An optical vortex beam, which carries orbital angular momentum (OAM), has aroused increasingly the research interest because of its wide potential applications, ranging from optical manipulation^[1,2] to optical communication in free-space.^[3] The spatial distribution wave front of a vortex beam can be described as , where p is the topological charge (TC) and φ represents the azimuth angle. Various diffractive optical elements, such as vortex encoding gratings,^[4] holographic gratings,^[5] binary phase Fresnel zone plates,^[6] spiral phase plate (SPP),^[7,8] and spiral zone plate (SZP)^[9,10] have been used to generate optical vortices. In particular, SZP offers great advantages of compactness, light weight, high alignment, and high degree of design flexibility. Optical vortex beam generated by a computer-generated SZP has been demonstrated in Ref. [11]. However, the focused vortex diverges rapidly after passing through the focal plane since a conventional zone plate provides only one focal spot for one shot of laser beam, which greatly limits the scope of applications. Therefore, many different kinds of generators for beams with multiple optical vortices have been proposed recently, such as a vortex grating^[12] and a three-dimensional (3D) twister superlattice embedded with vortex spirals.^[13]

Fractal zone plate (FZP) is a meaningful method of producing a sequence of focal planes and the internal structure of each focus exhibits a characteristic fractal structure.^[14] In 2006, Tao et al.^[15] and Cheng et al.^[16] proposed a new family of zone plate named spiral fractal zone plates by combining the phase modulation function of helical phase structure and the properties of the FZP to generate a series of focused optical vortices along the propagation direction. In addition, it has been reported that different aperiodic zone plates combined with a vortex lens could produce a chain of vortices with tunable separation, strength and transverse section.^[17] However, the fabrication of a perfect helical phase structure with the required surface variation is a challenge for the current production technology. More recently, Yu et al.^[18] presented a kind of spiral Dammann zone plate with a binary structure to generate a coaxial dipole vortex array. However, in an optical system, a focusing objective must be placed after this zone plate, which may be difficult for working in the extreme ultraviolet or x-ray regions.

In this paper, we present a technique for producing a series of optical vortices along the optical axis based on a simple compact system by using binary fractal spiral zone plates (FSZP). The key idea of the proposed FSZP is to combine the advantages of SZP and Cantor fractal structure along the square of the radial coordinates together. Unlike the phase-only zone plate loaded onto the SLM in Ref. [19], the design and fabrication by applying an electron beam lithography to the FSZP are presented and the corresponding axial irradiance is examined. An FSZP with 6.4-mm diameter and 0.2-m focal-length is fabricated by photo-etching technology. We also demonstrate the observation of generating optical vortices with different TCs. This ZP is meaningful for various areas, and the typical binary structure will be beneficial to multi-wavelength application.

The SZP can be considered as a variation of Fresnel zone plate combined with radial Hilbert transform (RHT) filtering operation. Therefore, the phase function of the SZP can be expressed as

Fig. 1

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	Fig. 1 Schematic view of zone plate in (r 2, θ) coordinate system and (x, y) coordinate system (a) SZP with p = 1, m = 14; (b) FSZP with p = 1, N = 2, S = 3.

The fabrication process of FSZP by using the photo etching technique is similar to that of conventional binary SZP. To obtain a better focusing performance, in this work a specific FSZP with p = 1, N = 2, and S = 4 was fabricated with the processing parameters in Table 1 based on the photo-etching technique. As illustrated in Fig. 2(a), the fabrication process consists of four main steps. First, a 100-nm-thick chromium layer was spin-coated on a 2.25-mm-thick soda-lime glass substrate, sequentially photoresist (AZ1500, AZ electronic materials, USA) with a thickness of 500-nm was spin-coated onto it. Then, the FSZP pattern formed on the photo resist which was placed in a direct writing laser at a wavelength of 413 nm (DWL 2000, Heidelberg Instruments, Germany) and developed in a 6-wt% sodium hydroxide solution. After exposure, the opening parts of the chromium layer were removed by using a chromium etchant to transfer the pattern from the photoresist to the chromium layer. Finally, we removed the remaining resist by using chemicals. The microstructure image of the obtained FSZP is shown in Fig. 2(b).

