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Photonics
, Volume 9 (1) – Jan 1, 2022

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hv photonics Article Impact of Mode-Area Dispersion on Nonlinear Pulse Propagation in Gas-Filled Anti-Resonant Hollow-Core Fiber 1 , 2 1 Ying Wan *, Md Imran Hasan and Wonkeun Chang School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore; wonkeun.chang@ntu.edu.sg Optical Sciences Group, Research School of Physics, The Australian National University, Canberra, ACT 2600, Australia; imran.hasan@anu.edu.au * Correspondence: WANY0015@e.ntu.edu.sg Abstract: We numerically investigate the effect of mode-area dispersion in a tubular-type anti- resonant hollow-core ﬁber by using a modiﬁed generalized nonlinear Schrödinger equation that takes into account the wavelength-dependent mode area in its nonlinear term. The pulse evolution dynamics with and without the effect of mode-area dispersion are compared and analyzed. We show that strong dispersion of the mode area in the proximity of the cladding wall thickness-induced resonances has a signiﬁcant impact on the soliton pulse propagation, resulting in considerable changes in the conversion efﬁciencies in nonlinear frequency mixing processes. The differences become more prominent when the pump has higher energy and is nearer to a resonance. Hence, the mode-area dispersion must be accounted for when modeling such a case. Keywords: anti-resonant hollow-core ﬁber; nonlinear optics; optical soliton 1. Introduction The anti-resonant hollow-core ﬁber (AR-HCF) is gaining growing popularity owing to Citation: Wan, Y.; Hasan, M.I.; its ability to guide light through a hollow channel [1,2]. Nonlinear optics is among the areas Chang, W. Impact of Mode-Area where AR-HCF is ﬁnding interesting applications [3–5]. The tubular-type AR-HCF, which Dispersion on Nonlinear Pulse consists of several thin-wall dielectric cladding tubes surrounding the central hollow core Propagation in Gas-Filled has a simple structure that is relatively easy to fabricate, yet achieves reasonably low loss [6]. Anti-Resonant Hollow-Core Fiber. Photonics 2022, 9, 25. https:// It offers a wide transmission window and a high power-damage threshold. It can also be doi.org/10.3390/photonics9010025 ﬁlled with gas for tunable dispersion and nonlinearity. The long light-matter interaction length in gas-ﬁlled AR-HCF offers an ideal platform for developing coherent broadband Received: 9 November 2021 light sources using nonlinear effects [7]. The light guidance in AR-HCF can be described Accepted: 28 December 2021 by the anti-resonant reﬂection model [8], where the light in the hollow core is tightly Published: 1 January 2022 conﬁned via strong reﬂection at the core-cladding interface at wavelengths that satisfy the Publisher’s Note: MDPI stays neutral anti-resonant condition. In AR-HCF, the guiding properties such as the transmission loss, with regard to jurisdictional claims in dispersion, and mode-area, exhibit smooth variation across the anti-resonant (transmission) published maps and institutional afﬁl- bands, while they change rapidly near the resonant wavelengths. Such characteristics have iations. been exploited in several recent studies for generating multi-octave-spanning supercontin- uum [9,10] and mediating nonlinear frequency down-conversion [11]. The rapid change in the dispersion near the resonance enables various combinations of phase-matched nonlin- ear frequency mixing processes to be achieved [12]. Moreover, since the locations of the Copyright: © 2022 by the authors. resonances are solely determined by the cladding wall thickness, it is possible to induce Licensee MDPI, Basel, Switzerland. emission at a desired wavelength by tailoring the cladding wall thickness [9,11]. This article is an open access article While there are a handful of studies reporting the effect of fast variation in the dis- distributed under the terms and persion proﬁle near the cladding-wall thickness-induced resonances on nonlinear pulse conditions of the Creative Commons propagation in gas-ﬁlled AR-HCF, less attention has been paid to the impact of the rapid Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ mode area change around these resonances. The effect of mode-area dispersion on nonlin- 4.0/). ear pulse propagation was ﬁrst theoretically studied in single-mode photonic band-gap Photonics 2022, 9, 25. https://doi.org/10.3390/photonics9010025 https://www.mdpi.com/journal/photonics Photonics 2022, 9, 25 2 of 15 ﬁber [13]. It turned out that the mode-area dispersion is an important factor in dictating femtosecond pulse propagation, and, therefore, it has been commonly incorporated in numerical studies involving photonic bandgap ﬁbers [14,15]. On the other hand, the effect of mode-area dispersion has not been thoroughly considered in the past numerical studies that involve nonlinear pulse propagation in gas-ﬁlled AR-HCFs. A constant mode area evaluated at the pump wavelength was assumed in most cases. However, such an assump- tion may become invalid in small-core ﬁbers, as well as when the pump is close to the resonance, due to strong mode-area dispersion. In this work, we present a numerical study on the effect of mode-area dispersion in a small-core AR-HCF together with an insightful analysis in detail. As we shall see, this becomes non-negligible when the pump of sufﬁcient energy is launched near the resonance. Such a clear impact of the mode-area dispersion in gas-ﬁlled AR-HCF has not been observed until now. 2. Modeling Pulse Propagation in Gas-Filled AR-HCF with Mode-Area Dispersion To accurately model the pulse propagation near the resonance of AR-HCF, we use a modiﬁed generalized nonlinear Schrödinger equation. We present the equation in the frequency domain, rewriting the nonlinear coefﬁcient g in a form that includes the full wavelength-dependence of the effective mode area. This is given as [13,14,16]: ¶C w w 0 0 0 ˆ e L(w)C(z, w)= ig(w) 1 + F C(z, t) R T C z, T T dT , (1) ¶z w 0 ¥ where: n n w 2 0 0 g(w) = q , (2) cn (w) A (w)A (w ) e f f e f f e f f 0 a(w) L(w) = i[b(w) b b (w w )] , (3) 0 1 0 " # 1 A (w) e f f e e FfC(z, t)g = C(z, w) = A(z, w). (4) A (w ) e f f 0 w is the angular frequency with w denoting the carrier, and z is the propagation length; g(w) is the wavelength-dependent nonlinear coefﬁcient that includes the dispersions of the effective index, n (w), and effective mode area, A (w); n and n are the linear and e f f e f f 0 2 nonlinear refractive indices, respectively; C(z, w) is the normalized form of the spectral e ˆ amplitude, A z, w , which sets the unit to be consistent within the equation [16]; L w ( ) ( ) represents the linear operator which includes the full frequency dependent wavevector b(w) and attenuation a(w); b and b are the zeroth- and ﬁrst-order Taylor series expansion 0 1 coefﬁcients of b(w) around w , which are related to the phase and group velocities at the carrier, respectively; T = t b z is then the time frame moving at the group velocity of the pulse; and R( T ) represents the Raman response function of the nonlinear medium. This term can be removed when AR-HCF is ﬁlled with atomic gas. The pulse can also be subject to self-induced photoionization. However, for the set of parameters considered in this numerical study, we can neglect this. The pulse propagation simulation is carried out by solving the modiﬁed generalized nonlinear Schrödinger equation using a split-step Fourier method in MATLAB. We consider a silica-based AR-HCF with the core diameter D = 20 m surrounded by eight cladding tubes each with inner diameter d = 7 m and wall thickness t = 570 nm. The index of silica is obtained from a Sellmeier equation [17], and it ranges between 1.435 and 1.538 in the 0.2–2.2 m wavelength region. An idealized cross-section of the ﬁber is presented in the inset of Figure 1b. Optical properties of the evacuated AR-HCF, i.e., the effective index, mode area, and conﬁnement loss for the fundamental core mode, are obtained using the ﬁnite-element method (FEM) in COMSOL Multiphysics. We use the pressure-dependent dispersion and nonlinear index of the ﬁlling gas as measured in Photonics 2022, 9, x FOR PEER REVIEW 3 of 15 1.435 and 1.538 in the 0.2–2.2 µm wavelength region. An idealized cross-section of the fiber is presented in the inset of Figure 1b. Optical properties of the evacuated AR-HCF, i.e., the effective index, mode area, and confinement loss for the fundamental core mode, are obtained using the finite-element method (FEM) in COMSOL Multiphysics. We use the pressure-dependent dispersion and nonlinear index of the filling gas as measured