1. INTRODUCTION

Although 2011 is the fiftieth anniversary of nonlinear optics, it is perhaps worth noting that 2012 is both the fortieth anniversary of nonlinear fiber optics [1, 2] and the tenth anniversary of the first time hollow-core photonic crystal fiber (HC-PCF) was used to achieve ultralow-threshold stimulated Raman scattering in hydrogen [3]. In 2002, Downer, writing about this last result in the journal Science [4], expressed the view that “a new era in the nonlinear optics of gases, and maybe even plasmas, is about to begin”—a prophetic statement, because this year the first reports of controlled plasma generation in gas-filled hollow-core PCFs have emerged [5].

The remarkable success of solid-core fiber (conventional or photonic crystal) as a vehicle for discovering, observing, and exploring a wide range of different nonlinear optical effects [6, 7, 8], is due to ultralong effective path lengths, a consistent and predictable dispersion, configurable birefringence, and a nonlinear figure of merit (the balance between nonlinearity and loss) that, for silica fiber, is the highest of any medium. These characteristics led to the observation of optical solitons [9, 10], the discovery of the soliton self-frequency shift [11, 12], the development of ultrafast all-optical switches [13], and the generation of octave-spanning supercontinua [14, 15] and even continuous-wave pumped supercontinua [16], as well as advancing our fundamental understanding of four-wave mixing [17], modulation instability (MI) [18], self-phase-modulation [19], Raman [1] and Brillouin [2] scattering, and self-organized second-harmonic generation [20].

Extending nonlinear fiber optics beyond interactions with a solid-core material, to cores filled with gases and plasmas, cannot be achieved using conventional fiber designs based on total internal reflection. This is because the core refractive index must be higher than that of the cladding, but it is close to unity for gases—lower than any dielectric material at optical frequencies. In cases where use of a gas is essential, for example to avoid optical damage when broadening the spectrum of millijoule-level femtosecond-duration $800\mathrm{nm}$ pulses, the glass capillary fiber provides a partial solution. Although such capillaries do not support bound modes, the loss of the fundamental ${\mathrm{HE}}_{11}$ leaky mode scales as the inverse cube of the bore diameter, and it can be as low as $1.2\mathrm{dB}/\mathrm{m}$ for diameters of $\sim 200\mathrm{\mu m}$ at the $800\mathrm{nm}$ wavelength. Even though many higher order leaky modes are supported in bores of this size, it is possible to excite only the ${\mathrm{HE}}_{11}$ mode if careful attention is paid to the launching optics. Such large-bore capillary fibers have been used with great success in the compression of very high energy pulses [21, 22, 23], and the generation of both low- [24] and high-order harmonics [25].

An additional attractive feature of a gaseous core, impossible in solid-core fiber, is that the nonlinearity and the group velocity dispersion (GVD) can be varied by changing the gas pressure. The GVD of the ${\mathrm{HE}}_{11}$ mode in an evacuated large-bore capillary is very weakly anomalous, whereas the GVD of noble gases is normal at optical frequencies. It turns out that the zero-dispersion points in a gas-filled capillary are shifted far into the infrared (IR) at useful gas pressures, and the overall GVD is strongly normal in the ultraviolet (UV) to near-IR (NIR) spectral region, greatly restricting the range of accessible ultrafast nonlinear dynamics.

HC-PCFs [26, 27, 28, 29, 30] overcome these limitations, offering ultralong single-mode interaction lengths ($\sim 10$ or even $\sim 1000\mathrm{m}$), high intensities per watt (due to the small core diameter), a high optical damage threshold, and exquisite control of the GVD. These features have been utilized in a wide range of experiments [31, 32], including clean demonstrations of solitary waves and self-similarity in stimulated Raman scattering [33, 34], well-controlled coherent atomic physics experiments [35, 36], delivery of intense, high-energy optical solitons [37], and particle guidance [38], to name just a few.

In this paper we discuss ultrafast nonlinear dynamics in HC-PCF. We briefly introduce the linear properties of the two main types of HC-PCF in Section 2, and then we concentrate on the advantageous properties of kagomé PCF for ultrafast applications. In Section 3 we present a generalized analysis, based on the soliton order and zero-dispersion wavelength (ZDW), which simplifies the description of the highly tunable properties of gas-filled kagomé PCF. This is followed by an overview of techniques to compress pulses to a few optical cycles with kagomé PCF (Section 4). A recently reported, highly efficient, compact, and tunable deep-UV source is discussed in Section 5. It has also recently been shown that soliton self-compression can lead to intensities above the ionization threshold of the filling gas in a kagomé PCF, allowing, for the first time, controllable and extended single-mode laser-plasma interactions in fiber. We review this new area of nonlinear fiber optics in Section 6. Finally, in Section 7 we offer a perspective on the use of kagomé PCF for high-harmonic generation (HHG).

2. PROPERTIES OF HC-PCF

HC-PCF comes in two main varieties [26, 27, 28, 29, 30]. The first, reported in 1999 [26], confines light by means of a full two- dimensional photonic bandgap (we call it PBG-guiding HC-PCF). The most common structure is formed by a hexagonal lattice of air holes, as shown in Fig. 1a. It can provide extremely low transmission loss ($<1\mathrm{dB}/\mathrm{km}$ at $1550\mathrm{nm}$ in the best case) [39], guiding a tightly confined single mode over a restricted spectral range. This feature makes it useful for spectral filtering, for example, to suppress unwanted Stokes bands in Raman scattering [40]. The GVD of PBG-guiding HC-PCF has a steep slope, passing through zero inside the transmission window and attaining very large values at its edges. To illustrate this, Fig. 1b shows the results of finite-element modeling (FEM) calculations [41] for an ideal structure designed to operate at $800\mathrm{nm}$. The calculated loss around $800\mathrm{nm}$ is $0.01\mathrm{dB}/\mathrm{m}$. The ZDW is at $768\mathrm{nm}$, and the dispersion changes considerably across the narrow transmission band, with a dispersion slope greater than $2000{\mathrm{fs}}^3/\mathrm{cm}$ at the zero- dispersion point. The very low loss makes these fibers useful for a wide range of applications, but the limited transmission windows and extreme dispersion slopes prevent application to extreme ultrafast pulse experiments.

The second type of HC-PCF, first reported in 2002 [3], has a kagomé-lattice cladding, characterized by a star-of-David pattern of glass webs, as shown in Fig. 1c. It provides ultrabroadband (several hundred nanometers) guidance at loss levels of $\sim 1\mathrm{dB}/\mathrm{m}$, and it displays weak anomalous GVD ($|{\beta }_2|<15{\mathrm{fs}}^2/\mathrm{cm}$, when evacuated) over the entire transmission window, with a low dispersion slope. FEM calculations, for one example structure ($30\mathrm{\mu m}$ core diameter, designed for operation in the UV and around $800\mathrm{nm}$), are shown in Fig. 1d, illustrating the broadband guidance windows ($200–350\mathrm{nm}$ and $600–850\mathrm{nm}<2\mathrm{dB}/\mathrm{m}$), with relatively small all-anomalous dispersion magnitude (below $15{\mathrm{fs}}^2/\mathrm{cm}$) across the whole band, excluding the anticrossing with a strong cladding resonance at $\sim 380\mathrm{nm}$. Such cladding resonances disrupt the transmission window but are usually quite narrow and so have little influence on the global dispersion properties. Also, they can be tuned away from the wavelength bands required in specific applications by varying the web thickness. As we shall see, unlike for capillary fibers, the normal GVD of a noble gas can be balanced against the anomalous GVD of the kagomé PCF, allowing the ZDW to be tuned across the UV, visible, and NIR spectral regions, simply by varying the pressure. This makes the system ideal for ultrafast applications [42]. We now take a closer look at kagomé PCF.

