Abstract

We review the use of hollow-core photonic crystal fibers (PCFs) in the field of ultrafast gas-based nonlinear optics, including recent experiments, numerical modeling, and a discussion of future prospects. Concentrating on broadband guiding kagomé-style hollow-core PCF, we describe its potential for moving conventional nonlinear fiber optics both into extreme regimes—such as few-cycle pulse compression and efficient deep ultraviolet wavelength generation—and into regimes hitherto inaccessible, such as single-mode guidance in a photoionized plasma and high-harmonic generation in fiber.

© 2011 Optical Society of America

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 800nm pulses, the glass capillary fiber provides a partial solution. Although such capillaries do not support bound modes, the loss of the fundamental HE11 leaky mode scales as the inverse cube of the bore diameter, and it can be as low as 1.2dB/m for diameters of 200μm at the 800nm wavelength. Even though many higher order leaky modes are supported in bores of this size, it is possible to excite only the HE11 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 HE11 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 (10 or even 1000m), 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 (<1dB/km at 1550nm 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 800nm. The calculated loss around 800nm is 0.01dB/m. The ZDW is at 768nm, and the dispersion changes considerably across the narrow transmission band, with a dispersion slope greater than 2000fs3/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 1dB/m, and it displays weak anomalous GVD (|β2|<15fs2/cm, when evacuated) over the entire transmission window, with a low dispersion slope. FEM calculations, for one example structure (30μm core diameter, designed for operation in the UV and around 800nm), are shown in Fig. 1d, illustrating the broadband guidance windows (200350nm and 600850nm<2dB/m), with relatively small all-anomalous dispersion magnitude (below 15fs2/cm) across the whole band, excluding the anticrossing with a strong cladding resonance at 380nm. 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]:

nmn(λ,p,T)=ngas2(λ,p,T)umn2k2a21+δ(λ)p2p0T0Tumn22k2a2,
where k is the vacuum wave vector, ngas is the refractive index of the filling gas, δ(λ) is a Sellmeier expansion for ngas2 [46], p is the pressure, p0 is the atmospheric pressure, T is the temperature, T0=273.15K, and umn is the nth zero of the mth-order Bessel function of the first kind, where m=n=1 corresponds to the HE11 mode of the fiber. Several core radii, a, can be defined for the circle used to approximate the kagomé fiber core in Eq. (1), including one that preserves core area (aA) and one based simply on the flat-to-flat distance of the hexagonal core (af), as shown in the inset of Fig. 2a. Comparison with full finite-element simulations shows that Eq. (1) is highly accurate [Figs. 2a, 2b] [42, 45], aA providing the best agreement when comparing effective mode index and af providing the best agreement with the GVD calculations. The validity of Eq. (1) has been further confirmed by phase-matched third-harmonic generation in a higher order guided mode [43].

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 30μm (required for anomalous dispersion at reasonable levels of nonlinearity—see below), the loss in a capillary exceeds several 100dB/m, whereas that of a kagomé PCF can be below 1dB/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 (30μm 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 500nm [Fig. 3b]. Whereas solid-core PCF permitted soliton propagation at 800nm 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 1000nm, allowing normal dispersion at 800nm—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 J02(u11r/a) for the fundamental HE11 mode, with r the radial coordinate, the conventional definition of effective mode area [7] yields Aeff1.5a2; i.e., a kagomé PCF with a 30μm core diameter will have Aeff=338μm2. 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 30μm core diameter, the nonlinear coefficient γ is (at 10bar): 1.5×107m1W1 (Ne), 2×106m1W1 (Ar), 5.3×106m1W1 (Kr), and 1.5×105m1W1 (Xe). These are over 4 orders of magnitude smaller than in a solid-core PCF (0.24m1W1 for a core diameter of 1μm) but much higher than in a gas-filled capillary with a bore diameter of 200μm (5×108m1W1 at 10bar 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 N2 or SF6, 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 800nm, 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 380nm 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 4570μm, guided intensities over 1016W/cm2 have been demonstrated [52]; guidance of similar intensities in kagomé PCF can be expected. Our group has coupled ultrafast pulses (65fs) with over 12μJ into a kagomé PCF with a 26μm core diameter, with efficiencies up to 80%. After self-compression (Subsection 4B), we have reached intensities of over 1014W/cm2 [5]. By careful mode matching, another group reported 98% launch efficiency of 40fs pulses into a PBG-guiding HC-PCF (core diameter 6.8μm), reaching a transmitted energy of 2μJ [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 <1min 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]:

zU=isgn(β2)2LDτ2U+sgn(β3)6LDτ3U+ieαzLNL(U|U|2+isτ(U|U|2)),
where U(τ,z) is the normalized optical field envelope and α is the power loss coefficient. Note that the Raman effect is absent in the monatomic gases considered here, and although dispersion orders β4 and higher can play an important role in the dynamics, they do not alter the results in this section. Defining τ0 as the pump pulse duration, ω0 its central angular frequency, and P0 its peak power, the following normalized parameters fully define the propagation dynamics:
LD=τ02|β2|,LD=τ03|β3|,βn=ωnβ|ω0,LNL=1γP0,s=1ω0τ0.
Without loss of generality, we assume a pump pulse propagating at 800nm with a duration of 30fs full-width at half- maximum (FWHM) intensity. The length-scales in Eq. (3) can then be reduced to just two parameters—the ZDW λ0 and the soliton order N:
(LD,LD)λ0(LD,LNL)N=γP0τ02|β2|.
Note that although bright solitons only occur in regions of anomalous dispersion, this definition of the soliton order can also be used in the normal dispersion regime, providing a convenient means of comparing the relative nonlinearity in all cases. The reduction of the full parameter space to (λ0, N) is useful for extracting the global dynamics and understanding the scaling behavior of a given system. For example, if it is known that λ0=600nm and N=9 leads to UV generation, then any combination of core diameter, gas species, filling pressure, and pulse energy that yields the same values of (λ0, N) can also be expected to generate UV, even though the physical system parameters may be drastically different. Note that it makes no sense to compare λ0 or N on their own. For example, although a nearly evacuated capillary fiber will exhibit anomalous dispersion well below 200nm, it will have close to zero nonlinear coefficient.

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 (λ0, N), we can select a gas species and, for a wide choice of kagomé PCF diameters, find the correct filling pressure to fix λ0. 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 (λ0, N) for some selected gas species and kagomé PCF diameters, illustrating that, for example, the dynamics of λ0=600nm and N=9 can be accessed with pulse energies ranging from 100nJ to 10μJ.

