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Realization of double uniform line self-focusing of elliptical Airyprime beams

Jian He, Jiahao Chen, Yimin Zhou, Fei Wang, Yangjian Cai, and Guoquan Zhou

Open Access Open Access

Abstract

Double line self-focusing characteristics of elliptical Airyprime beams (EAPBs) with different elliptical vertical-axis factor β are investigated by varying the main ring radius r0. Overly large or small r0 results in the inhomogeneous distribution of light intensity at one linear focus of the double line self-focusing. Only when r0 is appropriate and β is within a certain range, can double uniform line self-focusing happen to the EAPB. Moreover, the self-focusing ability of the second line self-focusing is weaken than that of the first line self-focusing. Under the premise of our selected values of beam parameters, the EAPB can achieve double uniform line self-focusing when r0 = 0.3 mm and β = 0.58∼0.71. The focal length of the first line self-focusing, the lengths of double linear focus, and the self-focusing abilities of the double uniform line self-focusing can be regulated by varying β within the range of 0.58∼0.71. If β is smaller than 0.58 or larger than 0.71, it will lead to nonuniform line self-focusing. An explanation of the physical mechanism behind the double uniform line self-focusing of the EAPB is proposed. Finally, the experimental measurements of the line self-focusing of the EAPB confirm the validity of the above conclusions. This research provides a new solution on how to generate double uniform line self-focusing and new insights into the practical application of elliptical self-focusing beams.

© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

A circular Airy beam (CAB) is a fascinating beam whose greatest attraction is its ability to self-focus [1,2]. Based on the CAB, researchers have investigated the variation of the corresponding self-focusing ability by modifying the CAB [3,4], blocking the initial distribution of the CAB [5], constructing an incompletely coherent CAB [6,7], implanting optical vortices into the CAB [8,9], and introducing chirps into the CAB [10,11]. Self-focusing beams have numerous application scenarios, such as optical trapping [12,13], multiphoton polymerization [14], optical dynamic imaging [15,16], and terahertz emission [17]. The CAB can be evolved into an elliptical Airy beam (EAB), which has the property of double self-focusing [18,19], The EAB has a greater application potential than the single self-focusing CAB. The double self-focusing property of laser beams can be used not only for generating a wider bandwidth supercontinuum [20], but also for medical scenarios that require fine manipulation, such as bifocal radiation therapy [21].

As a deepening of a CAB, a circular Airyprime beam (CAPB) has been introduced [22]. Whether propagating in free space or chiral media, the CAPB exhibits much stronger self-focusing ability than the CAB under the same conditions [22,23]. Any positive second-order chirped factor can improve the self-focusing ability of the CAPB, but the cost is the shortening of its focal length [24]. When the self-focusing ability induced by a linear chirped factor is enhanced, the focal length of the CAPB will vary accordingly based on the exponential attenuation factor and the dimensionless main ring radius [25]. If an exponential attenuation factor is fixed, the self-focusing ability of a CAPB can be maximized by choosing the optimum dimensionless main ring radius [26]. When an optical vortex is introduced into the CAPB, the self-focusing ability decreases significantly with increasing topological charge or radial position of the vortex singularity [27]. When the CAPB carries vortex pairs, the self-focusing ability is enhanced with increasing topological charge or decreasing radial position of the vortex pair [28]. If the CAPB is masked on the starting plane, the outcome is the decrease of self-focusing ability [29]. Both the CAPB and the CAB only undergo self-focusing once, and both are point-focusing. Although the EAB undergoes self-focusing twice, it is also point-focusing. What happens when expanding from the CAPB to the elliptical Airyprime beam (EAPB)? This is what this paper aims to investigate. In this paper, double uniform line self-focusing of the EAPB is realized. The line-focusing with a large area is more meaningful compared to the traditional point-focusing, which can improve the transmission efficiency of the laser communication system [30]. By focusing the energy of the laser beam more intensively, the optical loss in the transmission process can be reduced, and the data transmission rate and the transmission distance can be improved. Therefore, laser beams capable of double line self-focusing have obvious advantages in free space optical communication. The structure of this paper is as follows. Section 2 presents the theoretical description of the EAPB propagating in free space. In Section 3, numerical simulations are performed to find the conditions for the generation of double uniform line self-focusing of EAPBs. The line self-focusing of EAPBs is experimentally measured in Section 4. Finally, the above research is summarized in Section 5.

2. Propagation equation of the EAPB in free space

Referring to the definition of EABs [18], the characterization of EAPBs is as follows. The electric field distribution of the EAPB on the starting plane z = 0 is given by

$$E(x,y,0)\textrm{ = }A\exp \left[ {a\left( {\frac{{{r_0} - \sqrt {{{(x/\alpha )}^2} + {{(y/\beta )}^2}} }}{{{w_0}}}} \right)} \right]Ai^{\prime}\left( {\frac{{{r_0} - \sqrt {{{(x/\alpha )}^2} + {{(y/\beta )}^2}} }}{{{w_0}}}} \right),$$
where x and y are the horizontal and the vertical coordinates in a right-angle coordinate system, respectively; A is a amplitude adjustment coefficient that makes the initial peak light intensity I0p = Max[|E(x, y, 0)|2]= 1 in any case; Max[.] represents taking the maximum value; r0 is the main ring radius; w0 is a radial scale; a is an exponential attenuation factor; α and β are the elliptic transverse and the elliptic vertical factors of the EAPB; and Ai′(.) is an Airyprime function. When the EAPB propagates in free space, its electric field is characterized by the Collins integral [31]:
$$E(x,y,z) = \frac{k}{{2\pi iz}}\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {E(x^{\prime},y^{\prime},0)} \exp \left\{ {\frac{{ik}}{{2z}}[{{(x - x^{\prime})}^2} + {{(y - y^{\prime})}^2}]} \right\}dx^{\prime}} dy^{\prime},$$
where z is the longitudinal coordinate of the right-angle coordinate system; k = 2π/λ is the wave number; λ is the wavelength; and i is the unit of the imaginary number. The analytical solution for Eq. (2) cannot be obtained. Therefore, Eq. (2) is calculated by using numerical methods, and here the numerical simulation of Eq. (2) is carried out by using the fast Fourier transform algorithm [32].

