## Abstract

An air-core fiber imposed by torsion is investigated in this paper. We refer to this kind of fiber as twisted air-core fiber (TAF). It has been demonstrated that the eigenstates of the TAF consist of guided optical vortex waves with different propagation constants of a different effective index. With the increase of the twist rate, TAF could separate the OAM modes which are near degenerate or degenerate in the air-core fiber. The separation of OAM modes in TAF is conductive to ultralong distance propagation with low crosstalk. TAF could be considered as an ideal candidate fiber for OAM based optical communication. Moreover, we investigated the twisted air-core photonic crystal fiber (TAPCF) which can improve the relative energy distribution of the OAM modes. Compared with TAF, more energy is located in the ring shaped core, which is conductive to ultralong distance propagation. TAF and TAPCF are of potential interest for increasing channel capacity in optical telecommunications, and the result is also of interest to the photonic crystal community.

© 2016 Optical Society of America

## 1. Introduction

Orbital-angular-momentum (OAM) light has been considered to be the effective means of increasing the capacity of communication [1,2 ]. Its inherent unlimited orthogonal eigenstates provides an extra degree of freedom for multiplex technique which leads to the exploration of OAM communication [3,4 ]. Comparing with the communication in free space, fiber-optic transfer network promises advantages of large bandwidth, ultralong distance communication, freedom from crosstalk and other types of interference. Thus the OAM propagation in the fiber with low crosstalk and long propagation distance is the primary consideration and fundamental research. Recent years, ring-fiber and air-core fiber were proposed for supporting the propagation of the OAM modes [5–8 ]. In these fibers the OAM modes could be considered as the superposition of two degenerate vector modes of the same azimuthal numbers.

Moreover, in the air-core fiber the effective index difference between two near-degenerate modes HE_{l}_{+1,1} and EH_{l}_{-1,1} is only 10^{−4} order magnitude (*l* is the topological charge). The inevitable interactively coupling among OAM modes will cause severe crosstalk problems. On the other hand, the superposition of two degenerate vector modes with ± π/2 phase difference could generate two OAM modes with same topological charge number and opposite topological sign. The two OAM modes with opposite topological sign have nearly identical effective index which results in the separating and crosstalk problems. Therefore, separating and lifting the near degeneracy and degeneracy of OAM modes in the air-core fiber are the key points for enhancing channel capacity. And some different techniques have been suggested to optimize OAM modes transmission in fibers [9–11
]. Since the helically waveguides could generate optical vortices [12] and helical twisted photonic crystal fiber could excite OAM resonances [13], the theoretical and experimental investigations were carried out, such as chirally-coupled-core fibers [14,15
], solid-core photonic crystal fibers [16], twisted anisotropic fibers [17]. However, the optical performance of TAF for exciting and separating the OAM modes has never been demonstrated.

In this letter, we demonstrated the excitation and separation of the OAM modes in TAF. In the normal air-core fiber the OAM modes could be considered as the superposition of two degenerate vector modes (even mode ± *j* odd mode) of the same azimuthal number. In TAF, instead, the OAM modes could be considered as eigenstates with different modes effective index and topological charges. As increasing of the twist rate, TAF could separate the OAM modes which are near degenerate or degenerate in the air-core fiber, but at the cost of losing high order modes. Therefore, there must be a trade-off between the maximum number of OAM modes and the twist rate of TAF. The separation of OAM modes is conductive to ultralong distance propagation with low crosstalk. In addition, we also investigated the twisted air-core photonic crystal fiber (TAPCF) which could improve the relative energy distribution of the OAM modes. Comparing with TAF, more energy is located in the ring shaped core which is conductive to ultralong distance propagation.

