Circular butterfly light beam period self-focusing transmission method

Abstract

The invention discloses a circular butterfly light beam period self-focusing transmission method. The method is provided based on the fractional Schrodinger equation with the parabolic waveguide, so that the transmission of the circular butterfly beam is prolonged. And the parabolic potential well waveguide is introduced, so that parameters such as periodic recovery and vortex position of the circular butterfly light beam can be realized, and the application of the light beam in related fields such as particle capture and microscopic imaging is further expanded. According to the method provided by the invention, periodic self-focusing transmission of the circular butterfly light beam can be realized, parameters such as parabolic waveguide potential well depth and circular butterfly light beam waist width factor can be changed, so that the focusing position, the focusing period and the intensity at the focusing point of the circular butterfly light beam can be flexibly controlled, and the control on the circular butterfly light beam can be realized.

Description

Technical Field

[0001] The invention belongs to the field of optical technology, and in particular relates to a method for periodic self-focusing transmission of a circular butterfly light beam. Background Art

[0002] In May 1960, the world's first ruby ​​solid-state laser was produced, allowing people to obtain lasers in experiments for the first time. The invention of the laser has had a significant impact on nonlinear optics. Lasers can be described by Gaussian functions, that is, Gaussian beams. The structural invariance of Gaussian beams during propagation has promoted people's continued research on the properties of lasers, and laser technology has ushered in rapid development. With the application and development of laser technology in hospitals, agriculture, communications and other fields, the single propagation property of Gaussian beams has gradually become a limitation to the development of laser technology. People have begun to devote themselves to researching and developing other properties of lasers, and some new beams with unique propagation properties have been developed.

[0003] In 2010, Efremidis et al. first proposed a sudden self-focusing beam, which can be generated by radially symmetric Airy beams or by properly superimposing Airy beams. The self-focusing beam can keep the intensity almost unchanged during propagation, while the intensity suddenly increases by several orders of magnitude near the focus. The unique focusing characteristics of the self-focusing beam enable it to concentrate energy and form a self-focusing light field during propagation without any external conditions. In 2018, Chen et al. developed another self-focusing beam, the circular Pierce beam, based on the Pearcey function. Due to the particularity of the function, the circular Pierce beam has stronger self-focusing performance than the circular Airy beam. Higher-order sudden beams, such as butterfly beams, have more control parameters and stronger focusing capabilities. In 2023, Deng Dongmei et al. discussed the virtual source problem of butterfly beams and theoretically realized how to generate butterfly beams with virtual sources. The structure of the circular butterfly beam shows flexible adjustability and can be used to generate various arrays of light field structures. The focusing performance can be further improved by introducing vortices. These results show that for high-dimensional self-focusing beams, more adjustment parameters can be used to control the focusing intensity and self-focusing behavior. Compared with circular Airy beams and circular Pierce beams, circular butterfly beams have stronger focusing capabilities.

[0004] In addition, research results in recent years have shown that the introduction of an optical potential well can change the propagation trajectory of a light beam, providing a new method for the manipulation of light beams. Recently, based on the fractional Schrödinger equation, Tan Chao et al. discussed the trajectory-controlled transmission of circular Airy beams under three potential wells, and proved that the transmission trajectory of the beam can be controlled by adjusting the Lévy index and various parameters. In 2023, He Shangling et al. realized the periodic evolution of a chirped symmetric Pierce-Gaussian vortex beam by adjusting the Lévy index based on the fractional Schrödinger equation. Summary of the invention

[0005] Based on the fractional Schrödinger equation with parabolic waveguide, the present invention proposes a method for realizing periodic self-focusing of circular butterfly beam, thereby extending the transmission of circular butterfly beam. By changing the parameters such as the potential well depth of the parabolic waveguide, the waist width factor of the circular butterfly beam, and the vortex position, the periodic restoration of the circular butterfly beam can be realized, further expanding the application of the beam in related fields such as particle capture and microscopic imaging.

