## Abstract

We present a multi-orbital-angular-momentum (OAM) multi-core supermode fiber (MOMCSF) to transmit OAM modes. The MOMCSF consists of equally-spaced and circularly-arranged multiple cores, in which the core pitch is small enough to support strong coupling OAM supermodes. The characteristics of OAM modes in MOMCSFs with different core pitches and core numbers are analyzed. The performances of mode coupling and nonlinearity are optimized by designing multiple degrees of freedom of the supermode fiber. The obtained results show that the designed MOMCSF can transmit multiple OAM modes with favorable performance of low mode coupling, low nonlinearity, and low modal dependent loss.

© 2015 Optical Society of America

## 1. Introduction

Angular momentum (AM) is one of the fundamental quantities in physics, along with energy and linear momentum. In optics, the total AM of a light beam can contain a spin contribution associated with polarization, and an orbital contribution associated with the spatial profile of the light intensity and phase. When a light beam is circularly polarized, each of its photons carries a spin angular momentum (SAM) of *sħ*, where *ħ* is the reduced Planck constant and *s* = ± 1 corresponds to left- or right-circular polarization [1]. While for a light beam with spiral phase front exp(*ilφ*), each of its photons carries an orbital angular momentum (OAM) of *lħ*, where *φ* is azimuth angle and *l* is an integer called topological charge number [2]. An OAM carrying beam is twisted like a corkscrew around its axis of travel and the light waves at the axis itself cancel each other out resulting in a doughnut intensity profile. Due to the special phase structure (spiral phase front), intensity structure (doughnut intensity), and dynamic characteristic (orbital angular momentum), OAM beams have been applied to explore for a variety of novel natural phenomena. For instance, OAM beams can be used for particle trapping, imaging, quantum information processing, and so on [3–6
]. Very recently, OAM beams have also shown their potential use in communication systems to overcome the capacity crunch [7]. The unlimited topological charge values of OAM and the inherent orthogonality between different OAMs provide the great potential to multiplex a large number of OAM beams to tremendously increase the total capacity. In the recent years, OAM multiplexing has witnessed remarkable advancements in both free-space and fiber communication systems [7–12
].

In order to mitigate the influence of external environment and realize long-distance or flexible bending-route OAM transmission, it is believed that using fibers for OAM transmission is preferable. A lot of theoretical and experimental efforts have been devoted for OAM transmission in fibers [8, 13–21
]. Remarkably, most of the previous works adopt small-size high-contrast-index ring-core fiber structures. Such structures can reduce the inter-mode crosstalk, restrict radially higher order modes and stably support multiple OAM modes. However, high-contrast-index and small core area will result in a small mode effective area (A_{eff}), i.e. a large nonlinear coefficient. From the view of OAM transmission, high nonlinearity is not expected. For long-distance OAM transmission, nonlinear effects could cause nonlinearity-induced power loss and destroy the purity of the transmitted OAM spectrum. Nonlinear effect is considered to be a fundamental limit on fiber transmission capacity, especially for long-distance OAM multiplexing transmissions. Moreover, small-size high-contrast-index structure is also not conducive to fiber fabrication and mode excitation.

In this paper, we propose a multi-OAM multi-core supermode fiber (MOMCSF) for OAM transmission. Multi-core supermode fibers offering both large mode A_{eff} and high core density, have been used in high power fiber transmission, sensing, high power laser, and space-division multiplexing [22–26
]. Here, we focus on using multi-core supermode fibers for low-crosstalk and low-nonlinearity OAM transmission. Although uncoupled multi-core fibers have been used for OAM transmission [27, 28
]. However, to the best of our knowledge, there have been limited research efforts on supermode fibers for OAM transmission. We study in detail the characteristics of OAM modes guided by MOMCSFs with different core numbers and core pitches. By optimizing the design of the core pitch, relative refractive index difference, and core radius, a 6-core MOMCSF has been presented with low nonlinearity, low mode coupling, and low modal dependent loss.

