## Abstract

We describe theoretically a new technique for quasi-phase-matched generation of high harmonics and attosecond pulses in a gas medium, in a high ionization limit. A corrugated hollow-core fiber modulates the intensity of the fundamental pulse along the direction of propagation, resulting in a periodic modulation of the harmonic emission at wavelengths close to the cutoff. This leads to an increase of the harmonic yield of up to three orders of magnitude. At the same time the highest harmonics merge in a broad band that corresponds to a single attosecond pulse, using 15-fs driving pulses.

© Optical Society of America

## 1. Introduction

High-order harmonic generation (HHG) in gases is a useful source of coherent radiation throughout the extreme ultraviolet and soft x-ray regions of the spectrum [1–3]. In HHG, an intense laser is focused into a gas jet [1–3] or a gas-filled fiber [4]. Due to the extreme nonlinear interaction between the laser and the atoms in the gas, high-order harmonics of the fundamental frequency are radiated in the forward direction as a coherent, low-divergence, beam. For HHG in a gas jet, the main limitation on the efficiency stems from the short interaction region at the laser focus, which is on order of the confocal parameter. In addition, the Guoy phase shift experienced by the fundamental laser pulse when it passes through the focus results in reduced harmonic output due to interference effects [3]. As a result, for HHG in a gas jet the conversion efficiency of laser light into each harmonic order is typically 10^{-7}. The use of a guiding geometry in a hollow-fiber waveguide makes it possible both to increase the interaction length, and also to phase match the process by matching the phase velocity of the laser with that of the generated harmonics [4]. This increases the conversion efficiency by factors of 100–1000 for mid-plateau harmonics around 30-70 eV. Further improvements in conversion efficiency are possible by using optimally shaped laser pulses [5]. However, the conversion efficiency for higher energy photons (>100eV) is significantly lower, falling to about 10^{-11} per harmonic at photon energies of ~500eV close to the cut-off for HHG using helium. This is because the higher energy photons are generated at high levels of ionization, where plasma-induced variation in the refractive index destroys the phase matching. At low levels of ionization (<10%), phase matching is achieved by balancing the negative dispersion due to the hollow fiber with the positive dispersion due to the quasi-neutral gas [4,6]. Phase matched generation of higher-order harmonics under even higher levels of ionization is more difficult because of the large negative dispersion due to the generated plasma. This problem may be ameliorated in certain regimes by using a structured waveguide to periodically modulate the phase of the fundamental pulse to match that of the harmonics [7,8].

In general, the use of hollow waveguides for high-harmonic generation allows more flexible control on the experimental conditions. By shaping the waveguide, the intensity of the laser can be varied along the fiber axis to achieve additional control over the conversion process. For example, the use of a tapered fiber maintains a high laser intensity along the fiber, despite the loss experienced by the laser pulse due to the leakage of energy through the walls, and due to the ionization of the gas [7,8]. This is especially important for longer fibers because the efficiency of the harmonic generation is very sensitive to the peak intensity of the laser pulse. In this paper we analyze the performance of a tapered hollow waveguide for high harmonic generation under conditions of high ionization. We show that the conversion efficiency for the harmonics close to the cutoff can be significantly improved if the walls of the waveguide are properly corrugated so that the harmonic emission occurs only in certain zones along the fiber, due to periodic focusing of the laser pulse. This scheme is, in effect, a quasi-phasematching technique, complementary to that reported in [7,8]. In that work, the phase of the fundamental light was modulated; here the generation of the high harmonics is modulated. We would also like to point-out that other methods for obtaining such a periodic intensity modulation, such as launching a non-stationary spatial mode in a simple hollow fiber, are possible. Other techniques in addition to those mentioned above for improving the efficiency of HHG include phase matching by difference frequency mixing [9], quasi-phasematching by using counter-propagating beams in a focusing geometry [10], modulating the atomic density [9], using thin glass plates [11], and “nonadiabatic” phase matching [12].

## 2. Results of the model

Our approach is based on fact that the highest harmonic orders (close to cutoff) are generated during only few cycles of the laser field, close to the peak of the pulse. If the pulse intensity is sufficiently high that the gas is almost fully ionized near the peak, these harmonics propagate in a nearly stationary plasma, and therefore experience regular coherent ringing due to a constant plasma dispersion. This coherent ringing of the harmonic signal can be manipulated by changing the peak intensity of the laser pulse periodically, so that even a small reduction in the laser intensity can periodically quench the harmonic emission in the cutoff region. A quasi-phasematched geometry can thus be attained for those harmonics. Here we consider one simple way to realize this idea, by using a tapered, weakly corrugated, hollow waveguide, whose radius changes as a sine function, as shown in Fig.1(c). We compare the harmonic emission of this corrugated waveguide with that from an un-corrugated fiber (Fig.1(a)), which we call henceforth a “flat fiber”.

The analysis in this paper is based on three dimensional propagation of an optical pulse in a hollow-core waveguide. Our method includes a numerical solution of the three-dimensional scalar wave equations written for the laser field *E*(*r*,*t*) and for the harmonic field *E _{h}*(

*r*,

*t*) in a local frame of reference, moving with the speed of light along the fiber axis (z) [13]:

where Δ_{⊥} is the transverse Laplacian. The first term in the right-hand side of Eq.(1a,b) describes the dispersion that is due to the transient plasma, the second term in Eq.1(a) accounts for the ohmic power dissipation that is due to the ionization, and the third term describes the dispersion by the neutrals. *n*(*p*) is the index of refraction as a function of the pressure of the neutral atoms [14]: *p*=*p0*[*1*-*P*(*E*)] where *p0* is the initial pressure of a gas of atoms with concentration *N*, and with an ionization potential *I _{p}*.