For soliton orders 2N15, 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]:

Lfiss=LDN.
Nonlinear propagation will be limited if power losses are too high over this length scale (here we take a guideline value of 1dB over one soliton fission length). For capillary fibers, nonlinear interaction would be prohibited over the entire param eter space in all three cases presented in Figs. 7a, 7b, 7c; although the loss in a capillary decreases with larger core diameter, the concomitant increase in nonlinear interaction length more than cancels this out (due to lower dispersion and lower nonlinearity). This means that none of the dynamics we discuss below are accessible in capillary fibers. Conversely, most of the parameter space in Figs. 7a, 7b, 7c is accessible to kagomé PCF.

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, 1mJ of energy is required to create an eighth- order soliton in a kagomé PCF with a core diameter of 50μm filled with He so as to yield a ZDW of 250nm. 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 10μm, He is replaced with Xe, and the pulse energy is lowered to 10μJ—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 0.2 for λ0=600nm 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 (λ0, 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 1MW, 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 (0.1100μJ) 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 4 for 1ps 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 10μJ, 30fs pulse in a kagomé PCF with a core diameter of 70μm filled with 43bar of Ar. This causes the ZDW to shift to 1300nm so that the pulse (at 800nm) 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 8cm 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 8cm of propagation, the pulse can be compressed to 3fs 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 3GW. 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 30fs pulses to shorter than 10fs. In these experiments, the PCF was much longer than the optimum compression length. Numerical simulations show that pulses as short as 4fs 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:

Fc4.6N,Qc3.7/N,
implying that a 30fs pulse at 800nm can be compressed to less than a single optical cycle for N3. Such extreme compression is inevitably limited by higher order effects, for example, by higher order dispersion [71]. Figure 9 shows numerical results for propagation through a kagomé PCF (30μm core diameter) filled with Ar. Figure 9a shows the shortest self- compressed pulse duration as a function of (λ0, N). As N increases, the compressed pulse gets shorter until it saturates between N=4 and 6. The duration of the compressed pulse depends weakly on λ0; as λ0 moves closer to the pump wavelength (800nm), higher order dispersion starts to perturb the self-compression, causing premature pulse fission. As λ0 moves further away from 800nm, which requires lower gas pressures and hence higher pump intensities, ionization of the gas starts to occur, decreasing the effectiveness of the self- compression process. Figure 9b shows the compressed pulse quality factor as a function of (λ0, N). As stated in Eq. (6), this degrades significantly with increasing N and also degrades as λ0 shifts far from 800nm. There is a trade-off between the compressed pulse duration and quality factor. A good choice is λ0=500nm, N=3.5 [corresponding to point (iii) in Figs. 7, 15]; the spectral and temporal evolution of a 30fs pulse for these parameters is shown in Fig. 9c. In contrast to propagation in the normal dispersion regime [Fig. 8a], the temporal profile is seen to dramatically sharpen upon propagation. At the point of optimum compression, a 2fs pulse is produced [Fig. 9c]. For higher values of N (9), the self-compressed pulses can reach subcycle durations, though with a significant loss in quality, and the pulse spectrum extends into the UV. This is the initiation of the UV dispersive-wave emission discussed in Section 5.

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 τc in the following form:

τc=τ0Fcτ0N|β2|γP0,
which shows that τc remains the same if the peak power of the pump pulse is maintained; i.e., the soliton order is increased in proportion to the pulse duration τ0. Equation (6) shows, however, that the quality factor significantly decreases with increasing soliton order, leading to worse compression as τ0 gets longer.

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 β2, 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 0.3dB/m limit the usable fiber length to 10m.

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 27μm core diameter, pumped with 45fs pulses at 800nm [77]. Each spectrum was obtained by tuning the pulse energy (0.53μJ) and filling pressure (412bar) to demonstrate the range of clean UV output tunable from 200300nm. The spectra have relative widths of 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 5080nJ for a kagomé PCF with λ0600nm, N9 [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., β(ω)=βsol(ω), where β(ω)=nmnk is the linear mode propagation constant and

βsol(ω)=β(ωsol)+β1(ωsol)[ωωsol]+γPc2,
where Pc is the self-compressed pulse peak power and ωsol is the soliton frequency. In Fig. 11a we plot the propagation constant mismatch [β(ω)β(ωsol)] for a range of Ar pressures in a 30μm diameter kagomé PCF. We see that the phase-matched points occur in the UV spectral region [82]. Using the results of Subsection 4B on pulse compression, PcP0Fc [70], and expanding β(ω) around ωsol, the phase-matching condition reduces to
n2(ωωsol)nn!βn(ωsol)=γPc22.3γP0N.
In Figs. 11b, 11c we plot this phase-matching condition for two distinct systems (a 10μm diameter Xe-filled and a 50μm diameter He-filled kagomé PCF) as a function of (λ0, N). For an 800nm pump pulse, phase-matched solutions can be found at least from 150 to 500nm. While the dependence on λ0 (and hence gas pressure) is strong, the dependence on N is relatively weaker in the UV region, although it becomes more significant as λ0 approaches 800nm and the nonlinear contribution to Eq. (9) becomes more dominant. The similarity between these two plots shows the utility of the scaling analysis presented in Section 3. Noticeable disagreement is only found for extreme phase matching to around 150nm, where higher order dispersion becomes important.

Further evidence of this scaling can be seen in Figs. 12a, 12b, which show that both 50nJ and 10μJ pulses can experience essentially identical spectral evolution along the fiber, including clear UV dispersive-wave emission at 230nm, 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 10fs pulse with 20μJ energy propagating in a 50μm kagomé PCF filled with 10bar He (corresponding to λ0=380nm, N=3). After the characteristic pulse self-compression at around 5cm, a UV dispersive wave at 150nm is emitted, containing over 3.5μJ (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 350nm) 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 QUV=PFWHM/PUV, where PFWHM is the spectral power within the FWHM of the strongest UV peak and PUV is the total spectral power in the UV region. Figure 13b shows that QUV is strongly dependent on the pump pulse duration. For 15fs, 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 120fs pulse. This is clearly evident in Fig. 13c, which shows the spectral evolution of 15, 30, 60, and 90fs pulses along the fiber for a normalized soliton order of S=N/τFWHM=0.26 (N=7.8 for 30fs pulse duration). For the 15fs pulse, a high-quality UV band emerges approximately at the soliton fission length [Eq. (6)]. For the 30fs pump, the UV band is still of relatively high quality, but at 60fs, it degrades considerably, and at 120fs, 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 30fs 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 150nm. Whether a further extension deeper into the UV is possible is an open question. Phase matching could be achieved to well below 150nm if the pump wavelength is shifted to 400nm. 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 100nJ and 10μJ pulses, with similar efficiencies, promises wide applicability of this technique. Energies of >100nJ 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 10μJ pumped system should enable the generation of deep- UV pulses with energies in excess of 1μJ. 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 300nm is sufficient to achieve saturation (i.e., exhaust the exponential FEL gain) [88].

