1. Introduction

Hollow–core photonic crystal fibers (HC-PCF) [1,2] with a hexagonal arrangement of holes in the cladding guide light by the photonic bandgap mechanism which offer low-loss transmission with loss coefficients in the range of 1 dB/km for small fiber diameters, enabling long effective interaction lengths for nonlinear processes and high-power delivery of fs pulses. Due to these guiding properties a number of applications in medicine and in nonlinear optics has been demonstrated such as in Raman studies and frequency conversion. However, the main drawback of these fibers is their intrinsically narrow transmission bandwidth determined by the bandgaps, which excludes its implementation in a large number of applications in ultrafast nonlinear optics requiring broadband guidance or guidance in the visible and UV.

An alternative HC-PCF design replaces the hexagonal lattice cladding with a kagome lattice [3–6]. These fibers do not have bandgaps, rather, the photonic guidance relies on a mechanism akin to Von Neumann-Wigner bound states of the Schrödinger equation within a continuum [7]. The fiber-guided modes in these fibers only weakly interact with the cladding modes due to the rapid phase modulation of the latter. Due to this guiding mechanism kagome lattice HC-PCFs exhibit broad transmission regions with relatively low loss covering the spectral range from the infrared up to the UV.

Standard hollow waveguides with a cladding from fused silica [8] have important applications in ultrafast nonlinear optics, such as the generation of few-cycle mJ pulses [9], high-order harmonic generation [10] and four-wave mixing for UV [11] and VUV [12,13] femtosecond pulse generation. Unfortunately, they provide a tolerable level of loss only for diameters larger than 100 μm at which the waveguide contribution to dispersion is relatively week. However, besides the desired broad transmission range the control of the group velocity dispersion is the key factor for interesting applications in ultrafast nonlinear optics and supercontinuum generation. Recently, we studied dielectric-coated metallic hollow waveguides [14] and chirped multilayer hollow waveguides [15] as an option for the combination of broad-band transmission and anomalous dispersion at optical frequencies. However, the challenges in the fabrication of these fibers require to proceed with the search for a viable type of hollow waveguides enabling dispersion control and ultrabroadband transmission.

The aim of the present paper is to study the dispersion properties, the loss and the optimum design of kagome lattice HC-PCFs filled with argon for the purpose of possible applications in ultrashort-pulse delivery and in ultrafast nonlinear optics. As will be shown numerically and by using an approximate analytical formula, these fibers exhibit anomalous dispersion for visible or UV wavelengths both for a 1-cell-core as well for a 3-ring-core HC-PCF for diameters of 25 μm and 80 μm, respectively. The loss of a kagome lattice HC-PCF is mainly determined by the cladding structure such as the strut thickness and not greatly influenced by the core diameter and remain a few orders of magnitude below that of a hollow silica waveguide with the same core-size. These results show that kagome lattice HC-PCFs can have interesting applications in ultrafast nonlinear optics. By choosing an optimized pressure, zero-GVD parameters and small third-order dispersion (TOD) at the input wavelength can be realized, which offer the possibility to combine small loss and low dispersion and allow to guide few-cycle pulses over extended distances without significant temporal broadening and pulse distortion. Besides, the anomalous dispersion in the visible at relatively high gas pressure is promising for the generation of soliton-induced supercontinuum generation [16] with more than one octave broad spectra and a much higher peak power density than in microstructure fibers. Such waveguides also enable soliton-effect compression of fs pulses without technically demanding chirp compensation. Finally, anomalous dispersion in the visible or UV allows phase-matching for four-wave mixing [10–13] at much higher pressures than in fused-silica hollow waveguides and therefore higher efficiencies for frequency conversion.

2. Loss characteristics and mode distributions

We consider a straight translationally invariant fiber consisting of a hollow core, a kagome lattice cladding and a bulk fused-silica outer region, as depicted in Fig. 1 . The commercial finite-element Maxwell solver JCMwave was utilized to calculate the complex-valued effective refractive index n_eff, from which the propagation constant β = ωRe(n_eff)/c and the loss coefficient α = 2ωIm(n_eff)/c were determined. The dispersion of fused silica as well as that of argon was described by the Sellmeyer formula for the corresponding dielectric function. In practice, there are always variations of the fiber geometry in the longitudinal direction due to manufacturing imperfections. The change of the size of the structure leads to shift of the resonance frequencies and to widening of the resonance peaks in the loss and the group velocity dispersion (GVD) profiles. We have accounted for this effect by smoothing the loss and GVD profile, using a 5% wavelength interval for averaging.

