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Abstract
We introduce what we believe to be a new family of abruptly autofocusing waves named autofocusing Bessel beams (ABBs). Since the beams only strongly influence the area near the focus, it holds promise for medical laser treatment and optical tweezers. By the angular spectrum method, ABBs are proved to be a class solution for the Helmholtz equation. The focal length is well-defined and easily tuned in our mathematical description. Under the finite energy limitation, the abruptly autofocusing and vortex characteristics of Gaussian-modulated ABBs are studied. Interestingly, we found a kind of abruptly autofocusing waves focusing twice on the propagation axis, which is formed by an ABB passing through a focusing lens. Dual-focus ABBs make it possible for a single laser to manipulate two particles on the propagation axis simultaneously. In the experiment, the autofocusing of ABBs and the dual focus of ABBs passing through a focusing lens are observed. This article provides a theoretical model and experimental protocol for studying abruptly autofocusing waves.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Abruptly autofocusing waves have application prospects in controlled filament generation at particular spatial locations [1,2], optical capture and manipulation of dielectric particles [3–5], and medical laser treatment [6,7]. Such beams have a unique property: their maximum intensity changes slowly during propagation and suddenly increases by orders of magnitude near the focus. It means that the beams strongly influence only the area near the focus. 2010, circular Airy beams with the abruptly autofocusing property were first proposed [6], and the beams can be achieved by computed holography [1]. Then, by using the Fast Fourier Transform to analyze the propagation process, circle Pearcey beams have also been shown to achieve abruptly autofocusing [8]. Until now, abruptly autofocusing waves as an exact solution to the Helmholtz equation has not been reported.
Diffraction theory is a very successful tool for studying laser propagation. Many structure beams are solutions of the Helmholtz equation, such as Bessel beams [9], Laguerre-Gauss beams [10], Hermite-Gauss beams [11], and Airy beams [12] etc. Angular spectrum theory is an essential method for solving the wave equation. The solution of the wave equation is constructed as the superposition of coherent plane waves, which are weighted in amplitude by angular spectrum function [13]. Recently, some new solutions to the Helmholtz equation have been reported, such as incomplete gamma beams [14], radial carpet beams [15], perfect vortex beams [16], and asymmetric Laguerre-Gaussian beams [17]. Structured beams greatly enrich the application of optical field modulation [18–22].
We obtain a family of solutions for the scalar Helmholtz equation named autofocusing Bessel beams (ABBs). ABBs will automatically focus without an initial focusing phase. The focal length is a well-defined physical quantity in the mathematical expression of AABs. Compared with other types of abruptly autofocusing waves [6,23,24], ABBs can maintain their profile shape (the Bessel function of r2) unchanged during the focusing process. Using the angular spectrum method, we derive the analytical solution for this beam propagating in free space. It is important to note that ABBs differ from non-diffracting Bessel beams in their physical properties and mathematical forms. In addition, optical tweezer techniques generally use focusing beams to manipulate particles [25–27]. ABBs passing through a focusing lens have two focuses on the propagation axis, and both focuses have abruptly autofocusing property. Therefore, dual-focus ABBs will significantly enrich the dimensions of optical manipulation. The propagation dynamics in these processes are analyzed. Finally, the correctness of the theoretical model is verified by experiment. The beams have practical value for medical laser treatment and optical tweezers.
2. ABBs – solutions to the scalar Helmholtz equation
We consider $U(x,y,z)$ as the complex amplitude of monochromatic light, which satisfies the following differential equation called the scalar Helmholtz equation [28]:
k is the wave number. Under the paraxial approximation, the solution of the plane wave superposition form of Eq. (1) can be expressed as [13]Using the Jacobi-Anger identity and the following standard integral [29]
Substituting Eq. (6) into Eq. (2) and employing the following standard integrals [29]
According to the definition of intensity $I = E{E^\ast },$ the intensity of ABBs is
It is evident that the beams have a circular symmetry profile, and the profile shape (the Bessel function of r2) remains unchanged during propagation. Unlike the well-known non-diffracting Bessel beams depending on the Bessel function of r, ABBs depend on the Bessel function of r2. Both k and f determine the size of the beam profile. Unless otherwise specified in this article, the calculation parameter is k = 106m−1, f = 0.1 m.
