XUV frequency combs via femtosecond enhancement cavities

It is important to develop an intuition about how an optical resonator responds to pulsed excitation of a mode-locked laser, and when maximum intracavity buildup occurs. In the time domain, one can imagine that the envelope of the pulse circulating in the fsEC must overlap with that of each subsequent pulse input from the oscillator. A second requirement is that the optical carrier under the envelope must also constructively interfere. In other words, the round-trip carrier-envelope phase shift introduced by the cavity must match the pulse-to-pulse carrier-envelope phase shift from the oscillator. In the frequency domain, these two requirements correspond to matching the pulse repetition rate f_rep of the mode-locked oscillator [3] to the fsEC and setting equal the relative carrier-envelope-offset frequencies of the fsEC and oscillator, that is, Δf_ceo = 0. Practically speaking, active stabilization of f_rep and possibly Δf_ceo is required, as is described in section 3.3. The theory of optical resonators is well known, and what follows in this section is a brief summary of results that we find particularly relevant for fsEC work. Similar material has also been presented in recent reviews of cavity-enhanced direct frequency comb spectroscopy [22, 23].

Physical insight can be gained by examining the resonance condition of an fsEC excited by the mode-locked laser optical spectrum, or frequency comb. To approach this problem we begin with the general resonance condition of an optical resonator,

$$\begin{equation} 2\pi m = \frac{\omega }{c}L+\Phi (\omega ), \end{equation} \tag{ 1 }$$

which requires that the phase accumulated by an optical wave of frequency ω in one round trip of a resonator of length L is equal to 2πm, where m is an integer, and c is the speed of light in vacuum. In addition to the phase shift due to propagation in free space, an additional frequency-dependent phase Φ(ω), or dispersion, is added by elements within the cavity. Dispersion leads to a nonuniform spacing (as a function of frequency) of the resonances, which cannot be exactly aligned with the equal spacing of the modes produced by the mode-locked oscillator, as depicted in figure 2. Excessive dispersion leads to a reduction in bandwidth and longer pulses, and the amount of filtering depends on the finesse of the cavity and the spectral bandwidth of the mode-locked oscillator. In a bare fsEC in vacuum, Φ(ω) is solely due to the dielectric mirrors that make up the resonator, but when a gas target is present, additional terms arise to account for the dispersion of the gas and the plasma generated.

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Following the standard analysis of a Fabry–Perot resonator, the steady state intensity circulating in the cavity is given by,

$$\begin{equation} I_c(\omega ,d)=\frac{t^2_iI_0(\omega )}{(1-r_ir_{\rm cav})^2+4r_ir_{\rm cav}\sin ^2(\phi /2)}, \end{equation} \tag{ 2 }$$

where I₀(ω) is the incident intensity, t_i and r_i are the magnitude of the field transmission and reflection coefficients of the input coupler, respectively, and r_cav is the magnitude of the field reflection coefficient for the remaining cavity mirrors, combined. The field and intensity reflection coefficients are related by, R_i = r²_i, and likewise, T_i = t²_i. Generally, very low loss mirrors are available such that R + T≅1, for visible and near-infrared optical frequencies. The resonance condition information is included in the phase, $\phi =\frac{\omega }{c}d+\Phi (\omega )$. Here we have written only the incident electric field and the cavity phase explicitly as functions of frequency, but strictly speaking the mirror reflectivity and transmission are also functions of optical frequency.

Equation (2) can be used to derive an expression for the cavity finesse, $\mathcal {F}$, which is defined as the ratio of the cavity free spectral range (FSR) to the cavity full width at half maximum (FWHM). For sufficiently large values ($\mathcal {F}\gg 1$), the cavity finesse is given by,

$$\begin{equation} \mathcal {F}=\frac{\pi \sqrt{r_ir_{\rm cav}}}{1-r_ir_{\rm cav}}. \end{equation} \tag{ 3 }$$

The cavity enhancement B(ω, d) = I_c(ω, d)/I₀(ω) of the intensity can also be expressed in terms of the finesse and the resonance condition phase,

$$\begin{equation} B(\omega ,d) = \left(\frac{t_{i}}{1-r_{i}r_{\rm cav}}\right)^{2}\frac{1}{1+\big(\frac{2\mathcal {F}}{\pi }\big)^2\sin ^2(\phi /2)}. \end{equation} \tag{ 4 }$$

The maximum enhancement is achieved when the input coupler intensity transmission is equal to all other losses in the cavity, and in this case the resonator is said to be impedance matched. In this limit, the enhancement factor is approximately $\mathcal {F}/\pi$. When there is a lossy element in the cavity, such as in intracavity second harmonic generation, impedance matching can be particularly important. From (4), one can see that fluctuations in the intracavity phase, ϕ, are magnified by a factor of $\mathcal {F}^2$. As a result, for intracavity HHG in the fsEC, it is often desirable to limit or spoil the finesse by increasing the input coupler transmission. When the finesse is reduced in this way, and the net cavity loss is dominated by the input coupler, the cavity enhancement is $B\approx 2\mathcal {F}/\pi$. In summary, in these two cases the cavity enhancement is,

$$\begin{eqnarray} &&B_{\rm max}(\omega ,d) \approx \frac{\mathcal {F}}{\pi } \quad ({\rm impedance\ matched}) \end{eqnarray} \tag{ 5a }$$

$$\begin{eqnarray} &&\hphantom{B_{\rm max}(\omega ,d) }\approx \frac{2\mathcal {F}}{\pi } \quad ({\rm input\ coupler\ limited}). \end{eqnarray} \tag{ 5b }$$

The expressions presented in this section are derived for one single-frequency mode resonating in the cavity and typically the frequency dependence is not explicitly written. As long as nonlinearities do not arise anywhere in the cavity we can simply apply this theory to the case of excitation with the frequency comb generated by a femtosecond mode-locked oscillator by keeping track of the resonance condition and enhancement for each comb element. We expand this notion in the next section, and we discuss the nonlinear effects that arise when a plasma is generated in the cavity in section 4.2.

We have found it useful to visualize the behaviour of the fsEC using a resonance map, which is a plot of the cavity enhancement B(ω, d). For a CW laser source, a single row (column) of this map is the well-known Fabry–Perot resonance structure that is observed when the cavity length (laser frequency) is scanned. The situation is slightly more complicated when the cavity is simultaneously excited with ∼10⁵ modes typical in a mode-locked oscillator; in this case it is easiest to compute the cavity enhancement numerically and plot the enhancement on a heat map type plot.

The seed pulse train is taken to be a frequency comb with modes separated by the repetition rate of the oscillator, with a spectral envelope function to simulate the bandwidth of the oscillator. We then compute the cavity intensity on a 2D grid representing the resonance map. We show an example resonance map in the top panel of figure 3 for the case of zero cavity dispersion, Φ(ω) = 0, the relative carrier-envelope frequency difference between the oscillator and cavity is zero, a spectral bandwidth of 30 nm centred at 800 nm and relatively low finesse, $\mathcal {F}=50$, to introduce its utility. Resolving the individual comb elements, which are separated by several tens to hundreds of MHz, is not necessary for the resonance map. On the other hand, we do require high resolution for the cavity length axis. We plot three cavity resonances in figure 3, spanning a cavity length change of about 2 μm. We see that there is only one cavity length, d = d₀, where all modes of the input spectrum are resonant simultaneously in the cavity. This central fringe has the largest enhancement, B, and occurs at the cavity length where the round-trip time equals the oscillator pulse period. The adjacent fringes (d − d₀ = ±λ) are tilted because the resonance condition is not satisfied for all frequency comb modes at the same cavity length simultaneously. The bottom panel of figure 3 represents the cavity transmission as a function of the cavity length, where the transmitted intensity, at a given cavity length, is proportional to the sum of the intensity of all frequency modes stored in the fsEC. The amplitude of the modes adjacent to the central fringe are considerably reduced due to the tilt of the corresponding fringes in the resonance map.

