Ring-core fiber with negative curvature structure supporting orbital angular momentum modes

Compared to glass walls with a positive curvature, those with a negative curvature have been proven to have stronger confinement of light. Therefore, we change the multi-layered air holes in a photonic crystal fiber into several negative curvature tubes. As a result, the confinement medium is shifted from a low-index cladding material into a special structure. The theoretical analysis shows that each vector eigenmode has a corresponding threshold value for the outer tube thickness. It means that we can confine the target modes and filter the unnecessary modes by shifting the outer tube thickness. After substantial investigation on this fiber, we obtain the appropriate values for each structural parameter and then fabricate this negative curvature ring-core fiber under the guidance of the simulation results. Firstly, we draw the central cane under vacuum condition, then stack the cane and six capillaries to form the preform, and finally draw the ring-core fiber by using vacuumization method. The fiber test experiment indicates that the fiber length should be at least 15 m∼20 m to form the donut facula, and the tested losses of OAM+1,1, OAM+2,1, OAM+3,1, and OAM+4,1 are 0.30 dB/m, 0.36 dB/m, 0.37 dB/m, and 0.42 dB/m, respectively. © 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

. (a)Circular type photonic crystal fiber, (b) equivalent Bragg fiber, (c) loss comparison between multi-layered annular fiber and negative curvature fiber, where the radius of the air-core (a), the distance of the tube (d), and the thickness of the tube (t) in the annular fiber are 15 µm, 10 µm, and 0.24 µm, respectively, and the radius of the air-core (a ), the distance of the tube (d ), and the thickness of the tube (t ) in the negative curvature fiber are 14.33 µm, 16 µm, and 0.24 µm, respectively, and (d) improved ring-core fiber with negative curvature tubes. glass wall, which is also known as the anti-resonant effect. We will explain the design process, introduce the OAM modes control method, and finally present the fabricated fiber.

Fiber design and fabrication
The cross-section of the proposed negative curvature ring-core fiber (NC-RCF) is shown in Fig.  2, where r 0 and r are the inner and outer radii of the ring-core, respectively, d and Λ are the inner diameter of the air hole and the distance between the adjacent air holes, respectively, t and R are the thickness and inner radius of the cladding tubes, respectively, and D is the outer diameter of the fiber. In this simulation work, the tube outside the air-hole has the same thickness as that of the six outer cladding tubes, which is also denoted by t. All the simulation results are obtained through the finite element method (FEM) by using the commercial software COMSOL.
Designing fiber involves figuring out how the structural parameters affect the fiber properties and then deciding the design region for each parameter based on the target function of the fiber. For the proposed NC-RCF, the structural parameters include core index (n co ), cladding index (n cl ), wavelength (λ), thickness of the ring-core, and thickness of the cladding tube (t). Here, n co , n cl , and λ can be summarized in the effective normalized frequency (V eff ), while the thickness of the ring-core can be normalized as the ratio of the inner radius to the outer radius (ρ = r 0 /r). Therefore, the design steps for this NC-RCF are as follows: A. decide the expression of V eff by analyzing the corresponding n cl for different air-hole sizes and wavelengths; B. investigate how V eff , ρ, and t influence the fiber properties and then decide the design region for these parameters; C. present the fiber characteristics, such as the separation degree between the vector eigenmodes in one OAM group, that between adjacent OAM groups, confinement loss, bending loss, nonlinear efficient, and chromatic dispersion.

Expression of the effect normalized frequency V eff
Normalized frequency plays an important role in evaluating the propagation characteristics of a fiber, because it includes the fundamental structure parameters of the fiber. The effective normalized frequency (V eff ) of this proposed fiber is expressed as where λ is the operating wavelength, n co is the core index (which is set as 1.444), and n cl is the cladding index. For this NC-RCF with a special cladding structure, n cl is an unknown parameter. The relative hole diameter (d/Λ) and the normalized wavelength (λ/Λ) are two key values for deciding the cladding index n cl . First, we obtain the effective index (n eff ) of the NC-RCF for different d/Λ and λ/Λ. Figure 3(a) shows the calculated n eff of the NC-RCF as a function of λ/Λ with d/Λ ranging from 0.2 to 0.8 in steps of 0.1, where r and r 0 /r are set to be 7 µm and 0.6, respectivley. Subsequently, we calculate the corresponding cladding index of the step index ring-core fiber (SI-RCF) as shown in Fig. 3(b) for each n eff in Fig. 3(a), and this SI-RCF has the same ring-core size as that of NC-RCF. Figure 3(c) shows the n cl dependence on λ/Λ with d/Λ ranging from 0.2 to 0.8 in steps of 0.1, and this cladding index can be approximately treated as that of the NC-RCF. After getting the n cl , the V eff values for each d/Λ and λ/Λ can be obtained. The fitting curves are shown in Fig. 3(d), and the fitting equation is given by Eq. (2). In Eq. (2), the fitting parameters A i (i = 1 to 3) depend on d/Λ only. The data are well described by Eq. (3), and the coefficients a i to d i are given in Table 1. According to Eq. (2) and Eq. (3), we can easily understand the V eff value of the NC-RCF . -