The maximum intensity Imax of zero-order ABBs is always on the z-axis:
Equation (11) indicates that Imax satisfies the Lorentzian distribution. Figure 1 shows the dynamics of the ABB with l = 0. As is apparent from Fig. 1(a), the beam radius of the ABB first increases and then decreases. Thus, the beam has an apparent focusing behavior, and the energy is concentrated near the focus. As shown in Fig. 1(b), Imax changes slowly during propagation but suddenly increases by orders of magnitude near the focus, resulting in an extremely strong abruptly autofocusing effect. The beam's intensity is a singularity with an infinite magnification at the focal point. These features are unique to the family of abruptly autofocusing waves. Unlike the self-focusing effect produced by the Kerr medium, the abruptly autofocusing of ABBs, which results from the optical field structure itself, is a purely linear process. ABBs can be regarded as circularly symmetric self-accelerating beams. Near the focus, the energy rushes in an accelerating manner toward the focal point, resulting in a sudden increase in the maximum intensity. As ABBs propagate beyond the focus, the energy accelerates away from the focal point, and the maximum intensity decreases. Figure 1(c) shows the variation of the angular spectrum with the radial spectral component ρ. Clearly, the angular spectrum oscillates between positive and negative values, and its envelope decreases with ρ.
Fig. 1. Dynamics of the ABB. (a) Variation of I with r and z; (b) variation of Imax with z; (c) angular spectrum.
The abruptly autofocusing behavior can be explained by the variation of the radial phase distribution. It is commonly believed that the focusing phase is proportional to -r2, and the diverging phase is proportional to r2. The radial phase of Eq. (9) is
For the convenience of comparing the radial phase, $\mathrm{\Psi }$ is not taken as the remainder of $2\pi$. Figure 2 shows the difference in ABBs before and after the focus, l = 0. Obviously, the beam has a bright spot at the center, extending outward as a series of increasingly tightly arranged concentric intensity rings. By comparing Figs. 2(a) and (c), it is found that the beam profile increases with f. $\mathrm{\Psi }$ before the focal point, which is proportional to -r2, is similar to a focusing lens phase [Figs. 2(a) and (c)]. From Eq. (12), the closer the beam is to the focus, the larger the focusing phase. It explains the abruptly autofocusing behavior near the focus. In contrast, $\mathrm{\Psi }$ after the focal point, which is proportional to r2, is similar to a negative lens phase [Figs. 2(b) and (d)], causing the diverging of the ABB behind the focus.
Fig. 2. Variation of I and $\mathrm{\Psi }$ of the ABBs with r. (a) f = 0.1 m, z = 0.5f ; (b) f = 0.1 m, z = 1.5f ; (c) f = 0.2 m, z = 0.5f ; (d) f = 0.2 m, z = 1.5f.
3. Transformation of ABBs passing through the lens
This section provides two cases of the transformation of ABBs passing through the lens. In the first case, an ABB described by Eq. (3) is located in the front focal plane of a Fourier lens with the focal length ft. The back focal plane is the Fourier plane. Using the Fourier transform relation [16], we calculated the electric field distribution of ABBs in the Fourier plane as follows:
Comparing Eq. (13) with Eq. (3) reveals that the electric field in the Fourier plane turns into the ABB with the focal length ft2/f. In other words, ABBs can be expanded or contracted by a Fourier lens. We know that non-diffracting beams become perfect vortex beams after the Fourier transform. In contrast, ABBs maintain their own profile shape after the Fourier transform.
In the second case, we consider placing a focusing lens on the initial plane of ABBs. The initial field of this case can be expressed as
By repeating the derivation process from Eq. (2) to Eq. (9), we use the angular spectrum method to obtain the electric field distribution of ABBs passing through a focusing lens at z
For the condition of f > ff, ABBs passing through a focusing lens have two focuses, which are located at f1 and f2, respectively. We refer to the beams described in Eq. (15) as dual-focus ABBs. For l = 0, the expression for the maximum intensity Imax is
Imax of dual-focus ABBs are singularities with infinite intensity at f1 and f2. Figure 3 shows the dynamics of the dual-focus ABB, l = 0. From Fig. 3(a), it is clear that the beam focuses twice during propagation. By analyzing Imax variation with z, the changing of the focal point is investigated. Comparing Fig. 3(b) with Fig. 1(b), we find that the ABB passing through a focusing lens has two focal points with abruptly autofocusing property, located at f1 = 0.033 m and f2 = 0.1 m, respectively. The dual focus can be explained by the radial phase of Eq. (15). In Fig. 3(b), we use - to denote the range where the radial phase is the focusing phase and + to denote the range where the radial phase is the diverging phase. Between 0-f1, the radial phase is the focusing phase, resulting in the first focusing of the dual-focus ABB. The radial phase between f1-zt is the diverging phase, causing the beam to diverge. Then, the radial phase between zt-f2 turns into the focusing phase, resulting in the second focusing of the dual-focus ABB. Optical tweezers typically use focusing beams to manipulate particles. The dual focus of ABBs passing through a focusing lens makes it possible for a single laser to manipulate two particles on the propagation axis simultaneously.