in Refs. [18] and [19], respectively. The blue line in Figure 1a is the group-velocity dispersion (GVD or 𝛽 , 𝜕 𝛽 (𝜔 )⁄𝜕𝜔 ) of the fundamental mode when it is filled with 20 bar xenon. The first- to fourth-order resonant bands located at 900–1400 nm, 550–630 nm, 390–420 nm, and 300–320 nm, are highlighted in yellow. The red-dashed line represents the GVD profile of an equivalent dielectric capillary having the same core size. It agrees well with that of AR-HCF in the transmission bands, especially towards the short wavelength side. The difference between the capillary and AR-HCF grows from 1650 nm onwards. The confinement loss of the AR-HCF in dB per unit length is shown in grey shade. Figure 1b presents the effective area of the fundamental core mode as a function of the wavelength. The mode area varies smoothly in the transmission windows but changes rapidly near the resonances. A large increase in the mode area in the resonant bands leads to a substantial reduction in the nonlinearity since they are inversely proportional to each other. We observe that the effective area of the fundamental core mode increases with the Photonics 2022, 9, 25 3 of 15 decreasing order of the transmission window, whereas within each transmission band, it decreases with the wavelength. A similar observation has been reported in [20]. Noting that the effective mode area in a solid-core fiber grows monotonically with wavelength, Refs. [18,19], respectively. The blue line in Figure 1a is the group-velocity dispersion (GVD we attribute this unintuitive effect in AR-HCF to the fact that the guidance originates from 2 2 or b , ¶ b(w)/¶w ) of the fundamental mode when it is ﬁlled with 20 bar xenon. The the presence of microstructured cladding. As the wavelength increases, the relative cross- ﬁrst- to fourth-order resonant bands located at 900–1400 nm, 550–630 nm, 390–420 nm, and sectional size of the AR-HCF structure with respect to the wavelength shrinks, resulting 300–320 nm, are highlighted in yellow. The red-dashed line represents the GVD proﬁle of an in the longer wavelength light experiencing a stronger effect of the waveguide. This leads equivalent dielectric capillary having the same core size. It agrees well with that of AR-HCF to tighter confinement of the mode and a smaller effective mode area at a longer wave- in the transmission bands, especially towards the short wavelength side. The difference length within a given transmission band. However, the general tendency of the mode between the capillary and AR-HCF grows from 1650 nm onwards. The conﬁnement loss of spreading over a larger area when the wavelength increases as seen in solid-core fiber is the AR-HCF in dB per unit length is shown in grey shade. Figure 1b presents the effective also in effect. Due to the above two effects competing against one another, the variation area of the fundamental core mode as a function of the wavelength. The mode area varies of the mode area across transmission bands in AR-HCF, excluding the regions around smoothly in the transmission windows but changes rapidly near the resonances. A large resonances, is relatively small (e.g., the difference is only 1.8% between the mode at 0.5 increase in the mode area in the resonant bands leads to a substantial reduction in the µm and 2 µm) nonlinearity since they are inversely proportional to each other. Figure 1. (a) Group-velocity dispersion (GVD) of the fundamental core mode calculated using the Figure 1. (a) Group-velocity dispersion (GVD) of the fundamental core mode calculated using the finite-element ﬁnite-element method (FEM) method (FEM) (blue line) (blue line) whe when n the the anti-reso anti-resonant nant hollow-co hollow-cor re fiber ( e ﬁberA (AR-HCF) R-HCF) is fi is lle ﬁlled d with 20 bar xenon. The red-dashed line is the GVD of the fundamental core mode in the dielectric capillary of the same core size. The conﬁnement loss obtained from the FEM is shown in grey shade. (b) Effective mode area calculated using FEM. Inset shows the idealized cross-section of the AR-HCF. It has eight dielectric cladding tubes of diameter d = 7 m and wall thickness t = 570 nm, surrounding the hollow core of diameter D = 20 m. We observe that the effective area of the fundamental core mode increases with the decreasing order of the transmission window, whereas within each transmission band, it decreases with the wavelength. A similar observation has been reported in [20]. Noting that the effective mode area in a solid-core ﬁber grows monotonically with wavelength, we attribute this unintuitive effect in AR-HCF to the fact that the guidance originates from the presence of microstructured cladding. As the wavelength increases, the relative cross- sectional size of the AR-HCF structure with respect to the wavelength shrinks, resulting in the longer wavelength light experiencing a stronger effect of the waveguide. This leads to tighter conﬁnement of the mode and a smaller effective mode area at a longer wavelength within a given transmission band. However, the general tendency of the mode spreading over a larger area when the wavelength increases as seen in solid-core ﬁber is also in effect. Due to the above two effects competing against one another, the variation of the mode area across transmission bands in AR-HCF, excluding the regions around resonances, is relatively small (e.g., the difference is only 1.8% between the mode at 0.5 m and 2 m). Photonics 2022, 9, x FOR PEER REVIEW 4 of 15 with 20 bar xenon. The red-dashed line is the GVD of the fundamental core mode in the dielectric capillary of the same core size. The confinement loss obtained from the FEM is shown in grey shade. (b) Effective mode area calculated using FEM. Inset shows the idealized cross-section of the AR- Photonics 2022, 9, 25 4 of 15 HCF. It has eight dielectric cladding tubes of diameter 𝑑= 7 µm and wall thickness 𝑡= 570 nm, surrounding the hollow core of diameter 𝐷= 20 µm. 3. Discussion 3. Discussion 3.1. 3.1. Pulse Pulse P Propagation ropagation a at 1430 t 1430nm nm We ﬁrst investigate the case where the femtosecond pump is at 1430 nm wavelength We first investigate the case where the femtosecond pump is at 1430 nm wavelength as as an an ex example, ample, which which isis close close to the first re to the ﬁrst resonant sonant b band and and w and within ithin the first transmission the ﬁrst transmission window window. We la . We launch unch a a 30 fs, 30 f 0.8 s, 0. J8 pulse µJ pu center lse cent edered at at 1430 14 nm 30 into nma int 5 cm-long o a 5 cm AR-HCF -long AR ﬁlled -HCF with 20 bar xenon. We simulate the pulse propagation using two different models. One filled with 20 bar xenon. We simulate the pulse propagation using two different models. accounts for the wavelength-dependent mode area (WDA model) as outlined in Equations One accounts for the wavelength-dependent mode area (WDA model) as outlined in (1–4), and the other one assumes a constant mode area for all wavelengths, i.e., wavelength- Equations (1–4), and the other one assumes a constant mode area for all wavelengths, i.e., independent mode area model (WIA model). The spectra obtained with the two models wavelength-independent mode area model (WIA model). The spectra obtained with the at the output of the ﬁber are shown in Figure 2. They are normalized with respect to the two models at the output of the fiber are shown in Figure 2. They are normalized with maximum peak power obtained with the WDA model. We note that the spectral output of respect to the maximum peak power obtained with the WDA model. We note that the the WDA model is ﬂatter, and it exhibits a spectral peak at 330 nm generated via dispersive spectral output of the WDA model is flatter, and it exhibits a spectral peak at 330 nm wave emission. This is absent in the WIA model. generated via dispersive wave emission. This is absent in the WIA model. Figure 2. Output spectra for the wavelength-dependent area (WDA, black line) and wavelength- Figure 2. Output spectra for the wavelength-dependent area (WDA, black line) and wavelength- independent area (WIA, red line) models with the femtosecond pump at 1430 nm. independent area (WIA, red line) models with the femtosecond pump at 1430 nm. To identify the origin of the differences in the two cases, we compare the spectral To identify the origin of the differences in the two cases, we compare the spectral evolutions under the two models shown in Figure 3a,d. In both models, the pump at evolutions under the two models shown in Figure 3a,d. In both models, the pump at 1430 1430 nm initially experiences weak normal dispersion. Due to strong nonlinearity coming nm