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Abstract
A perfect vortex beam has been attracting tremendous attention due to the fact that its ring radius is independent of the topological charge. Taking advantage of the superposition principle of phase in Fourier space, we proposed to generate perfect vortex beam using propagation-phase-based dielectric metasurface, which exhibits production efficiency larger than 83.5%. Due to the sensitivity of propagation phase to the polarization of incident beam, two sets of phase profiles can be imposed on a single dielectric metasurface, enabling the simultaneous generation of dual perfect vortex beams. Based on this property, convenient control to the radius and/or topological charge of perfect vortex beam is achieved by switching the incident polarization between two orthogonal polarizations, without redesigning metasurface or changing optical path. What’s more important, the crosstalk of these two channels is low, less than 4%. Thus, the propagation-phase method of producing perfect vortex beam will find significant applications in optical communication, particle trapping, particle manipulation and holographic display.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Vortex beam is a kind of special light beam carrying orbit angular momentum (OAM) because of the existence of helical phase factor ${\rm{exp}} ({im\theta } )$(m is topological charge (TC), an integer number; $\theta $ is azimuthal angle) in its expression [1]. Possessing the feature of carrying OAM, vortex beam has pivotal applications in optical communication [2,3], optical trapping [4,5], and holographic display [6,7]. However, in some application, say optical communication, coupling multiple OAM beams simultaneously into single optical fiber to realize multiplexed communication is not so easy because the ring radius of vortex beam changes with the variation of topological charge it carries. To overcome this issue, perfect vortex (PV) beam with its ring radius independent of TC was proposed [8,9]. Traditionally, PV beam is generated using Gaussian beam by means of a series of bulky optical components, such as axicon, spiral phase plate, Fourier transformation lens and spatial light modulation, which hinders the device integration of OAM-based photonic circuits.
Metasurface, composed by the array of optical elements, has been paid increasing attention because of its capability of shaping the incident beam with great degree of freedom by means of manipulating the phase profile through each element [10–14]. Undoubtedly, metasurface can be utilized to generate PV beam. Different from traditional method, the phase profile of PV beam is discretized and each element is coded with desired phase, thus, when light beam is incident upon the surface of metasurface, the transmitted beam is characterized by the desired OAM due to the imposition of the coded phase profile [15,16]. This kind of producing perfect vortex beam is beneficial to scale down the dimension of PV-generator, which is advantageous to the integration of OAM-related devices. Till now, there are some reports on the generation of vortex beam using metasurface [17–22]. For instance, Yue et al [23] designed geometric-phase plasmonic metasurface, which produces vortex beam with the incident of circularly polarized light. In 2019, Yan et al [24] proposed all-dielectric metasurface to generate vortex beam, whose intensity profile is controllable by the polarization of incident light beam. As for perfect vortex beam, Zhang et al [16] in 2018 proposed to generate perfect vortex beam by geometric-phase metasurface that was fabricated on Au-film with 50-nm thick. Due to the usage of Au-film, the transmission of incident beam through metasurface is very low. In addition, the conversion of incident circularly polarized beam to the orthogonal polarization is a bit low, 58% or so [16]. In other words, the transmitted beam still contains large part of original polarization, which is unprofitable to application.
In contrast to geometric-phase plasmonic metasurface, the present paper designs dielectric metasurface by means of propagation phase, through which PV beam can be generated. As we know, propagation phase is sensitive to the orientation of linear polarization of incident beam, i.e. the same element exhibits distinct phases for horizontal and vertical polarizations; so two completely independent sets of phase profiles can be patterned on a single metasurface, one for horizontal polarization and the other for vertical polarization [25,26]. In this way, the metasurface can be utilized to generate two different perfect vortex beams simultaneously or individually, depending on the incident beam polarization. Furthermore, with proper design, convenient control to the radius and/or topological charge of perfect vortex beam can be achieved. Thus, the propagation-phase method of generating perfect vortex beam will find significant applications in optical communication, particle manipulation and holographic display.
2. Theory and design of metasurface
Theoretically, the perfect vortex beam is obtained from the Fourier transformation of an ideal Bessel beam, which reads [9]
Here, ${J_l}$ is the first kind of l-th order Bessel function, $k = \sqrt {k_r^2 + k_z^2} = 2\pi /\lambda$ is the wave vector with $\lambda$ being incident wavelength ( = 0.915 µm in the following numerical simulation), $r = ({\rho ,\theta } )$ is the polar coordinates in beam cross section and $({{k_r},{k_z}} )$ are the radial and longitudinal wave vectors respectively. In practice, however, the accessible Bessel beam is actually Bessel-Gaussian (BG) beam of the form [8]
Fig. 1. the formation of the metasurface phase profile by superposing the phase profiles of axicon, spiral phase plate, and Fourier transformation lens.
