Spectral anti-crossings between the fundamental guided mode and core-wall resonances alter the dispersion in hollow-core anti-resonant-reflection photonic crystal fibers. Here we study the effect of this dispersion change on the nonlinear propagation and dynamics of ultrashort pulses. We find that it causes emission of narrow spectral peaks through a combination of four-wave mixing and dispersive wave emission. We further investigate the influence of the anti-crossings on nonlinear pulse propagation and show that their impact can be minimized by adjusting the core-wall thickness in such a way that the anti-crossings lie spectrally distant from the pump wavelength.
© 2018 Chinese Laser Press
Gas-filled capillary fibers have been extensively used for nonlinear optics experiments over the last half-century . Nowadays, they are routinely used for spectral broadening and temporal compression of ultrashort millijoule-level laser pulses , frequency conversion , and even phase-matched high-harmonic generation to the X-ray spectral region . The last 15 years have seen the emergence of hollow-core photonic crystal fibers that guide by anti-resonant reflection (ARR-PCFs). ARR-PCFs come in two main varieties: kagomé-style  and single-ring (SR) [6–8]. In contrast to wide-bore capillaries, ARR-PCFs provide low-loss broadband guidance even for core diameters of tens of micrometers, reducing the pulse energies required for strong nonlinear light–matter interactions from the millijoule to the microjoule level and allowing scaling from kilohertz to megahertz repetition rates [9–11]. In particular, soliton-related and even strong-field dynamics can be accessed in gas-filled ARR-PCFs pumped with infrared ultrashort laser pulses owing to their weak anomalous dispersion across the visible and infrared regions for a wide range of gas pressures [12–15]. As a result, ARR-PCFs provide a convenient platform for studies of nonlinear dynamics as well as for applications such as ultrashort pulse compression to the single-cycle limit , efficient generation of broadband deep and vacuum ultraviolet (UV) radiation via both dispersive wave (DW) generation and four-wave mixing (FWM) [11,16–19], and impulsive excitation of Raman coherence for multi-octave supercontinuum generation . All these processes rely not only on the broadband guidance offered by ARR-PCFs, but crucially on their weak, spectrally flat dispersion, which can be accurately approximated using a simple hollow-capillary model [21,22]. In reality, the broad transmission bands exhibited by kagomé-PCF and SR-PCF are interrupted by localized high-loss regions caused by anti-crossings between the core mode and resonances in either the core walls for kagomé-style PCFs or in the capillary walls for SR-PCFs. The wavelength of the th-order anti-crossing can be predicted to good accuracy by the expression , where is the speed of light in vacuum, is the wall thickness, is the refractive index of the glass, and is an integer number [23,24].
Here we study the effects of these anti-crossings on the nonlinear propagation of ultrashort laser pulses in gas-filled ARR-PCFs. We find that the strong, spectrally localized modification of the real part of the modal effective refractive index creates new phase-matching routes for parametric nonlinear processes such as FWM and DW generation, leading to the emission of strong narrowband light peaks localized in the spectral vicinity of the resonances, despite the high confinement loss. In addition, we investigate how soliton self-compression and ultraviolet DW generation are impaired when the anti-crossings lie close to the pump wavelength, and demonstrate that by appropriate optimization of the fiber structure, in particular tailoring the core-wall thickness, these detrimental effects can be mitigated to a great extent.
In the spectral vicinity of an anti-crossing, the capillary model is no longer valid. The influence of the resonance on the real part of the effective refractive index can, however, be captured by introducing a phenomenological Lorentzian response function centered at the anti-crossing frequency . This allows us to incorporate the anti-crossing dispersion for the resonances into the capillary model:1). The amplitude and damping of the th resonance are estimated by fitting the modal index calculated through finite-element modeling (FEM) to Eq. (1). In general, both coefficients depend on the precise fiber structure and in particular on the uniformity of the core-wall thickness: the larger the non-uniformity, the broader the resonance. The approximation used to derive Eq. (1) allows us to conveniently add the Lorentzian contributions to the dispersion.
Figure 1 shows the wavelength-dependent effective refractive index (blue dots) of a kagomé-PCF with and , and a SR-PCF with and , calculated by FEM in the vicinity of the resonance. These fibers were used in the experiments; their scanning electron micrographs are shown as insets in Fig. 1.
Predictions of Eq. (1) are in excellent agreement with the FEM data obtained for the two fiber structures, both in the vicinity of and far away from resonances (Fig. 1, solid-green curve). Note that the non-circular core boundary increases the effective core diameter of the SR-PCF , so that the best fit of Eq. (1) to the FEM data occurs for a core diameter larger than the distance between two diametrically opposite capillaries.
