Efficient third-harmonic generation (THG) in a gas-filled hollow-core photonic bandgap fiber requires three building blocks: guidance of the pump, guidance of the third harmonic (TH), and phase matching between them. Montz and Ishaaya (Opt. Lett. 40, 56, 2015) designed hybrid-cladding hollow-core photonic bandgap fibers (HC-PBGFs) with two bandgaps whose central frequencies differ by a factor of ~3, so that a Gaussian-like fundamental mode and a Gaussian-like TH mode are each guided in their own bandgap with good modal overlap. However, phase matching between the two bandgap-guided modes is not achieved, and its feasibility has never been analyzed or demonstrated: by 2020 the authors themselves stated it is 'unclear if phase-matching of these modes is feasible'. The open problem is to determine whether dispersion engineering of the hybrid cladding, combined with gas dispersion in the hollow core, can achieve phase matching between the bandgap-guided pump and TH modes while keeping both wavelengths inside their bandgaps, maintaining low confinement loss and high modal overlap, with fabricable thin-strut structures (~19-38 nm) and acceptable surface-mode and bandwidth (~30 nm higher-order bandgap) behavior. No prior work resolves this question: the all-solid analog was demonstrated (Cavanna et al., Optica 3, 952, 2016) and pressure-tuned phase-matched THG exists in kagome fibers (Nold et al., Opt. Lett. 35, 2922, 2010), but neither addresses the hollow-core dual-bandgap geometry.

Background

Third-harmonic generation (THG) is the nonlinear optical process in which three photons at a pump frequency combine into one photon at triple the frequency. In an optical fiber, the generated TH grows coherently with propagation distance only under phase matching, i.e. when the propagation-constant mismatch β(3ω)−3β(ω)\beta(3\omega)-3\beta(\omega) vanishes; otherwise the harmonic generated along the fiber destructively interferes with itself. A hollow-core photonic bandgap fiber (HC-PBGF) is a microstructured silica fiber whose cladding contains a periodic lattice of air holes that forbids light in certain wavelength bands (photonic bandgaps), so light in those bands is confined to a central hollow core. Filling the core with a gas provides the nonlinear medium while the silica structure guides the light, allowing pressure- and gas-tunable dispersion. A kagome HC fiber is a non-periodic hollow-core fiber with a lattice of interlaced triangles that guides by inhibited coupling rather than a true bandgap; it can guide any wavelength but with less strict mode filtering. A surface mode is a mode localized at the core boundary that can couple to the guided core mode and degrade it; confinement loss is the residual leakage of a nominally guided mode through the finite cladding. Nold et al. (2010) demonstrated pressure-controlled phase-matched THG in an argon-filled kagome HC fiber: by tuning gas pressure, gas dispersion counterbalances fiber dispersion so that β(3ω)=3β(ω)\beta(3\omega)=3\beta(\omega). However, the generated TH appeared in a higher-order, non-Gaussian mode with low pump-harmonic overlap, suffered UV attenuation, and exhibited walk-off. Betourne et al. (2008) proposed phase matching a total-internal-reflection (TIR) guided mode to a bandgap-guided harmonic in an all-solid doped photonic crystal fiber (PCF), but their modified-TIR scheme has no counterpart in hollow-core fibers. To obtain Gaussian-like TH output with high modal overlap, Montz and Ishaaya (2015) proposed hybrid-cladding HC-PBGFs in which two different periodic microstructures (hybrid tilings) create two photonic bandgaps with central-frequency ratio of about 3 at the air line (the dispersion curve of light propagating in air, on which hollow-core bandgaps are located): one bandgap guides the ~1.5 µm pump, the other the ~0.5 µm TH, both as Gaussian-like fundamental modes with good spatial overlap. Their companion analysis (arXiv:1407.3688) verified dual-bandgap guidance and modal overlap, and identified these as two of the three building blocks for efficient THG in gas-filled HC-PBGFs; 'still missing is the phase-matching between the pump and the high harmonic, which may be achieved with careful dispersion engineering of the fiber combined with gas dispersion'. The design's practical prerequisites are themselves unresolved: the hybrid cladding requires strut thicknesses of roughly 19 and 38 nm with interstitial air holes likely to collapse during fiber drawing unless a dual-pressurizing system is used; the influence of the core wall thickness on surface modes and confinement loss needed study; and the higher-order bandgap is only about 30 nm wide, with its central wavelength shifted by less than 11 nm from the exact TH wavelength.

Problem Statement

Determine whether efficient third-harmonic generation can be realized in a gas-filled dual-bandgap hollow-core photonic bandgap fiber of the hybrid-cladding type proposed by Montz and Ishaaya (2015), where a Gaussian-like fundamental pump mode and a Gaussian-like third-harmonic mode are each guided by their own photonic bandgap (central-frequency ratio ~3) in the same hollow core. The load-bearing question is phase matching: find a combination of hybrid-cladding dispersion engineering (element sizes, including strut thicknesses on the order of tens of nanometers, and lattice layout) and gas species/pressure in the hollow core such that the propagation-constant mismatch β(3ω)−3β(ω)\beta(3\omega)-3\beta(\omega) between the two bandgap-guided modes vanishes, or establish that no such solution exists within this fiber class. The answer must respect the coupled constraints the source itself identifies as prerequisites of the same objective: both wavelengths remain inside their respective bandgaps with low confinement loss; the higher-order bandgap width (~30 nm) and its <11 nm shift from the exact third-harmonic wavelength are accounted for; surface modes and confinement loss introduced by the core wall thickness are controlled so the predicted Gaussian-like modal overlap is preserved; and the required structures (including ~19-38 nm struts and interstitial air holes) are fabricable. A resolution of the question is a design of this fiber class in which the propagation-constant mismatch vanishes between the two bandgap-guided modes with both wavelengths inside their respective bandgaps — or a rigorous demonstration that no such design exists; the remaining coupled constraints are to be estimated and reported for that design rather than optimized to a fixed threshold.

References

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