initially experiences weak normal dispersion. Due to strong nonlinearity coming from from the high-density xenon, it undergoes rapid spectral broadening induced by self-phase the high-density xenon, it undergoes rapid spectral broadening induced by self-phase modulation in both models, and a large chunk of the spectrum tunnels into the anomalous modulation in both models, and a large chunk of the spectrum tunnels into the anomalous dispersion regime located above 1450 nm. The presence of the ﬁrst resonant band near the dispersion regime located above 1450 nm. The presence of the first resonant band near the pump prevents the broadening of the spectrum towards the shorter wavelength side. A pump prevents the broadening of the spectrum towards the shorter wavelength side. A similar phenomenon has also been observed in Ref. [14], where the authors referred to it as similar phenomenon has also been observed in Ref. [14], where the authors referred to it a frequency locking effect. The strong effect of third-order dispersion near the resonance as a frequency locking effect. The strong effect of third-order dispersion near the reso- decelerates further frequency shift. As a result, the spectrum broadens asymmetrically nance decelerates further frequency shift. As a result, the spectrum broadens asymmetri- towards the longer wavelength and the spectral component in the anomalous dispersion cally towards the longer wavelength and the spectral component in the anomalous dis- regime forms a soliton. In both cases, the maximum spectral broadening is achieved at persion regime forms a soliton. In both cases, the maximum spectral broadening is around 0.75 cm, where the pulse is compressed to its shortest duration. After reaching the achieved at around 0.75 cm, where the pulse is compressed to its shortest duration. After maximum compression, soliton ﬁssion accompanied by emission of a dispersive wave (DW) reaching the maximum compression, soliton fission accompanied by emission of a disper- at 330 nm is observed in the WDA model, which arises due to the phase-matching between sive wave (DW) at 330 nm is observed in the WDA model, which arises due to the phase- the soliton and linear wave in the presence of higher-order dispersion [21]. The broadened matching between the soliton and linear wave in the presence of higher-order dispersion pump spectrum overlaps with the phase-matching point initiating the onset of DW [22–24]. [21]. The broadened pump spectrum overlaps with the phase-matching point initiating Nevertheless, as shown in Figure 3d, this is absent in the case of the WIA model. Since the onset of DW [22–24]. Nevertheless, as shown in Figure 3d, this is absent in the case of the DW generation is mediated by the soliton, we extract the green highlighted spectral the WIA model. Since the DW generation is mediated by the soliton, we extract the green components in Figure 3b,e. We note that the two highlighted spectral components are both highlighted spectral components in Figure 3b,e. We note that the two highlighted spectral in the anomalous dispersion regime. However, they are centered at different wavelengths. components are both in the anomalous dispersion regime. However, they are centered at In the WDA model, the center wavelength—wavelength with the highest intensity—is at 1591 nm, while in the WIA model, this is at 1559 nm. The temporal proﬁles of the green highlighted spectra are presented in the respective insets. Namely, the peak power is 1.32 GW and the full width at half maximum (FWHM) duration is 21.7 fs in the WDA model, corresponding to the soliton number, N = 2.8. In the WIA model, the peak power is slightly higher at 1.36 GW and FWHM duration is longer at 23.5 fs, amounting to the Photonics 2022, 9, x FOR PEER REVIEW 5 of 15 different wavelengths. In the WDA model, the center wavelength—wavelength with the highest intensity—is at 1591 nm, while in the WIA model, this is at 1559 nm. The temporal profiles of the green highlighted spectra are presented in the respective insets. Namely, the peak power is 1.32 GW and the full width at half maximum (FWHM) duration is 21.7 fs in the WDA model, corresponding to the soliton number, 𝑁= 2.8 . In the WIA model, the peak power is slightly higher at 1.36 GW and FWHM duration is longer at 23.5 fs, amounting to the soliton number, 𝑁= 3.4 . Figure 3c,f shows the DW and degenerate four-wave mixing (DFWM) dephasings for the two solitons using their respective center wavelengths of the green-highlighted spectral band as the pump. The DW and DFWM Photonics 2022, 9, 25 5 of 15 phase-matching conditions are given by [25]: 𝛾𝑃 sol ∆𝛽 (𝜔 ) =𝛽 (𝜔 ) −𝛽 (𝜔 ) +𝛽 (𝜔 )(𝜔− 𝜔 ) + , (5) soliton number, N = 3.4 DW . Figure 3c,f shows the solDW an d degenerate sol sol four-wave mixing (DFWM) dephasings for the two solitons using their respective center wavelengths of ∆𝛽 =𝛽 (𝜔 ) +𝛽 (𝜔 ) −2𝛽𝜔 + 2𝛾𝑃. (6) the green-highlighted spectral band DFWMas the pump. s The i DW and pDFWM phase-matching conditions are given by [25]: In Equation (5), 𝜔 is the center angular frequency of the soliton and 𝑃 is the sol sol peak power at its maximum compression point. In Equation (6 gP), 𝜔 , 𝜔 , and 𝜔 are the sol Db (w) = b(w) b(w ) + b (w )(w w ) + , (5) DW sol 1 sol sol angular frequencies of the pump, idler, and signal in the DFWM process, and we use 𝜔 = 𝜔 in plotting Figure 3e,f. 𝑃 is the peak power of the pump soliton as indicated in the sol Db = b(w ) + b(w ) 2b w + 2gP. (6) DFWM s i p insets of Figure 3b,e. Figure 3. First row: Spectral evolutions of a 1430 nm pump propagating in a 5 cm-long AR-HCF ﬁlled with 20 bar xenon calculated using (a) WDA and (d) WIA models. Second row: Spectra at z = 0.75 cm calculated using (b) WDA and (e) WIA models. The green highlighted band in each corresponds to a higher-order soliton with the duration and peak power as indicated in the inset. Last row: DW (blue) and DFWM (red) dephasings of the highlighted soliton for (c) WDA and (f) WIA models. PM: phase-matching. Photonics 2022, 9, 25 6 of 15 In Equation (5), w is the center angular frequency of the soliton and P is the peak sol sol power at its maximum compression point. In Equation (6), w , w , and w are the angular p s frequencies of the pump, idler, and signal in the DFWM process, and we use w = w in sol plotting Figure 3e,f. P is the peak power of the pump soliton as indicated in the insets of Figure 3b,e. In Figure 3c, the DW emission at 330 nm for the WDA model is accurately predicted. Some other phase-matching points, such as those located at 1.06 m, 1.18 m, and 1.23 m in the dephasing plot are also manifested by radiations that appear at these wavelengths in Figure 3a. Radiation induced by the rest of the phase-matching points is either too weak to be observed or being overlapped by the main pulse spectral. Observations are similar in the WIA model shown in Figure 3d,e. However, we note that the phase-matching at around 330 nm is not reﬂected in the spectral evolution shown in Figure 3d. We attribute this to weaker spectral overlap between the soliton and DW phase-matching point as shown in the subsequent analysis. The differences between the dephasings in both models are small, which is because the inclusion of the mode-area dispersion only slightly affects the nonlinear correction term, which has only a small contribution to the dephasing in both DW emission and DFWM process. Figure 4a,b presents the spectrograms at different positions in AR-HCF with the WDA and WIA models. They are obtained using the cross-correlation frequency-resolved optical gating with a 35 fs Gaussian pulse as the gate. The pulse initially undergoes SPM-induced symmetric spectral broadening. After propagating for a short distance, the blue edge of the spectrum enters the highly dispersive resonant region, where the pulse experiences rapid temporal broadening and a decrease in the peak intensity. Moreover, the loss and mode area also increase substantially in the resonant band. The combination of these Photonics 2022, 9, x FOR PEER REVIEW 7 of 15 effects prohibits the spectrum’s further extension toward the shorter wavelength, making it highly asymmetric. Figure 4. Simulated spectrograms at different positions in the ﬁber in (a) the WDA and (b) WIA Figure 4. Simulated spectrograms at different positions in the fiber in (a) the WDA and (b) WIA models. They are obtained using the cross-correlation frequency-resolved optical gating technique models. They are obtained using the cross-correlation frequency-resolved optical gating technique with a Gaussian gate pulse of duration 35 fs. The yellow highlighted regions correspond to the with a Gaussian gate