Fig. 2

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	Fig. 2 (a) Schematic flow chart of fabrication process of FSZPs which consists of 1 chromium layer and photoresist coatings, 2 exposure, 3 chromium etching, and 4 resist removal; (b) microscopic image of fabricated FSZP sample.

Table 1.

Table 1.

Table 1. Fabrication parameters. . Focal distance 200 mm Wavelength 632 nm Last zone width Diameter 6.4 mm Number of zones 41 Thickness 2.25 mm Membrane chromium layer Substrate soda-lime glass

Table 1. Fabrication parameters. .

The FSZP can be used as a single element objective to optically implement the radial Hilbert filtering operation and the Fresnel zone plate focusing operation in one step. Therefore, numerical simulations for the focusing characteristics of FSZP can be performed by using the Fresnel–Kirchhoff diffraction equation based on the scalar diffraction theory. The distribution of complex amplitude after diffractive plane can be expressed as

To verify our design and evaluate the advantage of FSZPs by increasing the depth of focus, we compare the axial intensity distribution of an FSZP with a corresponding SZP under the same circumstance. Here, these parameters are chosen to be λ = 632 nm, f = 200 mm, p = 1, N = 2, and S = 4. As shown in Figs. 3(a) and 3(b), we can see that doughnut-shape patterns are achieved in the primary focal plane for both SZP and FSZP. The difference is that a series of subsidiary vortices in the other focal planes (180, 193, 207, and 225 mm, respectively) for FSZP can be clearly observed, which is considered as splitting from the primary focus. With the increase of the s value, the number of subsidiary foci increases, with the whole axial range unchanged, which is times higher than the number of subsidiary foci in the previous fractal structure. Other major foci that are also surrounded by subsidiary foci appear at distances of f/7, f/5, and f/3 have also been observed in this process. Figure 3(c) and 3(d) show the diffraction patterns of the SZP and FSZP at calculated focal planes, from which we can clearly see that the SZP beam is divergent quickly at positions before or after focal length. In contrast, the vortices of FSZP beam have almost the same size at the plane of the primary focus as those for the cases of the other four subsidiary foci.

Fig. 3

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	Fig. 3 Intensity distribution of (a) SZP with p = 1, (b) FSZP with p = 1, N = 2, S = 4, and calculated cross-section irradiance of (c) FSZP and (d) SZP.

The central point of the optical vortex is called a singular point because of the undefined phase and zero intensity.^[20] Around the dark core, each focus of FSZP beam as shown in Fig. 3(c) exhibits the variation of a helical phase that is an integer multiple of 2π. Therefore, the phase distribution of optical vortices can be expressed as under the paraxial approximation, and the intensity of OAM is . The value of TC has a great influence on the characteristics of optical vortices. Figure 4 shows the FSZP with different TCs and the calculated intensities of light fields on the first subsidiary focal plane, the main focal plane and the second subsidiary focal plane, respectively. Noticeably, here we normalized the intensity patterns for better exhibiting the results. In the case of p = 1, the foci present bright rings with uniformity and symmetry at the main and subsidiary focal plane. Generations of higher order TC p = 2 are shown in the middle line of Fig. 4, where relatively large bright intensity rings are captured in the main focal plane. However, the two subsidiary foci have a tendency to break down into two lobes due to the particular helical phase structures of optical vortices. As the value of TC increases to p = 3, the size of main focus keeps growing and the phase variations of subsidiary foci are more obvious. Such optical vortex beams can trap or guide particles in their dark center or drive particles to spin by transferring the angular momentum to them.^[21]

Fig. 4

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	Fig. 4 Diffraction patterns of the main and subsidiary foci for FSZP with different TCs.

