Since the Bessel beam was proposed as an exact solution of propagation-invariant mode to the Helmholz equation^[1,2] by Durnin in 1987, it has attracted a great deal of research interest. Due to its properties of nondiffraction and self-healing^[3], the Bessel beam has been widely used in the fields of free-space optical interconnects and communications^[4,5], optical capture and particle manipulation^[6,7], optical micro-nanomachining^[8], optical microscopic imaging^[9,10], and femtosecond laser processing^[11]. In addition, the Bessel beam shows more potential for applications in spatiotemporal light bullets, beam alignment, plasma channels, to name but a few.

Recently, the manipulation of Bessel beams has become an issue of fundamental importance. Many novel behaviors have been developed successively based on the properties of Bessel-like beams. Specifically, employing the transverse-to-longitudinal and spectrum-to-distance mapping of Bessel beams, researchers have proposed some methods to generate Bessel-like beams with propagation-variant polarization^[12] or arbitrary trajectories^[13–17]. Taking advantage of the dependence of the beam on-axis evolution on the spatial spectrum, it is possible to engineer the axial intensity^[18] and polarization^[19] of Bessel beams. The linear Gouy phase shift^[20] of Bessel beams has also been used to realize propagation-variant polarization^[21,22]. In addition, many special structured beams such as helical beams^[23–25] have been constructed by superposing multiple Bessel beams.

On the other hand, researchers have also worked to obtain the self-similar solutions of propagating modes in free space, where they can maintain their transverse profiles under specific stretching during propagation. The most two typical examples are Laguerre–Gaussian beams^[26] and nondiffracting Bessel beams^[1,2]. A few years ago, Xie et al. proposed a class of self-similar beams with different scaling factors by solving the paraxial wave equation^[27,28]. The exact solutions of self-similar Bessel beams can be acquired, but there are still restrictions on the scaling factors. Recently, Efremidis et al. showed the self-similar arbitrary-order Bessel-like beams under tunable stretching transformations based on the Fresnel integral^[29]. The related works either have restrictions in scaling factors or require complicated integral calculations.

In this paper, we propose a new method to construct an arbitrary self-similar Bessel-like beam more straightforwardly by employing the transverse-longitudinal mapping. This method enables the predesigning of the beam width during propagation for both zeroth- and higher-order Bessel-like beams by radially tuning the wave vector cone of the Bessel beam. Through experimentation, we observe these self-similar Bessel-like beams with different width-changing functions, and the experimental results meet our expectations. The proposed method is more intuitive and easier to realize and supports the point-to-point controlling of the beam width. It would be of benefit for exploring applications in optical manipulation of microparticles and optical imaging.

Let us start with the general scalar expression for propagation-invariant Bessel fields, U(r,φ,z)=exp(ikzz)Jm(krr)exp(imφ),(1)where (r,φ,z) denotes the cylindrical coordinate, m is the topological charge, and kr and kz denote the transverse and longitudinal wave vectors, meeting kr2+kz2=k02 and k0=2π/λ. In k-space, the wave vector locates on a cone, with the radius and the slant height being kr and k0, respectively^[30]. We first focus on the case of m=0 [i.e., the zeroth-order Bessel mode J0(krr)]. The beam width is mainly determined by the transverse wave vector kr. The full width at half-maximum (FWHM) of the central peak is approximately W≈2.25/kr^[31]. Thus, kr can be regarded as the proportion coefficient of the Bessel beam. To generate a self-similar beam of which the beam width is stretched during propagation, it is ideal to change the transverse wave vector along propagation, kr(z)≈2.25W(z).(2)

To freely control the transverse wave vector of a beam during propagation, we can take advantage of the transverse-longitudinal mapping of the Bessel beams^[13,32]. Considering that the Bessel beam can be generated by an axicon, we assume that there exists a special axicon segmented by a series of annulus with radius ri (i is the serial number), each of which has a different apex angle [corresponding to transverse wave vector kr(ri)], as shown in Fig. 1(a). Light rays from the annulus (ri→ri+1) focus to the band (zi) on the axis and generate a Bessel beam with width 2.25/kr(ri). If the axicon has a continuously varied apex angle of which the transmission function is expressed by exp[−iΦ(r)], the transverse wave vector would become a continuous function of the radial coordinate, and then we would get a Bessel beam with longitudinally varied beam width [see Fig. 1(b)]. Note that the kr(z) at distance z is determined by kr(r) at the corresponding radius r on the input plane. There is a one-to-one relationship between the radius r and distance z according to the transverse-longitudinal mapping, meeting rz=kr(r)k02−kr2(r).(3)