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Figure 4 illustrates how the cavity transmission is modified when the offset frequency, f_ceo, is varied. In general the cavity contributes its own carrier-envelope phase and the relevant parameter for the fsEC is the relative difference, Δf_ceo, between the oscillator f_ceo and the carrier-envelope-offset of the cavity. Figure 4 shows several cavity transmission fringes for a non-zero relative offset frequency, Δf_ceo, of the incident field for a cavity finesse, $\mathcal {F}=500$, and Δf_ceo = f_rep/8, and still no cavity dispersion, Φ(ω) = 0. We see the cavity resonances are shifted relative to the resonances for Δf_ceo = 0, and the peak of the central fringe does not occur at d = d₀. In fact, as Δf_ceo is varied from zero, the peak of the fringes map out an envelope function, shown as a dotted red line that reaches a maximum at d = d₀. The peak of the envelope function corresponds to Δf_ceo = 0. This envelope function has been described previously [24, 25]. The narrow width of the envelope (dotted line in figure 4) illustrates the limited range over which the cavity length and offset frequency can vary while maintaining maximum transmission (and therefore maximum cavity enhancement, B). Increasing the optical bandwidth or the cavity finesse further narrows this range.

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As the offset frequency is changed, the resonances as seen in the resonance map undergo a continuous evolution as well. For example, as Δf_ceo is increased from its value of zero in figure 3, the fringes shift from right to left and also change their tilt angle. This is illustrated in a resonance map in figure 5, for two sets of fringes: (1) for Δf_ceo = 0, which has a central fringe centred at d − d₀ = 0 and (2) for Δf_ceo = f_rep/3. When Δf_ceo is further increased, the tilted mode that started near d − d₀ = 800 nm ends at d = d₀ perfectly vertically aligned when Δf_ceo = f_rep.

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There is another feature of the resonance map that is interesting to note. If one examines the behaviour of cavity fringes much farther away from the central fringe (|d − d₀| ≫ λ), it is seen that the tilt of the fringes becomes greater as |d − d₀| increases. Eventually, multiple fringes will have some frequency range resonant in the fsEC for the same cavity length. For large |d − d₀|, ranging from 100 μm to a few mm, the cavity transmission consists of a series of distinct transmission peaks, each corresponding to a cavity fringe coming into resonance in a different portion of the optical spectrum. These fringes become increasingly more dense and lower in intensity as the cavity length moves farther from the location of the central fringe. This is a very useful feature when performing the rough length-matching of the fsEC during the initial setup. Of course, this behaviour has been observed previously in the context of cavity-enhanced absorption spectroscopy [26], but we point it out here to illustrate the power of the fsEC resonance map as a means of unifying the variety of behaviour observed in the fsEC.

The resonance map is also a powerful tool to visualize the effect of dispersion, which is simply a variation in the resonance condition as a function of frequency, represented as Φ(ω) in (1).

For high-finesse cavities ($\mathcal {F}>1000$) dispersion of the cavity mirrors will limit the enhancement if the residual group delay dispersion (GDD) is more than a few fs². Such high-finesse cavities prove to be difficult to work with for a number of reasons that will be discussed further in section 4. The dispersion requirements for cavities with a finesse of a few hundred are significantly relaxed, but even modest second and third order dispersion can significantly restrict the spectrum supported by the fsEC. In figure 6, we plot the resonance map for two functional forms of the cavity phase: (1) Φ(ω) is a quadratic spectral phase with a magnitude of 20 fs² (left), and (2) Φ(ω) is a cubic spectral phase with a magnitude of 200 fs³. The inset on the left panel shows the spectrum that is stored in the cavity when the cavity length is stabilized to d − d₀ = −0.5 nm. This spectrum is a vertical slice of the resonance map, indicated with the dotted white line.

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We show a set of experimental spectra for a pulse stored in the fsEC and the Ti:sapphire seed oscillator in figure 7. This figure shows that the majority of the optical spectrum can be enhanced in a high-finesse cavity ($\mathcal {F}>2000$) with commercially available low GDD mirrors. Higher power seed sources enable lower finesse cavities to be used, and therefore a further increase in the spectral bandwidth stored in the fsEC is possible.

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Several similar methods of measuring the dispersion of the fsEC have been developed, each of which can be useful depending on technical details of the individual frequency comb source and fsEC combination. Each of the schemes involves interrogating the spectral dependence of the resonance condition of (1) in one way or another. The least technically demanding, if somewhat time consuming, method [27] involves scanning across the cavity resonances and measuring the differential change in the resonance cavity length, Δd, as a function of optical frequency relative to a reference optical frequency. The repeatability and accuracy of this measurement depends on the stability of the oscillator's f_ceo, which may need to be stabilized for the measurement of small cavity dispersion and high-finesse cavities. Systems with a reasonably stable offset frequency (typically fibre lasers) do not need to be stabilized, but in either situation the numerical value of f_ceo is not needed for the dispersion measurement.

In another cavity dispersion measurement technique [28], the enhancement cavity is locked to the oscillator at a fixed, narrow band of optical frequencies and then f_ceo of the oscillator is varied. For each value of f_ceo, the optical spectrum stored in the enhancement cavity, which is a sensitive function of the cavity phase, Φ(ω), is recorded. This technique requires that the step size of the f_ceo scan be known, but its absolute value is not necessary. This method is very convenient when an acousto-optic modulator already exists in the system for other purposes, such as carrier-envelope-offset frequency stabilization described in section 3.3.

Finally, in a third technique [29] the average frequency of a narrow band of optical frequencies of the laser is locked to the cavity and then the repetition rate of the laser is scanned. For each value of f_rep the optical spectrum stored in the cavity is recorded, and as in the previously described technique, the resulting 2D map depends on the cavity phase, Φ(ω).

Each of the previous three techniques were developed to measure the dispersion characteristics of a bare cavity, and therefore requires linear behaviour of the cavity to obtain this information. It may also be desirable to study the dispersion properties of the cavity at higher intensities where nonlinearities are important, or for a nonlinear medium present in the cavity. For this purpose spatial spectral interferometry (SSI) has been applied to fsECs [30]. This technique utilizes the interference between a reference beam and a second beam passed through a dispersive 'device under test' (i.e. the fsEC) to determine the frequency-dependent phase introduced by the fsEC. A small angle between the two beams is introduced in one dimension (e.g. vertical), while a grating is used to disperse the beams spectrally in the other direction (horizontal). The fringes that appear contain the spectral phase information desired, for example, parabolic shaped fringes appear for a purely parabolic spectral phase (GDD) added by the device under test. This measurement can be completed while the cavity is loaded, meaning that it is locked to the seed oscillator with high intracavity intensity, and is thus capable of measuring the dispersion resulting from nonlinear phenomena in the fsEC.

