---
title: He-Filled Stretched Hollow-Core Fiber
url: https://www.emergentmind.com/topics/he-filled-stretched-hollow-core-fiber
type: topic
---

# He-Filled Stretched Hollow-Core Fiber

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 [1806.10416][2511.09522].

## 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 [1806.10416]).

- **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 $f_{\mathrm{FEM}} \sim 10^{-2} \ldots 10^{-3}$, adjustable per design.

- **Impedance Approach:** The boundary is more accurately replaced by effective impedances $Z_{\mathrm{TE}}$, $Y_{\mathrm{TM}}$, giving
  \[
  n_{\mathrm{eff}}(\lambda) = 1 - \frac{u_{mn}^2}{2a^2 k_0^2} - \frac{u_{mn}^2}{2a^3k_0^3}\Im\left[\frac{1}{2}(Z_{\mathrm{TE}}+Y_{\mathrm{TM}})\right]
  \]
  \[
  \alpha(\lambda) = f_{\mathrm{FEM}} \frac{2u_{mn}^2}{a^3 k_0^2}\Re\left[\frac{1}{2}(Z_{\mathrm{TE}}+Y_{\mathrm{TM}})\right]
  \]

When gas is introduced, the unity in the formula for $n_{\mathrm{eff,MS}}^2$ is replaced by $n_{\mathrm{gas}}^2(\lambda,p,T)$:
\[
n_{\mathrm{gas}}^2(\lambda,p,T) = 1 + \delta(\lambda)\frac{p}{p_0}\frac{T_0}{T}
\]
where $\delta(\lambda)$ 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 $\approx 100\ \mu$m, OD $\approx 165\ \mu$m). The capillary is heated and stretched following protocols such as Nagy et al. (2008), with the following outcomes [2511.09522]:

- **Stretching:** The capillary is elongated to lengths of $0.63$ m or greater, with a stretching ratio of $2$–$3\times$. This reduces wall thickness from $\sim$5 μm to $0.5$–$1\ \mu$m, while the core diameter remains nearly unchanged at $100\ \mu$m.
- **Geometric Effects:** Thin walls reduce both bend and confinement loss (measured $<0.1$ dB/m at $343$ nm), and shift anti-resonance wavelengths as $\lambda_{AR}\propto t$. Stretching enables engineering the locations of low-loss/minima and the overall dispersion.
- **Adiabatic Taper:** Variations in $a(z), t(z)$ are assumed adiabatic, preserving modal structure. Theoretical propagation, dispersion, and loss at each axial position $z$ are treated locally, then path-integrated for total transmission and net group-delay dispersion (GVD).

| Parameter               | Typical Value (Unstretched) | After Stretching               |
|-------------------------|-----------------------------|-------------------------------|
| Inner Diameter ($a$)    | $100\ \mu$m                 | $100\ \mu$m                   |
| Wall Thickness ($t$)    | $5\ \mu$m                   | $0.5$–$1\ \mu$m               |
| Length ($L$)            | $0.3$ m                     | $0.63$ m (after $2$–$3\times$ 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 $P=3.08\ \mathrm{bar}$. Helium plays a dual role:

- **Dispersion Tailoring:** Helium’s refractive index, $n_{\mathrm{gas}}(\omega,P)$, 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:
  \[
  \Delta k \equiv 2\beta(\omega_p) - \beta(\omega_s) - \beta(\omega_{\text{out}})
  \]
  \[
  \Delta k(P) = P \cdot \Delta k_{\text{gas}} - \Delta k_{\text{mode}}
  \]
  where modal and gas terms are explicitly defined by wavelength, pressure, and geometry [2511.09522].
- **Suppression of Nonlinear Loss:** Helium’s high ionization potential ($24.6$ eV) strongly suppresses nonlinear absorption processes (multiphoton ionization), particularly at DUV–VUV wavelengths ($\lambda \gtrsim 206$ 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 $\lambda_p=343$ nm (pump) and $\lambda_s=1030$ nm (seed), their interaction yields $\lambda_{\text{out}}=206$ nm (fifth harmonic of Yb), obeying $2\omega_p-\omega_s=\omega_{\text{out}}$.
- **Phase-Matching:** Achieved by tuning helium pressure so that modal and gas-induced terms combine to yield $\Delta k=0$. At $P=3.08$ bar for the specified geometry, simulations and experiment confirm phase matching for 206 nm.
- **Measured Performance:** At $f_{\text{rep}}=100$ kHz, 206 nm, output pulses with $>16\,\mu$J energy, $96$ fs duration (autocorrelation deconvolved), and $0.73$ nm bandwidth (transform limit $\approx85$ fs) are achieved. The conversion efficiency from 343 nm pump to 206 nm DUV reaches $\sim$30%, with fiber transmission $>90\%$ (mode-matched to EH$_{11}$ mode), and total average DUV output power of $1.6$ 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 $\mu$J/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 [2511.09522].

## 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 $f_{\mathrm{FEM}}$ matched to a reference FEM simulation or measurement [1806.10416]. Gas-filled, stretched geometry is handled by replacing core unity with $n_{\mathrm{gas}}$, and integrating local properties over the fiber length $L$.
- **Nonlinear FWDFM Simulation:** Simple analytic expressions (e.g., $I_{5\omega} \propto N^2|\chi^{(3)}|^2 I_s I_p^2 L^2 \mathrm{sinc}^2(\Delta k L/2)/...$) 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 $n_{\mathrm{eff}}(\lambda),\ \alpha(\lambda)$ 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 $f_{\mathrm{FEM}}$ adjustment. Pump–probe or spectral interferometry directly measures group delay dispersion, confirming integration-of-local-$\beta_2(z)$ predictions [1806.10416].

## 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 $a(z), t(z)$ 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 ($>$90% coupling to EH$_{11}$) 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) [2511.09522]. 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 ($a$)     | $\sim$100 $\mu$m                     | Determines modal area, A$_{\rm eff}$       |
| Wall thickness ($t$)     | $0.5$–$1\,\mu$m (after stretch)      | Sets anti-resonance, loss minima           |
| Fiber length ($L$)       | $0.63$ m                             | Nonlinear gain, quadratic efficiency       |
| He pressure ($P$)        | $3.08$ bar                           | Phase-matching, dominant GVD source        |
| Loss ($\alpha$)          | $<0.1$ dB/m at 343 nm                | DUV transmission, scaling to meter lengths |
| Pulse energy (DUV)       | $\sim$16 $\mu$J (at 206 nm, 100 kHz) | High-efficiency frequency conversion       |
| Conversion efficiency    | $\sim$30% (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 [1806.10416][2511.09522].

Source: https://www.emergentmind.com/topics/he-filled-stretched-hollow-core-fiber