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Spin-dependent Optical Geometric Transformation for Cylindrical Vector Beam Multiplexing Communication Juncheng Fang, Zhenwei Xie, Ting Lei, changjun min, Luping Du, Zhaohui Li, and Xiaocong Yuan ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.8b00680 • Publication Date (Web): 03 Aug 2018 Downloaded from http://pubs.acs.org on August 3, 2018
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ACS Photonics
SpinSpin-dependent Optical Geometric Transformation for Cylindrical Vector Beam Multiplexing Communication Juncheng Fang,†,|| Zhenwei Xie,†,§,|| Ting Lei,†,* Changjun Min,† Luping Du,† Zhaohui Li,‡,* Xiaocong Yuan†,* †Nanophotonics
Research Centre, Shenzhen Key Laboratory of Micro-Scale Optical Information Technology, Shenzhen University, 518060, China. ‡State Key Laboratory of Optoelectronic Materials and Technologies and School of Electronics and Information Technology, Sun Yat-sen University, Guangzhou 510275, China. §School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798. KEYWORDS: optical communication, cylindrical vector beam, optical geometric transformation, P-B phase.
ABSTRACT: Given by the Shannon theorem, the data rate in a single mode fiber is approaching the capacity limit of 100 Tbit/s, which even applies to all existing wavelength division multiplexing and advanced modulation formatting techniques. Optical vortex beams, including orbital angular momentum (OAM) beams with phase singularities and cylindrical vector beams (CVBs) with polarization singularities, are orthogonally structured light beams providing new degrees of freedom for multiplexing optical communication, for which the multiplexer is the key component. Although there are various OAM detection approaches such as the optical geometric transformation and vortex grating, CVB sorting with high efficiency and large dynamic range has not been demonstrated before. In this work, we propose and demonstrate an efficient approach for multiple coaxial CVB sorting based on the spin-dependent optical geometric transformation using the Pancharatnam–Berry optical element (PBOE) device fabricated with the photo-aligned liquid crystal. We demonstrate a CVB multiplexing communication system in both free space and few-mode optical fiber. The CVB sorter is compatible with wavelength division multiplexing and shows the potential to further increase the communication capacity by 1–2 orders of magnitude.
Cylindrical vector beams (CVBs), as one class of singular optical beams, have hollow-shaped intensities and rotational-symmetric polarization distributions, i.e., polarization singularities1. Because the focusing and polarization properties are unique2-5, CVBs have been widely investigated in regard to, for example, optical measurements6, optical trapping7, and microscopic imaging8. Multiplexing communication is an important application of CVBs that have been attracting much attention recently9. The multiplexing of singular optical beams such as the orbital angular momentum (OAM) has shown rapid capacity increases in both free-space and optical-fibre communication. Also, numerous methods are proposed for the OAM mode generation and multiplexing10-13. As eigenmodes of the optical fibre, CVB modes have the advantage of stable propagation in multiplexing optical communication compared with OAM modes14-16. There are several CVB generation approaches including using liquid-crystal (LC) spatial light modulators17, 18, meta-hologram nanostructures19, 20, and spot-defect mirrors21, as well as coherent beam combining22. The CVB encoding/decoding in free-space and optical-fibre communication have also been demonstrated16, 23, 24. However, the sorting of the CVB modes is still a great challenge preventing the practical application of CVB mode division multiplexing.
