2.1 Simulation settings

We used the SISYFOS package, which has excellently modeled multiple MPC experiments^[ Reference Seidel, Balla, Li, Arisholm, Winkelmann, Hartl and Heyl^32, Reference Viotti, Li, Arisholm, Winkelmann, Hartl, Heyl and Seidel^33], and numerically studied spectral broadening of 300 ps (full width at half maximum, FWHM), 100 mJ pulses over 1300 MPC passes. This number of passes can be implemented by a system with 11 spherical mirrors arranged in a circle, as explained in Section 3. The pulse parameters were chosen because they are readily accessible by cryogenically cooled Yb:YAG and commercially available Nd:YAG lasers.

The MPC was modeled under the following approximations: the beam was always on the optical axis of the system. The input beam was a fundamental Gaussian beam ( ${M}^2=1$ ). Reflectivity and dispersion of the MPC mirrors, essentially SiO ${}_2$ /Ta ${}_2$ O ${}_5$ quarter-wave stacks, were taken from the theoretical data provided by the high-reflectivity mirror supplier. The MPC mirrors had 500 mm radii of curvature ( $\mathrm{\mathcal{R}}$ ) and were separated by 975 mm. Their reflectivity of 99.99% resulted in 87% transmission after 1300 bounces. The MPC geometry, neglecting slight astigmatism at this point, resulted in linear mode-matching beam radii of 1.01 mm on the mirrors and 160 μm in the focus. The fluence on the mirrors was below 6 J/cm². Highly reflective mirrors are clearly the best option for achieving large spectral broadening factors and compact setups. Dispersive mirrors exhibit lower reflectivities and laser-induced damage thresholds (LIDTs). The Kerr medium was krypton. Its nonlinear refractive index was derived from Ref. [Reference Travers, Chang, Nold, Joly and Philip St.34] ( ${n}_2=1.85\times {10}^{-19}\ {\mathrm{cm}}^2/\mathrm{W}\times p/1\kern0.1em$ bar, where $p$ is pressure) and the dispersion from Ref. [Reference Börzsönyi, Heiner, Kovacs, Kalashnikov and Osvay35]. Up to ${2}^{14}$ points along the time axis were used. This was required due to the large broadening factors for the highest gas pressures studied. The simulations automatically included self-steepening by solving the nonlinear wave equation in the frequency domain. However, due to the sub-ns pulse durations, self-steepening was not observed.

2.2 Input pulse dependent spectral broadening

Following the pictorial scheme of Figure 1, we investigated different input pulse shapes and spectra. For this purpose, simulations with cylindrical symmetry and a grid with 256 points along the radial coordinate were conducted. We first considered an FT-limited Gaussian pulse with the complex electric field:

(3) $$\begin{align}{E}_\mathrm{G}(t)={E}_0\exp \left[-2\ln 2{\left(t/\Delta {t}_\mathrm{p}\right)}^2+i{\omega}_\mathrm{c}t\right],\end{align}$$

and a positively chirped Gaussian pulse with 6 ps FT limit, described by the following:

(4) $$\begin{align}{E}_\mathrm{sG}(t)&=\frac{E_0}{\sqrt{1+ is}}\exp \left[-\frac{2\ln 2\kern0.1em {t}^2}{\Delta {t}_{\mathrm{p}0}^2\left(1+ is\right)}+i{\omega}_\mathrm{c}t\right], \end{align}$$

where $s=\sqrt{\frac{\Delta {t}_\mathrm{p}^2}{\Delta {t}_{\mathrm{p}0}^2}-1}$ .

The field amplitude ${E}_0$ was set to obtain 100 mJ pulse energy, $\Delta {t}_\mathrm{p}=300\kern0.1em$ ps, ${\omega}_\mathrm{c}=2\pi c/1030$ nm with $c$ the speed of light in vacuum and $\Delta {t}_{\mathrm{p}0}=6\kern0.1em$ ps. Input pulses and spectra for the simulations are shown in Figures 2(a) and 2(b).

Figure 2

Simulated spectral broadening of 300 ps, 100 mJ input pulses in a 1300-pass MPC for Fourier-transform (FT) limited pulses with a Gaussian beam (black lines) and parabolic intensity envelope (red lines) as well as chirped Gaussian pulses with 6 ps FT limit (blue lines). (a) Input spectra of the simulations. (b) Input pulses of the simulations. The pulse shape is virtually preserved during propagation. (c) Simulated output spectra for a Kr pressure of $p=4.5\kern0.1em$ bar.

