## Abstract

With growing data demands, the capacity of the single-mode fiber will reach its limit. Cylindrical vector beams (CVBs) as the eigenmodes of fiber have been demonstrated to increase the capacity in multiplexing optical communication. As a key component of CVB multiplexing, the sorter based on spin-dependent optical geometric transformation has the advantages of high efficiency and a large channel number. However, the demonstrated spatial resolution of the CVB sorter was not sufficient to separate the adjacent CVB orders. Therefore, there were still serious crosstalk issues for the CVB multiplexing channels. Here, we propose and demonstrate the high-resolution CVBs sorting by introducing fan-out structures in the spin-dependent optical geometric transformation. The crosstalk among the CVBs are suppressed compared with the original design. In addition, a cylindrical lens is used to modify the output beams to Gaussian shape. We experimentally demonstrate the sorting of 4 coaxial CVB channels to standard multimode fiber array. In the numerical modeling, we also design the CVB sorter for free space to integrated waveguide array demultiplexing.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The single mode fiber has a capacity limit of 100 Tbit/s given by the Shannon theorem [1]. With the increasing information, the capacity of single mode fiber is approaching this limit. Mode division multiplexing is a potential solution to further increase the capacity by transmitting orthogonal modes parallelly [2]. Orbital angular momentum (OAM) beams are a set of orthogonal modes with helical phase profile of exp(*ilϕ*), where *ϕ* is the azimuthal angular and *l* is an integer number indicating the topological charge [3]. In principle, OAM beams with different *l* are mutually orthogonal as information channels to for data transmission [4–6]. Cylindrical vector beams (CVBs) as another set of orthogonal modes are the eigenmodes in optical fiber [7]. Due to the unique polarization properties, CVBs have been widely used in optical trapping [8], microscopic imaging [9–11], and optical measurements [12]. Recently, CVBs multiplexing optical communication have been demonstrated in both free space and fiber communication systems [13,14]. In our previous work, multiplexing of 4 coaxial CVBs with the orders of ± 2 and spatially orthogonal polarization states have been demonstrated in few-mode fiber communication [15]. However, the high efficiency and large dynamic range detection of CVBs is still a hindrance preventing the CVB multiplexing communication from practical applications [16]. The CVBs can be divided into left-handed and right-handed circular polarized OAM beams with conjugated topological charges. The beam shaping of CVBs are typically based on the Pancharatnam-Berry optical element (PBOE) devices, which could modulate the left-handed and right-handed circular polarized components of the CVB individually. As the widely used PBOE, the Q-plates can only detect the CVBs with certain topological charge. Although the Dammann grating was developed to detect multiple CVBs, it had a loss proportional to the number of the multiplexing channels [17,18]. Inspired by the near-perfect OAM sorting [19], Gary F. Walsh proposed and demonstrated a PBOE device for sorting spin and OAM simultaneously [20,21]. In our previous work, we demonstrated the CVB sorting with 60% efficiency and 20 channels using the spin-dependent optical geometric transformation [22] enabled by liquid crystal (LC) [23,24]. The spatial resolution of the CVB sorter was not sufficient to separate the adjacent CVB modes. There was still serious crosstalk among the CVB channels. It is possible to improve the CVBs sorting resolution and reduce the crosstalk by fan-out elements for the optical geometric transformation [25–32].

In this work, we propose and demonstrate a high-resolution mode sorter for CVBs to Gaussian modes conversions based on the spin-dependent optical geometric transformation. With the fan-out structures and cylindrical lens, the sorted light spots are sufficiently separated spatially and coupled to multimode fiber array for simultaneous detection with improved efficiency. High-resolution CVB sorter requires two PBOE devices fabricated by the photo-aligned LC with a size of 9mm × 9mm and a total pixel number of 768 × 768. We demonstrate detections of CVB modes from −5 order to + 5 order using the fabricated PBOE devices at 1550nm wavelength. We also demonstrate the demultiplexing of 4 coaxial propagating CVB channels to fiber array with the high-resolution CVB sorter. The crosstalk between the adjacent CVB channels and CVB channels with an interval of 2 are suppressed to −7dB and −12dB, respectively. The cylindrical lens converts the sorted light spots to Gaussian shape and reduces the fiber coupling loss by 7 dB.