pulse of duration 35 fs. The yellow highlighted regions correspond to the clad- cladding tube wall thickness-induced resonant bands of the AR-HCF. ding tube wall thickness-induced resonant bands of the AR-HCF. The reason for the shorter compressed duration and stronger spectral broadening in the WDA model despite the smaller soliton number is due to the wavelength-dependent nonlinear coefficient, 𝛾(𝜔) . Figure 5 shows 𝛾(𝜔) in the spectral region of our interest, and this is what is used in the pulse propagation simulation in the WDA model. The ver- tical red-dotted line marks the pump at 1430 nm, and the horizontal red-dashed line indi- cates the nonlinear coefficient at the pump wavelength, which is the value that is used in the WIA model. The soliton appears centered in the red shaded area, and hence the non- linear coefficient experienced by the soliton in the WIA model is underestimated. The higher nonlinear parameter shifts the soliton in the WDA model further to the longer wavelength (center wavelength of 1591 nm at 0.75 cm) than the WIA model (center wave- length of 1559 nm at 0.75 cm) due to larger nonlinear phase shift induced by the stronger nonlinearity. As seen from the GVD profile in Figure 1(a), further away from the zero- dispersion wavelength at 1450 nm alleviates the impact of higher-order dispersion which deteriorates the soliton effect compression [26], resulting in a shorter compressed duration for the soliton in the WDA model that supports a wider spectral broadening, as shown in the second column in Figure 4a,b. Moreover, a higher nonlinear coefficient also enables more efficient energy transfer via the DFWM process [27] to the intermediate wavelength at 1.06 µm in the WDA model, as evident in Figure 3a,d. Photonics 2022, 9, x FOR PEER REVIEW 7 of 15 Photonics 2022, 9, 25 7 of 15 Figure 4. Simulated spectrograms at different positions in the fiber in (a) the WDA and (b) WIA At 0.75 cm, the pulse reaches the maximum compression point, as indicated by the models. They are obtained using the cross-correlation frequency-resolved optical gating technique black dashed lines in Figure 3a,d. In the case of the WDA model, the spectrum broadens with a Gaussian gate pulse of duration 35 fs. The yellow highlighted regions correspond to the clad- signiﬁcantly and overcomes the high-loss in the resonant region (highlighted in yellow in ding tube wall thickness-induced resonant bands of the AR-HCF. Figure 4) and seeding sufﬁcient photons at the DW phase-matching point at 330 nm. In the case of the WIA model, the spectral broadening is weaker, and its tail does not reach the phase-matching point in the ultraviolet region as shown in Figure 4b middle panel, thus The reason for the shorter compressed duration and stronger spectral broadening in no emission is observed at 330 nm. A bigger spectral broadening is achieved in the WDA the WDA model despite the smaller soliton number is due to the wavelength-dependent model than in the WIA model despite the lower soliton number. nonlinear coefficient, 𝛾(𝜔) . Figure 5 shows 𝛾(𝜔) in the spectral region of our interest, The reason for the shorter compressed duration and stronger spectral broadening in and this is what is used in the pulse propagation simulation in the WDA model. The ver- the WDA model despite the smaller soliton number is due to the wavelength-dependent tical red-dotted line marks the pump at 1430 nm, and the horizontal red-dashed line indi- nonlinear coefﬁcient, g(w). Figure 5 shows g(w) in the spectral region of our interest, and cates the nonlinear coefficient at the pump wavelength, which is the value that is used in this is what is used in the pulse propagation simulation in the WDA model. The vertical red-dotted line marks the pump at 1430 nm, and the horizontal red-dashed line indicates the WIA model. The soliton appears centered in the red shaded area, and hence the non- the nonlinear coefﬁcient at the pump wavelength, which is the value that is used in the WIA linear coefficient experienced by the soliton in the WIA model is underestimated. The model. The soliton appears centered in the red shaded area, and hence the nonlinear coefﬁ- higher nonlinear parameter shifts the soliton in the WDA model further to the longer cient experienced by the soliton in the WIA model is underestimated. The higher nonlinear wavelength (center wavelength of 1591 nm at 0.75 cm) than the WIA model (center wave- parameter shifts the soliton in the WDA model further to the longer wavelength (center length of 1559 nm at 0.75 cm) due to larger nonlinear phase shift induced by the stronger wavelength of 1591 nm at 0.75 cm) than the WIA model (center wavelength of 1559 nm at nonline 0.75 cm) ardue ity. As seen from to larger nonlinear the phase GVD shift profile in Fig induced by u the re str 1(a), onger furthe nonlinearity r away. from the As zero- seen from the GVD proﬁle in Figure 1a, further away from the zero-dispersion wavelength dispersion wavelength at 1450 nm alleviates the impact of higher-order dispersion which at 1450 nm alleviates the impact of higher-order dispersion which deteriorates the soliton deteriorates the soliton effect compression [26], resulting in a shorter compressed duration effect compression [26], resulting in a shorter compressed duration for the soliton in the for the soliton in the WDA model that supports a wider spectral broadening, as shown in WDA model that supports a wider spectral broadening, as shown in the second column the second column in Figure 4a,b. Moreover, a higher nonlinear coefficient also enables in Figure 4a,b. Moreover, a higher nonlinear coefﬁcient also enables more efﬁcient energy more efficient energy transfer via the DFWM process [27] to the intermediate wavelength transfer via the DFWM process [27] to the intermediate wavelength at 1.06 m in the WDA model, as evident in Figure 3a,d. at 1.06 µm in the WDA model, as evident in Figure 3a,d. Figure 5. Nonlinear coefﬁcient proﬁle across the spectrum. Vertical red- and black-dotted lines indicate locations of the pump (1430 nm) and soliton in the WDA model, respectively. The horizontal red-dashed line is the nonlinear coefﬁcient at 1430 nm. The red shaded area is the nonlinear coefﬁcient where the soliton forms in the WDA model. The coefﬁcient values are higher in this region than than assumed in the WIA model. Figure 6a,b shows the color density plot of the spectrum at the output of the 5 cm-long AR-HCF for varying pump pulse energies in the WDA and WIA models, respectively. The two results are indistinguishable at low energy levels (below 350 nJ), indicating that the effect of mode-area dispersion is small in this regime. As the pump energy increases, notable differences appear, implying growing importance of the mode-area dispersion. We also notice the threshold pump energy required for the onset of DW in the WIA model is higher (850 nJ) than the WDA model (740 nJ). Photonics 2022, 9, x FOR PEER REVIEW 8 of 15 Figure 5. Nonlinear coefficient profile across the spectrum. Vertical red- and black-dotted lines in- dicate locations of the pump (1430 nm) and soliton in the WDA model, respectively. The horizontal red-dashed line is the nonlinear coefficient at 1430 nm. The red shaded area is the nonlinear coeffi- cient where the soliton forms in the WDA model. The coefficient values are higher in this region than than assumed in the WIA model. Photonics 2022, 9, x FOR PEER REVIEW 8 of 15 Figure 6a,b shows the color density plot of the spectrum at the output of the 5 cm- long AR-HCF for varying pump pulse energies in the WDA and WIA models, respec- Figure 5. Nonlinear coefficient profile across the spectrum. Vertical red- and black-dotted lines in- tively. The two results are indistinguishable at low energy levels (below 350 nJ), indicating dicate locations of the pump (1430 nm) and soliton in the WDA model, respectively. The horizontal that the e red-dashed line is the ffect of mode-are nonlinear coeff a dispe icient atr 1430 sion nm. The is small in red shaded area this regime is the. nonline As the pump en ar coeffi- ergy in- cient where the soliton forms in the WDA model. The coefficient values are higher in this region creases, notable differences appear, implying growing importance of the mode-area dis- than than assumed in the WIA model. persion. We also notice the threshold pump energy required for the onset of DW in the WIA model is higher (850 nJ) than the WDA model (740 nJ). Figure 6a,b shows the color density plot of the spectrum at the output of the 5 cm- long AR-HCF for varying pump pulse energies in the WDA and WIA models, respec- tively. The two results are indistinguishable at low energy levels (below 350 nJ), indicating that the effect of mode-area dispersion is small in this regime. As the pump energy in- creases, notable differences appear, implying growing importance of the mode-area dis- Photonics 2022, 9, 25 8 of 15 persion. We also notice the threshold pump energy required for the onset of DW in the WIA model is higher (850 nJ) than the WDA model (740 nJ). Figure 6. Color density plots of the spectrum at the output of the 5-cm long AR-HCF with the 1430 nm pump at different pump pulse energies in the (a) WDA and (b) WIA models. The spectral inten- sity is normalized to the maximum in the WDA model. Figure Figure 6. 