In most CVB communication work, the Q-plate is used to detect the CVB modes16, 23, 24. However, the Q-plate only detects the CVB of a specific mode order and does not satisfy the requirement of parallel sorting of multiple CVB modes in communication applications. Moreno et al. proposed a CVB detection method using a circular-polarized and a vortex sensitive diffraction grating encoded on a parallel-aligned LCD25, 26. Nevertheless, this work only demonstrated the detection of low-order CVB with radial and azimuthal polarizations. Therefore, there is a need to develop a CVB sorting approach with large dynamic range and low crosstalk as well as high power efficiency. Inspired by the well-demonstrated near-perfect OAM sorting27-29, we propose a CVB sorting scheme based on the spindependent optical geometric transformation30. The CVB can be considered as a combination of two OAM beams with opposite-handed circular polarization and conjugate topological charges31, 32. As the Pancharatnam–Berry optical element (PBOE) device brings a spin-dependent response, the left-handed and right-handed circularpolarized components of the CVB can be individually modulated by the PBOE. In our study, the PBOE devices are fabricated using the photo-alignment LC in a thin film with a total pixel number of 768×768 and a pixel size of 11.7 µm. Through the spin-dependent optical geometric transformation enabled by the PBOE devices, the CVB is con-
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verted from a doughnut shape to two straight lines. During the spin-dependent geometric transformation, the LCP and RCP components are unwrapped along opposite directions, which is essentially different from the conventional optical geometric transformation (see Supporting Information for the details). After phase correction, the CVB is finally focused onto a single light spot with a lateral displacement proportional to the input CVB orders. In a proof-of-concept experiment, we demonstrated CVB sorting with orders from −10 to 10 at an efOiciency of up to 61.7%. We also demonstrated multiple coaxial CVBs demultiplexing for both free-space and optical-fibre communication systems. The CVB sorting approach shows potential applications in the next generation high-capacity optical communication with multiplexing technology. The polarization state of the mth-order CVB is defined by the Jones matrix Jm,
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specifically enables the design of the spin-dependent geometric transformation device.
1 i(m ϕ +φ 0 ) − i(m ϕ +φ 0 ) e +e } cos(mϕ + φ0 ) 2 { Jm = = sin(mϕ + φ0 ) 1 {ei(m ϕ +φ 0 ) − e − i(m ϕ +φ 0 ) } (1) 2i 1 1 1 1 = ei(m ϕ +φ 0 ) + e − i(m ϕ +φ 0 ) 2 −i 2 i where m is an integer number, φ the azimuthal angle, and ϕ0 the initial phase angle. From Eq. (1), the mth-order CVB decomposes into a pair of optical vortex (OV) beams, one right-handed circular-polarized (RCP), the other lefthanded circular-polarized (LCP). The phase terms indicate that the RCP(LCP) component carries an OAM with a topological charge of +m (−m). That is, a CVB combines two OV beams of opposite-circular polarizations and conjugate topological charges [Fig. 1(a)]. Fig. 1(b) shows the Pancharatnam–Berry (PB) phase introduced by the microstructures where CVBs interact with the LC molecules. The thickness of the LC layer is carefully adjusted to achieve a half-wave retardation of the incident light. At any local position of the CVB, there is a fixed angle between the fast axis of the LC and the light polarization direction. The Jones matrix of the PB phase is expressed as
cos 2θ (x, y) sin 2θ (x, y) M (x, y) = , sin 2θ (x, y) − cos 2θ (x, y)
(2)
where θ (x, y) is the angle between the fast-axis orientation of the LC and the incident polarization. When the LCP or RCP light passes through the LC layer, the electric fields are derived as
M ( x, y ) ELCP = E0 M ( x, y ) [1; i ] = E0 ei 2θ1 ( x , y ) [1; −i ]
=e
u = −a ⋅ ln( x 2 + y 2 / b) , where a , b are scale parameters of the transformation along the u, v directions, respectively. The corresponding phase profile of the geometric transformation element is written as
(3)
M ( x, y ) ERCP = E0 M ( x, y ) [1; −i ] − i 2θ 2 ( x , y )
The optical geometric transformation refers to a geometric coordinate transformation from Cartesian to logpolar33. We use (x, y) and (u, v) to indicate the coordinates of the input and output planes, respectively. For a conformal transformation33, 34, we have v = a ⋅ tan −1 ( y / x) and
φ ( x, y ) =
= ei 2θ1 ( x , y ) ERCP , = E0 e − i 2θ2 ( x , y ) [1; i ]
Figure 1. Schematic of the interaction between the CVB and the liquid crystal molecule. (a), A mth-order CVB comprised of an LCP OV beam with a topological charge of −m and an RCP OV beam with a topological charge of +m. (b), Pancharatnam– Berry (PB) phase introduced by the liquid crystal molecule. The axis of the liquid crystal (LC) molecule has an angle of θ with respect to the polarization direction of the incident light. With a half-wavelength retardation controlled by the LC layer thickness, the LCP and RCP light interact with the LC and result in a PB phase modulation with opposite sign.
