He-Filled Stretched Hollow-Core Fiber

• He-filled stretched hollow-core fibers are optical waveguides made by elongating glass capillaries and filling them with high-purity helium, which enables efficient nonlinear frequency conversion.
• They allow independent tuning of modal and gas dispersion, achieving precise phase matching for high-harmonic and deep-ultraviolet generation with optimized pressure control.
• Comprehensive modeling, simulation, and experimental validation confirm high conversion efficiency, low-loss performance, and robust phase-matching in these fibers.

A helium-filled stretched hollow-core fiber (He-HCF) is an optical waveguide formed by elongating a glass capillary to achieve a thin-walled, large-bore geometry, subsequently filled with high-purity helium at controlled pressure. This architecture leverages modal confinement within the hollow core and the tailored phase-matching capability provided by gas dispersion, enabling efficient nonlinear optical processes such as high-harmonic and deep-ultraviolet (DUV) generation. The combination of longitudinal stretching and gas filling allows independent tuning of geometric (modal) and material (gas) contributions to propagation constant and dispersion, underpinning phase-matched nonlinear conversion schemes with high efficiency and spectral purity (Bache et al., 2018, Forbes et al., 12 Nov 2025).

1. Capillary Fiber Model, Modal Structure, and Anti-Resonance Loss

The propagation behavior in hollow-core fibers is governed by the Marcatili–Schmeltzer (MS) capillary waveguide model, modified to include realistic glass wall thickness and gas filling. For a capillary of core radius $a$ and wall thickness $t$:

• Vacuum MS Model: The fundamental (EH$_{11}$ or HE$_{01}$) mode satisfies

$$n_{\mathrm{eff,MS}}^2(\lambda) = 1 - \frac{u_{mn}^2}{a^2 k_0^2}$$

where $u_{11} \approx 2.405$ is the first zero of $J_0(x)$, and $k_0=2\pi/\lambda$.

• Anti-Resonant Loss: Real fibers have finite wall thickness. Loss and anti-resonance features arise due to imperfect reflection:

$$\alpha_{\text{BR}}(\lambda) \approx [1 - |r|^2]\frac{u_{mn}}{2a^2 k_0}$$

where $|r|^2$ is the thin-wall reflection coefficient (see Eqs. (6–8) in (Bache et al., 2018)).

• Single Parameter Model: Loss and dispersion, including anti-resonance peaks, are accurately reproduced by supplementing the MS model with Lorentzian terms at each wall resonance, all scaled by a single fit parameter $t$0, adjustable per design.
• Impedance Approach: The boundary is more accurately replaced by effective impedances $t$1, $t$2, giving

$t$3

$t$4

When gas is introduced, the unity in the formula for $t$5 is replaced by $t$6: $t$7 where $t$8 is the known dispersive increment for helium. Under typical conditions (1–10 bar), gas dispersion dominates far from resonance, while wall-induced loss is nearly independent of gas pressure to leading order.

2. Fabrication, Stretching, and Geometry Control

Fabrication begins with a commercial glass capillary (e.g., Polymicro TSP100665, ID $t$9m, OD $_{11}$0m). The capillary is heated and stretched following protocols such as Nagy et al. (2008), with the following outcomes (Forbes et al., 12 Nov 2025):

• Stretching: The capillary is elongated to lengths of $_{11}$1 m or greater, with a stretching ratio of $_{11}$2–$_{11}$3. This reduces wall thickness from $_{11}$45 μm to $_{11}$5–$_{11}$6m, while the core diameter remains nearly unchanged at $_{11}$7m.
• Geometric Effects: Thin walls reduce both bend and confinement loss (measured $_{11}$8 dB/m at $_{11}$9 nm), and shift anti-resonance wavelengths as $_{01}$0. Stretching enables engineering the locations of low-loss/minima and the overall dispersion.
• Adiabatic Taper: Variations in $_{01}$1 are assumed adiabatic, preserving modal structure. Theoretical propagation, dispersion, and loss at each axial position $_{01}$2 are treated locally, then path-integrated for total transmission and net group-delay dispersion (GVD).