5D. UV Supercontinuum Generation

Raising the soliton order to N245 by launching, e.g., 10μJ 600fs 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 λ0=750nm (25bar Ar, 30μm 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 1500nm. 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 400nm in a fiber designed for zero dispersion at 350nm (8.8bar Ar, 10μm core diameter), we obtain a very flat and smooth, MI-based, high-energy supercontinuum extending from 1401000nm [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 2×1014W/cm2 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 (<10μJ), 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 Ne in a 30μm 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 1022m3. 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 1023 to >1024m3, 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]:

R=|ΔnKerrΔnplasma|=n2I2n0ω02ωp2,
where I is the optical intensity; n0 is the linear refractive index; ω0 is the pump frequency; and ωp is the plasma frequency ωp=[Nee2/(meεo)]0.5, where ε0 is the vacuum permittivity, e is the electronic charge, and me is the electron mass.

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 1022m3 the Kerr effect is over 10 times stronger almost independently of the ZDW (or gas pressure). However at Ne1023m3, the effects become comparable, and the ratio R1 [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 65fs pulses in a 26μm diameter kagomé PCF filled with 1.7bar Ar (corresponding to a ZDW at 380nm) leads to the generation of blueshifted sidebands. The first one occurs at a pump energy of 2μJ. 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 ψ(z,t) [96]:

zψ=i[m2βm(it)mm!+γR(t)|ψ(t)|2ωp22ω0c+iAeffIptNe2|ψ|2]ψ,
where z is the longitudinal coordinate along the fiber, t is the time in a reference frame moving at the group velocity, R(t) is the normalized Kerr and Raman response function of the gas, Ip is the ionization energy of the gas, and Ne(t) is the temporally varying free-electron density that is modeled with an additional coupled equation, Eq. (A1) (Appendix A).

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 30μm diameter Ar-filled kagomé PCF with a ZDW at 380nm, corresponding to the parameters of the experimental results shown in Fig. 16. A clear blueshift is seen in the spectrum after 12cm 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 SF6, 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 15μm) filled with 30mbar of Xe [110]. A conversion efficiency of 109 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 200nJ, 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 105 have been obtained to individual harmonics with photon energies of 45eV (28nm) [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 1.6μm (an easily attainable laser wavelength), one could phase match to the 99th harmonic using a driving laser at 800nm 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 0.130μJ range. With the correct dispersion profile, self-compression can lead to a highly efficient and compact, tunable deep-UV source, tuning between 200350nm having been demonstrated, and 150500nm being realistically feasible. UV energies >1μJ, 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]

Net=W(t)NaηNeβrNe2,
where W(t) is the ionization rate of the atoms in the presence of the time varying electric field E(z,t), Na is the density of neutral atoms, and η and βr are the electron attachment and recombination rates, respectively, both of which are negligible for durations <1ps [92]. The calculation of the free- electron density Ne(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.

Tables Icon

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)

 

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 800nm, with 11μm core diameter and 2.1μm pitch (Λ), (c) SEM of a kagomé PCF designed for operation in the UV and around 800nm, and (d) GVD and loss calculated using the FEM of an idealized kagomé PCF structure with 30μm core diameter, 15μm pitch (Λ), and 0.23μm 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 (af) and area preserving radius (aA); A is the hexagonal core area. The kink at 380nm 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 30μm, (b) dispersion curves for silica-core PCFs (silica strands) with the shortest ZDWs, with core diam eters of 0.47 to 2.86μm (0.2μm steps), and (c) the dispersion of a kagomé PCF with a 30μm core diameter, filled with 0 to 20bar Ar (2bar 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 70μm (steps of 10μm), filled with 0 to 60bar of (a) Xe, (b) Kr, (c) Ar, and (d) He.

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Fig. 5 FEM calculations for an idealized kagomé PCF with 30μm core diameter, 15μm pitch, and 0.23μm web thickness—designed for operation in the UV and around 800nm: (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 (380nm), and (c) the same as (b) at 800nm.

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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 (μJ) required for formation of 30fs solitons at 800nm, plotted against soliton order and ZDW, the ZDW being adjusted by varying the gas pressure: (a) 10μm diameter filled with Xe, (b) 30μm diameter filled with Ar, and (c) 50μm 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 (30μm 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 10μJ, 30fs pulse at 800nm through a kagomé PCF with 70μm core diameter filled with 43bar of Ar, placing the ZDW at 1300nm; (b) the broadened temporal shape of the intensity, and the smooth parabolic temporal phase after 8cm 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 800nm in a kagomé PCF with a 30μm diameter core filled with Ar: (a) dependence of the compressed pulse duration on λ0 (horizontal axis) and N (numbered curves), (b) corresponding quality factor dependence, (c) temporal and spectral evolution of self-compression of a pulse for λ0=500nm, 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, 27μm core diameter kagomé PCF (45fs pulses at 800nm with 0.53μJ pulse energy and 412bar filling pressure) and (b) reported UV conversion efficiencies [49].

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Fig. 11 (a) Propagation constant mismatch between solitons (N=9 at 800nm) and dispersive waves at a given wavelength for a 30μm kagomé PCF filled with 2 to 20bar Ar (2bar steps); (b), (c) phase-matching wavelengths for (b) a 10μm diameter Xe-filled fiber, and (c) for a 50μm diameter He-filled fiber, assuming pressure- tunable ZDW λ0.

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Fig. 12 (a), (b) Spectral evolution of two systems, both with λ0=563nm and N=7.5, over two fission lengths [Eq. (5)]: (a) a 50nJ, 30fs pulse through 0.68cm of 6μm diameter kagomé PCF filled with 38bar Xe and (b) 10μJ 30fs pulse through 68cm of 60μm diameter fiber filled with 36bar He. (c), (d) Temporal and spectral evolution of a 10fs pulse with 20μJ energy propagating in a 50μm kagomé PCF with 10bar He (corresponding to λ0=380nm, N=3).

