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Abstract
Independently tailoring two or more focused vortex beams and controlling their intensity distributions in one device is indispensable in optical trapping and manipulation fields. We propose the phase equation for a helicity-multiplexing metalens generating dual-focused vortex beams based on the propagation phase and geometry phase. The metalens can independently manipulate dual-focused vortex beams, i.e., their focal lengths, lateral displacements, and topological charges could be independently tailored. Moreover, we can adjust their relative focal intensity by modulating the ellipticity of the incident beam. Such metalenses allow for advanced research and applications in trapping and rotating microparticles.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Vortex beams carrying orbital angular momentum (OAM) have become a research hotspot since they were discovered by Allen et al. in 1992 [1] for their numerous applications in optical tweezers and spanners [2,3], optical communication [4,5], nonlinear optics [6], and quantum information processing [7,8], etc. For example, to form tunable gradient force in optical trapping [9] and manipulation [10], two or multiple focused vortex beams with independent focal lengths, lateral displacements, topological charges, and controllable relative focal intensity are highly desired. These focused vortex beams can usually be gained by multiple lenses and conventional vortex beam generators, such as spiral phase plates [11], space light modulators [12], Q-plates [13], and computer-generated holograms [14], etc. However, these conventional vortex beam generators limit their practical applications because of bulk mass and single function. Therefore, researchers are encouraged to seek novel vortex beam generators.
Metasurfaces, composed of specifically designed subwavelength units in a two-dimensional plane, have tremendous power in manipulating light [15–18]. A single metasurface can combine the functions of several conventional optical components [17–21], which is beneficial for realizing the miniaturization of the entire optical system [22–26]. In these regards, a metasurface is an ideal vortex beam generator at miniature scales [27–30], especially for multiple focused vortex beams [9,10,31–35], which can replace multiple spiral phase plates, space light modulators, Q-plates, and other optical components. For instance, Tian et al. [33] proposed a nonlinear metasurface that can focus the incident beam with various helicities and wavelengths into three diverse vortex beams. The focal lengths of the three vortex beams are mathematically relevant. If the focal length of one vortex beam is designed, the properties of the other vortex beams will be determined accordingly. Meanwhile, Teng et al. [34] created a spatial multiplexing metalens with dual-focused vortex beams. Such a metalens was divided into dual-ring bands. Each ring band was used to generate a single vortex beam independently. Chen et al. [35] demonstrated a metasurface with multiple focused vortex beams by introducing the holographic principle into the metasurface design. The vortex beams generated by both designed metalens [34] and metasurface [35] above carry independent focal lengths and topological charges while the relative focal intensity is invariable. If the relative focal intensity must be changed, the metalens and metasurface need to be repatterned. Therefore, to our knowledge, a single metasurface generating two or more focused vortex beams with independent focal lengths, lateral displacements, topological charges, and tunable relative intensity has not yet been deeply studied and demonstrated. Fortunately, the metalens with intensity-adjustable longitudinal dual-foci reported in Ref. [24] provide possibilities for independent focal lengths and tunable relative intensity. The two independent off-axis foci generated by single metalens [36] offer opportunities to achieve independent lateral displacements.
In this paper, inspired by Refs. [24] and [36], we design a series of helicity-multiplexing metalenses with dual-focused vortex beams and propose their corresponding phase equation by combing the propagation phase and geometry phase. The incident beam with different helicities could independently tailor two sets of phase profiles, which lead to the focal lengths, lateral displacements, and topological charges of the generated vortex beams being independent. And the relative focal intensity of which can also be easily adjusted. The analysis of the principle and design of our metalenses is addressed in section 2. The corresponding simulated results using the FDTD method and characteristic analysis of the generated vortex beams are displayed in section 3.
