Open Access
Add to BibSonomy
Abstract
Based on the negative curvature structure, we design a graded-index photonic crystal fiber (GI-PCF) supporting the orbital angular momentum (OAM) mode transmission and discuss its optimization strategy. The core of the designed GI-PCF is sandwiched by three-layer inner air-hole arrays with gradually decreasing air-hole radii and a single outer air-hole array, where the inner side of the annular core forms a graded refractive index distribution. All these structures are clad with negative-curvature tubes. By optimizing characteristic structural parameters, including the air-filling fraction of the outer array, the air-hole radii of the inner arrays, and the thickness of the tubes, the GI-PCF can support 42 OAM modes and most of them have a purity greater than 85%. Compared with conventional structures, the present design of GI-PCF has better properties on an overall level, which can stably transmit multiple OAM modes with high mode purity. These results inject new interest in the flexible design of PCF and have potential applications in various fields, including but not limited to the mode division multiplexing system and terabit data transmission.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Achieving a higher capacity of data transmission is always one of the major tasks in optical fiber communications. With a series of technological breakthroughs and development, such as multiplexing technology, etc., the transmission capacity of optical fiber has grown rapidly, and the expansion of the transmission capacity of single-mode optical fiber has approached a maximum limit [1,2]. Spatial division multiplexing, especially mode division multiplexing (MDM) based on orbital angular momentum (OAM) mode, has been considered a potential technique to relieve future bandwidth issues [3–5]. OAM-carrying beam is characterized by a helical phase structure expressed by Hilbert factor $\textrm{exp}({il\varphi } )$, where l is the topological charge and $\varphi $ is the azimuthal angle. OAM modes with different l are orthogonal to each other. Therefore, techniques combining MDM and OAM modes show great potential for improving the capacity of fiber links [6–9].
In 1992, Allen et al. [10] proposed that the Laguerre-Gaussian beam could carry OAM, which provides a theoretical basis for subsequent OAM-related applications. To efficiently transmit the OAM modes, a series of special fibers were designed and applied successively, including ring fibers [11,12], air-core fibers [13], and photonic crystal fibers (PCFs, also known as microstructure fiber) [14–21]. Different from conventional optical fibers, PCFs enable large effective refractive index contrast between fiber core and cladding without doping. Besides, PCFs have more adjustable parameters and flexible design freedom, thereby having potential applications in multiple OAM modes transmission. In 2012, Yang et al. first proposed a PCFs to support the transmission of OAM modes [14], while the cladding hexagonal structure of the air hole is not suitable for the stable transmission of OAM modes. Then, circular PCFs and their design strategies [22–25] were proposed successively to support OAM modes as many as possible. In addition, the improvement of fiber properties, such as dispersion, confinement loss, etc., has also been extensively investigated [19,26–30]. Meanwhile, the negative curvature fiber [31], whose surface normal vector of the core boundary is directionally opposite to the radial unit vector, was proposed during the research on anti-resonance and Kagome fiber. In 2011, researchers found that the Kagome fiber with negative curvature of the core boundary has a lower loss than that with a straight core boundary [32]. Afterward was reported that the negative-curvature structures have an enhanced ability to confine light than the positive-curvature ones [33–35]. In addition, the graded-index profile is beneficial for improving the purity of modes [36].
In this work, we focus on graded-index photonic crystal fibers (GI-PCFs) with negative curvature structures and their relevant properties. We begin in Sec. 2 with the structure design and mode properties of the GI-PCFs. Next, in Sec. 3, we analyze the optimization strategy. In Sec. 4, we compare the typical modes properties and transmission characteristics of the proposed GI-PCF with other structures. Section 5 concludes this work.
2. Structure design and mode properties of GI-PCFs
2.1 Structure design
The schematic diagram of the GI-PCF we designed is shown in Fig. 1(a), in which the blue area is the background material filled with SiO2, whose refractive index is 1.444 at 1550 nm [37]; the white area is filled with air whose refractive index ${n_{\textrm{air}}} = 1$; the purple area is set to the perfect matching layer (PML) functioned to absorb additional electromagnetic radiation. The GI-PCF can be roughly divided into three regions from outside to inside, the outermost tube, the middle six ring-shaped negative curvature tubes, and the innermost tube, which is the region that we are interested in, where the fiber ring-core is sandwiched by three-layer inner air hole arrays and a single-layer outer air hole array [inside the dashed square of Fig. 1(a)].
