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We numerically investigate the effect of ionization on ultrashort high-energy pulses propagating in gas-filled kagomé-lattice hollow-core photonic crystal fibers by solving an established uni-directional field equation. We consider the dynamics of two distinct regimes: ionization induced blue-shift and resonant dispersive wave emission in the deep-UV. We illustrate how the system evolves between these regimes and the changing influence of ionization. Finally, we consider the effect of higher ionization stages.
©2011 Optical Society of America
1. Introduction
Recent progress in the development of the hollow-core photonic crystal fiber (HC-PCF) has led to the investigation of areas of physics inaccessible in solid-core fibers or in bulk gas cells. One of the main advantages of HC-PCF is that light is guided in the hollow core, providing a highly efficient means for studying light-matter interactions when it is filled with a gas or liquid [1]. Moreover, the dispersion properties of the HC-PCF can be easily engineered by altering the fiber design parameters or even simply by changing the gas pressure, allowing the control of many important nonlinear processes in the gas. Indeed, numerous experiments have been performed in gas-filled HC-PCFs to examine various nonlinear effects such as stimulated Raman scattering [2], four-wave mixing [3], third harmonic generation [4] and soliton dynamics [5].
One notable experiment is the efficient generation of femtosecond deep-UV pulses (200-300 nm) in an argon-filled HC-PCF by pumping it with ultrashort pulses in the sub-µJ range [6]. In this experiment, a kagomé-lattice HC-PCF was used [7], providing broadband guidance [8]. The transmission spectrum of such fibers covers a large bandwidth, ranging from the near-IR to the UV with losses of the order of few dBm−1, making it an ideal candidate for applications in various nonlinear experiments that involve propagation of ultrashort optical pulses. Numerical simulations using the generalized nonlinear Schrödinger equation (GNLSE) reveal that these deep-UV pulses are generated by processes familiar from nonlinear optics in solid-core fibers, i.e., through soliton effect compression to the single-cycle regime followed by the emission of resonant radiation [9]. However, unlike in the case of supercontinuum generation in a solid-core photonic crystal fiber, gas-filled HC-PCFs have much lower material absorption in the UV range, which, added to the unique dispersion landscape of gas-filled HC-PCF, has enabled resonant radiation to be generated efficiently in the deep-UV region by shifting the zero dispersion wavelength to higher frequencies far from the pump wavelength. One of the most remarkable features of this system is that it allows the wavelength of the UV pulses to be tuned easily by changing the gas pressure or the pump pulse energy. This is due to the dependence of the dispersion and the nonlinearity on these system parameters.
An important and unique aspect of pulse propagation in hollow-core fibers is that the power damage threshold of the interacting medium in these fibers is much higher than that of solid-core fibers, and hence one can easily reach intensity levels where partial ionization of the gas plays an important role in the pulse dynamics. This extends the boundaries of nonlinear fiber optics into regions where plasma generation must be taken into account. The influence of ionization on ultrashort optical pulse propagation has been investigated mostly in the context of gas-jets [10] and filaments [11]. There have previously been limited studies on propagation of optical pulses in gas-filled HC-PCFs with ionization: in one case propagation was considered in a HC-PCF which had a highly restricted guidance window, limiting the dynamics to the immediate vicinity of the pump wavelength [12]; in another case, numerical studies were performed using a propagation equation that includes the ionization terms, but no detailed investigation of the ionization dynamics was made [13].
In this paper, the propagation of ultrashort high-energy optical pulses in kagomé-lattice HC-PCFs filled with gas is investigated numerically, focusing on the role of ionization on the pulse dynamics. A suitable mathematical model, which closely follows the full-field propagation approach with the field ionizing term introduced by Geissler et al. [14], is presented. This model is an essential tool for understanding the novel nonlinear optical processes observed in recent experiments, and for exploring the possibilities of this new and interesting system.