2. Principle and design of the helicity-multiplexing metalenses
2.1 Overall design of the metalenses
Three typical helicity-multiplexing metalenses are designed and schematically shown in Fig. 1. The metalenses are composed of numerous titanium dioxide nanobricks with diverse rotation orientation angles and diverse cross sizes sitting on the fused silica substrate. Such metalenses can generate different dual-focused vortex beams under the illumination of an x-polarized (XLP) incident beam. The focused left circularly polarized (LCP) vortex beams generated by helicity-multiplexing metalenses, i.e., Metalens 1∼ Metalens 3, have different focal lengths and topological charges. Still, the focused right circularly polarized (RCP) vortex beams remain unchanged. i.e., the focal lengths and topological charges of the dual-focused vortex beams generated by our helicity-multiplexing metalenses could be independently manipulated. In addition, as shown in the right part of Fig. 1, the relative focal intensity of the dual-focused vortex beams can be efficiently allocated by controlling the ellipticity of the incident beam. The specific design method and various design cases corresponding to Fig. 1 will be described later. And the lateral displacements of the dual-focused vortex beams can also be independently tailored, which will be shown later.
Fig. 1. Schematic of the helicity-multiplexing metalenses with independent and intensity-adjustable dual-focused vortex beams. The patterns and letters in the lower-left corner represent the polarization states of incident beams and their corresponding abbreviations, respectively. REP and LEP stand for right and left elliptical polarization, respectively. The letters in the dotted box at the lower right represent the polarization states of the monochromatic incident beams. Their corresponding monochromatic focused vortex beams generated by Metalens 3 are shown in the dotted box at the higher right, where pseudo-color mapping is adopted to indicate changes in intensity.
2.2 Principle of the metalenses
Theoretically, to focus an incident beam into a single-focused vortex beam, the required phase profile can be respectively calculated via the following expression [37]
For designing a helicity-multiplexing metalens that can focus incident beams with different helicities into different focused vortex beams independently, the phase profile for this metalens should contain two different phase profiles of the metalens with a single-focused vortex beam. Moreover, we introduce the helicity of the incident beam into the phase profile, more accurately, into the focal lengths, lateral displacements, and topological charges of the dual-focused vortex beams. Hence, the phase equation for the helicity-multiplexing metalens with independent dual-focused vortex beams can be deduced like this:
Then, we can divide the total phase of the helicity-multiplexing metalens into the propagation phase and the geometry phase, which can be respectively expressed [38] with the phase profiles in Eq. (2) as:
where δx and δy are the propagation phase distributions for a linear polarization incident beam in x-and y-directions, respectively; 2θ is the geometric phase distributions; ${\varphi _1}$ and ${\varphi _{ - 1}}$ represent independent phase profiles for σ=±1.2.3 Design of the metalenses
The specifically designed unit cell is shown in Fig. 2(a) and (b) to achieve the required propagation and geometric phase distributions, consisting of a titanium dioxide nanobrick sitting on a fused silica substrate. These nanobricks are periodically arranged with a fixed square lattice constant Px = Py = 360nm and a height H = 600nm. The propagation phase can be modulated by changing the length L and width W of the nanobrick. To cover 0∼2π, the range of L and W of the nanobricks covers 80 to 280nm, with 2 nm increments of each geometric variable. The simulated phase δx for an XLP incident beam as a function of L and W is shown in Fig. 2(c), and the corresponding transmission coefficient distribution is exhibited in Fig. 2(d). Note that the phase δy and transmission coefficient distributions for y polarized incident beam as a function of L and W can be obtained through rotation of Figs. 2(c) and 2(d), respectively. Differently, the geometric phase is twice the rotation orientation angle θ. So, it can easily cover 0∼2π if the rotation angle of the nanobricks can range from 0 to π. In addition, each nanobrick is designed to work as a half-wave plate for maximizing the polarization conversion efficiency. Accordingly, any phase combination (δx, δy, 2θ) can be obtained by choosing an appropriate size and a rotation orientation angle (L, W, θ) of the nanobrick.
Fig. 2. (a) and (b) are a side view and vertical view of a typical unit cell of the helicity-multiplexing metalens with the period (Px, Py), height (H), varying cross sizes (L and W), and different rotation angles (θ). (c) and (d) are simulated propagation phase and transmission coefficient distributions for an XLP incident beam.