6. Color Color density density plot plots of the s ospectr f the spectrum at the ou um at the output of the tput o 5-cmflong the 5-cm long AR-HCF with the 143 AR-HCF with the 1430 nm 0 nm pump at different pump pulse energies in the (a) WDA and (b) WIA models. The spectral inten- pump at different pump pulse energies in the (a) WDA and (b) WIA models. The spectral intensity is 3.2. Pulse Propagation at 800 nm normalized sity is normaliz to the ed to maximum the maximum in the WDA mod in the WDA model. el. We carry out another study with the pump now at 800 nm to observe the effect of 3.2. Pulse Propagation at 800 nm mode-area d 3.2. Pulse Propaga ispersion tion at 80 whe 0 nm n the system is pumped in the second transmission window. We carry out another study with the pump now at 800 nm to observe the effect of The other system parameters are kept the same as before. Figure 7 shows the spectral out- We carry out another study with the pump now at 800 nm to observe the effect of mode-area dispersion when the system is pumped in the second transmission window. mode-area dispersion when the system is pumped in the second transmission window. puts obtained with the two models. Enhanced emissions at around 1300, 1430, 1620, and The other system parameters are kept the same as before. Figure 7 shows the spectral The other system parameters are kept the same as before. Figure 7 shows the spectral out- outputs obtained with the two models. Enhanced emissions at around 1300, 1430, 1620, 1840 nm are observed in the WDA model. and 1840 nm are observed in the WDA model. puts obtained with the two models. Enhanced emissions at around 1300, 1430, 1620, and 1840 nm are observed in the WDA model. Figure 7. Output spectrum for the WDA (black line) and WIA (red line) models when the same Figure 7. Output spectrum for the WDA (black line) and WIA (red line) models when the same Figure 7. Output spectrum for the WDA (black line) and WIA (red line) models when the same system is pumped at 800 nm. system is pumped at 800 nm. system is pumped at 800 nm. Unlike in the previous case where there is a broad spectral region of anomalous Unlike in the previous case where there is a broad spectral region of anomalous dis- dispersion close to the pump that can facilitate the formation of solitons, the pump is persion close to the pump that can facilitate the formation of solitons, the pump is Unlike in the previous case where there is a broad spectral region of anomalous dis- bounded by resonant bands on both sides at 630 and 900 nm. Soliton can be formed in the bounded by resonant bands on both sides at 630 and 900 nm. Soliton can be formed in the persion close to the pump that can facilitate the formation of solitons, the pump is spectral region between 800 and 900 nm, i.e., within the anomalous dispersion regime of bounded by resonant bands on both sides at 630 and 900 nm. Soliton can be formed in the the second transmission window. Outside this range, the dispersion and loss become very high, severely perturbing the pulse and suppressing the spectral advance into these regions. Figure 8a,d shows the color density plots of the spectral evolutions of the 800 nm pump in the two models. A soliton appears in both the WDA and WIA models with different central wavelengths of 889 and 895 nm, respectively at z = 1.65 cm, as indicated in Figure 8a,d. Figure 8b,e shows the spectra at z = 1.65 cm for the two models. The part of the spectrum that forms the soliton is highlighted in green. In the time domain, this part translates to a pulse with the peak power of 170 MW and FWHM duration of 49.6 fs amounting to Photonics 2022, 9, x FOR PEER REVIEW 9 of 15 spectral region between 800 and 900 nm, i.e., within the anomalous dispersion regime of the second transmission window. Outside this range, the dispersion and loss become very high, severely perturbing the pulse and suppressing the spectral advance into these re- gions. Figure 8a,d shows the color density plots of the spectral evolutions of the 800 nm pump in the two models. A soliton appears in both the WDA and WIA models with dif- ferent central wavelengths of 889 and 895 nm, respectively at z = 1.65 cm, as indicated in Figure 8a,d. Figure 8b,e shows the spectra at z = 1.65 cm for the two models. The part of the spectrum that forms the soliton is highlighted in green. In the time domain, this part translates to a pulse with the peak power of 170 MW and FWHM duration of 49.6 fs Photonics 2022, 9, 25 9 of 15 amounting to a higher-order soliton with N = 4.64 in the case of the WDA model, and peak power of 160 MW and FWHM duration of 40.5 fs amounting to N = 3.54 soliton in the WIA model. Figure 8c,f shows the DW and DFWM dephasing induced by the soliton high- a higher-order soliton with N = 4.64 in the case of the WDA model, and peak power of lighted in Figure 8b,e as the pump. Multiple DW and DFWM phase-matching wave- 160 MW and FWHM duration of 40.5 fs amounting to N = 3.54 soliton in the WIA model. lengths are achieved in both models. Some of them can be identified in the spectral evo- Figure 8c,f shows the DW and DFWM dephasing induced by the soliton highlighted in lutions in Figure 8a,d, such as the discrete narrow radiation bands in the first resonant Figure 8b,e as the pump. Multiple DW and DFWM phase-matching wavelengths are region as well as the radiation at 1.68 𝜇 m, as indicated by the arrows. Their spectral loca- achieved in both models. Some of them can be identiﬁed in the spectral evolutions in tions match well with those predicted in the dephasing plots in Figure 8c,f. Other radia- Figure 8a,d, such as the discrete narrow radiation bands in the ﬁrst resonant region as well tions such as the band at around 2 µm are induced through four-wave mixing as we shall as the radiation at 1.68 m, as indicated by the arrows. Their spectral locations match well with see later those. predicted in the dephasing plots in Figure 8c,f. Other radiations such as the band at around 2 m are induced through four-wave mixing as we shall see later. Figure 8. Top row: Color density plots of the spectral evolutions of the 800 nm pump propagating in the 5 cm-long AR-HCF ﬁlled with 20 bar xenon. The simulation results are obtained with the (a) WDA and (d) WIA models. Middle row: The pulse spectra at z = 1.65 cm for the (b) WDA and (e) WIA models. The green highlighted parts correspond to higher-order solitons with the peak powers and FWHM duration as indicated in the insets. Bottom row: DW (blue) and DFWM (red) dephasing plots for the green highlighted pulse for the (c) WDA and (f) WIA models. Photonics 2022, 9, x FOR PEER REVIEW 10 of 15 Figure 8. Top row: Color density plots of the spectral evolutions of the 800 nm pump propagating in the 5 cm-long AR-HCF filled with 20 bar xenon. The simulation results are obtained with the (a) WDA and (d) WIA models. Middle row: The pulse spectra at z = 1.65 cm for the (b) WDA and (e) WIA models. The green highlighted parts correspond to higher-order solitons with the peak powers and FWHM duration as indicated in the insets. Bottom row: DW (blue) and DFWM (red) dephasing plots for the green highlighted pulse for the (c) WDA and (f) WIA models. We plot the spectrogram evolutions for the two cases using the same approach as in Figure 4. These are presented in Figure 9. In both the WDA and WIA models, the pulse initially experiences self-phase modulation induced spectral broadening. At z = 0.85 cm, the red-edge of the spectrum reaches the wavelength of around 1.17 µm, which is in the first resonant band (wide yellow area), while the blue edge of spectrum continues to move to the shorter wavelength, passing through the second resonant band (narrow yellow Photonics 2022, 9, 25 10 of 15 area), and eventually appears in the third transmission window. At the same time, weak radiation emerges at 2 µm in both cases. At z = 1.05 cm, the red edge of the spectrum is pushed further into the first resonant We plot the spectrogram evolutions