Generation of isolated attosecond pulses with enhancement cavities—a theoretical study

HHG in ECs has been an active research topic in the last years, and many prerequisites to generate high-repetition-rate IAPs have already been established. First, XUV radiation is emitted collinearly with the strong driving beam, it needs to be separated without introducing too much loss to the driving field. Several approaches have been demonstrated, offering output coupling efficiencies between 5 and around 20% [24]. Most suitable for the generation of IAPs seem geometrical output coupling techniques [25, 26], which are power-scalable and do neither angularly disperse nor spectrally alter the XUV beam because it leaves the resonator without reflection or diffraction at an optical element.

Further, formation of plasma on the time scale of one pulse leads to a blueshift limiting the overlap of input and intracavity spectrum and thus the power enhancement. For high intensities, plasma lensing can be expected to affect the spatial overlap. This effect is quantitatively understood [17–19], and approaches to alleviate the limitations arising from the blueshift have been suggested [19].

The problems of thermal lensing, mirror damage and resonator stability were addressed in [12, 16, 27], identifying ways of progressively scaling the intracavity power. Thanks to these results, a state-of-the-art experiment demonstrated a enhancement-cavity-based 250 $\mathrm{MHz}$ HHG source reaching photon energies in excess of $100\,\mathrm{eV}$ and a photon flux of $9\times {10}^{7}\,\mathrm{photons}\,{{\rm{s}}}^{-1}$ in a 2% bandwidth around $94\,\mathrm{eV}$ [13], which indicates that intracavity HHG has come to a point where it is potentially useful for ultrafast photoelectron spectroscopy and microscopy experiments. For this, 30 $\mathrm{fs}$ pulses at $1040\,\mathrm{nm}$ with a pulse energy of $0.7\,\mu {\rm{J}}$ were power-enhanced a factor of 60 and focused down to ${w}_{0,x}\cdot {w}_{0,y}={(13.4\mu {\rm{m}})}^{2}$, reaching peak intensities around $3\times {10}^{14}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ in a 200 $\mu {\rm{m}}$ long neon gas target with an atomic density of $5{n}_{\mathrm{std}}$ placed 0.5 Rayleigh ranges before the focus, where n_std is the atomic density of an ideal gas at IUPAC standard temperature and pressure and ${w}_{0,x}$ and ${w}_{0,y}$ are the beam waists in x and y direction. The XUV radiation was coupled out through a 120 $\mu {\rm{m}}$-inner-diameter hole in the cavity mirror right after the focus, leaking and scattering approximately 1% of the intracavity power and coupling out 5% of the XUV radiation.

For waveform-stable IAPs, first of all it is a requirement that the enhanced comb has an offset frequency of zero. This can be accomplished by phase-stabilizing the seeding comb [22, 28, 29] and using tailored cavity mirrors [30]. Then, the XUV emission must be confined to only one attosecond burst per pulse. The bandwidth of currently available highly reflective (HR) mirrors does not allow to enhance pulses short enough to reach the single-cycle limit. However, recent work in our group [20] shows that a power enhancement of 75 is still possible with mirrors supporting pulses shorter than $20\,\mathrm{fs}$ at a central wavelength of $1050\,\mathrm{nm}$, corresponding to 5.4 cycles.

Considering the aforementioned state-of-the-art HHG experiment [13] and the new results regarding the mirror bandwidth, for our study we assume the ability to enhance 17.5 $\mathrm{fs}$ pulses at $1040\,\mathrm{nm}$ in an empty cavity with zero offset frequency and $\leqslant 0.8 \%$ round trip losses (corresponding to a power enhancement of 125 in the impedance-matched case). We presume that it is possible to generate phase-stable 17.5 fs pulses with a pulse energy of $0.7\,\mu {\rm{J}}$ as seed for the EC. High-repetition-rate pulses with 17 fs and similar pulse energy have already been reported in [22].

Apart from the seeding pulse parameters and the round-trip losses, there are several other technical restrictions for intracavity HHG: the gas flowing through the nozzle deteriorates the vacuum causing XUV reabsorption amongst other effects, so there is a technical limit on the gas flux, which is proportional to n · L², where n is the atomic density and L the target diameter. There is also a lower limit on the beam waist due to EC alignment sensitivity and astigmatism, and on beam diameter on the curved mirrors due to truncation. Finally, the peak intensity on the curved mirrors is limited by mirror damage. In our study, we restrict ourselves to parameters similar to the ones demonstrated in the reference experiment [13]:

• gas flux ${n}\cdot {{L}}^{2}\leqslant 5\,{{n}}_{\mathrm{std}}\cdot {(150\mu {\rm{m}})}^{2}$,
• beam waist ${w}_{0}\geqslant 14\,\mu {\rm{m}}$,
• beam diameter $w\lt 0.2D$ on curved mirrors, where $D=25.4\,\mathrm{mm}$,
• peak intensity on the curved mirrors $\leqslant {3.6\times 10}^{9}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$.

