# Impact of Mode-Area Dispersion on Nonlinear Pulse Propagation in Gas-Filled Anti-Resonant Hollow-Core Fiber

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## Abstract

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## 1. Introduction

## 2. Modeling Pulse Propagation in Gas-Filled AR-HCF with Mode-Area Dispersion

## 3. Discussion

#### 3.1. Pulse Propagation at 1430 nm

#### 3.2. Pulse Propagation at 800 nm

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Pryamikov, A.D.; Biriukov, A.S.; Kosolapov, A.F.; Plotnichenko, V.G.; Semjonov, S.L.; Dianov, E.M. Demonstration of a waveguide regime for a silica hollow-core microstructured optical fiber with a negative curvature of the core boundary in the spectral region > 35 μm. Opt. Express
**2011**, 19, 1441–1448. [Google Scholar] [CrossRef] [PubMed] - Yu, F.; Wadsworth, W.J.; Knight, J. Low loss silica hollow core fibers for 3–4 μm spectral region. Opt. Express
**2012**, 20, 11153–11158. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Adamu, A.I.; Habib, S.; Petersen, C.R.; Lopez, J.E.A.; Zhou, B.; Schülzgen, A.; Bache, M.; Amezcua-Correa, R.; Bang, O.; Markos, C. Deep-UV to Mid-IR supercontinuum generation driven by Mid-IR ultrashort pulses in a gas-filled hollow-core fiber. Sci. Rep.
**2019**, 9, 4446. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Köttig, F.; Schade, D.; Koehler, J.; Russell, P.S.J.; Tani, F. Efficient single-cycle pulse compression of an ytterbium fiber laser at 10 MHz repetition rate. Opt. Express
**2020**, 28, 9099–9110. [Google Scholar] [CrossRef] [Green Version] - Fu, J.; Chen, Y.; Huang, Z.; Yu, F.; Wu, D.; Pan, J.; Zhang, C.; Wang, D.; Pang, M.; Leng, Y. Photoionization-induced broadband dispersive wave generated in an AR-filled hollow-core photonic crystal fiber. Crystals
**2021**, 11, 180. [Google Scholar] [CrossRef] - Abu Hassan, M.R.; Yu, F.; Wadsworth, W.; Knight, J. Cavity-based mid-IR fiber gas laser pumped by a diode laser. Optica
**2016**, 3, 218–221. [Google Scholar] [CrossRef] - Russell, P.; Hoelzer, P.; Chang, W.; Abdolvand, A.; Travers, J. Hollow-core photonic crystal fibres for gas-based nonlinear optics. Nat. Photon.
**2014**, 8, 278–286. [Google Scholar] [CrossRef] - Duguay, M.A.; Kokubun, Y.; Koch, T.L.; Pfeiffer, L. Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures. Appl. Phys. Lett.
**1986**, 49, 13–15. [Google Scholar] [CrossRef] [Green Version] - Sollapur, R.; Kartashov, D.; Zürch, M.; Hoffmann, A.; Grigorova, T.; Sauer, G.; Hartung, A.; Schwuchow, A.; Bierlich, J.; Kobelke, J.; et al. Resonance-enhanced multi-octave supercontinuum generation in antiresonant hollow-core fibers. Light. Sci. Appl.
**2017**, 6, e17124. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Qi, X.; Schaarschmidt, K.; Chemnitz, M.; Schmidt, M.A. Essentials of resonance-enhanced soliton-based supercontinuum generation. Opt. Express
**2020**, 28, 2557. [Google Scholar] [CrossRef] [PubMed] - Gavara, T.; Hasan, I.; Abu Hassan, M.R.; Deng, A.; Chang, W. Band-edge mediated frequency down-conversion in a gas-filled anti-resonant hollow-core fiber. Opt. Lett.
**2020**, 45, 6815–6818. [Google Scholar] [CrossRef] - Tani, F.; Köttig, F.; Novoa, D.; Keding, R.; Russell, P.S. Effect of anti-crossings with cladding resonances on ultrafast nonlinear dynamics in gas-filled photonic crystal fibers. Photon. Res.
**2018**, 6, 84–88. [Google Scholar] [CrossRef] - Laegsgaard, J. Mode profile dispersion in the generalised nonlinear Schrödinger equation. Opt. Express
**2007**, 15, 16110–16123. [Google Scholar] [CrossRef] - Vanvincq, O.; Kudlinski, A.; Bétourné, A.; Quiquempois, Y.; Bouwmans, G. Extreme deceleration of the soliton self-frequency shift by the third-order dispersion in solid-core photonic bandgap fibers. J. Opt. Soc. Am. B
**2010**, 27, 2328–2335. [Google Scholar] [CrossRef] - Pureur, V.; Bétourné, A.; Bouwmans, G.; Bigot, L.; Kudlinski, A.; Delplace, K.; Le Rouge, A.; Quiquempois, Y.; Douay, M. Overview on solid core photonic bandgap fibers. Fiber Integr. Opt.
**2009**, 28, 27–50. [Google Scholar] [CrossRef] - Travers, J.C.; Frosz, M.H.; Dudley, J.M. Nonlinear fibre optics overview. In Supercontinuum Generation in Optical Fibers; Dudley, J.M., Taylor, J.R., Eds.; Cambridge University Press: Cambridge, UK, 2010; pp. 32–51. [Google Scholar]
- Malitson, I.H. Interspecimen comparison of the refractive index of fused silica. J. Opt. Soc. Am.
**1965**, 55, 1205–1209. [Google Scholar] [CrossRef] - Börzsönyi, A.; Heiner, Z.; Kalashnikov, M.P.; Kovács, A.P.; Osvay, K. Dispersion measurement of inert gases and gas mixtures at 800 nm. Appl. Opt.
**2008**, 47, 4856–4863. [Google Scholar] [CrossRef] - Lehmeier, H.; Leupacher, W.; Penzkofer, A. Nonresonant third order hyperpolarizability of rare gases and N2 determined by third harmonic generation. Opt. Commun.
**1985**, 56, 67–72. [Google Scholar] [CrossRef] [Green Version] - Hasan, I.; Akhmediev, N.; Chang, W. Empirical Formulae for Dispersion and Effective Mode Area in Hollow-Core Antiresonant Fibers. J. Light. Technol.
**2018**, 36, 4060–4065. Available online: http://jlt.osa.org/abstract.cfm?URI=jlt-36-18-4060 (accessed on 15 December 2019). [CrossRef] - Husakou, A.V.; Herrmann, J. Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers. Phys. Rev. Lett.
**2001**, 87, 203901. [Google Scholar] [CrossRef] [Green Version] - Wai, A.; Menyuk, C.R.; Lee, Y.C.; Chen, H.H. Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers. Opt. Lett.
**1986**, 11, 464–466. [Google Scholar] [CrossRef] - Wai, A.; Chen, H.H.; Lee, Y.C. Radiations by ‘‘solitons’’ at the zero group-dispersion wavelength of single-mode optical fibers. Phys. Rev. A
**1990**, 41, 426–439. [Google Scholar] [CrossRef] - Karpman, V.I. Radiation by solitons due to higher-order dispersion. Phys. Rev. E
**1993**, 47, 2073–2082. [Google Scholar] [CrossRef] - Agrawal, G. Nonlinear Fiber Optics, 4th ed.; Academic Press: Cambridge, MA, USA, 2007. [Google Scholar]
- Wan, Y.; Chang, W. Effect of decreasing pressure on soliton self-compression in higher-order modes of a gas-filled capillary. Opt. Express
**2021**, 29, 7070–7083. [Google Scholar] [CrossRef] - Yang, Y.; Xu, Z.; Jiang, X.; He, Y.; Guo, X.; Zhang, Y.; Qiu, C.; Su, Y. High-efficiency and broadband four-wave mixing in a silicon-graphene strip waveguide with a windowed silica top layer. Photon Res.
**2018**, 6, 965–970. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**(