Fig. 1. (a) Cross-section of the designed GI-PCF, where the blue area is the background material SiO2, the white area is filled with air and the purple area is set to the perfect matching layer (PML). The graded-index profile is formed inside the dashed square, and the outside six bigger tubes construct a negative curvature structure. (b) Zoomed-in part of the GI-PCF taken from the dashed square in (a), where t is the thickness of tubes, ${r_i}\textrm{}({i = 1,\textrm{}2,\textrm{}3} )$ are the radii of the air holes for inner air-hole arrays (marked with red dashed circles), d is the distance between adjacent arrays, ${r_n}$ are the radii of the air holes for the outer air-hole array (marked with the black dashed circle), $5{d_n}$ is the distance from the fiber center to the outer air-hole array.
Different from traditional PCFs, the presented one is based on the negative curvature structure and at the same time introduces radius-decreasing air-hole arrays to construct a graded-index profile. Details of the enlarged innermost region are illustrated in Fig. 1(b), which includes four layers of air-hole arrays. The inner three layers (red dashed circles) are constructed by graded-index profile with decreasing radii along the radial direction, as labeled by ${r_i}\; ({i = 1,\; 2,\; 3} )$ and assumed ${r_1} > {r_2} > {r_3}$, while the radii of the air holes in each layer are identical. For the inner air-hole arrays, the number of component small air holes in each layer is respectively set to be 6, 12, and 18, they are uniformly distributed along each layer. The layer-to-layer distance of the adjacent inner layers is set to be d. The outer array (black dashed circle) consisting of 30 air holes of radiuses ${r_n}$ is relatively far away from the inner air-hole array next to the fiber core, which is characterized by a distance of $5{d_n} - 3d$. The fiber core and these air hole arrays are encapsulated in the tube with the thickness of t, which is the same for the six negative curvature tubes (only four minor arcs are left in the zoomed-in view). Clearly, the fiber ring-core sandwiched by inner and outer air hole arrays forms a high-index area, where the modes of the optical field are confined by the total internal reflection.
2.2 Mode properties
An OAM mode can be superposed by even and odd modes of the same vector eigenmodes (HE or EH modes) with a π/2 phase difference [38]. The transverse electric field ${\vec{e}_{t;l,m}}$ ($l > 1$ and we have set $m = 1$) is expressed as
In the weakly guiding approximation, the HE and EH modes constituting the OAM mode family possess the same propagation constant and therefore cannot exist independently. They are easy to couple with each other to form a linearly polarized (LP) mode in the transmission process, resulting in the desired pure OAM mode cannot transmit stably [39]. To address this, the effective refractive index difference ($\mathrm{\Delta }{n_{\textrm{eff}}}$) between adjacent HE and EH modes in the same OAM mode family, expressed by
3. Optimization of the GI-PCFs
To ensure the designed GI-PCF has potential applications in optical communication systems, the following conditions need to be considered: 1) The designed GI-PCF should support OAM modes as many as possible and avoid generating higher-order radial modes [45–47] at the same time. 2) All modes supported by the structure should have a high mode purity, flat and low dispersion, low confinement loss, and low nonlinear coefficient. Therefore, the dependency of modes’ properties on structural parameters, including the air-filling fraction of the outer air-hole array, the air-hole radii of the inner air-hole arrays, and the thickness of negative-curvature tubes, is discussed to find the optimization strategy. The discussion in this section is carried out at a wavelength of 1550 nm.
The air-filling fraction of the outer air-hole array, not that in the whole innermost tube, is defined by $f = 2{r_n}/{d_n},$ with the fixed radii of the outer air holes ${r_n} = 1.1\, \mu \textrm{m}$. And it influences the effective refractive index distribution on the cross-section of the GI-PCF by changing the area of the fiber core. Figure 2(a) illustrates the effective refractive index difference $\mathrm{\Delta }{n_{\textrm{eff}}}$ tends to increase as f increasing, which means the fiber will obtain the ability to support more OAM modes. Nearly all eigenmodes supported by this structure could be effectively separated when $f > 0.915$, shown in the inset of Fig. 2(a). Besides, when $f > 0.96$, the eigenmodes with $l = 11$ cannot exist in the fiber in that its cut-off condition appears, manifested as the missing data point in the inset.
Fig. 2. At 1550 nm, (a) $\mathrm{\Delta }{n_{\textrm{eff}}}$ and (b) purity dependency of modes supported by the GI-PCF on f, the horizontal dashed line in (a) marks the condition $\mathrm{\Delta }{n_{\textrm{eff}}} > {10^{ - 4}}$ discussed by Eq. (2); (c) Intensities and phase of OAM11,1 composed by EH10,1 supported by the GI-PCF with $f = 0.84,\; \; 0.96$, respectively. The higher the value of f, the higher the deformation degree. The mode field distribution experiences an obvious deformation at $f = 0.96$.