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Fig. 13 Dependence of UV generation on pulse duration (15, 30, 60, and 120fs) and soliton order for a fiber with λ0=600nm (kagomé PCF with 30μm core filled with 9.8bar Ar): (a) UV conversion efficiency (to wavelengths shorter than 350nm) as a function of normalized soliton order S=N/τFWHM, corresponding to N=3 to 9 for a 30fs 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 N245 (600fs, 10μJ) pulse at 800nm in a kagomé PCF with λ0=750nm (30μm core diameter filled with 25bar Ar): (a) temporal evolution of one shot and (b) spectral evolution of an ensemble average of 30 simulations. (c), (d) Propagation of an N429 (600fs, 10μJ) pulse at 400nm in a kagomé PCF with λ0=350nm (10μm core diameter filled with 8.8bar 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 (m3) in a 30μm 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 26μm diameter kagomé PCF filled with 1.7bar 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 30μm diameter kagomé PCF with 1.7bar Ar filling pressure (ZDW at 380nm).

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2011 (9)

P. Hölzer, W. Chang, J. C. Travers, A. Nazarkin, J. Nold, N. Y. Joly, M. Saleh, F. Biancalana, and P. St. J. Russell, “Femtosecond nonlinear fiber optics in the ionization regime,” Phys. Rev. Lett. 107, 203901 (2011).
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F. Benabid and P. J. Roberts, “Linear and nonlinear optical properties of hollow core photonic crystal fiber,” J. Mod. Opt. 58, 87–124 (2011).
[CrossRef]

N. Y. Joly, J. Nold, W. Chang, P. Hölzer, A. Nazarkin, G. K. L. Wong, F. Biancalana, and P. St. J. Russell, “Bright spatially coherent wavelength-tunable deep-UV laser source using an ar-filled photonic crystal fiber,” Phys. Rev. Lett. 106, 203901 (2011).
[CrossRef] [PubMed]

J. Limpert, S. Hädrich, J. Rothhardt, M. Krebs, T. Eidam, T. Schreiber, and A. Tünnermann, “Ultrafast fiber lasers for strong‐field physics experiments,” Laser Photon. Rev. 5, 634–646 (2011).
[CrossRef]

M. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Joly, P. St. J. Russell, and F. Biancalana, “Theory of photoionization-induced blueshift of ultrashort solitons in gas-filled hollow-core photonic crystal fibers,” Phys. Rev. Lett. 107, 203902 (2011).
[CrossRef] [PubMed]

X. Jiang, T. G. Euser, A. Abdolvand, F. Babic, F. Tani, N. Y. Joly, J. C. Travers, and P. St. J. Russell, “Single-mode hollow-core photonic crystal fiber made from soft glass,” Opt. Express 19, 15438–15444 (2011).
[CrossRef] [PubMed]

C. Rodríguez, Z. Sun, Z. Wang, and W. Rudolph, “Characterization of laser-induced air plasmas by third harmonic generation,” Opt. Express 19, 16115–16125 (2011).
[CrossRef] [PubMed]

O. H. Heckl, C. J. Saraceno, C. R. E. Baer, T. Südmeyer, Y. Y. Wang, Y. Cheng, F. Benabid, and U. Keller, “Temporal pulse compression in a xenon-filled kagome-type hollow-core photonic crystal fiber at high average power,” Opt. Express 19, 19142–19149 (2011).
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W. Chang, A. Nazarkin, J. C. Travers, J. Nold, P. Hölzer, N. Y. Joly, and P. St. J. Russell, “Influence of ionization on ultrafast gas-based nonlinear fiber optics,” Opt. Express 19, 21018–21027 (2011).
[CrossRef] [PubMed]

2010 (6)

S.-J. Im, A. Husakou, and J. Herrmann, “High-power soliton-induced supercontinuum generation and tunable sub-10 fs VUV pulses from kagome-lattice HC-PCFs,” Opt. Express 18, 5367–5374 (2010).
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J. Nold, P. Hölzer, N. Y. Joly, G. K. L. Wong, A. Nazarkin, A. Podlipensky, M. Scharrer, and P. St. J. Russell, “Pressure-controlled phase matching to third harmonic in Ar-filled hollow-core photonic crystal fiber,” Opt. Lett. 35, 2922–2924(2010).
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P. J. Mosley, W. C. Huang, M. G. Welch, B. J. Mangan, W. J. Wadsworth, and J. C. Knight, “Ultrashort pulse compression and delivery in a hollow-core photonic crystal fiber at 540 nm wavelength,” Opt. Lett. 35, 3589–3591 (2010).
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T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent x-ray generation,” Nat. Photon. 4, 822–832 (2010).
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J. C. Travers, “Blue extension of optical fibre supercontinuum generation,” J. Opt. 12, 113001 (2010).
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A. Nazarkin, A. Abdolvand, A. V. Chugreev, and P. St. J. Russell, “Direct observation of self-similarity in evolution of transient stimulated Raman scattering in gas-filled photonic crystal fibers,” Phys. Rev. Lett. 105, 173902 (2010).
[CrossRef]

2009 (7)

P. Londero, V. Venkataraman, A. R. Bhagwat, A. D. Slepkov, and A. L. Gaeta, “Ultralow-power four-wave mixing with Rb in a hollow-core photonic band-gap fiber,” Phys. Rev. Lett. 103, 043602 (2009).
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A. Abdolvand, A. Nazarkin, A. V. Chugreev, C. F. Kaminski, and P. St. J. Russell, “Solitary pulse generation by backward Raman scattering in H2-filled photonic crystal fibers,” Phys. Rev. Lett. 103, 183902 (2009).
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J. M. Dudley and J. R. Taylor, “Ten years of nonlinear optics in photonic crystal fibre,” Nat. Photon. 3, 85–90 (2009).
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O. H. Heckl, C. R. E. Baer, C. Kränkel, S. V. Marchese, F. Schapper, M. Holler, T. Südmeyer, J. S. Robinson, J. W. G. Tisch, F. Couny, P. Light, F. Benabid, and U. Keller, “High harmonic generation in a gas-filled hollow-core photonic crystal fiber,” Appl. Phys. B 97, 369–373 (2009).
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S.-J. Im, A. Husakou, and J. Herrmann, “Guiding properties and dispersion control of kagome lattice hollow-core photonic crystal fibers,” Opt. Express 17, 13050–13058 (2009).
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A. A. Ishaaya, C. J. Hensley, B. Shim, S. Schrauth, K. W. Koch, and A. L. Gaeta, “Highly-efficient coupling of linearly- and radially-polarized femtosecond pulses in hollow-core photonic band-gap fibers,” Opt. Express 17, 18630–18637 (2009).
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J. Lægsgaard and P. J. Roberts, “Theory of adiabatic pressure-gradient soliton compression in hollow-core photonic bandgap fibers,” Opt. Lett. 34, 3710–3712 (2009).
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2008 (10)