For an accurate theoretical description of HHG in gas targets, our model considers all relevant effects affecting the propagation of the driving field and the generation/propagation of the XUV field in the target: both fields experience linear refraction in transverse and in longitudinal direction, as well as absorption. Nonlinear effects on the driving field, most importantly plasma effects resulting in a spectral blue shift, defocusing and loss, as well as the Kerr effect causing focusing and self-phase modulation, are accounted for. For the XUV emission, the dipole response of an individual atom to the strong driving field is modeled, including polarization-dependent effects and depletion of the ground state.

A standard approach for HHG simulations is to employ a semiclassical model: the Maxwell equations are solved classically, whereas the dipole response is modeled quantum-mechanically. This approach is described in detail in [31]. Our model follows the standard approach, but in contrast to many computational models for HHG, our implementation is not limited to configurations with rotational symmetry. The model is also valid for polarized fields if vector quantities are used for the electric field and the polarization.

2.3.1. HHG model

For the description of XUV and driving field propagation through the gas target, we employ the forward wave equation [31], a first-order propagation equation obtained by applying the slowly evolving wave approximation [32] to the scalar wave equation in Fourier domain, using coordinates co-moving at vacuum speed of light (z is the propagation direction):

$$\begin{eqnarray}&&{\partial }_{z}{E}_{\mathrm{IR}/\mathrm{XUV}}=-\displaystyle \frac{{\rm{i}}{c}}{2\omega }{{\rm{\Delta }}}_{\perp }{E}_{\mathrm{IR}/\mathrm{XUV}}-\displaystyle \frac{{\rm{i}}\omega }{2c{\epsilon }_{0}}{P}_{\mathrm{IR}/\mathrm{XUV}}.\end{eqnarray} \tag{ 1 }$$

Here, E_IR is the driving field and E_XUV the generated high harmonic radiation, and ${P}_{\mathrm{IR}}={P}_{\mathrm{IR},\mathrm{lin}}+{P}_{\mathrm{Kerr}}+{P}_{\mathrm{plasma}}$ is the source term for the driving field and composed of the linear response ${P}_{\mathrm{IR},\mathrm{lin}}(t)={\epsilon }_{0}{\chi }_{c}(t){E}_{\mathrm{IR}}(t)$, the Kerr contribution [33]

$$\begin{eqnarray}&&{P}_{\mathrm{Kerr}}(t)={\epsilon }_{0}{\chi }^{(3)}(t)| {E}_{\mathrm{IR}}(t){| }^{2}{E}_{\mathrm{IR}}(t)\end{eqnarray} \tag{ 2 }$$

and the plasma contribution [34]

$$\begin{eqnarray}&&{\partial }_{t}{P}_{\mathrm{Plasma}}(t)={I}_{p}{n}_{0}\ \displaystyle \frac{{\partial }_{t}\eta (t)}{| {E}_{\mathrm{IR}}(t){| }^{2}}\ {E}_{\mathrm{IR}}(t)+\displaystyle \frac{{e}^{2}{n}_{0}}{{m}_{e}}{\int }_{-\infty }^{t}\eta (t^{\prime} ){E}_{\mathrm{IR}}(t^{\prime} ){\rm{d}}t^{\prime} .\end{eqnarray} \tag{ 3 }$$

The XUV source term ${P}_{\mathrm{XUV}}={P}_{\mathrm{XUV},\mathrm{lin}}+{P}_{\mathrm{dipole}}$ consists of the linear response ${P}_{\mathrm{XUV},\mathrm{lin}}(\omega )={\epsilon }_{0}\chi (\omega ){E}_{\mathrm{XUV}}(\omega )$ and the HHG term ${P}_{\mathrm{dipole}}(t)={n}_{0}d(t)$ computed with the strong-field approximation (SFA), accounting for elliptic polarization [35, 36] and ground state depletion [37]. Here, I_p is the ionization potential of the atom, e, m_e denote the electronic charge and mass, respectively, ${\chi }_{c}(t)=n(t){\alpha }_{c}$ the susceptibility at the driving field's carrier frequency, $\chi (\omega )={n}_{0}\alpha (\omega )$ the complex XUV susceptibility, ${\chi }^{(3)}(t)=n(t){\alpha }^{(3)}$ the third-order susceptibility. Furthermore, $n(t)={n}_{0}(1-\eta (t))$ denotes the time-dependent atomic density of neutrals with n₀ being the total atomic density, $\eta (t)=1-\exp \left[-{\int }_{-\infty }^{t}w({E}_{\mathrm{IR}}(t^{\prime} )){\rm{d}}t^{\prime} \right]$ the ionization fraction, ${\alpha }_{c}$, $\alpha (\omega )$ and ${\alpha }^{(3)}$ the first- and third-order polarizabilities, and w(E) the static ionization rate. The values for ${\alpha }_{c}$, $\alpha (\omega )$ and ${\alpha }^{(3)}$ are taken from [38–40], respectively. We use static ionization rates w(E) [41], obtained with the approach described in [42, 43]. We interpolate $\mathrm{log}(w(E))$ for lower intensities and obtain reasonable agreement with recently published rates [44] in the relevant intensity regime.