The light intensity of this propagating EAPB reads as

$$I(x,y,z) = {|{E(x,y,z)} |^2}.$$

The self-focusing phenomenon will happen to the EAPB during free space propagation, and its self-focusing ability is characterized by Ifp/I0p. Ifp = Max[|E(x, y, zf)|2] is the peak light intensity on the focal plane. The focal length zf is the longitudinal propagation distance at which self-focusing occurs.

3. Theoretical simulations and analyses

This paper mainly analyzes the characteristics of the double line self-focusing of the EAPB. The initial electric field of the EAPB is governed by five beam parameters: a, r0, w0, α and β. The fixed parameters are chosen as a = 0.1, w0 = 0.1 mm, α = 1, and λ = 532 nm. The emphasis is on studying the effects of r0 and β on the double line self-focusing of the EAPB.

3.1 Propagation characteristics of EAPBs

Figure 1 shows the variation of the intensity contrast Izp/I0p of the propagating EAPBs with different r0 and β with respect to the axial propagation distance z. Izp represents the peak intensity of the EAPB on the observing plane z. The red dashed and the blue dashed lines in the Fig. 1 represent the focal lengths of the first and the second line self-focusing, namely zf1 and zf2, respectively. Due to the fact that double line self-focusing occurs on the x-axis or the y-axis, we define self-focusing with a linear distribution of light intensity along the axis of the focus as line self-focusing, and line self-focusing with approximately equal light intensity at various positions along the line segment as uniform line self-focusing. Contrary to traditional notions, the first line self-focusing of the EAPB does not occur at the location of the maximum intensity contrast, but rather before or after the maximum intensity contrast. zf1 and zf2 are obtained by simulating the intensity of light along the axis of the linear focus at different positions along the line segment. For example, when r0 = 0.2 mm and β = 0.50, the intensity contrast of the EAPB reaches its maximum value at z = 0.106 m, but the first line self-focusing occurs at z = zf1 = 0.092 m. When β = 0.50, the focal lengths of the first line self-focusing of EAPBs with r0 = 0.3 mm and 0.4 mm are zf1 = 0.107 m and 0.121 m, respectively. When β = 0.60, the focal lengths of the first line self-focusing of EAPBs with r0 = 0.2 mm, 0.3 mm, and 0.4 mm are zf1 = 0.140 m, 0.155 m, and 0.173 m, respectively. When β = 0.70, the focal lengths of the first line self-focusing of EAPBs with r0 = 0.2 mm, 0.3 mm, and 0.4 mm are zf1 = 0.195 m, 0.212 m, and 0.236 m, respectively. When β = 0.80, the focal lengths of the first line self-focusing of EAPBs with r0 = 0.2 mm, 0.3 mm, and 0.4 mm are zf1 = 0.270 m, 0.288 m, and 0.315 m, respectively. As β increases, the self-focusing ability of the first line self-focusing of the EAPB becomes stronger, and the corresponding focal length increases. The variation of β has no effect on the focal length zf2 of the second line self-focusing of the EAPB. The focal lengths of the second line self-focusing of EAPBs with r0 = 0.2 mm, 0.3 mm, and 0.4 mm are zf2 = 0.395 m, 0.450 m, and 0.505 m, respectively. However, the self-focusing ability of the second line self-focusing of the EAPB is enhanced by increasing β. Obviously, the self-focusing ability of the second line self-focusing of the EAPB is weaker than that of the first line self-focusing. zf1 and zf2 of EAPBs with r0 = 0.2 mm, 0.3 mm, and 0.4 mm at four cases of β = 0.50, 0.60, 0.70 and 0.80 are listed in Table 1.

Tables Icon

Table 1. zf1 and zf2 of EAPBs with r0 = 0.2 mm, 0.3 mm, and 0.4 mm at four cases of β = 0.50, 0.60, 0.70 and 0.80

View Table
 figure: Fig. 1.

Fig. 1. Variation of intensity contrast Izp/I0p of the propagating EAPBs with different r0 and β with respect to the axial propagation distance z: (a) r0 = 0.2 mm; (b) r0 = 0.3 mm; (c) r0 = 0.4 mm.

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As shown in Fig. 1, the self-focusing ability of the bifocal spacing region of the EAPB first dreases and then rebounds, which is interpreted as follow. After the first line self-focusing, the energy along the vertical axis begins to diverge outward, and the energy along the horizontal axis continues to converge inward. At first, the rate of weakening of the self-focusing ability in the direction of vertical axis is greater than the rate of enhancement of the self-focusing ability in the direction of horizontal axis, and the self-focusing ability of the EAPB first decreases. After continued propagation, the rate of enhancement of the self-focusing ability in the direction of horizontal axis begins to be greater than the rate of weakening of the self-focusing ability in the direction of vertical axis, and the self-focusing ability of the EAPB begins to rebound.