[0006] In order to solve the technical problem that a circular butterfly light beam cannot periodically self-focus during the transmission process, the present invention provides a method for periodically self-focusing transmission of a circular butterfly light beam.

[0007] The technical solution adopted by the present invention is:

[0008] A method for periodic self-focusing transmission of a circular butterfly beam comprises the following steps:

[0009] S1, input index-truncated circular butterfly beam, use radially symmetric butterfly beam with spatial light modulator to generate circular butterfly beam with limited initial energy and no vortex Select the circular butterfly beam waist width factor w0 = 0.2 and inject it into the optical waveguide for transmission;

[0010] S2, under the conditions of the above initial beam, according to model (1) and selecting the parabolic waveguide potential well parameter p = 0, the dynamic evolution characteristics of the circular butterfly beam can be obtained; the intensity of the circular butterfly beam increases when it is transmitted to z = 2.4L, and as the transmission distance continues to increase, the intensity decreases and gradually dissipates to zero;

[0011] S3, according to model (1) and selecting the parabolic optical waveguide potential well parameter p = 1; under the action of the parabolic potential well, the circular butterfly beam is periodically focused during the transmission process; in each focusing period, the beam shape remains unchanged and the period is restored, thereby extending the transmission distance of the beam;

[0012] S4, change the optical waveguide parameters, increase the parabolic optical waveguide potential well parameters; select L = 10, parabolic optical waveguide potential well parameters p = 1, 1.5, 2, the focusing period of the circular butterfly beam increases during transmission, the focusing intensity is enhanced, the focusing period varies in the range of 1.6L-3.1L, and the normalized focusing intensity varies in the range of 37-78;

[0013] S5, change the parameters of the initial circular butterfly beam, increase the waist width factor w0 of the circular butterfly beam; select the circular butterfly beam width factor w0 = 0.2, 0.3, 0.5, the focusing intensity of the circular butterfly beam increases during the transmission process, the period remains unchanged, and the position of the focal point changes; the normalized focusing intensity changes in the range of 37-146, and the position of the focal point changes in the range of 0.6L-1.2L;

[0014] S6, adjust the initial phase, introduce vortex into the circular butterfly beam, and generate a circular butterfly beam containing vortex with the help of spatial light modulator Select the topological charge number of the vortex m = 1 and change the position of the vortex r s ; Select the vortex position vortex r s =1, 2, 3, the change of the vortex position will not affect the focusing period and focusing position of the circular butterfly beam during the transmission process, and can improve the intensity of periodic focusing; the normalized focusing intensity varies in the range of 151-485.

[0015] Compared with the prior art, the present invention has the following beneficial effects:

[0016] The present invention uses a parabolic optical waveguide to propose a method for realizing periodic self-focusing transmission of a circular butterfly beam. The method proposed by the present invention requires an external potential well to realize periodic self-focusing transmission of a circular butterfly beam. By changing the parameters such as the potential well depth of the parabolic waveguide, the waist width factor of the circular butterfly beam, and the vortex position, the focusing position, focusing period, and intensity at the focusing point of the circular butterfly beam can be flexibly controlled to realize the manipulation of the circular butterfly beam. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] Figure 1 is the transmission characteristic diagram of circular butterfly beam without parabolic waveguide;

[0018] Figure 2 It is the transmission characteristic diagram of circular butterfly beam introduced into parabolic waveguide;

[0019] Figure 3 is the transmission characteristic diagram of circular butterfly beam under different potential well depths;

[0020] Figure 4 is the transmission characteristic diagram of circular butterfly beam under different waist width factors;

[0021] Figure 5 It is the transmission characteristic diagram of circular butterfly beam under different vortex positions. DETAILED DESCRIPTION

[0022] The present invention will be described in further detail below with reference to the accompanying drawings and embodiments.

