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Abstract
Lommel beam is a non-diffractive vortex beam of high concern recently, widely used in communication and turbulence studies. However, conventional methods of generating Lommel beams, such as using spatial light modulators (SLMs), are limited by their low resolution, poor phase manipulation, and small numerical aperture (NA). Here, non-diffractive Lommel beams based on all-dielectric metasurfaces are proposed. Using the Pancharatnam-Berry (PB) phase arrangement, the focal depth of the main lobe of the generated beam can reach 75 µm (∼119λ). Additionally, the broadband characteristics of the designed metasurface between 550 and 710 nm are observed. The resulting beam is demonstrated to show excellent self-healing properties by placing up obstacles. We also combine the phase of the Dammann grating with that of the Lommel beam to create a metasurface capable of producing a 1 × 4 Lommel beam array; the generated beams are still characterized by uniformity and non-diffraction. This study provides a new idea for Lommel beam generation with promising applications in optical communication, optical tweezers, and laser fabrication.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Non-diffractive beams are a kind of special beams with their cross-section intensity distributions unchanged with propagation distance. In 2015, Kovalev et al. proposed the Lommel beam, a non-diffractive beam with the same axial projection of the wave vector [1]. It is essentially an infinite linear superposition of Bessel modes carrying orbital angular momentum (OAM). Compared with the Bessel mode, this beam has a narrower radius of the central spot; its transverse intensity distribution is symmetrical with respect to the Cartesian coordinates, and its OAM is continuously variable. Zhao et al. first verified and generated non-diffractive Lommel beams by a digital binary amplitude mask in 2015 [2]. Belafhal et al. investigated the propagation of non-diffracted Lommel beams in a medium containing a sphere. They concluded that irradiating a Lommel beam on a rigid sphere would produce asymmetric scattering [3]. Hui et al. studied the propagation of Lommel beams in a gradient-index medium and specified the effects of asymmetry factors and topological charges on the propagation characteristics of the Lommel beams [4]. Recently, research on the communication and turbulence of Lommel beams have made great progress. For example, Yu et al. explored the effect of atmospheric turbulence on the vortex mode propagation of Lommel beams [5], contributing to designing free-space OAM-based optical communication links. Mert Bayraktar analyzed the scintillation and bit error rate of Lommel beams and revealed the better advantages of Lommel beams than Gauss beams in communication [6].
Conventionally, non-diffractive beams can be obtained by a combination of complex optical paths and multiple optical components, such as spatial light modulators (SLMs) [7,8], axicon [9], optical fibers [10], and lasers. Thanks to the rapid advancement in optical metasurfaces formed by the periodic arrangement of subwavelength materials, the above problems have been addressed in some way. The concept of metasurface was first proposed by Cappaso et al. [11].
Compared with traditional bulky and heavy optical devices, eye-catching metasurface has the advantages of low cost, small size, high integration, and versatility. Advances have been made in light-beam shaping [12–15], metasurface holographic imaging [16–21], polarization control [22–24], and spin Hall effects [25–29]. In recent years, metasurfaces comprising different meta-atom arrangements have been employed to generate various non-diffractive beams [30–36]. However, the metasurface-based application of non-diffracted Lommel beams has not been attempted.
To generate the Lommel beam array, an optical diffraction element, namely, Dammann grating, was used [37,38]. Dammann grating is a binary phase grating to generate one-dimension (1D) or 2D arrays of equal intensity spots. In 1971, Dammann and Görtler initiated the idea of Dammann grating to obtain multiple images from one input. The method of generating point arrays by Dammann gratings is interesting for two reasons. First, it is based on using phase gratings, precisely arranging the phase transition values to generate arbitrarily arranged lattices, and the grating uniformity is independent of the incident light wave, which allows high fluxes of light energy. Second, these gratings are binary, making it possible to use planar fabrication techniques such as photolithography and ion beam etching. Wen et al. utilized a combination of Dammann gratings and non-diffractive beams to generate various metasurfaces with uniform non-diffractive beam arrays [39,40]. Here, we followed this approach to generate the Lommel beam arrays.