Because of the anti-crossing, wavelengths on opposite sides of the resonance can have the same effective refractive indices. Through nonlinearity, this can lead to an energy transfer between these wavelengths provided the overall photon energy is conserved. When the fiber is filled with noble gas, the nonlinearity is given mainly by the optical Kerr effect. As a result, new spectral components can be generated or amplified in the vicinity of the resonance via FWM or DW emission, which can also be described as cascaded FWM . In the case of degenerate FWM, two pump photons at frequency generate an idler photon at frequency and a signal photon at frequency . For this interaction, energy and momentum conservation require and , where is the modal propagation constant, is the FWM dephasing rate, and is the nonlinear correction due to the optical Kerr effect ( is the nonlinear fiber parameter and the pump peak power) . Figure 2(a) plots for the kagomé-PCF when filled with 35 bar Ne and pumped with pulses of peak power . (The parameters are obtained from the numerical simulation corresponding to the experiment described later.) Under these conditions the anti-crossing-related refractive index change strongly alters the landscape of allowed FWM interactions, introducing new narrowband solutions indicated by the curves originating at , where phase matching is satisfied for many different combinations of pump and idler frequency. As a result, a pulse with a spectrum broad enough to contain the pump and idler frequencies will transfer part of its energy to the signal in the vicinity of .
The second mechanism—DW emission—can be understood as light shed by a soliton to a linear wave at a different frequency. Within this picture, the dephasing rate of the interaction is [29,30], where is the propagation constant, is the inverse group velocity at the soliton central frequency , is the peak power of the self-compressed th-order soliton, and is the input power , with , where is the dispersion length, is the nonlinear length, is the initial soliton duration, and is the second-order dispersion. Figure 2(b) plots calculated for the same conditions as the FWM case above, and shows that phase matching is satisfied in the deep UV at , in the anti-crossing-free case (). Using the effective refractive index in Eq. (1), including higher-order resonances, just as for FWM, the DW dephasing rate is altered by the anti-crossings and an additional DW can be phase-matched at the wavelength corresponding to the anti-crossing.
3. EXPERIMENTAL RESULTS AND DISCUSSION
As already discussed, anti-crossings that are spectrally far away from the pump mainly result in emission of narrow spectral peaks in their vicinity, where the loss is high. These peaks are observed in experiments under many different conditions and can be caused by FWM and/or DW emission, depending on the experimental parameters. Figure 3(a) shows the measured output spectrum of a 17 cm long kagomé-PCF, when filled with 35 bar Ne and pumped with 8.4 μJ, 23 fs pulses in the anomalous dispersion region at 1030 nm. During propagation through the fiber, the pulses experience soliton self-compression followed by DW emission at . Narrow peaks are clearly visible in the vicinity of anti-crossings at for the fundamental, and for the first higher-order, core-wall resonances. The amplitude of these peaks was found to be strongly pump-energy-dependent. In particular, we observed a threshold-like behavior, in which the peaks appear only after the tail of the continuum extends to wavelengths shorter than . We attribute the additional spectral peaks at and to inhomogeneities in the thickness of the core walls in the kagomé-PCF. Similar peaks were observed with several different kagomé and SR-PCFs, their spectral position always coinciding with the anti-crossing wavelengths, irrespective of the specific experimental parameters.
To further confirm the relationship between the narrow spectral peaks observed in the experiments and the anti-crossing-related refractive index change, we performed numerical simulations using a unidirectional field equation . The input pulses were characterized using second harmonic generation frequency-resolved optical gating and used in the simulations (the measured traces are available in Ref. ). The fiber dispersion was modeled using the real part of the effective index in Eq. (1) including higher-order resonances, while the loss was deduced from the imaginary part (in general, this gives higher loss values than FEM). The simulation in Fig. 3(b) agrees well with the experiment with respect to the emission of narrow spectral peaks at the anti-crossing wavelength (). These peaks are absent when the fiber is replaced by an anti-crossing-free capillary dispersion model in the simulations.
Depending on their spectral position, the anti-crossings can have a strong impact on nonlinear pulse propagation, impairing effects such as soliton self-compression and DW emission. Using numerical simulations, we investigated this dependence for the case of the kagomé-PCF filled with 35 bar Ne. To be general, we simulated the propagation of a 23 fs (full width at half-maximum, FWHM) Gaussian pulse with 6 μJ energy (this corresponds roughly to the energy contained within the main peak of the pulse used in the experiments) [Fig. 3(c)].
