Generation of high-quality and tailored specific optical beams is a perpetual requirement and pivotal challenge in optics and photonics. Generalized cylindrical vector beams (G-CVBs) characterized by their axially symmetric spatial variation of optical polarization and vortex beams (VBs) possessing orbital angular momentum (OAM) associated with helical wavefronts have attracted increasing attention in various fields of science and technology. The polarization of a G-CVB in the beam’s cross-sectional plane can be expressed as E→/|E→|=(cosϕ,sinϕ), where E→ is the optical electric field, and ϕ(φ)=Pφ+ϕ0. The integer P is the polarization order number (a topological charge), φ is the azimuthal angle, and ϕ0 is the polarization orientation relative to the radial direction^[1]. G-CVBs with different values of P or ϕ0 are known for their unusual focusing and combined properties^{[24" target="_self" style="display: inline;">–4]}. Vectorial VBs (VVBs), which possess the properties of G-CVBs and VBs simultaneously, provide the extra degree of freedom for manipulating the light wavefront flexibly.

Due to their novelty features, specific optical beams have been extensively investigated for possible use in various optical systems, ranging from optical trapping and manipulation^[5,6], high-efficiency plasmonic structures^[7,8], and direct femtosecond laser surface structuring^[9,10], to quantum information protocols in quantum communications^[11,12]. However, broad-scale practical applications of specific optical beams are at present still restrained by the relatively complicated methods for generating high-quality beams and realizing transformation of different types of optical beams. The engendered approaches can be categorized into active and passive, depending on whether an active gain medium is needed or not^[13]. Currently, passive generation methods that provide contrivable and tunable properties are achieving increased attention^[14]. Passive units can be divided into four classes: spatially variant phase retarders^{[15–18" target="_self" style="display: inline;">–18]}, spatially variant polarization converters^{[1,19–22" target="_self" style="display: inline;">–22]}, spatial light modulators (SLMs) based on interferometric arrangements^{[23–26" target="_self" style="display: inline;">–26]}, and liquid crystal (LC) q-plates^{[27–31" target="_self" style="display: inline;">–31]}. Interconversion of an arbitrary state of polarization (SOP) to other polarization states on the low or high-order Poincaré sphere can be obtained by additional wave plates^[32,33]. Among the listed units, LC q-plates are particularly promising due to their simple construction and relatively low cost. A q-plate consists of a nematic LC layer sandwiched between two glass plates, and the LC is manipulated so that the in-plane director field n→=(cosα,sinα) (n→ is a unit vector describing local preferential orientation of the LC molecules and is associated with the local direction of the optical axis of the material) varies with the azimuthal angle φ as α(φ)=qφ+α0, where q is an integer, and α0 is the orientation at φ=0. Such LC configurations can simultaneously affect both the polarization direction and the phase of an optical beam.

An important advantage of LC q-plates is their tunability with relatively low applied voltages. An external electric field in combination with boundary conditions at the interfaces of the LC medium with its confining scaffold determines a spatial profile of the LC director field n→(r→) and, consequently, also the optical properties of the entire structure^{[34–40" target="_self" style="display: inline;">–40]}. As a result, the local SOP of the transmitted light can be attained by suitably adjusting the external voltage applied to the predesigned LC configuration. As was recently demonstrated, this can be achieved either by segmented LC structures^[41,42] or by segmented electrodes^[31].

