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Effect of chirped factors on the abrupt autofocusing ability of a chirped circular Airyprime beam

Xiang Zang, Wensong Dan, Yimin Zhou, Fei Wang, Yangjian Cai, Zhangrong Mei, and Guoquan Zhou

Open Access Open Access

Abstract

Recently, a new type of abruptly autofocusing beam called circular Airyprime beam (CAPB) has been reported. Its abrupt autofocusing ability has been proven to be approximately seven times that of a circular Airy beam under the same conditions. Further improving the abrupt autofocusing ability of the CAPB without changing the beam parameters is a concern in optical research. In this study, we investigated the effect of introducing first- and second-order chirped factors on the abrupt autofocusing ability of the CAPB. When the positive first-order chirped factor was below the saturated chirped value, the abrupt autofocusing ability of the chirped CAPB was stronger and the focus position was smaller compared with those of the conventional CAPB. Regarding the abrupt autofocusing ability, there was an optimal value for the first-order chirped factor. At the optimal value, the abrupt autofocusing ability of the chirped CAPB was the strongest. On the other hand, a positive second-order chirped factor promoted the abrupt autofocusing ability of the CAPB and shortened the focus position. The introduction of such value was more effective than the introduction of a positive first-order chirped factor in promoting abrupt autofocusing of the CAPB. The abrupt autofocusing ability of the CAPB was further improved by combining the optimal first-order chirped factor and a positive second-order chirped factor. Finally, the chirped CAPB was experimentally generated, and the corresponding abrupt autofocusing behaviors were measured, validating the theoretical results. Overall, we provide an approach for improving abruptly autofocusing CAPBs.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

When Efremidis and Christodoulides theoretically investigated the characteristics of circular Airy beams propagating in free space, they observed an abrupt autofocusing phenomenon and introduced the corresponding concept [1,2]. Soon after, abrupt autofocusing was successfully observed experimentally [3,4]. The auto-defocusing Airy beam can be converted into an autofocusing circular Airy beam by using a cone-frustum-shaped fiber end [5]. In addition, when the first few rings of the circular Airy beam are blocked on the initial plane, the abrupt autofocusing ability is enhanced [6]. Moreover, when the spatial coherence of partially coherent circular Airy beams decreases, the focal spot size increases and the focal intensity decreases [7]. As the partially coherent radially polarized circular Airy beam inherits the autofocusing ability, it creates an optical potential well in the beam center [8].

After mastering the abrupt autofocusing characteristics of the circular Airy beam, researchers investigated the corresponding autofocusing performance by modifying Airy beams. A symmetric Airy beam autofocuses in the initial stage of free-space propagation and collapses promptly after passing through the focal plane [9]. The multiple Airy beam formed by superposition of two and four partially coherent Airy beams does not reduce the acceleration properties. The shape and peak intensity of the interference spot can be controlled by the transverse width of the beam [10]. The abrupt autofocusing ability of a modified circular Airy beam can be substantially improved by appropriately choosing the apodization parameters [11]. The autofocusing effect of conical circular Airy beams is considerably strengthened under positive cone angles [12]. A modulated circular Airy beam exhibits dual abrupt focusing behaviors at given focal intensity and focal spot size [13]. When a ring Airy beam approaches the wavelength limit, a counterintuitive and strong enhancement of the focal peak intensity occurs [14]. The abrupt autofocusing ability of circular Airy beams with a Gaussian envelope is notably enhanced by suitably adjusting the two parameters of the Gaussian function [15].

The effect of the insertion of an optical vortex on the abrupt autofocusing ability has attracted considerable attention. The influences of an on-axis optical vortex, off-axis optical vortex, and multiple optical vortices on abrupt autofocusing of circular Airy vortex beams have been analyzed [16]. The propagation properties and the radiation forces of Airy Gaussian vortex beams in a harmonic potential have been investigated by analytical and numerical methods [17]. Spiral focusing of Airy beams carrying power-exponent-phase vortices has been experimentally demonstrated [18]. The number of topological charges considerably affects the focal intensity and focal length of ring Airy Gaussian vortex beams [19]. With more vortex arrays, the focal intensity of annular arrayed Airy beams carrying vortex arrays can be increased by two orders of magnitude [20].

