A light beam carrying orbital angular momentum (OAM) with a helical wavefront, which can be represented as $\exp (-il\phi )$ with a phase singularity in the center, is called an optical vortex^[1–3]. The unique optical properties of optical vortices have inspired both fundamental and emerging fascinating applications, such as nano/microparticle trapping^[4] and manipulation^[5], microscopic imaging^[6]. So far, many mature methods have been carried out to produce light with designed OAM modes, such as spatial light modulators, fork gratings^[7], spiral phase plates^[8], and computer generated holograms^[9]. Such traditional devices play significant roles in diverse optical systems. However, these thick and bulky devices are unsuitable for system integration and miniaturization; on the other hand, they either have limited working bandwidth or can only generate vortices with quite limited modes. Recently, the abundant practical applications of optical vortices call for more complex generation and manipulation of vortex beams. For example, under the scenes in which different OAM modes need to be fed into systems within one component^[10,11], multiple/continuous ordered vortex beams are demanded; for trapping of tiny particles and stimulated emission depletion microscopy^[12], non-diffracting vortex beams are required. Therefore, in this work, we propose a broadband multifocal graphene vortex generator, which can produce broadband angular momentum cascade containing continuous integer non-diffracting vortex modes. Our multifocal graphene vortex generator is composed of a multi-turn spiral slit structure, which naturally embodies a continuous spiral slit vortex generator and a zone plate. Thus, the device enables the generation of broadband high-quality continuous vortex modes with multiple foci of deep depths. Meanwhile, the generated vortex modes can be simultaneously tuned through changing the incident wavelength or position of the focal plane. The elegant structure of our multifocal vortex graphene generator largely improves the design efficiency and can be fabricated using facile laser nanofabrication in a single step^[13], which provide possibilities for future scalable production. Moreover, thanks to the exceptional optical properties of grapheme^[14], our proposed broadband multifocal vortex generator (BMVG) may provide new possibilities in enormous practical applications such as dark-field imaging, nanoparticle’s manipulation, and optical information processing, even in harsh and/or extreme environments^[15], such as aerospace applications.

The distribution of our BMVG can be simply expressed as $$F(\theta )=\sqrt{r_0^2+2l_0{z}_0{\lambda }_0·\frac{\theta }{2\pi }},\theta \in [0,M·2\pi ],(M\ge 3,r_0\ll \frac{z_0}{b}),$$(1)where $\theta$ denotes the azimuthal angle, $r_0$ denotes the starting radius of the structure, $z_0$ is the pre-designed focal plane, $l_0$ is the pre-designed topological charge at focal plane $z_0$, and ${\lambda }_0$ is the pre-designed working wavelength. Instead of a normal spiral structure where the $\theta$ extends from 0 to $2\pi$, we make $\theta$ extend from 0 to $M·2\pi$ ($M\ge 3$) to obtain a multi-turn ($M$ turns) spiral structure, as shown in Fig. 1(a). When dissembling the structure, our BMVG can be expressed as follows: $$F(\theta )=\sqrt{F_1{(\theta )}^2+F_2{(\theta )}^2},\theta \in [0,2\pi ].$$(2)

Here, $$F_1(\theta )=\sqrt{N{\lambda }_0{z}_0},\theta \in [0,2\pi ],$$(3)where $N=2l·(n-1)$, $n=1,2,\dots ,M$, and $$F_2(\theta )=\sqrt{r_0^2+2l_0{z}_0{\lambda }_0·\frac{\theta }{2\pi }},\theta \in [0,2\pi ],(r_0\ll z_0).$$(4)

As is shown in Fig. 1(a), the zone plate structure defined by Eq. (3) can generate multiple foci with multiple deep depths of foci (DoFs) at focal planes of $z=z_0/m$, $m=1,2,3,\dots$, since light penetrating from the ring areas always has constructive interference at $z=z_0/m$, $m=1,2,3,\dots$, as long as the Fresnel approximation can be satisfied. In the meantime, the Fermat spiral slit structure defined by Eq. (4) can produce a Bessel-like beam carrying a series of continuous OAM modes along the propagation direction, and the topological charges carried by the light beams are inversely proportional to the propagation distance. Since our BMVG possesses the properties of the above two devices simultaneously, our BMVG can produce a series of non-diffracting continuous integer OAMs at $z=z_0$, $z_0/2,\dots ,z_0/m$ with topological charges of $l=l_0$, $2l_0,\dots ,m{l}_0$, $m=1,2,\dots$, respectively.