2. Model
The model is based on the uni-directional optical field propagation equation. It does not require the slowly varying envelope approximation, the pulse being represented by the fast-oscillating field which gives a more intuitive picture of pulse structure and chirp. A rigorous derivation of this basic model is available in a paper by Kinsler [15], and is not shown here. Using this approach, the equation describing the propagation of ultrashort pulses in fibers is given by:
where z is the propagation distance in the fiber; t is the time in a frame moving at a suitable reference velocity v; ω is the frequency in rad.s−1; Ẽ(z,ω) is the field in the spectral domain which is given by taking the Fourier transform of the real electric field strength, i.e., Ẽ(z,ω) = {E(z,t)}; and α(ω) is the wavelength dependent fiber loss. ε0 and c are the vacuum permittivity and the speed of the light in vacuum. The linear dispersion is given in β(ω), while PNL(z,t) describes the nonlinear response.The axial wavevectors β(ω) of the modes of a gas-filled kagomé-lattice HC-PCF can be obtained by either modeling the structure numerically using, for example, the finite element method, or simply by approximating them to modes of a hollow dielectric waveguide having the same core area [4,16]. By using the latter approach, the axial wavevectors are given by:
where ngas is the pressure and temperature dependent refractive index of the filling gas, which can be calculated using, for example, a Sellmeier equation with experimentally obtained coefficients [17]; a is the effective core-radius calculated assuming a circular core of the same area as the real fiber structure; and umn is the nth zero of mth order Bessel function of the first kind, where m = n = 1 corresponds to the fundamental (HE11) mode of the fiber. The hollow dielectric waveguide model assumes no overlap between the modes and the glass, which is a good approximation for the modes of the kagomé-lattice HC-PCFs where less than 1% of the light is in the glass when we are away from cladding resonances [2]; indeed, no nonlinear response can be detected in an evacuated fiber at pulse energies where strong UV light is generated in an argon-filled fiber [6]. A comparison of the effective refractive index, neff, given by Eq. (2) with finite element solutions of Maxwell’s equations are in good agreement, apart from the resonances of the cladding structure [8], as shown in Fig. 1(a) . It should be noted that the accuracy of Eq. (2) for approximating the dispersion properties of kagomé-lattice HC-PCFs has been firmly established in an experiment that involves the phase-matching between the pump in the fundamental mode and the third-harmonic in a higher-order mode [4].
Fig. 1 (a) A comparison between the wavelength dependent effective refractive indices of the fundamental mode calculated using the approximate analytical model, Eq. (2), and the finite element method for an evacuated ideal structure having the effective core-diameter of 25 μm. The fundamental mode profile is shown in the inset. (b) Variation of β2 with wavelength for a kagomé-lattice HC-PCF, with an effective core-diameter of 25 μm, filled with argon at different pressures.
One of the greatest advantages of the gas-filled system is that its optical properties can be easily tuned by changing the gas pressure, enabling precise control of the nonlinear process under investigation. This is particularly interesting in kagomé-lattice HC-PCFs, since their waveguide dispersion is much smaller in magnitude compared to that of photonic-bandgap-guiding HC-PCFs. This allows for large changes in dispersion landscape for relatively small changes in gas pressure, making the system easily tunable as shown in Fig. 1(b).
The nonlinear term PNL depends on the type and density of the filling gas and on the field intensity. It is given by:
where χ(3) is the pressure and temperature dependent third-order susceptibility of the gas and fR is the fractional Raman contribution – zero for noble gases. R(t) is the Raman response function of the gas medium. The first term describes the nonlinear response of the bound electrons, while the second term is responsible for the molecular response of the medium, giving rise, e.g., to intra-pulse Raman scattering. The inclusion of the suggested higher-order nonlinear terms, which describe the saturation of nonlinear susceptibility [18], is relatively straightforward, and requires the corresponding χ(n) values. The last term accounts for the effect of ionization due to high intensity optical fields.The nonlinear polarization response of the field ionizing medium is given by [14]:
where Ne is the free electron density; Ip is the ionization energy; e and me are the elementary charge and the mass of an electron. The first term in Eq. (4) is responsible for the loss of pulse energy due to the ionization process, and the second term describes mainly the phase effect due to free electrons following the Drude model. Considering only the first ionization, the free electron density is given by:where W(t) is the ionization rate that depends on the time varying electric field E(z,t). Calculation of the free electron density Ne(t) requires the use of a suitable model for the ionization rate W(t), the choice of which is still a matter of research and debate. For the cases we consider here, the peak intensity inside the HC-PCF can be higher than 1013 Wcm−2 and can even approach 1015 Wcm−2, and it has been shown experimentally that, at these intensity levels, tunnel ionization is dominant over multi-photon ionization for noble gases [19,20]. In what follows, we choose an approach by Ammosov, Delone, and Krainov (ADK) [21]:where ωp = Ip/ħ is the transition frequency, n* = Z(Iph/Ip)0.5 is the effective principal quantum number, the tunneling frequency ωt = e|E(z,t)|[2meIp]−0.5 and |Cn*|2 = 22n*[n* Γ(n*+1) Γ(n*)]−1. Z is the atomic charge after ionization and Iph is the ionization energy of the hydrogen atom. The recombination time for ionized noble gases is on a much longer time scale than the few hundred fs that we consider [22], and therefore it can be neglected in the model.If required, it is straightforward to compare different models in the simulations. For example, multi-photon ionization based Keldysh-Faisal-Reiss (KFR) model [23] was used in [13]. The hybrid model suggested by Perelomov, Popov, and Terent’ev (PPT) [24], or the well-established Yudin-Ivanov model [25], could also be implemented to calculate the ionization rate W(t). However, it has been shown experimentally that Eq. (6) describes well the qualitative features of the field ionizing pulse dynamics in HC-PCFs, while maintaining a simple calculation of the ionization rate for an arbitrary pulse [26].