3.2 Double uniform line self-focusing of EAPBs

The theoretical simulation results of the intensity distribution of EAPBs with different r0 and β on the starting plane and the double line self-focusing planes are presented in Fig. 2. When r0 = 0.2 mm, the first line self-focusing of the EAPB exhibits a uniform line-shaped intensity distribution on the horizontal axis, with several weak elliptical outer rings around the linear focus. The theoretical peak intensities of the first line self-focusing for β = 0.60 and 0.70 are 6.71 and 8.87, respectively. Compared with β = 0.60, the length of the linear focus with β = 0.70 is shortened by 11.11%, and the corresponding focal length is shifted back by 39.29%. However, the linear focus of the second line self-focusing shows higher intensity at the upper and lower ends, and lower intensity in the middle, accompanied by weak symmetric outer rings on the sides. The theoretical peak intensities of the second line self-focusing for β = 0.60 and 0.70 are 2.72 and 5.52, respectively. The length of the linear focus with β = 0.70 is shortened by 32.35% compared with β = 0.60. When r0 = 0.4 mm, the first line self-focusing of the EAPB exhibits an uneven line-shaped distribution with higher intensity at the ends and lower intensity in the middle on the horizontal axis, along with several weak elliptical outer rings around the linear focus. The theoretical peak intensities of the first line self-focusing for β = 0.60 and 0.70 are 7.96 and 9.86, respectively. For β = 0.60 and 0.70, the length of the linear focus with r0 = 0.4 mm is respectively extended by 44.44% and 39.47%, compared with r0 = 0.2 mm. Also, the focal length is respectively shifted back by 10.71% and 8.72%, compared with r0 = 0.2 mm. The linear focus of the second line self-focusing forms a uniform intensity line distribution on the vertical axis, with symmetric weak outer rings on the sides. The theoretical peak intensities of the second line self-focusing for β = 0.60 and 0.70 are 3.37 and 5.36, respectively. For β = 0.60 and 0.70, the length of the linear focus with r0 = 0.4 mm is respectively increased by 28.26% and 15.22%, compared with r0 = 0.2 mm. Compared to the first line self-focusing, however, the self-focusing ability of the second line self-focusing is reduced. As r0 increases, the lengths of the double linear focus increase, and the self-focusing ability of the first line self-focusing increases, while the self-focusing ability of the second line self-focusing sometimes increases and sometimes decreases, which is because the second line self-focusing occurs in a region where the self-focusing ability recovers and fluctuates (as shown in Fig. 1). As β increases, the lengths of the double linear focus decrease, and the self-focusing ability of the double line self-focusing increases. As r0 increases, the length of the horizontal axis of the EAPB grows more than the length of the vertical axis, and the energy along the direction of the horizontal axis pools more slowly. When the energy along the vertical axis is pooled on the horizontal axis, the energy along the horizontal axis is farther away from the center of the circle, the more the first linear focus tends to be inhomogeneous, and the size of the energy at the two ends of the linear focus is gradually larger than that in the middle part, resulting in the outer ring of the linear focus starting to split to the left and the right ends.

 figure: Fig. 2.

Fig. 2. Theoretical simulation results of the light intensity distribution of EAPBs with different r0 and β on the starting plane and the double line self-focusing planes: (a)-(d) z = 0; (e) zf1 = 0.140 m; (f) zf1 = 0.195 m; (g) zf1 = 0.173 m; (h) zf1 = 0.236 m; (i) and (j) zf2 = 0.395 m; (k) and (l) zf2 = 0.505 m.

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When r0 = 0.2 mm, the first line self-focusing is uniform, while the second line self-focusing is nonuniform. Whereas for r0 = 0.4 mm, the first line self-focusing is nonuniform, while the second line self-focusing is uniform. Is there an appropriate r0 where the double line self-focusing of the EAPB have a uniform intensity distribution? The answer is yes. When r0 = 0.3 mm and β = 0.58∼0.71, the EAPB can achieve double uniform line self-focusing. Let’s first look at the line self-focusing process of the EAPB with r0 = 0.3 mm and β = 0.60 or 0.70 during free space propagation. Figure 3 presents the theoretical simulation results of the intensity distribution of the EAPB with r0 = 0.3 mm and β = 0.60 on the observing planes z = 0, 0.100 m, 0.155 m, 0.200 m, 0.300 m, and 0.450 m. The peak intensity at the observing plane z = 0.100 m appears at the ends of the inner ring, with an intensity of 1.65. When reaching z = zf1 = 0.155 m, the energy along the vertical direction uniformly converges onto the horizontal axis first, and the focus takes on a uniform intensity line distribution on the horizontal axis, with a theoretical peak intensity of 7.26. After passing zf1, the energy along the horizontal direction continues to gather inward while the energy along the vertical direction begins to diverge outward. At the same time, the energy at both ends symmetrically splits along the horizontal axis. After a brief split, the intensity distribution on the horizontal axis is compressed, and the intensity distribution on the vertical axis is stretched. The energy along the vertical axis quickly diverges to both ends. The theoretical peak intensities at z = 0.200 m and 0.300 m are 5.09 and 2.77, respectively. After a certain distance of propagation, the EAPB undergoes the second line self-focusing at z = zf2 = 0.450 m, where the energy along the horizontal direction uniformly converges onto the vertical axis, forming a uniform intensity line distribution, with a theoretical peak intensity of 2.96.

 figure: Fig. 3.