In simulation, the phase profile of metasurface is discretized and each phase at the discretized point is implemented by an element of metasurface. As shown in Fig. 2(a), the metasurface is composed by numerous elliptical silicon cylinders patterned in hexagonal lattice, which are sitting on silica substrate. Generally, two kinds of phase are involved in the art of metasurface, propagation phase and geometric phase [25,27,28]. Geometric phase, also named as Pancharatnam-Berry (PB) phase [29], stems from the rotation of elliptical cylinder around the propagation axis relative to, say, x axis when the element of metasurface is optically effective to a quarter-wave plate. Normally, it acts with respective to circularly polarized beam, and has the form of ${\varphi _{PB}} ={\pm} 2\sigma \theta $, where $\theta $ is rotation angle of cylinder relative to x axis and $\sigma $ is helicity of incident beam. On the other hand, the propagation phase originates from the phase delay when the incident beam transmits through cylinder [26]. Each cylinder in the array of metasurface can be taken as an optical resonator, which confines the optical energy and then reradiates it. Namely, each cylinder imposes a phase delay to the incident light field, and cylinders with different cross section exhibit different phase delay even with same height. Therefore, by scanning the length of major and minor axes of cylinder we get the phase and transmittance distributions with respective to Dx and Dy for, say, x-polarized incident beam [Fig. 2(c) and 2(d)]. Note that the phase and transmittance distributions for y-polarized incident beam can be obtained through π/2 rotation of Figs. 2(c) and 2(d), respectively.
Fig. 2. (a) Top view of the proposed implementation of metasurface, which is composed by elliptical silicon cylinders with same height H (=715 nm), but with different diameters (Dx and Dy). The cylinders are located at the centers of the hexagonal unit cells with period P (=650 nm). (b) Schematic side view of generating dual perfect vortex beams using a single metasurface through the incidence of circular polarization. (c) and (d) are the phase and transmittance distributions, respectively, as functions of Dx and Dy for x polarization.
Due to the orthogonality of x and y polarizations, it is possible to pattern two sets of phase profiles (one for x polarization and the other for y polarization) on a single metasurface [27,28]. Then, x-polarized incident beam evokes x-polarization-related phase profile of metasurface and generates one PV beam, while y-polarized incident beam provokes y-polarization-related phase profile of metasurface and generates another PV beam. In this way, a single metasurface can be designed to generate two distinct perfect vortex beams [Fig. 2(b)]. If the two PV beams to be generated by metasurface differ in topological charge but with same ring radius, then, the topological charge of generated PV beam can be modified by switching the incident beam polarization, without redesigning metasurface or changing optical path, which is useful in optical trapping [5], because, under the condition of constant ring radius, the optical torque is linearly proportional to the topological charge of the beam. In addition to that, the PV beam with polarization-controllable topological charge also finds applications in holographic display [30]. Similarly, the ring radius of the generated PV beam can be also altered using the same approach with the condition of constant topological charge.
3. Results and discussion
Now we numerically demonstrate above theoretical prediction through the commercial software-FDTD Solutions, where perfectly matching layers (PMLs) were used in x, y, and z directions, and plane-wave sources were used in all simulations. Notice that all the results were obtained by directly simulating the metasurface structure out to 140µm, no other technique, like far-field projection or Fourier transformation of near field, was utilized. First of all, we designed two sets of dielectric metasurfaces with theoretical focal length f = 100µm: one set for x-polarization with axicon period d = 4µm and different topological charges (m = 2, 3, 5, 7) and the other set for y-polarization with axicon period d = 6µm and different topologic -al charges (m = 2, 3, 5, 7). Figure 3(a) presents the intensity profiles of the generated PV beams at distinct longitudinal positions, which shows that the transmitted beam first focuses at z = 38.40µm, then diverges and forms annular ring profile at z = 80µm. The radius of ring increases gradually with further propagation, but keeping circular shape. Figure 3(b) and Fig. 3(c) show the intensity profiles at z = 100µm for different topological charges, where Fig. 3(b) is for x polarization and Fig. 3(c) for y polarization. We can see that no matter for the incidence of x or y polarization, the ring radius almost not varies with the change of topological charges [ Fig. 4(a) and 4(b)], indicating that PV beam is generated by the metasurface. In our simulation, all the conversion efficiencies (defined as the ratio of the intensity of bright ring to the intensity of incident beam) of incident beam into PV beams were all larger than 83.5%, which is slightly different for x and y-polarization. The loss of energy may come from ①reflection at the metasurface surface, ②scattering by individual cylinder, ③the phase difference of realistic phase and theoretical phase at each lattice location (which is within 3.5 degrees in our simulations), and ④ the material absorption. Nevertheless, the comparison of Figs. 3(b) and 3(c) reveals that the ring radius decreases with the increase of axicon period. To get the concrete relation, we designed ten metasurfaces with f = 100µm and m = 2 but for different topological charges. The simulation results show that the ring radius is an inverse function of TC [Fig. 4(c)], and the fitting function is
where the unit of all variables is µm. Note that above relation of ring radius and axicon period is obtained for fixed incident wavelength (0.915 µm); in fact, the ring radius is also related to the incident wavelength [16].Fig. 3. (a) The intensity profiles of generated PV beam at different longitudinal positions z. (b) The intensity profiles of generated PV beam at z = 100µm for different topological charges. The polarization of incident beam is along x axis and the axicon period d = 4µm. (c) The intensity profiles of generated PV beam at z = 100µm for different topological charges. The polarization of incident beam is along y axis and the axicon period d = 6µm.
