A Hybrid Cladding Ring-Core Photonic Crystal Fibers for OAM Transmission with Weak Spin–Orbit Coupling and Strong Bending Resistance

Abstract

A hybrid cladding ring-core photonic crystal fiber (PCF) for transmitting orbital angular momentum (OAM) modes is proposed, which breaks the circular symmetry of the fiber structure to suppress the spin–orbit coupling and promotes bending resistance. Through the optimization of fiber structure parameters, the designed fiber can support 22 OAM modes (6 OAM mode groups) over a 200-nm wide bandwidth (covering the whole C + L band) with large effective refractive index separation between adjacent modes (>10⁻⁴) and mode groups (>3.6 × 10⁻³), low confinement losses (<3.5 × 10⁻⁹ dB/m), and high mode purity (>98.3%). Meanwhile, the phase of the OAM modes varies periodically and uniformly with an increase in the azimuth angle, and the polarization of OAM modes maintain nearly circular polarization in the designed fiber, which also demonstrates that the fiber has weak spin–orbit coupling. Moreover, the confinement losses of all vector modes are less than 10⁻⁷ dB/m when the bending radius is larger than 0.8 mm, indicating strong bending resistance. Furthermore, the fiber also exhibits large differential group delay, relatively low and flat dispersion, and low nonlinear coefficients (<2.0 W⁻¹/km). Therefore, the novel fiber structure has great potential in the application of mode division multiplexing (MDM) based on OAM modes.

1. Introduction

The current communication architecture is built based on a single-mode optical fiber communication system. However, such communication systems are approaching their capacity limits after employing time-division multiplexing (TDM), wavelength-division multiplexing (WDM), polarization-division multiplexing (PDM), and an advanced modulation scheme [1,2]. A promising solution to this capacity crunch is space-division multiplexing (SDM) using multiple spatial channels to increase the transmission capacity and the spectral efficiency [3,4]. Mode-division multiplexing (MDM) is one SDM which adopts different types of fiber modes as information channels carrying independent data streams. In general, there are several mode basis sets used in MDM communication systems such as linearly polarized (LP) modes, orbital angular momentum (OAM) modes, and vector modes. In recent years, the MDM based on OAM modes has received extensive research interest due to its compatibility with the existing mature multiplexing technologies as well as the special properties of OAM modes [5,6,7,8,9,10,11,12,13,14,15].

The light carrying the OAM is a type of vortex beam with a spatial phase factor exp(ilθ) (l is the topological charge and θ is the azimuthal angle), which can be formed by using a linear combination of the odd and even modes of the same vector mode HE_l+1,m or EH_l−1,m with a phase difference of ±π/2 [16,17]. These OAM modes with different l are orthogonal to each other, and the value of l is unlimited in theory. Therefore, the MDM technology based on OAM modes can greatly increase the transmission capacity of the fiber communication system, which has been demonstrated in Ref. [15]. The OAM modes exhibit some special mode properties such as the ring intensity distribution, helical phase wavefront, and circular polarization state [18,19]. Compared to LP modes, OAM modes have three advantages: (1) OAM modes in the fiber show a highly similar annular intensity profile, which makes it easy to achieve small differential modal gain (DMG) to support all OAM modes in an erbium-doped fiber amplifier (EDFA), and different modes have almost the same loss. (2) The first-order radial OAM modes are used to implement an MDM system, which helps to simplify multiplex and demultiplex schemes. (3) Free or few multiple input multiple output (MIMO) digital signal processing technologies can be implemented in the receiver because of low coupling between adjacent OAM modes or among OAM mode groups, which will greatly simplify the complexity and reduce the cost of the MDM system.