Parameter	Typical Value (Unstretched)	After Stretching
Inner Diameter ($_{01}$3)	$_{01}$4m	$_{01}$5m
Wall Thickness ($_{01}$6)	$_{01}$7m	$_{01}$8–$_{01}$9m
Length ($$n_{\mathrm{eff,MS}}^2(\lambda) = 1 - \frac{u_{mn}^2}{a^2 k_0^2}$$0)	$$n_{\mathrm{eff,MS}}^2(\lambda) = 1 - \frac{u_{mn}^2}{a^2 k_0^2}$$1 m	$$n_{\mathrm{eff,MS}}^2(\lambda) = 1 - \frac{u_{mn}^2}{a^2 k_0^2}$$2 m (after $$n_{\mathrm{eff,MS}}^2(\lambda) = 1 - \frac{u_{mn}^2}{a^2 k_0^2}$$3–$$n_{\mathrm{eff,MS}}^2(\lambda) = 1 - \frac{u_{mn}^2}{a^2 k_0^2}$$4 stretching)

3. Helium Gas Filling and Phase-Matching Physics

The fiber core is filled with high-purity helium (99.999%) at a pressure optimized for nonlinear phase-matching; an example optimal value is $$n_{\mathrm{eff,MS}}^2(\lambda) = 1 - \frac{u_{mn}^2}{a^2 k_0^2}$$5. Helium plays a dual role:

• Dispersion Tailoring: Helium’s refractive index, $$n_{\mathrm{eff,MS}}^2(\lambda) = 1 - \frac{u_{mn}^2}{a^2 k_0^2}$$6, is a linear function of pressure, contributing a positive dispersion term. This is essential for compensating the (negative) waveguide dispersion, enabling momentum conservation for high-order harmonic or DUV generation. The net phase-mismatch for four-wave mixing (FWM) is:

$$n_{\mathrm{eff,MS}}^2(\lambda) = 1 - \frac{u_{mn}^2}{a^2 k_0^2}$$7

$$n_{\mathrm{eff,MS}}^2(\lambda) = 1 - \frac{u_{mn}^2}{a^2 k_0^2}$$8

where modal and gas terms are explicitly defined by wavelength, pressure, and geometry (Forbes et al., 12 Nov 2025).

• Suppression of Nonlinear Loss: Helium’s high ionization potential ($$n_{\mathrm{eff,MS}}^2(\lambda) = 1 - \frac{u_{mn}^2}{a^2 k_0^2}$$9 eV) strongly suppresses nonlinear absorption processes (multiphoton ionization), particularly at DUV–VUV wavelengths ($u_{11} \approx 2.405$0 nm), allowing high energy throughput without plasma-induced damage.

A plausible implication is that alternative gases with lower IP would introduce stronger nonlinear absorption and thus degrade conversion efficiency and beam quality in analogous processes.

4. Nonlinear Optical Applications and Phase-Matched Conversion

He-filled stretched HCFs enable phase-matched nonlinear optics with engineered dispersion and minimized loss. One major application is four-wave difference-frequency mixing (FWDFM) for DUV generation:

• FWDFM Process: With drive wavelengths at $u_{11} \approx 2.405$1 nm (pump) and $u_{11} \approx 2.405$2 nm (seed), their interaction yields $u_{11} \approx 2.405$3 nm (fifth harmonic of Yb), obeying $u_{11} \approx 2.405$4.
• Phase-Matching: Achieved by tuning helium pressure so that modal and gas-induced terms combine to yield $u_{11} \approx 2.405$5. At $u_{11} \approx 2.405$6 bar for the specified geometry, simulations and experiment confirm phase matching for 206 nm.
• Measured Performance: At $u_{11} \approx 2.405$7 kHz, 206 nm, output pulses with $u_{11} \approx 2.405$8J energy, $u_{11} \approx 2.405$9 fs duration (autocorrelation deconvolved), and $J_0(x)$0 nm bandwidth (transform limit $J_0(x)$1 fs) are achieved. The conversion efficiency from 343 nm pump to 206 nm DUV reaches $J_0(x)$230%, with fiber transmission $J_0(x)$3 (mode-matched to EH$J_0(x)$4 mode), and total average DUV output power of $J_0(x)$5 W.
• Power Scaling and Roll-Over: DUV output is linear in seed energy, quadratic in pump energy, with pump depletion and self-focusing in windows limiting scaling above 58 $J_0(x)$6J/pulse.

Advantages over photonic crystal/kagomé fibers include simpler modal structure (no annular resonances), higher damage thresholds, and absence of solid-core-like absorption or color-center damage at wavelengths below 200 nm. Compared to crystals, no two-photon absorption or color-center formation occurs (Forbes et al., 12 Nov 2025).