Experiments were carried out by using the fabricated zone plate and the experimental arrangement is shown schematically in Fig. 5(a). The incident light with a wavelength of 632 nm was expanded and collimated by a collimator lens to illuminate the FSZP sample. The uniform parallel light was then modulated by the FSZP and then focused on the focal planes. A charge-coupled device (CCD) (Lumenera Infinity, 2048 × 2448, /pixel) camera was located on a sliding guide behind the FSZP to record the intensity distributions of reconstructed beam at different positions. It should be noted that a filter should be placed in front of the CCD to avoid the camera from becoming saturated. To make a direct comparison with simulation, we recorded the cross-section intensities of the FSZP beam away from 150 mm to 250 mm for 100 images at different positions. Figure 5(b) shows the recovered axial irradiances of the FSZP along the propagation direction. As expected, a beam with sequence of vortex foci was generated by the fabricated FSZP. To further validate the experimental data, Figure 5(c) presents the comparison curves of the intensity distributions obtained from the experiment and simulation with the same parameters. Typically, we take the curves crossing over the most intense center of the vortex in the focal plane of the primary focus to represent the distribution of foci due to the same size of sequence vortices along the propagation direction at different focal positions. Overall, the experimental characteristics of generation accord with simulations considering that the only difference in the similar diffraction feature (such as the same positions and sizes of the main and subsidiary foci) is a relatively high background noise, which may be caused by the interference of the external light in the laboratory.

Fig. 5

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	Fig. 5 (a) Schematic diagram of experimental setup; (b) longitudinal intensity distribution on meridian plane; (c) measured normalized peak intensity distribution of vortex ring along e optical axis.

The captured intensity distributions for focused vortices at different positions of the FSZP beam are shown in Fig. 6(a). As expected, the main focus that is located at 200 mm exhibits a uniform intensity distribution while the other four subsidiary foci are founded at 178, 194, 206, and 227 mm, respectively. As the beam spreads along the propagation direction, the sizes of the optical vortices stay almost the same within the whole focal volume, which can be explained as an extension of the depth of focus. The detailed characteristics of the main focus of the FSZP are shown in Fig. 6(b), the marks represent each pixel of the CCD. As we can see, the two peaks are completely symmetric and the radius of the doughnut pattern is about . A comparison shows that the experimental results including the number of foci and size of the vortices are in good agreement with the simulations, in which the limitation of the CCD resolution ( /pixel) is taken into account. In the calibration, experimental errors can emerge, which are related to the instability of the light source, the accuracies of the instruments, and the recovery of the vortex beam. Figure 6(c) shows the curves of normalized intensity profiles of the obtained main focus and subsidiary foci along the longitudinal axis. Obviously, the curves are fitted well with each other, implying that the sequence optical vortices generated by FSZP exhibit almost constant lateral dark core sizes. Therefore, we anticipate that FSZP can provide a complementary and versatile high-resolution nondestructive tool for particle manipulation and microscope imaging.^[22]

Fig. 6

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	Fig. 6 (a) Intensity distribution of the main and subsidiary dark cores of FSZP, (b) detailed characteristics of main focus, (c) intensity profiles along x axis on main and subsidiary focal planes.

We propose FSZP as an alternative diffractive element for the generation of optical vortex beams with multiple foci. The focal depths and cross-section irradiances at different positions of the FSZP are simulated and the influences on the generated vortices with different TCs are discussed. As presented in this work, an FSZP with p = 1, N = 2, and S = 4 is fabricated by using the electron beam lithography. According to this, we also experimentally demonstrated the remarkable ability of FSZP to generate the sequence optical vortices with a larger depth of focus. One potential application of this new design with an increased depth of focus is its integration in a three-dimensional optical alignment system.^[23] In addition, the FSZP may become a potential alternative to replace the conventional vortex array generators in the x-ray,^[24] extreme ultraviolet,^[25] and microwave^[26] applications.