We introduced an all-dielectric metasurface based on the Pancharatnam-Berry (PB) phase to produce non-diffractive, high-quality Lommel beams. The Lommel beam was generated by incidenting left-handed circularly polarized (LCP) light with wavelength $\lambda $= 630 $\textrm{nm}$ onto a metasurface with numerical aperture $NA$= 0.25, radius $r$ =24 ${\;\ \mathrm{\mu} \mathrm{m}}$, complex parameter $c$ = 0.8, and topological charge $n$= 4. The focus depth of the main lobe of the generated Lommel beam could reach 75 ${\mathrm{\mu} \mathrm{m}}$ (∼119 $\lambda $). In addition, the broadband characteristics of the designed metasurface between 550 $\textrm{nm}$ and 710 $\textrm{nm}$ were also observed, and the full width at half maximum (FWHM) was 0.05 ${\mathrm{\mu} \mathrm{m}}$ at the height of 50 ${\mathrm{\mu} \mathrm{m}}$. Then, different sizes of perfect electrical conductor (PEC) spheres were placed on the propagation path of the generated Lommel beam. We observed that the self-healing property of the Lommel beam appeared after a certain-distance transmission, and the light intensity distribution of the healed beam was almost identical to that of the Lommel beam without obstructions, verifying the excellent self-healing property of the Lommel beam. Finally, the phase of the Lommel beam was combined with the phase of the optimized Dammann grating to create a composite phase generating a metasurface of 1 × 4 Lommel beams. When the LCP light with $\lambda $= 630 $\textrm{nm}$, $NA$ = 0.167, $c$= 0.8, and $n$= 4, the 1 × 4 Lommel beam array was generated by the metasurface. The average depth of focus was measured to be 120.7 ${\mathrm{\mu} \mathrm{m}}$ (∼192 $\lambda $), and four groups of light spots with the same size and uniform energy can be observed, proving the excellent performance of the Lommel beam array generated by the designed array metasurface.
2. Theory and design
The Lommel mode is essentially a linear superposition of Bessel modes, and thus the complex amplitude expression of the Lommel beam along the optical axis transmission direction can be expressed by the Bessel function in cylindrical coordinates [1]:
Moreover, the Lommel function ${U_n}(w,\zeta )$ can be expressed as:
Then the Lommel function ${U_n}(w,\zeta )$ can be applied in Eq. (1)
Converting Eq. (4) from cylindrical to Cartesian coordinates, Eq. (4) can be written as:
Here we designed a metasurface capable of generating a 1 × 4 Lommel beam array using a synthetic phase approach. The Dammann grating phase comprised $\textrm{six}$ supercells; each supercell contained 32 rows of silicon nanopillars. Nanopillars corresponding to the rows of 7, 14, 16, 23, and 30 in each supercell underwent a phase change of 0 or $\mathrm{\pi }$, and these rows correspond to transition values of 0.22057, 0.44563, 0.5, 0.72057 and 0.94563 [41,42]. As shown in Fig. 2, the composite phase of the Dammann grating phase and the Lommel beam phase is expressed as follows:
where $\varphi (x,y)$ is the phase of generating a Lommel beam, and $\varphi DG(x,y)$ is the phase of the designed 1D Dammann grating. By combining the above two phases, the phase of a 1 × 4 Lommel beam array with uniformly distributed diffraction orders could be generated.3. Result
The metasurface design was optimized and simulated by the finite-difference time-domain (FDTD) simulation software of Lumerical Solutions. Metasurface nanopillars were made of silicon material, and the base material was fused silica substrate. Periodic boundary conditions were set in the x and y directions, and a perfect match layer (PML) was set in the z-direction and absorbed the electromagnetic field incident on it, indicating the field propagating unobstructed to infinity. As shown in Fig. 1, the optimized silicon nanopillar had a height (H) of 300 $\textrm{nm}$, a length (L) of 130 $\textrm{nm}$, and a width (W) of 80 $\textrm{nm}$. The interval