Beyond Two-Octave Coherent OAM Supercontinuum Generation in Air-Core As2S3 Ring Fiber

In this work, we designed and simulated a novel air-core As2S3 ring fiber that supports orbital angular momentum (OAM) modes. By optimizing the structure parameters of the designed fiber to effectively tailor its chromatic dispersion, a near-zero flat dispersion profile with a total of < ±30 ps/nm/km variation over 3380-nm bandwidth from 2025 nm to 5405 nm is achieved for OAM1,1 mode. After launching a 100-fs 70-kW hyperbolic secant pulse into an 8-mm air-core As2S3 ring fiber, a light-carrying OAM supercontinuum is numerically formed beyond two-octave range, covering 5717nm bandwidth from 1182 nm to 6897 nm at -30dB. Furthermore, the generated supercontinuum is highly coherent across the whole spectral range. This can serve as an effective manner to expand the spectral coverage of the OAM beams for various applications.

Supercontinuum has been an active field for decades due to its numerous applications. Among different characteristics, the spectral range of the supercontinuum is one of its critically important factors. Recently, supercontinuum generation The associate editor coordinating the review of this manuscript and approving it for publication was Sukhdev Roy. of OAM light states in nonlinear fiber has been reported, which could expand the spectral coverage of the ensuing optical vortex beams. One pioneer research has shown that an octave-spanning supercontinuum of light-carrying OAM was generated using specially designed optical fibers [15]. Another latest work used a 100-kW 60-fs input pulse to obtain a stable supercontinuum of optical vortex beams spanning from 696 to 1058 nm at −20 dB power level [16]. However, the spectral range of the generated OAM supercontinua is limited to around one octave. Proper material choice and advanced dispersion engineering are highly desirable to further expand the OAM spectrum with improved coherence.
Numerous researches have been performed toward various fiber or fiber-based devices suitable for near-and midinfrared (IR) wavelengths. One of the promising material platforms is chalcogenide glass (including S, Se, and Te), which exhibits a wide transmission window in IR band [17]. Consequently, it becomes an enabling material for various IR VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ devices. A number of previous researches have been investigated in chalcogenide waveguides and fibers, and suggested that they may be suitable for supercontinuum generation with both broad bandwidth and flat spectrum profile [18]- [23].
In this paper, we propose a novel air-core ring fiber with a high-index As 2 S 3 ring, which could better support OAM modes. The annular high-index profile of the ring fiber makes it suitable for preserving OAM modes, as the intensity profile of the OAM beams is also ring-shaped. By optimizing the structural parameters of the designed fiber to 8-µm air-core and 1-µm ring width, we could theoretically achieve flattened dispersion with < ±30 ps/nm/km variation over a 3380-nm optical bandwidth ranging from 2025 to 5405 nm. After launching a 100-fs 70-kW pulse train centered at 3400 nm into the 8-mm long designed air-core As 2 S 3 ring fiber, a >2.5 octave supercontinuum spectrum carrying the OAM 1,1 mode is generated with the wavelength from 1182 nm to 6897 nm at -30 dB of power level. These results are validated through the numerical solution of the generalized nonlinear Schrodinger equation. Figure 1(a) illustrates the concept of the OAM supercontinuum generation, in which the spectrum is broadened when a short pulse train is incident into a nonlinear transmission medium. The phenomenon is mainly due to the interaction of the nonlinear and dispersion effects. Here, we propose a novel fiber design with a low-index air core, a high-index As 2 S 3 ring, and a SiO 2 cladding, as shown in Figure 1(b). The ring fiber's annular high-index profile makes it suitable for preserving OAM modes, as the OAM beams also has a ringshaped intensity profile. The large material index-contrast among the cladding, the air core, and the ring region enables a large effective-index separation among the HE l+1,1 or EH l−1,1 mode of the same |l| family, which significantly reduces the intermodal coupling and potentially reduces the modal crosstalk. Therefore, this design choice on fiber material and structure can better support and conserve OAM modes. Moreover, the As 2 S 3 chalcogenide material, in which most of the OAM mode resides, has high nonlinear coefficient over wide transparency window, which can potentially realize efficient OAM supercontinuum generation over a wide spectral range. Besides, we choose a 125-µm cladding diameter, the same as the standard single-mode optical fiber (SMF). Because of the large index contrast, the fiber could preserve sufficient effective refraction index difference between adjacent modes, which enables mode separation and help avoid the modal crosstalk.