Fig. 3. Theoretical simulation results of the light intensity distribution of the EAPB with r0 = 0.3 mm and β = 0.60 on different observing planes: (a) z = 0; (b) z = 0.100 m; (c) z = zf1 = 0.155 m; (d) z = 0.200 m; (e) z = 0.300 m; (f) z = zf2 = 0.450 m.

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Figure 4 shows the theoretical simulation results of the intensity distribution of the EAPB with r0 = 0.3 mm and β = 0.70 on different observing planes z = 0, 0.100 m, 0.212 m, 0.250 m, 0.350 m, and 0.450 m. The theoretical peak intensities on these observing planes are 1.00, 1.24, 9.17, 8.43, 4.83, and 4.98, respectively. The evolution of the intensity distribution for the EAPB with r0 = 0.3 mm and β = 0.7 during free space propagation follows the same trend as for the EAPB with r0 = 0.3 mm and β = 0.6, and the double line self-focusing are uniform. Figure 5 presents the theoretical simulation results of the EAPB with r0 = 0.3 mm on the starting plane and the double line self-focusing planes where β = 0.58 and 0.71. The theoretical peak intensities of the double linear focus for β = 0.58 are 6.82 and 2.72, and the length of the double linear focus is respectively shortened by 4.05% and 1.03%, compared with β = 0.60. While for β = 0.71, the theoretical peak intensities are 9.38 and 5.74, and the length of the double linear focus is respectively shortened by 25.68% and 25.77%, compared with β = 0.60. When β = 0.58 and 0.71, the focal lengths zf1 of the first line self-focusing are 0.144 m and 0.209 m, respectively. Under the condition of double uniform line self-focusing, the self-focusing ability, the lengths of the double linear focus, and the focal length of the first line self-focusing can be adjusted by varying β.

 figure: Fig. 4.

Fig. 4. Theoretical simulation results of the light intensity distribution of the EAPB with r0 = 0.3 mm and β = 0.70 on different observing planes: (a) z = 0; (b) z = 0.100 m; (c) z = zf1 = 0.212 m; (d) z = 0.250 m; (e) z = 0.350 m; (f) z = zf2 = 0.450 m.

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 figure: Fig. 5.

Fig. 5. Theoretical simulation results of the light intensity distribution of the EAPB with r0 = 0.3 mm and β = 0.58 (the top row) and 0.71 (the bottom row) on the starting plane and the double line self-focusing planes: (a) z = 0; (b) zf1 = 0.144 m; (c) zf2 = 0.450 m; (d) z = 0; (e) zf1= 0.209 m; (f) zf2 = 0.450 m.

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For EAPBs with r0 = 0.3 mm and β = 0.58∼0.71, the double line self-focusing are characterized by uniform intensity distribution. However, what happens to the line self-focusing when β further increases or decreases? Let’s first consider the case of β further increasing. Figure 6 shows the theoretical simulation results of EAPBs with β = 0.80 and different r0 on the starting plane and the double line self-focusing planes. As β approaches 1, the intensity distribution of the EAPB on the starting plane gradually approximates that of the CAPB, and the outer rings around the linear focus reduce. The length of the first linear focus of the EAPB with β = 0.80 and r0 = 0.3 mm is shortened by 48.01% compared with β = 0.60 and r0 = 0.3 mm. When β = 0.80, the theoretical peak intensities of the first line self-focusing of EAPBs with r0 = 0.2 mm, 0.3 mm, and 0.4 mm are 13.44, 14.53, and 14.96, respectively. With increasing r0, the peak intensity of the first line self-focusing of the EAPB with β = 0.80 also increases. When β = 0.80, however, the second linear focus of EAPBs with r0 = 0.2 mm, 0.3 mm, and 0.4 mm all exhibit discontinuous and uneven intensity distribution, with higher intensity at the upper and lower ends and lower intensity in the middle. When β = 0.80, the theoretical peak intensities of the second line self-focusing of EAPBs with r0 = 0.2 mm, 0.3 mm, and 0.4 mm are 13.55, 12.73, and 12.22, respectively. With increasing r0, the peak intensity of the second line self-focusing of EAPB with β = 0.80 decreases. Finally, let’s examine the case of β further decreasing. The theoretical simulation results of EAPBs with β = 0.50 and different r0 on the starting plane and the double line self-focusing planes are demonstrated in Fig. 7. Compared with β = 0.60 and r0 = 0.3 mm, the length of the linear focus during the first self-focusing event is extended by 9.43% when β = 0.50 and r0 = 0.3 mm. When β = 0.50, the first linear focus of EAPBs with r0 = 0.2 mm, 0.3 mm, and 0.4 mm all exhibit uneven intensity distribution, and the corresponding theoretical peak intensities are 5.47, 6.38, and 7.29, respectively. During the second line self-focusing of EAPBs with β = 0.50, the outer rings around the linear focus bend inward, forming elliptical outer rings, and the second linear focusing exhibits uneven intensity distribution for r0 = 0.2 mm, 0.3 mm, and 0.4 mm, with theoretical peak intensities of 1.64, 1.89, and 2.13, respectively. With increasing r0, the peak intensities of the first and second linear focus of EAPBs with β = 0.50 both increase. The above analysis indicates that for EAPBs with r0 = 0.3 mm, β = 0.58∼0.71 will result in double uniform line self-focusing. If β further increases or decreases, it may only be uniform line self-focusing once, or even nonuniform line self-focusing twice.

 figure: Fig. 6.