Fig. 3. Dynamics of the dual-focus ABB, ff = 0.05 m, zt = 0.067 m. (a) Variation of I with r and z; (b) variation of Imax with z.
Compared to chirped annular Bessel Gaussian beams [23], ABBs will automatically focus without requiring a chirped term. After adding a chirped term (focusing lens phase), ABBs have two focal points with abruptly autofocusing property. In addition, the maximum intensity of zero-order ABBs is always on the propagation axis, while the maximum intensity of chirped Pearcey beams [24] is on the self-accelerating curve.
4. Finite energy ABBs
Since the integral $\int_0^\infty {\textrm{Ir}\textrm{d}r}$ of Eq. (10) is divergent, ABBs have infinite energy. Therefore, ABBs cannot be fully realized physically, but the analysis of the beams is of theoretical value. Further, we consider ABBs under the finite energy limitation. Adding Gaussian term modulation to ABBs, the initial field of finite energy ABBs is
The electric field distribution for finite energy ABBs at z is
Figures 4 and 5 show the propagation of the finite energy ABBs, l = 0. The comparison between Fig. 4(b) and Fig. 1(b) reveals that the beam is a finite intensity value at the focal point rather than the singularity. As shown in Fig. 4(a), for large w, the finite energy ABB, which preserves the center and part of the sidelobes, is close to the ideal ABB. The beam has an explicit abruptly autofocusing behavior during propagation, and the maximum intensity is still located at the focal point f, as shown in Figs. 4(b) and (c). Figure 5(a) shows that for small w, the finite energy ABB mainly affected by the Gaussian term can be approximately regarded as a Gaussian beam. In this case, the beam has no focusing behavior during propagation [Fig. 5(c)], and Imax decreases with increasing z [Fig. 5(b)]. Autofocusing of finite energy ABBs occurs only when the waist of the Gaussian modulation term can contain many sidelobes.
The topological charge significantly affects the optical field distributions. The propagation of the high-order finite energy ABB is shown in Fig. 6, l = 1. In the initial plane, the high-order finite energy ABB has a hollow intensity distribution and a vortex phase, as shown in Figs. 6(a) and (b). From Fig. 6(c), intensity on the z-axis is always a zero value throughout the propagation due to the phase singularity at the beam center. In Fig. 6(d), the high-order finite energy ABB at the focal point is a bright ring. This high-power density vortex beam affecting only a specific region has potential applications for optical trapping and rotation.
Fig. 6. Intensity and phase distributions of the high-order finite energy ABB, w = 0.7 mm. (a) (c) (d)Intensity; (b) (e)phase.
5. Experiment
Our method of generating ABBs is as follows: the phase mask is illuminated by a collimated beam to form the ABB as first-level diffraction; by filtering with a small aperture behind the Fourier lens, only the ABB is present in the Fourier plane. Since the phase mask size is limited, the beams generated in the experiment are finite energy ABBs. The experimental setup for generating finite energy ABBs is shown in Fig. 7. The pulsed laser (wavelength 1064 nm) is collimated and expanded by a collimating lens. A polarizer is used to obtain linearly polarized beams. According to the computational holography method [3], we obtain the expression of the phase masks ${|{\textrm{exp} (imx) + 2E({\textrm{r},\varphi ,0} )} |^2}$ based on Eq. (3). The screen of SLM (UPOLABS, HDSLM80R) loads the phase mask shown in Fig. 7, m = 10000. When a collimated linearly polarized beam illuminates the phase mask, the first-level diffraction of the reflected wavefront is a finite energy ABB with the focal length 0.4 m. Then, the reflected wavefront is Fourier transformed via a lens L1 with the focal length 0.2 m. The aperture placed behind the L1 is used for blocking undesired zero-level diffraction. Selecting the Fourier plane (back focal plane) of L1 as the initial plane, the intensity is recorded by moving a CCD camera (Thorlabs, SP928) along the beam propagation direction.