period (P) of the adjacent unit cells was 250 $\textrm{nm}$. When the Nyquist sampling criterion $(P < \lambda /2NA)$ was satisfied, it was sufficient to generate Lommel beams at 550-710 $\textrm{nm}$ wavelength. For incident circularly polarized light, the PB phase produced a local phase shift by rotating anisotropic rectangular nanopillars acting as half-wave plates, $\varphi (x,y) = 2\theta (x,y)$, where $\theta $ is the rotation angle of the nanopillars, as shown in Fig. 1(a). The polarization conversion efficiency (PCE) of the nanopillar is important. Figure 3(a) shows the PCE of the nanopillar under the incident LCP light at wavelengths of 550-710 $\textrm{nm}$. PCE reached 82$\% $ at 600 $\textrm{nm}$ and more than 60$\% $ at 550-630 $\textrm{nm}$. The black curve in Fig. 3(b) indicated the linear relationship between the abrupt phase change of the transmitted light at an incidence of 630 $\textrm{nm}$ and the rotation angle $\theta $. The abrupt phase change generated by the nanopillar rotation achieved a continuous change from 0 to 2 $\mathrm{\pi }$. Additionally, as $\theta $ changed, the transmission efficiency of the nanopillars was roughly consistent, and the average transmission efficiency reached 88$\%$, as indicated by the red curve in Fig. 3(b).
Fig. 1. Schematic diagram of the metasurface generating Lommel beam and the meta-atom. (a) Top view of the unit cell. (b) Side view of the unit cell, silicon nanopillars on a glass substrate, length (L) = 130 nm, width (W) = 80 nm, height (H) = 300 nm, and period (P) = 250 nm. (c) Left-handed circularly polarized (LCP) light incident on the metasurface generates a non-diffractive Lommel beam.
Fig. 2. The Lommel beam phase $\varphi (x,y)$ is combined with the Dammann grating phase $\varphi DG(x,y)$ to generate a composite phase $\varphi ^{\prime}(x,y)$.
Fig. 3. Design of metasurface nanopillars. (a) The polarization conversion efficiency (PCE) of the unit structure at different wavelengths. (b) The relationship between the rotation angle change of the silicon nanopillar and the transmission efficiency and phase.
To investigate the effect of different incident light on the metasurface, we incident RCP and LCP light respectively. Figure 4 (a) and (b) show the longitudinal y-z plane of the Lommel beam generated by RCP and LCP light at wavelength $\lambda $ = 630 $\textrm{nm}$ incidenting on a metasurface with $NA$ = 0.25, $\textrm{r}$= 24 ${\;\ \mathrm{\mu} \mathrm{m}}$, $c$ = 0.8, and $n$ = 4. Two very energetic main lobes and side lobes could be seen on both sides, and the focus depth of the two main lobes was 75 ${\mathrm{\mu} \mathrm{m}}$ (∼119 $\lambda $). The ratio of the maximum intensity of Lommel beam main lobes to the incident light intensity was as high as 50.3%. Therefore, there was no difference between the results of RCP and LCP light incidence, in this paper we have chosen to use LCP light incidence. Figure 4 (c-e) presents the focal planes at different white dotted lines ($z$= 30, 50, and 70 ${\mathrm{\mu} \mathrm{m}}$) in the longitudinal y-z plane, and two symmetrical and bright central spots could be observed. Compared with the central ring of the first-order Bessel beam, the Lommel beam represented obvious ring fractures and split into two spots with relatively concentrated energy. According to the light field distribution intensity in the transverse x-y plane, we calculated FWHM values at different altitudes (the FWHM here was the distance between the two peaks), as shown in Fig. 4(f-h). The FWHM values at heights of 30, 50, and 70 ${\mathrm{\mu} \mathrm{m}}$ were 0.48, 0.51, and 0.55 ${\mathrm{\mu} \mathrm{m}}$, respectively, indicating little difference in the FWHM values at different distances and relatively uniform and highly-concentrated energy change during the beam transmission along the z-axis, proving the excellent non-diffraction of the resulting Lommel beam.