In this Letter, we demonstrate a new versatile method for fabrication of electrically tunable micro-patterned LC structures. The method is based on our recent finding^{[3436" target="_self" style="display: inline;">–36]} that the arbitrary tailored azimuthal distributions of LC director orientations can be enabled by utilizing sparse out-of-plane polymeric ribbons created by two-photon polymerization-based direct laser writing (TPP-DLW). We investigated in detail a radial micro-patterned LC structure, which is the same as an LC q-plate with q=1. With the aid of applied voltage, tunable phase retardance between the extraordinary and ordinary waves δ in the range of 0–4π was obtained for an optical wavelength of 633 nm. A left circularly polarized (LCP) incident beam with a Gaussian cross-sectional intensity profile (TEM00) in combination with either δ=2π, 3π/2, π/2, or π was used to generate an LCP beam with the analog Gaussian profile, the VVBs with the helicity l=−1 and a more flat intensity profile, and a pure right circularly polarized (RCP) optical VB with the helicity l=−2 and a doughnut-shaped intensity profile, respectively. Furthermore, the SOP of the generated specific optical beams obtained by δ=3π/2, π, or π/2 was converted to different states on the first- and the second-order Poincaré spheres by changing polarization directions of the incident beam or by modifying the angle between the fast axes of the two additional half-wave plates (HWPs).

A radial assembly of out-of-plane polymer ribbon [Fig. 1(a)] was fabricated on an ITO-coated glass plate by the TPP-DLW method as described previously^[34]. The diameter of the pattern was 2 mm, and the angle between two neighboring ribbons at the outer circle was 1.4°. The discontinuous ribbon pattern was designed so that the distance between two neighboring ribbons never exceeds 30 μm. This ensures good alignment of the LC director along the local direction of the ribbons and also reduces the total exposure of the central part of the pattern, consequently preventing its damage. After chemical developing, the ribbon assembly was covered with another ITO-coated glass plate (without any surface alignment layer) to form an LC cell. Then, the cell was filled with the commercial nematic LCs E7 (Shijiazhuang Chengzhi Yonghua Display Material Co., TIN=61.1°C, Δε=13.7, Δn=0.22, and K22=6.5pN) via capillary flow in the isotropic phase and then slowly cooled to the nematic phase at room temperature. Figures 1(b)–1(e) show the transmission polarized optical microscopy (POM) images of the obtained LC structure as observed under crossed polarizers at different orientations. A characteristic “Maltese cross” pattern is present in all images, which confirms that a radial LC alignment pattern, as illustrated in Fig. 1(f), is achieved, i.e., a q-plate with q=1, and α0=0.

Fig. 1. (a) Radial structure of polymer ribbons. Transmission POM images of the ribbon structure filled with a nematic LC under crossed polarizer and analyzer captured at different orientations of the polarizer with respect to the x direction: (b) 0°, (c) 45°, (d) 90°, and (e) 135°. (f) Sketch of the LC alignment induced by the ribbon pattern.

下载图片 查看所有图片

The setup for generating and analyzing G-CVBs is depicted in Fig. 2(a). A linearly polarized incident laser beam with a TEM00 transversal profile and a wavelength of 633 nm is at first transformed into an LCP beam by a combination of a Glan–Taylor polarizer (P) with its transmission axis parallel to the x axis and a quarter-wave plate (QWP1) with its fast axis oriented at 135° with respect to the x axis. The LCP beam then passes through the q-plate. Based on the Jones formalism^[43], the effect of a q-plate with q=1 and α0=0 can be expressed by the Jones matrix^[41], Mq(δ)=(cosδ2+isinδ2cos2φisinδ2sin2φisinδ2sin2φcosδ2−isinδ2cos2φ).(1)

Fig. 2. (a) Setup for generating and analyzing G-CVBs. (b) Normalized transmission intensity after a circular polarizer placed behind the sample measured at different voltages.

下载图片 查看所有图片

For an incident LCP polarization corresponding to the Jones vector (1/2)(1−i), the polarization outgoing from the q-plate can be expressed as^[27,41]E→/|E→|=cosδ2×12(1−i)+isinδ2×e−i2φ×12(1i).(2)

Namely, it can be decomposed into the superposition of a homogenous (non-vectorial) LCP and an RCP component. The important property of the RCP component is its helical phase given as e−i2φ, corresponding to an optical vortex with a helicity of l=−2. Because of its vortex nature, which is associated with a phase singularity in the center of the beam, the outgoing RCP component exhibits a doughnut-shaped cross-sectional profile.