for the two cases using the same approach as in band and more energy is transferred to the 1.9–2.2 µm region. As it will be shown in the Figur subse e 4 q.uent section, this r These are presented adiat in ion is c Figura eused by 9. In both a fo the ur-wave mix WDA and ing WIA process in models, duc the ed by the pulse initially pump p experiences hotons in tself-phase he first reson modulation ant band at induced around spectral 1.17 µm br. At oadening. z = 1.65 At cm z , t = h 0.85 e sol cm, iton the red-edge of the spectrum reaches the wavelength of around 1.17 m, which is in the indicated by the arrows in the last column in Figure 9 generates a new wavelength com- ﬁrst resonant band (wide yellow area), while the blue edge of spectrum continues to move ponent centered at 1.68 µm through the DFWM process. The newly generated band at 1.68 to the shorter wavelength, passing through the second resonant band (narrow yellow area), µm overlaps with the pump soliton in the time domain in both models, as shown in the and eventually appears in the third transmission window. At the same time, weak radiation last columns in Figure 9a,b. Similarly, the 2 µm radiation also overlaps temporally with emerges at 2 m in both cases. the pump located in the first resonant band in both models. Figure 9. Spectrograms at three different positions in the fiber were calculated using (a) WDA and Figure 9. Spectrograms at three different positions in the ﬁber were calculated using (a) WDA and (b) WIA models. (b) WIA models. At Due to the z = 1.05 cm, complex evolutionary the red edge of the dyn spectr am um ics is in the pushed presence of sev further intoethe re perturb ﬁrst resonant ation by band resonant and b mor ane ds, t ener he pha gy is s transferr e-matchin edg co to the ndit1.9–2.2 ions for t m he mult region. iple r As a itdiwill ation b be a shown nds in t inhe the firsubsequent st transmission w section, indow, this radiation such as the is 2 caused 𝜇𝑚 radiation, c by a four-wave annot be exp mixing lpr icitly ocess determ induced ined. by We veri the pump fy our previ photonsoin us the assumpti ﬁrst resonant on that th band ese r at adiat around ions are 1.17 gene m.rated v At z =ia DF 1.65 WM pro cm, the - soliton indicated by the arrows in the last column in Figure 9 generates a new wavelength cesses from the pump photons in the wavelength range between 1170 and 1180 nm. We component centered at 1.68 m through the DFWM process. The newly generated band at 1.68 m overlaps with the pump soliton in the time domain in both models, as shown in the last columns in Figure 9a,b. Similarly, the 2 m radiation also overlaps temporally with the pump located in the ﬁrst resonant band in both models. Due to the complex evolutionary dynamics in the presence of severe perturbation by resonant bands, the phase-matching conditions for the multiple radiation bands in the ﬁrst transmission window, such as the 2 m radiation, cannot be explicitly determined. We verify our previous assumption that these radiations are generated via DFWM processes from the pump photons in the wavelength range between 1170 and 1180 nm. We estimate the actual peak power of the soliton to be within 0.2 to 15 MW. Based on the relation given in Equation (6), we calculate the phase-matching points by scanning through different values of pump photon wavelengths and soliton peak powers within these speciﬁed ranges. Since there may be multiple phase-matching wavelengths due to the rapid variation in the dispersion near the resonances, we select only the phase-matching point with the longest wavelength. The results are plotted in Figure 10. Photonics 2022, 9, x FOR PEER REVIEW 11 of 15 estimate the actual peak power of the soliton to be within 0.2 to 15 MW. Based on the relation given in Equation (6), we calculate the phase-matching points by scanning through different values of pump photon wavelengths and soliton peak powers within these specified ranges. Since there may be multiple phase-matching wavelengths due to the rapid variation in the dispersion near the resonances, we select only the phase-match- Photonics 2022, 9, 25 11 of 15 ing point with the longest wavelength. The results are plotted in Figure 10. Figure 10. DFWM-induced phase-matching wavelength (the longest idler wavelength) for varying Figure 10. DFWM-induced phase-matching wavelength (the longest idler wavelength) for varying peak peak power (0.2–15 MW) power (0.2–15 MW) in the in nonlinear the nonlinear correctioncorrection term term in Equation (6). in Equation The wavelength (6). The of the wavelength of the pump photon is indicated next to each line. pump photon is indicated next to each line. As we can see in Figure 10, the phase-matching wavelength increases with decreasing As we can see in Figure 10, the phase-matching wavelength increases with decreas- pump photon wavelength and increasing peak power P . Since the pump is in the ﬁrst ing pump photon wavelength and increasing peak power 𝑃 . Since the pump is in the first resonant band where the dispersion varies rapidly, even a small difference in its wavelength results in a large shift in the phase-matching wavelength. As the pump moves to the longer resonant band where the dispersion varies rapidly, even a small difference in its wave- wavelength, deeper into the ﬁrst resonant band, the phase-matching wavelength changes length results in a large shift in the phase-matching wavelength. As the pump moves to more rapidly. The radiation located in the spectral region between 1.9 m and 2.2 m the longer wavelength, deeper into the first resonant band, the phase-matching wave- is highlighted in pink. Therefore, we can conclude; the pump that is responsible for the length changes more rapidly. The radiation located in the spectral region between 1.9 𝜇𝑚 generation of the abovementioned DFWM-induced radiation bands is located between 1173 and 2. and21177 𝜇𝑚 nm, is hig conﬁrming hlighted our in p earlier ink. There assertion fore that , we thecradiation an conclude; the p at around 2 um m p is that is responsi- generated via DFWM with the pump at around 1.17 m. We verify that no other pump ble for the generation of the abovementioned DFWM-induced radiation bands is located wavelengths can achieve the radiation at 2 m via DFWM. We also observe some weaker between 1173 and 1177 nm, confirming our earlier assertion that the radiation at around radiations in the spectral region between 1.4 and 1.6 m in both models as shown in 2 µm is generated via DFWM with the pump at around 1.17 µm. We verify that no other Figure 8a,d. This is highlighted in cyan in Figure 10. Similarly, we can conclude that these pump wavelengths can achieve the radiation at 2 µm via DFWM. We also observe some radiations are also caused by DFWM, with the pump photon slightly longer at around 1177 to 1179 nm. As the pulse propagates, its spectrum continues to broaden towards the weaker radiations in the spectral region between 1.4 and 1.6 µm in both models as shown red side, expanding deeper into the ﬁrst resonant band at around 1177 to 1179 nm and seed in Figure 8a,d. This is highlighted in cyan in Figure 10. Similarly, we can conclude that series of DFWM-induced phase matching radiations ranges between 1.4 and 1.6 m. As these radiations are also caused by DFWM, with the pump photon slightly longer at the wavelength of the pump photon increases, the phase-matching wavelength shifts to around 1177 to 1179 nm. As the pulse propagates, its spectrum continues to broaden to- a shorter wavelength. However, due to exponentially increasing loss and substantially reduced nonlinearity around the resonance, further spectral broadening into the longer wards the red side, expanding deeper into the first resonant band at around 1177 to 1179 wavelength is largely impeded and DFWM-induced conversion becomes inefﬁcient. This nm and seed series of DFWM-induced phase matching radiations ranges between 1.4 and also explains why the 2 m radiation band appears earlier and the radiation bands with 1.6 µm. As the wavelength of the pump photon increases, the phase-matching wavelength shorter wavelengths emerge later in the propagation with weaker intensities. shifts to a shorter wavelength. However, due to exponentially increasing loss and sub- By comparing Figure 9a,b, we observe that the 2 m radiation at z = 1.05 cm in the stantially reduced nonlinearity around the resonance, further spectral broadening into the WDA model is weaker than that in the WIA model. On the contrary, the 1.68 m radiation that appears at z = 1.65 cm in the WDA model is stronger than that in the WIA model. This longer wavelength is largely impeded