2A. Dispersion and Loss of Gas-Filled Kagomé HC-PCF

To a good approximation, the modal refractive index of kagomé HC-PCF closely follows that of a capillary fiber [42, 43, 44, 45]:

Given that kagomé PCF has dispersion properties so similar to those of a simple capillary, why bother to use it at all? The reason is that it has much lower loss than is theoretically possible in a capillary fiber [Fig. 3a]. For core diameters of $\sim 30\mathrm{\mu m}$ (required for anomalous dispersion at reasonable levels of nonlinearity—see below), the loss in a capillary exceeds several $100\mathrm{dB}/\mathrm{m}$, whereas that of a kagomé PCF can be below $1\mathrm{dB}/\mathrm{m}$.

Apart from broadband guidance with sufficiently low loss, another key feature that makes kagomé PCF particularly useful in ultrafast applications is its pressure-tunable dispersion. Figure 3c shows how the ZDW of an Ar-filled kagomé PCF ($30\mathrm{\mu m}$ core diameter) can be tuned across the whole UV- visible spectral region, allowing, for example, solitons to form in the deep UV. It is interesting to compare this behavior with that of solid-core PCF. Although known for the wide flexibility of its dispersion, the strong material dispersion of the glass limits the shortest attainable ZDW to $\sim 500\mathrm{nm}$ [Fig. 3b]. Whereas solid-core PCF permitted soliton propagation at $800\mathrm{nm}$ and into the visible spectral region [47, 48], kagomé PCF brings anomalous dispersion to the UV for the first time. Additionally, the dispersion magnitude is much smaller in kagomé PCF, which means that ultrafast pulses broaden much less quickly, and pulses at discrete frequencies walk off less rapidly, increasing the nonlinear interaction length compared to solid-core PCF. The relatively smaller dispersion slope also means that perturbations to soliton propagation are reduced, leading, for example, to the generation of shorter self-compressed pulses (see Subsection 4B).

Figure 4 summarizes how the ZDW can be extensively tuned by varying the gas type, pressure, and core diameter. Not only can it be pushed deep into the UV, but it also can be tuned into the IR beyond $1000\mathrm{nm}$, allowing normal dispersion at $800\mathrm{nm}$—useful for applications such as the fiber-grating/mirror compressors discussed below.

2B. Nonlinearity of Gas-Filled Kagomé PCF

Assuming a modal intensity distribution in the form $J_0^2(u_{11}r/a)$ for the fundamental ${\mathrm{HE}}_{11}$ mode, with r the radial coordinate, the conventional definition of effective mode area [7] yields $A_{\text{eff}}\sim 1.5a^2$; i.e., a kagomé PCF with a $30\mathrm{\mu m}$ core diameter will have $A_{\text{eff}}=338{\mathrm{\mu m}}^2$. Experiments and numerical modeling show that the contribution of the glass to the nonlinearity of kagomé PCF is usually negligible because the fraction of light in glass is $<0.01\%$ away from the cladding resonances [Fig. 5a] [49, 50].

Gas nonlinearity scales linearly with gas density, or with pressure at constant temperature [51]. For a kagomé PCF with a $30\mathrm{\mu m}$ core diameter, the nonlinear coefficient γ is (at $10\mathrm{bar}$): $1.5\times {10}^{-7}{\mathrm{m}}^{-1}{\mathrm{W}}^{-1}$ (Ne), $2\times {10}^{-6}{\mathrm{m}}^{-1}{\mathrm{W}}^{-1}$ (Ar), $5.3\times {10}^{-6}{\mathrm{m}}^{-1}{\mathrm{W}}^{-1}$ (Kr), and $1.5\times {10}^{-5}{\mathrm{m}}^{-1}{\mathrm{W}}^{-1}$ (Xe). These are over 4 orders of magnitude smaller than in a solid-core PCF ($0.24{\mathrm{m}}^{-1}{\mathrm{W}}^{-1}$ for a core diameter of $1\mathrm{\mu m}$) but much higher than in a gas-filled capillary with a bore diameter of $200\mathrm{\mu m}$ ($\sim 5\times {10}^{-8}{\mathrm{m}}^{-1}{\mathrm{W}}^{-1}$ at $10\mathrm{bar}$ Ar).

Raman scattering is absent in monatomic gases such as Ar, allowing for the study of soliton dynamics and pulse compression in the absence of the soliton self-frequency shift. However, by choosing molecular gases such as ${\mathrm{N}}_2$ or ${\mathrm{SF}}_6$, Raman scattering can also be studied in kagomé PCF. For ultrafast applications, one intriguing possibility is to mix gases with differing Raman contributions and hence add to our pressure-tuned system the ability to continuously adjust the fractional Raman contribution.

Apart from perturbative bound electron nonlinearities, the intensities reached in kagomé PCF at few-microjoule energy levels can be sufficient to ionize the gas and produce a partial plasma. Although the field-plasma interaction itself is usually a linear process at the free-electron densities and optical intensities encountered in kagomé PCF, the electron generation process is both ultrafast and highly nonlinear, leading to new nonlinear effects such as a soliton self-frequency blueshift, which we discuss in Section 6.

2C. Energy Handling of Kagomé HC-PCF

Finite-element calculations of the power-in-glass fraction, shown in Fig. 5a, show that, at around $800\mathrm{nm}$, over 99.99% of the light can be guided in the hollow regions of a kagomé PCF, meaning that optical damage thresholds are much higher than in glass-core fibers. The power-in-glass fraction significantly increases at cladding resonances (there is one at $\sim 380\mathrm{nm}$ in Fig. 5), where light couples into the glass webs [compare Figs. 5b, 5c]. Optical damage is far more likely at these resonances; thus, it is important to locate them away from operating wavelengths by tuning the web thickness. Additionally optimizing launch efficiency is important because slight input coupling misalignment excites cladding modes, also leading to optical damage.

In capillary fibers with core diameters of $45–70\mathrm{\mu m}$, guided intensities over ${10}^{16}\mathrm{W}/{\mathrm{cm}}^2$ have been demonstrated [52]; guidance of similar intensities in kagomé PCF can be expected. Our group has coupled ultrafast pulses ($65\mathrm{fs}$) with over $12\mathrm{\mu J}$ into a kagomé PCF with a $26\mathrm{\mu m}$ core diameter, with efficiencies up to 80%. After self-compression (Subsection 4B), we have reached intensities of over ${10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ [5]. By careful mode matching, another group reported 98% launch efficiency of $40\mathrm{fs}$ pulses into a PBG-guiding HC-PCF (core diameter $6.8\mathrm{\mu m}$), reaching a transmitted energy of $\sim 2\mathrm{\mu J}$ [53].

3. ULTRAFAST NONLINEAR DYNAMICS IN KAGOMÉ PCF

The experimental setup typically used for ultrafast pulse propagation experiments in kagomé PCF filled with gas is illustrated in Fig. 6. It consists of two gas cells, one at each fiber end, both equipped with a high-quality window for launching and extracting light. The fiber is first evacuated, purged several times, and then filled with the gas to the desired pressure. Pressure gradients can be easily achieved, and gas filling times are $<1\min$ for the pressures and core sizes used [54]. The simplicity of this setup belies its versatility, as we shall see in the next sections.