Figure 2 Long description

Panel a at left plots normalized spectral power on the y axis versus wavelength in nanometers on the x axis, ranging from 1029.7 to 1030.3. Three curves are shown: a narrow black line for transform-limited Gaussian, a broader blue line for chirped Gaussian, and a red dashed line for parabolic pulse, all peaking near 1030 nanometers. Panel b in the center plots instantaneous power in gigawatts on the y axis versus time in picoseconds on the x axis, from negative 400 to positive 400. The black and blue curves are nearly identical and centered at zero, while the red dashed curve is wider. Panel c at right plots normalized spectral power on the y axis versus wavelength from 1027 to 1033 nanometers. The black and blue curves show broad, structured output spectra with dips near the center, while the red dashed curve forms a smooth, wide parabola. The legend below identifies black as transform-limited Gaussian, blue as chirped Gaussian, and red dashed as parabolic pulse.

The cell was filled with 4.5 bar of krypton, and thus the peak power was about 17% of the critical self-focusing power ${P}_\mathrm{cr}$ . The broadened spectra plotted as black and blue lines in Figure 2(c) correspond to sub-picosecond FT limits of 896 and 837 fs, respectively, but they are strongly modulated. Only the spectral range of the input spectrum remains smooth in the output spectrum. The non-quadratic phase that a Gaussian pulse acquires from SPM limits the peak power enhancement and pulse contrast at high compression factors. This problem can be circumvented by shaping the input pulses to a parabolic intensity profile, in our simulations described by the following:

(5) $$\begin{align}{E}_\mathrm{P}(t)={E}_0\kern0.1em \mathrm{{Re}}\left\{\sqrt{1-2{\left(t/\Delta {t}_\mathrm{p}\right)}^2}\right\}{e}^{i{\omega}_\mathrm{c}t},\end{align}$$

and depicted by the red dashed lines in Figure 2. Owing to the quadratic intensity envelope, SPM induces quadratic phase in analogy to the dominant dispersion term of pulse stretchers in CPA schemes. The red line in Figure 2(c) shows the corresponding smooth output spectrum with a 546 fs FT limit. Several approaches to generate parabolic pulses have been reported, for example in Refs. [Reference Finot, Dudley, Kibler, Richardson and Millot36–Reference Lemons, Hirschman, Zhang, Durfee and Carbajo39]. Investigating their applicability to our high-power pulse generation concept will be subject of follow-up research.

It is important to emphasize that effects different from SPM, for example, optical wave-breaking, self-steepening or modulational instabilities (MIs), are absent even for spectral broadening factors bigger than 1000 that are reached at higher krypton pressures. This is enabled by the combination of long input pulses and MPCs as the spectral broadening platform.