The influence of introducing chirps on the abrupt autofocusing ability is also of interest. When the second-order chirped factor is above a critical value, autofocusing of chirped circular Airy beams is eliminated [21]. The chirped factors of a chirped circular Airy Gaussian vortex beam affect the focus position, focal intensity, and focal spot size [22]. The lengths and sizes of the optical bottles in second-order chirped symmetric Airy vortex beams can be adjusted by the chirped factor and topological charge, respectively [23]. A polycyclic tornado ring Airy beam, whose phase is an annular spiral zone with a second-order chirped factor, exhibits controllable multi-focus, multi-optical bottles, and rotation [24].

The abrupt autofocusing feature has broad applications in biomedical treatments [1], optical trapping/guiding [25–28], generation of light bullets [29], atom manipulation [30], multi-photopolymerization [31], emission of terahertz waves [32], crosstalk reduction [33,34], optical manipulation [35], nonlinear manipulation [36], dynamic imaging [37], and three-dimensional laser manipulation [38].

Recently, we devised a new type of abruptly autofocusing beam called circular Airyprime beam (CAPB), which was theoretically and experimentally demonstrated [39]. The abrupt autofocusing ability of the CAPB has been proven to be approximately seven times that of a circular Airy beam under the same conditions. Still, the abrupt autofocusing ability of the CAPB should be further improved without changing the beam parameters. The intensity profile of the CAPB in the focal plane is hollow if an optical vortex is introduced into the CAPB. Moreover, the abrupt autofocusing ability of the CAPB with an optical vortex is weaker than that of the conventional CAPB [40]. On the other hand, introducing a chirp maintains a solid intensity profile on the focal plane. Fortunately, studies on chirped circular Airy beams have shown that a positive chirped factor can remarkably enhance the abrupt autofocusing ability [21]. Therefore, we investigated the influence of the introduction of first- and second-order chirped factors on the abrupt autofocusing ability of the CAPB through theoretical derivations and experiments. Overall, we aimed to provide an approach to improve the abrupt autofocusing ability of the CAPB.

2. Free-space propagation of chirped CAPB

The electric field of a chirped CAPB on initial plane z = 0 is described by

$$U(r,0)\textrm{ = }A\,\exp \left[ {a\left( {\frac{{{r_0} - r}}{{{w_0}}}} \right)} \right]Ai'\left( {\frac{{{r_0} - r}}{{{w_0}}}} \right)\exp \left[ {i{c_1}\left( {\frac{{{r_0} - r}}{{{w_0}}}} \right) - i{c_2}\frac{{{r^2}}}{{w_0^2}}} \right],$$
where r denotes the radial coordinate, the z axis is consistent with the direction of beam propagation, A is a control parameter for the light intensity and ensures that the peak light intensity on the initial plane is always 1, r0 is the radius of the primary ring, w0 is a scaling factor, a is an exponential decay factor, Ai′ is the Airyprime function [41], and c1 and c2 are the first- and second-order chirped factors, respectively. c1 is the linear chirp related to the incident angle of the optical beam. c2 is the quadratic chirp related to the ratio of the Rayleigh length to the focal length of a spherical lens.

The electric field of the chirped CAPB propagating in free space is governed by [42]

$$U(r,z) = \frac{{ - ik}}{{2\pi z}}\int_0^\infty {\int_0^{2\pi } {U(r^{\prime},0)\exp \left\{ {\frac{{ik}}{{2z}}[{{r^{\prime}}^2} + {r^2} - 2rr^{\prime}\cos (\varphi - \varphi^{\prime})]} \right\}r^{\prime}dr^{\prime}} } d\varphi ^{\prime},$$
where k = 2π/λ is the optical wavelength and φ is the azimuthal angle. Although an analytical expression for U(r, z) is difficult to find, Eq. (2) can be simulated numerically using the fast Fourier transform [43]. The light intensity of the chirped CAPB on transverse plane z is given by I(r, z) = |U(r, z)|2. The abrupt autofocusing ability can be assessed by the intensity contrast defined by Izp/I0p, where I0p and Izp are the peak intensities of the beam on the initial plane and observation plane of z, respectively. Because the peak intensity on the initial plane is 1, the intensity contrast is exactly equal to the peak intensity on the observation plane of z.

3. Theoretical calculations and analyses

We analyzed the abrupt autofocusing characteristics of the chirped CAPB through concrete examples. For convenience of comparison, the beam parameters were set as in [39]: r0 = 1 mm, a = 0.1, w0 = 0.1 mm, and λ = 532 nm.