By combining Eqs. (2) and (3), we can easily obtain the one-to-one correspondence between the beam width W(z) along the propagation axis and the transverse wave vector distribution kr(r) at the input plane, while for kr(r), it denotes the transverse phase gradient, meeting kr(r)=dΦ(r)/dr. Then, we can infer the initial axicon phase Φ(r) to generate the self-similar Bessel-like beam by the integration Φ(r)=∫kr(r)dr.(4)

Thus, the input zeroth-order Bessel-like beam can be expressed as U(r)=J0[Φ(r)].(5)The subsequent self-similar field can be evolved. It is important to point out that the subsequent propagation field at distance z follows the Bessel profile J0[kr(z)r], rather than the expression of Eq. (5).

Notably, the precondition of the above one-to-one correspondence is that the rays from the expanding concentric circles cannot intersect with each other. Namely, the focal distance z of the rays from a circle should increase with the circle radius r, meeting r′(z)>0. Combined with Eqs. (2) and (3), we obtain a constraint on the beam width function of the self-similar Bessel-like beam, W(z)−zW′(z)>0,(6)which is consistent with the conclusion in Ref. [29]. Moreover, according to the mapping, there is a limitation on the self-similar propagation length due to the finite aperture of the input beam. Theoretically, for a circular aperture R, the propagation length zmax can be given by zmax=Rk02−kr(R)2kr(R).(7)

First, we provide a simple example to demonstrate the feasibility of this method. Assuming a linearly varying beam width with W(z)=Az+B, Eq. (6) is satisfied as long as B>0. The axicon phase can be directly calculated by Eqs. (2)–(4), Φ(r)=2.25Br−Ak02Br2.(8)

To verify the self-similar propagation of this beam, we employ the experiment setup containing a spatial light modulator to generate a phase-type hologram, which can encode both the amplitude and the phase into the input beam^[33,34]. A collimated linearly polarized laser beam (He–Ne laser, 632.8 nm) is set as the input beam, and the propagation process of the beam is detected in the same way as in our previous work^[13].

By setting the parameters A=2×104 and B=50μm in Eq. (8), we demonstrate the zeroth-order Bessel-like beam J0[F(r)] with linearly varying FWHM as depicted in Fig. 2, where Figs. 2(a) and 2(b) depict the simulated and experimental propagation processes in the x–z plane, respectively. The predesigned beam width is marked by the black dashed lines, which match well with the beam profile. Specifically, the green dashed lines in Fig. 2(a) display the schematic rays to manifest the transverse-to-longitudinal mapping. It can be seen that the angle of the ray decreases during the propagation and produces an increasingly larger spot. Figure 2(c) shows the comparison of the experimentally measured FWHM (error bar) with the predesigned line (red dashed line) and the simulation result (blue dashed line). The experimental data are derived from the average FWHM of four different cross sections, as shown in the inset of Fig. 2(c). From these results, we can see that the experimentally measured FWHM (50–90 µm) has an absolute error of less than 3 µm. This error is acceptable, given the inevitable distortion of the beam profile generated in experiment. Figures 2(d)–2(f) show the intensity cross sections at different propagating distances, referring to the theoretical Bessel mode (red solid lines). They obviously show that the beam maintains classic zeroth-order Bessel modes with the FWHM changes during propagation. These results perfectly represent the self-similar characteristics of the produced beam.