II. CONCEPT AND FIBER STRUCTURE
From the fabrication point of view, the material selection and fiber structure design are feasible [24] and fibers composed by silica and chalcogenide have been manufactured in practice [25]. The detail fabrication process is as the following. Firstly, the bulk chalcogenide glass should be grinded into fine powder using a ceramic mortar. This process should be in a Nitrogen environment to avoid the material oxidation. Then the chalcogenide glass powder is dissolved in ethylenediamine with >99% purity to prevent the solvent from evaporation. In the next step, the initial silica fibers are infiltrated by means of the capillary forces. Finally, the samples can be further annealed close to the transition temperature (Tg) of the chalcogenide glass in order to remove the solvent and leave only the glass layer. Noted that using high purity chalcogenide glass prepared based on the melt-quenching technique could effectively improve the glass quality in the experiments [25].
The intensity and phase distributions of the OAM 1,1 mode (OAM 1,1 =HE even 2,1 + i×HE odd 2,1 ) are depicted in Fig. 1(c), which are obtained by analytical theory and validated by full-vector finite-element-method (FEM). Since the HE even 2,1 and HE odd 2,1 eigenmodes are nearly degenerate, the walk-off between them within 8-mm length is very small. We note that the shape of the ring-like intensity distribution remains and the phase distribution of the OAM 1,1 mode has a 2π change azimuthally. For the higher-order OAM l,m modes (l ≥ 2), they can be composed by HE l+1,m even + i×HE l+1,m odd or EH l−1,m even +i×EH l−1,m odd . Here, we focus our investigation of supercontinuum generation to HE 2,1 , corresponding to the l = 1 mode.

III. FIBER PROPERTIES WITH OPTIMIZED DISPERSION
The chromatic dispersion plays an important role, in which a flat and low dispersion is significantly desirable for achieving supercontinuum generation over a wide bandwidth. The dispersion D in the units of ps/km/nm can be calculated by where c is the velocity of light in free space and n eff is the effective refractive index of the propagating mode in the designed fiber. In order to obtain flat and near-zero dispersion in the proposed air-core ring fiber, we have investigated the structure parameters including the ring width ( r) and air-core radius (r 1 ). We first change the ring width ( r) and the dispersion curve of HE 2,1 mode is shown in Figure 2(a). One can see that the smaller ring width induces a faster-changing dispersion under conditions of the same air-core radius and the dispersion curve moves up with the larger ring width. We further optimize the air-core radius (r 1 ), which has less impact on the dispersion as illustrated in Figure 2(b). The dispersion curve comes down slowly by enlarging the air-core radius. According to the above simulated results, we can obtain the desired dispersion curve by carefully choosing the structure parameters of the fiber.
Nonlinearity is another important parameter to affect the efficiency of nonlinear process. Figure 2(c) shows the nonlinear coefficient (γ ) and the dispersion from 1000 nm to 7500 nm of the HE 2,1 mode in the designed waveguide with optimized geometric parameters (r 1 = 8 µm, r = 1 µm). The structure is dispersive in the short wavelength range until 2 µm, which is mainly induced by the material dispersion. In the mid-infrared region, the fiber has small normal dispersion, with a total dispersion variation of < ±30 ps/nm/km over a 3380-nm bandwidth from 2025 nm to 5405 nm. It is obvious that the optimized structure has a smooth dispersion profile over a wide wavelength range. Moreover, the chromatic dispersion is negative up to 5300 nm, reaching a local maximum of -6.5 ps/nm/km at 2685 nm. The simulation results show that the proposed special air-core ring fiber offers γ as approximately 707 /W/km at 1.0 µm, and it decreases with the wavelength due to the increased effective mode size.