In summary, dispersion in the fsEC is a factor that must be considered carefully in optimizing a fsEC-based XUV source, and one must select low dispersion mirrors for the cavity. Beyond the requirement of low GDD mirrors, the use of relatively lowfinesse cavities ($\mathcal {F}<500$) significantly reduces the necessity to manage dispersion. More care must be taken to control the dispersion in the cavity if one wishes to increase the bandwidth supported by the cavity, use a higher finesse, or use a dispersive intracavity element (such as a Brewster plate for an XUV output coupler).

HHG is a well-known nonlinear process that has experienced rapid, sustained development since the earliest experiments over two decades ago [1, 31]. HHG is usually explained classically in terms of the three step model in which the Coulomb potential that binds the electron to the atomic nucleus is perturbed by an intense laser field. If the external field is great enough, the electron can tunnel through the potential barrier whereupon it is accelerated away from the nucleus by the external laser field. After a short time, the oscillating electric field reverses direction and accelerates the electron towards the nucleus. If the nucleus and the electron recombine, a high energy photon is released with an energy equal to the sum of the ionization potential of the atom and the ponderomotive energy the electron acquires due to accelerating in the laser field. In general, a series of odd harmonics of the fundamental laser frequency is produced.

Theoretical classical models and quantum mechanical calculations of the HHG process have been developed that very accurately describe the effects seen in experiments. These calculations determine the intensity spectrum of the generated harmonics by single atoms and the phase of the radiators, which must be incorporated to predict the harmonic emission from a macroscopic ensemble of atoms. An integral part of the picture is the continuum dynamics of the electron after it is released from the atom. Classical models reveal that above a certain threshold intensity two pathways are possible, a long and a short trajectory, corresponding to the extent to which the electron travels in the continuum. Quantum mechanically, an electron can take many pathways through the continuum. For high intensities such that the Coulomb potential is negligible compared to the electron energy gained by the acceleration of the laser field (the ponderomotive energy), the quantum pathways follow closely the classical trajectories in generating the high harmonics. Interestingly, harmonics with photon energy near the ionization potential also show multiple pathways when the Coulomb potential is included in the model [32]. The case of HHG in a fsEC currently falls into this latter category, and optimizing harmonic generation in a fsEC inevitably relies on developing an understanding of the HHG process for the conditions present near the tight focus of the fsEC.

In nearly all optical nonlinear processes, including HHG, the phase evolution between the fundamental and generated field is a key parameter. Addressing the phase mismatch between the fundamental and HHG fields is necessary to optimize the XUV flux. Shortly after the discovery of HHG, it was realized that the focus geometry played a large role in the efficiency of the harmonic conversion process [33, 34]. In the tight-focus regime, it was found that the harmonic flux scales as b³ for a fixed intensity [35], where b is the confocal parameter of the focused Gaussian beam. Improved technology enabled further investigation into the loose focus regime and HHG within gas-filled hollow fibre waveguides [36–38]. Through better understanding of the waveguide and plasma dispersion it became possible to incorporate these parameters into a phase matching procedure, and thereby improving the yield and beam shape of certain harmonics. The use of waveguides was further revolutionized through quasi-phase matching by modulating the diameter of the waveguide [39]. This improvement enables phase matching up to several hundred eV [2]. Furthermore, counterpropagating pulses have been used in the loose focus geometry [40, 41] to turn off the generation of harmonics during phase-mismatched events. This idea was expanded to gas-filled waveguides [42, 43], where the effect of counterpropagating pulse trains were used to further increase the XUV flux through improved phase matching.

There is likewise a push to maximize the harmonic flux generated in fsECs. In this section we present an overview and optimization of the HHG process via a numerical model, while keeping in mind the constraints imposed by the fsEC. In order to reach the necessary peak intensity, fsEC HHG systems are required to have a tight focus. In this regime the Gouy phase experienced by the driving fundamental field severely limits phase matching. Other factors that are requisite to fsECs include the fact that the fundamental field is very sensitive to strong nonlinear phase shifts (such as due to plasma creation) and loss (as in the case of hollow-fibre coupling). These factors highlight the need to optimize the gas nozzle geometry in fsEC systems, so as to maximize the harmonic generation from the minimum required amount of gas/plasma.

In fsECs the optimal gas nozzle geometry concentrates the gas target to a much smaller interaction region, and at much higher pressures than used in some previous studies. Nozzles used in these systems produced relatively long interaction lengths and to keep plasma phase shifts small, low gas pressure was used. We address effects of different gas nozzle designs in section 3.4. In this low-pressure, long-interaction-length regime the phase matching can vary significantly for different harmonic orders, particularly for near-threshold harmonics that are generated with fsECs. On the other hand the optimal harmonic flux requires high pressure and tight confinement of the gas to a region very close to the peak of the intracavity intensity. To study both of these regimes, we applied previous theoretical work to develop an on-axis model of the harmonic generation process for the harmonic orders produced in fsEC XUV sources [44]. In what follows we briefly review some of the key points of the theory here, but full details of the numerical model are provided elsewhere [44, 45].

To develop a numerical model we first consider the amplitude of the harmonics generated on-axis. The field amplitude of the qth harmonic E_q can be found by solving,

$$\begin{eqnarray} &&\fl \frac{\partial }{\partial z} E_q(z) =-\frac{\rho (z) \sigma (\omega _{q})}{2} E_q(z)\nonumber\\ &&+\, \frac{{\rm i} \mu _0 q^2 \omega _1^2}{2 k_q} \rho (z) |\tilde{d}(\omega _q,z)|\, {\rm e}^{{\rm i} \Delta \phi (z)}, \end{eqnarray} \tag{ 6 }$$

where the gas density is ρ(z) and σ(ω_q) is the absorption cross-section of the qth harmonic. The dipole response, $\tilde{d}(\omega _q,z)$, is calculated by solving the time-dependent Schrödinger equation. In [44] the dipole response is not deconstructed into magnitude and phase contributions from separate quantum trajectories directly from the calculated dipole moment, as discussed elsewhere [46, 47]. Instead, long and short trajectory phases are added phenomenologically as described below. This approach has its limitations; for example, it would not be able to compute the relative contributions to the harmonic intensity from long and short trajectories, nor would it predict the interference that arises between them. However, the model does capture much of the physics we are interested in, primarily the variation of the generated harmonics with the geometry and density of the target gas for a given intensity within the fsEC.