In addition, the transverse-longitudinal mapping of the Bessel beam shows a bigger advantage in controlling the beam width because the points along the propagation axis are independently modulated to a certain extent. This allows for creating an arbitrary function of the beam width variation, including the piecewise function spliced with several different sections. In practical applications, it is essential to obtain a beam with a fixed width and stable propagation. Thus, we can stop changing the beam width after it reaches a certain value, by setting the beam width variation as W(z)={Az+B(0≤z<z0)Az0+B(z≥z0),(9)where z0 is the turning point. The corresponding phase can be obtained, Φ(r)={2.25Br−Ak02Br2(0≤r<r0)kr0r+Φr0(r≥r0),(10)where kr0=(2.25−Ak0r0)/B, Φr0=Ak0r02/2B, and r0 is given from Eq. (3) by substituting z=z0. In our experiments, we set the parameters A=2×104, B=50μm, and z0=10cm in Eq. (9). Figure 3 presents the experimental results, which carry the same representation as Fig. 2. In Fig. 3(b), the white dashed line marks the turning point z0. It is observed that the FWHM no longer changes after z0. Hence, the beam is restored to the nondiffracting Bessel beam J0(kr0r). Importantly, the nondiffracting distance of this beam is equal to that of the Bessel beam with transverse wave vector kr0. Figure 3(c) shows the measured FWHM, along with the corresponding simulated and theoretical results. These results coincide with each other. Additionally, the beam retains a high-quality Bessel mode during propagation, as illustrated in Figs. 3(d)–3(f).

In theory, the phase Φ(r) can be calculated by Eqs. (2)–(4) according to the beam width function. However, it is hard to obtain the analytical solutions for a complicated width function W(z). Nevertheless, it is convenient to numerically solve the phase. Here, we select a typical sinusoidal function W(z)=Asin(2πz/T)+B, which represents an oscillated beam width. According to Eq. (6), this type of self-similar beam is only valid within some distance. According to Eq. (6), there is an upper limit on the amplitude A of the sinusoidal width. For instance, for given T=20cm and B=55μm, the beam width should be less than Amax=8.75μm to guarantee the stable propagation in one period. Considering the edge diffraction effect, we conduct our investigations on the oscillated self-similar beam by setting A=5μm, T=20cm, and B=55μm. The theoretical and experimental results are shown in Fig. 4. The variation of the FWHM of the beam basically follows the theoretical and simulated curves during propagation. Even though there are some local deviations, the beam width follows a sinusoidal variation overall. Notably, the simulated result also deviates from the ideal sinusoidal curve [see Fig. 4(c)], of which the maximum relative error is ∼4.5%. This indicates that a more complicated beam width variation has a larger error caused by the nonlinearity of the transverse-longitudinal mapping. In specific calculations, we improve the computing accuracy, and obtain similar simulated results, while the errors still are inevitable. Additionally, as shown in Figs. 4(d)–4(f), the intensity cross sections at different propagating distances demonstrate that the produced beam can persist with the Bessel profile during propagation. Note that the beam width oscillation is only shown in one period. To achieve the oscillated beam width at a longer distance, we need increase the parameters T or B, or reduce A, according to Eq. (6).

Finally, we show that the proposed method of constructing a self-similar beam is applicable not only for zeroth-order Bessel beams (m=0), but also for higher-order ones (m≠0). Thus, we can generate a vortex Bessel-like beam with any integer order and adjustable beam width. Specifically, for a vortex Bessel-like beam, the beam width can be expressed as the radius of the first ring lobe (i.e., the distance from the origin to the location of the peak intensity) described by W(z)=Xm/kr(z). Here, Xm is a proportional coefficient determined by the order of the Bessel function m. Generally, X1=1.841, X2=3.054, and X3=4.201. We experimentally demonstrate first- to third-order self-similar vortex Bessel-like beams Jm[F(r)] with a sinusoidally varied beam width. The variation functions of the beam widths are identical to that in Fig. 4. The corresponding phases Φ(r) are calculated with Eqs. (2)–(4), but replacing 2.25 by Xm in Eq. (2). The simulated and experimental results are shown in Fig. 5 and are basically consistent with the theory. Notably, the asymmetry intensity profile of the experimental results is attributed to the errors of the experimental system arising from generation and detection. Yet the measured beam width [see Figs. 5(g)–5(i)] coincides well with the theory.

In conclusion, based on transverse-longitudinal mapping, we show the generation of self-similar Bessel-like beams of arbitrary orders with tunable stretching profiles during propagation and the constraint on the beam width function. We experimentally demonstrate three types of typical self-similar Bessel-like beams with width variations that can be described by linear, piecewise, and periodic functions of propagation distance, respectively. The experimental results match well with the theoretical predictions when the beam width function satisfies the constraint. The proposed approach also enables the generation of self-similar higher-order vortex Bessel-like beams. We expect the proposed method can be also applicable to other nondiffraction beams containing the conical wave vector. The above results would be of benefit for exploring applications in optical manipulation of microparticles and optical imaging.