(4)
ELCP ,
Eq. (3) and (4) show that the LCP (RCP) light is converted into RCP (LCP) light with an additional PB phase of 2θ ( −2θ ) [Fig. 1(b)]. This spin-dependent phase response
x2 + y2 2π a y [ y tan −1 ( ) − x ln( ) + x], λf x b
(5)
Under this transformation, the azimuthal phase profile of an OAM mode is mapped to a tilted planar wave front28, 29. For a given CVB mode, Eq. (1), a pure phase modulated geometric transformation device would result in two inverse-tilted planar wave fronts. Here, we use a LC-based spin-dependent geometric transformation device, so that the LCP and RCP components have an inverse phase modulation given by
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ACS Photonics device, the LCP and RCP components undergo opposite phase responses and are converted from a ring to a strip through different unwrap processes at the Fourier plane (see Supporting Information for the details). Thus, the helical phase shifts along the original OAM beams with topological charges of ±m are also converted into a gradient phase of 2mπ along the line direction. Because the LCP and RCP components receive opposite phase responses, the unwrap direction of these components are also different [Fig. 2(b)]. After the transformation, the LCP beam carrying an OAM state of −m and the RCP beam carrying OAM state of +m are converted into two strips with identical phase gradients. For the LCP component, the geometric transformation maps (x, y) to (u, v), and for the RCP component, the geometric transformation maps (x, y) to (−u, −v). In the output plane, these two components are mapped to two identical tilted planar wave fronts. The corresponding phase correction is calculated using the stationary phase approximation34,
Figure 2. Schematic of CVB sorting based on the spindependent optical geometric transformation enabled by liquid-crystal-based geometric transformation devices. (a), Specific geometric transformation procedure for sorting a single CVB mode. P1 is the geometric transformation device. P2 is the phase correction device. L1 and L2 are optical lenses. (b), Spindependent geometric transformation of the LCP and RCP components comprising one CVB mode. The color map from red to blue indicates the phase gradient after the phase correction device. (c), Multiple coaxial CVB modes sorted by the optical geometric transformation. The colors are used to label the CVBs with different topological charges.
x2 + y 2 2π a y φ1LCP ( x, y ) = [ y tan −1 ( ) − x ln( ) + x] λf x b −y 2π a RCP [ − y tan −1 ( ) φ1 ( x, y ) = −x λf (6) 2 (− x) + (− y ) 2 + x ln( ) − x] b
α1 ( x, y ) =
x2 + y2 y 2π a [ y tan −1 ( ) − x ln( ) + x], λf x b
where α1 is the orientations of the fast-axis of the LC molecules for the spin-dependent geometric transformation. Fig. 2 shows a schematic of CVB sorting using the spindependent geometric transformation devices. Through the spin-dependent phase modulation by the LC, the ringshaped CVB is converted into two straight light beams with identical linear phase gradient and focused onto a single light spot after phase corrections [Fig. 2(a)]. Fig. 2(b) shows the specific beam-shaping procedure. The incident CVB of mth-order can be considered as two OV beams with opposite OAM topological charges and circular handed polarization states. When the CVB interacts with the LC
2π ab u v LCP φ2 (u , v) = − λ f exp[− a ]cos( a ) φ RCP (u , v) = − 2π ab exp[− −u ]cos(− v ) 2 λf a a
α (u , v ) = − 2
(7)
π ab |u | v exp[− ]cos( ), λf a a