To illustrate our idea, we design our BMVG according to Eq. (1) by choosing ${\lambda }_0=633\mathrm{nm}$, $r_0=20\mathrm{\mu m}$, $l_0=1$, $z_0=3600\mathrm{\mu m}$, and $M=8$ as a proof-of-concept example. The slit is set as 0.4 µm considering the balance between fabrication convenience and the device’s efficiency while maintaining the phase difference induced by the slit width to be as small as possible. To verify the function of the device, we simulated the intensity and phase profiles of the light field passing through the BMVG in the Fresnel far-field region at $z=z_0/m$, $m=1,2,\dots ,10$, respectively, and the results are shown in Fig. 1(b). According to Fig. 1(b), it can be seen that at the pre-designed focal plane $z_0$, the intensity profile of the light field shows a “donut” shaped ring, while the phase profile of the light field shows phase change of $2\pi$ around the central singularity point; both indicate an OAM mode of $l_0=1$. When we capture the light field at $z_0/2$ along the propagation direction, the phase profile together with the intensity profile indicates an OAM mode of $l=2=2l_0$. So on and so forth, we can totally obtain 10 clear OAM modes along the propagation direction. Therefore, such a single device can produce a series of high-quality continuous integer OAMs.

To demonstrate that the generated OAMs from our BMVG have deep DoFs, we simulated the intensity profile of the light field in the $x–z$ plane after our designed BMVG from $z=300\mathrm{\mu m}$ to $z=4000\mathrm{\mu m}$, and the result is shown in Fig. 2(a). It can be seen that due to the conical wavefront induced by the zone plate structure concealed in our device, the light field after our BMVG becomes a continuous variant-order Bessel-like beam, and the modes of each section in the Bessel-like beam, i.e., the non-diffracting region, vary with the propagation distance. In order to explore more details, we extracted every mode in the beam from $l=1$ to $l=10$ and showed their zoom-in $x–z$ intensity profiles in Fig. 2(b). When $l=1$, the non-diffracting region of the beam spans from $z=3200\mathrm{\mu m}$ to $z=4000\mathrm{\mu m}$, which is more than 1260 times the length of ${\lambda }_0$. As the topological charge $l$ increases, the length of the non-diffracting region decreases. But, even at $l=10$, the non-diffracting region still spans almost 16 times the length of ${\lambda }_0$, which proves that our BMVG can generate continuous high-quality OAMs with long DoFs.

To further verify the function of our device, we experimentally fabricated and characterized the designed BMVG. We fabricated our designed BMVG on a 1-µm-thick reduced graphene oxide (rGO) film^[15] so that the film is thick enough to block the impinging light while guaranteeing fabrication convenience using direct laser ablation^[16]. In our sample, the slit part is ablated so that incident light can transmit through the slit. In order to read the order of the generated OAMs directly from the measured results, a circular hole with a radius of 10 µm was drilled through the center of the sample, as shown in the scanning electronic microscopy (SEM) image in Fig. 3(a). The total area of the device is within $400\mathrm{\mu m}\times 400\mathrm{\mu m}$. Figure 3(b) shows the topographic profile (top) of our rGO sample measured by an optical profiler and the cross-sectional profile (bottom) marked by the white dashed line in the top image of Fig. 3(b). The Gaussian shaped surface profile is due to the intensity distribution of the laser focal spot during the laser ablation fabrication process. The measurement results confirm that the multi-turn spiral slits are thoroughly ablated through the rGO film.

Figure 3(c) shows a series of interference patterns captured before and after the centers of the non-diffracting regions of different modes. According to the interference patterns, we can directly read the modes of the generated OAMs from the number of arms in the intensity profiles. For each mode, the handedness of the interference patterns before and after the center of the non-diffracting region is opposite. This is because before the beginning of the non-diffracting region of each mode the wavefront of the light wave coming from our designed BMVG is convex [as is shown in Fig. 2(a)]. After the non-diffracting region, it will then diverge, and the wavefront will become concave. Such a phase change in the beam leads to the change of handedness of the interference patterns. But, the handedness of the OAMs never changes during this process. From Fig. 3(c), the interference patterns from $l=1$ to $l=9$ can be clearly observed. For $l=10$, the interference pattern after the center of the non-diffracting region can still be captured, though it is slightly messy. We did not capture the interference pattern of $l=10$ before the center of the non-diffracting region. This is because when the propagation distance gets too short, the Fresnel approximation cannot be satisfied anymore. Our fabricated BMVG has a diffraction efficiency that varies as the focal position changes. At $z=3600\mathrm{\mu m}$ ($l=1$), the diffraction efficiency is around 10.53%, and, at $z=1800\mathrm{\mu m}$ ($l=2$), the diffraction efficiency is around 28.88%. The diffraction efficiency increases as the focal position gets closer to the device. At $z=360\mathrm{\mu m}$ ($l=10$), the diffraction efficiency is around 65%. We can improve the diffraction efficiency by increasing the number of turns of the spiral and the slit width.