Spatial lensing effects, such as self-focusing caused by the optical Kerr effect or the self-defocusing resulting from the enhanced ionization near the propagation axis, may have a considerable influence on the propagation dynamics for exceptionally high energy pulses [27,28]. However, the (1+1)-dimensional model presented here gives excellent agreement with the experimental results showing that these spatial effects are insignificant for the parameter regimes under consideration [6,26]. The model can be generalized to incorporate cross-coupling to higher order modes [29], or a full transverse spatial dependence, in cases where such spatial effects become important.
3. Pressure dependent dynamics
One of the most important features of the gas-filled kagomé-lattice HC-PCFs is that by keeping the core-size and gas pressure relatively small (< 50 μm in diameter and < 10 bar argon pressure), the zero dispersion wavelength can be shifted well into the visible-UV range as shown in Fig. 1(b). This allows the system to be anomalously dispersive in the wavelength region where ultrashort pulse sources are readily available. The combined effect of anomalous dispersion and self-phase modulation enables the formation of solitons, enabling the investigation of soliton dynamics in the presence of ionization.
In order to study the pressure dependent dynamics of the system, we numerically propagated a 30 fs Gaussian pulse centered at 800 nm, with an energy of 2.5 μJ, through a 10 cm long argon-filled kagomé-lattice HC-PCF with an effective core-diameter of 20 μm. Figure 2(a) shows the pressure dependent output spectra calculated using the complete model presented in Section 2, while Fig. 2(b) shows the results when the term describing the effect of ionization is omitted. By comparing Figs. 2(a) and 2(b), one can identify the dynamics that are unique to the creation of free electrons in the fiber.
Fig. 2 Pressure dependent output spectra from 10 cm of argon-filled kagomé-lattice HC-PCF (20 μm effective core diameter) pumped with a 30 fs Gaussian pulse at 800 nm with an energy of 2.5 μJ (a) with the ionization term, and (b) without ionization. The zero dispersion wavelength is shown by the dashed curve. A and N indicate the regions of anomalous and normal dispersion. Detailed numerical studies on two different regimes, denoted by the horizontal lines labeled (i) and (ii), are given in Sections 4 and 5.
At low pressures (1 to 2 bar), the ionization induces a spectral blue-shift of up to 200 THz from the pump frequency, as can be seen by comparing Figs. 2(a) and 2(b). Although such plasma blue-shifts are well known in free-space and capillary geometries, they are novel in the context of fiber optics where the system exhibits anomalous dispersion. Several important features of these dynamics are discussed in Section 4 below.
At higher pressures, a UV band emerges between ~160 and ~200 nm. This is due to resonant emission of dispersive waves by compressed higher order solitons. The soliton order increases for higher pressures, and hence the initial soliton compression and spectral expansion both increase. Also, the zero dispersion wavelength is brought closer to the pump wavelength at higher pressures, enabling phase-matching at wavelengths accessible to the compressed soliton. As shown in Fig. 2, the ionization does have an effect on the UV emission, reducing the tuning range and generating a small amount of additional radiation at longer wavelengths than the main UV band. A detailed description of this process is given in Section 5.
4. In-fiber plasma formation
The tight single-mode confinement of the light and the long interaction lengths provided by HC-PCF enables the formation of a plasma at much lower power levels than in other systems. In particular, theinterplay between the optical Kerr effect provided by the gas and the uniquely small-magnitude anomalous dispersion of the kagomé-lattice HC-PCF allows the launched pulse to be compressed to peak intensities that may be large enough to partially ionize the medium [26]. The ability to study light-plasma interactions in a single-mode guiding geometry allows one to access parameter ranges beyond the reach of previous experiments, and is likely to lead to the observation of new physical effects.