The phase mismatch is Δϕ(z) = qϕ₁ − ϕ_q with,

$$\begin{eqnarray} &&\fl q \phi _1(z) = q k_1 n(z,\omega _1) z-q\tan ^{-1}\left(\frac{z}{n(z,\omega _1)z_R}\right)+ \Phi (z) \end{eqnarray} \tag{ 7a }$$

$$\begin{eqnarray} &&\fl \phi _q(z) = k_q n(z,\omega _q) z - \tan ^{-1}\left(\frac{z}{n(z,\omega _q) z_R} \right), \end{eqnarray} \tag{ 7b }$$

where z_R = b/2 is the Raleigh range of the fundamental beam (z_R ∼ 0.5 mm in tight-focus fsEC systems). The phase of the atomic response is given by Φ(z) = −α_iI(z), where α_i parameterizes the different trajectories. In [44] the short and long trajectory phase coefficients values of α_i = 1 × 10⁻¹⁴ and 25 × 10⁻¹⁴ cm² W⁻¹ were used, respectively, consistent with the classical phases of the electron for 800 nm excitation. The model allows us to calculate the effect of different gas jet geometries on the harmonic generation process through the refractive index n(z, ω_q), which is related to the gas density and plasma by,

$$\begin{equation} n(z,\omega _{q}) = 1 + \frac{\rho (z)}{\rho _0} \left[\left(1-\eta \right)\delta _{q} - \frac{\omega _p^2}{2 \omega _{q}^2} \eta \right], \end{equation} \tag{ 8 }$$

where ρ₀ is the gas density at 1 ATM, η is the ionization fraction, δ_q is the xenon index correction from vacuum and ω_p is the plasma frequency. The ionization fraction includes a dynamic ionization component that changes during each pulse and a steady state component that arises due to the accumulation of plasma resulting from multiple pulses. The ionization fraction is estimated by combining calculations for the ionization rate [48] with estimates of the transit time of atoms through the interaction region based on gas velocity and plasma expansion dynamics. Different gas jets will lead to very different ionization fraction, which affects the harmonic yield and beam shape of the generated harmonics. We discuss these effects in section 3.4.

The results of these simulations are consistent with previous work on above-threshold harmonics in the tight-focus regime with respect to fundamental intensity and gas density [49–51]. A notable difference exists in the higher intensity regime used in these studies, in that the high pulse energy leads to a significantly higher ionization fraction at the focus. This manifests itself in a quantitative difference in the harmonic yield for different gas nozzle positions relative to the focus. Thus, in the tight-focus regime within a fsEC the optimal position for the HHG process occurs much nearer to the focus with a very high gas density. Interestingly, the tight-focus regime is also relevant in recent development of high repetition rate single-pass HHG systems operating near 100 kHz with several μJ pulses. A very recent work [52] investigates the role of phase matching in this regime, and considers how the phase matching is modified when the focal length of a lens used to achieve the tight focus is used as a scaling parameter.

The role of multiple quantum pathways is particularly important for the development of XUV frequency combs, where the coherence of the generated light is critical, and the coherence of long trajectory harmonic generation is known to be reduced compared to that of the short trajectories. The role of coherence has already been discussed in the context of XUV direct frequency comb spectroscopy [53] and will also be important in XUV dual comb [54] spectroscopy applications. It is therefore appealing to further develop numerical modelling to study the macroscopic aspects of HHG. The on-axis simulations of harmonic yield that we have performed show that phase matching affects long and short trajectory harmonic generation differently at low gas pressure (several tens of torr) and longer interaction lengths (300–500 μm) at the driving field focus. Moreover, the phase matching is harmonic dependent and the long and short trajectories are favourably phase-matched under different experimental conditions [44]. Extension of this on-axis theory to compute the far-field beam profile will help to connect theoretical predictions with the measurements, such as for the data shown in figure 8. This data illustrates that when the gas jet is positioned before the focus, the long and short trajectories are largely separated, and when positioned after the focus, the trajectories overlap. This feature may be useful in some XUV frequency comb applications. Preliminary results of a model including phase matching in three dimensions to compute the far-field beam profiles shows qualitative agreement with the measurements, and they indicate that the relative contributions of the different generation pathways in the experimental observables, such as beam shape and harmonic flux, are very sensitive to the parameters used in a particular experiment. In this way, the low pressure regime can be utilized to explore the role of separate quantum trajectories in the HHG process for near-threshold harmonics [46].

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In the HHG process, the XUV is generated colinearly with the optical driving field. Thus, one of the fundamental challenges to fsEC development is the design of a high-efficiency XUV output coupler. The design criteria for the XUV output coupler in fsEC applications are covered comprehensively in [55]. The technical requirements for the XUV output coupler are extremely demanding as it must efficiently couple the XUV out of the cavity while withstanding the high average power and peak intensity in the fsEC, and at the same time it must contribute negligible loss and dispersion as well as low nonlinearities for the circulating cavity field. The two output couplers that have been used most successfully in fsEC's are the Brewster plate and diffraction grating output coupler, and are illustrated in figure 9.

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The first demonstrations of fsEC-based XUV generation used Brewster plates to extract the harmonics from the cavity [19, 20]. This method relies on the different refractive indices for the fundamental and harmonic wavelengths. The plate is aligned at Brewster's angle for the fundamental beam to eliminate Fresnel reflection, but the harmonics have different reflection coefficients and a portion of each harmonic is coupled out of the cavity. Thin Brewster plates are chosen to limit the dispersion added to the cavity, and for a moderate finesse fsEC ($\mathcal {F}<500$) the dispersion can be compensated without the use of chirped mirrors [54]. Normally, a thin plate (e.g. 250 μm) of sapphire is used, although other appropriate materials also exist [56]. The efficiency of the Brewster plate output coupler depends on wavelength and material used for the plate but varies from 5–15% for the harmonics of interest. The output-coupled XUV beam in this case consists of many harmonics superimposed on one another. This situation may be desirable for some applications, while for others a diffraction grating must be used to separate the harmonics which causes additional losses for the XUV. One experimental difficulty with this approach is that over time the sapphire plates suffer irreversible optical damage, leading to excessive losses for the optical driving field, and the plates must be replaced.

For applications requiring spectrally resolved harmonics, an XUV diffraction grating output coupler can be employed. First reported in [57], this method utilizes a small-period (e.g. λ_g = 420 nm) diffraction grating etched in the outermost layer of the dielectric stack mirror to couple the harmonics out of the cavity without significantly affecting the circulating fundamental field. The grating structure is considered in the design of the dielectric mirror, and does not reduce the high reflectivity of the fundamental light. An important aspect of the grating design is that the sub-wavelength grating can facilitate coupling between the cavity mode and a slab waveguide mode created by the dielectric stack. Fortunately, the coupling can be eliminated by increasing the depth of the top layer without degrading the performance [58]. The XUV is coupled out of the fsEC with the diffraction grating with an efficiency ranging from a few per cent to 10%, depending on the harmonic order. More recently, a blazed diffraction grating has been developed that achieves a diffraction efficiency of about 15–20% from 35 to 80 nm [59].

Another out-coupling method involves passing the harmonics through a small aperture in one of the focusing mirrors [60], however this method has been met with limited success. This approach takes advantage of the fact that the divergence of the harmonics is less than that of the fundamental beam. The aperture must be small to mitigate loss and diffraction due to the high-finesse requirement for intracavity HHG, making alignment of the harmonic through the hole difficult. It was proposed that a higher order intracavity mode, such as TEM₀₁, could be used so that the fundamental beam has negligible field amplitude along the axis of propagation, and thus could avoid the aperture. However, due to the increased spot size and increased Gouy phase shift for higher order modes, the HHG efficiency decreases. Conversely, intracavity mode tailoring has been proposed to remedy these limitations [61]. Here, degenerate cavity modes are used, which allow for designing the aperture to be large without causing considerable loss, while the field is at a maximum along the beam axis near the focus. To achieve mode degeneracy within a non-confocal cavity, the focusing mirror separation is carefully tuned such that the increased phase shift experienced by higher order modes can be compensated, allowing for a degeneracy of several mode orders.