## 2. Principle

The polarization state of the *m*-th order CVB is described by the Jones matrix as [14,20]:

*m*is an arbitrary integer, $\phi $is the azimuthal angle in polar coordinates, and $\varphi 0$is the initial polarization angle. From Eq. (1), the

*m*order CVB can be decomposed into right-handed circular-polarized (RCP) and left-handed circular-polarized (LCP) OAM components with opposite topological charges. Therefore, the spin-dependent optical geometric transformation of CVBs requires the PBOE with polarization responses. The PBOE can induce a patterned Pancharatnam-Berry (PB) phase to the polarized light by controlling the rotations of the LC molecules. The Jones matrix of the PB-phase can be written as:

Equations (3) and (4) show that the LCP (RCP) light is converted into RCP (LCP) light with additional PB-phase of $2\alpha $$(-2\alpha ).$ This spin-dependent phase response device can independently modulate different polarization states.

Figure 1(a) shows schematic of the traditional optical geometry transformation method without fan-out structures. Due to the limited phase gradient, the adjacent CVBs cannot be spatially separated. Therefore, it is necessary to develop the high-resolution optical geometry transformation to efficiently sort the adjacent CVB orders with low crosstalk. Figure 1(b) shows the schematic of the high-resolution CVB sorting based on spin-dependent fan-out optical geometric transformation. There are two PBOE devices required in the sorting system. The first device is designed for CVB transformation from the Cartesian to the log-polar coordinates and the fan-out functionality (P3). We use (*x, y*) and (*u, v*) to represent the input and output coordinates, respectively. For the coordinates transformation, we have$v=\frac{d}{2\pi}\cdot {\mathrm{tan}}^{-1}(y/x)$ and $u=-\frac{d}{2\pi}\cdot \mathrm{ln}(\sqrt{{x}^{2}+{y}^{2}}/b),$ where d and b are the scale parameters. The optical geometric transformation is capable of converting the ring-shaped CVBs into straight light beams with linear phase gradients. The incident *m*-th order CVB can be considered as two OAM modes with opposite topological charges and circular handed polarization states. When CVB interacts with PBOE, the LCP and RCP components experience opposite phase responses. The LCP and RCP components undergo different expansion processes from ring shaped to strip light at the Fourier plane as shown in Fig. 1(a). For LCP components, the geometric transformation maps the coordinates from (*x, y*) to (*u, v*), and for RCP component, the geometric transformation maps the coordinates from (*x, y*) to (-*u,* -*v*). Therefore, the spiral phase distributions of OAM with a topological charge of ± *m* are converted into linear gradient of 2*m*π. During this process, the fan-out structures also duplicate the unwrapped beam into 2*N* + 1 copies. Therefore, the gradient phase of light beam is increased to $\left(2N+1\right)2m\pi .$ The larger phase gradient gives the sorted light spots higher spatial resolution [23,30]. The fan-out geometric optical transformation is expressed as:

Where $\varphi \text{1}(x,y)$is the orientation of the fast-axis of the PBOE devices for fan-out optical geometric transformation, 2*N* + 1 is the number of the copies, *θ* is the angular separation between the adjacent copies, *c _{k}* and

*a*are the adjustment parameters to optimize the light beams. In the output plane, the light beams have distorted phase from the optical geometric transformation and the phase jumps generated from the fan-out structures, so the second PBOE device of phase corrector (P4) is necessary. The expression of the phase corrector is given by

_{k}Where $\varphi 2(u,v)$is the orientation of the fast-axis of the PBOE devices for phase corrector, *f*_{1} is the focal length of the Fourier-transforming lens. The phase jumps between the adjacent copies is compensated by ${\phi}_{k}\left(k\right)$ to smooth the wavefront. After phase corrector, 2*N* + 1 strip shaped light beams are further converted into 2*N* + 1 rectangular-shaped light beams. 2*N* + 1 rectangular-shaped light beams are focused into a Gaussian like spot by a combination of a lens and a cylindrical lens with an offset position proportional to the phase gradient. Therefore, the coaxial CVBs are converted into light beams with different phase gradients according to their orders. The offset of the lateral position of the sorted light spots is proportional to the order of the CVB. High-resolution CVB sorter can spatially separates adjacent order CVBs at the focal plane, just as in Fig. 1(b), taking m = 1 and m = 2 order CVB as examples.