Figures 3(a) and 3(b) show the temporal and spectral evolution of the pulse along the fiber at 2 bar pressure – at the slice position (i) marked in Fig. 2(a). At this pressure, the entire bandwidth of interest lies in the anomalous dispersion regime. It can be seen from Fig. 3(c) that the compressed pulse ionizes a significant fraction of the gas (~0.3%, amounting to a free electron density of ~1.6×1017 cm−3) thus locally altering the temporal phase. The polarizability of the free electrons is comparable to that of the optical Kerr effect in such cases, so that plasma formation causes the refractive index to drop locally in the time domain, resulting in the momentary acceleration of the peak that ionized the medium. In Fig. 3(a), this peak corresponds to the pulse traveling faster than the moving time-frame. It can be seen clearly in Fig. 3(d) that the local decrease in refractive index causes the pulse peak to move faster than the tails, resulting in a steepening of the pulse’s leading edge. In the spectral domain, a blue-shift characteristic of plasma formation [30], is observed as shown in Fig. 3(e).
Fig. 3 Light-plasma interaction at 2 bar; (a) shows the temporal evolution on a linear scale, while (b) shows the spectral evolution on a logarithmic scale. The vertical dashed-line in (b) corresponds to the zero-dispersion wavelength. The ionization fraction is plotted against fiber length in (c). The temporal and spectral slices of the pulse after propagating 6.3 cm are presented in (d) and (e). The initial spectrum is shown with the green dashed-line in (e).
The ionization process results in an intensity-dependent loss of pulse energy, described by the first term in Eq. (4). This loss eventually reduces the pulse peak intensity below the critical value, preventing further ionization from taking place.
Figure 3(b) indicates a breathing effect in the spectrum between 6 and 10 cm. We propose that this is due to interference between ionization blue-shifted solitons and solitons remaining at the pump wavelength. In such a case the breathing period should be shorter than for a collection of fundamental solitons at a single wavelength (i.e. a higher order soliton).
5. Influence of ionization on UV emission
The generation of UV light in kagomé-lattice HC-PCFs was first shown numerically by Im et al. [13], and its experimental observation was reported soon after [6]. In the latter case however, the effect of ionization was insignificant as the peak intensity in the fiber was not sufficiently high to ionize enough gas to have any noticeable impact on the propagation dynamics. This is evident from the excellent agreement between the experimental and numerical results reported in [6], where the model did not include the contribution of ionization. The numerical results of Im et al. [13], on the other hand, were for very different parameter ranges than the experiment in [6], or those discussed here. The ionization terms were included in their model, but no analysis of its effect on the pulse dynamics was given. Figure 4 shows the result at a pressure of 4 bar – at the slice position (ii) marked in Fig. 2(a).
Fig. 4 UV light generation in kagomé-lattice HC-PCF filled with argon at 4 bar; (a) shows the temporal evolution in linear scale, while (b) shows the spectral evolution in logarithmic scale; (c) and (d) show the temporal and spectral evolution when the ionization term is not included; (e) – (h) show the time domain slices of the pulse at positions indicated by the dashed lines in (a). The vertical lines in (b) and (d) indicate the zero-dispersion wavelength.
When the pump pulses are launched into the fiber, they initially experience spectral broadening and temporal compression due to the combined effects of self-phase modulation and anomalous dispersion as shown in Figs. 4(a), 4(b) and 4(e). This symmetric broadening of the pulse spectrum continues until it reaches the few-cycle regime. As the pulse duration becomes shorter, the rapid temporal variation in pulse intensity leads to a large nonlinear refractive index difference between the pulse center and the tails, causing the trailing edge of the pulse to steepen as shown in Figs. 4(f) and 4(g), in the opposite sense compared to the plasma blue-shift case shown in Fig. 3(d). This steepening of the pulse edge enhances the generation of high frequency components in the spectral domain, beyond those produced from SPM alone, leading to asymmetric broadening of the spectrum towards the high frequency side [31]. The spectral reach (far into the deep-UV region), together with the compressed high peak intensity, allows efficient transfer of pump energy to one or more high frequencies in the UV range that satisfy the nonlinear phase matching condition. This process, known resonant radiation [32], takes place over a short distance (a few cm), and the UV pulse quickly decouples from the pump due to group velocity mismatch as shown in Fig. 4(h). No further efficient transfer of the energy occurs after this point.