IV. SUPERCONTINUUM GENERATION
The supercontinuum generation is investigated using the generalized nonlinear Schrodinger equation (GNLSE), which considers the contributions of both the linear effects (loss and chromatic dispersion) and the nonlinear effects (selfphase modulation, stimulated Raman scattering, and selfsteepening) [26].
where α represents for the loss coefficient, ω 0 is the input pulse frequency, τ is the present time frame, and f R is the fractional contribution due to delayed Raman function R(τ ) [27]. Besides, β n means the nth-order dispersion and up to tenth order of dispersion is taken into consideration for this simulation. The nonlinear refractive index n 2 for As 2 S 3 is 3 × 10 −18 m 2 /W and we used a full-vector model to obtain the Kerr nonlinear coefficient γ in the simulation [17], [27].
In the simulation, we set the fiber loss as 1 dB/m from 1000 nm to 6000 nm, and then gradually increased to 10 dB/m at wavelength larger than 6000 nm, according to the As 2 S 3 material loss [17] and the previous experiment results of As 2 S 3 ring fiber [28]. The generated supercontinua in the designed fiber with 8-µm air-core radius and 1-µm ring width are shown in Figure 3, where the influence of various parameters including the pump center wavelength, pump peak power and pulse width have been analyzed. We obtained the material refractive indices of SiO 2 and As 2 S 3 using the Sellmeier equations in our model [29], [30].
First, a 100-fs 70-kW secant hyperbolic pulse with different center wavelength λ 0 is launched into the 8-mm fiber with different peak powers. The simulation results are shown in Figure 3(a). We pumped the fiber at 2500 nm, 3400 nm and 4000 nm, respectively, considering that 2500-nm optical parametric amplifier pump lasers are practically available [31] and the emission band of future praseodymium (Pr 3+ )-doped chalcogenide fiber lasers have broadband emission in the 3000-5000 nm wavelength region [32], [33]. We noted that the spectrum obtained with the pump peak power at 3400 nm is much broader than at 2500 nm and much flatter than that at 4000 nm, covering the wavelength range from 1.2 µm to 7.1 µm. This can be explained that the spectrum is easily extended to short wavelength range as the nonlinear coefficient (γ ) in the short wavelength range is larger than that of long wavelength range. Therefore, pumping at longer wavelength is preferable to further expand the supercontinuum to a longer wavelength region. The supercontinuum pumped at 4000 nm is not effectively broadened and becomes poorer since the pump wavelength was far away from local minimum of the chromatic dispersion. The influence of the input peak power is illustrated in Figure 3(b). It can be clearly seen that higher peak power results in larger supercontinuum bandwidth, while it could degrade the flatness and induce spectral fluctuation if the input power is too high. We choose 70-kW input pulse to achieve a tradeoff between the spectral broadness and its flatness. Moreover, symmetric broadening can be observed for the pump with 20-kW power due to self-phase modulation (SPM), whereas four-wave mixing rises in the long wavelength and the spectral broadening tends to be asymmetric at higher pump power.
Finally, Figure 3(c) illustrates the effect of the full-width at half maximum (T FWHM ) of the input pulse. We use a 70-kW short pulse as the input to the designed fiber, with different pulse durations of 50 fs, 100 fs, 200 fs, respectively. One can clearly see that there is more fluctuation in the output spectrum for an input pulse with a longer duration. The input pulse with a longer pump pulse duration has more energy and narrower spectrum when the peak power is fixed. Thus, the nonlinear effect is more pronounced, which is mainly responsible for the roughness. Moreover, the output spectrum of the 50-fs input pump pulse is narrower than the others. The reason behind is that different pulses has the same peak power, and thus shorter pulse has lower total input power and wider spectrum. As a result, shorter input pulse has a smaller power spectral density, and thus not enough power is provided at the long wavelength region besides the low γ . Additional simulation shows that one can enhance the pump power to further expand the supercontinuum range for 50-fs input pump pulse case. Figure 4 illustrates that supercontinuum generation using the 70-kW 100-fs input pulse centered at 3400 nm after the propagation of 0, 0.6, 1, 2, 4 and 8-mm fiber lengths, respectively. The result shows that the broadening of the output pulse can be obtained in only a few millimeters propagation length, which is mainly due to the strong nonlinearity and low dispersion. Here, the nonlinear length (L NL ) and dispersion 96546 VOLUME 8, 2020 length (L D ) can be expressed as where γ is the nonlinear coefficient, P 0 is the peak power of input pulse, and β represents the propagation constant.
The L NL and L D for the proposed air-core ring fiber are 8.4 × 10 −2 mm and 260 mm, respectively. The nonlinear effect is stronger than the dispersion effect, and play a leading role in the process of the spectrum broadening. Moreover, although the loss increases to 10 dB/m at wavelength larger than 6000 nm, this only results in small total insertion loss for the 8-mm length designed fiber. Thus, the spectrum in the long wavelength region holds and the shape of the two-octave supercontinuum remains. By further properly adjusting the designed fiber structure, the supercontinuum can be potentially obtained for other OAM modes. Figures 5(a) and (b) present the temporal and spectral evolutions of the corresponding supercontinuum over propagation distance. One can see that the spectrum is expanded to approximately 5000-nm wavelength range after propagating through a 6-mm designed fiber. Firstly, the spectrum broadens uniformly around the input pulse wavelength due to SPM. Then, optical wave breaking (OWB), which can also be explained as a degenerate four-wave mixing (FWM) process, leads to the generation of new spectrum components [34]. After 2-mm propagation length, the accumulated dispersion leads to the walk-off effect, as shown in Figure 5(b). After 4-mm propagation length, the spectrum stops broadening and get smoother, which is illustrated in Figure 5(a). The simulated results show that >2.5 octave supercontinuum can be generated, covering 5715-nm bandwidth from 1182 nm  to 6897 nm at -30 dB. We further compare our result with some previous researches as shown in TABLE 1.
Finally, we analyzed the coherence of the corresponding generated supercontinuum. The quantum noise of the input source can influence the coherence property of the supercontinuum spectrum [27]. A series of simulated spectra is formed and the result of the simulation is shown in the Figure 6. Here, we took random quantum noise into consideration and executed 40 times. Thanks to the normal dispersion design for the <5300 nm wavelength regime, the generated spectra are highly coherent.

V. CONCLUSION
In summary, 8-mm designed air-core As 2 S 3 ring fiber is used for OAM supercontinuum generation with a 100-fs 70-kW secant hyperbolic pump pulse. Simulation results show that a >2.5 octave supercontinuum can be generated of lightcarrying OAM 1,1 mode in the mid-infrared region, expanding from 1182 nm to 6897 nm. The generated OAM spectrum can maintain >0.99 coherence across the above 6-µm spectral range. The new type of ring fiber we designed can be a good candidate for generating supercontinuum carrying OAM modes which is applicable for various nonlinear applications. Furthermore, it is expected that the supercontinuum carrying higher-order OAM modes can be achieved with new design of the fiber.