**a**) Group-velocity dispersion (GVD) of the fundamental core mode calculated using the finite-element method (FEM) (blue line) when the anti-resonant hollow-core fiber (AR-HCF) is filled 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 $d=7$ µm and wall thickness $t=570$ nm, surrounding the hollow core of diameter $D=20$ µm.

**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.

**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.

**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 with a Gaussian gate pulse of duration 35 fs. The yellow highlighted regions correspond to the cladding tube wall thickness-induced resonant bands of the AR-HCF.

**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 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 intensity is normalized to the maximum in the WDA model.

**Figure 7.**Output spectrum for the WDA (black line) and WIA (red line) models when the same system is pumped at 800 nm.

**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.

**Figure 9.**Spectrograms at three different positions in the fiber were calculated using (

**a**) WDA and (

**b**) WIA models.

**Figure 10.**DFWM-induced phase-matching wavelength (the longest idler wavelength) for varying peak power (0.2–15 MW) in the nonlinear correction term in Equation (6). The wavelength of the pump photon is indicated next to each line.

**Figure 11.**Nonlinear coefficient versus wavelength. Vertical red- and black-dotted lines indicate the 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, and Section 3 covers the third and fourth resonant bands. The vertical grey-dotted line marks another wavelength with the nonlinear coefficient value that is equal to that at the pump, which separates Section 2 and Section 3. The horizontal red-dashed line indicates the nonlinear coefficient evaluated at 800 nm. Red (green) shade marks the region in Section 3 where the nonlinear coefficient in the WDA model is higher (lower) than that assumed in the WIA model.

**Figure 13.**Output spectra when the system is pumped with an 800 nm pulse of different input energies in the (

**a**) WDA and (

**b**) WIA models. The plots are normalized to the maximum spectral intensity recorded in the WDA model.

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**MDPI and ACS Style**

Wan, Y.; Hasan, M.I.; Chang, W.
Impact of Mode-Area Dispersion on Nonlinear Pulse Propagation in Gas-Filled Anti-Resonant Hollow-Core Fiber. *Photonics* **2022**, *9*, 25.
https://doi.org/10.3390/photonics9010025

**AMA Style**

Wan Y, Hasan MI, Chang W.
Impact of Mode-Area Dispersion on Nonlinear Pulse Propagation in Gas-Filled Anti-Resonant Hollow-Core Fiber. *Photonics*. 2022; 9(1):25.
https://doi.org/10.3390/photonics9010025

**Chicago/Turabian Style**

Wan, Ying, Md Imran Hasan, and Wonkeun Chang.
2022. "Impact of Mode-Area Dispersion on Nonlinear Pulse Propagation in Gas-Filled Anti-Resonant Hollow-Core Fiber" *Photonics* 9, no. 1: 25.
https://doi.org/10.3390/photonics9010025