and DFWM-induced conversion becomes ineffi- is also shown in Figure 8a,d. We investigate the reason for the difference by looking at cient. This also explains why the 2 µm radiation band appears earlier and the radiation the nonlinear coefﬁcient across the spectrum in Figure 11. Note that this is the same plot bands with shorter wavelengths emerge later in the propagation with weaker intensities. as shown in Figure 5, but with the constant nonlinear coefﬁcient used in the WIA model By comparing Figure 9a,b, we observe that the 2 µm radiation at z = 1.05 cm in the evaluated at the new pump wavelength of 800 nm. To facilitate our analysis, we separate the nonlinear coefﬁcient proﬁle into three WDA model is weaker than that in the WIA model. On the contrary, the 1.68 µm radiation sections as indicated in Figure 11. Each of the ﬁrst two resonant bands sits in Sections 1 that appears at z = 1.65 cm in the WDA model is stronger than that in the WIA model. This and 2, respectively. The third and fourth resonant bands are in Section 3. In Section 1, the is also shown in Figure 8a,d. We investigate the reason for the difference by looking at the red (green) shaded area indicates the region where the nonlinear coefﬁcient in the WDA nonlinear coefficient across the spectrum in Figure 11. Note that this is the same plot as Photonics 2022, 9, 25 12 of 15 model is higher (lower) than that in the WIA model. The solitons highlighted in Figure 8b,e are located in the red shaded region while the DFWM pump photons at around 1175 nm are in the green shaded region. Since the nonlinear coefﬁcient at 1175 nm in the WDA model is below that of the WIA model as shown in Figure 11, there is a less efﬁcient conversion of the pump to the signal and idler in the DFWM process, resulting in weaker radiation at 2 m compared to that in the WIA model. This is evident from the spectrograms in the Photonics 2022, 9, x FOR PEER REVIEW 12 of 15 middle column of Figure 9a,b. The nonlinear coefﬁcient at 889 nm, which the soliton is centered at in the WDA model, is higher than that perceived by the soliton in the WIA model. Hence, a larger amount of energy is transferred to the 1.68 m band in the WDA model by the soliton through a more efﬁcient DFWM process, as seen in the last column shown in Figure 5, but with the constant nonlinear coefficient used in the WIA model of Figure 9a,b. evaluated at the new pump wavelength of 800 nm. Figure Figure 11. 11. Nonlinear Nonlinear coef coeffi ﬁcient cient versu versusswavelength. wavelength. Vertic Vertical alr red- ed- and and b black-do lack-dotted lines in tted lines indicate dicatethe the positions of the source wavelength (800 nm) and the pump wavelength for the DFWM (1.175 um), positions of the source wavelength (800 nm) and the pump wavelength for the DFWM (1.175 um), respectively. Section 1 includes the first resonant band, Section 2 includes the second resonant band, respectively. Section 1 includes the ﬁrst resonant band, Section 2 includes the second resonant band, and Section 3 covers the third and fourth resonant bands. The vertical grey-dotted line marks an- and Section 3 covers the third and fourth resonant bands. The vertical grey-dotted line marks another other wavelength with the nonlinear coefficient value that is equal to that at the pump, which sep- wavelength with the nonlinear coefﬁcient value that is equal to that at the pump, which separates arates Sections 2 and 3. The horizontal red-dashed line indicates the nonlinear coefficient evaluated Sections 2 and 3. The horizontal red-dashed line indicates the nonlinear coefﬁcient evaluated at 800 at 800 nm. Red (green) shade marks the region in Section 3 where the nonlinear coefficient in the nm. Red (green) shade marks the region in Section 3 where the nonlinear coefﬁcient in the WDA WDA model is higher (lower) than that assumed in the WIA model. model is higher (lower) than that assumed in the WIA model. To facilitate our analysis, we separate the nonlinear coefficient profile into three sec- Figure 12a–c shows the simulation results when the mode-area dispersion is assumed tions as indicated in Figure 11. Each of the first two resonant bands sits in Sections 1 and only in either Section 1, Section 2, or Section 3, respectively while keeping the coefﬁcient 2, respectively. The third and fourth resonant bands are in Section 3. In Section 1, the red constant with the value evaluated at 800 nm in the other Sections. This is to separately (green) shaded area indicates the region where the nonlinear coefficient in the WDA study the impact of mode-area dispersion for different resonant bands. We ﬁnd that when model is higher (lower) than that in the WIA model. The solitons highlighted in Figure we consider the mode-area dispersion only in Section 1, the spectral evolution is almost 8b,e are located in the red shaded region while the DFWM pump photons at around 1175 identical to that of the WIA model as shown in Figure 8a,d. From this, we can conclude nm are in the green shaded region. Since the nonlinear coefficient at 1175 nm in the WDA that the mode-area dispersion in the third and fourth resonant bands has a negligible model is below that of the WIA model as shown in Figure 11, there is a less efficient con- effect on the pulse propagation dynamics. This is because they are far from the pump version of the pump to the signal and idler in the DFWM process, resulting in weaker wavelength at 800 nm and, therefore, have a negligible effect on the main pulse. The radiation at 2 µm compared to that in the WIA model. This is evident from the spectro- inclusion of the mode-area dispersion around the ﬁrst or second resonant bands affects grams in the middle column of Figure 9a,b. The nonlinear coefficient at 889 nm, which the the pulse propagation dynamics in the ﬁrst transmission window (above 1400 nm) and soliton is centered at in the WDA model, is higher than that perceived by the soliton in second transmission window (550–900 nm), as can be seen from the comparison between the WIA model. Hence, a larger amount of energy is transferred to the 1.68 µm band in Figure 12a–c. Some of the notable differences lie in the ﬁrst transmission window. While the WDA model by the soliton through a more efficient DFWM process, as seen in the last all three cases show relatively strong radiation at around 2 m, other radiations in the column of Figure 9a,b. wavelength between 1.3 to 1.6 m are totally different. We attribute these differences to Figure 12a–c shows the simulation results when the mode-area dispersion is assumed the change in DFWM phase-matching conditions and conversion efﬁciency when using only in either Section 1, Section 2, or Section 3, respectively while keeping the coefficient different nonlinear coefﬁcient proﬁles. constant with the value evaluated at 800 nm in the other Sections. This is to separately Figure 13a,b shows the output spectra of the 800 nm pump after propagating the 5-cm study the impact of mode-area dispersion for different resonant bands. We find that when long xenon-ﬁlled AR-HCF with different pump energies for the WDA and WIA models, we consider the mode-area dispersion only in Section 1, the spectral evolution is almost respectively. Similar to the case of pumping at 1430 nm, the differences between the WDA identical to that of the WIA model as shown in Figure 8a,d. From this, we can conclude and WIA models are small at low energy levels (below 400 nJ). As the energy increases, that the mode-area dispersion in the third and fourth resonant bands has a negligible ef- the differences become more pronounced. The radiations in the ﬁrst transmission window fect on the pulse propagation dynamics. This is because they are far from the pump wave- mediated by the DFWM process are completely different between the WDA and WIA models length at at 800 all ener nm and, ther gy levels, efore, have indicating that a neglig the pr ib ocess le effe is cthighly on the ma sensitive in puto lse. the The mode-ar inclusion ea of the mode-area dispersion around the first or second resonant bands affects the pulse propagation dynamics in the first transmission window (above 1400 nm) and second transmission window (550–900 nm), as can be seen from the comparison between Figure 12a–c. Some of the notable differences lie in the first transmission window. While all three cases show relatively strong radiation at around 2 µm, other radiations in the wavelength between 1.3 to 1.6 µm are totally different. We attribute these differences to the change in DFWM phase-matching conditions and conversion efficiency when using different non- linear coefficient