F. Gérôme, P. Dupriez, J. Clowes, J. C. Knight, and W. J. Wadsworth, “High power tunable femtosecond soliton source using hollow-core photonic bandgap fiber, and its use for frequency doubling,” Opt. Express 16, 2381–2386 (2008).
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A. R. Bhagwat and A. L. Gaeta, “Nonlinear optics in hollow-core photonic bandgap fibers,” Opt. Express 16, 5035–5047 (2008).
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E. E. Serebryannikov, D. von der Linde, and A. M. Zheltikov, “Broadband dynamic phase matching of high-order harmonic generation by a high-peak-power soliton pump field in a gas-filled hollow photonic-crystal fiber,” Opt. Lett. 33, 977–979(2008).
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J. Henningsen and J. Hald, “Dynamics of gas flow in hollow core photonic bandgap fibers,” Appl. Opt. 47, 2790–2797 (2008).
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A. Börzsönyi, Z. Heiner, M. P. Kalashnikov, A. P. Kovács, and K. Osvay, “Dispersion measurement of inert gases and gas mixtures at 800 nm,” Appl. Opt. 47, 4856–4863 (2008).
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H. Ren, A. Nazarkin, J. Nold, and P. St. J. Russell, “Quasi-phase-matched high harmonic generation in hollow core photonic crystal fibers,” Opt. Express 16, 17052–17059 (2008).
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E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320, 1614–1617(2008).
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G. Lambert, T. Hara, D. Garzella, T. Tanikawa, M. Labat, B. Carre, H. Kitamura, T. Shintake, M. Bougeard, S. Inoue, Y. Tanaka, P. Salieres, H. Merdji, O. Chubar, O. Gobert, K. Tahara, and M.-E. Couprie, “Injection of harmonics generated in gas in a free-electron laser providing intense and coherent extreme-ultraviolet light,” Nat. Phys. 4, 296–300 (2008).
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A. A. Voronin and A. M. Zheltikov, “Soliton-number analysis of soliton-effect pulse compression to single-cycle pulse widths,” Phys. Rev. A 78, 063834 (2008).
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2007 (6)

E. E. Serebryannikov and A. M. Zheltikov, “Ionization-induced effects in the soliton dynamics of high-peak-power femtosecond pulses in hollow photonic-crystal fibers,” Phys. Rev. A 76, 013820 (2007).
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A. B. Fedotov, E. E. Serebryannikov, and A. M. Zheltikov, “Ionization-induced blueshift of high-peak-power guided-wave ultrashort laser pulses in hollow-core photonic-crystal fibers,” Phys. Rev. A 76, 053811 (2007).
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L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
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F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and photonic guidance of multi-octave optical-frequency combs,” Science 318, 1118–1121 (2007).
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F. Gérôme, K. Cook, A. K. George, W. J. Wadsworth, and J. C. Knight, “Delivery of sub-100 fs pulses through 8 m of hollow-core fiber using soliton compression,” Opt. Express 15, 7126–7131(2007).
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J. C. Travers, J. M. Stone, A. B. Rulkov, B. A. Cumberland, A. K. George, S. V. Popov, J. C. Knight, and J. R. Taylor, “Optical pulse compression in dispersion decreasing photonic crystal fiber,” Opt. Express 15, 13203–13211 (2007).
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2006 (4)

F. Couny, F. Benabid, and P. S. Light, “Large-pitch kagome-structured hollow-core photonic crystal fiber,” Opt. Lett. 31, 3574–3576 (2006).
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P. St. J. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24, 4729–4749 (2006).
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J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184(2006).
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G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulses,” Science 314, 443–446 (2006).
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2005 (5)

S. Naumov, A. Fernandez, R. Graf, P. Dombi, F. Krausz, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators,” New J. Phys. 7, 216(2005).
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A. Suda, M. Hatayama, K. Nagasaka, and K. Midorikawa, “Generation of sub-10 fs, 5 mJ-optical pulses using a hollow fiber with a pressure gradient,” Appl. Phys. Lett. 86, 111116(2005).
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S. Ghosh, J. E. Sharping, D. G. Ouzounov, and A. L. Gaeta, “Resonant optical interactions with molecules confined in photonic band-gap fibers,” Phys. Rev. Lett. 94, 093902 (2005).
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P. Roberts, F. Couny, H. Sabert, B. Mangan, D. Williams, L. Farr, M. Mason, A. Tomlinson, T. Birks, J. Knight, and P. St. J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express 13, 236–244 (2005).
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D. Ouzounov, C. Hensley, A. Gaeta, N. Venkateraman, M. Gallagher, and K. Koch, “Soliton pulse compression in photonic band-gap fibers.,” Opt. Express 13, 6153–6159 (2005).
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2004 (4)

F. Benabid, G. Bouwmans, J. C. Knight, P. St. J. Russell, and F. Couny, “Ultrahigh efficiency laser wavelength conversion in a gas-filled hollow core photonic crystal fiber by pure stimulated rotational Raman scattering in molecular hydrogen,” Phys. Rev. Lett. 93, 123903 (2004).
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C. J. S. de Matos, S. V. Popov, A. B. Rulkov, J. R. Taylor, J. Broeng, T. P. Hansen, and V. P. Gapontsev, “All-fiber format compression of frequency chirped pulses in air-guiding photonic crystal fibers,” Phys. Rev. Lett. 93, 103901 (2004).
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E. E. Serebryannikov, D. von der Linde, and A. M. Zheltikov, “Phase-matching solutions for high-order harmonic generation in hollow-core photonic-crystal fibers,” Phys. Rev. E 70, 066619 (2004).
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V. S. Popov, “Tunnel and multiphoton ionization of atoms and ions in a strong laser field (Keldysh theory),” Phys. Usp. 47, 855–885 (2004).
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2003 (5)

P. St. J. Russell, “Photonic crystal fibers,” Science 299, 358–362(2003).
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J. C. Knight, “Photonic crystal fibres,” Nature 424, 847–851(2003).
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D. G. Ouzounov, F. R. Ahmad, D. Müller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, “Generation of megawatt optical solitons in hollow-core photonic band-gap fibers,” Science 301, 1702–1704 (2003).
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C. J. S. de Matos, J. R. Taylor, T. Hansen, K. Hansen, and J. Broeng, “All-fiber chirped pulse amplification using highly-dispersive air-core photonic bandgap fiber,” Opt. Express 11, 2832–2837 (2003).
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J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, and A. Tünnermann, “All fiber chirped-pulse amplification system based on compression in air-guiding photonic bandgap fiber,” Opt. Express 11, 3332–3337 (2003).
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2002 (5)