2.3 Role of sub-critical self-focusing

As both ${\overline{L}}_\mathrm{D}$ and ${\overline{L}}_\mathrm{NL}$ scale inversely with gas pressure, high pressures are favorable to keep the propagation length as short as possible (Equation (2b)). In Ref. [Reference Viotti, Seidel, Escoto, Rajhans, Leemans, Hartl and Heyl16], the pressure limit was assumed to be set by the equality of peak power ${P}_\mathrm{p}$ and critical self-focusing power ${P}_\mathrm{cr}$ . In practice, MPC experiments with pressures up to ${P}_\mathrm{p}=0.4\kern0.1em {P}_\mathrm{cr}$ have been reported^[ Reference Rajhans, Escoto, Khodakovskiy, Velpula, Farace, Grosse-Wortmann, Shalloo, Arnold, Põder, Osterhoff, Leemans, Hartl and Heyl^40]. For krypton as a nonlinear medium and the laser parameters we investigated, a pressure of ${p}_\mathrm{cr}=26.6\kern0.1em$ bar results in ${P}_\mathrm{p}={P}_\mathrm{cr}$ . However, sub-critical self-focusing changed the shape of the SPM broadened spectra significantly also for $p={p}_\mathrm{cr}/2=13.3\kern0.1em$ bar and $p={p}_\mathrm{cr}/4=6.15\kern0.1em$ bar, as the red and violet lines in Figure 3(a) show. For comparison, the dashed black line results from a plane wave simulation with $p={p}_\mathrm{cr}/2$ , that is, without spatial degrees of freedom. Its bell shape is in clear contrast to the red solid line obtained with identical nonlinear and dispersive lengths, but by additionally including the radius $r$ as a transversal coordinate in the simulation. The broadened spectrum for $p={p}_\mathrm{cr}/10=2.66\kern0.1em$ bar (Figure 3(a), green line) does not exhibit significant features of self-focusing, but resembles the shape of a plane wave simulation. With reduced numbers of passes, comparable broadening factors were simulated with $p={p}_\mathrm{cr}/4$ and $p={p}_\mathrm{cr}/2$ . These spectra are represented by the blue and brown lines in Figure 3(a). They exhibit signatures of nonlinear space–time couplings: along with the change of spectral shape, a change of spectral phase was observed. Figure 3(b) shows that the compressibility of the pulses may suffer from space–time couplings. Self-phase modulated pulses could be compressed close to their FT limit of 153 fs in the case of plane wave simulations (black line). A GDD of −11.9 ps ${}^2$ and a third-order dispersion (TOD) of −0.036 ps ${}^3$ resulted in pulses with 176 fs duration and excellent contrast. The result was sensitive to TOD optimization. In contrast, only a pulse duration of about 400 fs and a bad pulse contrast could be computed if a transversal degree of freedom was included (red line). A GDD of −7.4 ps ${}^2$ and a TOD of −0.021 ps ${}^3$ led to the shortest attainable pulses. The compression result was not very sensitive to TOD. Higher-order dispersion terms were not considered in either case. The impact of spatio-temporal couplings becomes even more pronounced when the astigmatism caused by the non-zero angle of incidence on the spherical mirrors is taken into account, which is discussed explicitly in Appendix A.1. On the other hand, astigmatism has been recognized as a means to strongly increase the number of passes in Herriott-type MPCs^[ Reference Herriott and Schulte^41] and is also essential for the MM-MPCs, which will be discussed in the following sections. To our knowledge, phase distortions due to self-focusing in the critical regime have been investigated for bulk-based MPCs^[ Reference Seidel, Pressacco, Akcaalan, Binhammer, Darvill, Ekanayake, Frede, Grosse-Wortmann, Heber, Heyl, Kutnyakhov, Li, Mohr, Müller, Puncken, Redlin, Schirmel, Schulz, Swiderski, Tavakol, Tünnermann, Vidoli, Wenthaus, Wind, Winkelmann, Manschwetus and Hartl^42], but comparable observations in the sub-critical regime, that is, for gas-filled MPCs, are lacking. In contrast, in studies with ${P}_\mathrm{p}>0.4{P}_\mathrm{cr}$ , 24-fold compression to a near FT-limited pulse duration of 50 fs with a good pulse contrast has been reported^[ Reference Rajhans, Escoto, Khodakovskiy, Velpula, Farace, Grosse-Wortmann, Shalloo, Arnold, Põder, Osterhoff, Leemans, Hartl and Heyl^40]. Therefore, the numerical predictions require additional experimental investigations.

Figure 3

The simulations presented in this figure use the parabolic input pulse described by Equation (5). (a) The solid lines show a comparison between output spectra after 1300 passes for $p={p}_\mathrm{cr}/2$ (red), ${p}_\mathrm{cr}/4$ (violet) and ${p}_\mathrm{cr}/10$ (green). Transverse effects are included in the simulations under the assumption of cylindrical symmetry. The black solid line results from a simulation with only a single spatial grid point (i.e., plane waves) and krypton pressure $p={p}_\mathrm{cr}/2$ . Spatio-temporal couplings scale with the peak-to-critical power ratio, as the comparison among the green solid, blue dashed and brown dashed lines shows. These three curves were obtained by adjusting the number of passes and krypton pressures, such that comparable B-integrals were acquired in all cases. (b) Optimized pulse compression from simulations using a Gaussian beam and applying cylindrical symmetry (red line) and from simulations with plane waves (black line).