Cutoff condition for OAM modes
The OAM modes are formed by a linear combination of the vector eigenmodes with ±π/2 phase shift, which can be expressed as where σ + and σ − represent the left and right circular polarization, respectively, F l,m is the radial field distribution, l denotes the topological charge number, m is the radial order, which indicates the intensity profile of the mode in the radial direction, the sign +/− in front of l indicates the right or left rotation of the wavefront, and i represents a π/2 phase shift. When l = 0, the σ ± OAM ±0,m mode is formed by the combination of two fundamental modes HE 1,m with left or right circular polarization, but it cannot be treated as an OAM mode since they carry no OAM [11]. For l = 1, OAM has two states, σ ± OAM ±1,m mode, which is composed of HE even 2,m and HE odd 2,m with the same directions of polarization and wavefront rotation. When l > 1, 4 OAM states can be obtained.
To decide the OAM mode number, we firstly investigate the cutoff condition of each vector eigenmode. Besides the above-mentioned V eff , the ratio of the inner radius to the outer radius of ring-core ρ (in other words, the thickness of the ring-core) also has a significant impact on the OAM characteristics. Figure 4 illustrates the cutoff condition of V eff and ρ for each vector eigenmode, where d/Λ = 0.8. In order to ensure the quality of the OAM modes, the higher radial order modes should be suppressed, because the phase distributions of higher radial order modes (m > 1) are different at different areas of different rings, which makes the multiplexing and demultiplexing of OAM modes very difficult. This fact indicates that m of OAM should be 1. As shown in Fig. 4, ρ should be at least larger than 0.6 to prevent higher radial order modes. However, the thickness of the ring core cannot be reduced infinitely, because it is hard for the very thin ring core to support the OAM modes with good quality. After considering this trade-off, the range of ρ is set to be 0.6∼0.8, which is shown as the blue area in Fig. 4, while the selected V eff depends on the target number of OAM modes.
However, when V eff becomes larger than 11, ρ of > 0.6 is not enough to suppress the higher radial order mode any more. Fortunately, there is another way to control the OAM modes, that is, controlling the thickness of the outer negative curvature cladding tubes t. Figure 5 shows the dependence of the confinement loss (CL) of each vector eigenmode on t, where r = 7 µm, ρ = 0.65, λ = 1.55 µm, and d/Λ = 0.8. For each vector eigenmode, there is a threshold value of t to make the light confinement condition of the negative curvature tubes change from anti-resonance to resonance. It means that the mode number can be controlled and some specified modes can be filtered by tuning the parameter t. Here, t of 1.2 µm is the critical value that can filter the higher radial order modes TE 02 and HE 12 while confining the OAM component vector eigenmodes HE 21 , HE l+1,1 and EH l−1,1 (i = 2, 3, 4, 5, 6,7,8). Moreover, we also calculate the surface scattering loss (SSL) based on the expressions in [21,22] and the estimated SSL is about 10 −4 dB/m. Thus, we can conclude that if t is smaller than the threshold value, the SSL is the dominated loss, while when t becomes larger than the threshold value, the CL becomes the dominated one. Here, t of 1.2∼2.5 µm can be regarded as an appropriate design region for the designed NC-RCF. In order to prove the effect of those negative curvature tubes, we investigated four cases with In these Figs. 6(b)-6(e), red color represents the silica and white color represents the air. In Fig. 6(a), the red and blue lines illustrate the confinement losses of HE 31 mode for ring-core fiber with silica cladding and air cladding, respectively. Since the refractive index of air is much lower than that of the silica, the air cladding helps the ring-core fiber obtain the ultra-low confinement loss. The green and black lines in Fig. 6(a) show the confinement losses of ring-core fiber with negative curvature tubes when t=2.5 µm, and t=2.0 µm, respectively. It is obvious that the thickness of the negative curvature tubes has very big impact on the confinement of mode. When t is 2.0 µm, the confinement degree of negative curvature tubes is very close to the air cladding, however, when t is increased to be 2.5 µm, there is an obvious increase in the confinement loss. This phenomenon can be explained by the mode coupling and anti-resonant effect. When t=2.0 µm, the thickness of the negative curvature tube makes the ring-core fiber work under the anti-resonant condition and the mode in the core will not be coupled out into the tubes, which can be illustrated by the electrical field distribution of HE 31 mode shown at the left side of Fig. 6(a). However, when t= 2.5 µm, anti-resonance and inhibited coupling condition of the tubes have been broken, the ring-core fiber works in the resonant status and a lot of energy are coupled into the tubes, which can be understood by observing the electrical field distribution of HE 31 mode shown at the right side of Fig. 6(a). Therefore, it has been proven that the negative curvature structure is a good cladding candidate to further confine the mode. For the designed NC-RCF, we take V eff of 17.5 as an example to discuss the fiber properties.