What does an MDM system based on OAM modes require for a fiber link? Multiplexers, optical fibers, and amplifiers are three critical online devices. Especially, the optical fiber determines the transmission distance of the optical fiber link. In this paper, we focus on fiber design and characterization. To transmit OAM modes in a high number and with stability, some important properties such as low mode coupling, low confinement and bending loss, and low nonlinearity, etc., should be considered when designing an OAM fiber [20,21]. The fiber with a ring core structure can closely match the annular intensity profile of the OAM mode, in which the ring core area of the mode field distribution has a higher refractive index, and the inner and the outer cladding have lower indices. Generally, an effective index separation (>10⁻⁴) between the adjacent eigenmodes or an effective index separation (>10⁻³) among the OAM mode groups is targeted as one of the requirements during the process of the fiber design [10,11,12]. Moreover, the thinner ring core design is adopted to suppress the generation of high radial order modes and to simplify the demultiplexing systems [22,23,24,25]. Therefore, OAM fibers are generally characterized by a thin ring core and large refractive index contrast (Δn) between ring core and cladding. To expand the separation between adjacent OAM modes or mode groups to ensure the stability of the OAM mode transmission, there are two main ways to increase Δn. One method is up-doping the fiber core and down-doping fiber cladding, which is limited by optical fiber manufacturing technology. Moreover, the fiber fabricated by this method has been reported to support a maximum of 36 OAM modes [23]. However, heavily up-doping/down-doping may result in higher fiber loss. Another approach without any doping is to introduce air holes into the microstructure in the fiber cladding to obtain large Δn, which has attracted extensive attention [20,21,26,27,28,29,30,31,32,33,34,35]. Compared with traditional ring core fibers, photonic crystal fibers (PCF) are characterized by multiple flexible and adjustable structure parameters, which help to optimize the desired fiber transmission performance [36]. Therefore, PCF has greater potential to transmit more high-quality OAM modes. Researchers have undergone much work in designing an OAM-PCF.

However, larger refractive index contrast and a thinner ring core give rise to severe spin–orbit coupling, causing high transmission losses. The spin–orbit coupling comes from the interaction of light and matter at the boundary of the core-cladding, which enhances with an increase in Δn [37]. Moreover, the refractive index profile of a rotationally symmetric fiber will aggravate this effect [38,39,40]. In addition to the dominant component, OAM modes in the fibers also have a second component with opposite polarization and a topological charge difference of ±2 [41]. Therefore, the fiber with a large index contrast will greatly increase the intrinsic crosstalk of OAM modes to deteriorate the communication quality, which limits the transmission distance and increases the demultiplexing complexity of the OAM-MDM system in the receiver. Therefore, it is of great significance to design an OAM fiber with weak spin-orbit coupling and good transmission performance.

In this paper, we propose a hybrid cladding (air holes arranged in a combination of triangular and ring lattice) OAM-PCF with a thicker ring core, which matches the OAM mode ring intensity distribution and breaks the circular symmetry of the fiber structure to weaken the spin–orbit coupling. Numerical results show that the optical fiber can support 22 high-quality OAM modes (six OAM mode groups) with a large effective index difference (>10⁻⁴ between adjacent modes, >10⁻³ among OAM mode groups), low confinement loss (<3.5 × 10⁻⁹ dB/m), high mode purity (>98.6%), weak spin–orbit coupling (nearly circular polarization), good bending resistance (the minimum bending radius is 0.8 mm) covering a 200-nm wide bandwidth (over the whole C + L band), relatively flat dispersion (dispersion variation is lower than 50.0 ps/nm/km over 200 nm), large differential group delay, and a low nonlinear coefficient (<2.0 W⁻¹/km). Therefore, the designed fiber has great potential in the application of an MDM system based on OAM modes.

2. Fiber Design

Based on the concept of weakening the spin–orbit coupling effect introduced using a circular symmetrical structure and the feasibility of fiber drawing, a ring core OAM-PCF with hybrid cladding is proposed. Air holes of the first ring must be arranged as a circular ring to match the ring intensity distribution of OAM modes and ensure the OAM modes’ excitation. To lift the index separation and reduce the spin–orbit coupling, triangular lattice is introduced to break the circular symmetry. Therefore, a combination of the circular and triangular lattices is designed as the fiber outer cladding, as shown in Figure 1. The fiber base material is made of pure silica (refractive index is 1.444 at the wavelength 1.55 μm). The cross-section of the designed optical fiber consists of the inner cladding of a large central air hole, the annular core region supporting OAM modes, and the combination outer cladding of one-layer ring lattice air holes and the four-layer triangular lattice air holes [20]. r_a is the radius of the central air hole, and r_b and r₁ are the distance from the center of the fiber to the center of the annular air holes and the second ring air holes with triangular lattice in the cladding, respectively. d₀ is the diameter of the air hole in the innermost ring and d₁–d₄ are the diameters of the four layers of air holes arranged in a triangular lattice from inside to outside. The diameter of air holes at the six corners of the second ring arranged in a triangular lattice is d₅, which can further break the circular symmetry of the optical fiber structure and weaken the spin–orbit interaction between OAM modes.$\Lambda$ is the pitch between the adjacent air holes in the triangular lattice. Λ_C is the pitch between adjacent air holes in the ring lattice. Generally speaking, the larger the air filling fraction in the cladding is, the higher the index contrast between the fiber core and cladding. However, the stable structure of the fiber simultaneously needs to be considered; hence, the sizes of the air holes in the cladding should be a trade-off value. The structure parameters of the fiber are presented in