According to Eqs. (5) and (6), we design and fabricated the spin-dependent fan-out optical geometric transformation and phase corrector devices using the photoaligned LC. Here, we used a sulfonic acid azo dye SD1 (Dai-Nippon Ink and Chemicals, Japan) as an aligning agent. Further, a 0.5% solution SD1 in dimethylformamide was spin-coated on an indium tin oxide (ITO) glass substrate. The 6μm spacers were spurted over one SD1-coated substrate, and then the counter substrate was covered. They were assembled together and sealed with epoxy glue to form the cell. The collimated UV beam is then reflected onto the DMD and the designed hologram is loaded. After focusing through the tunable objective, the beam is polarized by an electrically rotating polarizer and then projected onto the LC cell. When the SD1 molecule absorbs UV photons, the dye molecules are isomerized and eventually tend to be perpendicular to the local polarization orientation due to their dichroic absorption. The polarization dependent hologram is fabricated by repeating the above process multiple times with desired polarized incident UV beam [23,24]. PBOE can be considered as half-wave plates with different orientations that shifts the polarization direction of the linearly polarized light. Therefore, the intensity distributions of the linearly polarized light through the PBOE placed between two polarizers indicate the directions of the fast axis. Figures 1(b) and 1(c) show the calculated polarization images of the devices. The devices are fabricated by photoaligned LC with the pixel number of 768 × 768 and the pixel size of 11.7 μm. The polarization pattern exhibits a periodic variation of intensity with the linearly polarized light changing from 0° to 90° or180° to 90°. we characterize the fast axis distributions of the PBOEs at 1550 nm wavelength using a microscopic imaging system. Figures 1(d) and 1(e) show the corresponding microscopic images of the fabricated devices in the area as labeled in Figs. 1(b) and 1(c). The devices parameters are set as d = 2 mm, b = 4mm, *f*_{1} = 200mm, λ = 1550nm and N = 1. The parameters of three copies generated from the fan-out are set as $\theta =\frac{d}{{f}_{1}}$, ${\phi}_{k}\left(k\right)$ = (0, $-\frac{3\pi}{2}$, 0), *c _{k}* = (1.32, 0, 1.32) and

*a*= ($-\frac{\pi}{2}$, 0, $-\frac{\pi}{2}$) for k = −1, 0 and 1. The focal length of the second lens (L2) determine the size of the light spot. After the fan-out geometric transformation, the CVBs are converted to long strips with aspect ratios less than 1:10. From the modeling, the cylindrical lens with less than 1/10 focal length of the second lens is sufficient to compress the light spots into Gaussian like light spots.

_{k}## 3. Experiment results and analysis

We experimentally demonstrate high-resolution CVB sorting based on spin-dependent fan-out optical geometric transformation. In the experimental setup (Fig. 2(a)), the collimated light beams at the wavelength 1550 nm pass through vortex wave plates (VWP, Thorlabs WPV10L-1550 and WPV10-1550) to generate CVBs. With the combination of a VWP and a half-wave plate, CVBs from + 5-th orders to - 5-th orders are generated, respectively. Figure 2(b) shows the incident CVB from −5 to + 5 orders passing through the liner polarizer captured by a near-infrared camera. Then, the light beams are focused by a lens (L2) with a focal length of 150mm, and a cylindrical lens with a focal length of 3.9 mm placed at 146 mm behind the L2. We characterize the CVBs sorting results using a microscopic imaging system. Figure 2(c) shows the calculated light spots converted from the CVBs after the high-resolution spin-dependent fan-out optical geometric transformation. The light spots show Gaussian like shapes which are suitable for free-space to fiber coupling. Taking the 0 order CVB as a reference, the CVBs with positive (negative) orders are offset to the right (left). The experimental results of the CVB sorting are consistent with the modeling results as shown in Fig. 2(d). The intensity distributions of the sorted light spots are extracted from the images both in the calculation and experiment. Figures 2(e) and 2(f) show the calculated and measured intensity distributions of the CVB sorting results from the −5 order to + 5 order. The sorted light spots have a diameter of 70μm. The crosstalk among the CVB sorting results mainly come from the sidelobes. The sidelobes in the sorted CVB beams come from high order diffractions of the devices. In principle, the sidelobes as well as the crosstalk can be reduced by increasing the number of the fan-out copies [27,29,33]. Due to the resolution limit of the LC fabrication process, we only demonstrated three fan-out copies in this work. It is possible to demonstrate devices with five fan-out copies to effectively suppress the crosstalk between adjacent CVB orders. And then we couple the sorted light spots to multimode fiber with a core diameter of 62μm. With the cylindrical lens, the sorted light spots are converted to Gaussian shape beams and show 7 dB coupling loss improvements compared to the case without the cylindrical lens as shown in Fig. 2(g). The insertion loss of the CVB sorter from the free space to multimode fiber is around 8dB, which mainly comes from the sidelobes and slightly size mismatch between the light spots and the multimode fiber core.