Characteristics of OAM mode
In the simulation, r = 7 µm, ρ = 0.65, t = 1.2 µm, and d/Λ = 0.8. In order to ensure that the HE and EH vector modes are not coupled into the LP mode again, ∆n eff should be larger than 10 −4 , which is the modal birefringence of the polarization-maintaining fibers. Therefore, we investigate the n eff of each vector eigenmode and calculate the value of ∆n eff . Figure 7(a) shows the dependence of n eff of each vector eigenmode on λ for OAM mode groups 1∼8, where the solid lines and dashed lines stand for the HE mode and EH mode, respectively. It is obvious that the gap between the HE mode and EH mode in an OAM mode group reduces as the order of the OAM mode group increases. Figure 7(b) shows the dependence of ∆n eff between the adjacent vector eigenmodes in an OAM mode group on λ. It is seen that the ∆n eff between HE mode and EH mode in OAM mode groups #2, #3, #4, #5, #6, #8 can meet the requirement of > 1×10 −4 in the wavelength band 1.4∼1.8 µm. Figure 7(c) illustrates the minimum ∆n eff between the adjacent OAM mode groups over λ from 1.2∼2.0 µm. The accurate modes used to calculate the minimum ∆n eff between the OAM mode groups are summarized in the table shown in Fig. 7(d). In Fig.  7(c), it can be found that the n eff gap between OAM mode groups increases as the order of the OAM mode group increases, which is the opposite trend to that observed for the ∆n eff between HE mode and EH mode in one OAM mode group. As λ increases, the n eff gap between OAM mode groups will also increase and all the ∆n eff between OAM mode groups become larger than 1×10 −3 when λ > 1.2 µm. Therefore, the proposed NC-RCF can theoretically support 26 OAM states in the telecommunication band as follows: OAM mode group #1 (2 states), #2 (4 states), #3 (4 states), #4 (4 states), #5 (4 states), #6 (4 states), and #8 (4 states). The mode intensity distribution, electric field distribution, and phase are all included in Fig. 8. Subsequently, we analyze the different fiber characteristics, such as confinement loss, bending loss, nonlinear efficiency (γ), and chromatic dispersion (CD). Here, confinement loss is expressed by Eq. (5) in unit of dB/m, and the imaginary part of the effective index is obtained by adding a perfect match layer (PML) to the fiber model in COMSOL. In Eq. (5), k is the wavenumber in vacuum and k = 2π/λ. The bending loss is calculated by using the equivalent refractive index, which is expressed by Eq. (6). The parameters γ and CD are given by Eq. (7) and Eq. (8), respectively. In Eq. (7), n 2 is the nonlinear index for fused silica and is set as 2.6×10 −20 m 2 W −1 , and A eff represents the effective area. In Eq. (8), c and n eff are the velocity of light in vacuum and the effective index, respectively. Firstly, we swept the wavelength and confinement loss spectra of the 13 vector eigenmodes are shown in Fig. 9(a). From Fig. 9(a), we can find that all the modes have ultra-low confinement loss, thanks to the anti-resonant effect of the outer cladding tubes, and if λ is larger than about 1.64 µm, the losses of HE 91 and EH 71 increase immediately due to that the longer wavelength breaks the anti-resonance condition and light cannot be suppressed anymore. Figure 9(b) shows the bending loss dependence on the bending radius (R c ) at λ of 1.55 µm, and it is known that the bending loss vibrates only slightly when R c is changing. Figure 9(c) and 9(d) illustrate the γ and CD characteristics over the C+L wavelength bands. In Fig. 9(c) and Fig. 9(d), we can find that γ is within 1.3∼1.8 over the C+L bands, which is close to that of a single-mode fiber, and the CD curves are significantly flattened with a maximum dispersion slope of 0.2051 ps/(km· nm 2 ) (HE 91 ).