For an incident RCP polarization corresponding to the Jones vector (1/2)(1i), the outgoing polarization from the q-plate is analogously given as^[41]E→/|E→|=cosδ2×12(1i)+isinδ2×ei2φ×12(1−i).(3)

By using Eqs. (2) and (3), one can calculate the outgoing field for any polarization state of the incident beam.

In order to validate the electrical tunability of the phase retardance δ, a circular polarizer (CP) consisting of QWP2 and a linear polarizer (A) was placed after the radial micro-patterned LC structures. In order to pick out only the LCP component of the outgoing beam, the fast axis of QWP2 should be along the x axis, the transmission axis of the linear polarizer was oriented at 135° to the fast axis of the QWP2, and the intensity of the LCP component was measured by a CCD camera. According to Eq. (2), this intensity is proportional to cos2(δ/2). A square wave-form voltage with a frequency of 1 kHz was applied to the ITO electrodes. The dependence of the normalized intensity on the voltage amplitude is shown in Fig. 2(b) and demonstrates that our q-plate possesses a tunable phase retardance between the extraordinary and ordinary waves in the range 0<δ<4π, corresponding to wave retardation in the interval 0<dλ<2λ for λ=633nm.

By suitably adjusting the value of δ, a homogeneous (non-vectorial) LCP incident beam can be converted into a desirable specific beam without the need of mechanically changing the optical path. Figure 3 shows transversal profiles of the outgoing beams obtained for four voltages (7.7, 9.0, 10.8, and 12.8 V), corresponding to δ=2π, 3π/2, π, and π/2. The SOPs and the phase fronts of the generated specific beams behind the q-plate are illustrated in the first and the second columns, respectively. The same setup as shown in Fig. 2(a) was used, but in one case we removed both the QWP2 and the analyzer (A) (column 3 in Fig. 3); in other cases only QWP2 was removed, while A was rotated to different orientations (columns 4-7 in Fig. 3). The cross-sectional intensity profile of the outgoing beam was detected by the CCD camera. By this method, the azimuthal dependence of polarization of generated VBs was resolved.

Fig. 3. SOP, phase fronts, and generated beam profiles detected with/without analyzer for four different optical retardations: (a) δ=2π, (b) δ=3π/2, (c) δ=π, and (d) δ=π/2. The orientation of the black arrows in dashed circles (see top of columns 4-7) represents the transmission axis of the analyzer.

下载图片 查看所有图片

In Fig. 3(a), one can see that the q-plate acting as a full wave plate (δ=2π) neither affects the SOP nor the intensity profile of the beam. So, the intensity profile retains its Gaussian shape. For phase retardances δ=3π/2 and π/2, the VVBs with P=1, ϕ0=±π/4, and l=−1 are obtained [Figs. 3(b) and 3(d), respectively]. The outgoing beam from the q-plate corresponds to a 1:1 superposition of a non-vortex type (l=0) LCP beam and a vortex type (l=−2) RCP beam. Hence, its intensity profile (column 3 in Fig. 3) is a superposition of a Gaussian and a doughnut-shaped profile. The intensity patterns obtained with the A have an “S” shape consistent with the combined effect of the SOP and the phase front as depicted in the left two columns^[18]. For δ=π [Fig. 3(c)], an RCP VB with the helicity l=−2 is generated. In this case, the q-plate acts as an angular momentum converter that converts an LCP beam with the spin s=−1 and zero OAM to an RCP beam with the spin s=+1 and an OAM associated with the helicity l=−2. Consequently, no momentum transfer to the LC medium takes place. Because of its vortex nature, the outgoing beam exhibits a doughnut-shaped form with zero intensity in the center^[27]. The above experimental results show that the q-plate can realize the interconversion of conventional Gaussian modes and specific beams conveniently by adjusting the applied voltage.