The second-order chirped factor, c2, was first set to 0 to investigate the influence of the first-order chirped factor, c1, on the abrupt autofocusing ability. Owing to the symmetry of the initial beam pattern, the position of the peak on-axis intensity was the focus position, and the peak on-axis intensity represented the abrupt autofocusing ability. Figure 1 shows the on-axis intensity distribution of the chirped CAPB for different c1 values according to axial propagation distance z. With increasing positive c1, the focus position moved toward the initial plane, and the peak on-axis intensity first increased and then decreased. With a decrease in negative c1, the focus position moved along the positive z-axis direction, and the peak on-axis intensity always decreased. Figure 2 shows focus position zf and peak on-axis intensity If (0, z) of the chirped CAPB according to first-order chirped factor c1. The dotted lines indicate relevant c1 values. When c1 increased from a negative value, focus position zf decreased and peak on-axis intensity If (0, z) first increased and then decreased. For c1 = 1.1, peak on-axis intensity If (0, z) reached 417.78 at focus position zf of 0.562 m. Therefore, c1 = 1.1 was considered as the optimal chirped value (red dotted line). When 0 < c1 < 2.72, the abrupt autofocusing ability of the chirped CAPB was stronger than that of the conventional CAPB, and its focus position was smaller than that of the conventional CAPB. Accordingly, c1 = 2.72 (purple dotted line) was defined as the saturated chirped value. When c1 < 0 (to the left of the green dotted line), the abrupt autofocusing ability of the chirped CAPB was weaker than that of the conventional CAPB, and the focus position was greater than that of conventional CAPB. When c1 > 2.72, the abrupt autofocusing ability and focus position of the chirped CAPB were smaller than those of the conventional CAPB.

 figure: Fig. 1.

Fig. 1. On-axis intensity distribution of chirped CAPBs for c2 = 0 according to axial propagation distance z for (a) c1 ≥ 0 and (b) c1 ≤ 0.

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 figure: Fig. 2.

Fig. 2. (a) Focus position zf and (b) peak on-axis intensity If (0, z) of chirped CAPBs for c2 = 0 according to first-order chirped factor c1.

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The first-order chirped factor, c1, was then set to 0 to investigate the effect of the second-order chirped factor, c2, on the abrupt autofocusing ability. Figure 3 shows the on-axis intensity distribution of chirped CAPBs with different c2 values according to axial propagation distance z. Any positive second-order chirped factor could improve the abrupt autofocusing ability, while a negative value reduced this ability. Moreover, abrupt autofocusing was completely suppressed for negative second-order chirped factor c2 < $- \sqrt {{w_0}/{r_0}} /4$ = −0.079 [21] (not shown in Fig. 3). Figure 4 shows focus position zf and peak on-axis intensity If (0, z) of the chirped CAPB according to second-order chirped factor c2. When c2 increased, zf decreased, whereas If (0, z) increased. Therefore, a positive second-order chirped factor promoted abrupt autofocusing. By comparing Figs. 3 and 4, the abrupt autofocusing ability of the CAPB was more sensitive to the second-order chirped factor, c2, than to the first-order chirped factor, c1.

 figure: Fig. 3.

Fig. 3. On-axis intensity distribution of chirped CAPBs for c1 = 0 according to axial propagation distance z with (a) c2 ≥ 0 and (b) c2 ≤ 0.

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 figure: Fig. 4.

Fig. 4. (a) Focus position zf and (b) peak on-axis intensity If (0, z) of CAPBs for c1 = 0 according to second-order chirped factor c2.

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Figure 5 shows the intensity distribution of the chirped CAPB for c1 = 1.1 and c2 = 0 on various observation planes during free-space propagation. The intensity pattern on the initial plane, z = 0, was composed of a series of concentric rings. When the chirped CAPB left the initial plane, energy flowed from the outer ring into the inner ring, gradually reducing the number of outer rings and contracting the inner ring. The inner ring disappeared and evolved into a bright solid on-axis spot at the focus position (Fig. 5(c)), and the number of outer rings was minimized. Therefore, the beam energy at the focus position was mainly concentrated at the on-axis point, and the intensity of the outer ring was very weak. Under the same calculation accuracy, the peak intensity contrast and focus position of the conventional (unchirped) CAPB were 358.85 and 0.778 m, respectively, while those of the chirped CAPB were 417.78 and 0.562 m, respectively. Compared with the unchirped CAPB, the peak intensity contrast of the chirped CAPB for c1 = 1.1 and c2 = 0 increased by 16.4%. Moreover, the focal spot size of the chirped CAPB was smaller than that of the unchirped CAPB. When the beam passed through the focal plane, the energy flowed from the center to the outer rings, resulting in the emergence of the inner ring and a gradual increase in the number of outer rings. Therefore, the beam spot size after the focus position expanded.

 figure: Fig. 5.