It is worth noting that the increase of f leads to the reduction of the fiber-core area, thus the number of OAM modes supported by the GI-PCF increases, and the mode purity decreases because the optical fiber refractive index profile change becomes steeper [36]. As shown in Fig. 2(b), the purity of the OAM modes decreases with increasing f, and lower-order modes are more sensitive to the change of f than higher-order ones. When the value of f is too large (i.e., the fiber-core area is too small), higher-order eigenmodes will encounter deformation. As an example, the mode field distribution and phase distribution of OAM11,1 composed by EH10,1 shown in Fig. 2(c) occur obvious deformation when $f = 0.96$ compared with that when $f = 0.84$. For $f = 0.96$, the mode field tends to transform into a higher-order radial mode, which would add the difficulty of processing and the cost of multiplexing.
Here, only eigenmodes HE3,1 ($l = 2$) and HE11,1 ($l = 11$) are selected as specific examples to demonstrate the dependence of eigenmodes’ transmission properties on the air-filling fraction f. This selection is not arbitrary but follows the fact that the two eigenmodes are respectively the lowest and highest order among the accessible eigenmodes that can construct OAM modes with $l > 1$, thereby they could reflect the overall property. Some typical transmission properties of the eigenmodes, including dispersion, confinement loss (CL), effective mode field area (${\textrm{A}_{\textrm{eff}}}$) and nonlinear coefficient ($\mathrm{\gamma }$), are defined by
The choosing of inner air hole radii $({{r_1},{r_2},\; {r_3}} )$ is another important parameter for the GI-PCF. It also will influence the effective refractive index distribution. Given $f = 0.93$, a family of GI-PCFs with different radii differences are designed, as shown in Fig. 3(a). Specifically, four different configurations are considered, A# with $({{r_1},{r_2},\; {r_3}} )= ({1.2,\; 1.1,\; 1.0} )\, \mu m$, B# with (1.2, 1.0, 0.8) $\mu \textrm{m}$, C# with (1.2, 0.9, 0.6) $\mu \textrm{m}$, and D# with (1.2, 0.8, 0.4) $\mu \textrm{m}$. From configuration A# to D#, the radii difference between adjacent inner air-hole arrays gradually increases, and the effective refractive index contrasts between the fiber-core area and both sides decrease. Figure 3(b) illustrates that the $\mathrm{\Delta }{n_{\textrm{eff}}}$ at a fixed l decrease from configuration A# to D#. For example, for eigenmodes with $l = 7$, $\mathrm{\Delta }{n_{\textrm{eff}}}$ is $5.26 \times {10^{ - 4}}$ in GI-PCF A#, while $\mathrm{\Delta }{n_{\textrm{eff}}}$ is $9.88 \times {10^{ - 5}}$ in GI-PCF D# where the eigenmodes cannot be separated effectively in that they are below the horizontal dashed line. In addition, the dependence of $\Delta {n_{\textrm{eff}}}$ on topological charges shows some anomalous increasing changes at $l = 8,9,10$, this may be related to the special performance of these modes. Fortunately, these anomalous changes do not affect the trends we focus on, and they are not of interest here. Besides, in GI-PCF D#, the eigenmodes with $l > 10$ ($l > 11$, in GI-PCF C#) cannot exist in the fiber for its cut-off condition appears, manifested as the missing data point in the inset.
Fig. 3. (a) Schematic diagrams for four different configurations of GI-PCFs. The gradient change of the radii of the three-layer inner air-hole arrays $({{r_1},{r_2},\; {r_3}} )$ are explicitly given at the bottom of each panel; (b) Dependence of $\Delta {n_{\textrm{eff}}}$ between HE and EH modes in the same family on the difference of inner air-hole arrays. The horizontal dashed line marks the condition $\mathrm{\Delta }{n_{\textrm{eff}}} > {10^{ - 4}}$ discussed by Eq. (2); (c) Take the modes with the topological charge from 2 to 7 as representative examples, purity dependency of modes supported by the GI-PCF on the radii difference between adjacent inner air-hole arrays. Left (right) for SO anti-aligned (aligned) OAM mode.