 

Fig. 1 Cross sections of a 1-cell-core (a) and a 3-ring-core (b) kagome lattice HC-PCF. The outer region is made of fused silica.

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Figure 1 shows the cross-sections of a kagome lattice HC-PCFs, with a 1-core (a) and a 3-ring-core (b). Our calculations below show that the guiding properties of the fiber do not depend critically on the termination of the cladding at the core boundary.

Figure 2 shows the loss of a kagome lattice HC-PCFs with a pitch of Λ = 16 μm and a strut thickness of t = 0.2 μm for a 1-cell-core (a) and a 3-ring-core (b). Here the blue crosses are the results of the numerical simulations, the red solid curves are the averaged one over longitudinal inhomogeneous variations of the fiber parameters. The green dashed lines present the loss of a hollow silica waveguide with a core size equal to D = 1.6Λ (a) and D = 5Λ (b). We can see that the loss of the kagome lattice HC-PCF is by few orders of magnitude lower than the one of the hollow silica waveguide with a difference being more outstanding at the shorter wavelength. Although the core-size in (b) is three times lager than in (a), the losses in the 3-ring and 1-cell-core fibers are close over almost the whole considered spectral range, except for the long-wavelength region where the loss of the 3-ring-core fiber becomes lower than that for the 1-cell-core fiber. This behavior is in contrast to the case of hollow core waveguides with a cladding made from fused silica, for which the loss drops as D ⁻³ with the diameter [8]. The low loss of the kagome lattice HC-PCF is explained by the highly oscillating transverse structure of the strut-localized mode, hindering its coupling to the spatially smooth core-localized mode [5]. The above condition, however, does not hold for the set of wavelengths ${\lambda }_m=2t{(n^2-1)}^{1/2}/m$ (strut resonances) where t is the strut thickness and m = 1,2… is an integer. Under this condition matching of the core and cladding mode distribution, high coupling and high loss are predicted [5]. In Fig. 2, these resonances are visible as the loss peaks at the wavelengths ${\lambda }_m$ indicated by the black vertical dotted lines.

 

Fig. 2 Losses of kagome lattice HC-PCFs with a pitch of 16 μm and a strut thickness of 0.2 μm for a 1-cell-core (a) and a 3-ring-core (b). The blue crosses represent the direct numerical simulations, the red solid are averaged results over longitudinal inhomogeneity using 5% of the wavelengths interval, the green dashed line is the loss of a hollow silica waveguide with the same core-size as the one of the kagome lattice HC-PCF and the black vertical dotted lines indicate the strut-resonance.

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Additionally, light in the core can couple to the modes localized in the crosses of the struts (so-called node modes). The spectral position and the density of states of these modes are determined by the geometry of the crosses; they are responsible for the additional peaks far from the strut resonances and for the oscillatory character of the loss curve. Note however, that this guiding mechanism does not necessarily imply that the loss decreases with increasing waveguide diameter.

In Fig. 3 , we can see that the loss of the kagome lattice HC-PCF with a pitch of 24 μm is at the same wavelengths significantly below the loss of a hollow silica waveguide with the same core-size and lower than the one of the kagome lattice HC-PCF with a pitch of 16 μm, shown in Fig. 2. Although the core-size in the case of Fig. 2(b) is larger than the one in Fig. 3, the loss is even higher in this case. We assume that this can be related to the lower density of the crosses in the large-pitch kagome structure, which results in a low density of states for node modes and a reduced loss outside of the strut resonances. Again we see that the cladding structure seems to be much more important for the loss characteristics than the core diameter of the waveguide.

 

Fig. 3 Loss of a kagome lattice HC-PCFs with a pitch of 24 μm and a strut thickness of 0.2 μm for a 1-cell-core. The meanings of colors are same as in Fig. 2.

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Figure 4 shows the distribution of the square of the electric field in the fiber cross-section for the wavelengths of 420 nm (a) and 600 nm (b). It can be seen that at the strut resonance the field is delocalized and spreads over the whole cladding region and is guided by the reflection at the outer-region boundary with bulk fused silica. On the other hand, off-resonance field localization (and therefore the waveguide contributions to dispersions) are determined by the core-cladding boundary.