5. Modeling, Simulation, and Characterization

Analytical and numerical models underpin both the design and performance optimization:

• “Poor-Man’s” Model: Dispersion and loss, including anti-resonances, are predicted by the capillary plus Lorentzian or impedance-perturbative approach, using a single empirical parameter $J_0(x)$7 matched to a reference FEM simulation or measurement (Bache et al., 2018). Gas-filled, stretched geometry is handled by replacing core unity with $J_0(x)$8, and integrating local properties over the fiber length $J_0(x)$9.
• Nonlinear FWDFM Simulation: Simple analytic expressions (e.g., $k_0=2\pi/\lambda$0) predict scaling. Full numerical models (e.g., in Luna.jl) incorporate GVD, SPM, XPM, ionization, and pump depletion, accurately reproducing phase-matching curves, power scaling, and roll-over thresholds.
• Validation: FEM (COMSOL, JCMwave) on reference z-sections is used to extract $k_0=2\pi/\lambda$1 for calibration. Experimental cut-back and interferometric measurements retrieve loss and GVD versus wavelength and pressure; measured and predicted anti-resonance minima match well after $k_0=2\pi/\lambda$2 adjustment. Pump–probe or spectral interferometry directly measures group delay dispersion, confirming integration-of-local-$k_0=2\pi/\lambda$3 predictions (Bache et al., 2018).

6. Limitations, Practical Considerations, and Applications

Although the modeling approaches provide rapid parameter scanning and reliable phase-matching prediction, several constraints and limitations apply:

• Model Accuracy: Perturbative and Lorentzian expansions for anti-resonance become inaccurate near loss peaks due to coupling to the cladding continuum. Single-parameter fit requires recalibration for each distinct geometry (wall tubes, nesting, etc.).
• Adiabaticity: The assumption of slow (adiabatic) variation in $k_0=2\pi/\lambda$4 may fail for abrupt tapers, possibly inducing mode coupling and localized loss spikes.
• Nonlinearity: Gas-phase nonlinearities (Kerr, ionization) and higher-order dispersion are not included in the baseline “poor-man’s” model.
• Experimental Issues: Mode matching ($k_0=2\pi/\lambda$590% coupling to EH$k_0=2\pi/\lambda$6) provides robust single-mode operation, but self-focusing, window damage, or levels of purity can restrict achievable energies. Routine use demands a high degree of geometric and pressure control.

He-filled stretched HCFs are deployed for DUV/VUV pulse generation, time-resolved spectroscopy, ARPES, ultrafast micromachining, and tunable DUV sources (via seed wavelength tuning between 208–234 nm) (Forbes et al., 12 Nov 2025). These fibers enable a regime of efficient, high-repetition-rate, high-peak-power nonlinear optics not accessible in bulk or conventional waveguide platforms.

7. Summary Table of Key Parameters and Features

Feature/Parameter	Typical Value/Effect	Role/Significance
Inner diameter ($k_0=2\pi/\lambda$7)	$k_0=2\pi/\lambda$8100 $k_0=2\pi/\lambda$9m	Determines modal area, A$$\alpha_{\text{BR}}(\lambda) \approx [1 - |r|^2]\frac{u_{mn}}{2a^2 k_0}$$0
Wall thickness ($$\alpha_{\text{BR}}(\lambda) \approx [1 - |r|^2]\frac{u_{mn}}{2a^2 k_0}$$1)	$$\alpha_{\text{BR}}(\lambda) \approx [1 - |r|^2]\frac{u_{mn}}{2a^2 k_0}$$2–$$\alpha_{\text{BR}}(\lambda) \approx [1 - |r|^2]\frac{u_{mn}}{2a^2 k_0}$$3m (after stretch)	Sets anti-resonance, loss minima
Fiber length ($$\alpha_{\text{BR}}(\lambda) \approx [1 - |r|^2]\frac{u_{mn}}{2a^2 k_0}$$4)	$$\alpha_{\text{BR}}(\lambda) \approx [1 - |r|^2]\frac{u_{mn}}{2a^2 k_0}$$5 m	Nonlinear gain, quadratic efficiency
He pressure ($$\alpha_{\text{BR}}(\lambda) \approx [1 - |r|^2]\frac{u_{mn}}{2a^2 k_0}$$6)	$$\alpha_{\text{BR}}(\lambda) \approx [1 - |r|^2]\frac{u_{mn}}{2a^2 k_0}$$7 bar	Phase-matching, dominant GVD source
Loss ($$\alpha_{\text{BR}}(\lambda) \approx [1 - |r|^2]\frac{u_{mn}}{2a^2 k_0}$$8)	$$\alpha_{\text{BR}}(\lambda) \approx [1 - |r|^2]\frac{u_{mn}}{2a^2 k_0}$$9 dB/m at 343 nm	DUV transmission, scaling to meter lengths
Pulse energy (DUV)	$|r|^2$016 $|r|^2$1J (at 206 nm, 100 kHz)	High-efficiency frequency conversion
Conversion efficiency	$|r|^2$230% (pump to DUV)	Validated by analytic and numeric models

These characteristic parameters define the viability of He-filled stretched HCFs for current high-intensity and high-repetition-rate DUV/VUV photonics applications. The architecture continues to be optimized through a combination of capillary analytic modeling, full-field simulation, and direct experimental benchmarking (Bache et al., 2018, Forbes et al., 12 Nov 2025).