profiles. Photonics 2022, 9, x FOR PEER REVIEW 13 of 15 Photonics 2022, 9, 25 13 of 15 dispersion. This is expected since the spectral components in the ﬁrst transmission window Photonics 2022, 9, x FOR PEER REVIEW 13 of 15 are induced by the pump photons in the vicinity of the ﬁrst resonant band, where the mode-area undergoes a dramatic change in the WDA model. Figure 12. (a–c) show the spectral evolutions obtained with the mode-area dispersion included only in Sections 1, 2, and 3, respectively, while the other sections are assumed to have a constant nonlin- ear coefficient evaluated at 800 nm. Figure 13a,b shows the output spectra of the 800 nm pump after propagating the 5- cm long xenon-filled AR-HCF with different pump energies for the WDA and WIA mod- els, respectively. Similar to the case of pumping at 1430 nm, the differences between the WDA and WIA models are small at low energy levels (below 400 nJ). As the energy in- creases, the differences become more pronounced. The radiations in the first transmission window mediated by the DFWM process are completely different between the WDA and WIA models at all energy levels, indicating that the process is highly sensitive to the mode-area d Figure 12. (a–ci)spersion show the. This is expected spectral evolutions obtained w since the spectra ith the mode-area l components i dispersion in n the f clir uded only st trans- Figure 12. (a–c) show the spectral evolutions obtained with the mode-area dispersion included in Sections 1, 2, and 3, respectively, while the other sections are assumed to have a constant nonlin- mission window are induced by the pump photons in the vicinity of the first resonant only in Sections 1–3, respectively, while the other sections are assumed to have a constant nonlinear ear coefficient evaluated at 800 nm. band, where the mode-area undergoes a dramatic change in the WDA model. coefﬁcient evaluated at 800 nm. Figure 13a,b shows the output spectra of the 800 nm pump after propagating the 5- cm long xenon-filled AR-HCF with different pump energies for the WDA and WIA mod- els, respectively. Similar to the case of pumping at 1430 nm, the differences between the WDA and WIA models are small at low energy levels (below 400 nJ). As the energy in- creases, the differences become more pronounced. The radiations in the first transmission window mediated by the DFWM process are completely different between the WDA and WIA models at all energy levels, indicating that the process is highly sensitive to the mode-area dispersion. This is expected since the spectral components in the first trans- mission window are induced by the pump photons in the vicinity of the first resonant band, where the mode-area undergoes a dramatic change in the WDA model. Figure 13. Output spectra when the system is pumped with an 800 nm pulse of different input en- Figure 13. Output spectra when the system is pumped with an 800 nm pulse of different input ergies in the (a) WDA and (b) WIA models. The plots are normalized to the maximum spectral energies in the (a) WDA and (b) WIA models. The plots are normalized to the maximum spectral intensity recorded in the WDA model. intensity recorded in the WDA model. 4. 4. Con Conclusions clusions In this work, we numerically investigate the effect of mode-area dispersion on pulse In this work, we numerically investigate the effect of mode-area dispersion on pulse prp opagation ropagation dynamics dynamics in in a xenon-ﬁlled a xenon-filled AR-HCF AR-HC.F W . W e study e studthe y thcases e caseof s o pumping f pumping atat two two dif different ferent wavelengths, wavelengths, one one at at 1430 1430 nm nm and and the the other other at 800 at 800 nm, close to nm, close to resonant resonbands, ant bands, to understand how the dynamics differ when the pump is launched in different transmission to understand how the dynamics differ when the pump is launched in different transmis- windows in the AR-HCF. We compare the results obtained using two different models; sion windows in the AR-HCF. We compare the results obtained using two different mod- one els; on thate that take takes into s into account accothe unt the wav wavelength-dependent elength-dependent mode are mode area, and a, and the the other other that that Figure 13. Output spectra when the system is pumped with an 800 nm pulse of different input en- assumes a constant mode area evaluated at the pump wavelength. For the same set assumes a constant mode area evaluated at the pump wavelength. For the same set of ergies in the (a) WDA and (b) WIA models. The plots are normalized to the maximum spectral of simulation parameters, a strong dispersive wave is observed at 330 nm in the WDA simulation parameters, a strong dispersive wave is observed at 330 nm in the WDA model intensity recorded in the WDA model. model while this is not visible in the WIA model when the pump is at 1430 nm. We while this is not visible in the WIA model when the pump is at 1430 nm. We identify that identify that the differences are coming from the nonlinear parameter perceived by the the differences are coming from the nonlinear parameter perceived by the soliton that 4. Conclusions soliton that induces the dispersive wave. In the WDA model, the soliton experiences a In this work, we numerically investigate the effect of mode-area dispersion on pulse smaller effective mode area and thus higher nonlinearity, which causes larger nonlinear propagation dynamics in a xenon-filled AR-HCF. We study the cases of pumping at two phases shifts, moving the soliton to a longer wavelength further from the zero-dispersion different wavelengths, one at 1430 nm and the other at 800 nm, close to resonant bands, wavelength. This suppresses the effect of higher-order dispersion and results in achieving to understand how the dynamics differ when the pump is launched in different transmis- a better pulse compression, which facilitates the spectral broadening that extends further into sion w the ultraviolet indows in the AR-HCF. We comp region at 330 nm. This enables are the re the sults obtained onset of DW emission using two differ with the ent mod- pump ener els; on gy that e that take is lower s into thanacc what ount the wav is required elength-d in the WIA epenmodel. dent mode are When the a, and the pump is other that at 800 assumes a constant mode area evaluated at the pump wavelength. For the same set of simulation parameters, a strong dispersive wave is observed at 330 nm in the WDA model while this is not visible in the WIA model when the pump is at 1430 nm. We identify that the differences are coming from the nonlinear parameter perceived by the soliton that Photonics 2022, 9, 25 14 of 15 nm, the WDA and WIA models exhibit signiﬁcantly different spectral outputs in the ﬁrst transmission window. We identiﬁed that the radiation which appears at 1.68 m in both models is generated through a DFWM process with the pump photon at 891 nm in the WDA model and 895 nm in the WIA model. The rest of the radiation bands in the ﬁrst transmission window are mediated by DFWM processes with the pump photons in the ﬁrst resonant region. The efﬁciency of the DFWM process and thus the strength of the DFWM- induced radiations is directly related to the nonlinear coefﬁcient at the wavelength of the corresponding pump photons. In the WIA model, the nonlinear coefﬁcient at the DFWM pump of 895 nm is underestimated while that in the ﬁrst resonant band is overestimated with respect to the WDA model. Therefore, we observe lower intensity radiation at 1.68 m and stronger radiation at 2 m in the WIA model as compared to the WDA model. Our results suggest that the inclusion of mode-area dispersion is important when investigating ultrafast dynamics near the resonant bands of AR-HCF and at high pump energy levels, such as in the case of generating resonance-enhanced supercontinuum [9,10]. Author Contributions: Conceptualization, W.C.; methodology, W.C. and Y.W.; software, W.C., Y.W. and M.I.H.; validation, Y.W.; formal analysis, Y.W.; investigation, Y.W.; resources, W.C.; data curation, W.C. and Y.W.; writing—original draft preparation, Y.W.; writing—review and editing, W.C. and M.I.H.; visualization, Y.W.; supervision, W.C.; project administration, W.C.; funding acquisition, W.C. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Ministry of Education—Singapore, AcRF Tier 1 RG135/20. 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Photonics – Multidisciplinary Digital Publishing Institute

**Published: ** Jan 1, 2022

**Keywords: **anti-resonant hollow-core fiber; nonlinear optics; optical soliton

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