## 2. Fiber structure and OAM characteristics

The schematic three dimensional (3D) structure and cross-section of a typical MOMCSF are shown in Fig. 1
. The MOMCSF consists of multiple identical cores and each of them supports only two degenerate polarization modes. The cores are arranged equally-spaced and circularly from the fiber center. In Fig. 1, the number of cores, as an example, is assumed to be 6.The radius of each core is *a*, the core pitch between adjacent cores is $\Lambda $, and the relative refractive index difference between core and cladding is defined as $\Delta $ = (*n _{1}*-

*n*)/

_{2}*n*. When the core-to-core distance $\Lambda $ becomes shorter, isolated modes in each core will undergo strong mode coupling. During light propagation, evanescent field coupling occurs resulting in the formation of so-called supermodes. By combining the high-order eigensupermodes with appropriate phase difference, one can get OAM modes. As an example, the calculated intensity and phase profiles of OAM

_{2}_{-1}guided by the 6-core MOMCSF are depicted in the insets of Fig. 1.

A 6-core MOMCSF can support 12 eigensupermodes (including degenerate polarization modes) which can be divided into four groups (0-order, 1-order, 2-order, 3-order) according to the propagation constants (modes of each group have the same propagation constant). The 0- and 3-order supermodes are double degenerate, while the 1- and 2-order supermodes are quadruple degenerate [25]. Figure 2(a) illustrates the intensity and polarization distributions of the 2-order supermodes. OAM states exist as coherent combinations of the quadruple degenerate supermodes. By combining two supermodes with a phase difference of $\pm \pi /2$, one can get two OAM modes with opposite sign of topological charge numbers, as shown in Fig. 2(b). From the phase profiles, one can believe that such a supermode fiber structure can support OAM transmission. Meanwhile, the obtained OAM modes are circularly polarized. Remarkably, the possible combinations forming an OAM beam can be varied. By combining all four degenerate supermodes together with appropriate phase difference, arbitrarily polarized OAM beams can also be obtained.

To further evaluate the characteristics of OAM modes in a MOMCSF, we analyze the quality of OAM modes with different core pitches and different core numbers. The core radius *a*, wavelength $\lambda $, and relative refractive index difference $\Delta $ are set to 4µm, 1.55µm, and 0.3%, respectively. For a 6-core MOMCSF with different Λ (9µm, 12µm, 16µm), the intensity profiles, phase distributions and azimuthal phase variations for OAM_{-1} and OAM_{2}are shown in Fig. 3(a)
. Moreover, the calculated results for OAM_{-1}, OAM_{2}, OAM_{-3} and OAM_{4} of a 10-core MOMCSF are displayed in Fig. 3(b). The maximum order of the OAM modes guided by an N-core MOMCSF is less than N/2 which means that a 6-core MOMCSF can support the 2-order OAM (the max |*l*| is 2) transmission and a 10-core MOMCSF can support the 4-order OAM (the max |*l*| is 4) transmission. From Fig. 3 it can be clearly seen that the OAM supermodes have helical phase fronts.

To estimate the purity of the OAM supermodes, we calculate the OAM power weight. Assuming that the phase function of an OAM supermode is$f(\phi )$. Expanding $f(\phi )$ in Fourier series, one can obtain

*l*[29]. Using Eq. (1), we calculate the mode purity of the 6-core and 10-core MOMCSF as a function of Λ. The results are depicted in Figs. 4 and 5 . From the figure, we can find that the purity of OAM modes almost linear decrease with the increase of Λ. And the purity of high-order OAM modes decrease faster than low-order OAM modes. Moreover, for the same Λ and the same order OAM modes, the more cores we put in the structure, the purer OAM mode with higher purity we can achieve. For instance, the purity of OAM

_{1}in the 10-core MOMCSF is larger than the one in the 6-core MOMCSF. For a larger Λ (e.g. 16um), the purity of high-order (e.g. OAM