Table 1. Take the modes HE3,1 (top) and HE11,1 (bottom) as representative examples, each row lists the variation of the effective refractive index ${{\boldsymbol n}_{\textrm{eff}}}$, the dispersion D, the effective mode field area ${{\boldsymbol A}_{\textrm{eff}}}$, and the nonlinear coefficient ${\boldsymbol \gamma }$ of eigenmode under different air-filling fraction ${\boldsymbol f}$
In addition to the number of OAM modes, changing the radii difference between the adjacent inner air-hole arrays also affects the purity of the modes supported by the GI-PCF. Here, only modes with a topological charge from 2 to 7 are taken as representative examples to show the dependence of purity on the difference of inner air-hole arrays. From configuration A# to D#, the radii difference between the inner air-hole arrays gradually increases, and the optical fiber refractive index profile change becomes slower. As shown in Fig. 3(c), the purity of most modes is less than 80% in GI-PCF A#, but the purity of most modes has been improved to 80% in GI-PCF C# and 90% in GI-PCF D#. To sum up, with the increasing of the radii difference between adjacent inner air-hole arrays from configuration A# to D#, the purity of modes supported by the GI-PCF increase, however, it will lead to the number of modes supported by GI-PCF decrease.
It is a natural result that all the fiber properties discussed here cannot be improved simultaneously because of intrinsic constraints between various structural parameters. The performance gains of properties of most eigenmodes are pursued by optimizing the structure. In addition to the number of supported modes, the OAM-mode purity, and other transmission properties, the feasibility of manufacturing should be considered carefully. For example, the spacing between adjacent air holes is an important aspect, it should be at least 0.1 μm [48]. Therefore, the air-filling fraction f of the outer air-hole array could be selected from 0.915 to 0.945. For the arrangement of inner air hole arrays, the purity of modes supported by the configuration C# or D# could reach a relatively high purity (most modes have a purity greater than 80% or 90%), while the configuration C# is a better choice considering the number of modes supported by the GI-PCF.
Figure 4(a) shows how confinement loss (CL) changes with the negative curvature tubes’ thickness t. It is worth noting that the CL for certain modes, such as eigenmodes with $l = 11$ (i.e., HE12,1 and EH10,1), there are no clear loss peaks when the negative curvature tubes’ thickness is lower than 1.65 μm, where the optical confinement condition of the secondary-protecting negative-curvature structure works in the anti-resonance regime; while clear loss peaks appear once the negative curvature tubes’ thickness exceed 1.65 μm, where the negative-curvature structure works in the resonance regime. It may indicate that a threshold value of t theoretically exists and the optical confinement condition of the negative-curvature structure could switch from the anti-resonance to resonance. Within a certain range of t, the CL of most modes has a relatively low value and varies smoothly, suggesting that there is a certain fault tolerance rate in the designed structure against a slight change of t in the real pulling process. To illustrate this aspect, we enumerate the CL variations for three eigenmodes in Figs. 4(b)–4(d), corresponding $l = 2$ (low order), $l = 6$ (intermediate order) and $l = 11$ (high order), respectively. When t fluctuates in the range of 1.25∼1.45, the CL nearly does not change (overall within the same magnitude). Apart from the CL, the effect of t on the transmission characteristics, such as the dispersion, could be negligible. To sum up, the optimized range of the parameter t could be chosen from 1.25 μm to 1.45 μm.
Fig. 4. (a) Confinement loss dependency of eigenmodes on t. (b)-(d) The confinement loss as a function of t for eigenmodes with $l = 2,6,11$, respectively.
4. Results and discussions
4.1 Properties of the designed GI-PCF
According to the results of optimization, the parameters of the GI-PCF are set as: ${r_1} = 1.2\,\mu \textrm{m}$, ${r_2} = 0.9\,\mu \textrm{m}$, ${r_3} = 0.6\,\mu \textrm{m}$, $d = 2.55\,\mu \textrm{m}$, ${r_n} = 1.1\,\mu \textrm{m}$, $f = 0.93$, $t = 1.4\,\mu \textrm{m}$. The eigenmodes’ effective refractive index (${n_{\textrm{eff}}}$) and the effective refractive index difference ($\mathrm{\Delta }{n_{\textrm{eff}}}$) varying with the wavelength in the wavelength band 1460∼1680 nm are shown in Figs. 5(a) and 5(b), respectively. It is clear that ${n_{\textrm{eff}}}$ decreases with the wavelength while $\mathrm{\Delta }{n_{\textrm{eff}}}$ increases. $\mathrm{\Delta }{n_{\textrm{eff}}}$ with topological charges $l \le 11$ except for $l \ne 7$ could meet the requirement of $\mathrm{\Delta }{n_{\textrm{eff}}} > 1 \times {10^{ - 4}}$, which means the structure could support 38 OAM modes transmission within 220 nm bandwidth. As for the eigenmodes with $l = 7$, when the wavelength λ is over 1480 nm, the $\mathrm{\Delta }{n_{\textrm{eff}}}$ is larger than 10−4. Therefore, the GI-PCF can support 42 OAM modes, and the intensities and phases of some modes supported by the GI-PCF are shown in Fig. 5(c).
