## Abstract

Precise knowledge of the group velocity dispersion in gas-filled hollow-core photonic crystal fiber is essential for accurate modeling of ultrafast nonlinear dynamics. Here we study the validity of the capillary approximation commonly used to calculate the modal refractive index in kagomé-style photonic crystal fibers. For area-preserving core radius ${a}_{\mathrm{AP}}$ and core wall thickness $t$, measurements and finite element simulations show that the approximation has an error greater than 15% for wavelengths longer than $0.56\surd ({a}_{\mathrm{AP}}t)$, independently of the gas-filling pressure. By introducing an empirical wavelength-dependent core radius, the range of validity of the capillary approximation is extended out to a wavelength of at least $0.98\surd ({a}_{\mathrm{AP}}t)$.

© 2014 Optical Society of America

Since the advent of hollow core photonic crystal fiber (HC-PCF) [1], gas-based nonlinear optics has entered a new era. Offering ultra-long gas-light interaction lengths at high intensities, HC-PCF allows convenient exploitation of the usually weak nonlinearities of gases. Kagomé-style HC-PCF [2] in particular offers an ideal system for ultrafast gas-based nonlinear optics through ultra-broadband guidance at relatively low loss levels ($\sim 1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}/\mathrm{m}$) and a pressure-tunable modal refractive index. These characteristics are key requirements for generating octave-spanning supercontinua and other spectacular effects such as efficient generation of tunable deep-UV light [3]. Moreover, the very low light-glass overlap guarantees a high optical damage threshold [4].

The wide applicability of kagomé PCF in nonlinear optics is apparent from recent publications on the generation of tunable UV radiation [5], plasma-influenced nonlinear optics [6], Raman-free nonlinear optics in very high pressure systems [7], and low threshold high-harmonic generation [8]. Furthermore, simulations have shown that such systems have the potential, given the appropriate dispersion, to generate supercontinua extending far into the deep-UV, covering the whole wavelength range from 140 to 1000 nm [4]. Kagomé PCF is also of great interest in the infrared spectral region.

All these applications rely on an accurate knowledge of the group velocity dispersion (GVD), in particular its dependence on wavelength. In this Letter we perform a detailed experimental and theoretical study of the GVD of evacuated and gas-filled kagomé PCFs, concentrating in particular on the visible to IR spectral ranges, where the capillary-based Marcatili–Schmelzer model (MSM) is found to deviate significantly from experimental measurements.

In gas-filled kagomé PCF the total GVD arises from two competing effects: the anomalous dispersion of the empty waveguide and the normal dispersion of the filling gas [3]. The MSM predicts that the refractive index ${n}_{mp}$ ($m$ being the azimuthal and $p$ the radial mode order) of the modes guided in a capillary can be written in the approximate form [9]

The GVD of the kagomé PCFs was measured over the wavelength range 525–1375 nm using a white-light Mach–Zehnder interferometer [11]. Since the presence of water can strongly influence the dispersion measurements, the fiber samples were placed in a vacuum oven for at least 24 h at 100°C to remove all remaining water without damaging the coating. Figure 2(a) shows the result of measurements on a 725 mm length of five-ring kagomé PCF (Fig. 1) with an area-preserving (AP) core diameter of 20.4 μm; the thickness of the core wall was 190 nm and the interhole spacing 13.7 μm. It can be seen that the data points become increasingly scattered at longer wavelengths. We attribute this to increasing penetration of the core light into the cladding, where it couples to localized inhomogeneous resonances that cause the GVD to fluctuate rapidly with wavelength. In order to extract the general trend of the dispersion, we therefore fitted a third-order polynomial using the least-square method.

Figure 2(b) compares the measured GVD curve with the values from different theoretical models. First, the GVD of an idealized one-ring kagomé PCF [inset of Fig. 2(b)] was computed using the finite element (FEM) method (JCM wave). The experimentally observed GVD fluctuations were also seen in the FEM simulations and have been reported previously [12]. Since however it is the overall trend of the GVD curve (not these fiber-dependent fluctuations) that is important in ultrafast experiments, the calculated $\beta $ values were smoothed using a fifth-order polynomial and the GVD derived by differentiation.

The hexagonal-shaped core of the kagomé PCF raises the question, since the MSM is based on a circular symmetry, of the appropriate value for the core radius. Two different conventions are in use [4]: the flat-to-flat (FF) radius and the AP radius. Figure 2(b) shows the GVD derived from the group-delay measurements along with the GVD calculated using the MSM for both FF and AP core areas. From the measurements and simulations we empirically find that the GVD of kagomé PCF is accurately predicted by the MSM for wavelengths below a certain critical value:

where ${a}_{\mathrm{AP}}$ is the AP radius, $t$ the core wall thickness, and ${\lambda}_{\mathrm{crit}}$ the wavelength above which the deviation between measurement and MSM with AP-radius is larger than 15%. For the fiber in Fig. 2 ${\lambda}_{\text{crit}}=800\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$, which is in good agreement with previous publications, where the MSM was successfully used to predict nonlinear dynamics in the UV [3,4]. At wavelengths longer than ${\lambda}_{\text{crit}}$ the difference between measurement and the MSM increases significantly for both AP and FF core areas, the measured GVD being much more anomalous. The error between model and measurement is already 64% at 1375 nm. The FEM data for the idealized kagomé PCF in Fig. 2, fitted to a fifth-order polynomial [Fig. 2(b)], reproduces the measured GVD much more accurately. The much better agreement between the simulated GVD of an idealized single ring kagomé PCF and the measurement is supported by previous publications studying the loss of kagomé-PCF [13,14]. In these papers it is pointed out that the first cladding ring has a huge effect on the total loss and, unlike in band-gap-guiding PCFs, the loss in kagomé PCF does not fall with the number of cladding rings [14–16]. Our results show additionally that the overall GVD of kagomé PCF is strongly affected by the structure of the first cladding ring. Separate FEM studies (not shown here) on kagomé PCFs with two cladding rings show that the influence of the second cladding ring is much smaller, making the GVD slightly more anomalous in the near infrared, in better agreement with the measurements.Next, we investigated the influence of finite core wall thickness (here 190 nm) on the dispersion of a silica capillary of inner diameter 20.4 μm. The dispersion of the thin capillary [Fig. 2(b)] was calculated using a transfer matrix method [17]. In contrast to FEM, this has the advantage that the computation time is reduced to a few seconds or even less. In Fig. 2(b), it may be seen that the dispersion of the thin-wall capillary matches the FEM simulation perfectly up to 1100 nm, the major deviations to the MSM coming from the finite thickness of the core wall. Beyond 1100 nm the deviations between the thin-wall capillary model and FEM become larger, indicating that the outer cladding structure is beginning to play a role. The thin-wall capillary provides an attractive trade-off between speed of calculation and accuracy, occupying the middle ground between FEM (accurate but slow) and the MSM (limited accuracy but very fast). It is much faster than FEM, while remaining accurate out to longer wavelengths than is true of the MSM. Additionally, it models the main loss windows of kagomé PCF (caused by resonances in the core wall) and their influence on the GVD [13].

We now introduce an empirical scaling law that can be used to increase the range of validity of the MSM. It is based on replacing the core radius in Eq. (1) with an effective radius that depends on the wavelength and the core wall thickness:

where $s$ is a dimensionless empirical parameter. From a least-squares fit to the calculated effective refractive indices we find that the value $s=0.047$ works well for all the FEM simulations. The measured GVD is however slightly more anomalous than that predicted by FEM for the simplified structure in Fig. 2(b). A value of $s=0.065$ was found to work best for the experimental data points. We checked this modified MSM (mMSM) experimentally for two five-ring kagomé PCFs with core diameters of 20.4 and 26.6 μm, core-wall thicknesses of 190 and 160 nm, and interhole spacings of 13.7 μm. The mMSM works well for the fundamental mode of both fibers out to a wavelength of 1400 nm—the limit of our measurement range. In Fig. 3 the GVDs predicted by FEM, the mMSM, and the AP MSM are compared with the experimental values. For the correct value of $s$, the mMSM fits the measurements and simulations much more closely, merging with the MSM for wavelengths shorter than 600 nm. In conclusion, it seems that the range of validity of the MSM can be extended out to a wavelength of $0.98\surd ({a}_{\mathrm{AP}}t)$ by introducing the wavelength-dependent radius in Eq. (3).Finally, we investigated how the GVD behaves if the fiber is filled with argon, as in many experiments [3,6,7,18]. Figure 4 shows the difference between the measured GVD at 5 bars and 10 bars and that measured in the evacuated fiber. GVD curves at 10 bars, obtained from both the MSM and the mMSM, are also shown. It is clear that the MSM for the pressurized fibers shows the same long-wavelength deviations as observed in the evacuated fiber, while the predictions of the mMSM match the measurement much closer at NIR wavelengths. Nevertheless, for gas-filled kagomé PCF the MSM predicts the correct GVD for wavelengths below ${\lambda}_{\text{crit}}$, which in this case is 800 nm.

In summary, an empirical scaling law allows one to distinguish two GVD regimes: a short-wavelength regime ($\lambda <{\lambda}_{\text{crit}}$) where the total GVD is predominantly determined by the size of the hexagonal core and a long-wavelength regime ($\lambda >{\lambda}_{\text{crit}}$) where the finite core-wall thickness and outer cladding structure become more and more important. The transfer-matrix method applied to a thin-wall capillary represents a compromise between very time-consuming FEM calculations and the MSM, although it must still be solved numerically. The most attractive model (which has the advantage of being analytical and accurate) is however the empirically modified MSM, in which the effective core radius is assumed wavelength dependent. Note that, despite these very helpful tools, it will still be necessary to carry out full FEM simulations, or measure the dispersion directly, if a highly accurate dispersion curve is needed, including the intrinsic GVD jittering caused by anticrossings with cladding states.

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