1. Introduction

Efficient generation of a coherent beam of femtosecond pulses at soft x-ray wavelengths may have applications such as high temporal and spatial resolution imaging of hydrated biological samples, photoemission spectroscopy and metrology of silicon wafers for next generation lithography. One approach for generating such beams is high harmonic generation (HHG) in gaseous targets [1, 2, 3]. To date, the highest harmonics observed have energies extending to >1keV [2], but the very low effective nonlinearity of the generating medium employed, He, coupled with the absence phase matching, limits the HHG signal to several hundred photons/harmonic peak/pulse. As a result, the most significant need for further advances remains in increasing the overall conversion efficiencies and pulse energies that can be achieved using high harmonic generation (HHG).

In general, the length over which the HHG signal will add constructively, known as the coherence length (L_c), is set by the wavevector mismatch, Δk, between the harmonic and driving laser E-fields, and is defined as

with k_q the wavevector of the q ^th harmonic order and k ₀ the laser wavevector.

For typical high intensity HHG experiments in which an ultrashort laser pulse is focused into a gas jet, there are several factors affecting Δk: geometric (Δk_g, net negative contribution); free electron, or ionisation, induced (Δk_e, net negative contribution); and neutral gas density contributions (Δk_a, net positive contribution) [4, 5]. Δk_g, due mainly to the Gouy shift, can be replaced by the more controllable dispersion of hollow gas filled capillaries [4, 5, 6]. The use of capillaries also allows high intensity conditions to be maintained over extended interaction lengths with near constant/controllable pressure. In general for HHG experiments, the contribution of Δk_e and Δk_g to the wavevector mismatch in the nonlinear medium dominates Δk under conditions of high average-ionisation (Z ^*=N_e/N_a ≥ ~1, where N_e and N_a are the number density of electrons and atoms respectively), and can be approximated simply as

In close analogy to the generation of optical harmonics of laser radiation in nonlinear crystals, high efficiency can be achieved by phase-matching the harmonic generation process throughout the non-linear medium i.e. setting Δk to 0. Phase matching of low energy (<50eV) HHG from capillaries has been demonstrated with dramatic improvements in the observed signal obtained by pressure tuning to balance the free electron, neutral gas density and capillary propagation contributions to Δk around specific atomic resonances [4]. However this scheme is limited to low levels of ionisation and hence low harmonic orders.

An alternative method for increasing the observed HHG signal is quasi-phasematching (QPM). In QPM no attempt is made to achieve Δk =0, instead harmonic production is suppressed for half of the QPM period

to prevent destructive interference between HHG from different regions of the medium, where m (odd only) indicates the order of the QPM process. In analogy to the phasematching condition given in Eq. 1, one can write for quasi-phasematching

where ΔK is the mismatch relevant to the QPM process.

Pioneering work has demonstrated the validity of the QPM scheme [5, 7, 8]. Paul, et al. [5] and Gibson, et al. [8], both using capillaries with a modulated bore diameter to vary the intensity of the driving laser in the capillary, show that enhancements in harmonic signal can indeed be achieved with some success. Paul, et al. [5] demonstrated enhancement by factor of order ~10 in the plateau region, while Gibson, et al. [8] demonstrated similar enhancements in the cut off regions of HHG spectra [8]. However, extending QPM to higher photon energies is difficult for current QPM schemes since L_c is then typically <50μm. Moving to higher order m (Eq. 4) does allow signal enhancement at short wavelengths, but at significantly reduced signal growth rates.

In this work a new method of QPM enhancement of the high harmonic signal generated in hollow gas filled capillaries is presented, with substantial improvements in the HHG signal extending to high photon energies. This QPM process takes advantage of mode beating [9, 10, 11, 12] in capillaries (multimode quasi-phase matching, or MMQPM), to achieve very significant periodic intensity modulations over the entire interaction length. Excitation of these higher order capillary modes, and their subsequent beating, forms significant spikes in the axial intensity, separated by extended regions of low ionisation (>10^-2 of peak), allowing the HHG signal to grow via QPM. This process is shown to allow the generation of bright soft x-ray harmonic radiation with enhancements of >10⁴ over the signal expected from a single coherence length and, as a consequence of the deep intensity modulations, to increase the quasi phase-matching period, L,at the highest harmonic orders.