Table 1. The structure parameters of the designed optical fiber.

Parameter	d0	d1	d2	d3	d4	d5	ra	rb	r1	Λ	Λc
Value(μm)	1.8	1.8	1.8	2	2	2.2	2.8	6.2	8.8	2.2	2.15

. The thickness of the silica ring is 2.5 µm, which is a result of balancing the excitation of the high-order radial modes and the mode field quality. Therefore, the fiber belongs to the thick ring structure, which can provide high mode field quality and suppress the spin–orbit coupling effect [14].

3. Fiber Characteristics

3.1. Number and Stability of OAM Modes

OAM modes are formed by the superposition of the vector eigenmodes by the following relations:

$$\{\begin{array}{c}{\mathrm{OAM}}_{\pm l,m}^{\pm }={\mathrm{HE}}_{l+1,m}^{even}\pm j{\mathrm{HE}}_{l+1,m}^{odd}\\ {\mathrm{OAM}}_{\pm l,m}^{\mp }={\mathrm{EH}}_{l-1,m}^{even}\pm j{\mathrm{EH}}_{l-1,m}^{odd}\end{array}\begin{array}{cc}& l\ge 2\end{array}$$

(1)

$$\{\begin{array}{c}{\mathrm{OAM}}_{\pm 1,m}^{\pm }={\mathrm{HE}}_{2,m}^{even}\pm j{\mathrm{HE}}_{2,m}^{odd}\\ {\mathrm{OAM}}_{\pm 1,m}^{\mp }={\mathrm{TE}}_{0,m}^{}\pm j{\mathrm{TM}}_{0,m}^{}\end{array} \begin{array}{cc}& l=1\end{array}$$

(2)

where superscript ‘±’ is the right or left circular polarization, and the subscript ‘±l’ denotes the right or left wave front rotation direction with a topological number of l, m is the radial order or the number of concentric rings in the intensity profile, j represents a π/2 phase shift. OAM modes formed by HE modes have the same directions of spin angular momentum (SAM) and OAM, while OAM modes from EH modes have the opposite directions of SAM and OAM. Moreover, the OAM modes for l = 1 composed by azimuthally polarized TE_0,m and radially polarized TM_0,m modes, as shown in Equation (2), cannot be used as OAM modes because they are unstable in the fiber. The OAM modes with the same l and m compose an OAM mode group, and each mode group supports four OAM modes except for the OAM_1,1 mode group (composed by even and odd HE_2,1). To simplify the multiplexing and demultiplexing of OAM modes, the radial order m is selected as 1.