## 2. Theory and principle

The cross-section of the air-core fiber is shown in Fig. 1(a)
. The proposed fiber is made up of Schott glasses (ring shaped core) and quartz (cladding) within the standard telecom single mode fiber dimensions (diameter of 125 μm). The geometric structure is characterized by an air-core of radius r_{1} = 2.2μm, a ring shaped core with the outer radius of r_{2} = 4.2μm. The refractive index are n_{1}, n_{2} and n_{3} respectively, n_{1} the index of air-core (n_{1} = 1), n_{2} the index of the ring shaped core (Schott-BAK1, n_{2} = 1.5552) and the outer-cladding (Quartz) n_{3} = 1.444 at the vacuum wavelength of 1550nm. The full-vector finite-element method with perfected matched layer (PML) at the outer boundary is used as the analysis tool to compute the effective mode index and the electromagnetic field distribution of the exact vector solution of the designed fiber. The schematic of TAF is shown in Fig. 1(b). Twist rate *α* is defined as the ratio of twist angle *φ* to twist length *L* (*α* = *φ /L*). The three-dimensional problem could be transformed to a two-dimensional one via applying equivalent materials which is related to geometrical transformation [13,18
]. In detail, the transformation has the effect of replacing the original material permittivity *ε* with equivalent inhomogeneous anisotropic tensors expressed as [*ε*`] = [*ε*]:**T**
^{−1}, where **T** is the transformation matrix:

**J**is the Jacobian for the transformation from helicoidal coordinates (

*ξ*,

_{1}*ξ*,

_{2}*ξ*) to Cartesian coordinates (

_{3}*x, y, z*) and the transformation could be written as [13]:

*ξ*,

_{1}*ξ*,

_{2}*ξ*are expressed as:

_{3}*p*is a coefficient related to photoelastic constant [19,20 ]. The photoelastic constant of quartz is 35.7 and

*ε*is the permittivity constant of ring and cladding of air-core fiber.

So the two-dimensional mode analysis is available to investigate the optical property of the TAF. The direction of twist is clockwise looking in the direction of propagation, + z.

## 3. Results and discussions

In the normal air-core fiber, each HE or EH vector mode has two degenerate variants (even and odd). The OAM modes are formed by these two basis of vector modes through coherent combinations which could be regarded as superposition of the exact vector modes based on the following formation rules:

where*l*is the topological charge. However, in TAF the physical phenomenon is quite different because of the changing of permittivity. For example, the axial component

*E*of the electric field are shown in Fig. 2(a) (HE

_{z}_{81}mode) and Fig. 2(b) (OAM mode “-7a”), where “-7” indicates the topological charge

*l*= −7 and “a” expresses the same direction of spin and vortex. Although the difference between Fig. 2(a) and Fig. 2(b) is not easy to distinguish, dynamic graph of OAM mode “-7a” with the twist rate 3.81 rad/cm shows continuous rotation, because of the coupling between longitudinal and transverse field components. Also, the coupling leads to the changing of propagation constants hence of different effective index. It is significant for the separation of OAM modes in TAF. Therefore, in TAF, the OAM modes could be considered as eigenstates with different modes effective index and topological charges. The OAM modes in TAF could be expressed by “±

*l*a” and “±

*l*t”, where the “±

*l*” expresses the topological charge, “a” (“t”) expresses the same (opposite) direction between spin and vortex. Another evidence of the excitation of the OAM modes is the spiral phase patterns which are shown in Fig. 3 . The corresponding OAM modes are “-1a, −2a, −6a and −7a”. The twist rate of TAF is 3.81 rad/cm.

For reducing the possibility of mode coupling and crosstalk between near-degenerate modes, the minimum effective index separation (*∆ _{neff}*) between two OAM modes should be larger than the threshold of 1 × 10

^{−4}. This criterion is demonstrated by [10]. In normal air-core fiber (TAF with zero twist rate), the effective index differences between near-degenerate OAM modes superposed by HE

_{l}_{+1,1}and EH

_{l}_{-1,1}are shown in Table 1 . It should be noticed that when the topological charge

*l*= 4, the

*∆*<1 × 10

_{neff}^{−4}and the OAM modes ( + 4a, + 4t, −4a, −4t) are unstable in the air-core fiber. The minimum effective index differences of OAM modes with the same topological charge in TAF (twist rate 3.81rad/cm) are also shown in Table 1. However, in TAF the

*∆*between two near-degenerate modes increases and the

_{neff}*∆*between OAM modes “+4a” and “+4t” is 0.0013 which makes the OAM modes more robust. Meanwhile, the

