Orbital angular momentum (OAM) modes have been widely investigated due to their unique spiral phase front given as exp(ilφ), where l is the topological charge number, and φ is the azimuthal angle [1]. This special phase feature could be utilized in many applications such as optical communications [2-5], quantum information [6, 7], and imaging [8]. In particular, the orthogonality and theoretically unlimited number (i.e. l could be infinity) of the OAM mode justifies it consideration as a new degree of freedom for mode division multiplexing (MDM), which is considered as an alternative to prevail over the increasing data capacity. OAM modes have been successfully demonstrated in free-space using holograms [9], a spiral phase plate [10], and metamaterials [11, 12]. However, the OAM modes based on free-space have limited applications because of the gradual enlargement of these modes with propagation. In this regard, the investigation of OAM modes in optical fibers has attracted significant interest due in part to the long propagation length and ease of mode generation, which could be realized by a combination of the even and odd modes of an identical eigenmode. Recently, terabit-data transmission based on OAM modes in a vortex fiber has been experimentally demonstrated [2, 3]. However, a conventional fiber supports only a few OAM modes.

In order to overcome this problem, OAM modes have been widely investigated in photonic crystal fibers (PCF) such as helically twisted PCFs [13], hexagonal lattice PCFs [14], Kagome lattice PCFs [15], circular PCFs [16-19], and PCFs with As₂S₃ background material [14, 17]. Recently, OAM modes were experimentally demonstrated in a helically twisted PCF [20]. PCFs have been utilized in various fields owing to their unique properties, such as high birefringence [21] and fiber communication [22], and the high index contrast between the air holes and the background material of the PCFs supports many OAM modes. In addition, the PCFs facilitate improved design flexibility compared with conventional fibers in the modification of the mode properties, such as the effective index, dispersion characteristics, and confinement losses. [23].

Although many investigations on OAM modes in PCF have been conducted, the reported results for PCFs indicate large dispersion and dispersion variation with wavelength, which causes signal distortions such as a temporal signal spreading [24]. Therefore, the realization of flat and low dispersion over a wide wavelength range is important for the exploitation of OAM modes for fiber-based high data transmission and multiplexing. In this paper, we propose a photonic quasi-crystal fiber (PQF) to accommodate low and flat dispersion OAM modes over a wide wavelength range. A photonic quasi-crystal is a structure composed of ordered non-periodic triangular and rectangular lattices. PQFs formed by the arrangement of quasi-crystal air holes in the cladding region possess unique properties that are not found in fibers with periodic array air holes [25, 26]. In particular, PQFs can support near zero ultra-flat dispersion over a wide wavelength range [27]. Based on this unique dispersion property of these fibers, we designed a PQF which supports 14 OAM modes with a flat dispersion characteristic over the C+L bands, where the variation of dispersion value for all modes is less than 5 ps/km/nm over C+L bands. Moreover, HE_2,1 facilitates almost zero dispersion and a very small dispersion variation of 0 to 0.22 ps/km/nm in the range from 1.52 µm to 1.625 µm. It also satisfies a radial single mode condition, a good effective index separation between eigenmodes, and low confinement loss. These properties are essential requirements for OAM multiplexing.

Figure 1 shows the schematic of the proposed PQF. It consists of a large hollow center (air hole) and quasi-periodic well-ordered 6-air hole layers in the cladding, where a is the period, r₀ is the radius of the center air hole, and r₁ to r₆ are the radii of the air holes in the cladding region. We chose silica as the background material due to its low loss and low nonlinear coefficient. All calculations were performed using a finite element method (FEM) with perfect matched layers (PMLs).

Similar to PCFs, the OAM modes can be supported in PQFs by linear combination of the even and odd symmetric modes of HE or EH, and the OAM_l,m mode can be described by the following relations [28].

where l is the topological charge number, m is the order of the mode for the radial direction, and the sign of +(−) denotes the right (left) phase rotation.

The PQF was designed to support low and flat dispersion by controlling the size and period of the air holes, and the optimized parameters are a = 1 µm, r₀ = 2 µm, r₁ = 0.353a, r₂ = 0.473a, r₃ = 0.42a, and r₄ = r₅ = r₆ = 0.9a. Figures 2(a) and 2(b) show the refractive indices of the supported eigen-modes and the effective index difference (Δneff) between the HE and EH modes which have the same order of the OAM mode as a function of wavelength. It should be noted that Δn_eff for OAM₁ denotes the effective index difference between the HE_2,1 mode and the TE_0,1 mode. As shown in Fig. 2(a), the proposed PQF can support up to HE_5,1 and EH_3,1 modes, and thus 14 OAM modes (|l| ≤ 4) are supported in total; the mode group from the HE_2,1 to HE_5,1 modes supports the OAM_l modes from l = 1 to l = 4, respectively, and the mode group from the EH_1,1 to EH_3,1 modes supports the OAM_l modes from l = 2 to l = 4, respectively. Note that each eigenmode supports right and left wave front rotation direction OAM modes. To exploit the OAM modes in the MDM system, a large effective index separation (Δn_eff > 10⁻⁴) between the same ordered OAM modes is necessary. As shown in Fig. 2(b), the Δn_eff of all OAM modes increases with an increase of the wavelength, and these are above 10⁻⁴ over the C+L bands. This means that the OAM modes in the designed PQF maintain a good separation, and thus the OAM modes can be stably transmitted without coupling into the LP modes. It should be noted that the designed fiber satisfies radial single mode (m < 2) condition to avoid exciting the higher radial order modes, which could make it difficult to multiplex and de-multiplex the OAM modes.