The number of OAM modes stably supporting the fiber is determined by the effective normalized frequency V_eff of the fiber and the degree of the separation of the mode effective index between adjacent vector modes. The most direct way to increase the number of OAM modes is to increase the radius of the fiber core. The stable transmission of OAM modes requires that the effective refractive index difference should be larger than 10⁻⁴ between adjacent modes for MDM or larger than 10⁻³ among mode groups for mode group division multiplexing, which can help to implement free or few MIMO digital signal processing in the receiver of MDM systems. This is the third advantage of the OAM modes mentioned above. The effective refractive index of the vector modes of the proposed fiber is shown in Figure 2a, in which the cut-off wavelength of HE_8,1 and EH_6,1 is 1.6 μm. In our design of an OAM fiber with a wide bandwidth and high stability, these two modes are not discussed. Figure 2a indicates that 22 vector modes (six OAM mode groups) can be guided over a wavelength range of 1.5–1.7 μm (covering the whole C + L band). The effective refractive index of vector modes decreases with an increase in mode order and wavelength. There is an interesting phenomenon in Figure 2a: when l ≤ 4, the effective refractive index of HE mode in the same mode group is larger than that of EH, while, when l > 4, the situation is the opposite. Moreover, the effective refractive index separation of HE and EH modes in the same mode group increases with an increase in the wavelength. Therefore, in Figure 2b, it is shown that the effective refractive index difference of the vector modes in the same OAM mode group increases with an increase in wavelength, and that the smallest effective refractive index difference is still greater than 10⁻⁴, which can ensure the stable transmission of an OAM mode without coupling into LP modes. The effective refractive index difference between adjacent the mode groups is defined as

$$\Delta n_{eff,MG}=\min (n_{H{E}_{l+1,1}},n_{E{H}_{l-1,1}})-\max (n_{H{E}_{l,1}},n_{E{H}_{l-2,1}})$$

(3)

In Figure 2c, the effective refractive index difference between different OAM mode groups is greater than 3.6 × 10⁻³. Therefore, the designed fiber can simultaneously ensure that the effective refractive index difference between the adjacent modes and mode groups are greater than 10⁻⁴ and 10⁻³, respectively, which can enable free and few MIMO to simplify the complexity of the MDM system.

The total propagation loss of optical fiber comes from the absorption loss of materials, Rayleigh scattering loss, defect loss caused by imperfect fabrication, infrared and ultraviolet absorption loss, and confinement loss caused by the fiber structure. Generally, the confinement loss of the conventional single mode and multimode fibers is so small that is ignored. However, for microstructure fiber, confinement loss due to the finite number of air holes is an important property to evaluate fiber performance. The confinement loss can be calculated by

$${\alpha }_{CL}=\frac{2\pi }{\lambda }\frac{20}{\ln 10}\mathrm{Im}(n_{eff}) (\mathrm{d}\mathrm{B}/\mathrm{m})$$

(4)

The confinement losses of all vector modes are less than 3.5 × 10⁻⁹, covering a wavelength range of 1.5–1.7 μm, as shown in Figure 2d, which indicates that the designed fiber can well confine the optical mode field into the annular core region and exhibits potential for long-distance transmission.

3.2. Weak Spin–Orbit Coupling

The purity can be used to express the spin–orbit coupling degree of the OAM modes supported in the designed fiber. The higher the mode purity is, the weaker the spin–orbit coupling. Here, the purity of OAM modes can be expressed as the power weight of the dominant component in all synthesized OAM beams [41]

$${\eta }_{Purity}=\frac{\max (P_{OAM,1},P_{OAM,2})}{P_{OAM,1}+P_{OAM,2}}$$

(5)

Figure 3 shows that the purity of all OAM modes decreases with an increase in wavelength, and the purity of all OAM modes is greater than 98.6%, covering a wavelength range of 1.2 μm to 1.7 μm. Here, A-OAM represents the OAM modes composed of HE modes, and AA-OAM represents the OAM modes composed of EH modes. The purity of AA-OAM modes in the same mode group is larger than that of A-OAM modes for l ≤ 4, while the opposite occurs for l > 4, which depends on the order in which the OAM modes appear.

The phase characteristics of different order-synthesized OAM modes are shown in Figure 4. The phase varies periodically and uniformly with an increase in azimuth angular θ, which indicates that the purity of each order mode is high, corresponding to weak spin–orbit coupling effect. Figure 5 shows the polarization characteristics of the synthesized OAM modes, which can also be used to qualitatively analyze the purity and spin–orbit coupling effect of the OAM modes. The black circles are the inner and outer core boundary of the designed fiber, and the red ellipse represents the polarization characteristics of the OAM modes. The size of the ellipse indicates the energy amount. The closer the ellipse is to the circle, the higher the purity of the mode. The purity of the synthesized OAM modes is very high, which indicates that the fiber can suppress spin–orbit coupling.