Fig. 5. (a) ${n_{\textrm{eff}}}$ and (b) $\mathrm{\Delta }{n_{\textrm{eff}}}$ of eigenmodes supported by the GI-PCF as a function of wavelength. The horizontal dashed line marks the condition $\mathrm{\Delta }{n_{\textrm{eff}}} > {10^{ - 4}}$ discussed by Eq. (2). (c) The first row shows a complex combination of the even and odd modes of HE3,1 giving rise to the OAM mode OAM + 2,1. The associated intensity distributions are displayed orderly in the left three panels, and the phase of the resulting OAM mode is displayed in the rightmost panel. The second row has the same layout as the first row, displayed however the combination of even and odd modes of HE6,1 gives rise to the OAM + 5,1. mode The selected wavelength is 1550 nm.
During transmission, the SO coupling affects the purity of modes, which is related to the rate between the maximum value of the radial and azimuthal components of the electric field, i.e., ${e_r}/{e_\varphi }$ [43]. For the SO-aligned OAM modes [red circles in Fig. 6(a)], ${e_r}/{e_\varphi }$ decreases with rising l, while it increases for SO anti-aligned OAM modes [blue squares in Fig. 6(a)]. Figure 6(a) shows that the value of ${e_r}/{e_\varphi }$ approaches 1 for both SO-aligned and anti-aligned OAM modes for higher $l,$ meaning that the influence of SO coupling on mode purity is weaker for high-order modes. The purity of SO-aligned and anti-aligned OAM modes at 1550 nm for different topological charges is shown in Fig. 6(b). One can see that for most OAM modes, the purity presents an increasing trend with increasing l. Besides, the purity of SO-aligned OAM modes is greater than that of SO anti-aligned. Importantly, the purity of 75% OAM modes is greater than 85%, which is advantageous for OAM mode communication. Note that the eigenmodes HE11,1 and EH9,1 have different degrees of leakage, they are not involved in the subsequent discussion about transmission characteristics.
Fig. 6. (a) The parameter ${e_r}/{e_\varphi }$ versus topological charge l. ${e_r}$ and ${e_\varphi }$ denote the maximum value of the radial and azimuthal components of the electric field, respectively. (b) Modes’ purity at 1550 nm. Different colors represent different OAM modes, blue (red) for SO anti-aligned (aligned) OAM mode.
The dispersion of the eigenmodes as a function of the wavelength is illustrated in Fig. 7(a). On the whole, the dispersion linearly increases with the increase of the wavelength. At a fixed wavelength, the dispersion increases as the increasing of eigenmode order l. Especially, for the eigenmodes with order $l \le 8$, their dispersions are lower than 150 ps/nm·km, which is at a relatively low level. All eigenmodes have a flat dispersion variation, and the EH10,1 mode has a maximum dispersion slope of 0.166 ps/(km·nm2). The confinement loss (CL) varying with wavelength ranging from 1460 nm to 1680 nm is shown in Fig. 7(b). The CLs of all eigenmodes are in the order of 10−7∼10−13. Particularly, the CL is less than 3.00 × 10−9 dB/m at 1550 nm, which is small enough to facilitate mode transmission. Figures 7(c) and 7(d) show the effective mode field area ${A_{\textrm{eff}}}$ and nonlinear coefficient $\mathrm{\gamma }$ of the eigenmodes varies with the wavelength in the interval 1460 nm∼1680 nm. ${A_{\textrm{eff}}}$ increases with the wavelength increasing, while the nonlinear coefficient changes in the opposite direction. At 1550 nm, HE12,1 mode has a maximum effective mode field area of 150.74 μm2 and a minimum nonlinear coefficient of 0.70 W-1·km-1, which are superior to the work of Ref. [30].
Fig. 7. (a) Dispersion, (b) Confinement loss, (c) Effective mode field area, and (d) Nonlinear coefficient as a function of wavelength within the wavelength band 1460∼1680 nm.
4.2 Comparison with previous designs
To highlight the advantages of the designed structure, an important aspect is to compare it with an appropriate reference group. Some properties of GI-PCF, including the number of OAM modes, the purity, the CD, and the nonlinear coefficient, are compared with those of previous works [22,30,49,50] in Table 2. It is necessary to emphasize that the overall property is relatively better than other work, although the present structure does not have a considerable improvement in terms of a specific feature. For instance, the proposed GI-PCF can support a relatively large number of OAM modes and meanwhile ensure most modes’ purity and dispersion at a relatively good level. The presented design of GI-PCF gains a perfect tradeoff between the supported number and the properties of modes. In this regard, it will probably give us a rather different perspective on the design of PCF.
Table 2. At 1550 nm, comparison of the number of modes properties and transmission characteristics for the proposed PCF with recently published papers.