2.3.2. Computational implementation and optimizations

The intracavity field in an EC must be an eigenmode of the resonator geometry. Therefore, compared to single-pass HHG, cavity-enhanced HHG permits less freedom in choosing the driving field incident on the gas target, narrowing down the number of viable gating methods. Apart from that, efficiently driving HHG in a passive resonator comes with a few more particularities: first, low round-trip losses are necessary to maintain the main advantage of an EC, i.e., a high enhancement. This is not only hindered by absorbing elements in the cavity, but also by energy coupling to non-resonant eigenmodes through perturbation of temporal/spectral or spatial features of the circulating pulse, e.g., while passing the gas target. Second, to enable such low losses, HR multilayer dielectric mirrors are used as cavity mirrors which can provide well-behaved reflectivity and phase only over a limited bandwidth, imposing a lower limit on the duration of the circulating pulse. Furthermore, at high peak intensities, intracavity optics can manifest undesired nonlinear and thermal effects and, ultimately, damage. In the following, we shortly explain each considered gating method, examine their compatibility with the EC geometry and analyze each scheme with respect to round-trip losses, necessary pulse duration and power scalability.

Amplitude gating [49] relies on the fact that the driving field intensity determines the high harmonic cutoff, so by spectrally filtering the XUV, the emission can be confined to a short time window around the peak of the driving pulse's field, which allows for the production of IAPs. This scheme does not impose further conditions on the spectral or the temporal shape of the pulse incident on the target and is therefore compatible with the standard EC geometry. However, efficient amplitude gating has only been shown with sub-two-cycle pulses [49], while the bandwidth of HR mirrors currently limits the intracavity pulse duration to $\gt 18\,\mathrm{fs}$ at a wavelength of $1050\,\mathrm{nm}$ [20], rendering this scheme unviable for intracavity IAP generation.

For ionization gating, there are two approaches: one is to fully ionize the gas within the first few cycles and therefore inhibit XUV emission from subsequent cycles [50]. Another possibility is to use phase matching in a high-density gas target: in the first few cycles, sufficient plasma is generated so that the critical free-electron density is reached at which phase matching becomes impossible [51]. Like amplitude gating, these approaches would be compatible with the standard EC geometry. Commonly, cavity-enhanced HHG is performed in a tight-focusing regime with a high gas density to achieve good phase matching, and a low enough peak intensity to limit ionization-related clamping effects due to blue-shifting and plasma lensing [19]. On the contrary, ionization gating implies higher ionization levels than typically achieved in intracavity HHG—even the phase matching variant of ionization gating requires an ionization of 5%, using three-cycle pulses [51]. Although the use of input couplers with a tailored transmission curve and of mirrors correcting for the nonlinear phase have been suggested as a countermeasure against blueshift-induced clamping [19], this cannot reduce spatial effects due to plasma lensing, which are to be expected at such high ionization fractions. Moreover, the necessary pulse duration is out of reach with state-of-the-art mirrors. Therefore, efficient production of IAPs in ECs with ionization gating does not seem a viable route.

Polarization gating takes advantage of the fact that the HHG efficiency drops considerably with increasing ellipticity of the polarization [36]. By shaping the ellipticity of a pulse in a time-dependent manner, the harmonic emission can be confined to a time window with a duration on the order of a single half-cycle. As a standard EC with small incidence angles and geometric output coupling is basically insensitive to polarization, it is possible to apply such a scheme to the seed without modifications to the geometry of the EC. A straightforward way to shape the ellipticity is to produce two delayed, perpendicularly polarized copies of an initial pulse by passing a linearly polarized pulse through a multi-order quarter-wave plate with its optical axis rotated by $45^\circ$ with respect to the polarization direction (see figure 1). The polarization of the resulting pulse then changes from linear to circular to linear. Then, circular and linear polarization are swapped by a zero-order quarter-wave plate with its optical axis parallel to the original pulse's polarization direction [52]. Interferometric polarization gating [53] produces a similarly shaped pulse by introducing the delays interferometrically, and offers the additional degree of freedom to choose the relative amplitudes of both polarization directions in the resulting pulse, permitting production of IAPs from multi-cycle driving pulses, at the expense of at least 50% loss to the driving pulse energy. In [54] a scheme called collinear many-cycle polarization gating (CMC-PG) based on the waveplate scheme is introduced, which adds the same degree of freedom using reflection off a silicon plate as a polarizer and achieves similar performance as interferometric gating, while it is easier to align and more stable. This scheme has been shown to produce IAPs from pulses as long as $33\,\mathrm{fs}$ at a wavelength of $800\,\mathrm{nm}$, corresponding to 12.4 optical cycles [54]. Efficient intracavity HHG with 30 $\mathrm{fs}$-pulses was already shown [13], approaching the optimum photon flux for time-resolved photoelectron emission experiments, and mirrors supporting even shorter pulses have been demonstrated [20], so CMC-PG is a viable candidate. To avoid damage at high intensities, the silicon plate can also be replaced by a broadband thin-film polarizer.