where α 2 is the orientation of the fast-axis of the LC molecules for phase correction. After phase correction by the second device, the two rectangular-shaped light beams are focused onto a single spot with an offset position corresponding to the phase gradient. Fig. 2(c) schematically shows the sorting of coaxial CVBs with orders m and n using the proposed approach. The coaxial CVBs are converted into straight beams with different gradient phases proportional to their orders. The shift in the transverse position of the spot is in direct proportion to the order of the CVB. Therefore, the CVBs of the mth and nth orders are separated spatially at the focal plane. In accordance with Eqs. (6) and (7), we designed and fabricated the spin-dependent optical geometric transformation and phase correction devices using the photoaligned LC (see Supplementary Information). The fast-axisorientation distributions of the LC for the spin-dependent optical geometric transformation device and phase correction device are denoted by α1 and α 2 , respectively. The LC device has a total number of 768×768 pixels with a 11.7µm pixel size. Fig. 3(a) and (b) show the fast-axis orientations of the LC molecules for the two designed devices. It is difficult to directly measure the fast axis distributions of the manufactured PBOEs. The PBOE can be considered as a set of half waveplates with different orientations, which shifts the polarization direction of linearly polarized light. Thus, an alternative way to investigate orientations of the fast axis is to measure the transmitted linearly polarized light after the PBOE with a cross-polarized polarizer (see Supplementary Information). The intensity is related to the polarization shift and indicates the orientation of the fast axis of the device. Fig. 3(c) and (d) show the simulated
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polarization images of the devices. Since the intensity distribution is same for the polarization variations from 0° to 90° and from 180° to 90°, the polarization patterns have doubled periods compared with the orientation patterns of the LC molecules. We characterize the fast axis distribution of the PBOEs at 1550 nm wavelength using a microscopic imaging system. Fig. 3(e–h) shows the corresponding microscopic images of the fabricated devices in the areas labelled as 1 to 4 in Fig. 3(c) and (d).
experiments, the spin-dependent geometric transformation shows superior performance in CVB sorting. In the experiment, we demonstrated a large dynamic sorting range of 20 different orders of CVBs limited by the aperture of the device and the fabrication precision of the Qplate for CVB modes generation. Limited by the pixel number (768 × 768) and size (11.7m) of the Q-plate used in our work, the largest topological charge of the generated CVB mode is up to ±10. Therefore, we only demonstrate CVBs sorting with a dynamic range of 20 channels in the proof-of-concept experiment. The measured sorting efficiency is as high as 61% and takes into account LC scattering and interface reflections. It is important that the power efficiency of the proposed method does not depend on the number of the multiplexed CVB modes, which is essentially different from the sorting system with multiple beamsplitters. The theoretical maximum power efficiency of the sorting system with multiple beam-splitters is 1/N, where N is the total number of the multiplexed modes. Fig. 4(d) shows the simulated intensity distribution of the sorted CVBs from the −10-th order to the +10-th order, assuming a lens focusing with focal length 250 mm after phase corrections. The separation between the adjacent sorted CVBs is around 40 µm at the focal plane. We also experimentally extracted the intensity distribution of the CVBs from their images after sorting [Fig. 4(d)]. The measured positions of the sorted CVBs with different orders are consistent with the simulated results. There is some crosstalk among the sorted CVBs that mainly come from the overlap of the sidelobes.
Figure 3. Designed and fabricated liquid crystal geometric transformation devices. Fast-axis orientation distributions of: (a), the spin-dependent geometric transformation device and (b), the phase correction device. (c) and (d), Simulated polarization images of the designed devices. (e-h), Corresponding microscopic polarization images of the fabricated LC devices in the areas labelled as 1 to 4 in (c) and (d). Scale bars in (c) and (d) are 1 mm. Scale bars in (e-h) are 0.5 mm.