5. Conclusion
In conclusion, we propose an interesting design of the ring-core GI-PCFs with a negative curvature structure. After optimizing, the GI-PCF can support a relatively large number of OAM modes up to 42 and meanwhile ensure most modes’ purity and dispersion at a relatively good level. The dispersion variation with wavelength is relatively flat with a maximum dispersion slope of 0.166 ps/(km·nm2) of EH10,1 mode. Due to the existence of the negative curvature structure, the confinement loss of the GI-PCF is as low as 10−7∼10−13 dB/m. Moreover, the fiber has small nonlinear coefficients and higher purities ($> 85{\%}$). It is worth noting that this work makes a great tradeoff between increasing the number of modes and optimizing properties of modes. We, therefore, hope that these results will bring new ingredients for the flexible design of PCFs and pave the way for the implementations in the near future.
In closing, we hereby give a short discussion on the fabrication of the proposed GI-PCF. Yu et al. have fabricated a ring-core fiber with a negative curvature structure (NC-RCF) by the stack-and-draw approach [30,51], which could provide a reference for GI-PCF. However, such kind of method is hard to be used to make circular patterns. Therefore, the inner air-hole arrays in our structure may pose a manufacturing challenge. In this aspect, the sol-gel method [52,53] may be helpful, which could change the air-hole size, spacing, and shape flexibly.
Funding
National Key Research and Development Program of China (2020YFB1805800); National Natural Science Foundation of China (No.12034010).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. A. D. Ellis, Z. Jian, and D. Cotter, “Approaching the Non-Linear Shannon Limit,” J. Lightwave Technol. 28(4), 423–433 (2010). [CrossRef]
2. R.-J. Essiambre and R. W. Tkach, “Capacity Trends and Limits of Optical Communication Networks,” Proc. IEEE 100(5), 1035–1055 (2012). [CrossRef]
3. D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013). [CrossRef]
4. P. J. Winzer, “Making spatial multiplexing a reality,” Nat. Photonics 8(5), 345–348 (2014). [CrossRef]
5. L. Feng, Y. Li, S. Wu, W. Li, J. Qiu, H. Guo, X. Hong, Y. Zuo, and J. Wu, “A Review of Tunable Orbital Angular Momentum Modes in Fiber: Principle and Generation,” Appl. Sci. 9(12), 2408 (2019). [CrossRef]
6. 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]
7. 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]
8. A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7(1), 66 (2015). [CrossRef]
9. Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019). [CrossRef]
10. L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef]
11. Y. Yang, Y. Yan, N. Ahmed, Y. Jeng-Yuan, Z. Lin, R. Yongxiong, H. Hao, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode Properties and Propagation Effects of Optical Orbital Angular Momentum (OAM) Modes in a Ring Fiber,” IEEE Photonics J. 4(2), 535–543 (2012). [CrossRef]
12. C. Brunet, B. Ung, L. Wang, Y. Messaddeq, S. LaRochelle, and L. A. Rusch, “Design of a family of ring-core fibers for OAM transmission studies,” Opt. Express 23(8), 10553–10563 (2015). [CrossRef]
13. 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]
14. Y. Yue, L. Zhang, Y. Yan, N. Ahmed, J.-Y. Yang, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Octave-spanning supercontinuum generation of vortices in an As2S3 ring photonic crystal fiber,” Opt. Lett. 37(11), 1889–1891 (2012). [CrossRef]
15. Z. A. Hu, Y. Q. Huang, A. P. Luo, H. Cui, Z. C. Luo, and W. C. Xu, “Photonic crystal fiber for supporting 26 orbital angular momentum modes,” Opt. Express 24(15), 17285–17291 (2016). [CrossRef]
16. H. Zhang, W. Zhang, L. Xi, X. Tang, X. Zhang, and X. Zhang, “A New Type Circular Photonic Crystal Fiber for Orbital Angular Momentum Mode Transmission,” IEEE Photon. Technol. Lett. 28(13), 1426–1429 (2016). [CrossRef]
17. X. Bai, H. Chen, and H. Yang, “Design of a circular photonic crystal fiber with square air-holes for orbital angular momentum modes transmission,” Optik 158, 1266–1274 (2018). [CrossRef]