As the anti-crossing approaches the pump wavelength (1030 nm), the maximum peak intensity along the fiber and the conversion efficiency to the UV DW degrade, indicating that soliton self-compression is strongly disrupted. This is because the resonance alters the dispersion landscape, introducing higher-order dispersion that leads to pulse break-up and the formation of side-pulses, which drain energy from the main pulse. As the peak intensity decreases, the shock effect—crucial for efficient UV emission—becomes weaker, resulting in lower conversion efficiency to the UV DW. In contrast, if the anti-crossing is far away from the pump wavelength, it has only a minor impact on the pulse propagation. The small drop in the conversion efficiency to the DW for the anti-crossing at 400 nm [Fig. 3(c)] is caused by the anti-crossing, which coincides with the DW wavelength. Note that the anti-crossing can also impair pulse compression and DW emission when the anti-crossing occurs at wavelengths longer than the pump, leading to a drop in both the maximum peak intensity and the conversion efficiency to the UV DW.
More generally, the impact of the resonances on the pulse propagation dynamics depends on the relative strength of the second- and third-order dispersion and on the pulse bandwidth. In order to quantify this, we introduce the parameter as the ratio of the second-order to the third-order dispersion lengths, . In the anomalous dispersion, when the soliton order , the pulse self-compresses by a factor that is inversely proportional to over a distance . As a result we expect the impact of the anti-crossings on the soliton compression to increase as increases. The maximum peak intensity along the fiber and the position of the anti-crossing are plotted in Fig. 4 as functions of , where is calculated using Eq. (1) with . The soliton order can be tuned either changing the gas pressure (which alters and ) or the input pulse energy. As the Ne pressure and increase, the self-compressing soliton reaches shorter durations and higher peak intensities. In the case of weak ionization, the balance between the compression ratio and the quality factor (which goes like for ) is optimal for , under which the self-compressed pulse reaches maximum peak intensity. As the compression ratio increases, the bandwidth becomes larger and a smaller value of (i.e., a spectrally more distant anti-crossing) is required for clean and efficient temporal compression [Fig. 4(a)]. For soliton order the pulse self-compresses over a shorter fiber length and as a result begins to drop for larger [brown curve in Fig. 4(a)].
In the absence of strong ionization, is solely a function of and . To illustrate this, in Fig. 4(b) we plot the ratios for two different combinations of gas pressure and pulse energy (6 μJ and 15 bar; 11.5 μJ and 8 bar) chosen so that . In the absence of ionization, the two curves coincide perfectly. When, however, ionization is included, the curves agree only at larger values of . This is because the additional polarization introduced by free electrons causes stronger temporal compression and a spectral blue-shift . As a result, the pulse wavelength moves closer to the anti-crossing and begins to drop at even smaller values of , as illustrated by the orange curve in Fig. 4(b). Thus, owing to the differing ionization-driven dynamics, the red and orange curves in Fig. 4(b) only coincide at large where the maximum intensity (and therefore the ionization) is reduced because of the anti-crossing.
To experimentally investigate the impact of the anti-crossing wavelength, we compared two different fibers: the previous SR-PCF with capillary-wall thickness and the same fiber after being etched with hydrofluoric acid so as to reduce the capillary-wall thickness to . In both cases, the fiber was 15 cm long and filled with 25 bar Ne. Figures 5(a) and 5(b) show the effective refractive index of the two fibers calculated via FEM. Etching the fiber shifts the anti-crossing from to . In Figs. 5(c) and 5(d), the output spectra of the fiber are shown for input energies of 11.3 μJ [panel (c), original fiber] and 9.7 μJ [panel (d), etched fiber]. Despite the lower input energy in the etched fiber, the amplitude of the DW at is more than 10 times stronger than in the original fiber, confirming that soliton self-compression is dramatically improved. Additionally, the spectral peaks at in the unetched fiber shift to in the etched fiber and become much weaker. Note that the anti-crossing is relatively wide because the capillary-wall thickness varies by from capillary to capillary.
In conclusion, anti-crossings between the fundamental core mode and resonances in the cladding membranes can strongly affect nonlinear propagation of ultrashort pulses in gas-filled anti-resonant-guiding hollow-core PCFs. The change in dispersion introduced by these anti-crossings leads to the emission of light in spectrally narrow bands through phase-matched FWM and DW emission. The impact of these anti-crossings depends on the soliton order of the pump pulse and can be minimized by etching the fiber so as to shift them far from the pump wavelength.