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Several methods have been suggested using multi-color collinear superpositions: in [55], an XUV continuum around $100\,\mathrm{eV}$ is generated by mixing the 6.7-cycles-long driving pulse with its detuned second harmonic. (Generalized) double optical gating [56] is a combination of two-color gating with polarization gating. In both cases, the second harmonic allows to suppress harmonic emission from every other half-cycle, allowing for the production of IAPs with multi-cycle driving pulses. These schemes could in principle also work with a standard EC geometry—however, both colors would need to be enhanced with the same mirror set, which imposes a serious technological challenge. Another possibility is to produce the second component inside the cavity. However, the portion η of energy that can be converted to a second radiation component limits the power enhancement to maximally $1/\eta$, limiting the practicability of this approach. Non-collinear combination would also be an option, but an angle large enough to afford spatial separation would angularly disperse the harmonic radiation strongly when combining two different wavelengths [57], which is disadvantageous for time-domain applications.

The angular streaking method uses a driving pulse with a wave front rotating over the time scale of a single driving pulse. The XUV bursts take over the instantaneous wavefront orientation and are therefore emitted in different directions, allowing to separate IAPs by spatial filtering in the far field. One way of achieving such a wave front rotation (WFR) is to impose a spatial chirp in one transverse direction on the pulse [58–60]. Such a spatially chirped pulse cannot propagate as an eigenmode of a standard resonator, because each frequency component has a different optical axis. Consequently, intracavity elements (e.g. wedges or gratings) would be necessary to introduce a WFR in the resonator, which, similar to the case of multi-color gating, come with significant technological challenges.

Another possibility to achieve WFR is non-collinear optical gating (NOG). Here, the idea is to cross two equally strong delayed pulses [61, 62]. Then, the wave front orientation will change continuously from the direction of the first pulse to the one of the second pulse, leading to an attosecond lighthouse effect as in the case of the spatially chirped driving pulse. For generating such a driving field using cavities, there exist several possibilities. The most obvious is to use two separate cavities for both pulses and cross their foci. An alternative approach is to use a single EC with two circulating pulses and two crossed foci [26] (see figure 2). It is preferable to have three focused arms instead of two for alignment sensitivity reasons [16]. A third approach to realize a continuous WFR in an EC is to exploit the similarity of two crossed beams to some higher-order transverse eigenmodes. For instance, the Gauss–Hermite mode ${\mathrm{GH}}_{01}$ (figures 3(a) and (b)) consists of two well-separated lobes. If a π phase mask is applied to one of the lobes far before the focus [26], a single lobe emerges in the focal plane (figure 3(c)). Then, by also delaying the pulse envelopes of the lobes with respect to each other (figure 3(d)), it is possible to achieve WFR in the focus (figure 3(e)). Such a delay can be introduced by depositing material onto one half of a cavity mirror before applying the coating. The π phase mask can be achieved by choosing the step height as $(n+0.5)/2$ times the wavelength, with an integer n, or by using different coatings for the two halves [30]. However, the resulting field distribution is not a resonator eigenmode. Thus, it is required to place the step mirror inside the cavity and compensate for the mode alteration after passing the focus (see figure 3(f)), which is possible with very low losses using a second step mirror (see appendix). In the following, we refer to this scheme as transverse mode gating (TMG). Alternatively, as suggested in [63], one can cross the driving pulse with a weaker but shorter pulse, which introduces a slight wave front tilt for the duration of the short pulse, resulting in temporally confined emission of harmonics in off-axis direction. This is possible without adaptation of the EC geometry.

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Production of IAPs by NOG was only experimentally demonstrated with sub-two-cycle pulses [62]. However, [61] predicts that separation of an IAP is still possible with $10$ $\mathrm{fs}$ pulses at $800\,\mathrm{nm}$, which corresponds to 3.75 cycles, assuming a harmonic beamlet divergence angle of $0.1{{\rm{\Theta }}}_{0}$, with ${{\rm{\Theta }}}_{0}$ denoting the divergence angle of the driving beam. As we show later, it is possible to obtain a significantly smaller beamlet divergence by placing the target before the focus, so non-collinear generation of IAPs at pulse durations realistic in cavities may come into reach.