2. Multimode quasi-phase matching

When a free space propagating laser beam is coupled into a hollow capillary waveguide a number of capillary eigenmodes are excited. For an axially centred, radially symmetric beam, azimuthal modes, EH_1,j, are dominantly excited, such that

where E_L(r) and E_1j (r) are the transverse, or radial, electric field (E-field) distributions of the incident laser beam and j^th eigenmode respectively. The relative peak on-axis E-field strength coupled into the j^th mode, is given by the coupling coefficient C_j [11] and the incident pulse intensity profile, A(r) = |E_L(r)|², in terms of the individual eigenmode profiles, is given as

where U _0,j is the j^th root of the zero order Bessel function, J ₀, a is the capillary radius and r is the radius of the profile. Best coupling (~98% incident energy) of a Gaussian TEM₀₀ free space mode into the fundamental mode, EH₁₁, of the hollow capillary is achieved for

where χ is referred to as the coupling parameter, w ₀ is the 1/e² radius of the incident beam, and a is the capillary radius [9]. For χ≠0.64 more energy is coupled into higher order modes. The fraction of the incident E-field coupled into each mode is determined by χ [9, 10], [11]. At a distance z into the capillary the axial intensity of the guided beam is given by

where L_z,j- = Exp(z/z^at_j and z^at_j(∝ a³) is the attenuation length for the j^th mode, and Δk_j = β ₁ - β_j and β_j, the capillary propagation wavevector of the j ^th mode, is given by [10].

3. Results and discussion

The characteristic high frequency beat length, L_B, of the intensity modulation, at a given point z in the capillary is set by the highest order mode, jmax, with a significant amplitude after propagating a distance equal to z_jmax^at [10]. Importantly, in the MMQPM schemes presented here, L_B at a given point z in the axial intensity profile, I(z), is not set by the number of modes, j ^*, included in the truncated calculation i.e. L_B is not a function of j ^*.

 

Fig. 1. Idealised multimode beating assuming two equal intensity modes (j=1 and j=20) excited in a capillary with a~90μm. Figure 1(a) shows the axial intensity as a function of z, I(z), normalised at z=0, over the first 0.2 cm of the capillary while Fig. 1(b) shows corresponding harmonic source term (assumed proportional to the ADK rate for tunnel ionisation [13]).

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In an ideal QPM scenario one would have perfectly periodic intensity modulations and this could be achieved for two excited modes with equal intensity (Eq. 8). This idealised case is shown in Fig. 1 for jmax=20 (giving L_B~200um), and assuming z_jmax^at → ∞, in an evacuated capillary with a~90μm.

 

Fig. 2. Normalised Intensity profiles for coupling to an evacuated capillary with a=90μm. The coupling coefficient for the Gaussian (solid line) and the central maximum of the Airy profile (dashed line) are χ=0.64 and χ=0.2 respectively.

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For realistic transverse profiles of the beam incident on the waveguide, a large number of waveguide modes - as opposed to two - will be excited. In this section we investigate MMQPM for the two possible transverse intensity profiles presented in Fig. 2: that of an ideal Gaussian beam profile and, more in line with that typical of many high power lasers with flattop/super Gaussian near-field distributions, an Airy profile.

The on axis intensity profiles (Eq. 8) in an evacuated capillary with a=90μm for values of C_j (Eq. 5) resulting from the Gaussian (χ=0.64) and Airy (χ=0.2 for the central maximum) profiles shown in Fig. 2 are presented in Figs. 3(a) and 3(b) respectively.

 

Fig. 3. Multimode quasi phase matching. Calculated axial intensity as a function of distance z into a capillary of radius 90μm for beams with the Gaussian (a) and Airy (b) incident transverse profiles shown in Fig. 2. The corresponding calculated harmonic source terms (normalised at peak intensity) are shown in Figs. 3(c) and 3(d). The resulting harmonic growth for q=201 (solid lines), normalised to the signal expected for one coherence length (dot-dashed line), is shown in Figs. 3(e) and 3(f). The dotted line in both 3(e) and 3(f) corresponds to the expected harmonic growth for the two mode axial intensity profile given in Fig. 1(a), while the dashed line is the signal expected for perfect phase matching. Harmonic growth is calculated using a 1-D code under optimal matching conditions for the normalised harmonic source terms given in Figs. 3(c) and 3(d). Optimal matching conditions were achieved in each case by varying intensity at z=0 and pressure (approx. 20mbar Ar) slightly to adjust Z ^* such that 2L_c ≈ L_B over extended interaction regions. It should be noted that the rapid growth of HHG over the first few mm of the capillary for the intensity profiles in Figs. 3(c) and 3(d) in comparison to that for the idealised MMQPM scheme from Fig. 2 is due to the constructive interference of multiple modes leading to increased on-axis intensity and hence greater HHG signal.