_{4}) OAM mode become relatively low (77.58%). In this case, one can use a shot length ring fiber at output end of the MOMCSF to improve the mode purity, just as ref [30]. does. Even though without the ring fiber filter, the characteristics of the OAM is still obvious which can meet most application situations. From Fig. 4 we can also find that crosstalk of OAM

*of an N-core MOMCSF are concentrated on the OAM*

_{l}

_{l-}_{N}and OAM

_{l+}_{N.}So when a pure free-space OAM

*is used to excite OAM*

_{l}*in a MOMCSF, it will not cause serious mode crosstalk. In addition, different OAM modes have similar intensity distributions giving similar confinement factors and leading to small modal dependent loss.*

_{l}## 3. Fiber optimization design for OAM modes transmission

It is noteworthy that, compared with conventional fiber structures, MOMCSFs have more degrees of design freedom, such as number of cores, core size, pitch length, relative refractive index difference and so on. Even compared to a ring fiber, the MOMCSF also has an additional degree, i.e. the number of cores which can be used to control the density of OAM states. The abundant design freedoms are very favorable for optimizing fiber performances. We consider two design goals to optimize the performance of the MOMCSF for OAM transmission: (1) large effective index difference (ΔN_{eff}) between adjacent higher order modes to avoid mode coupling; (2) large A_{eff} to reduce nonlinearity. In this optimization design, the number of cores is selected to be 6. The cladding diameter, wavelength and cladding refractive index n_{2}are assumed to be 125µm, 1.55µm and 1.444, respectively. We first fix the core radius *a* to be 4µm to optimize the fiber structure. Due to strong mode coupling, the number of eigensupermodes is not always a constant as Λ decreases. The number of supermodes (including spatial and polarization degrees of freedom) as variations of Λ and Δ is shown in Fig. 6(a)
.For a small Λ, large Δ can increase the number of supermodes, while small Δ can reduce the number of supermodes. In order to keep the number of supermodes of 12, only the parameters in light blue area are available. The minimum ΔN_{eff} among different order supermodes as a function of Λ and Δ is illustrated in Fig. 6(b) (only the 12 supermodes area (i.e. light blue area corresponding to Fig. 6 (a)) is effective).As shown in Fig. 6(b), it is found that minimum ΔN_{eff} increase with the decrease of Λ. While the variations of Δ has a small effect on minimum ΔN_{eff}, especially for a small Λ. The minimum ΔN_{eff} is larger than 1e-4 for most of the scan range which can guarantee a lower mode coupling. So considering large ΔNeff, small Λ is more favorable. Figure 7
shows A_{eff} as functions of Λ and Δ for the 0-, 1-, 2- and 3- order supermodes. For a smaller Λ, A_{eff} decreases with the increase of Δ and decrease of Λ. For a lager Λ and fixed Δ, the variations of A_{eff} become smooth. However, in most of the scan range, the A_{eff} is larger than 350 µm^{2} which is large enough to guarantee a small nonlinearity. From Figs. 6 and 7
, one can see that the design parameters have broad available scope, featuring favorable fabrication tolerance. Considering the practical fabrication process, we choose a group of optimized parameters as Δ = 0.33 and Λ = 12µm.The minimum ΔN_{eff} is 1.38e-4 and the A_{eff} of the 0-, 1-, 2- and 3-order supermodes are536.2 µm^{2}, 507.4 µm^{2}, 444.9µm^{2} and 392.2 µm^{2}, respectively. It is noted that even Δ and Λ deviate slightly from the designed values, the relatively large fabrication tolerance can still guarantee a favorable fiber performance.