 

Fig. 4 Mode field distributions at wavelengths 600 nm (a) and 420 nm (b) of a 1-cell-core kagome lattice HC-PCFs with a pitch of 24 μm and a strut thickness of 0.2 μm.

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In Figs. 5 and 6 the effect of the strut thickness on the energy fraction in the cladding (outside the core) f and the waveguide loss is studied. In Fig. 5 a shift of the strut resonances to the short wavelength with decreasing strut thickness can be seen. From the dependence of the loss on the strut thickness presented in Fig. 6 an optimum thickness of around 0.2 μm forthe wavelengths in the visible is found. The loss for a strut thickness t of 0.1 μm is higher than in the other both examples (ignoring the regions of extremely small and irrelevant loss lower than 0.01 dB/m), which suggest the existence of an optimum strut thickness. For a large strut thickness, there exist a large number of states with field distributions matched to the core modes connected with strut resonances and node resonances, resulting in the interactions between the core mode and the cladding mode and in high losses. In contrast, for a small strut thickness such that the first strut resonance wavelength is far below the considered wavelengths, the strut cannot eliminate the interaction between the core and cladding. In this limiting case, the kagome structure has no influence on the field for $t\to 0$. By comparison of Figs. 2, 3, 5 and 6 one can see that the optimum cladding structure is determined more sensitively by the strut thickness and the pitch than by increasing the core diameter.

 

Fig. 5 Energy fractions in the cladding of a 1-cell-core kagome lattice HC-PCFs with a pitch of 16 μm and strut sizes of 0.3 μm (a), 0.25 μm (b), 0.2 μm (c) and 0.1 μm (d) respectively. The blue crosses represent the direct numerical simulations, the red solid are averaged results over longitudinal inhomogeneity using 5% of the wavelengths interval and the black vertical dotted lines indicate the strut-resonance.

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Fig. 6 The effect of strut thickness t on the waveguide loss in a 1-cell-core kagome lattice HC-PCF. Here the black circles, blue squares and red crosses are the losses of a kagome lattice HC-PCF with a pitch of 16 μm and a strut thickness of 0.4 μm, 0.2 μm and 0.1 μm, respectively. The green dashed line is the loss of a hollow core silica waveguide with the same core diameter as to the one of the kagome lattice HC-PCF.

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3. Dispersion control and ultrashort-pulse delivery

In Von Neumann-Wigner bound states, the core-mode cohabitates with the cladding mode without notably interacting [5]. Therefore one can assume that the effective refractive index of the core mode depends only on the core properties, that is, on the refractive index of the material inside the core and the diameter of the core. Since the mode field is zero at the core boundary, the mode profile can be approximated by the zeroth-order Bessel function and the wave number can then be expressed similar as in the case of a hollow waveguide with a silica cladding [8] as follows:

Figure 7 shows the GVD of a 1-cell-core (a) and a 3-ring-core (b) kagome lattice HC-PCF with a pitch of 16 μm, a strut of 0.2 μm filled with Argon at 1 atm, while in Fig. 8 we compare the GVD of a 1-cell-core kagome lattice HC-PCF with a pitch of 24 μm, a strut thickness of 0.2 μm filled with Argon at 1 atm (a) and 2 atm (b). As can be seen in Fig. 7 und 8, outside of the strut resonances the analytical and numerical results agree well, while near the resonances the GVD curve strongly oscillates. However, these oscillations are reduced by the averaging of the dispersion due to the longitudinal inhomogeneities, as described above. One can expect that for many nonlinear processes, especially for frequency conversion but also for supercontinuum generation, the oscillations of the GVD will not have significant detrimental effects. As explained above the waveguide distribution to dispersion is determined by the diameter of the waveguide core, while the loss is determined predominantly by the strut thickness and, to a lesser degree, by the pitch, as shown in Figs. 1-3. This shows that in a certain range of parameters kagome lattice HC-PCFs allow to control the loss and the waveguide contribution to dispersion independently.

 

Fig. 7 GVD of kagome lattice HC-PCFs with a 1-cell-core (a) and a 3-ring-core (b) with a pitch of 16 μm and a strut thickness t of 0.2 μm filled with argon at a pressure of 1 atm. The blue circles represent numerical simulations and the red solid curve is calculated by the Eq. (1).

