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We describe theoretically the performance of a new design for quasi-phase matched harmonic generation in a gas medium. A hollow core fiber in which thin glass plates are periodically introduced allows good phase control of the fundamental light. The generated x-rays are transmitted through small holes in the plates. An increase of the harmonic yield of up to three orders of magnitude is predicted as a result of the structure. This design can be considered as a phase-locked travelling-wave x-ray laser.
©1998 Optical Society of America
1. Introduction
High harmonic generation (HHG) has become an attractive source of coherent x-ray radiation. More than 290 harmonic orders can be generated when an ultrashort laser pulse is focused in a gas jet, which corresponds to a wavelength of few nanometers for the highest harmonics [1,2]. This coherent x-ray source may find important applications in various field of science and technology if the efficiency of conversion into higher order harmonics is improved. The main limitation on the efficiency in the focused geometry stems from the short interaction region at the laser focus, which is of the order of the confocal parameter. In addition, the Guoy phase shift experienced by the fundamental laser pulse when it passes through the focus would result in reduced harmonic output due to interference, and therefore the gas jet must be positioned on one side of the focus [3]. As a result, the efficiency of conversion into the high-order harmonics is typically 10-7.
Recently, a new method to dramatically improve the conversion efficiency of high harmonic generation was demonstarted by phase-matching the conversion process within a hollow-core fiber [4]. In this case, the interaction length can increase significantly, and the plane wave geometry eliminates any phase mis-match due to focusing. The fiber diameters used are approximately 100 microns. The combination of the negative dispersion due to mode-dependent propagation in the fiber and the positive dispersion of the gas within the fiber allows the index of refraction of the fundamental light to be adjusted to match that of the harmonic field. Recent experiments have shown that good phase matching can be achieved for harmonic generation in near-neutral gases, where ionization is limited to below 10% [4]. By varying the gas pressure, the harmonic signal for orders 19–47 can increase by 2–3 orders of magnitude compared to that of a jet. However, generation of even higher harmonics is accompanied by further ionisation of the gas, which adds a large negative contribution to the index due to the presence of a plasma. For example, the generation of the highest harmonics > 290 corresponds to ionization levels >80%. This plasma contribution cannot easily be compensated for, and therefore phase-matching of high harmonics is limited to orders less than 50.
At first it appears difficult to phase match the generation of harmonics in the presence of negative dispersion due to a plasma by exploiting only geometrical considerations because the intrinsic wavelength dependence of diffraction leads to an advance of the phase (negative dispersion) for both focused and guided geometries. Theoretical schemes have been proposed based on the well known idea of quasi-phase matching, where the atomic density is modulated along the propagation direction in order to eliminate the harmonic emission from regions with destructive phase contribution [5]. An alternative approach proposed is to use counter propagating pulses in a focused geometry to disrupt the phase matching microscopically and hence suppresses the generation of out-of-phase harmonics [6]. Finally, recent experiments have used the dispersion of a thin silica glass plate placed in the generation medium to improve the phase shift between the fundamental and the third harmonic waves, and to increase the third harmonic signal by a factor of 3 [7]. However, this method cannot be used for phase-matching higher harmonics, since the glass is not transparent at short wavelengths. In this paper we propose a new scheme for guided-wave high harmonic generation, where the phase of the laser radiation is periodically shifted in order to better match the phase of the harmonic and the fundamental fields for an arbitrary harmonic order in the presence of a plasma. This scheme should make possible the extension of phase-matching techniques to very short wavelengths.
2. Results of the model
The schematic of the dispersion-controlled waveguide is shown in Fig.1. It represents a particular type of quasi-phase matching, where phase correction of the fundamental light is achieved by a periodic set of thin plates (with holes) of a material which is transparent in the visible, and absorbs in the x-ray region (e.g. glass). The laser beam propagates through a tapered waveguide in order to maintain high intensity in the presence of loss due to plasma generation and the internal structure of the waveguide. The key point in this design is that it is capable of matching the phase of the fundamental and the high harmonics despite the presence of holes along the axis. Moreover, the harmonics are generated and can propagate with low loss and with no significant phase distortion close to the axis of the waveguide where the laser intensity is high. Therefore, the set of plates acts as a resonator (spatial filter) for the harmonic field. Thus, this waveguide acts like a phase-locked travelling-wave x-ray laser. We note that “slow wave” structures have been used in the past for microwave propagating in travelling-wave tubes. However in this case the characteristic size of the structure is smaller than the wavelength - hence the phase velocity is controlled by the impedance of an equivalent LC-circuit (see e.g. [8]).
In order to model the operation of the dispersion-modulated waveguide shown in Fig.1, we used a simplified approach by solving the scalar nonlinear wave equation numerically for the transverse component of the electric field in a frame moving with the speed of light. The backward scattered radiation was neglected as well as the quantum contribution to the harmonic phase. The equation used was:
where Δ⊥ is the transverse Laplacian, N is the initial particle density, e is the electron charge, m is the electron mass, P(E) is the ionisation probability calculated by using a tunnelling-ionisation rate formula [9]. The first term on the right-hand side of Eqn. 1 is responsible for the generation of q-th harmonic order (assuming normalised susceptibility). The second term accounts for the plasma-induced blue-shift caused by the increasing ionisation on the leading front of the propagating pulse. This term is also a source of dispersion for both the fundamental and the harmonic field (we neglect the dispersion of the neutral gas). Since the Kerr nonlinearity of the ionized gas is much smaller than that of a neutral gas, we neglect the nonlinear index of refraction in our calculations. We also assume that the loss term which appears in Eqn. 1 is compensated for by the tapered fiber geometry, so that the peak intensity is kept constant along the beam axis.