The unique properties of our BMVG also facilitate it to work as a wavelength tunable and focal distance tunable device. Specifically, Eq. (1) can be rewritten in a different formation as $$\begin{gathered} F(\theta )=\sqrt{r_0^2+2(a·b{l}_0)\left(\frac{z_0}{a}\right)\left(\frac{{\lambda }_0}{b}\right)·\frac{\theta }{2\pi }} \\ \theta \in [0,M·2\pi ],(M\ge 3,r_0\ll \frac{z_0}{b}), \end{gathered}$$(5)where $a$ and $b$ can be any positive constant as long as $a·b$ is a positive integer, and the corresponding Fresnel approximation can be satisfied. According to Eq. (5), when $a=1$, our BMVG behaves as a wavelength tunable device. Supposing we change the working wavelength from ${\lambda }_0$ to ${\lambda }_0/b$, we can obtain an OAM with mode $l=b{l}_0$ at the focal plane of $z=z_0$. It means that at a fixed position, when we change the illumination wavelength, different modes can be obtained at the corresponding focal plane. Similarly, when $b=1$, our BMVG becomes a focal distance tunable device. It means that under the same working wavelength, we can achieve different OAM modes at different positions, which has already been demonstrated in the first part of this manuscript. Finally, when $a\ne 1$ and $b\ne 1$, but $a·b$ maintains a positive integer, under the required approximate condition, we can get an OAM mode of $l=a·b$ at focal plane $z=z_0/a$ under the working wavelength ${\lambda }_0/b$. This indicates that the device can be tuned by wavelength and focal distance simultaneously. Moreover, the slit structure enables our device to work under a broadband range, which extends from visible to near-infrared. To demonstrate this capability, we tested our device at several wavelengths with both simulation and measurement. The results are provided in Fig. 4 as proof-of-concept examples.

First of all, we tested the wavelength tunability of the device. We fixed the test position before the focal plane at $z=514\mathrm{\mu m}$ and changed the illuminating wavelengths from 886 nm, 739 nm to 633 nm. As a result, a series of different modes from $l=5$ to $l=7$ can be obtained at the same measured position. The simulated and measured intensity profiles captured before and after the center of the non-diffracting region are provided in Fig. 4(a). Next, to demonstrate that our device can be tuned simultaneously by both wavelength and focal distance, we randomly selected some working wavelengths and captured the same mode of $l=6$ near the corresponding focal positions. The working wavelengths we chose here are 633 nm, 760 nm, 800 nm, 850 nm, 886 nm, 900 nm, and 1000 nm. Their relative focal positions of $l=6$ are 720 µm, 600 µm, 570 µm, 536 µm, 514 µm, 506 µm, and 456 µm, respectively. The measured interference patterns are shown in Fig. 4(b), which proved that our BMVG can be simultaneously tuned by the wavelength and focal position. The above simulated and measured results span from visible to near-infrared. To further prove that our device can finely work in the near-infrared region, we tested our device at the working wavelength of 900 nm and obtained continuous interference OAM patterns from $l=1$ to 9, both before and after the focal planes. The results are shown in Fig. 4(c). The results presented in Fig. 4 well proved the multi-dimensional tunability of the proposed device.

In conclusion, we proposed a multi-turn spiral slit structure, which naturally embodies the structure of a continuous spiral slit vortex generator and a zone plate. Such a device can work as a broadband multifocal continuous vortex generator. We prove the function of our device with rigorous mathematical language and completely consistent simulation and experimental results. Compared with previous similar devices, our BMVG can work with both linearly polarized light and circularly polarized light at different wavelengths. In addition, it can be fabricated by a simple and scalable laser nanofabrication method and takes much less fabrication time. The flexibility of our design enables it to work under a broadband spectrum while working as a wavelength tunable and focal position tunable device. However, our device’s efficiency is low since a majority of the incident light is blocked by the rGO film. Nevertheless, overall, we believe the multi-functionalities and tuning flexibility of our graphene BMVG provide a new avenue for next generation scalable OAM generators.

Simulation: The simulation of this work is based on the Fresnel diffraction theory and the codes are implemented on MATLAB on a personal computer.

Fabrication: A 1-µm-thick graphene oxide (GO) film fabricated by the vacuum filtration method^[17–19] using aqueous GO solution is attached to a cover glass substrate. The aqueous GO dispersion was synthesized by chemical oxidization of graphite via the modified Hummers method.

The vortex graphene generator is fabricated in one step by a commercial laser nanoprinting setup (Special Edition, Nanoprint3D from Innofocus Photonics Technology Pty. Ltd.) with a femtosecond laser (Libra, 800 nm wavelength, 100 fs pulse, 10 kHz repetition rate). The computer-controlled system was used to control the parameters of the laser ablation process, including exposure time, laser power, scanning speed, and patterns. The scanning speed was 100 µm/s to ensure a smooth line fabrication. The laser focal spot has a full-width at half-maximum (FWHM) of 600 nm for presenting high resolution. The laser power is 100 µW.

Optical characterization: The focusing performance of the graphene generator is characterized by a home-made imaging system, which uses a $4f$ system composed of an objective lens and a tube lens. The focusing intensity distribution is recorded by a CCD camera (Watec 902H). The graphene generator is mounted on a scanning stage, which can move along the $z$ axis. A commercial supercontinuum laser source (NKT Photonics, SuperK Flanium) was used to illuminate the graphene generator.