We demonstrated in experiments CVB sorting using the spin-dependent geometric transformation approach. Fig. 4(a) shows the intensity distributions of the incident single CVBs for orders from −10 to +10. The intensity patterns of the CVBs are captured by a near infrared camera with a linear polarizer. Because the polarization states of the mth order CVB have a 2mπ continuous variation around the circle, there are 2m bright spots along the azimuthal direction. Fig. 4(b) shows the calculated focusing spots of the CVBs after the geometric transformations and phase corrections. Taking the 0-th CVB or Gaussian beam as a reference, a CVB with positive (negative) order is sorted with an offset to the right (left). Fig. 4(c) shows the corresponding experimental results of CVB sorting, which are consistent with those from simulations. Both from simulations and
Figure 4. Experimental results of CVB sorting using the spindependent optical geometric transformation approach. (a), Measured intensity distributions of the incident CVBs of order 10, 7, 3, 0, −3, −7, and −10 after the linear polarizer. (b), Calculated CVBs sorting results with displacements proportional to the incident CVBs orders. (c), Corresponding experimental results of CVBs sorting captured using an infrared camera. Sorted light spots positions extracted from: (d), simulation and (e), experiment for CVBs with order from −10 to +10.
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Figure 5. Fiber and free-space CVB-multiplexing optical communications. (a), Experimental setup with the spin-dependent optical geometric transformation sorting approach. Insets show the intensity distribution before and after sorting. (b), Coaxial CVBs of orders −4, −1, 2, 5 (upper) and −10, −5, 1, 10 (lower) and their corresponding sorting results (c), simulation and (d), measurement using the spin-dependent optical geometric transformation. The dash line denotes the forward error correction threshold. (e), Measured mode purities of the two CVB modes after propagation in a few-mode fiber for 2.8 km. Measured bit error rates of: (f), the multiplexed CVB channels and (g), the multiplexed coaxial CVB channels in free-space CVB communication.
We also demonstrated the sorting of the coaxial multiple CVBs. Fig. 5(a) shows two combinations of multiple coaxial CVBs: those with orders −4, −1, 2, 5 with a minimal order interval of 3, and those with orders of −10, −5, 1, 10 with a maximum order range of 20. Fig. 5(b) and (c) shows the calculated and experimentally measured sorting results for multiple coaxial CVBs, respectively. We implemented multiple CVB demultiplexing in free-space optical communication using these above two sets of coaxial CVBs. Fig. 5(d) shows the measured bit error rate for the CVBs as individually modulated communication channels, each carrying 10 Gbit/s signals. The data points labelled by “B2B” indicate the back-to-back signal transmission without any CVB multiplexing. For the multiplexing CVB communication results, there are two sets of data points for the coaxial CVBs of −4, −1, 2, 5 and of −10, −5, 1, 10. For the latter, the BER curves follow the same trend as the B2B curve with a power penalty of less than 4 dB. For the former, the BER curves are relatively far away from the B2B curve, which indicate large crosstalk among the communication channels. Fig. 5(e) shows the experimental setup of the CVBs multiplexing-based fibre communication using the isotropic geometric transformation at the receiver end. A 1550-nm laser beam is modulated to carry a 10 Gbit/s on-off keying signal that originates from a pattern generator, amplified by an erbium-doped fibre amplifier and split into two branches by a coupler. There is a 20 m length difference between the two branches to ensure the two signals are unrelated. Then the collimated output from the two branches are converted to CVBs with orders +2 and −2 using the vortex waveplates. These two CVBs are combined into a single coaxial beam using a beam splitter. They are coupled into the few-mode fibre and transmitted 2.8 km. The few-mode fibre used in the experiment sup-
ports CVB modes with orders up to ±2. The output CVBs from the fibre are demultiplexed by the isotropic geometric transformation and coupled into single mode fibre for error detection. Fig. 5(f) shows the mode purity measurement of the two CVB modes after propagation in the 2.8km few-mode fibre. With the output of the +2 and −2-order CVBs, we measured the power of the CVB modes from −4 to +4. The measured mode purities of the +2 and −2 CVB modes are 73.7% and 72.2%, respectively (see Supporting Information for the detailed definition of mode purity). Fig. 5(g) shows the measured BER curve of the +2 and −2 order CVBs with a power penalty of 6 dB.
Figure 6. Capacity of the CVB sorting scheme based on spindependent geometric transformation parameters. Maximum number of the sorted CVBs channels with: (a), aperture size from 9 to 15 mm and focal length from 5 to 40 cm and (b), aperture size from 0.8 to 1.5 mm and focal length from 0.5 to 20 cm. The scaling factors are set as 8 mm (a) and 0.8 mm (b), respectively.