3. Results and discussions
In section 3.1, to verify that the dual-focused vortex beams could be independently tailored according to the phase profile in Eq. (2), we design nine helicity-multiplexing metalenses and calculate their performance in detail. The designed metalens all consist of $123 \times 123$ nanobricks covering an area of 1539.7 square microns, and the working wavelength of the incident laser is 0.532µm. All the simulation results are calculated through the software- FDTD- solutions, where perfectly matching layers are used in x, y, and z directions, and plane-wave sources are utilized. Considering the simulation accuracy and time, we set mesh accuracy as 2. Then, in section 3.2, we investigate the intensity adjustment ability for one type of helicity-multiplexing metalens in section 3.1.
3.1 Generation of independent dual-focused vortex beams
For metalens with dual-focused vortex beams, the two focused vortex beams’ focal lengths and topological charges can be independently manipulated, which is vital in practical application. In our proposed phase equation in Eq. (2), if we set α=a=β=b = 0, the dual-focused vortex beams are distributed along the longitudinal direction. In addition, modifying the value of distance 2c between two focused vortex beams and the neutral distance f can arbitrarily design two independent focal lengths. Changing the value of n and d can arbitrarily tailor two independent topological charges. When the polarization of the incident beam is XLP, the independent dual-focused vortex beams are supposed to be simultaneously obtained.
Accordingly, we design five different helicity-multiplexing metalenses to demonstrate the independence of the dual-focused vortex beams. The corresponding theoretical focal lengths and theoretical topological charges of the dual-focused vortex beams generated by our designed helicity-multiplexing metalenses are displayed in Table 1. From Table 1, we can see that the theoretical focal lengths and topological charges of the designed Metalens 1∼ Metalens 3 keep unchanged under the LCP incident beam's illumination; the theoretical focal lengths and topological charges are different from one another under the RCP incident beam's illumination. To further demonstrate the independence of the dual-focused vortex beams, the theoretical focal lengths and topological charges of the designed Metalens 3 ∼ Metalens 5 keep unchanged under the RCP incident beam's illumination; the theoretical focal lengths and topological charges are different from one another under the LCP incident beam's illumination.
Table 1. The theoretical parameters and simulated results of the designed helicity-multiplexing metalensesa
The simulated results for Metalens 1∼ Metalens 5 are shown in Fig. 3. Figure 3(a1) ∼ (e1) respectively show the simulated intensity patterns of Metalens 1∼ Metalens 5 in the x-z plane when illuminated by XLP incident beam. Figure 3(a2) ∼ (e2) respectively display the simulated phase distributions of the focused RCP vortex beams generated by Metalens 1∼ Metalens 5 in the focal planes. Figure 3(a3) ∼ (e3) respectively display the simulated phase distributions of the focused LCP vortex beams generated by Metalens 1∼ Metalens 5 in another five focal planes. The simulated focal lengths and topological charges for Metalens 1∼ Metalens 5 are concluded and displayed in Table 1.
Fig. 3. (a1) ∼(e1) are respectively the simulated intensity distributions of the focused vortex beams generated by Metalens 1∼ Metalens 5 in the x-z plane. Note that the intensity distributions are all normalized using the same maximum value, and the label and scale of the axes in b1∼e1 are the same as that in a1 and omitted for conciseness. (a2) ∼(e2) are respectively the simulated phase distributions of the focused RCP vortex beams generated by Metalens 1∼ Metalens 5 in focal planes. The label and scale of the axes in b2∼e2 are the same as that in a2 and omitted for conciseness. (a3) ∼(e3) are respectively the simulated phase distributions of the focused LCP vortex beams generated by Metalens 1∼ Metalens 5 in another five focal planes. The label and scale of the axes in b3∼e3 are the same as that in a3 and omitted for conciseness.