3.3. Strong Bending Resistance

In practical applications, fiber bending is inevitable, which will cause a leakage in the mode field. As a result, bending reduces the number of modes supported by the fiber, thereby decreasing the transmission capacity. Therefore, as a fundamental characteristic, it is very necessary to research the fiber bending loss.

The bending model is bult as shown in Figure 6a. The fiber is bent around the Z-axis into an annular shape toward the X-axis with a curvature radius R, and the degree of bending gradually increases with the decreasing R. Figure 6b shows the cross-section of the designed fiber in Cartesian coordinates. The refractive index distribution of the fiber bent along the X-axis can be expressed as [42]

$$n_b^2\left(x,y\right)=n^2\left(x,y\right)\left(1+2x/R\right)$$

(6)

where, n(x, y) is the refractive index distribution of the fiber before bending, and R is the radius of curvature of the fiber.

The confinement loss of all vector modes varies with the bending radius at 1.55 μm, as shown in Figure 7. All curves show similar trends, and the confinement loss for all the modes gradually declines as the curvature radius increases until they are equal to those of the confinement losses in the straight fiber. The change rates of the confinement loss are fast in the small bending radius region and then slow down when the bending radius becomes large. For the same mode, the larger the degree of bending is, the higher the confinement loss. The confinement loss of all vector modes is still less than 10⁻⁷ dB/m when the bending radius is larger than 0.8 mm, indicating that the designed fiber has good bending resistance. At the same bending radius, the bending losses of the higher order modes are greater than that of the lower order modes, which implies that the high order modes will firstly leak out the fiber core when fibers are bent to an extent. Here, we exhibit the ultimate bending loss of the designed fiber. However, it is almost impossible for the optical fiber to reach such a bending radius in practical applications. Therefore, the bending loss of the fiber can be ignored in practical applications, which is an advantage of the designed fiber.

3.4. Transmission of OAM Modes

The waveguide dispersion of the vector modes supported by the designed fiber is calculated by

$$D_w=-\frac{\lambda }{c}\frac{d^2\mathrm{Re}(n_{eff})}{d{\lambda }^2}$$

(7)

where Re(n_eff) is the real part of the effective refractive index of the vector mode, λ is the wavelength of incident light, c is the velocity of light in the vacuum. The total dispersion is the sum of waveguide dispersion and material dispersion. The material dispersion is calculated using the Sellmeier dispersion equation. As shown in Figure 8a, the dispersion of all vector modes increases slowly with an increase in the wavelength, and the dispersion variation is less than 50.0 ps/nm/km in the wavelength range of 1.5–1.7 μm, indicating that the designed fiber has a relatively flat dispersion, especially in the C + L band.

The differential group delay (DGD) between different OAM mode groups is defined as follows [19]

$$\Delta \tau =\frac{n_{eff,l}-n_{eff,0}}{c}-\frac{\lambda }{c}\left(\frac{\partial n_{eff,l}}{\partial \lambda }-\frac{\partial n_{eff,0}}{\partial \lambda }\right)$$

(8)

where n_eff,l and n_eff,0 are the minimum effective refractive index in the OAM mode group with the order of l and the maximum effective refractive index in the OAM mode group with the order of 0, respectively. Figure 8b shows that DGD is well separated among different mode groups, which indicates that the OAM mode, as an independent information carrier, has little mutual influence at the receiver, which can reduce power consumption and the complexity of demultiplexing.

The equation of the effective mode field area (A_eff) can be expressed as

$$A_{eff}=\frac{{\left({\displaystyle {\iint }_A{\left|E\right|}^{2}dxdy}\right)}^2}{{\displaystyle {\iint }_A{\left|E\right|}^{4}dxdy}}$$

(9)

where $E$ is the electric field intensity and the integral region $A$ is the entire cross-section of the fiber. Figure 8c shows the effective mode field area A_eff as a function of the wavelength. The A_eff of all vector modes is greater than 47.8 μm², and the maximum A_eff can reach 54.5 μm², which corresponds to the HE₆₁ mode.

The nonlinearity has an inverse relationship with the effective mode field area, which is expressed as

$$\gamma =\frac{2\pi n_2}{\lambda A_{eff}}$$

(10)

where n₂ is the nonlinear refractive index of the material with a value of 2.3 × 10⁻²⁰ m²W⁻¹. As shown in Figure 8d, the nonlinear coefficients of all vector modes are less than 2.0 W⁻¹/km and decrease with an increase in wavelength.