Another intriguing method proposed for out-coupling the harmonics is non-collinear HHG [60, 62, 63]. Here, two pulses from the fundamental field interact at the focus at a small angle; the harmonics generated propagate in the direction bisectrix of the fundamental beams. The difference in the direction of the harmonics and fundamental beams allow for the harmonics to be out-coupled while minimally affecting the cavity finesse. Due to practical issues regarding the timing of colliding pulses, among others, this technique has yet to be experimentally demonstrated.

With better understanding of the limitations of the diffraction grating and Brewster plate as XUV output couplers, new beam separators have been developed. One method, in the vein of the diffraction output coupler, has a small angle top layer wedge on a highly reflecting mirror [55]. Because of the small angle, the wedge has minimal effect on the reflectivity of the mirror beneath for the fundamental beam, whereas the XUV beam is reflected by the wedge layer, allowing for a small divergence of the fundamental and harmonic beams. Although this output coupler has shown to have similar scalability and coupling efficiency to the diffraction grating, the harmonics are co-linear, which may have benefits for some applications. Another design uses the large Fresnel reflectivity at grazing angle incidence on an intracavity optic to increase the out-coupling to greater than 50% for XUV light down to 15 nm [64]. Here, the incident angle is increased to >75° to increase the XUV reflectivity, where the front and back of the plate have an anti-reflection coating for the fundamental beam at the grazing angle. The high reflectivity makes this a promising method of coupling out the XUV light, although the limitations of this design may be comparable to those of the Brewster plate (that is, dispersion and heating/optical damage for high-power scalability). In addition, these XUV output coupler designs will also be of use for lower repetition rate, traditional CPA single-pass HHG XUV systems that currently use gratings and aluminium filters to separate harmonics from the fundamental driving field as well as from each other. Indeed, for most applications of HHG generated XUV radiation, both of these filtering steps must be made.

The first demonstrations of XUV generation via fsEC employed Ti:sapphire mode-locked oscillators to seed the cavity [19, 20] and resulted in intracavity powers up to 480 W and pulse energies of 4.8 μJ. As HHG is highly nonlinear there is a large payoff for increasing these performance levels. An early effort to increase the intracavity pulse energies was made by decreasing the fsEC cavity length and seeding it with a chirped pulse oscillator operating at 10.8 MHz. While some success was reported [65], the XUV power did not substantially improve. Moreover, in our experience decreasing the repetition rate much below 50 MHz leads to difficulties locking the cavity length due to a decrease in the width of the fsEC's resonances and increased mechanical instability of such long cavities. In addition, these steps can be counterproductive for XUV frequency comb applications as high power per comb element with adequate spacing of the elements is desired. A second approach to increasing intracavity intensities is simply to increase the average power of the input seed by using either a CPA fibre-based system operating at its full repetition rate (100–150 MHz) or injection-locking a solid-state mode-locked oscillator to a secondary gain cavity. These latter two approaches are discussed in more detail in section 5. As a final note, while increasing the fsEC finesse is also capable (in theory) of boosting the intracavity intensity, it subsequently leads to experimental difficulties such as troubles in cavity length locking as well as higher sensitivity to intracavity dispersion (discussed in section 2.3) and mirror damage (see section 4.1). For these reasons, a finesse of approximately 500 is now typically used and intracavity intensities are obtained by using high-power seeds as discussed above.

A system producing XUV light must operate in a vacuum environment to avoid absorption of the light by atmospheric gases. While this might seem to suggest that only a rough vacuum is necessary, the low-loss demands of the optical coatings in the fsEC can result in sensitivity to hydrocarbons and other contaminants in the vacuum system and therefore high or ultra-high vacuum (UHV) is a desirable goal to minimize contaminants and degradation of the fsEC mirrors. Although not fully understood, degradation of the fsEC mirrors are most likely the result of optical and XUV photo-assisted surface hydrocarbon chemistry on the dielectric mirrors. We discuss these aspects in more detail in section 4.1. As a result of this mirror damage we attempt to use only high-vacuum-compatible components, and we thoroughly clean all components before introducing them to the vacuum system, including plasma cleaning of the fsEC optics. The first generation system built in our lab is a large, homemade aluminium vacuum chamber with o-ring seals that is pumped with a 500 L/s turbo pump. Next generation systems will use UHV compatible components and chamber designs to reduce contaminants as well as to provide better pumping of the residual gas used for the HHG process. We discuss technical details of this aspect in section 3.4.

Active stabilization of CW optical sources to optical reference cavities for frequency metrology and buildup cavities for intracavity harmonic generation has continuously improved over the past three decades [66–68]. Beyond the differences that arise in the character of the fsEC resonances, as described in section 2.1, the principles of stabilizing a mode-locked oscillator to a fsEC are much the same as stabilizing a CW laser to a high-finesse optical cavity: one can use a modulation-based method of generating an error signal, such as the Pound–Drever–Hall (PDH) method [67], or one can use a 'DC' approach such as the Hänsch–Couillaud method [66]. As discussed earlier in section 2.1, when exciting a fsEC there are two degrees of freedom that need to be stabilized. An appropriate, although perhaps not perfectly orthogonal, error signal for both the cavity length and the relative carrier-envelope-offset frequency can be derived from the same cavity reflection signal by spectrally resolving this reflection as discussed below.

3.3.1. Stabilization of cavity length

3.3.2. Stabilization of the relative carrier-envelope-offset frequency

In section 2.4 we reviewed the theory behind generation of harmonics in the tight-focus regime, which is required in a fsEC to achieve the intensity required for HHG (>10¹³ W cm⁻²). In this regime efficient generation of harmonics requires the gas target to be restricted to a small volume near the focus of the fundamental driving field in the fsEC. In this section, we review two nozzle geometries for the HHG gas target that have been used in previous demonstrations of fsEC-based HHG [19, 20, 65] (both are routinely employed in CPA-based HHG systems), and a third geometry that is a hybrid of the two nozzle geometries. We also discuss the results of computational fluid dynamics (CFD) simulations for these nozzle geometries that help with the evaluation and optimization of these nozzle designs for use in fsEC HHG systems.

In a previous demonstration of intracavity XUV generation, the authors of [65] used a through-nozzle geometry, in which the laser field is threaded through a small hole drilled through a stainless steel tube containing xenon gas. In the second geometry [19, 20], a nozzle is formed by a small hole in a gas line and the xenon gas is allowed to freely expand into vacuum. This end-fire nozzle is placed as closely as possible to the intracavity beam in order to maximize the density of gas interacting with the laser field. The geometry of these two gas nozzles is illustrated in figure 10.

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Each of the designs has attractive features as well as drawbacks. The end-fire nozzle is technically less challenging to implement, but the nozzle must be located sufficiently far from the intracavity focus so as not to introduce excessive loss in the cavity. Much higher gas density is possible for the through-nozzle, but the nozzle must be precisely positioned with respect to the intracavity field, and the apertures that the laser is threaded through must be large enough not to introduce significant loss. Furthermore, the apertures of the through-nozzle allow gas to exit the nozzle along the optical axis, which increases amount of plasma generated and increases the absorption of XUV by the neutral gas atoms. Both of these consequences are undesirable, but the excess plasma is a major problem for intracavity HHG, as we will discuss in section 4.2.