Fig. 6. Theoretical simulation results of the light intensity distribution of EAPBs with β = 0.80 and different r0 on the starting plane and the double line self-focusing planes: (a)-(c) z = 0; (d) zf1 = 0.270 m; (e) zf1 = 0.288 m; (f) zf1 = 0.315 m; (g) zf2 = 0.395 m; (h) zf2 = 0.450 m; (i) zf2 = 0.510 m.

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 figure: Fig. 7.

Fig. 7. Theoretical simulation results of the light intensity distribution of EAPBs with β = 0.50 and different r0 on the starting plane and the double line self-focusing planes: (a)-(c) z = 0; (d) zf1 = 0.092 m; (e) zf1 = 0.107 m; (f) zf1 = 0.121 m; (g) zf2 = 0.395 m; (h) zf2 = 0.450 m; (i) zf2 = 0.510 m.

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Lastly, a physical mechanism of the generation of the double linear focus of the EAPB is explained based on the energy flux distribution. As the EAPB propagates along the z-axis, the energy converges inward along the vertical and horizontal axes (as shown in Fig. 8(a)). Since the length of the light intensity distribution along the horizontal axis is longer than that along the vertical axis, the time required for energy to converge inward along the horizontal axis is also greater than that along the vertical axis. When the energy along the vertical axis converges onto the horizontal axis, a linear focus distributed along the horizontal axis is formed, known as the first line self-focusing (as shown in Fig. 8(b)). The distance from the starting plane to the first line self-focusing plane is defined as the focal length of the first line self-focusing zf1. After passing through zf1, as the energy along the vertical axis starts to diverge outward, the energy along the horizontal axis continues to converge inward, at which point the intensity contrast of the EAPB reaches its maximum. After the propagation distance z passes through the point with the maximum intensity contrast, the energy along the horizontal axis continues to converge inward, while the energy along the vertical axis diverges outward (as shown in Fig. 8(c)). When the energy along the horizontal axis converges onto the vertical axis, it is focused into a linear focus distributed along the vertical axis, which is the second line self-focusing (as shown in Fig. 8(d)). The distance from the starting plane to the second line self-focusing plane is defined as the focal length of the second line self-focusing zf2. The intensity distribution of the first linear focus becomes more uniform as r0 decreases, while the intensity distribution of the second linear focus becomes more uniform as r0 increases. As r0 increases, the light intensity distribution along the horizontal axis extends, requiring more time for energy convergence along the horizontal axis. When the energy along the vertical axis converges onto the horizontal axis, the energy along the horizontal axis still converges at the distant ends of the axis, and the length of the first linear focus becomes longer, bringing about a more nonuniform intensity distribution of the first linear focus with increasing r0. On the other hand, as r0 decreases, the light intensity distribution along the horizontal axis narrows, reducing the time required for energy convergence along the horizontal axis. This results in a more uniform intensity distribution of the first linear focus. After passing through zf1, the energy along the vertical axis that diverges outward increases, and the energy along the horizontal direction converges onto the vertical axis, causing an increase in the energy difference between the middle and two ends of the vertical axis. This leads to a more uneven intensity distribution of the second linear focus with decreasing r0. Therefore, only an appropriate r0 can ensure that double line self-focusing are uniform. If β is too small, the light intensity distribution along the vertical axis becomes too short. This reduces the time required for energy convergence along the vertical axis during the first line self-focusing, causing the energy along the horizontal axis to converge mainly at the distant ends. Therefore, too small β will result in a longer length of the first linear focus and an uneven intensity distribution. Moreover, due to the small aperture of the EAPB, it is difficult for all energy to enter and converge uniformly during the second line self-focusing, and the self-focusing ability of the second line self-focusing gradually disappears (as shown in Fig. 1), resulting in an uneven intensity distribution of the second linear focus. When β is too large, the distribution of the EAPB approaches that of the CAPB. As the first focusing plane is relatively close to the starting plane, the difference between the light intensity distributions along the horizontal and the vertical directions of the starting plane can still be reflected, forming the line self-focusing. Because the second focusing plane is relatively far from the starting plane, the difference between the light intensity distributions along the horizontal and vertical directions of the starting plane is less apparent. The phenomenon of line self-focusing almost disappears, resulting in a point-like distribution along the vertical axis. Therefore, only a certain range of β can result in the linear focus with uniform intensity distribution twice.

 figure: Fig. 8.

Fig. 8. Changes in the direction of energy flow at different propagation positions: (a) Starting plane; (b) The first line self-focusing; (c) Transition region between double-line self-focusing; (d) The second line self-focusing.

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4. Generation and recorded results

A EAPB is generate by constructing an experimental optical path, and its light intensity distribution on different observing planes is recorded. Figure 9 represent the experimental setup. Firstly, a raw Gaussian beam is generated by a laser (Ventus 532). The polarization direction of the raw Gaussian beam is adjusted by a half wave plate (HWP), and then the beam size of the raw Gaussian beam is increased by a beam expander (BE). Next, the raw Gaussian beam is divided into two equivalent parts by a beam splitter (BS), and the remaining half of the raw Gaussian beam is introduced into a spatial light modulator (SLM, Holoeye LETO-3). The pixel size of the SLM is 6.4 µm × 6.4 µm. The SLM modulates the incident raw Gaussian beam and reflects the modulated beam out. The modulated beam contains numerous diffraction orders, and only the first diffraction order is required to generate the EAPB. The modulated beam re-enters the BS and deviates by 90 degrees. A circular aperture (CA, GCM-5704 M) along the beam propagation path is used to remove other diffraction orders except for the first diffraction order. The only remaining first diffraction order is luckily transformed into the EAPB by the Fourier transformation of a lens (L). In order to make the optical path compact, the focal distance of the L is selected as 40 cm. As a result, the EAPB can be captured on the starting plane marked as z = 0 of Fig. 9. Moreover, the light intensity distribution of the EAPB on different observing planes can be recorded by the beam profiling analyzer (BPA, BGS-USB3-LT665), which is electrically driven along the direction of beam propagation.

 figure: Fig. 9.