Fabrication
Figure 10(a) shows the fabrication process of the NC-RCF, where two drawing stages were utilised to fabricate the designed fiber. This included four steps: 1) stack the center preform, 2) draw the center cane, 3) stack the fiber preform, 4) draw the fiber. In the first step, we used two types of fused silica tubes and 21 identical capillaries to stack the structure of the central ring-core. The inner and outer diameters of these silica tubes and capillaries are shown as tube 1 , tube 2 , and capillary 3 in Tab. 2. One end of tube 1 and that of capillaries 3 were sealed before stacking. The sealed capillaries and preform of the central ring-core are shown in Fig. 10(b) and 10(c), respectively. In the second step, the stacked preform with the structure of the central ring-core was drawn to a cane with a diameter of ∼1.3 mm, which is named as cane 5 , under vacuum; its microscopic cross-section is shown in Fig. 10(d). In the third step, six capillaries 3 and one of the canes were sealed at one end and then stacked in tube 4 to obtain the second preform, as shown in Fig. 10(f). Finally, the second stacked preform was drawn to the fiber with an outer diameter of ∼125 µm under vacuum of 60 kPa. The drawing temperature was about 1880 • C, which is relatively lower than that used for drawing an all-solid fiber to maintain the structure. Figure 10(g) shows the cross-sectional scanning electron microscopic (SEM) photo of the drawn fiber. Figure 11 shows the enlarged views of the fabricated fiber shown in Fig. 10(g). From Fig. 11, we can find that the ring-shape of the core is maintained very well, which is important to support OAM modes. From the SEM photos, the inner and outer radius of the ring-core were measured as r 0 = ∼4.5 µm, and r = ∼6.6 µm, respectively. The ratio of the inner radius to the outer radius of  the ring core is ρ = ∼0.68, and relative air-hole size is d/Λ = ∼0.78. These structural parameters of the fabricated fiber are in good agreement with the designed values used in the simulation, where r 0 = 4.55 µm, r = 7 µm, ρ = 0.65, and d/Λ = 0.8. Since the fiber was drawn with vacuum, the gaps between the adjacent outer tubes collapsed, resulting in slightly different values from that of the design. However, the negative curvature structures were formed successfully and the anti-resonant effect still confines the light. The characterization of the fabricated NC-RCF and its confinement capacity of OAMs are discussed in the following section. Figure 12 shows the experimental setup used to verify the transmission of OAM modes through the fabricated NC-RCF. At the input end, an erbium-doped fiber amplifier (EDFA) is used to increase the light power of the external cavity laser (ECL, 1550nm, 6 dBm) to 25.5 dBm. An The results are shown in Fig. 13 for comparison. The simulated intensity profiles agree well with the measured ones, and the phase indicates the orders of the OAM modes. The propagation loss of OAM in the fabricated fiber was measured by using the cut-back method. Taking OAM mode with l = +1 as an example, after the OAM +1,1 was coupled into the NC-RCF, we adjusted the position of the fiber very slightly to obtain the highest coupling power and recorded the output power (P1). Subsequently, we cut a 5 m-long fiber and then recorded the output power (P2) without changing the launching condition. Hence, the propagation loss of this NC-RCF can be calculated via 10log(P1/P2)/5 (dB/m). Since the fiber length of the fabricated NC-RCF is limited and proper length of the fiber remains at the input side to guarantee a single layer of facula, we only test the losses of the first four OAM modes by using 40 m NC-RCF with banding radius of ∼8 cm. The losses of OAM +1,1 , OAM +2,1 , OAM +3,1 , and OAM +4,1 were 0.30 dB/m, 0.36 dB/m, 0.37 dB/m, and 0.42 dB/m, respectively.

Results and discussion
We also compare the fiber properties for the reported ring-core fibers and the proposed negative curvature ring-core fiber. Although the measured loss of the fiber in this experiment is relative high compared to the other fabricated ring-core fibers, the novel principle of confining mode still have the potentiality to obtain lower loss and large mode separation. In the next fabrication, we will try the best to make the silica layer outside the air-hole thinner and thus enhance the confinement.