[0023] A method for periodic self-focusing transmission of a circular butterfly beam comprises the following steps:

[0024] S1, input index-truncated circular butterfly beam, use radially symmetric butterfly beam with spatial light modulator to generate circular butterfly beam with limited initial energy and no vortex Select the circular butterfly beam waist width factor w0 = 0.2 and inject it into the optical waveguide for transmission;

[0025] S2, under the conditions of the above initial beam, according to model (1) and selecting the parabolic waveguide potential well parameter p = 0, the dynamic evolution characteristics of the circular butterfly beam can be obtained; the intensity of the circular butterfly beam increases when it is transmitted to z = 2.4L, and as the transmission distance continues to increase, the intensity decreases and gradually dissipates to zero;

[0026] S3, according to model (1) and selecting the parabolic optical waveguide potential well parameter p=1; under the action of the parabolic potential well, the circular butterfly beam is periodically focused during the transmission process; in each focusing period, the beam shape remains unchanged and the period is restored, thereby extending the transmission distance of the beam;

[0027] S4, change the optical waveguide parameters, increase the parabolic optical waveguide potential well parameters; select L = 10, parabolic optical waveguide potential well parameters p = 1, 1.5, 2, the focusing period of the circular butterfly beam increases during transmission, the focusing intensity is enhanced, the focusing period varies in the range of 1.6L-3.1L, and the normalized focusing intensity varies in the range of 37-78;

[0028] S5, change the parameters of the initial circular butterfly beam, increase the waist width factor w0 of the circular butterfly beam; select the circular butterfly beam width factor w0 = 0.2, 0.3, 0.5, the focusing intensity of the circular butterfly beam increases during the transmission process, the period remains unchanged, and the position of the focal point changes; the normalized focusing intensity changes in the range of 37-146, and the position of the focal point changes in the range of 0.6L-1.2L;

[0029] S6, adjust the initial phase, introduce vortex into the circular butterfly beam, and generate a circular butterfly beam containing vortex with the help of spatial light modulator Select the topological charge number of the vortex m = 1 and change the position of the vortex r s ; Select the vortex position vortex r s =1, 2, 3, the change of the vortex position will not affect the focusing period and focusing position of the circular butterfly beam during the transmission process, and can improve the intensity of periodic focusing; the normalized focusing intensity varies in the range of 151-485.

[0030] like Figure 1 As shown, when the parabolic waveguide is not introduced, the beam intensity dissipates to zero as the transmission distance increases, and the beam cannot be transmitted over long distances.

[0031] After the parabolic waveguide is introduced, the circular butterfly beam shows a periodic focusing phenomenon along the z direction. The beam is first focused, then dispersed, and finally focused again, forming a periodic behavior. The high-intensity energy of the focus point is concentrated at the center of the axis y=0. Under the action of the parabolic waveguide, the high-intensity self-focusing characteristics can be restored periodically, thereby extending the transmission distance of the beam, such as Figure 2 shown.

[0032] Figure 3 It can be seen that as the depth p of the parabolic waveguide potential well increases, the intensity of the beam increases, forming a high-intensity beam. At the same time, the focusing period becomes shorter and the focal position also changes. In particular, after the circular butterfly beam is focused, a second higher focusing intensity point appears in one cycle, and the distance between the two focusing points is adjustable, that is, the focal length can be changed. Therefore, the increase in the depth of the parabolic waveguide potential well can not only increase the intensity at the focal point, but also increase the focusing period, shorten the focal length and change the focal position.

[0033] Figure 4 The relationship between the focus intensity of the circular butterfly beam and the transmission distance under different waist width factors is given. When the Lévy index is 1.5, the focus intensity decreases oscillatingly with the increase of the transmission distance. When the Lévy index increases to 2, the oscillation disappears and the focus intensity can be reproduced periodically. The increase of the waist width factor w0 enhances the focus intensity, the focus period remains unchanged, but the focus position changes.