Fig. 9. Experimental optical path diagram for generating a EAPB and measuring its light intensity distribution during free space propagation.

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Figure 10 presents the experimental recorded results of the light intensity distribution of EAPBs with different r0 and β on the starting plane and the double line self-focusing planes. The experimental measured values of the peak intensity on the double line self-focusing planes for the EAPB with r0 = 0.2 mm and β = 0.60 are 6.54 and 2.56, with deviations of 2.53% and 4.03% from the theoretical values, respectively. For the EAPB with r0 = 0.4 mm and β = 0.60, the experimental measured values of the peak intensity on the double line self-focusing planes are 8.57 and 5.28, with deviations of 3.38% and 4.35% from the theoretical values, respectively. The experimental measured values of the peak intensity on the double line self-focusing planes for the EAPB with r0 = 0.2 mm and β = 0.70 are 7.78 and 3.21, with deviations of 2.26% and 4.75% from the theoretical values, respectively. For the EAPB with r0 = 0.4 mm and β = 0.70, the experimental measured values of the peak intensity on the double line self-focusing planes are 9.55 and 5.10, with deviations of 3.14% and 4.85% from the theoretical values, respectively.

 figure: Fig. 10.

Fig. 10. Experimental recorded results of the light intensity distribution of EAPBs with different r0 and β on the starting plane and the double line self-focusing planes: (a)-(d) z = 0; (e) zf1 = 0.140 m; (f) zf1 = 0.195 m; (g) zf1 = 0.173 m; (h) zf1 = 0.236 m; (i) and (j) zf2 = 0.395 m; (k) and (l) zf2 = 0.505 m.

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Figure 11 represents the experimental recorded results of the light intensity distribution of the EAPB with r0 = 0.3 mm and β = 0.60 on observing planes z = 0, 0.100 m, 0.155 m, 0.200 m, 0.300 m, and 0.450 m. For the EAPB with r0 = 0.3 mm and β = 0.60, the experimental measured values of the peak intensity on the double line self-focusing planes are 7.04 and 2.84, with deviations of 3.03% and 4.05% from the theoretical values, respectively. Figure 12 presents the experimental recorded results of the light intensity distribution of the EAPB with r0 = 0.3 mm and β = 0.70 on observing planes z = 0, 0.100 m, 0.212 m, 0.250 m, 0.350 m, and 0.450 m. When r0 = 0.3 mm and β = 0.70, the experimental measured values of the peak intensity on the double line self-focusing planes are 8.83 and 4.75, with deviations of 3.71% and 4.62% from the theoretical values, respectively. Figure 13 provides the experimental recorded results of the light intensity distribution of EAPBs with r0 = 0.3 mm and different β on the starting plane and the double line self-focusing planes. For the EAPB with r0 = 0.3 mm and β = 0.58, the experimental measured values of the peak intensity on the double line self-focusing planes are 6.65 and 2.61, with deviations of 2.49% and 4.04% from the theoretical values, respectively. For the EAPB with r0 = 0.3 mm and β = 0.71, the experimental measured values of the peak intensity on the double line self-focusing planes are 9.07 and 5.46, with deviations of 3.30% and 4.88% from the theoretical values, respectively.

 figure: Fig. 11.

Fig. 11. Experimental recorded results of the light intensity distribution the EAPB with r0 = 0.3 mm and β = 0.60 on different observing planes: (a) z = 0; (b) z = 0.100 m; (c) z = zf1 = 0.155 m; (d) z = 0.200 m; (e) z = 0.300 m; (f) z = zf2 = 0.450 m.

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 figure: Fig. 12.

Fig. 12. Experimental recorded results of the light intensity distribution of the EAPB with r0 = 0.3 mm and β = 0.70 on different observing planes: (a) z = 0; (b) z = 0.100 m; (c) z = zf1 = 0.212 m; (d) z = 0.250 m; (e) z = 0.350 m; (f) z = zf2 = 0.450 m.

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 figure: Fig. 13.

Fig. 13. Experimental recorded results of the light intensity distribution of the EAPB with r0 = 0.3 mm and β = 0.58 (the top row) and 0.71 (the bottom row) on the starting plane and the double line self-focusing planes: (a) z = 0; (b) zf1 = 0.144 m; (c) zf2 = 0.450 m; (d) z = 0; (e) zf1 = 0.209 m; (f) zf2 = 0.450 m.