Fig. 7. Experimental setup: L, laser; CL, collimating lens; POL, polarizer; BS, beam splitter; SLM, spatial light modulator; PM, phase mask; L1, Fourier lens; A, aperture; L2, focusing lens; CCD, CCD camera.
According to the relevant theory in Section 2, the focal length of ABB obtained in the Fourier plane is 0.1 m. In the absence of L2, the intensity distributions of the finite energy ABBs are measured with a CCD camera, as shown in Fig. 8, f = 0.1 m. We provide theoretical results calculated based on Eq. (20) for comparison with experimental results, w = 1 mm. The finite energy ABBs consist of a central main lobe and a series of concentric intensity rings. At the focal point f, the zero-order finite energy ABB is a bright point [Fig. 8(a)]. Instead, the high-order finite energy ABB is a bright ring at f [Fig. 8(b)]. There is an apparent autofocusing process of the beams during propagation. The experimentally observed focal position of the finite energy ABBs is consistent with the theoretical description.
Fig. 8. Propagation of the finite energy ABBs. (a)l = 0, the experimental (upper) and theoretical (lower) results; (b)l = 1, the experimental (upper) and theoretical (lower) results.
A focusing lens L2 with ff = 0.05 m is placed in the Fourier plane of L1. The intensity distributions and the relative maximum intensity Imax of the finite energy ABB passing through a focusing lens are measured, as shown in Fig. 9, l = 0. From Fig. 9(a), the beam is focused as a bright point at f1 = 0.033 m and f2 = 0.1 m, respectively. Unlike the theoretical description in Section 2 (Fig. 3), the beam is not singularity at focal points f1 and f2 since the experimentally generated dual-focus ABB is of finite energy [Fig. 9(b)]. The ratio of peak intensity of the two focal points is ${{I_{_{\max }}^{{f_1}}} / {I_{_{\max }}^{{f_2}}}} = 2.9$. Axial multi-focus beams are usually formed by coaxial recombining focusing beams with different focal lengths [30]. When one of the sub-beams is focused at the specific focal point, the other sub-beams are not focused at this focal point. It means that the main energy of common axial multi-focus beams is not concentrated near the focal point. The most distinguished feature of dual-focus ABBs from other axial multi-focus beams is that, in each focal plane, the main energy of dual-focus ABBs beam is concentrated near the focal point. In contrast to the benefits of other axial multi-focus beams in high-resolution imaging, dual-focus ABBs have certain advantages in particle manipulation.
Fig. 9. Propagation of the finite energy ABB passing through a focusing lens in the experiment, ff = 0.05 m, f1 = 0.033 m, f2 = 0.1 m. (a) Intensity distributions; (b) variation of Imax with z.
6. Conclusions
We used the angular spectrum method to obtain a kind of mathematical description of abruptly autofocusing waves named autofocusing Bessel beams (ABBs). The complex amplitude of ABBs depends on the Bessel function of r2. In this mathematical expression, the focal length is a well-defined physical quantity that can be tuned conveniently. ABBs have a focusing lens phase in the near field, which increases as it approaches the focal point. This feature enables the beam to gain abruptly autofocusing ability. Ideally, an ABB has beam singularity at the focal point. In practical applications, we need to consider finite energy ABBs with Gaussian term modulation. For larger Gaussian beam widths, finite energy ABBs can maintain the abruptly autofocusing property. In addition, ABBs can be extended to high-order modes with orbital angular momentum. Interestingly, ABBs passing through a focusing lens have two focuses on the propagation axis. Dual-focus ABBs have potential applications in optical manipulation. We have experimentally obtained finite energy ABBs by irradiating the phase mask and observed the autofocusing and the dual focus of ABBs. This class solution of the Helmholtz equation also applies to other linear wave systems, such as elastic and sound waves.
Funding
Natural Science Foundation of Jilin Province (Grant No.20210101154JC).
Acknowledgments
We are very grateful for the aforementioned funding. We appreciate the reviewers for their valuable comments and suggestions.
Disclosures
The authors declare no conflicts of interest.
Data availability
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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