The maximum crossing angle for efficient NOG is only $\pi \cdot {{\rm{\Theta }}}_{0}$ [61]. This means that in the two-cavity and the three-focus approach, the curved mirrors next to the target have to be placed as close as possible to each other to avoid losses due to truncation of the mode. Reflection on a mirror cropped by a straight line in a distance of $\pi /2\cdot {{\rm{\Theta }}}_{0}$ from the center ideally leads to a round-trip loss of $0.64 \%$. The same holds for gating by an external pulse, but in that case one can choose to truncate the intracavity beam less and the external beam more to allow for a better power enhancement. TMG does not suffer from truncation losses, however care must be taken that the distortion introduced by the first delay mirror is compensated well by the second one without introducing too much loss. As it will be shown later, these losses can be kept small if the delay mirrors are placed appropriately.

Commonly, the resonator length and the seed repetition rate must be actively stabilized with respect to each other to maintain constructive interference of the carrier of the seed pulse with the carrier of the circulating pulse. Likewise, for NOG, sub-wavelength precision and stability of alignment are necessary for constructive interference of the two beams in the intersection point. From all presented NOG variants, TMG appears to be the most stable, because the delay between the beams is implemented monolithically. So, from the point of view of truncation losses and experimental effort, TMG can be considered the most promising variant for intracavity NOG.

In conclusion, we reviewed possible schemes for intracavity production of IAPs and identified two promising approaches (CMC-PG and TMG) and we found significant reasons to prefer them over the remaining schemes. In the remainder of the paper, we aim at performing a fair comparison of polarization gating with TMG. For this, we optimize both gating methods for optimum photon flux using the same target parameters and considering the same technical restrictions.

For a fair comparison of the two considered gating schemes for cavity-enhanced HHG, we need to optimize the parameters of both methods for optimum photon flux, demanding the same minimum intensity contrast ratio of the IAPs and taking into account the same technical restrictions, as discussed in section 2, i.e., limited seed pulse energy, cavity losses, gas flux, beam waist, beam diameter on the curved mirrors and damage intensity. We optimize for output coupling through a hole (for CMC-PG) and a slit (for TMG) in the mirror following the focus, because geometrical output coupling does not angularly disperse the harmonic radiation, works over a broad bandwidth and it is suitable for high photon energies.

For each scheme, we identify parameters affecting the contrast ratio and perform a broad scan on them, which is enabled by the approximations and optimizations of the implemented computational model. Based on these parameters, we optimize the phase matching conditions and the output coupling. Accounting for losses to the circulating pulse by nonlinear effects in the gas target, we obtain a complete set of optimum parameters, allowing us to simulate the optimum case with a minimum number of approximations.

To be consistent, we need to impose the same contrast ratio for both schemes. Having in mind time-resolved photoelectron emission spectroscopy (PES) and microscopy (PEEM) applications, in the broad parameter scan we optimize for IAPs with an intensity contrast ratio better than 10 after spectral filtering for harmonic orders between 74 and 84 (88.2–$100.1\,\mathrm{eV}$), which is about the minimum required photon energy for time-resolved PEEM experiments. In the following step of optimizing the phase matching, we only require a contrast ratio of 6.66 to make sure that promising results are not excluded due to numerical deviations, e.g., due to neglecting propagation effects in the broad parameter scan.

We restrict the study to HHG in neon, which was already shown experimentally in ECs and is a suitable choice for the targeted photon energy range due to its high ionization potential.

In the CMC-PG scheme, the contrast ratio is affected by the delay ${\rm{\Delta }}\tau$ of the multi-order quarter-wave plate, the CEP ${\varphi }_{\mathrm{CE}}$ of driving pulse and the incidence angle ${\varphi }_{{\rm{p}}}$ on the polarizing optic. Parameters affecting the photon flux are the target position z₀, the target length L, the atomic density n of the target gas, the driving pulse energy E_IR and the focusing w₀, which affect the phase matching, and the hole diameter d for the output coupling.

In a first step, we optimize the parameters affecting the contrast ratio. For this, we compute the single-atom dipole response, scanning the delay ${\rm{\Delta }}\tau$ from 0 to $25\,\mathrm{fs}$ in 1 $\mathrm{fs}$ steps, the carrier-envelope phase ${\varphi }_{\mathrm{CE}}$ from 0° to 180° in 5° steps and the incidence angle ${\varphi }_{{\rm{p}}}$ from  73.5° to 90° in 1.1° steps. This interval of incidence angles already covers all reflectance ratios and smaller angles would only result in higher losses to the p-polarized component (see figure 4). The long trajectory is suppressed by restricting the electron excursion time to values below 0.66 driving field periods, and the peak intensity is chosen to be $3\times {10}^{14}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ in each case. For each parameter set, we compute the conversion efficiency $| d(t){| }_{\max }^{2}/\int {I}_{\mathrm{IR}}(t)\,{\rm{d}}t$, where d(t) is the single-atom dipole response envelope after a bandpass from harmonic order 74–84 and I_IR is the driving field intensity. The contrast ratio, i.e. the ratio between the global and the secondary maximum of $| d(t){| }^{2}$, is also computed. Figure 5(a) shows that there is a tradeoff between the conversion efficiency and the contrast ratio. Choosing the parameter set with best efficiency and a contrast ratio $\geqslant 10$, we obtain an incidence angle ${\varphi }_{{\rm{p}}}=85.6^\circ$, a delay ${\rm{\Delta }}\tau =12\,\mathrm{fs}$ and a CEP of ${\varphi }_{\mathrm{CE}}=115^\circ$, resulting in a contrast ratio of 10.5 (see figure 5(b)). The energy loss at the polarizing optic is $60.1 \%$, i.e. $39.9 \%$ of the seed energy is available to the cavity.