C. M. Chen and P. L. Kelley, “Nonlinear pulse compression in optical fibers: scaling laws and numerical analysis,” J. Opt. Soc. Am. B 19, 1961–1967 (2002).
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F. Benabid, J. C. Knight, and P. St. J. Russell, “Particle levitation and guidance in hollow-core photonic crystal fiber,” Opt. Express 10, 1195–1203 (2002).
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F. Benabid, J. C. Knight, G. Antonopoulos, and P. St. J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298, 399–402 (2002).
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M. C. Downer, “A new low for nonlinear optics,” Science 298, 373–375 (2002).
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P. Sprangle, J. R. Peñano, and B. Hafizi, “Propagation of intense short laser pulses in the atmosphere,” Phys. Rev. E 66, 046418(2002).
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P. G. O’Shea and H. P. Freund, “Free-electron lasers: status and applications,” Science 292, 1853–1858 (2001).
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S. L. Voronov, I. Kohl, J. B. Madsen, J. Simmons, N. Terry, J. Titensor, Q. Wang, and J. Peatross, “Control of laser high-harmonic generation with counterpropagating light,” Phys. Rev. Lett. 87, 133902 (2001).
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G. L. Yudin and M. Y. Ivanov, “Nonadiabatic tunnel ionization: looking inside a laser cycle,” Phys. Rev. A 64, 013409(2001).
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2000 (3)

1999 (3)

V. M. Malkin, G. Shvets, and N. J. Fisch, “Fast compression of laser beams to highly overcritical powers,” Phys. Rev. Lett. 82, 4448–4451 (1999).
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R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539(1999).
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F. Dorchies, J. R. Marquès, B. Cros, G. Matthieussent, C. Courtois, T. Vélikoroussov, P. Audebert, J. P. Geindre, S. Rebibo, G. Hamoniaux, and F. Amiranoff, “Monomode guiding of 1016 W/cm2 laser pulses over 100 Rayleigh lengths in hollow capillary dielectric tubes,” Phys. Rev. Lett. 82, 4655–4658 (1999).
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A. Rundquist, C. G. Durfee, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Phase-matched generation of coherent soft x-rays,” Science 280, 1412–1415 (1998).
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N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607(1995).
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S. P. Le Blanc, R. Sauerbrey, S. C. Rae, and K. Burnett, “Spectral blue shifting of a femtosecond laser pulse propagating through a high-pressure gas,” J. Opt. Soc. Am. B 10, 1801–1809 (1993).
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P. K. A. Wai, H. H. Chen, and Y. C. Lee, “Radiations by ‘solitons’ at the zero group-dispersion wavelength of single-mode optical fibers,” Phys. Rev. A 41, 426–439 (1990).
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1985 (2)

E. M. Dianov, A. Y. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. F. Stel’Makh, and A. A. Fomichev, “Stimulated-Raman conversion of multisoliton pulses in quartz optical fibers,” JETP Lett. 41, 242–244 (1985).

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X. Jiang, T. G. Euser, A. Abdolvand, F. Babic, F. Tani, N. Y. Joly, J. C. Travers, and P. St. J. Russell, “Single-mode hollow-core photonic crystal fiber made from soft glass,” Opt. Express 19, 15438–15444 (2011).
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A. Nazarkin, A. Abdolvand, A. V. Chugreev, and P. St. J. Russell, “Direct observation of self-similarity in evolution of transient stimulated Raman scattering in gas-filled photonic crystal fibers,” Phys. Rev. Lett. 105, 173902 (2010).
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A. Abdolvand, A. Nazarkin, A. V. Chugreev, C. F. Kaminski, and P. St. J. Russell, “Solitary pulse generation by backward Raman scattering in H2-filled photonic crystal fibers,” Phys. Rev. Lett. 103, 183902 (2009).
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R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539(1999).
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J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809(2000).
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F. Benabid, J. C. Knight, G. Antonopoulos, and P. St. J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298, 399–402 (2002).
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F. Benabid, J. C. Knight, and P. St. J. Russell, “Particle levitation and guidance in hollow-core photonic crystal fiber,” Opt. Express 10, 1195–1203 (2002).
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N. Y. Joly, J. Nold, W. Chang, P. Hölzer, A. Nazarkin, G. K. L. Wong, F. Biancalana, and P. St. J. Russell, “Bright spatially coherent wavelength-tunable deep-UV laser source using an ar-filled photonic crystal fiber,” Phys. Rev. Lett. 106, 203901 (2011).
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M. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Joly, P. St. J. Russell, and F. Biancalana, “Theory of photoionization-induced blueshift of ultrashort solitons in gas-filled hollow-core photonic crystal fibers,” Phys. Rev. Lett. 107, 203902 (2011).
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P. Hölzer, W. Chang, J. C. Travers, A. Nazarkin, J. Nold, N. Y. Joly, M. Saleh, F. Biancalana, and P. St. J. Russell, “Femtosecond nonlinear fiber optics in the ionization regime,” Phys. Rev. Lett. 107, 203901 (2011).
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Birks, T. A.

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809(2000).
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Burnett, N. H.

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G. Lambert, T. Hara, D. Garzella, T. Tanikawa, M. Labat, B. Carre, H. Kitamura, T. Shintake, M. Bougeard, S. Inoue, Y. Tanaka, P. Salieres, H. Merdji, O. Chubar, O. Gobert, K. Tahara, and M.-E. Couprie, “Injection of harmonics generated in gas in a free-electron laser providing intense and coherent extreme-ultraviolet light,” Nat. Phys. 4, 296–300 (2008).
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P. Hölzer, W. Chang, J. C. Travers, A. Nazarkin, J. Nold, N. Y. Joly, M. Saleh, F. Biancalana, and P. St. J. Russell, “Femtosecond nonlinear fiber optics in the ionization regime,” Phys. Rev. Lett. 107, 203901 (2011).
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W. Chang, A. Nazarkin, J. C. Travers, J. Nold, P. Hölzer, N. Y. Joly, and P. St. J. Russell, “Influence of ionization on ultrafast gas-based nonlinear fiber optics,” Opt. Express 19, 21018–21027 (2011).
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M. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Joly, P. St. J. Russell, and F. Biancalana, “Theory of photoionization-induced blueshift of ultrashort solitons in gas-filled hollow-core photonic crystal fibers,” Phys. Rev. Lett. 107, 203902 (2011).
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N. Y. Joly, J. Nold, W. Chang, P. Hölzer, A. Nazarkin, G. K. L. Wong, F. Biancalana, and P. St. J. Russell, “Bright spatially coherent wavelength-tunable deep-UV laser source using an ar-filled photonic crystal fiber,” Phys. Rev. Lett. 106, 203901 (2011).
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N. Y. Joly, P. Hölzer, J. Nold, W. Chang, J. C. Travers, M. Labat, M.-E. Couprie, and P. St. J. Russell, “New tunable DUV light source for seeding free-electron lasers,” presented at the 33rd Free Electron Laser Conference (FEL 2011), Shanghai, China August 22–26 2011.