We then give a detailed analysis of the impact of coupling on A_{eff}. For a single mode fiber with core radius of 4 μm and relative refractive index difference Δ of 0.33%, the A_{eff} of fiber fundamental mode is about 81.9 µm^{2}, shown in Fig. 8
(blackdot dash line). To compare with a MOMCSF, we calculate average A_{eff} in one core as a function of Λ of different order supermodes. The average A_{eff} is defined as the A_{eff} of one supermode is divided by 6.The results are displayed in Fig. 8. From the curves, one can see that the mode coupling can influent A_{eff}. When Λ is relatively small, strong mode coupling can cause a small A_{eff}. However, for an appropriate Λ, the average A_{eff} in one core of 0- and 1- order supermodes can be larger than 81.9 µm^{2}. With the increase of core pitch Λ, the coupling between cores become very weak, and each core can be regard as an independent core. Then all the modes will have the same average effective area (approaches 81.9 μm^{2}).

To further assess the performance of the 6-core MOMCSF with different structure parameters, we fix the Δ to be 0.33% and analyze the number of supermodes, minimum ΔN_{eff} and A_{eff} as functions of the core radius *a* and Λ/*a*. As shown in Fig. 9(a)
, for a small Λ/*a*, the variation of *a* can also cause the change of the mode number. When the Λ/*a* is larger than 2.5, the number of supermodes remains 12 with *a* changing from 3.5 to 4.5 µm. The minimum ΔN_{eff} as functions of *a* and Λ/*a* is depicted in Fig. 9(b). The minimum ΔN_{eff} increases with the decrease of *a *and Λ/*a*. For most of the scan range, the minimum ΔN_{eff} is larger than 1e-4. A small *a *benefits large ΔN_{eff}, but goes against the A_{eff}, as shown in Fig. 10
. For a larger Λ and *a*, A_{eff} changes slightly and remains a larger value. Remarkably, even for a smaller Λ and *a*, A_{eff} larger than 350 µm^{2} is still available.

Moreover, if the MOMCSF is used for communication, compatible with wavelength division multiplexing(WDM) is very important. Here, we still consider a 6-core MOMCSF, the core radius a, core pitch Λ, wavelength λ, relative refractive index difference Δ are set to be 4 μm, 12 μm, 1.55 μm, and 0.3%, respectively. OAM purity as a function of wavelength is shown in Fig. 11(a)
. From 1525 nm to 1630 nm, the purity of OAM_{-1} and OAM_{2} are larger than 90% and the purity increase with the increase of wavelength. As shown in Fig. 11(b), ΔN_{eff} between adjacent order supermodes is larger than 1e-4 for all the wavelength scanning range. The A_{eff} is larger than 400 μm^{2} for all modes, shown in Fig. 11(c). From Fig. 11, it can be confirmed that the fiber is suitable for WDM transmission.

## 4. Conclusion

In summary, we have designed a MOMCSF featuring both low mode coupling and low nonlinearity for multiple OAM modes transmission. The proposed supermode fiber structure consists of multiple strong coupled cores which are arranged equally-spaced and circularly from the fiber center. The characteristics of OAM modes in a 6-core MOMCSF with different core pitches and core numbers have been studied in detail. OAM modes can be guided by such fiber structure with small distortion for a small core pitch. Even for a larger core pitch, one can still clearly see the distinct spiral phase profiles of OAM modes and only slight quality degradation of OAM modes is induced. Through the optimized design of the core pitch, relative refractive index difference, and core radius, the MOMCSF has featured superior properties such as low nonlinearity, low mode coupling, and large fabrication tolerance. Moreover, the MOMCSF is WDM compatible. The presented MOMCSF with favorable performance may find wide potential use in long-distance OAM multiplexing transmission systems and other OAM communication applications.

## Acknowledgments

This work was supported by the National Basic Research Program of China (973 Program) under grant 2014CB340004, the National Natural Science Foundation of China (NSFC) under grants 11274131, 61222502 and L1222026, the Program for New Century Excellent Talents in University (NCET-11-0182), the Wuhan Science and Technology Plan Project under grant 2014070404010201, the Fundamental Research Funds for the Central Universities (HUST) under grants 2012YQ008 and 2013ZZGH003, and the seed project of Wuhan National Laboratory for Optoelectronics (WNLO).

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