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After having determined the parameters for the optimum contrast ratio of the single-atom dipole response, the next step is to optimize the parameters affecting the phase matching (z₀, L, E_IR, w₀, n). The scaling law introduced in [64] states that a parameter set z₀, L, E_IR, w₀, n is equivalent to a parameter set ${z}_{0}{\eta }^{2}$, $L{\eta }^{2}$, ${E}_{\mathrm{IR}}{\eta }^{2}$, ${w}_{0}\eta$, $n/{\eta }^{2}$, where η is an arbitrary number and the XUV photon flux also scales with ${\eta }^{2}$. This allows us to eliminate one parameter by using the following scale-invariant parameters: relative target position ${z}_{0}/{z}_{{\rm{R}}}$, driving field peak intensity I, relative gas density $n\cdot {z}_{{\rm{R}}}$ and relative target length $L/{z}_{{\rm{R}}}$, where ${z}_{{\rm{R}}}=\pi {w}_{0}^{2}/\lambda$ is the Rayleigh range. The scale-invariant quantities corresponding to the pulse energy E and the XUV photon flux ${{\rm{\Phi }}}_{q}$ are the relative pulse energy ${E}_{\mathrm{IR}}/{\eta }^{2}$ and the relative XUV photon flux ${{\rm{\Phi }}}_{q}/{\eta }^{2}$.

To enable a parameter scan on such a broad parameter range, which requires computing the single-atom dipole response on a spatial grid for each parameter set, we resort to several approximations and optimizations: first of all, we exploit rotational symmetry. The relative target length $L/{z}_{{\rm{R}}}$ is successively increased and the far-field on-axis harmonic peak intensity is calculated. This intensity will typically first increase with increasing target length and then decrease when the phase matching length is exceeded. The calculations stop when the intensity drops to 75% of the maximum or when $L/{z}_{{\rm{R}}}$ exceeds 1. We only save the harmonic spectra from H74 to H84 and discard the rest of the spectrum. We vary ${z}_{0}/{z}_{{\rm{R}}}$ from −1 to 1 in steps of 0.1, the peak intensity from $2.0\times {10}^{14}$ to $4.0\times {10}^{14}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ in $0.2\times {10}^{14}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ steps, and $n\cdot {z}_{{\rm{R}}}$ from $0.5\times {10}^{-3}$ to 0.5 × 10^–3 n_stdm, using $0.5\times {10}^{-3}\,{{\rm{n}}}_{\mathrm{std}}{\rm{m}}$ steps. The step size in the direction of the optical axis is chosen as ${z}_{{\rm{R}}}/400$.

For each set of scale-invariant parameters, we compute the driving field and the XUV radiation incident on the output coupling mirror. Then we also scan the relative output coupling hole diameter d/w from 0.1 to 0.6 in steps of 0.05, where w is the driving beam radius at the output coupling mirror. This determines the output coupling efficiency as well as the losses of the circulating pulse at the pierced mirror ($0.36 \%$ for $d/w=0.6$).

Using the driving field on the output coupling mirror and the losses at the pierced mirror, we compute the achievable pulse-energy enhancement (see appendix). Together with the relative circulating pulse energy, this yields the relative seed pulse energy necessary to drive the cavity to reach the required peak intensity. The relative output coupled photon flux is determined from relative photon flux and output coupling efficiency.

As we are interested in the parameters with optimum photon flux, we apply the scaling law to each parameter set, increasing the focus size to the maximum without violating any of the technical restrictions (gas flux, beam waist, beam diameter, peak intensity), and without requiring more than the available seed pulse energy. Of the assumed input pulse energy of $0.7\,\mu {\rm{J}}$, a fraction of $60.1 \%$ is lost at the polarizing optics of the gating scheme, leaving a pulse energy of $0.28\,\mu {\rm{J}}$ to seed the cavity. We only allow target lengths with a resulting contrast ratio better than 6.66. The obtained optimum parameters are a peak intensity of $2.2\times {10}^{14}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$, an atomic density of $3.86{n}_{0}$, a beam waist of $14.64\,\mu {\rm{m}}$, a target length of $137.5\,\mu {\rm{m}}$, target position at $-0.4{z}_{{\rm{R}}}$ and a hole diameter of $0.045w$, leading to an estimated pulse-energy enhancement of 97.2 (considering nonlinearities). The curved mirrors must be placed at a distance of at least $70.6\,\mathrm{mm}$ from the focus in order to not exceed the assumed damage threshold.