Fig. 4. Simulation results of Lommel beam under the incident circularly polarized light with a wavelength of 630 nm. (a) and (b) are longitudinal field diagrams for right-handed circularly polarized (RCP) light and left-handed circularly polarized (LCP) light incident on a Lommel beam, respectively. (c-e) Transverse field distributions at longitudinal positions of $z$= 30 (c), 50 (d), and 70 µm (e). (f-h) Intensity maps of the extracted beams at the white dashed lines of Fig. 4(b).
Next, the characteristics of the designed metasurface device in the broadband range were also studied for the LCP light with incident wavelengths $\lambda $= 550, 590, 670, and 710 $\textrm{nm}$ at the metasurface. Figure 5(a-d) show the vertical planes under different wavelengths of incidence, where the Lommel beams generated by different wavelengths were set to the identical light intensity value. Specifically, the focal length and maximum intensity of the generated Lommel beam decreased with increasing wavelength induced by certain dispersion of different wavelengths in the metasurface device. Despite the dispersion effect, the FWHM values of the generated Lommel beams did not change significantly. As shown in Fig. 5(e), at $z$= 50 ${\mathrm{\mu} \mathrm{m}}$, the FWHM values of the x-y focal plane for a $\lambda $ of 550, 590, 630, 670, and 710 $\textrm{nm}$ were 0.53, 0.55, 0.51, 0.50, and 0.52 ${\mathrm{\mu} \mathrm{m}}$, respectively, demonstrating the broadband characteristics of our designed metasurface device at 550-710 $\textrm{nm}$.
Fig. 5. The simulation results of the Lommel beam generated by metasurface in the broadband range. (a-d) Longitudinal field distributions of Lommel beams generated by metasurface over a wide wavelength range (550, 590, 670, and 710 nm). (e) FWHM of the x-y focal plane at $z$= 50 µm at incident wavelengths of 550, 590, 630, 670, and 710 nm.
The Lommel beam showed a satisfying self-healing property during spatial transmission. When the beam was partially occluded by a small obstacle, it would self-heal after propagating behind the obstacle for a certain distance, recovering the spot shape and field distribution before occlusion. We investigated the self-healing ability of the metasurface generating Lommel beam by placing obstacles of different radii on the main lobes of the beam and calculating its energy flow density. Figure 6(a) shows the y-z longitudinal field of the Lommel beam generated by the metasurface in the absence of obstacles, and the white arrows in the figure indicate the energy flow direction.
Fig. 6. The simulation results of the self-healing performance of the Lommel beam generated by metasurface. (a) The y-z longitudinal field of the Lommel beam is generated by the metasurface without obstacles (the white arrows in the figure represent the energy flow direction). (b) and (c) represent the y-z longitudinal field of the Lommel beam when a sphere of the perfect electrical conductor (PEC) with a radius of 1 and 2.5 µm is placed at $z$= 25 µm, respectively.