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As can be seen, even for the optimal coupling of a transverse Gaussian beam profile to the fundamental mode, j=1, [Eq. (7)], significant intensity modulations (>50%) can exist in the capillary over z_j^at [10] of the higher order modes. The Airy profile couples more energy into higher order modes leading to increased on-axis beating, as can be seen in Fig. 3(b). The transverse intensity profile in Fig. 2 (dashed line) approaches that expected for perfect matching to eigenmode j=3 [Eq. (6)] giving enhanced low frequency modulation in the corresponding axial intensity profile while the high frequency modulation (L_B) is still set by the highest order significantly contributing mode. These profiles provide the platform for a significantly modulated on-axis harmonic source term as shown in Figs. 3(c) and 3(d) respectively.

The existence of sharp peaks in intensity, spaced by multiples of L_B, can lead to quasi-phasematching and consequently rapid growth in harmonic signal, as shown in Figs. 3(e) and 3(f) (solid line) for q=201. The signal growth is calculated using a 1-D code assuming 40fs pulse and ~20mbar Ar present in the capillary (the signal for q=201 is from Ar⁺). The harmonic source term is calculated for each spatial grid-point and its functional dependence on intensity is assumed to be proportional to the ionisation rate (approximated here the ADK rate [13]. This dependence can be easily understood in terms of the 3-step model of HHG, whereby the electron must first be ionised before it can generate harmonic radiation [1]. From the results presented in Fig. 3, the concept of MMQPM is clearly quite robust with regards to incident transverse intensity profiles i.e. as long as the intensity and pressure are chosen correctly it is possible to achieve substantial signal growth for a variety of input profiles. Low harmonic order MMQPM (low Δk) can be readily achieved by tuning pressure and intensity accordingly.

As can be seen (in this example) most of the growth for high Δk takes place over the first few mm of the capillary before the higher order modes are significantly damped and hence it might appear that MMQPM may not be readily scaled to longer lengths. It should be noted that I(z) has been calculated assuming evacuated capillaries (i.e. no gas present). The presence of gas in the capillary will cause continuous repartition of energy to higher order modes [10] thereby extending the distance over which MMQPM operates. Simulations performed assuming continuous transfer of energy into higher order modes by nonlinear coupling indicates increased harmonic enhancements. Ultimately the maximum effective length for our conditions is either limited by the absorption length of a given harmonic or the capillary mode attenuation length. For ultrashort pulses group velocity dispersion between the modes would need to be considered but is negligible here.

Previous work has shown that modifying the spatial profile of the incident intensity profile can lead to selectively enhanced harmonic generation [15, 16] which may prove suitable for extending the MMQPM scheme to higher energies. Furthermore, it may also be possible to force mode mixing suitable for MMQPM by slightly bending, or changing the bore radius of, the capillary.

We have recently demonstrated dramatic enhancements (>10³, conversion efficiencies >10^-6) at soft x-ray wavelengths (extending to 360eV, q=231) generated by high harmonic generation driven within an argon-filled straight bore capillary waveguide. This result, which we interpret as arising from MMQPM, will be presented in an imminent publication elsewhere [14].

Under conditions for which Eq. 4 can be satisfied, shown in Figs. 3(c) and 3(d), the rapid variations of I(z) correspond to imperfect m=1 QPM processes i.e. all the destructive contributions are ‘switched off’ but not all the constructive contributions are ‘switched on’. As a result of the strong intensity modulation, the degree of ionisation, Z ^*, is high at points of strong HHG and low everywhere else. This results in a much lower average Z ^* and hence, lower Δk, allowing a longer - and critically easier to achieve - QPM period to be employed (from Eq. 4). This is particularly important for QPM of very high-order harmonics, for example those beyond the C K-edge (~4.3nm, harmonic order, q, ~ 185 for ~800nm fundamental). In essence, MMQPM allows much longer L_c in comparison to a flat intensity case.

An important corollary of this work, in relation to HHG, is that even under optimal coupling conditions significant modulation of the on axis intensity can occur when HHG is driven within a waveguide. If these are to be avoided, such as when attempting true phase-matching, harmonic generation should only be undertaken at sufficiently long distances into the waveguide for all but the lowest-order mode to have been absorbed.

4. Conclusion

In conclusion, we have proposed a new scheme for quasi phase-matching which utilizes fluctuations in the axial intensity of a beam propagating in many modes of waveguide Numerical modeling shows that this approach can enhance the intensity of harmonics by a factor of order >10⁴ even for harmonics beyond the C K-edge, for which the coherence length can be very short. Given the inverse dependence of C_j and β_j with respect to capillary bore radius, a, the mode structure in the capillary therefore depends strongly on a. Our calculations indicate that moving to smaller capillary bore diameters will allow the generation of higher frequency mode beating profiles suitable for QPM at very high Δk.