*P*(

*E*) is the ionization probability. We have neglected the nonlinear index in our calculations because its effect is expected to be small for high degree of ionization. Since the laser pulse we use here is 15 fs in duration (λ=800nm), the average dipole acceleration 〈

*a*〉 in Eq.1(b) is calculated by using a quasiclassical approximation for the dipole moment [15]:

where ε is a positive regularization constant and we have neglected the bare atomic dipole moments (atomic units are used here). In Eq. 2 we assume that the electron is ionized at a time τ_{b} by the electric field E(t), and that it returns to the parent ion at a time τ after “free” motion in response to the laser field. Also, in Eq.2
$\gamma \left({\tau}_{b}\right)={\int}_{0}^{{\tau}_{b}}w\left(t\right)\mathit{dt}$, where *w*(*t*) is the Ammosov-Delone-Krainov tunneling ionization rate [16], and we have in Eq.1
*P*(*E*)=1-exp-*γ*(*τ _{b}*)].

It is important to point out that in order for the harmonic signal to experience regular coherent ringing, changes in the temporal profile of laser pulse due to the plasma dispersion should be weak. On the other hand, the absorption loss also affects the conversion efficiency for the higher harmonic orders and therefore a tapered geometry should be used. The energy loss due to the ionization can be assessed by using a simple formula *W*=*W _{0}*-

*NI*, where

_{p}VP_{f}*P*is the ionization after the pulse and

_{f}*V*is a measure for the volume of the guided mode. In our calculations we use a hollow waveguide with length 1 cm and input diameter 155 µm filled with argon at 1 Torr pressure. Figure 2 shows the superimposed laser pulses for different positions along the fiber axis, as well as the ionization, for a peak intensity 7×10

^{14}W/cm

^{2}. From Fig. 2, the ionization reaches near 100% close to the peak of the pulse, so after that point the phase shift experienced by the laser field becomes stationary.

Since the semiclassical cutoff for the above parameters corresponds to the 97-th harmonic, we calculate the propagation of both the laser pulse and the harmonics (Eqs.1,2), and monitor the energy of the 95-th harmonic. This harmonic exhibits coherent oscillations along the fiber axis with a coherence length of 0.5 mm (Fig.3(a)), that is determined by the net negative dispersion due to the plasma and the waveguide. We then apply a weak sine corrugation to the fiber walls, with a depth of 1.5 microns, which leads to a modulation of the peak intensity of the laser on the fiber axis by about 5% (see Fig.1(d)). We found that this intensity modulation is sufficient to periodically quench the generation of the highest harmonicorders. As a result their energy increases nearly parabolically, which is expected for quasi-phasematching (Fig.3(b)). We note that for longer propagation distances, the harmonic energy tends to saturate because the plasma reshapes the laser pulse, and the coherent oscillations in Fig.3(a) become irregular (chirped).

Figure 4 shows the harmonic spectra at the input of the fiber (curve 1), and at the output (curve 2) for the case of a flat fiber (a), and corrugated fiber (b). Four harmonic orders near the end of the plateau are seen to merge into a broad band, centered at the 95-th harmonic for which the quasi-phasematching is optimized. Therefore, the sine corrugation enhances the harmonic generation over a broad range, because the coherence lengths for the neighboring harmonics are similar. Moreover, the varying plasma-induced blue shift of the cutoff harmonic orders that is due to the periodic focusing of the laser pulse leads to continuum-like spectrum for these cut-off harmonics, rather than the usual discrete emission. This continuous spectrum corresponds to a single attosecond-duration pulse (Fig.5(b)), as opposed to the case of a flat fiber which results in a sequence of three pulses (Fig.5(a)). These simulations clearly show that high harmonic generation in a structured waveguide can lead to the generation of sub-femtosecond x-ray pulse, without the use of a single-cycle regime of interaction between the atom and the laser pulse [17]. Of course, the use of shorter laser pulses would further increase the efficiency of the harmonic conversion and produce even shorter attosecond pulses, as predicted previously [17,18]. Also, since here we use 15 fs driving pulses, the harmonic emission is less sensitive to the carrier-envelope phase offset than in the case of HHG by 5 fs pulses [19].

## 3. Conclusion

In conclusion, by employing a three-dimensional propagation model we describe theoretically the use of a corrugated hollow core fiber to implement quasi-phasematching for very high order harmonic generation. This scheme is designed for harmonics close to the end of the plateau, in a high-ionization limit. Our results indicate that the efficiency of HHG can be improved by almost three orders of magnitude for moderate values of the laser intensity, and for pulse durations of 15 fs. Quasi-phasematched generation within the corrugated hollow fiber can be further adjusted by modifying the profile of corrugation or by chirping the corrugation. This design could also be used effectively to generate and amplify attosecond-duration x-ray pulses. It is also well-suited for the efficient generation of high-energy harmonics near the cutoff, such as in the “water window”[1,2]. We note that since a weak intensity modulation is enough to suppress the generation of the harmonics of interest, the intensity variations across the beam propagation should be small. Such conditions can be achieved by choosing appropriate focusing optics so that the beam parameters at the input of the hollow fiber are close to those of the guided mode. In this case the laser beam is coupled to the waveguide mode after a shorter propagation distance. If the beam cross section is smaller than the optimum, oscillations of the beam profile may lead to a “spontaneous” quasi-phasematched regime of harmonic generation, even in a flat fiber.

## 4. Acknowledgment

The authors gratefully acknowledge support from the National Science Foundation and from the Department of Energy.

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