All of the numerical results presented in this paper are calculated using a unidirectional field propagation equation [45], including the full dispersion curve based on Eq. (1), the Kerr effect, photoionization, and the influence of free electrons created through photoionization (Appendix A).

3A. Interplay of Dispersion and Nonlinearity

Ultrafast nonlinear dynamics depend on a complicated interplay between the dispersion and nonlinearity. A combined analysis is therefore necessary. Here we use a simplified envelope equation to facilitate discussion of the physics underlying ultrafast processes in gas-filled kagomé PCF. Taking the guided mode to be linearly polarized (sufficient for comparing the global properties of both kagomé PCF and capillary fiber), the generalized nonlinear Schrödinger equation (GNLSE) takes the normalized form [7]:

3B. Energy Scaling of Solitons: Dependence on the Dispersion Landscape

The extraordinary scalability of the kagomé PCF system can be fully appreciated by considering again Fig. 4. For a given choice of (${\lambda }_0$, N), we can select a gas species and, for a wide choice of kagomé PCF diameters, find the correct filling pressure to fix ${\lambda }_0$. We then simply choose the correct pulse energy to obtain the given soliton order N. Figures 7a, 7b, 7c show the range of energies required for given (${\lambda }_0$, N) for some selected gas species and kagomé PCF diameters, illustrating that, for example, the dynamics of ${\lambda }_0=600\mathrm{nm}$ and $N=9$ can be accessed with pulse energies ranging from $\sim 100\mathrm{nJ}$ to $\sim 10\mathrm{\mu J}$.

For soliton orders $2\le N\le 15$, while pumping in the anomalous dispersion region, the propagation dynamics are usually dominated by soliton fission, which approximately occurs at a characteristic length scale [6, 15]:

From the discussion in Subsection 2C (on the energy- handling capabilities of kagomé PCF) it is clear that, for some of the parameter sets in Fig. 7, the energies required may be too high. Even so, all of the dynamics can be accessed by at least one system without exceeding the damage threshold. For example, $\sim 1\mathrm{mJ}$ of energy is required to create an eighth- order soliton in a kagomé PCF with a core diameter of $50\mathrm{\mu m}$ filled with He so as to yield a ZDW of $250\mathrm{nm}$. Such a high pulse energy may be beyond what such a kagomé PCF can handle; however, the same dynamics would result if the core diameter is reduced to $10\mathrm{\mu m}$, He is replaced with Xe, and the pulse energy is lowered to $10\mathrm{\mu J}$—within the capabilities of kagomé PCF.

A more difficult question is whether self-focusing due to the Kerr effect, or defocusing due to free-electron creation can significantly disturb the guided mode. In the case of large-bore capillary fibers, it was found that spatial perturbations to the leaky modes occurs at power levels similar to those required for critical self-focusing in bulk gas [55, 56]. Although the launched peak pulse power remains below this limit in every case in Fig. 7, the pulses can undergo strong temporal compression as they propagate (see Subsection 4B). To quantify this, Fig. 7d plots the ratio of the peak power after self- compression, to the critical self-focusing power at the relevant gas pressure. This ratio reaches a maximum value of $\sim 0.2$ for ${\lambda }_0=600\mathrm{nm}$ and $N=9$, which are approximately the parameters used in [49], where no degradation in modal quality was observed. This is probably because the peak self- compressed intensity is maintained over too short a distance for self-focusing to have any effect.

In the following sections we discuss the ultrafast dynamics in a number of exemplary systems, characterized by the (${\lambda }_0$, N) values summarized in Table 1 and indicated with colored dots and labeled (i) to (iii) in Figs. 7, 15.

4. PULSE COMPRESSION

4A. Fiber-Grating/Mirror Compression

A common pulse-compression technique is a two-stage process: in the first stage nonlinear spectral broadening is produced by self-phase modulation (SPM), which, in the presence of correctly tuned normal dispersion, produces a parabolic temporal phase, i.e., a linear chirp; in the second stage the parabolic phase is compensated for in an anomalously dispersive external compressor, such as a bulk grating or a chirped mirror. This is known as fiber-grating or fiber-mirror compression, depending on which dispersive compression element is used. It has been applied successfully to the generation of few-cycle pulses both in solid-core fiber [57, 58] and at very high pulse energies in gas-filled capillary fibers [21, 22, 23].

PBG-guiding HC-PCF can be designed to have very low nonlinearity and strong anomalous dispersion. These properties have been used to replace the grating/mirror in the compression stage of an all-fiber pulse-compression system [59, 60, 61] at peak powers of $\sim 1\mathrm{MW}$, well above what is possible in solid-core fibers because of nonlinear effects during compression and the onset of optical damage. However, the restricted guidance bandwidth and large dispersion slope limit the use of PBG HC-PCF in the nonlinear spectral broadening stage.

Kagomé PCF offers the opportunity of performing both stages of this pulse-compression technique in fiber because it has a broad guidance band, tunable GVD, and, most importantly, a small GVD slope. Additionally, the nonlinearity of kagomé PCF is intermediate between solid-core and hollow capillary fibers (Subsection 2B), uniquely enabling its use in the nonlinear spectral broadening stage at pulse energies ($0.1–100\mathrm{\mu J}$) much lower than possible with capillary fibers, but significantly exceeding what solid-core fiber can handle. Results on spectral broadening in Xe-filled kagomé PCF, and external compression, have been reported [62] at high average power, demonstrating a compression factor of $\sim 4$ for $\sim 1\mathrm{ps}$ pulses. However in that system, the dispersion was weakly anomalous, which is suboptimal for the production of clean parabolic pulses for optimal external compression to a few optical cycles.

4A1. Illustrative Design

Figure 8 shows numerical results for one potential system design using a kagomé PCF for the spectral broadening stage, and a chirped mirror as the linear compressor. Figure 8a shows propagation of a $10\mathrm{\mu J}$, $30\mathrm{fs}$ pulse in a kagomé PCF with a core diameter of $70\mathrm{\mu m}$ filled with $43\mathrm{bar}$ of Ar. This causes the ZDW to shift to $1300\mathrm{nm}$ so that the pulse (at $800\mathrm{nm}$) is propagating in the normal dispersion region. Both temporal and spectral broadening are observed, as expected for a combination of normal dispersion and SPM. Figure 8b shows the broadened temporal shape of the intensity and the smooth parabolic temporal phase after $8\mathrm{cm}$ of propagation. This phase profile can be compensated for with a linear compressor such as a chirped mirror. In Fig. 8c we plot the duration that would be achieved by linearly compressing the spectrally broadened output pulses from the given length of kagomé PCF. After $8\mathrm{cm}$ of propagation, the pulse can be compressed to $\sim 3\mathrm{fs}$ after the equivalent of approximately two bounces from commercially available chirped mirrors. The resulting compressed pulse shape is shown in Fig. 8d, showing a very clean few-cycle electric field, with a corresponding increase of peak power from 0.3 to $3\mathrm{GW}$. Note that the final peak power does not occur inside the fiber—where the pulse is still temporally broadened—but after a free-space compression stage, allowing much higher peak powers than can exist inside the gas-filled kagomé PCF. The quality of the fiber- grating compression scheme is quantified using a quality factor defined as the ratio of the energy within 1 FWHM intensity of the compressed pulse to that contained within 1 FWHM of the uncompressed pulse. In the example in Fig. 8d, this factor is over 80%, and in fact from Fig. 8c (right-hand axis) we see that it never falls below 70% in this particular system design.