In principle, the proposed approach has the capability to sort infinite numbers of CVBs. However, in practical, the maximum sorting numbers of the CVBs depend on various parameters of the device and communication system, such as the aperture size, pixel number, transformation scaling factors, and focal length of the lens. We analysed how these parameters influenced the maximum multiplexing channel
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number of the CVBs in the communication system. The sorting capacity is mainly determined by the scaling factor d, aperture size of the optical element A, and the focal length of the lens f. Within a certain aperture, the maximum order of the CVB that can be sorted with the proposed method is determined by m ≤ Ad / (2λ f ) . Therefore, to increase the multiplexing capacity, we prefer a large aperture, a large scaling factor, and a short lens focal length. For the experimental demonstration in this work, the maximum sorting capacity is calculated to be as high as 200 CVB channels (A=9 mm, d=8 mm, f=20 cm). Although in our proof of concept demonstration the largest number of sorted CVB is 20, there is no fundamental limit to a further increase in the capacity of CVB multiplexing by optimizing the fabrication procedures and enlarging the aperture of the device. As shown in Fig. 6(b), the device parameters can be scaled down and achieve CVB sorting for integrated photonic applications. To achieve a relative large CVB sorting capacity (~100) with a small aperture (~mm), we only need to adjust the focal length (~mm). Also the manufacture of the Q-plate need to be improved for generating high order the CVBs. The integrated CVB sorting devices can be designed and fabricated based on the dielectric metasurface. According to the space-bandwidth product and the scaling effect of the Fourier transformation, the separation efficiency (see Supporting Information for the detailed definition) of the sorted CVB channels stays constant under any variation of the parameters. The separation efficiency between the adjacent CVB modes under our conditions is around 0.7. Thus, we have demonstrated CVB communications with a mode interval of 3 to satisfy the crosstalk threshold (see Supporting Information for the detailed definition). The separation efficiency may be improved by the fan-out approach in the optical transformation to further increase multiplexing capacity 29, 35, 36.
ASSOCIATED CONTENT Supporting Information. The Supporting Information Available: Spin-dependent geometric transformation; Device fabrication; Numerical simulation and the measurements; Figures S1-S6. This material is available free of charge via the Internet at http://pubs.acs.org
AUTHOR INFORMATION Corresponding Author *
[email protected] (X. C. Yuan);
[email protected] (T. Lei);
[email protected] (Z. Li).
Author Contributions X.Y., Z. L. and T.L. developed the concept presented in this paper. J. F. and Z. X. carried out the analytical and numerical modeling and design the device. Z. X., T. L., Z. L. and J. F. conducted the measurements. X.Y. and T.L. supervised the entire project. Z. X., T. L., L. D. and C. M. wrote the manuscript. All authors discussed the results and commented on the article. || J. Fang and Z. Xie contributed equally to this work.
Notes The authors declare no competing financial interest.
ACKNOWLEDGMENT
This work was supported by the National Natural Science Foundation of China under Grant Nos. U1701661, 61427819, 61525502, 61435006, 61622504, 91750205, 11774240 and 11604218; Science and Technology Innovation Commission of Shenzhen under grant Nos. KQTD2015071016560101, KQJSCX20160226193555889, KQJSCX20170727100838364, ZDSYS201703031605029, KQTD2017033011044403. X.Y. appreciates the support given by the leading talents of Guangdong province program No. 00201505. Natural Science Foundation of Guangdong Province, China (2016A030312010, 2017A030313351); National Key Basic Research Program of China (973) under grant No. 2015CB352004.
ABBREVIATIONS PB, Pancharatnam–Berry; NIR, near infrared; FDTD, finite difference time domain; SPPs, surface plasmon polaritons; TE, transverse electric; TM, transverse magnetic; OAM, orbital angular momentum; CVB, cylindrical vector beam; LC, liquid crystal; PBOE, Pancharatnam–Berry optical element.
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Figure 7. TOC graphic
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