Fig. 4. The intensity curves of generated PV beams with different topological charges (m = 2, 3, 5, 7) at z = 100µm, (a) for x polarization and d = 4µm, (b) for y polarization and d = 6µm. (c) The relation of ring radius and axicon period, the common parameters are f = 100µm, m = 2.
Above simulations verify the generation of PV beam by dielectric metasurface designed using propagation phase. According to theoretical analysis, two sets of phase profiles are possible to be imposed to a single metasurface, generating two distinct PV beams. Here, we designed a metasurface with two sets of phase profiles, one set with d = 4µm and m = 3 for x-polarized incident beam, and the other set with d = 6µm and m = 2 for y-polarized incident beam. Figure 5(a) is the result when x- and y-polarized light illuminate simultaneously such metasurface, which clearly shows the simultaneous generation of dual PV beams with distinct ring radii. Figure 5(b) shows the intensity profiles and phase distributions for the incidence of only x- or y-polarized light (as indicated by the arrow).
Fig. 5. (a) The intensity profiles of generated dual PV beams at different longitudinal positions by dielectric metasurface with two sets of phase profiles, one set for x polarization and the other for y polarization. (b) The intensity profile and phase distribution when only x- (left two plots) or y-polarized (right two plots) beam is incident.
To tell the generation performance of the metasurface with two sets of phase profile, we also computed the crosstalk of these two channels. The results show that the crosstalk is less than 4% of the generated PV beam, implying excellent polarization-multiplexed performance of propagation-phase-based metasurface in producing PV beam, and this feature is critical practical applications. We would like to stress that the crosstalk has nothing to do with the relative phase difference between the incident x- and y-polarized beams.
In the following we verify that some features, such as ring radius, topological charge, of the generated PV beam is polarization-controllable. Here we designed two metasurfaces with two sets of phase profiles for each. The first metasurface is designed to generate two PV beams differing only in ring radius, so switching the incident polarization alters the ring radius of the generated PV beam, as shown in Figs. 6(a) and 6(b). This kind of convenient control to the ring radius of generated PV beam is very useful, especially in the application of particle trapping [5]. We know the optical intensity-gradient force is related to the field gradient, the more focused, the larger the intensity-gradient force [31]. From Fig. 6(a), we can see that by switching incident polarization from x-axis to y-axis, the radius of PV beam becomes smaller, leading to larger intensity gradient, thus exhibiting larger intensity-gradient force. In addition to controlling ring radius, we can also control the topological charge of generated PV beam by switching the incident polarization if the two generated perfect vortex beams differ only in topological charge, as shown by Figs. 6(c) and 6(d). This property can also find application in particle manipulation, because, besides intensity-gradient force, phase gradient also generates a kind of optical force [32], i.e. phase-gradient force, which is linearly proportional to the topological charge of vortex beam [33]. Notice that the phase distributions of the generated PV beams are very regular, as shown in Fig. 6(c), thus, leading to regular motion of particles.
Fig. 6. (a) The intensity profile and phase distribution of generated PV beam at z = 100um for different axicon periods but same topological charge. (b) The intensity curves of generated PV beam at z = 100µm for x (red) and y (blue) polarization. (c) The intensity profile and phase distribution of generated PV beam at z = 100µm for different topological charges but same axicon period. (d) The intensity curves of generated PV beam at z = 100µm for x (red) and y (blue) polarization.
4. Conclusion
In summary, we proposed to generate perfect vortex beam using dielectric metasurface designed by means of propagation phase. This kind of method of generating perfect vortex beam is characterized by high efficiency because of the high transmittance of dielectric materials. What’s more important, the sensitivity of propagation phase to the incident polarization enables the simultaneous generation of two perfect vortex beams by only a single dielectric metasurface with low crosstalk. On the basis of this excellent feature, the ring radius and/or topological charge of generated perfect vortex beams can be conveniently changed by switching the incident polarization between x and y polarizations, while without redesigning metasurface or altering optical path. The novel property has significant applications of particle trapping, optical communication and holographic display.
Funding
National Natural Science Foundation of China (61805141, 61975125).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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