The transmission characteristics of OAM modes over a wide wavelength range covering the whole C + L band indicate the compatibility of the designed fiber with the existing mature WDM technique.

4. Discussion on Mode Number and Fiber Fabrication

The designed fiber can support more OAM modes and maintain better performance by changing and optimizing the structural parameters of the fiber. As we all know, the mode number of the fiber depends on the V parameter ($V=\frac{2\pi }{\lambda }a\sqrt{n_{co}^2-n_{cl}^2}$) of the fiber. Therefore, two schemes can be used to increase the number of the OAM mode for a given wavelength range. One is to enlarge the radius of the fiber core, which is the most direct and effective method. The other is to increase the refractive index difference between the core and cladding, which is limited by the material and fiber structure. For the designed fiber in this paper, the number of the OAM mode can be increased by enlarging the radius of the ring core or by increasing the air filling fraction of the outer cladding structure (or up-doping in the ring core). After increasing the core radius and air filling fraction, it is necessary to optimize the structure parameters to achieve a fiber of good performance. In fact, the scheme to increase the mode number of the fiber and maintain good fiber performance by enlarging the core radius has been proved in Ref. [20].

The designed fiber is composed of the inner cladding of the central air hole with a large size, the annular core region supporting OAM modes, and a combination of the outer cladding of one-layer ring lattice air holes and the four-layer triangular lattice air holes. All air holes are circular in shape, and the outermost four layers of the air holes of the outer cladding are arranged as a hexagonal shape, which is similar to the structure of a standard PCF. Therefore, the designed fiber can be fabricated using the widely used stack-and-draw method. The specific manufacturing process can be divided into two steps. (1) Preform fabrication: Three types of capillaries and a large tube with a certain wall thickness as the fiber core are stacked into a bigger tube to form the preform of the center cane according to the fiber structure as shown in Figure 1. After stacking, the capillaries are tied together with fine thread and drawn to be the preform. (2) Fiber drawing: The preform and outer tube are stacked into the fiber preform. Then, the fiber preform is drew into PCF while precisely controlling the temperature and pressure. Especially, the pressure should be properly selected to prevent the collapse of the air holes and to maintain the circular shape of the air holes. In recent research, similar fiber structures have already been successfully fabricated and experimented on, as shown in Ref. [43].

5. Conclusions

To design an OAM mode supporting fiber for a weak spin–orbit coupling effect and strong bending resistance, we propose a hybrid cladding ring core photonic crystal fiber (PCF) with which to break the circular symmetry of the fiber structure, thereby ensuring the low loss and stable transmission of OAM modes. By optimizing the structural parameters, the fiber can support 22 OAM modes (six OAM mode groups) across a bandwidth of 200 nm (1.5–1.7 μm, including the whole C + L band). Moreover, the fiber can simultaneously ensure that the effective refractive index difference between adjacent modes and mode groups is greater than 10⁻⁴ and 10⁻³, respectively, which can facilitate free and few MIMO to simplify the complexity of the MDM system. The phase of the OAM modes changes periodically and uniformly with an increase in the azimuth angle, and the polarization of each point of the fiber core is nearly circular in polarization, which demonstrates the weak spin–orbit coupling characteristics of the designed fiber. Furthermore, the confinement loss of all vector modes is less than 10⁻⁷ dB/m when the bending radius is larger than 0.8 mm, indicating strong bending resistance. In addition, the fiber also shows some good properties such as low confinement loss (<3.5 × 10⁻⁹ dB/m), high mode purity (>98.3%), large differential mode delay, relatively low and flat dispersion, and low nonlinear coefficients (<2.0 W⁻¹/km), which provide an opportunity to be compatible with mature WDM technology. In addition, the designed fiber can support more OAM modes and maintain better performance by changing and optimizing the structure parameters of the fiber and can also be fabricated by the widely used stack-and-draw method. Thus, the fiber can be used to enlarge fiber communication capacity and simplify the system complexity in an MDM system based on OAM modes.