Figure 6(b) and (c) show the y-z longitudinal field of the Lommel beam when the PEC spheres with a radius of 1 ${\mathrm{\mu} \mathrm{m}}$ and 2.5 ${\mathrm{\mu} \mathrm{m}}$ are placed at a propagation distance $z$= 25 ${\mathrm{\mu} \mathrm{m}}$, respectively. As can be observed in Fig. 6(b), the light intensity of the main lobe could heal at 29 ${\mathrm{\mu} \mathrm{m}}$ after a short-distance transmission through the sphere. Moreover, it recovered to the same level as when there was no obstacle. We further increased the size of PEC spheres, as shown in Fig. 6(c), where PEC spheres with a radius of 2.5 ${\mathrm{\mu} \mathrm{m}}$ were placed on the main lobe propagation path at the same distance. It could be observed that the transmission distance required for self-healing became longer as the obstacle size grew, the obvious self-healing began at 34.5 ${\mathrm{\mu} \mathrm{m}}$. By looking at the energy flow in Fig. 6(b) and (c), it can be seen that when the main lobe region is blocked, energy from the main lobe region flows to the side lobes, and as the transmission distance increases, energy from the side lobes flows from the surrounding region to the blocked main lobe region. At this moment, the blocked main lobe region can repair itself, indicating that the Lommel beam generated by our designed metasurface showed a favorable self-healing performance.
To generate the 1 × 4 Lommel beam array, the Lommel beam phase $\varphi (x,y)$ with $NA$ = 0.167, $r$ = 24 ${\mathrm{\mu} \mathrm{m}}$, $c$= 0.8, and $n$= 4 was set based on the synthetic phase principle of Eq. (6), which was combined with the designed 1D Dammann grating phase $\varphi DG(x,y)$ to generate the array metasurface whose phase was $\varphi ^{\prime}(x,y)$. As shown in Fig. 7(a), the y-z longitudinal field distribution of the 1 × 4 Lommel beam array was generated at 630 $\textrm{nm}$ with LCP light incidence. Four groups of Lommel beams with complete shape and uniform intensity could be observed, and the measured average depth of focus was 120.7 ${\mathrm{\mu} \mathrm{m}}$ (∼192 $\lambda $). Figure 7(b) and (c) show the x-y transverse field distribution at 90 and 110 ${\mathrm{\mu} \mathrm{m}}$ at the location of the white dashed line in Fig. 7(a), where four spots of uniform size and energy could be seen. Figure 7(d) and (e) indicate the intensity distribution of the corresponding heights; at 90 ${\mathrm{\mu} \mathrm{m}}$, the average FWHM was calculated to be 5.03 ${\mathrm{\mu} \mathrm{m}}$, and at 110 ${\mathrm{\mu} \mathrm{m}}$ the average FWHM was 5.16 ${\mathrm{\mu} \mathrm{m}}$, reflecting the beam’s energy uniformity. These properties demonstrated the non-diffractive and uniformly-distributed performance of the Lommel beam array generated by combining our designed Lommel beam phase and the optimized Dammann grating phase.
Fig. 7. The simulation results of metasurface generating 1 × 4 Lommel beam array. (a) y-z longitudinal field distribution of 1 × 4 Lommel beam array. (b) and (c) are x-y transverse field distributions at $z$= 90 and 110 µm, respectively. (d) and (e) are intensity distribution maps of heights corresponding to $z$= 90 and 110 µm in Fig. 7 (a).
4. Conclusion
In summary, we proposed an all-dielectric metasurface based on the PB phase to generate a non-diffractive, high-quality Lommel beam. The main lobe of the Lommel beam produced by metasurface with LCP light of $\lambda $= 630 $\textrm{nm}$ had a focus depth up to 75 ${\mathrm{\mu} \mathrm{m}}$ (∼119 $\lambda $). The broadband characteristics of the designed metasurface were verified at 550-710 $\textrm{nm}$. Additionally, the excellent self-healing property of the Lommel beam was observed, and the light intensity distribution of the healed beams was almost consistent with that of the Lommel beam without obstructions. Finally, the composite phase was generated by combining the Lommel beam phase and the optimized Dammann grating phase to generate a 1 × 4 Lommel beam array. This method of achieving a non-diffractive Lommel beam in the visible broadband range using a fully dielectric metasurface showed promising applications in optical communications, optical tweezers, and optical imaging.
Funding
National College Students Innovation and Entrepreneurship Training Program (202011057038); National Natural Science Foundation of China (61975185).
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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