Figure 3 Long description

Panel a on the left plots normalized spectral power on the y-axis versus wavelength in nanometers on the x-axis, ranging from 1020 to 1041 nanometers. Multiple lines are shown: the input spectrum is a narrow gray curve centered at 1030 nanometers. The green solid line (p equals 2.66 bar, 1300 passes) is broader and peaks at the center. The blue dashed line (p equals 6.15 bar, 520 passes) is broader and peaks at the center. The violet solid line (p equals 6.15 bar, 1300 passes) is even broader. The red solid line (p equals 13.3 bar, 1300 passes) is the broadest, with steep edges. The brown dashed line (p equals 13.3 bar, 260 passes) is intermediate in width. The black solid line (p equals 13.3 bar, 1300 passes, plane waves) is broad and smooth, similar to the input but with sharper edges. The legend below the graph identifies each line by color, pressure, and number of passes. Panel b on the right plots instantaneous power in gigawatts on the y-axis (logarithmic scale) versus time in picoseconds on the x-axis, ranging from minus 300 to plus 300 picoseconds. The red line (cylindrical symmetry) shows a sharp central peak at zero time with broad wings. The black line (plane waves) also peaks at zero but is narrower and has more pronounced side lobes. The legend at the bottom right identifies the red line as cylindrical symmetry and the black line as plane waves. Both panels compare the effects of krypton pressure, number of passes, and spatial modeling on spectral broadening and pulse compression.

2.4 Pulse compression after spectral broadening in astigmatic MPCs

In regard of the observed space–time couplings, we used pulse compression simulations to derive an optimized critical-to-peak power ratio. Four compressor types were studied: two grating pair compressors using 1500 and 1000 lines/mm gratings in a Littrow configuration were simulated. The third compressor compensated the small TOD of the MPC mirrors ( $106\;{\mathrm{fs}}^3$ per reflection) and the linear chirp of the self-phase modulated pulses. Eventually, a phase shaper with control of up to the sixth-order dispersion term was applied. An overview of the compressors is presented in Table 1. Spectral broadening with krypton pressures of $p=2.66$ bar ( $p={p}_\mathrm{cr}/10$ ), $p=3.5$ bar, $p=4.5$ bar, $p=5.5$ bar and $p=6.15$ bar ( $p={p}_\mathrm{cr}/4$ ) was simulated including the astigmatism induced by the MPC mirrors (cf. Appendix A.1). The spectra after passing through the MPC are shown in Figure 4(a). To optimize the grating distances and the phase shaper, respectively, only the residual spectral phase of the x = 0, y = 0 simulation grid point was considered. For the grating compressors, the grating distance was optimized with respect to peak power. For the phase shaper, the spectral phase was fitted by a polynomial of the order of 6. Appendix A.2 shows the residual spectral phases for the compression of the $p=2.66$ bar and $p=6.15$ bar pressure cases. The derived pulse at the central beam position agreed very well with the beam averaged pulse, which indicates the excellent spatio-spectral homogeneity known from MPC spectral broadening.

Table 1

Simulated pulse compressors.

Table 1 Long description

Starting from the top row, the first compressor type is 1500 lines per millimeter gratings, with G D D of minus 34 p s squared per meter and T O D of 0.221 p s cubed per meter. The second row lists lines per millimeter gratings, G D D of minus 6.14 p s squared per meter and T O D of 0.0174 p s cubed per meter. The third row is T O D-free compressor, G D D marked as optimized, T O D is minus 1.4 times 10 to the minus 4 p s cubed. The fourth row is phase shaper, both G D D and T O D labeled best fit. Footnote a clarifies that meter refers to grating pair separation. Footnote b notes higher-order dispersion up to sixth order is fitted.

^a Unit $\mathrm{m}$ refers to grating pair separation.

^b Higher-order dispersion up to the sixth order is also fitted.

Figure 4

Simulated spectral broadening of pulses with a parabolic envelope in dependence of the krypton pressure in an 11-mirror MPC. Some 1300 passes were simulated, and the astigmatism originating from the angle of incidence on the mirrors is taken into account. (a) Simulated output spectra. (b) Post-compressed pulses after dispersion compensation with a 1500 lines/mm grating pair compressor. The pulses are offset by 15 GW for clarity. (c) Compressed pulses plotted on a logarithmic scale. (d) Overview of simulated output FWHM pulse durations for the different compressor types. (e) Overview of simulated output peak powers for the different compressor types.

.

Figure 4 Long description

The figure consists of five panels labeled a through e. A shared legend for the top row indicates pressure levels: 2.66 bar (black), 3.5 bar (red), 4.5 bar (blue), 5.5 bar (green), and 6.15 bar (purple).