Note added in proof: A recent publication reported a fully analytical model to calculate the effective refractive index of ARR-PCFs . Dispersive wave emission in ARR-PCFs in the vicinity of the fiber resonance was reported in a recent conference paper .
1. E. P. Ippen, “Low-power, quasi-cw Raman oscillator,” Appl. Phys. Lett. 21, 539–541 (1972). [CrossRef]
2. M. Nisoli, S. D. Silvestri, and O. Svelto, “Generation of high energy 10 fs pulses by a new pulse compression technique,” Appl. Phys. Lett. 68, 2793–2795 (1996). [CrossRef]
3. L. Misoguti, S. Backus, C. G. Durfee, R. Bartels, M. M. Murnane, and H. C. Kapteyn, “Generation of broadband VUV light using third-order cascaded processes,” Phys. Rev. Lett. 87, 013601 (2001). [CrossRef]
4. I. Durfee, G. Charles, A. R. Rundquist, S. Backus, C. Herne, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high-order harmonics in hollow waveguides,” Phys. Rev. Lett. 83, 2187–2190 (1999). [CrossRef]
5. F. Benabid, J. C. Knight, G. Antonopoulos, and P. St.J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298, 399–402 (2002). [CrossRef]
6. A. D. Pryamikov, A. S. Biriukov, A. F. Kosolapov, V. G. Plotnichenko, S. L. Semjonov, and E. M. Dianov, “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 >3.5 μm,” Opt. Express 19, 1441–1448 (2011). [CrossRef]
7. F. Yu, W. J. Wadsworth, and J. C. Knight, “Low loss silica hollow core fibers for 3–4 μm spectral region,” Opt. Express 20, 11153–11158 (2012). [CrossRef]
8. P. Uebel, M. C. Günendi, M. H. Frosz, G. Ahmed, N. N. Edavalath, J. Ménard, and P. St.J. Russell, “Broadband robustly single-mode hollow-core PCF by resonant filtering of higher-order modes,” Opt. Lett. 41, 1961–1964 (2016). [CrossRef]
9. F. Emaury, C. F. Dutin, C. J. Saraceno, M. Trant, O. H. Heckl, Y. Y. Wang, C. Schriber, F. Gerome, T. Südmeyer, F. Benabid, and U. Keller, “Beam delivery and pulse compression to sub-50 fs of a modelocked thin-disk laser in a gas-filled Kagome-type HC-PCF fiber,” Opt. Express 21, 4986–4994 (2013). [CrossRef]
10. M. Gebhardt, C. Gaida, S. Hädrich, F. Stutzki, C. Jaregui, J. Limpert, and A. Tünnermann, “Nonlinear compression of an ultrashort-pulse thulium-based fiber laser to sub-70 fs in Kagome photonic crystal fiber,” Opt. Lett. 40, 2770–2773 (2015). [CrossRef]
11. F. Köttig, F. Tani, C. Martens Biersach, J. C. Travers, and P. St.J. Russell, “Generation of microjoule pulses in the deep ultraviolet at megahertz repetition rates,” Optica 4, 1272–1276 (2017). [CrossRef]
12. G. Fan, T. Balciunas, S. Hässler, C. F. Dutin, T. Witting, A. A. Voronin, A. Zheltikov, F. Gérome, G. Paulus, A. Baltuska, and F. Benabid, “A compact single cycle driver for strong field applications based on a self-compression in a Kagome fiber,” in CLEO: Science and Innovations (Optical Society of America, 2015), paper SM1P.1.