With the optimum parameters, we repeat the simulation without suppressing the long trajectory. The resulting time-domain XUV intensity on output coupling mirror is shown in figure 6. The contrast ratio of the output coupled, spectrally filtered XUV radiation is 7.82.

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Like with polarization gating, the achievable contrast ratio is affected by the delay ${\rm{\Delta }}\tau$ and CEP ${\varphi }_{\mathrm{CE}}$. Moreover, the relative target position ${z}_{0}/{z}_{{\rm{R}}}$ affects the XUV beamlet divergence angle [65] and thus the separation of harmonic bursts in the far field, so it is also an important parameter for the contrast ratio. As before, the parameters affecting the photon flux are L, E_IR, w₀ and n, as well as the width d of the slit in the output coupling mirror.

As a first step, we again determine the optimum parameters for the contrast ratio. Here, it is not sufficient to only compute single atom dipole responses, because transverse effects (beamlet divergence) must be accounted for. Therefore we compute the XUV far field in the limit on an infinitesimally thin gas target, scanning ${\rm{\Delta }}\tau$ from 0 to $25\,\mathrm{fs}$ in 1 $\mathrm{fs}$ steps and the relative target position from −1.3 to 1.3 in steps of 0.1, again at a fixed peak intensity of $3\times {10}^{14}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$. To reduce computation time, we use the short-trajectory envelope approximation for the dipole response around H79, and only consider one transverse direction: the one along which the harmonic bursts are angularly separated. For each parameter set, we compute the time-dependent on-axis XUV far field. Because the envelope approximation only yields spectral components in a narrow bandwidth around the targeted H79, this time-dependent field does not exhibit the individual bursts of an attosecond pulse train, but rather represents an envelope over the pulse train. From this envelope E(t), we can compute the contrast ratio of the attosecond pulse train, i.e. the ratio between global maximum and secondary maximum, which is expected at least one half-cycle before or after the global maximum, as

$$\begin{eqnarray}&&c=\displaystyle \frac{| E({t}_{0}){| }^{2}}{{\max }_{| t-{t}_{0}| \gt T/2}| E(t){| }^{2}},\end{eqnarray} \tag{ 4 }$$

where t₀ is chosen so that $| E({t}_{0}){| }^{2}$ is maximal and $T/2$ the duration of a half-cycle. We also compute a conversion efficiency measure by relating the peak intensity of the XUV field with the necessary IR pulse energy:

$$\begin{eqnarray}&&\epsilon =\displaystyle \frac{| E({t}_{0}){| }^{2}}{\int \int | {E}_{{\rm{IR}}}(x,t){| }^{2}{\rm{d}}x\,{\rm{d}}t}.\end{eqnarray} \tag{ 5 }$$

Figure 7(a) shows the tradeoff between these two quantities. Choosing the parameter set with best efficiency and contrast ratio $\geqslant 10$ as before, we get a target position of $-0.9{z}_{{\rm{R}}}$ and a delay of $7\,\mathrm{fs}$, resulting in a contrast ratio of 19.3 (figure 7(b)).

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For the parameters of optimum contrast ratio, we optimize the phase matching parameters L, E_IR, w₀ and n by scanning the scale-invariant parameters $L/{z}_{{\rm{R}}}$, I and $n\cdot {z}_{{\rm{R}}}$. We compute in 3 + 1D, exploiting reflectional symmetry in one spatial direction, and use envelope approximations for the driving field source terms as well as the short-trajectory envelope approximation for the XUV. As in the case of polarization gating, we only consider relative target lengths $L/{z}_{{\rm{R}}}\leqslant 1$ and only compute until the photon flux (peak of the time-dependent on-axis far field) decreases to 75%. We vary the peak intensity and the relative gas density in the same parameter range.

For each set of scale-invariant parameters, we compute the driving field and the generated XUV radiation in the output coupling mirror plane. Then we scan the relative slit width d/w from 0.01 to 0.10 in steps of 0.01, where w is the radius of the driving beam on the mirror. For $d/w=0.10$, the round-trip loss is still below $0.07 \%$. Figure 3(c) shows that the on-axis minimum of the GH₀₁ mode in the far field is conserved if a phase shift is introduced to one lobe far from the focus, i.e. the circulating mode has an intensity minimum at the position of the slit, which explains the low losses. Thus, for these slit widths the loss is still not a limiting factor. However, the slit also serves to spatially separate one of the angularly dispersed harmonic beamlets. Increasing the slit size further would make it impossible to reach a good contrast ratio, as the results of the parameter scan will show later.