Further energy scaling may require the use of positive gas gradients, as demonstrated in capillary systems [23]. In this way, the impact of ionization is reduced at the input of the fiber, where the peak intensity is highest, due to low gas pressure. As the pulse temporally broadens on propagation, the intensity decreases [Fig. 8a], but with a positive pressure gradient, consistent SPM can be achieved. Alternatively, downtapered kagomé PCF may be used to reduce the intensity at the input end of the fiber, while preserving SPM broadening and normal dispersion throughout propagation.

4B. Soliton-Effect Compression

A different scheme involves soliton-effect self-compression, as commonly implemented in solid-core fiber [63, 64, 65] but ideally suited to kagomé PCF. In this process, the interplay of anomalous dispersion and SPM leads to pulse self-compression—the initial stages of higher order soliton propagation. The use of PBG-guiding HC-PCF for this process has been successfully demonstrated [66, 67, 68], allowing the generation of pulses with megawatt peak powers, beyond the damage limits of fused silica. Kagomé PCF is even more advantageous because of its inherently weaker higher order dispersion—often the limiting factor in extreme pulse compression—and its broader guidance band.

While exploring the efficient generation of deep-UV light in an Ar-filled kagomé PCF, Joly et al. [49] experimentally demonstrated soliton-effect self-compression of $\sim 30\mathrm{fs}$ pulses to shorter than $10\mathrm{fs}$. In these experiments, the PCF was much longer than the optimum compression length. Numerical simulations show that pulses as short as $\sim 4\mathrm{fs}$ must have occurred at the position of maximum compression, coinciding with the onset of UV emission (see Section 5).

4B1. Compression Ratio and Quality Factor Scaling

Detailed analysis of a large number of numerical simulations [69, 70] suggests the following scaling rules for compression ratio and quality factor in terms of soliton order:

4B2. Effect of Longer Pulses

What happens to the compression if we increase the pump pulse duration? From Eq. (6) we can write the compressed pulse duration ${\tau }_c$ in the following form:

4C. Adiabatic Pulse Compression

Adiabatic soliton compression [72, 73, 74] has also been demonstrated in tapered PBG-guiding HC-PCF [75]. Numerical analysis suggests that the shortest pulse durations can be reached if a pressure gradient is applied so as to cause the GVD to decrease with propagation distance [76], thus avoiding the need to taper the fiber structure itself. Simulations also show that the combination of decreasing dispersion and soliton-effect compression allows for higher compression ratios without impairment to the quality factor.

Kagomé PCF can be used for adiabatic compression, and a positive pressure gradient would lead to both increasing γ and decreasing ${\beta }_2$, perfect for this process. However, the fiber length must be many times the dispersion length if the compression is to remain adiabatic, so that this technique would be restricted to only the shortest input pulse durations because best-case losses of $\sim 0.3\mathrm{dB}/\mathrm{m}$ limit the usable fiber length to $\sim 10\mathrm{m}$.

5. DISPERSIVE-WAVE GENERATION IN THE UV

Recently, highly efficient deep-UV generation was reported using Ar-filled kagomé PCF [49]. The emitted UV wavelength was tunable by changing the gas pressure and hence the dispersion. To illustrate these results, Fig. 10a shows a series of experimentally achieved UV spectra obtained with an Ar-filled kagomé PCF with a $27\mathrm{\mu m}$ core diameter, pumped with $45\mathrm{fs}$ pulses at $800\mathrm{nm}$ [77]. Each spectrum was obtained by tuning the pulse energy ($0.5–3\mathrm{\mu J}$) and filling pressure ($4–12\mathrm{bar}$) to demonstrate the range of clean UV output tunable from $200–300\mathrm{nm}$. The spectra have relative widths of $\sim 0.03$ and can support ultrafast pulses. Figure 10b shows the experimentally achieved UV conversion efficiencies from [49], where 6% to 8% conversion was measured over a wide power tuning range, indicating output energies in the UV of around $50–80\mathrm{nJ}$ for a kagomé PCF with ${\lambda }_0\sim 600\mathrm{nm}$, $N\sim 9$ [corresponding to point (i) in Figs. 7, 15].

5A. Phase Matching to Dispersive Waves

The generation of UV light results from extreme soliton-effect pulse compression of the input pulse, resulting in a spectral expansion that overlaps with resonant dispersive-wave frequencies, which are consequently excited [78, 79, 80, 81]. Solitons are stable close to their central frequency, because they propagate with a higher phase velocity than dispersive waves with the same frequency [9, 10]. However, in the presence of higher order dispersion, they can phase match to dispersive waves at other frequencies [79, 80, 81]; i.e., $\beta (\omega )={\beta }_{\text{sol}}(\omega )$, where $\beta (\omega )=n_{mn}k$ is the linear mode propagation constant and

Further evidence of this scaling can be seen in Figs. 12a, 12b, which show that both $50\mathrm{nJ}$ and $10\mathrm{\mu J}$ pulses can experience essentially identical spectral evolution along the fiber, including clear UV dispersive-wave emission at $230\mathrm{nm}$, even though they have very different physical parameters.

To illustrate the (as-yet unexplored) degree to which this system may be extended in both wavelength and UV energy, Figs. 12c, 12d show the temporal and spectral evolution of a $10\mathrm{fs}$ pulse with $20\mathrm{\mu J}$ energy propagating in a $50\mathrm{\mu m}$ kagomé PCF filled with $10\mathrm{bar}$ He (corresponding to ${\lambda }_0=380\mathrm{nm}$, $N=3$). After the characteristic pulse self-compression at around $5\mathrm{cm}$, a UV dispersive wave at $150\mathrm{nm}$ is emitted, containing over $3.5\mathrm{\mu J}$ (17.5% of pump energy). Such high energies in the vacuum-UV region, coupled with the inherent wavelength tunability of this system, should lead to numerous applications (Subsection 5C).

5B. Efficiency of Deep-UV Generation

The efficiency of dispersive-wave generation depends strongly on the spectral power density of the compressed pulse at the phase-matched frequency. Tight temporal self-compression is therefore critical (see Subsection 4B) if the initial pump spectrum is to broaden far into the UV. In [49] it was found that in the kagomé PCF system, self-steepening and optical shock formation [83], which asymmetrically enhance the blue edge of an SPM broadened spectrum, were the keys to enhancing this process.

What effect does the input pulse duration have? In Fig. 13a we see that $>15\%$ conversion efficiencies to the UV (fraction of total power at wavelengths shorter than $350\mathrm{nm}$) are possible regardless of pulse duration and for a wide range of pump energies. We quantify the quality of the UV emission with the factor $Q_{\mathrm{UV}}=P_{\mathrm{FWHM}}/P_{\mathrm{UV}}$, where $P_{\mathrm{FWHM}}$ is the spectral power within the FWHM of the strongest UV peak and $P_{\mathrm{UV}}$ is the total spectral power in the UV region. Figure 13b shows that $Q_{\mathrm{UV}}$ is strongly dependent on the pump pulse duration. For $15\mathrm{fs}$, over 90% of the UV power is in the main UV peak for a wide range of pump parameters, compared to less than 30% for a $120\mathrm{fs}$ pulse. This is clearly evident in Fig. 13c, which shows the spectral evolution of 15, 30, 60, and $90\mathrm{fs}$ pulses along the fiber for a normalized soliton order of $S=N/{\tau }_{\mathrm{FWHM}}=0.26$ ($N=7.8$ for $30\mathrm{fs}$ pulse duration). For the $15\mathrm{fs}$ pulse, a high-quality UV band emerges approximately at the soliton fission length [Eq. (6)]. For the $30\mathrm{fs}$ pump, the UV band is still of relatively high quality, but at $60\mathrm{fs}$, it degrades considerably, and at $120\mathrm{fs}$, evidence of MI can be observed—the UV band developing a considerably fine structure, as is visible in the spectral slices in Fig. 13d. The reason for this quality degradation is the reduced quality factor of the pulse self-compression for high values of N. Pulse durations of $\sim 30\mathrm{fs}$ or shorter are necessary for obtaining high-quality UV spectra at high conversion efficiencies. High-quality conversion is possible with much longer pulses if we reduce the peak power but with significantly lower efficiency. A viable approach for the efficient generation of high-quality UV light from longer pump pulses would be to make use of the fiber-grating/mirror compression system proposed in Subsection 4A, with the UV system as a third stage.