[0034] Figure 5 The relationship between the focal intensity of the circular butterfly beam and the transmission distance at different vortex positions is given. Figure 5 It can be seen that the change of vortex position does not affect the focusing period and focusing position, but the peak intensity at the focus point can be increased by several hundred orders of magnitude. This significant enhancement of focusing ability helps to apply it to microscopic imaging, laser drilling, optical manipulation and other fields.

[0035] In summary, the present invention proposes a method for periodic self-focusing transmission of a circular butterfly beam. Based on the fractional Schrödinger equation, the dynamic evolution characteristics of the circular butterfly beam are studied using the split-step Fourier method, and the influence of various parameters on the transmission characteristics of the circular butterfly beam is discussed. The research results show that the butterfly beam exhibits high-intensity periodic focusing characteristics. The focusing intensity, period and focusing position are all controllable and have flexible adjustability. The increase in the Lévy index reduces the focusing intensity and improves the focusing stability. When the Lévy index increases to 2, the focusing intensity can be reproduced periodically. Multiple focusing points appearing in the transmission direction can periodically capture multiple particles. The increase in the depth p of the parabolic waveguide increases the focusing intensity, increases the focusing period and shortens the focusing length. At the same time, increasing the waist width factor will increase the focusing intensity, the focusing period remains unchanged, but the focusing position changes. After adding a vortex, the focusing performance can be further improved. The change in the vortex position will not affect the focusing period and position, but the peak intensity at the focusing point will increase by hundreds of orders of magnitude.

[0036] It is to be understood that the above embodiments are merely exemplary embodiments used to illustrate the principles of the present invention, but the present invention is not limited thereto. For those of ordinary skill in the art, various modifications and improvements can be made without departing from the spirit and essence of the present invention, and these modifications and improvements are also considered to be within the scope of protection of the present invention.

Claims

1. A method for periodic self-focusing transmission of a circular butterfly beam, characterized in that it comprises the following steps: S1, input index-truncated circular butterfly beam, use radially symmetric butterfly beam with spatial light modulator to generate circular butterfly beam with limited initial energy and no vortex Select the circular butterfly beam waist width factor w0 = 0.2 and inject it into the optical waveguide for transmission; S2, under the conditions of the above initial beam, according to model (1) and selecting the parabolic waveguide potential well parameter p = 0, the dynamic evolution characteristics of the circular butterfly beam can be obtained; the intensity of the circular butterfly beam increases when it is transmitted to z = 2.4L, and as the transmission distance continues to increase, the intensity decreases and gradually dissipates to zero; S3, according to model (1) and selecting the parabolic optical waveguide potential well parameter p = 1; under the action of the parabolic potential well, the circular butterfly beam is periodically focused during the transmission process; in each focusing period, the beam shape remains unchanged and the period is restored, thereby extending the transmission distance of the beam; S4, change the optical waveguide parameters, increase the parabolic optical waveguide potential well parameters; select L = 10, parabolic optical waveguide potential well parameters p = 1, 1.5, 2, the focusing period of the circular butterfly beam increases during transmission, the focusing intensity is enhanced, the focusing period varies in the range of 1.6L-3.1L, and the normalized focusing intensity varies in the range of 37-78; S5, change the parameters of the initial circular butterfly beam, increase the waist width factor w0 of the circular butterfly beam; select the circular butterfly beam width factor w0 = 0.2, 0.3, 0.5, the focusing intensity of the circular butterfly beam increases during the transmission process, the period remains unchanged, and the position of the focal point changes; the normalized focusing intensity changes in the range of 37-146, and the position of the focal point changes in the range of 0.6L-1.2L; S6, adjust the initial phase, introduce vortex into the circular butterfly beam, and generate a circular butterfly beam containing vortex with the help of spatial light modulator Select the topological charge number of the vortex m = 1 and change the position of the vortex r s ; Select the vortex position vortex r s =1, 2, 3, the change of the vortex position will not affect the focusing period and focusing position of the circular butterfly beam during the transmission process, and can improve the intensity of periodic focusing; the normalized focusing intensity varies in the range of 151-485.

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