The generation of UV light is significantly affected by the presence of the plasma [33]. The effects of ionization in this particular system are clear by comparing Figs. 4(a) and 4(b) with Figs. 4(c) and 4(d). In the time domain, ionization leads to an advancing compressed pulse as opposed to delayed compressed pulses. In addition, the UV emission is at 180 nm in the presence of ionization, as opposed to 140 nm without. Comprehensive numerical simulations have indicated that the presence of ionization generally prevents the wavelength of the dispersive waves from reaching far into the vacuum-UV range, although wavelengths as short as 125 nm are predicted for some parameters in a helium-filled fiber. Figure 4(b) shows that the presence of partial ionization causes the beating of two solitons with different wavelengths, as discussed in Section 4, which results in a breathing behavior in both the temporal and the spectral domains. The spectral beating in this case causes subsequent emission of resonant radiation at further compression stages along the fiber, leading to low level emission at longer wavelengths (200 – 300 nm) than the main UV band.
6. Higher-ionization stages
The density of the free electrons in Eq. (5) is calculated assuming that only one free electron is produced from each ionized atom. This is true in most cases, especially when the noble gases are used, since the second ionization energy is much higher than the first. For example, the first ionization energy of argon is 15.76 eV, whereas the second is 27.63 eV. This is however not the case for all gases. For example the first five ionization energies of SF6 are closely spaced due to its atomic structure: 15.34, 16.9, 18.3, 19.8, 22.7 eV [34]. In such cases the model needs to be extended to incorporate higher ionization stages. This can be easily achieved by calculating the ionization rate Wn+ with Z = n for each nth ionization process. The higher ionizations occur consecutively following:
where N0 denotes the initial density of the atoms and Nn+ is the density of the ions with charge n. This leads to a set of coupled rate equations for the populations of the successive ionization stages [35]:The density of free electrons can be readily obtained by solving these equations, and can then be substituted into Eq. (4) to determine the contribution from ionization processes up to any desired order.
The influence of higher ionization processes on pulse dynamics in gas-filled HC-PCF are now investigated numerically by simulating the propagation of a 3 μJ, 30 fs Gaussian pulse with wavelength centered at 800 nm through 20 cm of kagomé-lattice HC-PCF filled with a gas with closely spaced ionization levels. Since reliable data on the dispersion of SF6 is minimal, for the purpose of demonstrating the effect of higher order ionization we used the dispersion corresponding to a kagomé-lattice HC-PCF with an effective core diameter of 20 μm, filled with 1 bar of argon. Figures 5(a) and 5(b) show the results of simulations that include up to fifth-order ionization and those including only first-order ionization.
Fig. 5 Numerical propagation of 3 μJ, 30 fs pulse centered at 800 nm in kagomé-lattice HC-PCF with effective core diameter of 20 μm and filled with a gas with the dispersion of argon at 1 bar, but with the ionization levels of SF6. (a) Temporal (linear) and spectral (logarithmic) evolution of the pulse when up to fifth-order ionization processes are taken into account. (b) The results when only first-order ionization is included. (c) The electron density as a fraction of the initial gas density for both cases. (d) The evolution of the peak intensity. (e) The evolution of the pulse energy as it propagates. The blue solid lines are the numerical results where up to fifth-order ionization is included while the red dashed lines are results where only the first-order ionization is included.
The results clearly demonstrate that higher ionization processes have a major influence on the pulse dynamics in the system shown above. The presence of low-lying higher-ionization energies of SF6 tend to arrest ionization of the medium to lower fractions, but helps maintain propagation in the ionized medium over longer distances as can be seen from Figs. 5(c), 5(d) and 5(e). The extended length of the ionized medium results in continuous blue-shifting of the pulse spectrum as shown in Fig. 5(a). Further investigations are required for a complete understanding of the pulse dynamics under the influence of higher ionization stages.
7. Conclusion
The development of kagomé-lattice HC-PCFs has moved single-mode nonlinear fiber optics into hitherto inaccessible regimes. Numerical simulations, using a model based on a uni-directional field propagation equation which includes the effect of ionization, show that significant fractions of the gas can be ionized at pulse energies of a few μJ. This allows, for the first time, detailed and well-controlled investigation of optical pulse dynamics in a partially ionized medium in an optical fiber. At low pressures, when the zero dispersion wavelength is in the UV range, pumping with low order solitons at 800 nm leads to a clear plasma induced blue-shift. At higher pressures, resonant dispersive-wave emission can occur, leading to the emission of UV light. In this case ionization can restrict UV emission at the shorter edge of the UV spectrum and lead to faster spatial-spectral beating of the residual pump pulse. By extending the model to include higher ionization stages, the dynamics of pulse propagation in gases with closely spaced ionization potentials can be investigated, showing a continuous blue-shifting of the pulse spectrum over extended fiber lengths.
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