The emitted UV light can be a few optical cycles at the point of generation, but it propagates in a region of normal dispersion, resulting in group velocity broadening. The relatively flat and simple dispersion properties of the kagomé PCF should, however, make it straightforward to externally compress the chirped UV pulses at the output of the fiber. Compression may even be achievable by propagation through an evacuated kagomé PCF, which has anomalous dispersion at all wavelengths.

5C. Applicability of UV Source

These results suggest that an ultrafast coherent light source could be constructed that tunes continuously from 500 to $150\mathrm{nm}$. Whether a further extension deeper into the UV is possible is an open question. Phase matching could be achieved to well below $150\mathrm{nm}$ if the pump wavelength is shifted to $400\mathrm{nm}$. But nonlinear absorption and two-photon resonances will disrupt the process at certain wavelengths that depend on the filling gas (in the vacuum-UV for noble gases such as Ar and He). Furthermore, ionization becomes increasingly disruptive to phase-matching at shorter wavelengths [45].

The fact that the same tunable UV emission dynamics can be achieved with both $\sim 100\mathrm{nJ}$ and $\sim 10\mathrm{\mu J}$ pulses, with similar efficiencies, promises wide applicability of this technique. Energies of $>100\mathrm{nJ}$ are available from high-repetition-rate chirped oscillators [84] and ultrafast fiber lasers [85], enabling highly compact UV sources operating at repetition rates suitable for creating deep-UV frequency combs—with numerous applications in spectroscopy. Alternatively, the use of the $\sim 10\mathrm{\mu J}$ pumped system should enable the generation of deep- UV pulses with energies in excess of $1\mathrm{\mu J}$. Such a simple and compact source of spatially coherent, ultrafast deep-UV light has a wide range of potential applications, including femtochemistry [86] and UV-resonant Raman spectroscopy [87].

Another promising application is the seeding of free- electron lasers (FELs) [88], which have the major advantage over traditional laser systems of providing gain over the entire electromagnetic spectrum [89]. When seeded, the temporal coherence of the light emitted by an FEL is greatly improved and pulse-to-pulse energy fluctuations are reduced. Conventional lasers have been used to seed FELs in the visible and near IR, but practical seed lasers are unavailable in the UV. Instead, UV seed light has been generated in nonlinear crystals or by HHG in noble gases (see Section 7) [90]. Recent numerical simulations, using the kagomé-based UV source described above as the seed in a single-pass FEL, show that the energy available at $300\mathrm{nm}$ is sufficient to achieve saturation (i.e., exhaust the exponential FEL gain) [88].

5D. UV Supercontinuum Generation

Raising the soliton order to $N\sim 245$ by launching, e.g., $10\mathrm{\mu J}$ $600\mathrm{fs}$ pulses moves the system well into the regime of MI, where smooth and broad (although temporally incoherent) supercontinuum generation can be expected [15, 91]. To illustrate this, Fig. 14 shows propagation of such a long pulse in a kagomé PCF with ${\lambda }_0=750\mathrm{nm}$ ($25\mathrm{bar}$ Ar, $30\mathrm{\mu m}$ core diam eter). Figure 14a shows the temporal evolution of one shot, and Fig. 14b shows the spectral evolution of an ensemble average of 30 simulations (in the MI regime, each laser shot produces a different spectrum, modeled by including quantum noise and averaging over multiple shots). In the temporal picture, the splitting of the input pulse into a large number of ultrashort solitons, that subsequently undergo multiple collisions, can be clearly observed. In the spectral domain, this leads to a smooth and flat, high-energy supercontinuum spanning the range from 350 to $1500\mathrm{nm}$. Note that in the absence of Raman scattering, the MI supercontinuum shows a blue- enhanced asymmetry, in contrast to what is seen in glass-core fibers.

The unique opportunity to tune the ZDW in kagomé PCF into the UV enables us to shift these dynamics to shorter wavelengths. For example, if we pump with the same param eters as above, but at $400\mathrm{nm}$ in a fiber designed for zero dispersion at $350\mathrm{nm}$ ($8.8\mathrm{bar}$ Ar, $10\mathrm{\mu m}$ core diameter), we obtain a very flat and smooth, MI-based, high-energy supercontinuum extending from $140–1000\mathrm{nm}$ [Figs. 14c, 14d]; experimental realization of such a system would provide a unique source for metrology and spectroscopy.

6. PLASMA-INFLUENCED NONLINEAR FIBER OPTICS

The peak intensities attainable after self-compression of few-microjoule ultrafast pulses in a kagomé PCF can be sufficient to ionize the gas, thus creating a partial plasma inside the fiber. This allows, for the first time, controlled light–plasma interactions in an anomalously dispersive waveguide geometry. In recent experiments, estimated peak intensities of $\sim 2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ were reached, causing plasma generation and a soliton blueshift [5]. This experiment demonstrated that high-field effects can be observed in kagomé PCF at relatively low pump energies ($<10\mathrm{\mu J}$), compared to other systems.

6A. Comparison with the Kerr Effect

The influence of a partial plasma on pulse propagation is determined by the evolution of the free-electron density, which depends on the time-dependent interaction of the strong electric field with the atoms in the fiber core. Some insight into the strength of the plasma effects can be gained by considering the peak free-electron density throughout pulse propagation. Figure 15a shows the calculated peak free-electron density $N_e$ in a $30\mathrm{\mu m}$ diameter kagomé PCF as a function of ZDW and soliton order (see Appendix A). Under the conditions of deep-UV generation [marked with (i), Section 5], or of optimum soliton-effect compression [marked with (iii), Subsection 4B], the free-electron densities are below ${10}^{22}{\mathrm{m}}^{-3}$. But when the ZDW is shifted to shorter wavelengths, which requires a reduction in gas pressure, the intensity required for soliton formation increases, so that, after self-compression, solitons of order 4 to 10 experience free-electron densities of $\sim {10}^{23}$ to $>{10}^{24}{\mathrm{m}}^{-3}$, indicated by the vertical dashed arrow in Fig. 15a.