As for polarization gating, for each resulting parameter set we compute the necessary relative seed pulse energy and relative output-coupled photon flux, accounting for cavity round-trip losses, phase shift and output coupling efficiency. After that, we apply the scaling law to maximize the photon flux in each case without violating the same technical restrictions as in polarization gating, requiring a contrast ratio better than 6.66 and permitting $82.7 \%$ of $0.7\,\mu {\rm{J}}$ as seed pulse energy, where $82.7 \%$ is the maximum achievable overlap of a GH₀₀ seed with the GH₀₁ resonator eigenmode using a phase mask [66]. The resulting optimum parameters are a peak intensity of $2.6\times {10}^{14}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$, an atomic density of $4.6{n}_{0}$, a beam waist of $14.68\,\mu {\rm{m}}$, a target length of $156.3\,\mu {\rm{m}}$ and a slit width of $0.05w$, leading to an estimated pulse-energy enhancement of $108.3\times 82.7 \% =89.6$. The distance between focus and curved mirror must be larger than $220.9\,\mathrm{mm}$ to avoid damage.

For the optimum parameters, we repeat the simulation without envelope approximations and considering both trajectories. The resulting time-domain XUV intensity evolution on the output coupling mirror is shown in figure 8, where a clear lighthouse effect can be seen which allows the separation of an IAP by spatial filtering at the slit. We observe that the contrast ratio can be improved if not only the slit width but also the slit length is limited; when choosing a $0.05w\times 0.1w$ slit the contrast ratio of the output coupled, spectrally filtered XUV radiation is 7.0. This can most likely be attributed to the larger divergence angle of the long-trajectory contribution to the harmonic far field. It can be seen that the harmonic beamlet divergence is on the order of $0.025{{\rm{\Theta }}}_{0}$—this explains the good angular separation of harmonic bursts even with 17.5 $\mathrm{fs}$ pulses.

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Comparing figure 5(a) with figure 7(a), it can be seen that the trade-off between contrast ratio and conversion efficiency is much more critical in the CMC-PG scheme than in the TMG scheme. Also, the resulting XUV peak intensity is 135.7 times higher for TMG after optimizing under the previously specified contraints.

There are several reasons for this: in the CMC-PG scheme, only $39.9 \%$ of the seed energy is available, because one polarization direction must be suppressed, whereas in TMG up to $82.7 \%$ of the seed energy can be coupled into the cavity assuming optimum mode matching. Further, the resonator mode excited in TMG has an on-axis minimum in the middle of the output coupling mirror, leading to lower round-trip losses of the circulating pulse. Then, to reach the required contrast ratio, the delay for polarization gating must be chosen $\gt 12\,\mathrm{fs}$ while it is only $7\,\mathrm{fs}$ for TMG. Due to the larger delay, more pulse energy or a tighter focusing is needed to reach the same peak intensity in the focus, which results in a lower efficiency. Finally, the output coupling efficiency (fraction of the XUV power transmitted through the orifice to the whole XUV power incident on output coupling mirror, computed at the peak of the IAP) is $1/67.73=1.5 \%$ for CMC-PG, but $37.5 \%$ for TMG—for optimum beamlet divergence and thus optimum output coupling efficiency, the target has to be placed approximately one Rayleigh length before the focus, which is not possible in the case of CMC-PG because the necessary peak intensity can not be reached there due to the longer delay.

For these reasons, TMG can be regarded as the preferred method for implementation.

To estimate the photon flux that can be obtained with the TMG scheme, assuming the determined optimum parameters, we also simulate HHG with the parameters of the reference experiment [13], using the same approach applied to obtain figures 6 and 8, and compare. The number of XUV photons produced per pulse (in the spectral range from H74 to H84) can be obtained by integrating the harmonic power over time and is 1.8 times higher for the simualted TMG case than in the simulated reference experiment. From this we can conclude that a similar photon flux as in the reference experiment ($9\times {10}^{7}\,\mathrm{photons}\,{{\rm{s}}}^{-1}$) can be obtained with TMG. However, it is important to note that the photon flux in the reference experiment was strongly limited by cumulative effects in the gas target due to the high repetition rate of $250\,\mathrm{MHz}$ [13] and that these cumulative effects were not included in the model. Therefore, the predicted photon flux should be regarded as a rather strong underestimation of the flux attainable with implementing this scheme at a repetition rate at which each gas atom is hit by a single pulse only. With state-of-the-art lasers operating at the highest repetition rates in this regime [29] and a pulse-energy-scalable compression scheme (e.g. [67]), the seed pulse energy can be increased accordingly. The scaling law of [64] allows to change the geometry of the EC setup such that the same total photon flux can be obtained at a significantly lower repetition rate, without violating the constraints of beam waist and peak intensity as given in section 2. The beam diameter on the curved mirrors and therefore the size of the substrates may have to be increased, however. The maximum allowed gas flux is not exceeded because the flux is scale-invariant if the nozzle size is only scaled in transverse direction. Therefore, scaling to lower repetition rates is possible without reducing the XUV photon flux.

Decreasing the repetition rate to a value where cumulative effects do not play a role anymore promises considerably better photon flux than demonstrated in [13]. For instance, gas flow simulations predict that at $\sim 10\,\mathrm{MHz}$ and for typical beam waists, each atom is only hit by a single pulse.