Fig. 2. Time dependence of the electric field of a 10 fs laser pulse for different positions along the hollow fiber: (a)- with no phase control, pre-ionized gas; (b)- with a set of glass plates; (c)- with a set of glass plates, pulse-induced ionisation.
First, we note that unlike the full quantum description [10], the model used here allows us to separate the process of ionization from that of harmonic generation, which is important to better understand the dynamics of the process. In order to understand the role of phase mismatch for ultrashort pulses, we first consider the generation of the 5-th harmonic in a fully pre-ionised gas (P(E)=1 in Eqn.1) at a pressure of ~35 Torr (N~ 1.25×1018 cm-3) by a 10 fs laser pulse (λ=800 nm). For such a short pulse, both the phase and the envelope are strongly affected by dispersion in the medium as the pulse propagates. While the envelope is always delayed with respect to local time, the phase of the field advances with respect to the pulse peak for the case of negative dispersion. Figure 2(a) shows the time dependence of the fundamental field as it propagates through a hollow waveguide, with an average inner diamater of 100 μm, and a length of 1 cm filled with 100% pre-ionised Argon gas, for different positions along the z axis. Significant phase changes can be seen, which lead to coherent oscillations of the harmonic signal with a spatial period of ~470 μm (coherence length; also see the inset of Fig.3, curve 1). In order to control the phase of the fundamental, we introduce a periodic phase delay in the waveguide, equivalent to a set of 0.3 μm thick glass plates [11] with holes of diameter of 10 μm, placed each 320 μm along the waveguide. The result is that the fundamental pulse becomes better phase matched for different propagation distances (Fig.2(b)), which leads to a large increase of the output harmonic signal (curve 3 in Fig. 3). In fact, this scheme results in good quasi-phase matching for the harmonic generation process. If the phase correction of the fundamental is changed by introducing twice as much glass over two times longer distance between the glass plates, the phase matching is worse, and the harmonic signal exhibits deeper oscillations (Fig. 3, curve 2). These simulations indicate that even if the phase correction is applied at not exactly half the coherence length, or even it’s value is not exactly equal to π, there can be a significant increase in the harmonic yield of the output. Also, the thickness of the glass plates can be increased by an amount of mλ/(n-1) (m=1,2..), which introduces additional phase shift of 2mπ.
Fig. 3. Harmonic energy versus propagation distance for: no-phase control (curve 1, see Fig. 2(a)); 0.3 μm glass plates with holes (curve 3, see Fig. 2(b)); 0.6 μm glass plates with holes; 0.6 μm glass plates with holes, laser induced ionisation (curve 4, see Fig. 2(c)). The inset shows a part of curve 1 near the origin.
We now consider the more realistic case where the gas is ionized by the propagating laser pulse, with an intensity of ~3×1014 W/cm2. As expected, in this case the phase of the fundamental changes dramatically due to the plasma-induced blue shift. The phase rapidly increases along the ionization front and then decreases due to the plasma dispersion [10,12]. Therefore an appropriate phase correction can be introduced only for a given time “window” along the pulse front where the ionization is within certain limits. This case is shown in Fig. 2(c), where a phase correction is done by 0.6 μm glass plates, placed each 660 μm along the fiber, and the harmonic is assumed to be generated for levels of ionization above 80%. This ionization interval corresponds to our previous calculation [13], which demonstrated that for a very short laser pulse, the different higher harmonics in the plateau are generated during different cycles of the laser field. Therefore, the phase-matched time interval shown in Fig. 2(c) would lead to amplification of the harmonics near the end of the plateau where the ionization is above 80%. Also, Fig. 3 (curve 4) shows that the harmonic signal is weaker for the first 0.4 cm of propagation, because the harmonic is efficiently generated during a few cycles of the fundamental. However, the fundamental blue shift causes a transfer of energy from the lower to the higher harmonic orders (see also [13]) which leads to fast increase in the harmonic output for larger distances.
Fig. 4. Spatial distribution of the field (solid line) and the ionization probability (dashed line) across the hollow core fiber: (a) with no glass plates; (b) with glass plates with 10 μm holes.
Our calculations show that for 100 micron fiber diameters, there is a maximum allowed diameter of the hole of about 10 microns where the glass plates can still control the phase of the pulse along the fiber axis. Figure 4 shows the transverse profile of the laser beam in a guided regime, as well as the spatial distribution of the ionization probability. It can be seen from Fig. 4(a) that in the case of no glass plates, the steady-state spatial profile in the hollow-core fiber is a result of a balance of fiber-guiding effects and plasma-induced self-defocusing. In fact, the presence of plasma reshapes the spatial mode, which is given by a Bessel function for an empty fiber [14]. For the case of plates with 10 μm diameter holes, diffraction effects localise the plasma mostly within the frames of the holes, and the beam profile changes more significantly (Fig. 4(b)). We found that further increase of the hole diameter reduces the ability of the glass plates to control the phase of the fundamental close to the fiber axis, and therefore the size of the holes is of crucial importance for this type of quasi-phase matching.
3. Conclusion
In conclusion, we describe theoretically the performance of a dispersion-controlled hollow core fiber, capable of quasi-phase matching the process of high harmonic generation in a gas medium. Our results indicate that a good dispersion control of the fundamental can be achieved by using thin glass plates, which have central holes to transmit the x-rays. This design leads to an increase of the harmonic yield of up to three orders of magnitude. The dispersion within the hollow fiber can be further adjusted by varying the gas pressure, or by using appropriate multi-layer coated glass plates. This design could also be used effectively to generate and amplify attosecond-duration x-ray pulses [10,13].
4. Acknowledgments
The authors thank C. Durfee and Z. Chang for helpful discussions. The authors acknowledge support for this work from the National Science Foundation.
References and links
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