K. F. Mak, Max Planck Institute for the Science of Light, Günther-Scharowsky-Strasse 1, 91058 Erlangen, Germany, and J. C. Travers, W. Chang, P. Hölzer, J. Nold, N. Y. Joly, and P. St. J. Russell are preparing a manuscript to be called “Efficiency and tunability of UV-visible dispersive wave emission in gas-filled kagome photonic crystal fiber.”

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G. Lambert, T. Hara, D. Garzella, T. Tanikawa, M. Labat, B. Carre, H. Kitamura, T. Shintake, M. Bougeard, S. Inoue, Y. Tanaka, P. Salieres, H. Merdji, O. Chubar, O. Gobert, K. Tahara, and M.-E. Couprie, “Injection of harmonics generated in gas in a free-electron laser providing intense and coherent extreme-ultraviolet light,” Nat. Phys. 4, 296–300 (2008).
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A. Nazarkin, A. Abdolvand, A. V. Chugreev, and P. St. J. Russell, “Direct observation of self-similarity in evolution of transient stimulated Raman scattering in gas-filled photonic crystal fibers,” Phys. Rev. Lett. 105, 173902 (2010).
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A. Abdolvand, A. Nazarkin, A. V. Chugreev, C. F. Kaminski, and P. St. J. Russell, “Solitary pulse generation by backward Raman scattering in H2-filled photonic crystal fibers,” Phys. Rev. Lett. 103, 183902 (2009).
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F. Benabid, G. Bouwmans, J. C. Knight, P. St. J. Russell, and F. Couny, “Ultrahigh efficiency laser wavelength conversion in a gas-filled hollow core photonic crystal fiber by pure stimulated rotational Raman scattering in molecular hydrogen,” Phys. Rev. Lett. 93, 123903 (2004).
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G. Lambert, T. Hara, D. Garzella, T. Tanikawa, M. Labat, B. Carre, H. Kitamura, T. Shintake, M. Bougeard, S. Inoue, Y. Tanaka, P. Salieres, H. Merdji, O. Chubar, O. Gobert, K. Tahara, and M.-E. Couprie, “Injection of harmonics generated in gas in a free-electron laser providing intense and coherent extreme-ultraviolet light,” Nat. Phys. 4, 296–300 (2008).
[CrossRef]

N. Y. Joly, P. Hölzer, J. Nold, W. Chang, J. C. Travers, M. Labat, M.-E. Couprie, and P. St. J. Russell, “New tunable DUV light source for seeding free-electron lasers,” presented at the 33rd Free Electron Laser Conference (FEL 2011), Shanghai, China August 22–26 2011.

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F. Dorchies, J. R. Marquès, B. Cros, G. Matthieussent, C. Courtois, T. Vélikoroussov, P. Audebert, J. P. Geindre, S. Rebibo, G. Hamoniaux, and F. Amiranoff, “Monomode guiding of 1016 W/cm2 laser pulses over 100 Rayleigh lengths in hollow capillary dielectric tubes,” Phys. Rev. Lett. 82, 4655–4658 (1999).
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R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539(1999).
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C. J. S. de Matos, S. V. Popov, A. B. Rulkov, J. R. Taylor, J. Broeng, T. P. Hansen, and V. P. Gapontsev, “All-fiber format compression of frequency chirped pulses in air-guiding photonic crystal fibers,” Phys. Rev. Lett. 93, 103901 (2004).
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K. F. Mak, Max Planck Institute for the Science of Light, Günther-Scharowsky-Strasse 1, 91058 Erlangen, Germany, and J. C. Travers, W. Chang, P. Hölzer, J. Nold, N. Y. Joly, and P. St. J. Russell are preparing a manuscript to be called “Efficiency and tunability of UV-visible dispersive wave emission in gas-filled kagome photonic crystal fiber.”

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F. Benabid, G. Bouwmans, J. C. Knight, P. St. J. Russell, and F. Couny, “Ultrahigh efficiency laser wavelength conversion in a gas-filled hollow core photonic crystal fiber by pure stimulated rotational Raman scattering in molecular hydrogen,” Phys. Rev. Lett. 93, 123903 (2004).
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N. Y. Joly, J. Nold, W. Chang, P. Hölzer, A. Nazarkin, G. K. L. Wong, F. Biancalana, and P. St. J. Russell, “Bright spatially coherent wavelength-tunable deep-UV laser source using an ar-filled photonic crystal fiber,” Phys. Rev. Lett. 106, 203901 (2011).
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K. F. Mak, Max Planck Institute for the Science of Light, Günther-Scharowsky-Strasse 1, 91058 Erlangen, Germany, and J. C. Travers, W. Chang, P. Hölzer, J. Nold, N. Y. Joly, and P. St. J. Russell are preparing a manuscript to be called “Efficiency and tunability of UV-visible dispersive wave emission in gas-filled kagome photonic crystal fiber.”

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K. F. Mak, Max Planck Institute for the Science of Light, Günther-Scharowsky-Strasse 1, 91058 Erlangen, Germany, and J. C. Travers, W. Chang, P. Hölzer, J. Nold, N. Y. Joly, and P. St. J. Russell are preparing a manuscript to be called “Efficiency and tunability of UV-visible dispersive wave emission in gas-filled kagome photonic crystal fiber.”

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J. C. Travers, “Continuous wave supercontinuum generation,” in Supercontinuum Generation in Optical Fibers, J.M.Dudley and J.R.Taylor, eds. (Cambridge Univ., 2010), pp. 142–177.
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N. Y. Joly, P. Hölzer, J. Nold, W. Chang, J. C. Travers, M. Labat, M.-E. Couprie, and P. St. J. Russell, “New tunable DUV light source for seeding free-electron lasers,” presented at the 33rd Free Electron Laser Conference (FEL 2011), Shanghai, China August 22–26 2011.

K. F. Mak, Max Planck Institute for the Science of Light, Günther-Scharowsky-Strasse 1, 91058 Erlangen, Germany, and J. C. Travers, W. Chang, P. Hölzer, J. Nold, N. Y. Joly, and P. St. J. Russell are preparing a manuscript to be called “Efficiency and tunability of UV-visible dispersive wave emission in gas-filled kagome photonic crystal fiber.”

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Figures (17)

Fig. 1
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 nm , with 11 μm core diameter and 2.1 μm pitch (Λ), (c) SEM of a kagomé PCF designed for operation in the UV and around 800 nm , and (d) GVD and loss calculated using the FEM of an idealized kagomé PCF structure with 30 μm core diameter, 15 μm pitch (Λ), and 0.23 μm web thickness.