In order to better understand how the geometry of these different nozzle designs affect the generation of harmonics we performed CFD simulations to determine the gas density distribution for a variety of nozzle parameters, which is used in the computation of the harmonics reviewed in section 2.4. We presented a detailed summary of these results for the end-fire nozzle in [44]. We have also performed a similar analysis for the through-nozzle geometry, and a third geometry or hybrid nozzle that eliminates the stagnant flow at the centre of the through-nozzle by opening a hole at the end of the tube that allows gas to flow from the end as it does in the end-fire geometry [45]. We do not give a comprehensive summary of the simulations here, but we do summarize the properties of each of the nozzles in table 1 in terms of gas density, interaction length and the transit time or interaction time.

In figure 11 we present a series of CFD simulation results for the gas density distribution for three end-fire nozzles with 50, 100 and 150 μm diameters. The geometry simulated in this figure is identical to a capillary tube style nozzle, or simply a long cylindrical tube. This geometry is slightly modified from the geometries simulated in [44], but the results are only slightly different for the nozzle diameters of interest here. In the figure, the top left panel shows the 2D density distribution of gas, the lower left panel shows the variation in density along the laser propagation axis at a distance of 100 μm from the nozzle tip, the lower right panel shows the gas density drop as a function of distance from the tip of the nozzle, and the upper right panel shows the geometry of the gas nozzle relative to the intracavity laser field. Practically speaking, in a fsEC the nozzle can only be placed as close as a few tens of μm from the intracavity focus to avoid diffraction losses that spoil the cavity finesse.

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In our experience, the end-fire nozzle is the most suitable choice of nozzles for fsEC due to the elimination of gas flow along the optical axis and the decreased interaction time between the laser and the gas atoms. We have worked experimentally with the through-nozzle geometry, and we found it difficult to reproducibly deliver XUV beams with high quality spatial modes from the system, and plasma-induced oscillations were a serious limitation. The hybrid nozzle we tested represents a slight improvement over the through-nozzle, but about 10% of the gas still flows along the laser propagation direction and leads to significant plasma oscillation. The end-fire nozzle, on the other hand, reliably produces high quality beams, and while the harmonic generation is extremely sensitive to alignment of the nozzle, it is much more straightforward to align. Interestingly, very recent work using a 100 kHz single-pass Ti:sapphire CPA system in the tight-focus regime has measured the XUV flux for the 15th and 17th harmonics for the through-nozzle and end-fire nozzles [52]. Their results agree with the observations we have reported for the fsEC case, but the disadvantages of the through-nozzle geometry are not as severe in the single-pass geometry as they are in the fsEC. When singe-pass repetition rates beyond a few MHz are used, we expect the through-nozzle to become even more disadvantageous in this case.

We are currently using thin-walled quartz capillaries for our end-fire nozzles, with an inside diameter of 100–150 μm. Alignment of even smaller nozzles becomes very challenging, but a more fundamental limitation is that for smaller nozzle diameters, the gas density drops extremely quickly and the expansion of the gas begins to limit how small the interaction length can be in the laser field, some tens of micrometers away from the nozzle orifice [44]. Alignment of the 150 μm nozzles is simplified with the small outer diameter (250 μm) compared to thick-walled capillaries, or metal tubes of the same inside diameter. Similar gas nozzle geometries are also used in recent high XUV flux fsEC systems, where a 100 and 300 μm stainless steel nozzle [54], and 100 μm glass nozzle [58] have been used.

Finally, we note that excess gas that accumulates in the vacuum system due to imperfect pumping can cause a number of problems, but most troublesome of all is the reabsorption of the harmonics. Different methods have been employed to expel this gas, but both involve catching the gas before it can diffuse into the entire volume of the vacuum chamber. In the work of [54] a turbo pump is placed in the base of the vacuum system immediately below the gas nozzle, while in the work of [58], a special gas catch assembly is placed a small distance below the gas nozzle and connected to a vacuum roughing line [74]. We have implemented a similar gas catch arrangement in our systems, and we have observed significantly improved XUV flux when it is properly aligned.

Detecting the harmonics generated by the fsEC source can be challenging for the lower order harmonics, where detectors and filters must be able to detect the low harmonic power without detecting the on-axis scattered fundamental light. For wavelengths shorter than 80 nm, aluminium filters directly deposited on silicon photodetectors work very well, but we have not the same level of success filtering out the 800 nm fundamental light with directly deposited filters for the longer wavelength harmonics. To solve this problem, photomultiplier tubes with phosphor-coated windows can be used, and it is also common to use a phosphor screen to image the fluorescence to obtain beam profiles for the harmonics. The linearity and spectrally flat responsivity of phosphors such as sodium salicylate makes this method convenient for comparing relative power from the harmonics as well as measuring the beam profile of the harmonics. An example image is shown in figure 12.

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There is currently great interest in developing new detectors for XUV light. We are using our fsEC systems to evaluate the performance of diamond and selenium photoconductors. Diamond detectors are particularly interesting for detection of high energy photons for a variety of reasons, including their radiation hardness, low leakage current and visible light insensitivity (solar-blindness) [75]. This last factor is of particular interest for fsEC-based XUV generation where the scattered light from the fundamental near-infrared light in the cavity can be orders of magnitude brighter than the generated harmonics. Selenium is also used in digital x-ray radiography as direct, electronically readable x-ray detectors [76]. Development of selenium detectors for enhanced short wavelength visible radiation detectors with extremely low response in the near infrared is also progressing [77]. This recent success has sparked interest in evaluating the use of these photoconductors in the VUV and XUV spectral regions. Preliminary results with the fifth harmonic (158 nm) are promising, and demonstrate excellent extinction of scattered fundamental light at 800 nm.

As the mirrors within an fsEC must be highly reflective, even a slight degradation in their performance can be of concern. For the most part, at the intracavity average power and intensity achieved in fsEC systems the mirror reflectivity is often degraded rather than irreversibly damaged. In high-power Yb-doped fibre systems mirror damage is reported for high average power (18 kW) and pulse durations shorter than 640 fs [78]. The damage is not described in detail, but rather a subject for further investigation. In another Yb-doped fibre laser system [58] mirror damage is not a limiting factor for intracavity average power of >7 kW, and the mirror degradation can be reversed with an oxygen plasma [74]. The high-power Ti:sapphire system reported in [54] has issues with the permanent damage of the Brewster-plate XUV output coupler, which is located relatively close to the intracavity focus and is subject to much higher intensities than other optics in the cavity.

Optically induced degradation of the mirrors can arise via two sources. The first results from high (several kilowatt) intracavity average power and very high peak intensities of the optical driving fields on mirror surfaces. A second is degradation induced by the relatively high photon flux XUV generated in the fsEC. From work on synchrotrons and other advanced light sources, it is well known that XUV and x-ray optics suffer degradation by contaminant hydrocarbons in the vacuum that deposit on mirror surfaces and interact with XUV high energy photons (see, for example, [79] and references therein). Cracking of the hydrocarbons and attraction of new hydrocarbons to the surface from the surrounding vacuum leads to run-away buildup on and reduced reflectivity of the mirror surface. There is speculation that dangling bonds from the specific dielectric materials also play a role in the process. These effects can often be reversed with the introduction of oxygen gas or ozone near the mirror surface.