Figures 3(a) and 3(b) show the electric field intensity and phase distributions , respectively, of the HE_3,1 (OAM₂) mode. The electric field is well-confined in the silica region with a ring shape, and the phase changes smoothly by 4π in the azimuthal direction. Therefore, the designed PQF adequately supports OAM modes.

We calculated the dispersion characteristics of the eigenmodes as a function of wavelength and the results are shown in Fig. 4(a). The dispersion (D) is calculated using the following equation:

As shown in Fig. 4(a), all OAM modes have a low and flat dispersion characteristic over the C+L bands. As previously indicated, the photonic quasi crystal structure in the cladding region and the optimized diameters of r₁, r₂, and r₃ account for the flat and low dispersion characteristics, and the resulting variation of the dispersion for all modes is less than 5 ps/km/nm over the C+L bands. The enlarged dispersion value of the HE_2,1 mode is plotted in Fig. 4(b) to illustrate the specific dispersion variation with respect to wavelength. It can be seen that almost zero and flat dispersion of 0 to 0.22 ps/km/nm in the range from 1.52 µm to 1.625 µm are achieved. Figure 4(c) shows the enlarged dispersion variation of the HE_3,1 mode and EH_1,1 mode. The dispersion of these two modes is varied from 14 to 14.3 ps/km/nm and 17.8 to 18.3 ps/km/nm, respectively, in the range from 1.52 µm to 1.625 µm. It should be noted that these dispersion variations are very small compared with reported values for PCFs that support OAM modes [16, 17, 19]. For example, the dispersion values of the OAM modes in circular PCFs are above 100 ps/km/nm for |l| > 2 and these values are varied about 20 ps/km/nm over the C+L bands [16]. For a more quantitative investigation of dispersion variation, the wavelength derivative of dispersion is plotted in Fig. 4(d). The derivative value of all the eigenmodes is in the range of −0.06 to 0.04 ps/km/nm². In addition, The HE_2,1 mode shows almost a zero derivative value, and the derivative value of the HE_3,1 mode and the EH_1,1 mode are in the range of −0.01 to 0.01 ps/km/nm² .

We also calculated the confinement loss of the eigen-modes based on the imaginary part of the effective index. Figure 5 shows the confinement loss of eigenmodes as a function of wavelength. It increases with wavelength and the order of the mode due to mode leakage through the cladding. The confinement loss of the HE_5,1 mode, which has the largest value, is less than ~4 dB/km over the C+L bands. It should be noted that confinement loss can be reduced by increasing the number of air holes in the cladding.

The performance of the OAM modes at 1.55 µm is summarized in Table 1. The designed PQF provides outstanding dispersion characteristics without deterioration of the other mode properties. We also compared the dispersion characteristics of the PQF and the circular PCFs [16, 17], and the result is summarized in Table 2.

The fabrication process of the proposed PQF is more difficult than that of conventional fibers due to the complex structure of PQF. However, we believe that realization of the suggested fiber could be possible using the sol-gel method [29], because this method affords design flexibility in regard to the air hole size and the air hole structure by adjustment of the tuning, drawing speed, and gas pressure.

We propose a PQF to support OAM modes with low and flat dispersion characteristics. The unique structural property of the PQF and the optimized design of the air holes in the cladding region enable the realization of a very small variation of dispersion (<5 ps/km/nm) and its wavelength derivative (−60 ~ 40 ps/km/nm/µm) over the C+L bands. In particular, the HE_2,1 mode shows an almost zero and flat dispersion characteristics of D = 0 to 0.22 ps/km/nm, and the HE_3,1, and EH_1,1 modes show a small dispersion variation from 14 to 14.3 ps/km/nm and 17.8 to 18.3 ps/km/nm over the C+L bands, respectively. The designed fiber supports up to 14 OAM mode (|l| ≤ 4), satisfies the radial single mode condition, exhibits low confinement loss, and has good effective index separation between the eigenmodes. The proposed PQF may be applied to various OAM applications, especially in OAM multiplexing.