According to the simulated results performed above, we firstly compare the parameters and distributions of Metalens 1 ∼ Metalens 3. From Table 1 and Fig. 3(a1) ∼(c1), it is evident that the focal length of the upper vortex beam corresponding to the LCP incident beam keeps unchanged at around 11.8 µm, and the focal length of the lower vortex beam corresponding to the RCP incident beam can still change from 7.3 to 9.2µm. Similarly, comparing the simulated phase distributions in Fig. 3(a2) ∼ (c2) and Fig. 3(a3) ∼ (c3), we can see that the topological charges of the RCP vortex beam corresponding to the LCP incident beam keep 1 and unchanged, and the topological charges of the LCP vortex beam corresponding to the RCP incident beam can still change from 4 to 2. Then, we compare the simulated results of Metalens 3 ∼ Metalens 5. From Table 1 and Fig. 3(c1) ∼(e1), we can see that the focal length of the lower vortex beam corresponding to the RCP incident beam remains almost unchanged at around 9.1µm, and the focal length of the upper vortex beam corresponding to the LCP incident beam can transform from 11.8 to 13.6µm. Similarly, comparing the simulated phase distributions in Fig. 3(c2) ∼ (e2) and Fig. 3(c3) ∼ (e3), the topological charges of the LCP vortex beam corresponding to the RCP incident beam keep 2 and unchanged, and the topological charges of the RCP vortex beam corresponding to LCP incident beam can transform from 1 to 4.
We can preliminarily conclude that our helicity-multiplexing metalens can independently manipulate the focal lengths and topological charges of generated dual-focused vortex beams. Moreover, Table 1 shows that the theoretical and simulated topological charges are in good agreement; there is only a slight difference between the theoretical and simulated focal lengths. More exact simulated focal lengths can be obtained if we consider higher-order derivative terms of the focal lengths in the phase equation [24]. Therefore, we can summarize that the simulated results of the five different helicity-multiplexing metalenses are in basic agreement with the theoretical focal lengths and topological charges. We also calculate the focusing efficiencies of the focused vortex beams generated by Metalens 1 ∼ Metalens 5 in different focal planes. Table 1 shows that their focusing efficiencies for LCP and RCP incident beams are both from 19% to 26%.
In addition, the dual-focused vortex beams with independent lateral displacements are still essential. In Eq. (2), changing α and a can independently manipulate the lateral displacements along the x-direction. Accordingly, we design another four metalenses (Metalens 6 ∼ Metalens 9) to verify the independence of the lateral displacements. For simplicity, the focal lengths and topological charges of the dual-focused vortex beams generated by Metalens 6 ∼ Metalens 9 are the same as those generated by Metalens 3. Only the lateral displacements along the x-direction are variable. The theoretical lateral displacements of the focused vortex beams generated by Metalens 3, Metalens 6, and Metalens 7 for the RCP incident beam are all zero, and the theoretical lateral displacements for the LCP incident beam are 0, −1, and −2µm, respectively. To further demonstrate the independence of the lateral displacements, the theoretical lateral displacements of the focused vortex beams generated by Metalens 3, Metalens 8, and Metalens 9 for the LCP incident beam are all zero, and the theoretical lateral displacements for the RCP incident beam are 0, −1 and −2µm, respectively.
The simulated results for Metalens 6∼ Metalens 9 are shown in Fig. 4. From Fig. 3(c1), Fig. 4(a), and Fig. 4(b), we can see that the lateral displacement of the lower vortex beam corresponding to the RCP incident beam keeps unchanged near zero, and the lateral displacement of the upper vortex beam corresponding to the LCP incident beam can still change from 0 to around −2µm. From Fig. 3(c1), Fig. 4(c), and Fig. 4(d), it is also evident that the lateral displacement of the upper vortex beam corresponding to the LCP incident beam remains unchanged near zero, and the lateral displacement of the lower vortex beam corresponding to the RCP incident beam can transform from 0 to around −2µm. Therefore, we can conclude that our helicity-multiplexing metalenses can independently manipulate the lateral displacements by tailoring the value of α and a. A similar conclusion can also be gained by controlling the value of β and b, which we will not demonstrate for brevity.