Fig. 5. Intensity distribution of chirped CAPB for c1 = 1.1 and c2 = 0 on different observation planes during free-space propagation. (a) z = 0 m, (b) z = 0.3 m, (c) z = 0.562 m, and (d) z = 0.78 m.

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Figure 6 shows the intensity distribution of the chirped CAPB for c1 = 2.72 and c2 = 0 on various observation planes during free-space propagation. The peak intensity contrast of the chirped CAPB was 358.88, being equal to that of the conventional (unchirped) CAPB. The focus position of the chirped CAPB was 0.368 m, being less than half the focus position of the unchirped CAPB. Therefore, owing to the shorter focus position, the chirped CAPB with c1 = 2.72 and c2 = 0 seems more suitable for practical applications than the unchirped CAPB. By comparing Figs. 6(b) and 5(b), the contraction speed of the rings for c1 = 2.72 and c2 = 0 was faster than that for c1 = 1.1 and c2 = 0, resulting in early occurrence of abrupt autofocusing. However, the focal spot size for c1 = 2.72 and c2 = 0 was greater than that for c1 = 1.1 and c2 = 0. When z ≥ 0.78 m, the intensity contrast for c1 = 2.72 and c2 = 0 was larger than that for c1 = 1.1 and c2 = 0. In terms of the abrupt autofocusing ability, therefore, the optimal value for first-order chirped factor c1 was 1.1.

 figure: Fig. 6.

Fig. 6. Intensity distribution of chirped CAPB for c1 = 2.72 and c2 = 0 on different observation planes during free-space propagation. (a) z = 0 m, (b) z = 0.3 m, (c) z = 0.368 m, and (d) z = 0.78 m.

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Figure 7 shows the intensity distribution of the chirped CAPB for c1 = 3.0 and c2 = 0 on various observation planes during free-space propagation. By comparing Figs. 7(b) and 6(b), the contraction speed of the rings for c1 = 3.0 and c2 = 0 was faster than that for c1 = 2.72 and c2 = 0, resulting in a large intensity contrast and small beam spot size. As shown in Fig. 7(c), the focus position for c1 = 3.0 and c2 = 0 was 0.346 m, being slightly smaller than that for c1 = 2.72 and c2 = 0. The peak intensity contrast for c1 = 3.0 and c2 = 0 was 344.79, being smaller than that for c1 = 2.72 and c2 = 0. When z ≥ 0.78 m, the intensity contrast for c1 = 3.0 and c2 = 0 was greater than that for c1 = 2.72 and c2 = 0. The abrupt autofocusing ability for c1 = 2.72 and c2 = 0 was slightly better than for c1 = 3.0 and c2 = 0.

 figure: Fig. 7.

Fig. 7. Intensity distribution of chirped CAPB for c1 = 3.0 and c2 = 0 on different observation planes during free-space propagation. (a) z = 0 m, (b) z = 0.3 m, (c) z = 0.346 m, and (d) z = 0.78 m.

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Figures 5–7 show that not all the c1 values can improve the abrupt autofocusing ability. Only the appropriate selection of c1 can enhance this ability, as provided by the analysis above. However, the enhancement of the abrupt autofocusing ability is limited by using first-order chirped factor c1.

Figure 8 shows the intensity distribution of the chirped CAPB for c1 = 0 and c2 = 0.01 on various observation planes during free-space propagation. The abrupt autofocusing ability was substantially enhanced by adopting a positive second-order chirped factor c2. Compared with the unchirped CAPB, the peak intensity contrast of the chirped CAPB for c1 = 0 and c2 = 0.01 increased by 28.0%, and the focus position was shortened by 11.4%.

 figure: Fig. 8.

Fig. 8. Intensity distribution of chirped CAPB for c1 = 0 and c2 = 0.01 on different observation planes during free-space propagation. (a) z = 0 m, (b) z = 0.3 m, (c) z = 0.688 m, and (d) z = 0.78 m.