Such processes have been widely investigated, but recent work [79] that investigates the surface contamination of silica by toluene (C₇H₈), in the presence of 213 nm UV light is particularly reminiscent of the effects we observe in our fsEC systems (figure 13). The measurement relies on the detection of photoluminescence (PL) of the silica produced by UV light, and the decay of the PL signal via absorption within the layer of hydrocarbon contaminants as they accumulate on the silica surface. After a large reduction in PL signal is observed, the toluene gas is turned off and oxygen is introduced. In the presence of the 213 nm radiation, photolysis of the oxygen occurs and a photochemical reaction results in the formation of ozone (O₃), which etches the toluene-based photo-deposits and results in a revival of the PL signal. It would appear that the effects observed in fsEC systems is a multi-photon analogy of these experiments. This is not at all unreasonable given the high intensities present in the fsEC and bond energies involved in the hydrocarbon dissociation. The C–H and C–C bonds have dissociation energies of about 4.3 and 3.6 eV, respectively [79], each of which would the energy of three photons in the 800 nm Ti:sapphire system, or a two-photons for the 520 nm fsEC system described below.

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From our experience a similar hydrocarbon-based degradation process occurs in fsECs and can be mediated by both optical and XUV photons. Focusing first on degradation induced by the optical driving field, we have observed varying degrees of degradation in our fsEC systems operating at 800 and 1040 nm depending on the particular mirrors used (from different vendors) and the cleanliness of the vacuum environment. In general, we find the problem is much more pronounced at 800 nm. In our Ti:sapphire laser seeded system operating near 800 nm, different mirror coatings (i.e. different vendors) have different degradation characteristics, even when no intracavity plasma (and therefore no XUV light) is generated. Mirrors from one vendor exhibited a lifetime of several minutes to a few tens of minutes, depending on the intracavity intensity. We show an example of this degradation in figure 13 for mirrors from this vendor. Mirrors from another vendor have shown lifetimes in excess of several hours at the highest intracavity average power we can achieve with this particular mirror set (∼300 W).

Once XUV light is generated, the mirrors from this second vendor degrade from their optimum performance in a matter of minutes. XUV degradation is usually limited to the one or two mirrors following the gas target. It remains to be seen to what extent vacuum cleanliness can resolve these issues. Our Yb-doped fibre laser system operating near 1040 nm exhibits substantially slower degradation. The system shows signs of mirror degradation when the system has been operating at an average power of 2–3 kW for several hours. This system also maintains its performance levels for substantially longer periods of time during XUV production than our 800 nm system.

Similar to other XUV and x-ray optics, once degraded the fsEC mirrors can be rejuvenated by pressurizing the cavity with oxygen. This healing process can be significantly accelerated if the cavity is locked such that there is high intracavity optical intensity along with the oxygen atmosphere. The data shown in figure 13 shows this revival, where 6 torr of oxygen is introduced to the vacuum system at t = 8 min. We have not observed such dramatic degradation or revival with our 1040 nm system, but weekly removal of the mirrors from the vacuum and cleaning with an oxygen plasma cleaner is helpful. Interestingly, a fsEC operating near 520 nm was recently reported [80] that must be operated in an oxygen environment to avoid mirror degradation. Without the oxygen the cavity enhancement decreases exponentially with a time constant of about 20 s. In addition to the obvious change in photon energy for these three cases (1040, 800 and 520 nm), the specific dielectric materials and growth techniques used in the mirror construction at different wavelengths will also play a significant role in determining the mirror sensitivity to hydrocarbon contamination. Unfortunately, further quantification of the degradation mechanism from a material perspective is difficult given the unwillingness of mirror vendors to provide their mirror design specifications.

The plasma generated during the HHG process plays an important role in the phase matching process and must be managed to generate harmonics most efficiently. For kHz repetition rate CPA systems the harmonic generation becomes limited when the fractional ionization that occurs during a single femtosecond pulse exceeds a certain level, but issues related to the accumulation of plasma resulting from multiple pulses do not arise because the long time between pulses. Such is not the case for fsEC systems where the pulse period (∼10 ns) is shorter than the timescales relevant to plasma decay, and even the transit time for the room temperature gas atoms traversing typical focal volumes in the fsEC may exceed 10 pulse periods.

The generation of intracavity plasma and the limitations it imposes on XUV generation has been studied recently [81, 82]. Over short timescales (i.e. the duration of a pulse), nonlinear phase shifts (e.g. self-phase modulation) limit the peak intensity of the pulse circulating in the fsEC, and over longer timescales (tens of pulse periods and longer) the steady state plasma density also contributes a phase shift that significantly shifts the cavity resonance. There is little that can be done about the phase shift resulting from the dynamically generated plasma, and this factor may be one of the fundamental limits of fsEC-based XUV generation. The phase shift due to the steady state plasma can shift the fsEC resonance well in excess of the cavity linewidth. To demonstrate the effect of the plasma phase shift, we present a plot of the cavity lineshapes when the cavity length is scanned in opposing directions, in figure 14. For the solid curve, the change in cavity length partially compensates the increasing phase shift due to the plasma, and the cavity remains on resonance longer than it would without the plasma present. For the dot-dashed curve, the cavity scan is in the opposite direction, which adds with the plasma phase shift and causes the cavity to move away from resonance more rapidly. The servo control loop can compensate for the phase shift due to the steady state plasma, but even small perturbations in the intracavity intensity can cause a drop in the plasma density that shifts the resonance condition too rapidly for the control loop to follow. This condition initiates a bistable oscillation in which the servo loop acts together with the plasma buildup and decay to drive the system into and out of resonance. The mechanism for this oscillation was described in [58, 82], where it was also pointed out that locking slightly away from the peak of the cavity resonance improves the stability of the system and can prevent this bistable oscillation.

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We show an example of large plasma oscillations in figure 15, observed in our Ti:sapphire seeded fsEC system. When the cavity is locked to the transmission maximum, the on-resonance cavity length differs significantly from its value without plasma, due to the phase shift from the plasma. Because the ionization rate is a highly nonlinear function of intensity, even small perturbations of the intracavity intensity can cause a large and rapid decrease in plasma density, and likewise a rapid shift in the resonance detuning. Such an event can drive the system into the following oscillation sequence. Starting from the point of maximum cavity transmission (labelled point 1 in the inset of figure 15), the cavity is perturbed from resonance, which causes the plasma to disappear and the cavity resonance to shift moving the system to point 2. The servo begins to move the cavity length closer to resonance, corresponding in the figure to a decreasing cavity PZT signal. As the cavity length approaches resonance, the error signal is observed to acquire a slightly more positive signal and the cavity transmission begins to increase. Near point 3, the error signal abruptly changes sign and the cavity PZT changes direction indicating that the system has shifted from point 3 to point 4 as the growing plasma phase shift rapidly modifies the resonance detuning. As the plasma continues to grow, the cavity PZT moves to follow the shift in resonance, but the negative error signal suggests the loop cannot keep up. Near the peak of transmission (1) the plasma phase shift saturates and the cavity PZT overshoots, leaving the cavity slightly off of resonance and causing the plasma to disappear. The cycle begins anew with another rapid decrease in the cavity transmission. To prevent this oscillation, the cavity can be locked slightly off of resonance (point 5).