What is the effect of these free-electron densities? The strong negative polarizability of the free electrons causes a time-dependent reduction in the local refractive index, opposing the increase due to the optical Kerr effect of the noble gas in the visible-IR spectral region. The relative importance of the plasma and Kerr effects can be estimated by considering the ratio between the refractive index changes caused by the two processes [5, 92]:

Figure 15b shows R for the same parameters as in Fig. 15a. Over most of the parameter range, the Kerr effect dominates; for example, at free-electron densities of $\sim {10}^{22}{\mathrm{m}}^{-3}$ the Kerr effect is over 10 times stronger almost independently of the ZDW (or gas pressure). However at $N_e\sim {10}^{23}{\mathrm{m}}^{-3}$, the effects become comparable, and the ratio $R\sim 1$ [indicated by the black dashed curve in Fig. 15b]. The experiments by Hölzer et al. [5] operated in this regime [marked by the vertical dashed arrow in Fig. 15a].

As shown in Fig. 16a, the propagation of $65\mathrm{fs}$ pulses in a $26\mathrm{\mu m}$ diameter kagomé PCF filled with $1.7\mathrm{bar}$ Ar (corresponding to a ZDW at $380\mathrm{nm}$) leads to the generation of blueshifted sidebands. The first one occurs at a pump energy of $\sim 2\mathrm{\mu J}$. Detailed investigations have shown that extended, and more intricate, light–plasma interactions occur in this system [5, 45], for example, multiple compression points and the subsequent emission of additional blueshifted sidebands in Fig. 16a. Comparison with full propagation simulations [Fig. 16b] shows excellent agreement only when the ionization terms are included, and analysis of transmission losses also show excellent agreement only when photoionization- induced losses are accounted for [Fig. 16c].

6B. Soliton Blueshift

Because the creation of free electrons is highly nonlinear with pulse intensity, a fast transient change in the refractive index is created by few-cycle pulses, which does not recover on ultrafast time scales due to the slow rate of electron recombination (Appendix A). This creates a steep positive phase-shock across the optical pulse and induces a blueshift, as has been thoroughly investigated in bulk geometries [93, 94, 95]. Results by Saleh et al. [96] show that solitons propagating in a kagomé PCF undergo a self-frequency blueshift under these conditions, opposite in sign to the well-known redshift caused by intrapulse Raman scattering and that when combined, the two shifts can cancel out. To illustrate this, Saleh et al. derived an equation which extends the conventional GNLSE to account for free-electron effects on the propagation of the complex field envelope $\psi (z,t)$ [96]:

The first two terms on the right-hand side are the conventional dispersion and nonlinear operators, well known in nonlinear fiber optics [7]. The last two terms are novel in the context of fiber optics (although well known in other communities), and represent phase modulation caused by free electrons and photoionization-induced losses. It is worth noting that this is the first major addition to the GNLSE for over two decades, introducing significant new dynamics into the already rich field of nonlinear fiber optics. In a noble gas-based system without Raman scattering, Eq. (11) simplifies further, allowing the dynamics to be considered as weakly perturbed soliton solutions at certain peak power levels, allowing the derivation of analytical solutions and consequently improving physical insight [96].

One such analytical solution describes the continuous frequency blueshift of a soliton propagating with just sufficient peak intensity to ionize the medium. Figure 17 shows numerical simulations of an $N=4$ soliton propagating in a $30\mathrm{\mu m}$ diameter Ar-filled kagomé PCF with a ZDW at $\sim 380\mathrm{nm}$, corresponding to the parameters of the experimental results shown in Fig. 16. A clear blueshift is seen in the spectrum after $\sim 12\mathrm{cm}$ of propagation, confirming the analysis of Saleh et al. and allowing us to attribute the blueshifted sidebands in Fig. 16 to the presence of blueshifting solitons.

6C. Prospects for Plasma Generation in Fiber

Although the effects of partial plasmas have been studied in many other systems, the dispersion and guiding properties of kagomé PCF introduce unique possibilities. In capillary fibers, the lack of anomalous dispersion at reasonable pressures prevents the observation of soliton effects (see Section 2). In glass-core systems, ionization is synonymous with material damage. In bandgap fibers, although small plasma blueshifts (a few nanometers) have been observed [97, 98], the restricted guidance band and strong dispersion limit the soliton effects. Finally, in filaments in free space, stable self-guidance requires the free-electron density to be fixed relative to the Kerr nonlinearity [92, 99], again limiting the range of possible dynamics. In contrast, kagomé PCF offers an elegant and simple means to study light–plasma interactions in a well-controlled transverse mode and over extended length scales.

It is likely that the results described above are merely the beginning of a new chapter on nonlinear single-mode fiber optics in the presence of free electrons. A number of directions are possible at such high in-fiber intensities. It would be worthwhile to study ionization in kagomé PCF filled with different gas types. For example, the rate of ionization in a gas with closely spaced ionization energies, such as ${\mathrm{SF}}_6$, appears to modify the propagation dynamics [45]. Also, the possibility of a third-order nonlinearity resulting from the free electrons was recently proposed [100], and it could be studied as a means of enhancing third-harmonic generation in fiber. Alternatively, Raman-like interactions with plasmas [101], used for high-pulse-energy amplification and compression, could be transferred to a fiber context. A further degree of freedom in plasma-influenced propagation in kagomé PCF would result if the plasma was generated with an external excitation source, such as electrodes [102] or direct microwave excitation [103], effectively removing the need for very high optical intensities in the gas. This would make possible the study of longer pulses or even CW light in a plasma-filled kagomé PCF, when ponderomotive effects could become important.

7. HIGH-HARMONIC GENERATION

A key driver in the development of few-cycle pulsed lasers has been HHG [104, 105, 106], which offers a means of producing coherent extreme-UV and x-ray radiation in the laboratory and is the foundation technology of attosecond science [107, 108, 109]. Given the discussions above on compression to few optical cycles and the generation of free electrons, it is natural to ask what the prospects are for HHG in kagomé PCF.

7A. HHG in HC-PCF

One preliminary demonstration has been reported using a kagomé PCF (core diameter $15\mathrm{\mu m}$) filled with $30\mathrm{mbar}$ of Xe [110]. A conversion efficiency of ${10}^{-9}$ up to the thirteenth harmonic was observed, which is several orders of magnitude below the non-phase-matched conversion efficiencies observed in free space. The threshold energy, however, was only $200\mathrm{nJ}$, whereas millijoule pulses are usually used with gas jets—a result of the intensity and interaction length enhancement possible in kagomé PCF. Although this experiment is useful as a proof of principle, higher efficiencies are desirable. Additionally, no use was made of pump pulse self-compression and nonlinear phase matching—uniquely offered by kagomé PCF—to enhance the process.

It is well established that the conversion efficiency to high harmonics can be significantly enhanced by phase-matching techniques [106]; for example efficiencies of up to $\sim {10}^{-5}$ have been obtained to individual harmonics with photon energies of $\sim 45\mathrm{eV}$ ($\sim 28\mathrm{nm}$) [25]. The use of capillary waveguides, rather than gas jets, adds an extra degree of tunability to achieve phase matching [106] and naturally transfers to HC-PCF; indeed, several techniques have been proposed for phase-matched HHG in PBG-guiding HC-PCF [111, 112]. However, the principal difficulty of phase matching to high harmonics in any waveguide geometry arises from the strong free-electron dispersion, which is unavoidable at the intensities required for very high order harmonics.