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Figure 9 shows the intensity distribution of the chirped CAPB for c1 = 0 and c2 = −0.005 on various observation planes during free-space propagation. As shown in Fig. 9(b), the intensity contrast of the chirped CAPB for a small negative second-order chirped factor decreased slightly in the initial stage of free-space propagation, accompanied by an expansion of the beam spot size. Although the negative second-order chirped factor was very small, its effect on the abrupt autofocusing ability was substantial. In this case, the peak intensity contrast and focus position were 313.13 and 0.832 m, respectively. Thus, the abrupt autofocusing ability of the chirped CAPB with a small negative second-order chirped factor was weakened and accompanied by an increase in the focus position.

 figure: Fig. 9.

Fig. 9. Intensity distribution of chirped CAPB for c1 = 0 and c2 = −0.005 on different observation planes during free-space propagation. (a) z = 0 m, (b) z = 0.3 m, (c) z = 0.78 m, and (d) z = 0.832 m.

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The second-order chirped factor was more effective than the first-order one in improving the abrupt autofocusing ability of the CAPB. Therefore, the best approach to improve abrupt autofocusing is using a positive second-order chirped factor.

Figure 10 shows the intensity distribution of the chirped CAPB for c1 = 1.1 and c2 = 0.01 on various observation planes during free-space propagation. The abrupt autofocusing ability was strengthened by combining the optimal c1 value and a positive c2 value. For c1 = 1.1 and c2 = 0.01, the peak intensity contrast of the chirped CAPB was 500.92, and the focus position was 0.512 m.

 figure: Fig. 10.

Fig. 10. Intensity distribution of chirped CAPB for c1 = 1.1 and c2 = 0.01 on different observation planes during free-space propagation. (a) z = 0 m, (b) z = 0.3 m, (c) z = 0.512 m, and (d) z = 0.78 m.

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Figure 11 shows the intensity distribution of the chirped CAPB for c1 = 1.1 and c2 = −0.005 on various observation planes during free-space propagation. The focus position of this chirped CAPB was zf = 0.59 m. Compared with the chirped CAPB for c1 = 1.1 and c2 = 0, the abrupt autofocusing ability for c1 = 1.1 and c2 = −0.005 was attenuated owing to the small negative c2. However, the peak intensity contrast of the chirped CAPB for c1 = 1.1 and c2 = −0.005 was 379.00, being greater than that of the unchirped CAPB.

 figure: Fig. 11.

Fig. 11. Intensity distribution of chirped CAPB for c1 = 1.1 and c2 = −0.005 on different observation planes during free-space propagation. (a) z = 0 m, (b) z = 0.3 m, (c) z = 0.59 m, and (d) z = 0.78 m.

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4. Experimental results

We experimentally generated the chirped CAPB and measured its abrupt autofocusing ability. The experimental setup is illustrated in Fig. 12. A Gaussian beam with λ = 532 nm was generated using a solid-state laser (Ventus; Laser Quantum). The Gaussian beam was first expanded by a 50× beam expander, then reflected by a reflective mirror, and finally passed through a 50:50 beam splitter. Half of the light was incident onto a phase-only spatial light modulator (SLM; Holoeye LETO-3 with pixel size of 6.4 × 6.4 µm). The SLM worked in the reflective mode and was used to modulate the amplitude and phase of incident light. The modulated light reflected from the SLM passed through the beam splitter. A Fourier lens L1 with focal length f1 = 50 cm was used to perform the Fourier transform of modulated light on the SLM plane. A circular aperture was placed before L1 to block unwanted diffraction light originated from the hologram and SLM. The distances from the SLM to L1 and from L1 to the initial plane were f1 = 50 cm. Therefore, the chirped CAPB was obtained on initial plane z = 0, which was the rear focal plane of L1. A beam profile analyzer (BGS-USB3-LT665, Ophir, pixel pitch of 4.4 µm) mounted on the electric translation stage along the propagation z axis was used to record the intensity distribution of the chirped CAPB propagating in free space.

 figure: Fig. 12.

Fig. 12. Diagram of experimental setup for generating chirped CAPB and measuring its abrupt autofocusing ability (BE, beam expander; BPA, beam profile analyzer; BS, beam splitter; CA, circular aperture; RM, reflective mirror).