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These oscillation lineshapes are somewhat different that those seen in higher power systems due to the high finesse ($\mathcal {F}>2000$), which is responsible for the ringing in rising edge (and ringdown) of the cavity transmission. Also this behaviour is only observed for the through-nozzle gas geometry in the Ti:sapphire system, which has a considerable interaction length with xenon gas and plasma. We do not see these oscillations for the end-fire gas geometry at the intracavity pulse energy and gas pressures we can achieve with this system. In our Yb-doped fibre laser system, we find that oscillations similar to those in [58] arise for high gas pressures and average fsEC power above 1–2 kW. In this case, we are able to lock slightly off resonance to reduce the plasma oscillations, as first observed in [82]. The increased XUV output under these conditions appears to come at the cost of increased amplitude noise of the fundamental in the cavity, which could have implications in the ultimate linewidth of the resulting XUV frequency combs [46].

To curtail the limitations caused by the HHG-generated plasma, one must minimize its interaction time with the optical driving field. Use of a lower repetition rate oscillator gives the gas atoms and plasma more time to move through the interaction region, but this is highly undesirable for a frequency comb. In direct frequency comb spectroscopy, increasing the power per comb element increases the signal, and the increased comb spacing decreases the likelihood that multiple comb lines drive the same atomic or molecular transition simultaneously. The remaining option is to decrease the transit time of atoms through the interaction region. The velocity of the mixture flowing from the nozzle is, $v_{\rm gas} \propto \sqrt{T/m}$, where T is the gas temperature and m is the average gas mixture mass, thus indicating two practical methods to increase the velocity. The first is to increase the gas temperature, which can be achieved either through heating the gas at the source, or by heating the gas delivery nozzle. In the latter case, we have observed that when a small quartz nozzle is placed near the intracavity focus, substantial heating of the nozzle occurs without catastrophic diffraction losses introduced to the fsEC. However, it remains to be observed whether the additional nozzle heat can be transferred to the gas. The second method is to use a gas mixture, with the second gas type having a much lower mass than the harmonic generating medium (such as a xenon/helium mixture). The consequence of a mixture is that in order to maintain the same xenon density at the focus with a substantial increase in velocity, the gas flow must be considerably increased, potentially increasing the reabsorption of the generated harmonics and putting strain on the pumps for the vacuum system.

Of the four types of systems demonstrated, the first was a high-finesse cavity seeded with a Ti:sapphire mode-locked oscillator. In this first class of systems [19, 20, 65], the high peak intensity requirements of HHG were met by using large enhancement factors made possible through careful management of cavity dispersion. Although some variation exists in these systems, depending on the particular configuration (including pulse energy, repetition rate and the focusing geometry and enhancement factor achieved in the cavity) these types of systems typically generate harmonics with a plateau near the 11th or 13th harmonic of the 800 nm seed. Our version of such a system [44] utilizes a grating output coupler (λ_g = 200 nm) for the harmonics, which has enabled us to measure >1 μW (spectrally resolved) in the 11th harmonic (4 × 10¹¹ photons s⁻¹), and we have observed the 13th harmonic, which is about a factor of 5 lower in power. Given the simplicity of this approach, it is interesting to explore the uses of such a system.

An improvement over the single oscillator Ti:sapphire source was reported [88], wherein the authors demonstrated a high-power femtosecond amplification cavity. In this system, a 1 W output from a mode-locked oscillator is used to injection lock an actively pumped, Ti:sapphire amplifier cavity to achieve an average power of 7 W while maintaining fs pulses at 50 MHz when pumped with 18.5 W at 532 nm. The resulting pulse train can be compressed to its transform limit (∼70 fs), and is subsequently used to seed a fsEC. With an enhancement of 100–200 times, pulse energies as high as 25 μJ are produced, with an intensity at the cavity focus in excess of 10¹⁴ W cm⁻². XUV generation with this fsEC was recently reported [54], in which μW power levels are observed for harmonics as high as the 15th, and 77 (±15) μW of light at 72 nm (11th harmonic) was out-coupled from the fsEC. These impressive results are obtained at a significantly lower pulse energy (∼15 μJ) than the maximum operating conditions because of limitations due to the intracavity plasma. This system utilizes a Brewster plate output coupler that is about 10% efficient at this wavelength, which means that ∼0.8 mW of the 11th harmonic is generated intracavity. This is the highest power reported to date from a fsEC-based XUV system.

One of the most promising systems for extended-use, high flux fsEC-based XUV systems uses a seed from an amplified femtosecond Yb-doped fibre laser operating near 1 μm. These systems offer favourable power scaling and pulse durations compatible with fsEC use. The most advanced source of this type reported to date has a spectrum centred near 1070 nm and boasts 120 fs pulses, 80 W average power and 154 MHz repetition rate [58]. The 0.52 μJ pulses from this system only need modest enhancements to achieve the level needed for HHG. In fact the high power of this source exceeds the levels at which the fsEC exhibits nonlinear behaviour and possible mirror damage. Nevertheless, as high as 10 kW can be achieved in this system. Power scaling of a similar system has also been reported [78], in which 18 kW was achieved using 200 fs pulses and as high as 72 kW was achieved with pulses of 2 ps duration. The system reported in [58] uses an XUV grating output coupler (∼10% diffraction efficiency), and is capable of generating >200 μW while delivering ∼20 μW/harmonic. At 20 μW of measured power, this is currently the highest power reported that is spectrally resolved and can be used directly in an experiment.

A visible wavelength fsEC was recently reported [80]. In this work, the authors frequency-double the output of a Yb-doped fibre laser system to generate light at 520 nm, which is then used to seed a fsEC. This approach takes advantage of the relatively efficient (∼50%) second harmonic generation (SHG) process to gain access to the dramatically improved HHG efficiency that results from the shorter fundamental wavelength, which has been demonstrated to have a wavelength scaling that varies as λ^{−6.3 ± 1.1} [89]. Of course, a shorter wavelength also causes a reduction in the cut-off harmonic generated in the HHG process, although this is not an issue for some applications requiring a VUV frequency comb.

This demonstration illustrates the significant fundamental and technical challenges to working with the shorter fundamental wavelength. High reflectivity mirrors are not readily available at 520 nm, and while an enhancement of 200 was achieved for low input powers, the enhancement dropped to 75 for an input power of 4 W. To even maintain this latter level of enhancement, the fsEC must be operated in an oxygen environment to prevent mirror degradation that occurs in vacuum, as was discussed in section 4.1. This degradation phenomenon is similar in nature but dramatically more pronounced than that observed in 800 or 1040 nm based fsEC systems. Due to VUV absorption, the oxygen environment of the cavity is a major problem for the harmonics generated, and in order to extract the harmonics from the cavity a Brewster plate is used as the fsEC output coupler that directs the harmonics into a small tube towards a differentially pumped detection chamber to minimize the optical path length in oxygen. The system is reported to generate (out-couple) a power of 6.5 mW (3.1 μW) at 173 nm, 3.9μW (120 nW) at 104 nm and 150 nW (5 nW) at 74 nm. While the reported photon flux does not currently favour the SHG route to VUV generation (compared to using the same Yb-doped fibre source and an IR fsEC), this situation may improve greatly if the current limitations are better understood and resolved.