present invention shall be governed by the appended claims.

Claims

1. A method for constructing a self-focusing enhanced circularly symmetric Airy-like beam in frequency space, characterized in that: The steps include: S1. Select the working wavelength λ, and reasonably set the beam radius control factor r′0, scale factor x0, and attenuation factor a on the initial surface; S2. Based on the focal length of the lens used in the experiment and the size of the spatial light modulator (SLM), the frequency-space complex amplitude distribution matrix of the light beam is calculated on a computer according to the frequency-space beam construction model proposed in the present invention and the frequency spectrum range corresponding to the size of the SLM pixel matrix; S3, calculate the normalized amplitude matrix and phase matrix of the complex amplitude matrix in S2 (the phase value is retained between 0 and 2π); S4, using the numerical curve lookup table method of FIG1 to calculate the modified amplitude matrix of the normalized amplitude matrix in S3; S5, multiplying the corrected amplitude matrix in S4 by the phase matrix in S3 to obtain the phase plate matrix used to generate the beam; S6. Load the phase plate matrix in S5 onto the SLM and illuminate it with an expanded laser beam. Use a high-pass filter to block the 0th order diffraction at the focal plane (initial plane) of the Fourier lens to obtain a self-focusing enhanced circularly symmetric Airy-like beam. S7. By allowing the light beam to propagate freely, it can exhibit enhanced self-focusing effect and near-diffraction-free characteristics.

2. A method for constructing a self-focusing enhanced circularly symmetric Airy-like beam in frequency space according to claim 1, characterized in that The method for constructing a circularly symmetric Airy-like beam in frequency space proposed in the present invention, that is, the spectrum expression of the beam is: k is the distance from any point in the spectrum space to the origin, x0 is the beam scaling factor, which has the dimension of length, and a is the beam attenuation factor, which is dimensionless. 2 <<1, r′0 is the radius control factor of the main ring of the light beam on the initial plane (the ring where the maximum light intensity is located), which has the dimension of length and is used to control the radius R0 of the main ring of the light beam on the initial plane in real space.

3. A method for constructing a self-focusing enhanced circularly symmetric Airy-like beam in frequency space according to claim 2, characterized in that: The sign before the imaginary number in the exponential term exp(ir′0k) is positive, and the parameter r′0 has the dimension of length and is a number greater than 0. As a result, the beam generated on the initial surface in real space has a hollow ring structure similar to a circular Airy beam. The parameter r′0 can also be used to control the radius R0 of the ring of maximum intensity of the beam on the initial surface. A larger parameter r′0 increases the radius R0 of the ring of maximum intensity of the beam generated on the initial surface in real space. Numerical calculations show that R0 ≈ r′0 + 2x0 within a certain range. When the scale factor x0 << r′0, R0 ≈ r′0.

4. According to claim 2, the distance from the focusing position to the initial plane of the light beam, i.e., the focal length, can be controlled by the parameters r′0 and x0. λ is the wavelength of the laser in vacuum.

5. According to claim 2, the full width at half maximum of the self-focusing plane spot intensity distribution can be controlled by the parameter x0, and the full width at half maximum of the self-focusing plane spot intensity distribution is approximately equal to 2x0 / 3.

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