Radial LC alignment patterns with fixed phase retardance can also be utilized to generate arbitrary G-CVBs on the second-order Poincaré sphere. To demonstrate this property, QWP1 and QWP2 were removed from the setup [Fig. 2(a)]. Therefore, the incident light was linearly polarized. The phase retardance δ was adjusted to π, and the angle of incident polarization was set to 0°, 45°, 90°, and 135° with respect to the x axis, respectively. The cross-sectional profiles of the outgoing beam are shown in Fig. 4. The observed patterns rotate synchronously with the rotation of the A, which demonstrates that the SOPs of the generated G-CVBs agree well with the expected polarization patterns on the second-order Poincaré sphere. Namely, the q-plate with δ=π converts a linearly polarized Gaussian incident beam into a G-CVB with polarization order P=2, i.e., the polarization direction is given as ϕ(φ)=2φ+ϕ0^[44]. The helicity of the outgoing beam is in this case l=0, so the phase front of the beam is flat. Nevertheless, the beam exhibits a doughnut-shaped cross-sectional profile because the polarization vectors in the center sum up to zero.

Fig. 4. SOP, phase fronts, and generated beam profiles detected with/without analyzer for different polarization directions of the incident linearly polarized light. Black arrows in the first column indicate polarization states of the ingoing beam. The orientation of black arrows in dash circles represents the transmission axis of the analyzer.

下载图片 查看所有图片

Our last experimental demonstration was performed with δ set to 3π/2 and π/2. By replacing QWP2 (Fig. 2) with two HWPs and tuning the angle between their fast axes, it is possible to generate VVBs with P=1, l=−1, and an arbitrary value of ϕ0. For example, as shown in Fig. 5, an azimuthally polarized (ϕ0=π/2) and a radially polarized (ϕ0=0) VVB can be prepared with the fast axis of HWP1 being parallel to the x axis, and the fast axis of HWP2 being set at 22.5° and 67.5° with respect to the x axis, respectively. The cross-sectional profile of the outgoing beam corresponds to the superposition of a Gaussian and a VB with l=−2. Consequently, the intensity in the center is non-zero, which is similar to the cases shown in Figs. 3(b) and 3(d).

Fig. 5. SOP, phase fronts, and generated azimuthal and radial VVB profiles detected with/without analyzer in the measurement system in which QWP2 (see Fig. 2) was replaced by two HWPs.

下载图片 查看所有图片

Our results demonstrate that radial micro-patterned LC structures fabricated on the basis of the out-of-plane alignment technique provide electrically tunable generation of specific optical beams with a broad range of different SOPs. Phase retardances in the range of 0<δ<4π (at 633 nm) can easily be achieved, which facilitates generation of all theoretically possible conversions, as given by Eqs. (2) and (3). It should be noted that such q-plates can also work at other wavelengths provided that the bias voltage is tuned to corresponding values. The main advantage of our fabrication method is that the obtained LC director field n→(r→) is continuous and smooth, and, consequently, also the optical properties of the fabricated micro-patterned LC structures are varying continuously and smoothly. This enables beam transformations with a very high quality. Our q-plates also can be readily shrunk to the microscale, while, in contrast, the resolution of the LC SLMs, which are otherwise very suitable for construction of the beam converters, suffers from the pixelation of the LC orientational structure. On the other hand, ring-shaped segmented electrodes or radially or azimuthally segmented LC layers that are used in most of the tunable LC q-plates reported until now produce discontinuities that disturb the phase front of the beam.

Another advantage of our fabrication method is that it allows for the construction of micro-patterns with values of q that cannot naturally appear in the LC media, for example, q=1/4, or q=3/4. Such LC structures can be formed because polymer ribbons that provide the LC alignment can fully physically separate the neighboring areas of the LC from each other, so usual smoothing of the director field due to intramolecular orientational coupling is not present. This opens up interesting possibilities for non-standard conversions between non-vectorial and vectorial beams.