Fig. 4. (a) ∼(d) are respectively the simulated intensity distributions of the focused vortex beams generated by Metalens 6∼ Metalens 9 in the x-z plane. Note that the intensity distributions are all normalized using the same maximum value, and the label and scale of the axes in b∼ d are the same as that in a and omitted for conciseness.
We can conclude from the above nine cases that our helicity-multiplexing metalenses can generate dual-focused vortex beams with independent focal lengths, topological charges, and lateral displacements. In the future, such designed metalenses may generate independent multiple vortex beams by combing some new dynamic control methods [39,40] of the wavefront.
3.2 Generation of intensity-adjustable dual-focused vortex beams
Since a linear polarization or elliptical polarization incident beam can consist of any two orthogonal polarization states, such as LCP and RCP, we infer that the relative focal intensity of the generated two focused vortex beams can be modified by adjusting the proportion of LCP and RCP of the incident beam. Here, we take Metalens 3 as an example. Two different focused vortex beams with focal lengths of 11.8 and 9.2µm correspond to the LCP and RCP portion of the incident beam, respectively. Increasing the RCP portion of the incident beam can gain the intensity of the focused vortex beam with a focal length of 9.2µm. More portions of LCP in the incident beam will lead to more intensity of the focused vortex beam with a focal length of 11.8µm. We calculate their relative focal intensity distributions in the x-y plane for different ellipticity $\chi$ of the incident beam, which is defined as $\chi = {{({E_{RCP}} - {E_{LCP}})} / {({E_{RCP}} + {E_{LCP}})}}$ [24], where ${E_{LCP}}$ and ${E_{RCP}}$ respectively represent the intensity of the incident LCP and RCP beam. Theoretically, the ellipticity can be turned continuously so that the relative focal intensity also changes continuously. For simplicity, we exhibit three discrete representative results shown in Fig. 5. From Fig. 5(a) to Fig. 5(c), the ellipticity $\chi$ is set with respective to 1/5, 0, −1/5, and the corresponding LCP proportion of the incident beam increases gradually. We can see that the intensity of the focused vortex beam with a focal length of 11.8µm increases gradually, and the intensity of the focused vortex beam with a focal length of 9.2µm decreases accordingly.
Fig. 5. The intensity distributions of the focused vortex beams generated by Metalens 3 in different focal planes (top: z = 11.8µm; bottom: z = 9.2µm) when (a) ellipticity of the incident beam is 1/5, (b) ellipticity of the incident beam is 0, and (c) ellipticity of the incident beam is −1/5. The label and scale of the axes in b and c are the same as that in a and omitted for conciseness.
From these figures, it is clear that the relative focal intensity of the two vortex beams generated simultaneously can be flexibly modulated by changing the ellipticity of the incident beam, which can be easily achieved by rotating a quarter-wave plate before the helicity-multiplexing metalens in the experiment.
5. Conclusion
In summary, for forming tunable gradient force in optical trapping and manipulation, we have designed a series of helicity-multiplexing metalenses with independent and intensity-adjustable dual-focused vortex beams by combing the propagation phase and geometry phase. These helicity-multiplexing metalenses can focus one monochromatic visible wave with different helicities into cross-polarization focused vortex beams. The topological charges and focal lengths of the dual-focused vortex beams can be independently tailored by introducing the helicity of the incident beam to the phase profile of the metalens. i.e., if the topological charges and focal length of one vortex are designed, the properties of the other vortex will not be determined accordingly. Similarly, we demonstrate that the lateral displacements of the dual-focused vortex beams can also be independently manipulated using the method. Moreover, suppose the incident beam's polarization is linear or elliptical, the relative focal intensity of the generated dual-focused vortex beams can be easily adjusted by manipulating the ellipticity of the incident beam. Such designed metalenses may inspire the ideas for achieving novel optical gradient forces.
Funding
National Natural Science Foundation of China (51735002).
Acknowledgments
We thank Dr. Jing Wang and Rong Yan for their advice on this paper.
Disclosures
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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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