_{neff}*∆*between OAM modes “+6a” and “+6t” decreases and the OAM modes “±7t” can’t be found in TAF. There is only one high order OAM mode “-7a”. Then, we investigated the relationship between the number of OAM modes and the twist rate. From Fig. 4 we can see that as increasing of the twist rate the effective index differences also increase. The effect of twist also shows efficient separation between OAM modes with same topological charge and opposite topological sign (OAM modes “±

_{neff}*l*a” or “±

*l*t”, which have identical effective index in normal air-core fiber), which is another advantage of the TAF. The maximum effective index difference between the OAM modes with same topological charge and opposite topological sign even can reach a level of about 0.03 at the twist rate 5.04rad/cm. As increasing of the twist rate, the

*∆*also increases, but at the cost of losing high order modes such as “+7a”, “+7t” at 1.68 rad/cm; “+7a”, “+7t”, “-7t”at 3.81 rad/cm; “+7a”, “+7t”, “-7t”, “+6a”, “+6t” at 5.04 rad/cm. Therefore, there must be a trade-off between the separation of OAM modes and the twist rate of the fiber. The TAF with high twist rate could also be considered as an OAM mode filter.

_{neff}For improving the energy of the OAM modes in TAF, we designed a twisted air-core photonic crystal fiber (TAPCF) as shown in Fig. 5(a) . The diameter of air hole is d = 0.9μm, and the gap between them is g = 2.9μm. The TAPCF could support higher energy of the OAM modes in the ring. From Fig. 5(b), we can see that the relative energy of the OAM modes in TAPCF is higher than in the TAF. The relative energy is defined as:

where*E*is the time average energy in ring and the

_{R}*E*is time average energy of the whole cross section area of TAPCF or TAF. Comparing with TAF, the maximum increment of relative energy in TPCAF is about 1% as shown in Fig. 5(b). Moreover, it should be noticed that the electric field intensity distribution pattern of OAM mode “+6t” is two ring structure in TAPCF with twist rate 3.81rad/cm and the relative energy is about 80% which is not included in the profile. The electric field intensity distribution of OAM modes “-2a” in TAPCF and in TAF are shown in Fig. 5(c) and Fig. 5(d) respectively. The higher energy in the ring is conductive to ultralong distance propagation.

_{A}## 4. Conclusions

A kind of TAF was investigated to improve the robustness of OAM modes transmission. In the TAF, the vector modes HE_{l}_{+1, 1} and EH_{l}_{-1,1} could transform into the OAM modes via the coupling of longitudinal and transverse field components. The OAM modes are guided optical vortex waves with different propagation constants hence of different effective index. As increasing of the twist rate, the differences of effective index increase which could separate the near degenerate and degenerate OAM modes completely, but at the cost of losing high order modes. Therefore, the TAF could also be used as an OAM modes filter. Finally, we investigated TAPCF which could improve the relative energy distribution of the OAM modes. Comparing with TAF, more energy is located in the ring shaped core which is conductive to ultralong distance propagation.

## Acknowledgments

This work is supported by National Natural Science Foundation of China (Grant Nos. 11304064, 11304065 and 51506035), the Science and Technology Foundation of Shandong Province (Grant No. ZR2013AQ002), the Science and Technology Development Plan of Weihai (Grant No. 2013DXGJ10), the project (Grant No.201410030-03) supported by the Special Scientific Research Found of Public Service industry of Quality Inspection.

## References and links

**1. **J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics **6**(7), 488–496 (2012). [CrossRef]

**2. **N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science **340**(6140), 1545–1548 (2013). [CrossRef] [PubMed]

**3. **J. Du and J. Wang, “High-dimensional structured light coding/decoding for free-space optical communications free of obstructions,” Opt. Lett. **40**(21), 4827–4830 (2015). [CrossRef] [PubMed]

**4. **D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics **7**(5), 354–362 (2013). [CrossRef]

**5. **S. Ramachandran, S. Golowich, M. F. Yan, E. Monberg, F. V. Dimarcello, J. Fleming, S. Ghalmi, and P. Wisk, “Lifting polarization degeneracy of modes by fiber design: a platform for polarization-insensitive microbend fiber gratings,” Opt. Lett. **30**(21), 2864–2866 (2005). [CrossRef] [PubMed]