13. F. Tani, M. H. Frosz, J. C. Travers, and P. St.J. Russell, “Continuously wavelength-tunable high harmonic generation via soliton dynamics,” Opt. Lett. 42, 1768–1771 (2017). [CrossRef]
14. F. Köttig, D. Novoa, F. Tani, M. C. Günendi, M. Cassataro, J. C. Travers, and P. St.J. Russell, “Mid-infrared dispersive wave generation in gas-filled photonic crystal fibre by transient ionization-driven changes in dispersion,” Nat. Commun. 8, 813 (2017). [CrossRef]
15. T. Balciunas, C. F. Dutin, T. Witting, A. A. Voronin, A. Zheltikov, F. Gérome, G. Paulus, A. Baltuska, and F. Benabid, “A strong-field driver in the single-cycle regime based on self-compression in a kagome fibre,” Nat. Commun. 6, 6117 (2015). [CrossRef]
16. K. F. Mak, J. C. Travers, P. Hölzer, N. Y. Joly, and P. St.J. Russell, “Tunable vacuum-UV to visible ultrafast pulse source based on gas-filled Kagome-PCF,” Opt. Express 21, 10942–10953 (2013). [CrossRef]
17. A. Ermolov, K. F. Mak, M. H. Frosz, J. C. Travers, and P. St.J. Russell, “Supercontinuum generation in the vacuum ultraviolet through dispersive-wave and soliton-plasma interaction in a noble-gas-filled hollow-core photonic crystal fiber,” Phys. Rev. A 92, 033821 (2015). [CrossRef]
18. S. O. Konorov, A. B. Fedotov, and A. M. Zheltikov, “Enhanced four-wave mixing in a hollow-core photonic-crystal fiber,” Opt. Lett. 28, 1448–1450 (2003). [CrossRef]
19. F. Belli, A. Abdolvand, J. C. Travers, and P. St.J. Russell, “Ultrafast four wave mixing to the deep UV in gas filled kagomé PCF,” in Europhoton (2016), paper SS-1.5.
20. F. Belli, A. Abdolvand, W. Chang, J. C. Travers, and P. St.J. Russell, “Vacuum-ultraviolet to infrared supercontinuum in hydrogen-filled photonic crystal fiber,” Optica 2, 292–300 (2015). [CrossRef]
21. E. A. J. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964). [CrossRef]
22. J. C. Travers, W. Chang, J. Nold, N. Y. Joly, and P. St.J. Russell, “Ultrafast nonlinear optics in gas-filled hollow-core photonic crystal fibers,” J. Opt. Soc. Am. B 28, A11–A26 (2011). [CrossRef]
23. J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993). [CrossRef]
24. D. V. Skryabin, “Coupled core-surface solitons in photonic crystal fibers,” Opt. Express 12, 4841–4846 (2004). [CrossRef]
25. R. Sollapur, D. Kartashov, M. Zürch, A. Hoffmann, T. Grigorova, G. Sauer, A. Hartung, A. Schwuchow, J. Bierlich, J. Kobelke, M. Chemnitz, M. A. Schmidt, and C. Spielmann, “Resonance-enhanced multi-octave supercontinuum generation in hollow-core fibers,” Light Sci. Appl. 6, e17124 (2017). [CrossRef]
26. C. Wei, R. J. Weiblen, C. R. Menyuk, and J. Hu, “Negative curvature fibers,” Adv. Opt. Photon. 9, 504–561 (2017). [CrossRef]
27. M. Erkintalo, Y. Q. Xu, S. G. Murdoch, J. M. Dudley, and G. Genty, “Cascaded phase matching and nonlinear symmetry breaking in fiber frequency combs,” Phys. Rev. Lett. 109, 223904 (2012). [CrossRef]
28. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).
29. P. K. A. Wai, H. H. Chen, and Y. C. Lee, “Radiations by “solitons” at the zero group-dispersion wavelength of single-mode optical fibers,” Phys. Rev. A 41, 426–439 (1990). [CrossRef]
30. X. Liu, A. S. Svane, J. Lægsgaard, H. Tu, S. A. Boppart, and D. Turchinovich, “Progress in Cherenkov femtosecond fiber lasers,” J. Phys. D 49, 023001 (2016). [CrossRef]
31. F. Tani, J. C. Travers, and P. St.J. Russell, “Multimode ultrafast nonlinear optics in optical waveguides: numerical modeling and experiments in kagomé photonic-crystal fiber,” J. Opt. Soc. Am. B 31, 311–320 (2014). [CrossRef]
32. P. Hölzer, W. Chang, J. C. Travers, A. Nazarkin, J. Nold, N. Y. Joly, M. F. Saleh, F. Biancalana, and P. St.J. Russell, “Femtosecond nonlinear fiber optics in the ionization regime,” Phys. Rev. Lett. 107, 203901 (2011). [CrossRef]
33. M. Zeisberger and M. Schmidt, “Analytic model for the complex effective index of the leaky modes of tube-type anti-resonant hollow core fibers,” Sci. Rep. 7, 11761 (2017). [CrossRef]
34. F. Meng, S. Gao, Y. Wang, P. Wang, J. Liu, S. Wang, B. Liu, Y. Li, C. Wang, and M. Hu, “Efficient dispersive waves generation from argon-filled anti-resonant nodeless fiber,” in CLEO: Science and Innovations (Optical Society of America, 2017), paper STu3K.4.