18. L. Zhang, K. Zhang, J. Peng, J. Deng, Y. Yang, and J. Ma, “Circular photonic crystal fiber supporting 110 OAM modes,” Opt. Commun. 429, 189–193 (2018). [CrossRef]
19. A. Tandje, J. Yammine, M. Dossou, G. Bouwmans, K. Baudelle, A. Vianou, E. R. Andresen, and L. Bigot, “Ring-core photonic crystal fiber for propagation of OAM modes,” Opt. Lett. 44(7), 1611–1614 (2019). [CrossRef]
20. M. F. Israk, M. A. Razzak, K. Ahmed, M. M. Hassan, M. A. Kabir, M. N. Hossain, B. K. Paul, and V. Dhasarathan, “Ring-based coil structure photonic crystal fiber for transmission of Orbital Angular Momentum with large bandwidth: Outline, investigation and analysis,” Opt. Commun. 473, 126003 (2020). [CrossRef]
21. M. A. Kabir, M. M. Hassan, M. N. Hossain, B. K. Paul, and K. Ahmed, “Design and performance evaluation of photonic crystal fibers of supporting orbital angular momentum states in optical transmission,” Opt. Commun. 467, 125731 (2020). [CrossRef]
22. H. Zhang, X. Zhang, H. Li, Y. Deng, X. Zhang, L. Xi, X. Tang, and W. Zhang, “A design strategy of the circular photonic crystal fiber supporting good quality orbital angular momentum mode transmission,” Opt. Commun. 397, 59–66 (2017). [CrossRef]
23. H. Li, H. Zhang, X. Zhang, Z. Zhang, L. Xi, X. Tang, W. Zhang, and X. Zhang, “Design tool for circular photonic crystal fibers supporting orbital angular momentum modes,” Appl. Opt. 57(10), 2474–2481 (2018). [CrossRef]
24. W. Tian, H. Zhang, X. Zhang, L. Xi, W. Zhang, and X. Tang, “A circular photonic crystal fiber supporting 26 OAM modes,” Opt. Fiber Technol. 30, 184–189 (2016). [CrossRef]
25. S. Hong, Y. S. Lee, H. Choi, C. Quan, Y. Li, S. Kim, and K. Oh, “Hollow Silica Photonic Crystal Fiber Guiding 101 Orbital Angular Momentum Modes Without Phase Distortion in C + L Band,” J. Lightwave Technol. 38(5), 1010–1018 (2020). [CrossRef]
26. D. An and J. Tu, “Design of a low-dispersion ring-core photonic crystal fiber supporting 26 OAM modes,” presented at the International Conference on Optical Communications and Networks (2019).
27. X. Jiao, H. Zhang, X. Zhang, H. Li, J. Wei, Z. Wang, L. Xi, W. Zhang, and X. Tang, “Performance of circular photonic crystal fiber transmitting orbital angular momentum modes under macro-bending,” J. Opt. 21(6), 065703 (2019). [CrossRef]
28. W. Wei, Z. Zhi-Ming, T. Li-Qin, D. Lei, F. Wan-De, and L. Yi-Gang, “Transmission characteristics of vortex beams in a sixfold photonic quasi-crystal fiber,” Acta Phys. Sin. 68(11), 114209 (2019). [CrossRef]
29. H. Zhang, H. Li, X. Zhang, J. Yang, Z. Chen, L. Xi, and W. Zhang, “Design of a Hybrid Cladding Structure Photonic Crystal Fiber for OAM modes Transmission,” presented at the Asia Communications and Photonics Conference (2019).
30. J. Tu, Z. Liu, S. Gao, Z. Wang, J. Zhang, B. Zhang, J. Li, W. Liu, H. Tam, Z. Li, C. Yu, and C. Lu, “Ring-core fiber with negative curvature structure supporting orbital angular momentum modes,” Opt. Express 27(15), 20358–20372 (2019). [CrossRef]
31. N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. 27(18), 1592–1594 (2002). [CrossRef]
32. Y. Y. Wang, N. V. Wheeler, F. Couny, P. J. Roberts, and F. Benabid, “Low loss broadband transmission in hypocycloid-core Kagome hollow-core photonic crystal fiber,” Opt. Lett. 36(5), 669–671 (2011). [CrossRef]
33. Y. Wang and W. Ding, “Confinement loss in hollow-core negative curvature fiber: A multi-layered model,” Opt. Express 25(26), 33122–33133 (2017). [CrossRef]
34. C. Wei, R. Joseph Weiblen, C. R. Menyuk, and J. Hu, “Negative curvature fibers,” Adv. Opt. Photon. 9(3), 504–561 (2017). [CrossRef]
35. B. Debord, M. Alharbi, T. Bradley, C. Fourcade-Dutin, Y. Y. Wang, L. Vincetti, F. Gerome, and F. Benabid, “Hypocycloid-shaped hollow-core photonic crystal fiber Part I: arc curvature effect on confinement loss,” Opt. Express 21(23), 28597–28608 (2013). [CrossRef]
36. Z. Zhang, J. Gan, X. Heng, Y. Wu, Q. Li, Q. Qian, D. Chen, and Z. Yang, “Optical fiber design with orbital angular momentum light purity higher than 99.9%,” Opt. Express 23(23), 29331–29341 (2015). [CrossRef]
37. V. Brückner, “To the use of sellmeier formula,” Senior Experten Service (SES) Bonn and HfT Leipzig 42, 242–250 (2011).