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Figure 14 demonstrates the experimental recorded results of the light intensity distribution of EAPBs with different r0 and b = 0.80 on the starting plane and the double line self-focusing planes. When b = 0.80, the experimental measured values of the peak intensity on the first line self-focusing planes for EAPBs with r0 = 0.2 mm, 0.3 mm, and 0.4 mm are 12.97, 14.03, and 14.47, respectively, and the experimental measured values of the peak intensity on the second line self-focusing planes are 12.93, 12.16, and 11.63, respectively. Compared with Fig. 6, the maximum deviation between the experimental measured values and theoretical simulation values is 4.83%. Figure 15 presents the experimental recorded results of the light intensity distribution of EAPBs with different r0 and b = 0.50 on the starting plane and the double line self-focusing planes. When b = 0.50, the experimental measured values of the peak intensity on the first line self-focusing planes for EAPBs with r0 = 0.2 mm, 0.3 mm, and 0.4 mm are 5.31, 6.25, and 7.10, respectively, and the experimental measured values of the peak intensity on the second line self-focusing planes are 1.56, 1.81, and 2.03, respectively. Compared with Fig. 7, the maximum deviation between the experimental measured values and theoretical simulation values is 4.88%. To sum up, the experimental results of EAPBs match the theoretical map very well, and the maximum deviation of focus is only 4.88%, whch can be interptred as follow. Because the radius of the main ring of a EABA is small, the primary spot biased by the SLM is farther away from the zero-level spot, so it suffers from less diffraction interference from the zero-level spot, the saturated ring energy distribution is uniform. And the focal length is shorter, so the energy loss of the beam in the process of free-space propagation is less. Therefore, the experimentally measured value of the self-focusing ability is more consistent with the theoretical value.

 figure: Fig. 14.

Fig. 14. Experimental recorded results of the light intensity distribution of EAPBs with β = 0.80 and different r0 on the starting plane and the double line self-focusing planes: (a)-(c) z = 0; (d) zf1 = 0.270 m; (e) zf1 = 0.288 m; (f) zf1 = 0.315 m; (g) zf2 = 0.395 m; (h) zf2 = 0.450 m; (i) zf2 = 0.510 m.

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 figure: Fig. 15.

Fig. 15. Experimental recorded results of the light intensity distribution of EAPBs with β = 0.50 and different r0 on the starting plane and the double line self-focusing planes: (a)-(c) z = 0; (d) zf1 = 0.092 m; (e) zf1 = 0.107 m; (f) zf1 = 0.121 m; (g) zf2 = 0.395 m; (h) zf2 = 0.450 m; (i) zf2 = 0.510 m.

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By comparing Figs. 10–15 with Figs. 2–7, it can be observed that the experimental measurements effectively support the theoretical simulation results. Under the given parameters of the EAPB, including w0, a, a and l, appropriate selection of r0 and β can achieve line self-focusing with uniform intensity distribution in double linear focus. Furthermore, the focal length of the first line self-focusing, the lengths of the double linear focus, and self-focusing abilities of the double line self-focusing can be controlled by adjusting β.

5. Summary

This research investigates the influences of the main ring radius r0 and the elliptical vertical factor β on the double line self-focusing of the EAPB. An excessively large or small r0 will result in uneven intensity distribution of one linear focus of the EAPB. Only with an appropriate r0 and within a certain range of β, the EAPB can achieve double uniform line self-focusing, but the self-focusing ability of the second line self-focusing decreases compared to the first one. Under the premise of our selected beam parameters, when r0 = 0.3 mm and β = 0.58∼0.71, EAPB can achieve double uniform line self-focusing. Within the range of 0.58∼0.71, the focal length of the first line self-focusing, the lengths of the double linear focus, and the self-focusing abilities of the double line self-focusing can be controlled by varying β. The specific control rules are as follows: as β increases, the focal length of the first line self-focusing increases, the lengths of the double linear focus are shortened, and the self-focusing abilities of the double line self-focusing are enhanced. Excessively large or small values of β can cause the EAPB to lose linear focus with uniform intensity distribution. Based on the convergence speed of energy along the horizontal and vertical axes, the physical mechanism behind the generation of both uniform linear focus of the EAPB is explained. Finally, the EAPB is experimentally generated, and the experimental measurement results of its line self-focusing characteristics effectively support the theoretical predictions. Assuming the beam parameters of the EAPB including w0, a, α and λ are determined, selecting appropriate values for r0 and β can achieve double linear focus with uniform intensity distribution. This research provides a new approach for generating double uniform line self-focusing and offers new insights into the practical applications of elliptical self-focusing beams. The proposed EAPB has potential applications in fields such as free space optical communication and bio-medicine.

Funding

National Natural Science Foundation of China (12374281, 12274311).

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

References

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Data availability

No data were generated or analyzed in the presented research.

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Figures (15)
Fig. 1.
Fig. 1. Variation of intensity contrast Izp/I0p of the propagating EAPBs with different r0 and β with respect to the axial propagation distance z: (a) r0 = 0.2 mm; (b) r0 = 0.3 mm; (c) r0 = 0.4 mm.

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Fig. 2.
Fig. 2. Theoretical simulation results of the light intensity distribution of EAPBs with different r0 and β on the starting plane and the double line self-focusing planes: (a)-(d) z = 0; (e) zf1 = 0.140 m; (f) zf1 = 0.195 m; (g) zf1 = 0.173 m; (h) zf1 = 0.236 m; (i) and (j) zf2 = 0.395 m; (k) and (l) zf2 = 0.505 m.

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Fig. 3.
Fig. 3. Theoretical simulation results of the light intensity distribution of the EAPB with r0 = 0.3 mm and β = 0.60 on different observing planes: (a) z = 0; (b) z = 0.100 m; (c) z = zf1 = 0.155 m; (d) z = 0.200 m; (e) z = 0.300 m; (f) z = zf2 = 0.450 m.