7B. Phase Matching with a Counterpropagating Wave

One elegant solution to this problem is the use of quasi-phase-matching with a counterpropagating laser beam [113]. In this approach the counterpropagating beam modulates the driving laser intensity. As a result, the phase acquired by the freed electron—which is determined by its trajectory under the influence of the pump laser field before it recombines with the atom—is also modulated, leading to phase-modulation of the HHG emission. Tuning the frequency and intensity of the counterpropagating beam therefore provides a simple means of controlling the phase matching. In capillary waveguides, for phase matching to very high photon energies, this technique would, however, require an intense laser operating at wavelengths in the far-IR. A theoretical analysis of trans ferring this technique to HC-PCF was reported in [114]. It suggested that using a correctly tuned counterpropagating laser at $1.6\mathrm{\mu m}$ (an easily attainable laser wavelength), one could phase match to the $\sim 99$th harmonic using a driving laser at $800\mathrm{nm}$ with only a few microjoules of pulse energy.

7C. Pumping with Ultrashort Mid-IR Pulses

There is ongoing work to create compound-glass-based kagomé PCFs, which should allow for low-loss optical guidance in the mid-IR spectral region [115]. This region is well known as being advantageous for extending the range of high harmonics that can be produced to the kiloelectron volt region, because the longer optical half-cycle means that the released electron can be accelerated to higher energy before it recombines with the atom. Although single-atom HHG efficiencies are lower when pumped at longer wavelengths, the combination of reduced ionization due to relaxed intensity requirements (and hence easier phase-matching) and higher gas transparency for high-energy electrons, means that overall efficiencies can be enhanced [106]. The possibility of transferring the advantages of kagomé PCF to HHG driven by mid-IR pulses is very attractive. This might allow few-cycle pulses created through self-compression to be used for phase-matched HHG in kagomé PCF, thus providing an all-fiber table-top x-ray (and possibly attosecond) light source.

8. CONCLUSIONS

HC-PCFs allow one to do single-mode nonlinear fiber optics in cores made from gases, vapors, and plasmas. Gas-filled kagomé PCF, in particular, has almost perfect dispersion and loss properties for ultrafast nonlinear fiber optics along with the ability to guide high intensities without glass damage. This opens the door to the combination of high-field physics with more conventional nonlinear fiber processes. Varying the combination of filling gas, pressure, and PCF design provides flexibility in both the operation wavelength and the energy scale of the desired system. Additionally, kagomé PCF extends the region of fiber-optic operation (both low-loss and zero-dispersion points) deep into the UV.

Few-cycle pulse compression is possible using either a fiber-grating/chirped-mirror scheme, or by soliton self- compression, with energies covering at least the $0.1–30\mathrm{\mu J}$ range. With the correct dispersion profile, self-compression can lead to a highly efficient and compact, tunable deep-UV source, tuning between $200–350\mathrm{nm}$ having been demonstrated, and $150–500\mathrm{nm}$ being realistically feasible. UV energies $>1\mathrm{\mu J}$, together with a threshold energy low enough to allow direct pumping from a high-repetition-rate oscillator, will enable the creation of an all-fiber deep-UV frequency comb. Many applications seem possible in spectroscopy and metrology or even in the seeding of a free-electron laser.

At few-microjoule pump energies, ultrafast pulses can self-compress to intensities over the ionization threshold of the filling gas, allowing, for the first time, controllable, extended, single-mode laser-plasma interactions in an anomalously dispersive fiber. This breaks new ground in nonlinear fiber optics, dramatically increasing the range of phenomena possible in fibers beyond those possible in solid-core fibers, including, for example, a soliton self-frequency blueshift. Finally, phase-matched HHG in kagomé PCF may lead to the creation of an all-fiber table-top x-ray light source.

APPENDIX A

All of the numerical results presented in this paper are calculated using a unidirectional field equation [45], including the full dispersion curve based on Eq. (1), the Kerr effect, and the influence of free electrons created through photoionization.

In these calculations, the free electron density is given by [92]

(A1)$$\frac{\partial N_e}{\partial t}=W(t)N_a-\eta N_e-{\beta }_r{N}_e^2,$$

where $W(t)$ is the ionization rate of the atoms in the presence of the time varying electric field $E(z,t)$, $N_a$ is the density of neutral atoms, and η and ${\beta }_r$ are the electron attachment and recombination rates, respectively, both of which are negligible for durations $<\sim 1\mathrm{ps}$ [92]. The calculation of the free- electron density $N_e(t)$ requires the use of a model for the ionization rate $W(t)$. In this work, we used the Yudin–Ivanov modification of the Perelomov–Popov–Terent’ev technique [116, 117], which accounts for both tunnel- and multiphoton-based ionization; although it should be noted that simulations using the formulation in [45] have accurately recreated experimental results discussed in this review when using the Ammosov–Delone–Krainov (ADK) rate [118], indicating that the parameters chosen where within the tunnel ionization regime [5, 45].

The full free-electron density calculations using Eq. (A1) and the associated models for $W(t)$ are not straightforwardly amenable to analytical manipulation. Therefore, Saleh et al. [96] used a simplified model to derive their results, based on a linear approximation to the ionization rate, valid in a restricted range of intensities above the ionization threshold. This works surprisingly well due to the self-limiting of pulse peak intensities resulting from photoionization-induced optical loss.

ACKNOWLEDGMENTS

The authors thank K. F. Mak and P. Hölzer for providing some of the materials for the figures and for useful discussions and suggestions.

Table 1. Examples of Parameters Describing Certain Sets of Propagation Dynamics (as Described in the Text) Scaled across Different Core Diameters (d), Filling Gases and Gas Pressures (P), and Energies (E)

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Fig. 1 Scanning electron micrographs (SEMs) and FEM of the two main types of HC-PCF: (a) SEM of PBG-guiding HC-PCF, (b) GVD and loss calculated using FEM of an idealized PBG-guiding HC-PCF structure, designed for operation around $800\mathrm{nm}$, with $11\mathrm{\mu m}$ core diameter and $2.1\mathrm{\mu m}$ pitch (Λ), (c) SEM of a kagomé PCF designed for operation in the UV and around $800\mathrm{nm}$, and (d) GVD and loss calculated using the FEM of an idealized kagomé PCF structure with $30\mathrm{\mu m}$ core diameter, $15\mathrm{\mu m}$ pitch (Λ), and $0.23\mathrm{\mu m}$ web thickness.

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Fig. 2 Comparison between the FEM and the capillary model: (a) dispersion of the effective mode index and (b) the GVD. (a) Inset, definitions of the two core radii: flat-to-flat radius ($a_f$) and area preserving radius ($a_A$); A is the hexagonal core area. The kink at $\sim 380\mathrm{nm}$ in the FEM results is caused by an anticrossing with a cladding state, and it strongly affects the dispersion.

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Fig. 3 (a) Comparison between the lowest theoretical loss possible in a capillary fiber and the measured loss of a kagomé PCF with a core diameter of $30\mathrm{\mu m}$, (b) dispersion curves for silica-core PCFs (silica strands) with the shortest ZDWs, with core diam eters of 0.47 to $2.86\mathrm{\mu m}$ ($0.2\mathrm{\mu m}$ steps), and (c) the dispersion of a kagomé PCF with a $30\mathrm{\mu m}$ core diameter, filled with 0 to $20\mathrm{bar}$ Ar ($2\mathrm{bar}$ steps)—note the change of scale compared to (b).

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Fig. 4 ZDW scaling for kagomé PCFs with diameters from 10 to $70\mathrm{\mu m}$ (steps of $10\mathrm{\mu m}$), filled with 0 to $60\mathrm{bar}$ of (a) Xe, (b) Kr, (c) Ar, and (d) He.