We also demonstrated the high-resolution multiple coaxial CVBs sorting. Figure 3(a) shows the setup for sorting coaxial CVBs of −2, −1, 1, 2 orders using the spin-dependent fan-out optical transformation. A 1550 nm laser beam is amplified by erbium-doped fiber amplifier (EDFA) and divided into four branches. Then the collimated outputs from the four branches are converted to CVBs with ± 1 and ± 2 orders by the vortex wave plates. These four CVBs are combined into coaxial beams using three beam splitters. After free-space transmission of 1 meter, four coaxial CVBs channels are sorted by the spin-dependent fan-out optical transformation. Figure 3(b) shows modeling results of the 4 Gaussian shaped light spots sorted from the CVBs. Figure 3(c) shows the images of experimental sorting results captured by an infrared camera, which are consistent with the calculation. After the phase corrector, we use a lens of 150 mm focal length and a cylindrical lens of 3.9 mm focal length to obtain light spots with spacing of 127 μm and diameter of 70 μm. The spacings between the four channels match with the multimode fiber arrays. Thus, we couple these four channel signals into the fiber array simultaneously. Figure 3(d) shows the image of the multimode fiber array with spacing of 127 μm and core size of 62 μm. The coupling loss of four CVBs modes with the orders −1, 1, −2, −2 are 7.92 dB, 7.85 dB, 8.65 dB, 8.69 dB, respectively. Figure 3(e) shows the normalized mode crosstalk matrix of the four CVB channels. The crosstalk between the adjacent CVB modes is lower than −6dB, mainly come from side lobes. For the CVB modes with an interval of 2, the crosstalk could be further suppressed to less than −12dB.

## 4. Device design parameters optimization

The spatial offsets (*t _{m}*) of the converted spots are related to the wavelength (λ), scaling factor (d), focal length (

*f*

_{1}

*)*of the first lens and the CVB order (

*m*) as follow [17]:

The sorting resolution can be improved by increasing the number of fan-out elements. However, as the resolution of the sorting increases, the center position of the spot does not change. Therefore, the spatial separation is not related to the fan-out element number [25,27]. The second lens is used to control the spacing and spot size of the sorting light spots. We change the focal length of the second lens to calculate the distance of the adjacent spots under different parameters. Figures 4(a) and 4(b) show the distance of adjacent spot and the spot size respectively, with the incident beam waist diameter of 3 mm, devices size of 9 mm × 9 mm, *f*_{1} of 200 mm and the scaling factors from 0.8 mm to 4 mm. The red star in the figure indicates the parameters selected matching the multimode fiber array in the experimental demonstration. In order to miniaturize the device and reduce the distance between the two devices, we have to optimize the parameters. Figures 4(c) and 4(d) shows the distance of adjacent spot and the spot size respectively, with the incident beam waist diameter of 0.8 mm, devices size of 1.6 mm × 1.6 mm, *f*_{1} of 20 mm and the scaling factors from 0.8 mm to 4 mm. As shown in Figs. 4(c) and 4(d), with the scaling factors of 1.2mm, pixel size of 1.6μm and *f*_{2} of 40 mm, the spacing and the spot size could be scaled down to 6μm and 3μm labeled by the red dashed lines, which are suitable for the waveguide array coupling. Therefore, we can design the miniaturized CVB sorting device for integrated optical communication applications.

## 5. Conclusion

In summary, we propose and demonstrate high-resolution CVB sorting based on spin-dependent fan-out optical geometric transformation. The CVBs are converted from ring shape to Gaussian shaped spots by two PBOE devices. The PBOE devices are fabricated by the photo-aligned LC with insertion loss of 0.71 dB and 0.86 dB, respectively. We demonstrate CVB sorting from −5 order to + 5 order with improved spatial resolution and coupling efficiency enabled by the fan-out structure and the cylindrical lens. We also demonstrated −1, 1, −2 and −2 orders CVB modes sorting to multimode fiber array with matched spot size and spacing. Limited by the fabrication precision of the photoaligned LC, we only demonstrated CVB sorting with a dynamic range of 11 different CVB orders. The adjacent CVBs separation distance are related to the wavelength (λ), scaling factor (d) and focal length (*f*_{1}*)*. From the modeling, it is possible to design the compact devices with scale down parameters for CVBs sorting to integrated waveguide array.

## Funding

National Natural Science Foundation of China (NSFC) (U1701661, 61427819, 11774240, 11604218); Leading Talents of Guangdong Province Program (00201505); Science and Technology Innovation Commission of Shenzhen (KQTD2015071016560101, KQTD20170330110444030, KQJSCX20170727100838364); Natural Science Foundation of Guangdong Province, China (2016A030312010).

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