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Figures 13–15 show the measured intensity distribution of the chirped CAPB for c2 = 0 and different c1 values on various observation planes during free-space propagation. For c1 equal to the optimal value of 1.1, the abrupt autofocusing ability of the chirped CAPB was the best. For c1 equal to the saturated value of 2.72, the abrupt autofocusing ability was the same for both the chirped and conventional CAPBs. When c1 was above the saturated value (e.g., c1 = 3), the abrupt autofocusing ability of the chirped CAPB was weaker than that of the conventional CAPB, whereas the focus position of the chirped CAPB always decreased with increasing c1. The formation of three lobes in the inner rings in Figs. 14 (c) and 15 (c) is caused by insufficient pixel size and insufficient modulation accuracy of the spatial light modulator we used.

 figure: Fig. 13.

Fig. 13. Measured intensity distribution of chirped CAPB for c1 = 1.1 and c2 = 0 on different observation planes during free-space propagation. (a) z = 0 m, (b) z = 0.3 m, (c) z = 0.562 m, and (d) z = 0.78 m.

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 figure: Fig. 14.

Fig. 14. Measured intensity distribution of chirped CAPB for c1 = 2.72 and c2 = 0 on different observation planes during free-space propagation. (a) z = 0 m, (b) z = 0.3 m, (c) z = 0.368 m, and (d) z = 0.78 m.

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 figure: Fig. 15.

Fig. 15. Measured intensity distribution of chirped CAPB for c1 = 3.0 and c2 = 0 on different observation planes during free-space propagation. (a) z = 0 m, (b) z = 0.3 m, (c) z = 0.346 m, and (d) z = 0.78 m.

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The measured intensity distribution for c1 = 0 and different c2 values on various observation planes during free-space propagation are shown in Figs. 16 (positive c2) and 17 (negative c2). For c2 = 0.01, which is slightly greater than zero, the abrupt autofocusing ability of the chirped CAPB was substantially enhanced, and the corresponding focus position was shortened. When c2 was a very small negative number (e.g., c2 = −0.005), the abrupt autofocusing ability of the chirped CAPB was obviously reduced, and the focus position increased.

 figure: Fig. 16.

Fig. 16. Measured intensity distribution of chirped CAPB for c1 = 0 and c2 = 0.01 on different observation planes during free-space propagation. (a) z = 0 m, (b) z = 0.3 m, (c) z = 0.688 m, and (d) z = 0.78 m.

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 figure: Fig. 17.

Fig. 17. Measured intensity distribution of chirped CAPB for c1 = 0 and c2 = −0.005 on different observation planes during free-space propagation. (a) z = 0 m, (b) z = 0.3 m, (c) z = 078 m, and (d) z = 0.832 m.

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Figure 18 shows the measured intensity distribution of the chirped CAPB with optimal c1 and a small positive c2 on various observation planes during free-space propagation. The combination of optimal c1 and positive c2 enhanced the abrupt autofocusing ability of the CAPB. The focus position was greater than that for c1 = 1.1 and c2 = 0 and smaller than that for c1 = 0 and c2 = 0.01. The measured intensity distribution of the chirped CAPB with optimal c1 and small negative c2 on various observation planes during free-space propagation are shown in Fig. 19. Because c2 was very small, its negative effect could be offset by the positive effect of c1, strengthening the abrupt autofocusing ability compared with the conventional CAPB. Overall, the experimental results were consistent with the theoretical predictions.

 figure: Fig. 18.

Fig. 18. Measured intensity distribution of chirped CAPB for c1 = 1.1 and c2 = 0.01 on different observation planes during free-space propagation. (a) z = 0 m, (b) z = 0.3 m, (c) z = 0.512 m, and (d) z = 0.78 m.

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 figure: Fig. 19.

Fig. 19. Measured intensity distribution of chirped CAPB for c1 = 1.1 and c2 = −0.005 on different observation planes during free-space propagation. (a) z = 0 m, (b) z = 0.3 m, (c) z = 0.59 m, and (d) z = 0.78 m.