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Fig. 5 FEM calculations for an idealized kagomé PCF with $30\mathrm{\mu m}$ core diameter, $15\mathrm{\mu m}$ pitch, and $0.23\mathrm{\mu m}$ web thickness—designed for operation in the UV and around $800\mathrm{nm}$: (a) wavelength dependence of the fraction of power in glass, (b) axial component of the Poynting vector of the guided mode at the wavelength of a cladding resonance ($380\mathrm{nm}$), and (c) the same as (b) at $800\mathrm{nm}$.

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Fig. 6 Experimental setup typically used for ultrafast nonlinear experiments in gas-filled kagomé PCF.

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Fig. 7 (a)–(c) Contour plots of the pump energy ($\mathrm{\mu J}$) required for formation of $30\mathrm{fs}$ solitons at $800\mathrm{nm}$, plotted against soliton order and ZDW, the ZDW being adjusted by varying the gas pressure: (a) $10\mathrm{\mu m}$ diameter filled with Xe, (b) $30\mathrm{\mu m}$ diameter filled with Ar, and (c) $50\mathrm{\mu m}$ diameter filled with He. The entire parameter spaces would be severely loss-limited in a capillary fiber. The blue region (lower, shaded region) is where the estimated loss of the kagomé PCF would limit nonlinear interaction. (d) Contour plots of the ratio of the peak power to critical self-focusing power for pulses propagating in a kagomé PCF ($30\mathrm{\mu m}$ core diameter, filled with Ar), plotted against soliton order and ZDW, including pulse self- compression effects (discussed in Subsection 4B). The labeled dots correspond to the systems defined in Table 1.

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Fig. 8 Fiber-grating/mirror pulse compression with kagomé PCF; (a) density plots of the temporal and spectral evolution of a $10\mathrm{\mu J}$, $30\mathrm{fs}$ pulse at $800\mathrm{nm}$ through a kagomé PCF with $70\mathrm{\mu m}$ core diameter filled with $43\mathrm{bar}$ of Ar, placing the ZDW at $1300\mathrm{nm}$; (b) the broadened temporal shape of the intensity, and the smooth parabolic temporal phase after $8\mathrm{cm}$ of propagation in the kagomé PCF [dashed line in (a)]; (c) pulse duration after linear compression against the kagomé PCF length used for spectral broadening (left axis), and compression quality factor (right axis); (d) pulse after linear compression of (b).

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Fig. 9 Soliton-effect self-compression at $800\mathrm{nm}$ in a kagomé PCF with a $30\mathrm{\mu m}$ diameter core filled with Ar: (a) dependence of the compressed pulse duration on ${\lambda }_0$ (horizontal axis) and N (numbered curves), (b) corresponding quality factor dependence, (c) temporal and spectral evolution of self-compression of a pulse for ${\lambda }_0=500\mathrm{nm}$, $N=3.5$, and (d) field of the compressed output pulse at the point of optimum compression in (c).

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Fig. 10 Deep-UV dispersive-wave generation; (a) a series of experimentally achieved tunable output spectra [77] from an Ar-filled, $27\mathrm{\mu m}$ core diameter kagomé PCF ($45\mathrm{fs}$ pulses at $800\mathrm{nm}$ with $0.5–3\mathrm{\mu J}$ pulse energy and $4–12\mathrm{bar}$ filling pressure) and (b) reported UV conversion efficiencies [49].

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Fig. 11 (a) Propagation constant mismatch between solitons ($N=9$ at $800\mathrm{nm}$) and dispersive waves at a given wavelength for a $30\mathrm{\mu m}$ kagomé PCF filled with 2 to $20\mathrm{bar}$ Ar ($2\mathrm{bar}$ steps); (b), (c) phase-matching wavelengths for (b) a $10\mathrm{\mu m}$ diameter Xe-filled fiber, and (c) for a $50\mathrm{\mu m}$ diameter He-filled fiber, assuming pressure- tunable ZDW ${\lambda }_0$.

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Fig. 12 (a), (b) Spectral evolution of two systems, both with ${\lambda }_0=563\mathrm{nm}$ and $N=7.5$, over two fission lengths [Eq. (5)]: (a) a $50\mathrm{nJ}$, $30\mathrm{fs}$ pulse through $0.68\mathrm{cm}$ of $6\mathrm{\mu m}$ diameter kagomé PCF filled with $38\mathrm{bar}$ Xe and (b) $10\mathrm{\mu J}$ $30\mathrm{fs}$ pulse through $68\mathrm{cm}$ of $60\mathrm{\mu m}$ diameter fiber filled with $36\mathrm{bar}$ He. (c), (d) Temporal and spectral evolution of a $10\mathrm{fs}$ pulse with $20\mathrm{\mu J}$ energy propagating in a $50\mathrm{\mu m}$ kagomé PCF with $10\mathrm{bar}$ He (corresponding to ${\lambda }_0=380\mathrm{nm}$, $N=3$).

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Fig. 13 Dependence of UV generation on pulse duration (15, 30, 60, and $120\mathrm{fs}$) and soliton order for a fiber with ${\lambda }_0=600\mathrm{nm}$ (kagomé PCF with $30\mathrm{\mu m}$ core filled with $9.8\mathrm{bar}$ Ar): (a) UV conversion efficiency (to wavelengths shorter than $350\mathrm{nm}$) as a function of normalized soliton order $S=N/{\tau }_{\mathrm{FWHM}}$, corresponding to $N=3$ to 9 for a $30\mathrm{fs}$ pulse; (b) quality factor of UV conversion as defined in the text for the same parameters as (a); (c) spectral evolution along the fiber for each of the pulse durations at $S=0.26$; and (d) spectral slices at the position of optimum UV conversion for each of the pulse durations.

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Fig. 14 (a), (b) Propagation of an $N\sim 245$ ($600\mathrm{fs}$, $10\mathrm{\mu J}$) pulse at $800\mathrm{nm}$ in a kagomé PCF with ${\lambda }_0=750\mathrm{nm}$ ($30\mathrm{\mu m}$ core diameter filled with $25\mathrm{bar}$ Ar): (a) temporal evolution of one shot and (b) spectral evolution of an ensemble average of 30 simulations. (c), (d) Propagation of an $N\sim 429$ ($600\mathrm{fs}$, $10\mathrm{\mu J}$) pulse at $400\mathrm{nm}$ in a kagomé PCF with ${\lambda }_0=350\mathrm{nm}$ ($10\mathrm{\mu m}$ core diameter filled with $8.8\mathrm{bar}$ Ar): (c) temporal evolution of one shot and (d) spectral evolution of an ensemble average of 30 simulations.

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Fig. 15 Contour plot of (a) the calculated free- electron density (${\mathrm{m}}^{-3}$) in a $30\mathrm{\mu m}$ diameter kagomé PCF filled with Ar, as a function of ZDW and soliton order, taking into account the self-compression of the solitons upon propagation; the vertical dashed arrow indicates the operation region of Hölzer et al. [5] and (b) the nonlinear refractive index ratio R (defined in the text); the black dashed line indicates a ratio of 1.

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Fig. 16 Results of Hölzer et al. [5]: (a) experimental output spectra from a $26\mathrm{\mu m}$ diameter kagomé PCF filled with $1.7\mathrm{bar}$ Ar as a function of input pulse energy, (b) corresponding numerical simulations, and (c) comparison of the transmission between the experiment and the numerical simulations with and without the ionization terms included.

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Fig. 17 Numerical simulation of the propagation of an $N=4$ soliton through a $30\mathrm{\mu m}$ diameter kagomé PCF with $1.7\mathrm{bar}$ Ar filling pressure (ZDW at $380\mathrm{nm}$).

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