38. S. Ramachandran and P. Kristensen, “Optical vortices in fiber,” Nanophotonics 2(5-6), 455–474 (2013). [CrossRef]
39. J. Wang, S. Chen, and J. Liu, “Orbital angular momentum communications based on standard multi-mode fiber (invited paper),” APL Photonics 6 (2021).
40. 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]
41. S. Golowich, “Asymptotic theory of strong spin-orbit coupling in optical fiber,” Opt. Lett. 39(1), 92–95 (2014). [CrossRef]
42. P. Gregg, P. Kristensen, A. Rubano, S. Golowich, L. Marrucci, and S. Ramachandran, “Enhanced spin orbit interaction of light in highly confining optical fibers for mode division multiplexing,” Nat. Commun. 10(1), 4707 (2019). [CrossRef]
43. S. Ramachandran, P. Gregg, P. Kristensen, and S. E. Golowich, “On the scalability of ring fiber designs for OAM multiplexing,” Opt. Express 23(3), 3721–3730 (2015). [CrossRef]
44. Q. Lu, J. Tu, Z. Wang, S. Gao, J. Zhou, Q. Xiao, L. Shen, X. Lan, W. Liu, and Z. Li, “Spin-orbit coupling suppressed high-capacity dual-step-index ring-core OAM fiber,” Opt. Lett. 47(5), 1141–1144 (2022). [CrossRef]
45. Y. Wang, Y. Liu, W. Zhao, J. Yang, W. Geng, Y. Fang, C. Bao, Y. Ren, Z. Wang, Y. Liu, Z. Pan, and Y. Yue, “Multi-Ring-Air-Core Fiber Supporting Numerous Radially Fundamental OAM Modes,” J. Lightwave Technol. 40(13), 4420–4428 (2022). [CrossRef]
46. Y. Wang, W. Zhao, J. Hu, Z. Wang, Y. G. Liu, C. Bao, Y. Ren, Z. Pan, and Y. Yue, “19-ring-air-core fiber supporting thousands of OAM modes for spatial division multiplexing,” Opt. Lett. 47(9), 2206–2209 (2022). [CrossRef]
47. C. Brunet, B. Ung, P.-A. Belanger, Y. Messaddeq, S. LaRochelle, and L. A. Rusch, “Vector Mode Analysis of Ring-Core Fibers: Design Tools for Spatial Division Multiplexing,” J. Lightwave Technol. 32(23), 4648–4659 (2014). [CrossRef]
48. W. Wei, Y. Wang, H. Zhao, Z. Dong, L. Tang, L. Ding, and Y. Li, “Ring-Core Photonic Quasi-Crystal Fiber With 34 Polarization Multiplexing Modes,” IEEE Photonics J. 14(1), 1–7 (2022). [CrossRef]
49. W. Wei, N. An, Z. Zhang, L. Tang, L. Ding, and Y. Li, “Design of a microstructure optical fiber supporting 52 vortex beams,” Opt. Commun. 490, 126657 (2021). [CrossRef]
50. X. Zhang, Q. Ma, W. He, W. Hong, and Z. Deng, “Graded ring-core photonic crystal fiber for the transmission of OAM modes with high mode purity and quality in C + L bands,” Opt. Commun. 520, 128486 (2022). [CrossRef]
51. J. Tu, S. Gao, Z. Wang, Z. Liu, W. Li, C. Du, W. Liu, Z. Li, C. Yu, H. Tam, and C. Lu, “Bend-Insensitive Grapefruit-Type Holey Ring-Core Fiber for Weakly-Coupled OAM Mode Division Multiplexing Transmission,” J. Lightwave Technol. 38(16), 4497–4503 (2020). [CrossRef]
52. R. T. Bise and D. J. Trevor, “Sol-gel Derived Microstructured Fiber: Fabrication and Characterization,” presented at the Optical Fiber Communication Conference (2005).
53. H. E. Hamzaoui, Y. Ouerdane, L. Bigot, G. Bouwmans, B. Capoen, A. Boukenter, S. Girard, and M. Bouazaoui, “Sol-gel derived ionic copper-doped microstructured optical fiber: a potential selective ultraviolet radiation dosimeter,” Opt. Express 20(28), 29751–29760 (2012). [CrossRef]