**6. **B. Ung, P. Vaity, L. Wang, Y. Messaddeq, L. A. Rusch, and S. LaRochelle, “Few-mode fiber with inverse-parabolic graded-index profile for transmission of OAM-carrying modes,” Opt. Express **22**(15), 18044–18055 (2014). [CrossRef] [PubMed]

**7. **C. Brunet, P. Vaity, Y. Messaddeq, S. LaRochelle, and L. A. Rusch, “Design, fabrication and validation of an OAM fiber supporting 36 states,” Opt. Express **22**(21), 26117–26127 (2014). [CrossRef] [PubMed]

**8. **P. Gregg, P. Kristensen, and S. Ramachandran, “Conservation of orbital angular momentum in air core optical fibers,” Optica **2**(3), 2334–2536 (2015). [CrossRef]

**9. **N. Bozinovic, S. Golowich, P. Kristensen, and S. Ramachandran, “Control of orbital angular momentum of light with optical fibers,” Opt. Lett. **37**(13), 2451–2453 (2012). [CrossRef] [PubMed]

**10. **S. Ramachandran, P. Kristensen, and M. F. Yan, “Generation and propagation of radially polarized beams in optical fibers,” Opt. Lett. **34**(16), 2525–2527 (2009). [CrossRef] [PubMed]

**11. **B. Ung, P. Vaity, L. Wang, Y. Messaddeq, L. A. Rusch, and S. LaRochelle, “Few-mode fiber with inverse-parabolic graded-index profile for transmission of OAM-carrying modes,” Opt. Express **22**(15), 18044–18055 (2014). [CrossRef] [PubMed]

**12. **C. N. Alexeyev, T. A. Fadeyeva, B. P. Lapin, and M. A. Yavorsky, “Generation of optical vortices in layered helical waveguides,” Phys. Rev. A **83**(6), 063820 (2011). [CrossRef]

**13. **G. K. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science **337**(6093), 446–449 (2012). [CrossRef] [PubMed]

**14. **X. Ma, C. H. Liu, G. Chang, and A. Galvanauskas, “Angular-momentum coupled optical waves in chirally-coupled-core fibers,” Opt. Express **19**(27), 26515–26528 (2011). [CrossRef] [PubMed]

**15. **X. M. Xi, G. K. L. Wong, M. H. Frosz, F. Babic, G. Ahmed, X. Jiang, T. G. Euser, and P. S. J. Russell, “Orbital-angular-momentum-preserving helical Bloch modes in twisted photonic crystal fiber,” Optica **1**(3), 165–169 (2014). [CrossRef]

**16. **X. M. Xi, T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, M. J. Padgett, and P. S. J. Russell, “Optical activity in twisted solid-core photonic crystal fibers,” Phys. Rev. Lett. **110**(14), 143903 (2013). [CrossRef] [PubMed]

**17. **E. V. Barshak, C. N. Alexeyev, B. P. Lapin, and M. A. Yavorsky, “Twisted anisotropic fibers for robust orbital-angular-momentum-based information transmission,” Phys. Rev. A **91**(3), 033833 (2015). [CrossRef]

**18. **S. Guenneau, A. Nicolet, Y. O. Agha, and F. Zolla, “Geometrical transformations and equivalent materials in computational electromagnetism,” Compel - The international journal for computation and mathematics in electrical and electronic engineering **27**(4), 806–819 (2008). [CrossRef]

**19. **C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Optical vortices in twisted optical fibers with torsional stress,” J. Opt. A, Pure Appl. Opt. **10**(9), 095007 (2008). [CrossRef]

**20. **C. N. Alexeyev, E. V. Borshak, A. V. Volyar, and M. A. Yavorsky, “Angular momentum conservation and coupled vortex modes in twisted opitcal fibers with torsional stress,” J. Opt. A, Pure Appl. Opt. **11**(9), 094011 (2009). [CrossRef]