Fig. 2
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 380 nm in the FEM results is caused by an anticrossing with a cladding state, and it strongly affects the dispersion.

Fig. 3
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 μm , (b) dispersion curves for silica-core PCFs (silica strands) with the shortest ZDWs, with core diam eters of 0.47 to 2.86 μm ( 0.2 μm steps), and (c) the dispersion of a kagomé PCF with a 30 μm core diameter, filled with 0 to 20 bar Ar ( 2 bar steps)—note the change of scale compared to (b).

Fig. 4
Fig. 4

ZDW scaling for kagomé PCFs with diameters from 10 to 70 μm (steps of 10 μm ), filled with 0 to 60 bar of (a) Xe, (b) Kr, (c) Ar, and (d) He.

Fig. 5
Fig. 5

FEM calculations for an idealized kagomé PCF with 30 μm core diameter, 15 μm pitch, and 0.23 μm web thickness—designed for operation in the UV and around 800 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 nm ), and (c) the same as (b) at 800 nm .

Fig. 6
Fig. 6

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

Fig. 7
Fig. 7

(a)–(c) Contour plots of the pump energy ( μJ ) required for formation of 30 fs solitons at 800 nm , plotted against soliton order and ZDW, the ZDW being adjusted by varying the gas pressure: (a)  10 μm diameter filled with Xe, (b)  30 μm diameter filled with Ar, and (c)  50 μm 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 μm 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.

Fig. 8
Fig. 8

Fiber-grating/mirror pulse compression with kagomé PCF; (a) density plots of the temporal and spectral evolution of a 10 μJ , 30 fs pulse at 800 nm through a kagomé PCF with 70 μm core diameter filled with 43 bar of Ar, placing the ZDW at 1300 nm ; (b) the broadened temporal shape of the intensity, and the smooth parabolic temporal phase after 8 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).

Fig. 9
Fig. 9

Soliton-effect self-compression at 800 nm in a kagomé PCF with a 30 μm diameter core filled with Ar: (a) dependence of the compressed pulse duration on λ 0 (horizontal axis) and N (numbered curves), (b) corresponding quality factor dependence, (c) temporal and spectral evolution of self-compression of a pulse for λ 0 = 500 nm , N = 3.5 , and (d) field of the compressed output pulse at the point of optimum compression in (c).

Fig. 10
Fig. 10

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

Fig. 11
Fig. 11

(a) Propagation constant mismatch between solitons ( N = 9 at 800 nm ) and dispersive waves at a given wavelength for a 30 μm kagomé PCF filled with 2 to 20 bar Ar ( 2 bar steps); (b), (c) phase-matching wavelengths for (b) a 10 μm diameter Xe-filled fiber, and (c) for a 50 μm diameter He-filled fiber, assuming pressure- tunable ZDW λ 0 .

Fig. 12
Fig. 12

(a), (b) Spectral evolution of two systems, both with λ 0 = 563 nm and N = 7.5 , over two fission lengths [Eq. (5)]: (a) a 50 nJ , 30 fs pulse through 0.68 cm of 6 μm diameter kagomé PCF filled with 38 bar Xe and (b)  10 μJ 30 fs pulse through 68 cm of 60 μm diameter fiber filled with 36 bar He. (c), (d) Temporal and spectral evolution of a 10 fs pulse with 20 μJ energy propagating in a 50 μm kagomé PCF with 10 bar He (corresponding to λ 0 = 380 nm , N = 3 ).

Fig. 13
Fig. 13

Dependence of UV generation on pulse duration (15, 30, 60, and 120 fs ) and soliton order for a fiber with λ 0 = 600 nm (kagomé PCF with 30 μm core filled with 9.8 bar Ar): (a) UV conversion efficiency (to wavelengths shorter than 350 nm ) as a function of normalized soliton order S = N / τ FWHM , corresponding to N = 3 to 9 for a 30 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.

Fig. 14
Fig. 14

(a), (b) Propagation of an N 245 ( 600 fs , 10 μJ ) pulse at 800 nm in a kagomé PCF with λ 0 = 750 nm ( 30 μm core diameter filled with 25 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 429 ( 600 fs , 10 μJ ) pulse at 400 nm in a kagomé PCF with λ 0 = 350 nm ( 10 μm core diameter filled with 8.8 bar Ar): (c) temporal evolution of one shot and (d) spectral evolution of an ensemble average of 30 simulations.

Fig. 15
Fig. 15

Contour plot of (a) the calculated free- electron density ( m 3 ) in a 30 μm 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.

Fig. 16
Fig. 16

Results of Hölzer et al. [5]: (a) experimental output spectra from a 26 μm diameter kagomé PCF filled with 1.7 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.

Fig. 17
Fig. 17

Numerical simulation of the propagation of an N = 4 soliton through a 30 μm diameter kagomé PCF with 1.7 bar Ar filling pressure (ZDW at 380 nm ).

Tables (1)

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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)

Equations (12)

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n m n ( λ , p , T ) = n gas 2 ( λ , p , T ) u m n 2 k 2 a 2 1 + δ ( λ ) p 2 p 0 T 0 T u m n 2 2 k 2 a 2 ,
z U = i sgn ( β 2 ) 2 L D τ 2 U + sgn ( β 3 ) 6 L D τ 3 U + i e α z L NL ( U | U | 2 + i s τ ( U | U | 2 ) ) ,
L D = τ 0 2 | β 2 | , L D = τ 0 3 | β 3 | , β n = ω n β | ω 0 , L NL = 1 γ P 0 , s = 1 ω 0 τ 0 .
( L D , L D ) λ 0 ( L D , L NL ) N = γ P 0 τ 0 2 | β 2 | .
L fiss = L D N .
F c 4.6 N , Q c 3.7 / N ,
τ c = τ 0 F c τ 0 N | β 2 | γ P 0 ,
β sol ( ω ) = β ( ω sol ) + β 1 ( ω sol ) [ ω ω sol ] + γ P c 2 ,
n 2 ( ω ω sol ) n n ! β n ( ω sol ) = γ P c 2 2.3 γ P 0 N .
R = | Δ n Kerr Δ n plasma | = n 2 I 2 n 0 ω 0 2 ω p 2 ,
z ψ = i [ m 2 β m ( i t ) m m ! + γ R ( t ) | ψ ( t ) | 2 ω p 2 2 ω 0 c + i A eff I p t N e 2 | ψ | 2 ] ψ ,
N e t = W ( t ) N a η N e β r N e 2 ,

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