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Fig. 8 GVD of a 1-cell-core kagome lattice HC-PCF filled with argon at 1atm (a) and 2 atm (b) with a pitch of 24 μm and a strut thickness of 0.2 μm. The blue circles represent numerical simulations and the red solid curve is calculated by the Eq. (1).

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The above dispersion properties and its control by the waveguide parameters and the pressure of the gas filling combined with low attenuation in kagome lattice HC-PCFs is of interest for ultrashort pulse delivery and guiding of few-cycle pulses over extended distances without significant temporal spreading and distortion. Hollow waveguides with a glass cladding are intrinsically more lossy requiring larger diameters, leading to normal GVD. Bandgap HC-PCFs guide light with very low loss but much larger dispersion values of several 10⁵ fs²/m for the GVD and more than 10⁶ fs³/m for the TOD parameter [17]. Recently described chirped photonic crystal fibers [18] have reduced dispersion, with a GVD parameter of −500 fs²/m and a TOD parameter of −3000 fs³/m at 1 atm. In comparison, our simulations for a 3-ring core kagome lattice HC-PCF with a pitch of 16 μm, a strut thickness of 0.2 μm filled with argon at 1 atm, predict −60 fs²/m for the GVD and 100 fs³/m for the TOD parameter.

As shown in Fig. 9(a) , the GVD parameter can be controlled by the pressure of the gas filling and shifts towards zero as the pressure grows to the optimum value of 4 atm. As shown in Fig. 9(b), the TOD parameter remains small for this higher pressure. To demonstrate the improved guiding properties in Fig. 9(c) the temporal shape of a 13-fs input pulse is presented after linear propagation through the 1.05-m long kagome lattice HC-PCF filled with argon at different pressures. One can see that at 1 atm the pulse is broadened to 19 fs, but for the optimum pressure of 4 atm, the pulse duration is the same as the input duration despite higher-order dispersion effects while the maximum intensity is reduced by 21% due to loss. This results show that kagome lattice HC-PCFs have a promising potential for ultrashort pulse delivery because of its low dispersion and simultaneously low loss which can be manipulated independently by controlling the core properties such as gas pressure and core diameter, and the cladding properties such as the strut thickness.

 

Fig. 9 Dispersion control and ultrashort pulse delivery in a kagome lattice HC-PCF. In (a) and (b), the GVD and TOD parameters at 800 nm versus the argon pressure are shown, respectively. In (c), the linear temporal broadening of a 13 fs-input pulse (blue solid line) centered at the wavelength 800 nm is shown after propagation of 1.05 m through the kagome lattice HC-PCF filled with argon at pressures of 1 atm (red solid line), 2 atm (black dot-dashed line), 4 atm (green dashed line) and 6 atm (pink dotted line). The 3-ring core kagome lattice HC-PCF has a pitch of 16 μm and a strut thickness of 0.2 μm.

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Besides the improved properties in the guiding of few-cycle pulses over extended distances, the predicted anomalous dispersion at optical frequencies at high pressure of the noble gas filling opens the possibility to achieve phase matching for nonlinear processes, such as high harmonic generation (HHG) or four-wave mixing at a much higher pressure compared with the conditions in silica hollow waveguides as previously studied in [10–13]. Furthermore, anomalous dispersion at optical frequencies at a high pressure provide a promising potential for the generation of soliton-induced supercontinuum generation [16] with more than two octave broad spectra and up to five orders of magnitude higher power than in microstructure fibers.

5. Conclusion

In conclusion, we studied numerically and analytically the dispersion properties and the loss of kagome lattice HC-PCFs with a noble-gas filling and the ways to manipulate them. We have found that the loss depends mainly on the cladding structure such as the strut thickness and the waveguide contribution to dispersion on the core size. The reasons for this behavior are related to the coupling of the core-localized states to the resonances in the cladding. This fact provides the ability to control the loss and the dispersion properties independently by controlling the cladding structure and the core properties, respectively. The predictions show that the combination of dispersion control and anomalous dispersion in the visible with ultrabroadband transmission provide a potential for applications in ultrashort-pulse delivery and in ultrafast nonlinear optics enabling an improved efficiency in nonlinear spectral transformations or an increased power in supercontinuum generation.