Fig. 1. (a) Mode fields of the first four EH1n modes considered in this study. Intensities are depicted by colors, and electric fields are represented by arrows. (b) Ray-optical representation of hollow-fiber transmission. The wave vector k0 can be decomposed into a transverse component k⊥ and a longitudinal component β, which are connected by Pythagoras’ theorem [38]. (c) The propagation constants βn of the individual EH1n modes follow an approximate n2 dependence, whereas the k⊥,n underly a linear relationship with n. (d) Intramode dispersion, deviation of the propagation constants of the first 10 EH1n modes from the propagation constant of a plane wave, calculated at 800 nm wavelength for an HCF with a core radius of 100 μm; symbols, exact solution according to Eq. (3); curve, parabolic fit; the four modes of interest within this study are highlighted in red, indicating a dephasing of ≈8  rad/cm in the linear optical regime.

Fig. 2. Spatial soliton solution branches of Eq. (8). For normal modal dispersion (ncore>nclad), a single solution branch exists (green). In hollow fibers, two branches coexist (blue and red). The red branch is considered unstable (see discussion in text). (a) Radially integrated intensity ∫E(r)2rdr=P of the spatial cage soliton solutions versus effective nonlinearity. Powers have been normalized to the critical power of self-focusing Pcr in free space [9]. Insets show E(r) for parameters indicated by symbols. (b) Root mean square mode diameter of spatial solitons normalized to the HE11 mode; insets show spatial intensity profiles |E(r)|2.

Fig. 3. Three-dimensional visualization of the light bullet structure at the stability limit (≈1.4Pcr) in the anomalous modal dispersion regime. Equi-intensity surfaces are shown with colors red (80% peak intensity) to blue (10% peak intensity). In the center, a donut structure dominates, which evolves into an ellipsoidal shape in the temporal wings. The glass–gas interface of the hollow fiber is depicted in light gray for comparison.

Fig. 4. Comparison of model results with measured data. (a) Total losses (linear and nonlinear) versus ratio of a3 and λ2 (curve and red dots). Early measurements with relatively short hollow fibers exhibited significantly higher losses, whereas more recent measurements showed excellent agreement as indicated by the respective references [1921" target="_self" style="display: inline;">–21]. (b) Maximum beneficial length (solid curve and hollow triangles) and maximum compressibility (dashed line and solid triangles); cf. Eq. (13). This analysis confirms that superior compression can be reached with longer hollow fibers and larger core diameters.

Table 1. Nonlinear Mode-Coupling Factors^a