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The measured intensity contrast of the chirped CAPB according to the propagation distance is shown in Fig. 20. The corresponding theoretical result (solid line) was added for comparison. The measurements (dots) agreed with the theoretical results. The solid lines and dots in Fig. 20 confirm that the chirped CAPB exhibited abrupt autofocusing. The intensity contrast of the chirped CAPB maintained a low light intensity before reaching the focus position and suddenly increased when it reached the focus position. Beyond the focus position, the intensity contrast of the chirped CAPBs decreased rapidly with slight oscillations that quickly disappeared. The experimental value of Izp was 399.65 for c1 = 1.1 and c2 = 0 and 439.14 for c1 = 0 and c2 = 0.01.

 figure: Fig. 20.

Fig. 20. Intensity contrast of chirped CAPB according to axial propagation distance z for (a) c1 = 1.1 and c2 = 0 and (b) c1 = 0 and c2 = 0.01.

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5. Summary and conclusions

The effect of the first-order chirped factor on the abrupt autofocusing ability of the CAPB was first investigated. We found saturated and optimal values for the first-order chirped factor. When the positive first-order chirped factor was below the saturated value, the abrupt autofocusing ability of the chirped CAPB was stronger and the focus position was smaller than those of the conventional CAPB. A stronger abrupt autofocusing reduced the focal spot size of the chirped CAPB. At the optimal first-order chirped factor, the abrupt autofocusing ability of the chirped CAPB was the strongest, increasing by 16.4% compared with that of the conventional CAPB. In addition, the focus position of the chirped CAPB was reduced by 27.8% compared with that of the conventional CAPB. For a first-order chirped factor above 0 and not exceeding the optimal value, the abrupt autofocusing ability of the chirped CAPB also increased. Beyond the optimal value of the first-order chirped factor and up to the saturated chirped value, the abrupt autofocusing ability of the chirped CAPB decreased. Nevertheless, increasing the first-order chirped factor always decreased the focus position of the chirped CAPB.

The influence of the second-order chirped factor on the abrupt autofocusing ability of the CAPB was then examined. An arbitrary positive second-order chirped factor could improve the abrupt autofocusing ability of the CAPB. Increasing the positive second-order chirped factor enhanced the abrupt autofocusing ability of the chirped CAPB and reduced the focus position. The second-order chirped factor was more effective than the first-order one for promoting abrupt autofocusing of the CAPB. The abrupt autofocusing ability of the CAPB could be further improved by combining the optimal first-order chirped factor and a positive second-order chirped factor.

Finally, the chirped CAPB was experimentally generated, and the corresponding abrupt autofocusing behaviors were measured. The experimental results were consistent with the theoretical results. Our research can provide an approach to improve the abrupt autofocusing ability of the CAPB by properly choosing the first- and second-order chirped factors. Moreover, these factors retain a solid intensity profile in the focal plane. Owing to the stronger abrupt autofocusing and shorter focus position, the chirped CAPB seems more suitable for practical applications than the conventional unchirped CAPB.

Funding

National Natural Science Foundation of China (11974313, 11874046).

Disclosures

The authors declare no conflicts of interest.

Data availability

The data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

The data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (20)
Fig. 1.
Fig. 1. On-axis intensity distribution of chirped CAPBs for c2 = 0 according to axial propagation distance z for (a) c1 ≥ 0 and (b) c1 ≤ 0.

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Fig. 2.
Fig. 2. (a) Focus position zf and (b) peak on-axis intensity If (0, z) of chirped CAPBs for c2 = 0 according to first-order chirped factor c1.

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Fig. 3.
Fig. 3. On-axis intensity distribution of chirped CAPBs for c1 = 0 according to axial propagation distance z with (a) c2 ≥ 0 and (b) c2 ≤ 0.

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Fig. 4.
Fig. 4. (a) Focus position zf and (b) peak on-axis intensity If (0, z) of CAPBs for c1 = 0 according to second-order chirped factor c2.

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Fig. 5.
Fig. 5. Intensity distribution of chirped CAPB for c1 = 1.1 and c2 = 0 on different observation planes during free-space propagation. (a) z = 0 m, (b) z = 0.3 m, (c) z = 0.562 m, and (d) z = 0.78 m.

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Fig. 6.
Fig. 6. Intensity distribution of chirped CAPB for c1 = 2.72 and c2 = 0 on different observation planes during free-space propagation. (a) z = 0 m, (b) z = 0.3 m, (c) z = 0.368 m, and (d) z = 0.78 m.

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Fig. 7.
Fig. 7. Intensity distribution of chirped CAPB for c1 = 3.0 and c2 = 0 on different observation planes during free-space propagation. (a) z = 0 m, (b) z = 0.3 m, (c) z = 0.346 m, and (d) z = 0.78 m.