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Fig. 4.
Fig. 4. Theoretical simulation results of the light intensity distribution of the EAPB with r0 = 0.3 mm and β = 0.70 on different observing planes: (a) z = 0; (b) z = 0.100 m; (c) z = zf1 = 0.212 m; (d) z = 0.250 m; (e) z = 0.350 m; (f) z = zf2 = 0.450 m.

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Fig. 5.
Fig. 5. Theoretical simulation results of the light intensity distribution of the EAPB with r0 = 0.3 mm and β = 0.58 (the top row) and 0.71 (the bottom row) on the starting plane and the double line self-focusing planes: (a) z = 0; (b) zf1 = 0.144 m; (c) zf2 = 0.450 m; (d) z = 0; (e) zf1= 0.209 m; (f) zf2 = 0.450 m.

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Fig. 6.
Fig. 6. Theoretical simulation results of the light intensity distribution of EAPBs with β = 0.80 and different r0 on the starting plane and the double line self-focusing planes: (a)-(c) z = 0; (d) zf1 = 0.270 m; (e) zf1 = 0.288 m; (f) zf1 = 0.315 m; (g) zf2 = 0.395 m; (h) zf2 = 0.450 m; (i) zf2 = 0.510 m.

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Fig. 7.
Fig. 7. Theoretical simulation results of the light intensity distribution of EAPBs with β = 0.50 and different r0 on the starting plane and the double line self-focusing planes: (a)-(c) z = 0; (d) zf1 = 0.092 m; (e) zf1 = 0.107 m; (f) zf1 = 0.121 m; (g) zf2 = 0.395 m; (h) zf2 = 0.450 m; (i) zf2 = 0.510 m.

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Fig. 8.
Fig. 8. Changes in the direction of energy flow at different propagation positions: (a) Starting plane; (b) The first line self-focusing; (c) Transition region between double-line self-focusing; (d) The second line self-focusing.

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Fig. 9.
Fig. 9. Experimental optical path diagram for generating a EAPB and measuring its light intensity distribution during free space propagation.

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Fig. 10.
Fig. 10. Experimental recorded results of the light intensity distribution of EAPBs with different r0 and β on the starting plane and the double line self-focusing planes: (a)-(d) z = 0; (e) zf1 = 0.140 m; (f) zf1 = 0.195 m; (g) zf1 = 0.173 m; (h) zf1 = 0.236 m; (i) and (j) zf2 = 0.395 m; (k) and (l) zf2 = 0.505 m.

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Fig. 11.
Fig. 11. Experimental recorded results of the light intensity distribution the EAPB with r0 = 0.3 mm and β = 0.60 on different observing planes: (a) z = 0; (b) z = 0.100 m; (c) z = zf1 = 0.155 m; (d) z = 0.200 m; (e) z = 0.300 m; (f) z = zf2 = 0.450 m.

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Fig. 12.
Fig. 12. Experimental recorded results of the light intensity distribution of the EAPB with r0 = 0.3 mm and β = 0.70 on different observing planes: (a) z = 0; (b) z = 0.100 m; (c) z = zf1 = 0.212 m; (d) z = 0.250 m; (e) z = 0.350 m; (f) z = zf2 = 0.450 m.

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Fig. 13.
Fig. 13. Experimental recorded results of the light intensity distribution of the EAPB with r0 = 0.3 mm and β = 0.58 (the top row) and 0.71 (the bottom row) on the starting plane and the double line self-focusing planes: (a) z = 0; (b) zf1 = 0.144 m; (c) zf2 = 0.450 m; (d) z = 0; (e) zf1 = 0.209 m; (f) zf2 = 0.450 m.

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Fig. 14.
Fig. 14. Experimental recorded results of the light intensity distribution of EAPBs with β = 0.80 and different r0 on the starting plane and the double line self-focusing planes: (a)-(c) z = 0; (d) zf1 = 0.270 m; (e) zf1 = 0.288 m; (f) zf1 = 0.315 m; (g) zf2 = 0.395 m; (h) zf2 = 0.450 m; (i) zf2 = 0.510 m.

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Fig. 15.
Fig. 15. Experimental recorded results of the light intensity distribution of EAPBs with β = 0.50 and different r0 on the starting plane and the double line self-focusing planes: (a)-(c) z = 0; (d) zf1 = 0.092 m; (e) zf1 = 0.107 m; (f) zf1 = 0.121 m; (g) zf2 = 0.395 m; (h) zf2 = 0.450 m; (i) zf2 = 0.510 m.

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Tables (1)

Tables Icon

Table 1. zf1 and zf2 of EAPBs with r0 = 0.2 mm, 0.3 mm, and 0.4 mm at four cases of β = 0.50, 0.60, 0.70 and 0.80

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Equations (3)

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E ( x , y , 0 )  =  A exp ⁡ [ a ( r 0 − ( x / α ) 2 + ( y / β ) 2 w 0 ) ] A i ′ ( r 0 − ( x / α ) 2 + ( y / β ) 2 w 0 ) ,
E ( x , y , z ) = k 2 π i z ∫ − ∞ ∞ ∫ − ∞ ∞ E ( x ′ , y ′ , 0 ) exp ⁡ { i k 2 z [ ( x − x ′ ) 2 + ( y − y ′ ) 2 ] } d x ′ d y ′ ,
I ( x , y , z ) = | E ( x , y , z ) | 2 .
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