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Fig. 8.
Fig. 8. Intensity distribution of chirped CAPB for c1 = 0 and c2 = 0.01 on different observation planes during free-space propagation. (a) z = 0 m, (b) z = 0.3 m, (c) z = 0.688 m, and (d) z = 0.78 m.

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Fig. 9.
Fig. 9. Intensity distribution of chirped CAPB for c1 = 0 and c2 = −0.005 on different observation planes during free-space propagation. (a) z = 0 m, (b) z = 0.3 m, (c) z = 0.78 m, and (d) z = 0.832 m.

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Fig. 10.
Fig. 10. Intensity distribution of chirped CAPB for c1 = 1.1 and c2 = 0.01 on different observation planes during free-space propagation. (a) z = 0 m, (b) z = 0.3 m, (c) z = 0.512 m, and (d) z = 0.78 m.

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Fig. 11.
Fig. 11. Intensity distribution of chirped CAPB for c1 = 1.1 and c2 = −0.005 on different observation planes during free-space propagation. (a) z = 0 m, (b) z = 0.3 m, (c) z = 0.59 m, and (d) z = 0.78 m.

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Fig. 12.
Fig. 12. Diagram of experimental setup for generating chirped CAPB and measuring its abrupt autofocusing ability (BE, beam expander; BPA, beam profile analyzer; BS, beam splitter; CA, circular aperture; RM, reflective mirror).

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Fig. 13.
Fig. 13. Measured intensity distribution of chirped CAPB for c1 = 1.1 and c2 = 0 on different observation planes during free-space propagation. (a) z = 0 m, (b) z = 0.3 m, (c) z = 0.562 m, and (d) z = 0.78 m.

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Fig. 14.
Fig. 14. Measured intensity distribution of chirped CAPB for c1 = 2.72 and c2 = 0 on different observation planes during free-space propagation. (a) z = 0 m, (b) z = 0.3 m, (c) z = 0.368 m, and (d) z = 0.78 m.

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Fig. 15.
Fig. 15. Measured intensity distribution of chirped CAPB for c1 = 3.0 and c2 = 0 on different observation planes during free-space propagation. (a) z = 0 m, (b) z = 0.3 m, (c) z = 0.346 m, and (d) z = 0.78 m.

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Fig. 16.
Fig. 16. Measured intensity distribution of chirped CAPB for c1 = 0 and c2 = 0.01 on different observation planes during free-space propagation. (a) z = 0 m, (b) z = 0.3 m, (c) z = 0.688 m, and (d) z = 0.78 m.

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Fig. 17.
Fig. 17. Measured intensity distribution of chirped CAPB for c1 = 0 and c2 = −0.005 on different observation planes during free-space propagation. (a) z = 0 m, (b) z = 0.3 m, (c) z = 078 m, and (d) z = 0.832 m.

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Fig. 18.
Fig. 18. Measured intensity distribution of chirped CAPB for c1 = 1.1 and c2 = 0.01 on different observation planes during free-space propagation. (a) z = 0 m, (b) z = 0.3 m, (c) z = 0.512 m, and (d) z = 0.78 m.

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Fig. 19.
Fig. 19. Measured intensity distribution of chirped CAPB for c1 = 1.1 and c2 = −0.005 on different observation planes during free-space propagation. (a) z = 0 m, (b) z = 0.3 m, (c) z = 0.59 m, and (d) z = 0.78 m.

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Fig. 20.
Fig. 20. Intensity contrast of chirped CAPB according to axial propagation distance z for (a) c1 = 1.1 and c2 = 0 and (b) c1 = 0 and c2 = 0.01.

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Equations (2)

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U ( r , 0 )  =  A exp ⁡ [ a ( r 0 − r w 0 ) ] A i ′ ( r 0 − r w 0 ) exp ⁡ [ i c 1 ( r 0 − r w 0 ) − i c 2 r 2 w 0 2 ] ,
U ( r , z ) = − i k 2 π z ∫ 0 ∞ ∫ 0 2 π U ( r ′ , 0 ) exp ⁡ { i k 2 z [ r ′ 2 + r 2 − 2 r r ′ cos ⁡ ( φ − φ ′ ) ] } r ′ d r ′ d φ ′ ,
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