* Panel a: A line graph of Normalized spectral power versus Wavelength in n m from 1025 to 1035. As pressure increases, the spectral bandwidth broadens from a narrow peak at 1030 n m to a wider, more structured plateau. * Panel b: A line graph of Instantaneous power in G W versus Time in p s from minus 7 to 7. The pulses show a central peak at 0 p s with increasing peak power and more complex temporal pedestals as pressure rises. * Panel c: A semi-logarithmic line graph of Instantaneous power in G W versus Time in p s from minus 400 to 400. It shows the high-dynamic-range temporal contrast of the pulses, with the 6.15 bar purple line showing the highest pedestal levels. * Panel d: A scatter plot of Pulse duration in f s versus Pressure in bar from 2 to 6.5. Five data series (F T limit, Grating compressor 1500 lines per m m, Grating compressor 1000 lines per m m, T O D compensation, and Phase shaper) all show a downward trend in pulse duration as pressure increases, ranging from approximately 1200 f s down to 400 f s. * Panel e: A scatter plot of Peak power in G W versus Pressure in bar. The F T limit (black squares) and Phase shaper (purple circles) show a strong linear increase in peak power with pressure, reaching over 200 G W. The other methods (T O D compensation and grating compressors) show a much flatter or slightly increasing trend, plateauing below 100 G W.

An overview of the simulation results is presented in Figure 4. For all pressures, the pulses could be compressed relatively close to their FT limit (Figure 4(d)). However, the power in the pulse pedestals increased considerably with pressure unless the phase shaper was used for compression. As a consequence, the highest peak power enhancements were obtained at pressures of 4.5 or 5.5 bar if dispersion terms were only compensated up to the third order (Figures 4(b) and 4(e)). The compression was most sensitive to TOD for $p=5.5$ bar. In this case, the peak power could be increased from 64 to 79 GW by reducing the line density of the gratings from 1500 to 1000 lines/mm and to 98 GW if mirrors and gratings were TOD-free. Higher-order phase compensation can be addressed to some degree by tailored dispersive mirrors or prism–gratings pairs. Figure 4(c) shows that even for the 1500 lines/mm grating compressor pair, the instantaneous power dropped quickly on both sides of the main pulses to the ${10}^{-9}$ level relative to the peak. A broad pedestal as shown in Figure 3 was not visible. We also note that the amplified stimulated emission (ASE) background is strongly suppressed by the nonlinear peak power enhancement and the eigenmode properties of the MPCs. The ASE divergence is generally different from the laser beam divergence, that is, ASE is not mode-matched. Therefore, the MPC acts as a spatial filter of ASE.

In summary, our simulations predict that the best operation point for the MM-MPC experiments is at peak power levels around 20% of the critical power of the nonlinear gas. At this point, 500-fold compression from 300 ps to 600 fs is predicted if TOD is at least partially compensated. Higher-order phase control appears clearly favorable for further peak power enhancement and the reduction of pulse pedestals in the few-ps time range. These pedestals could also be cleaned by nonlinear ellipse rotation or cross-polarized wave (XPW) generation^[ Reference Seidel, Pressacco, Akcaalan, Binhammer, Darvill, Ekanayake, Frede, Grosse-Wortmann, Heber, Heyl, Kutnyakhov, Li, Mohr, Müller, Puncken, Redlin, Schirmel, Schulz, Swiderski, Tavakol, Tünnermann, Vidoli, Wenthaus, Wind, Winkelmann, Manschwetus and Hartl^42– Reference Kaur, Daniault, Cheng, Tourneur, Tcherbakoff, Réau, Hergott and Lopez-Martens^44] in a second nonlinear compression stage that would enable TW peak powers and pulse durations of 30–50 fs^[ Reference Kaumanns, Kormin, Nubbemeyer, Pervak and Karsch^18, Reference Pfaff, Barbiero, Rampp, Klingebiel, Brons, Teisset, Wang, Jung, Jaksic, Woldegeorgis, Trunk, Maier, Saraceno and Metzger^19].

3.1 Design of the multi-mirror multi-pass cell

The simulations relied on an MPC that supports 1300 passes. However, state-of-the-art Herriott-type MPCs for spectral broadening are operated with only 40–60 passes. Increasing the number of passes through MPCs is likewise of interest for applications such as high-sensitivity absorption spectroscopy^[ Reference Robert^45– Reference Kong, Liu and Zhou^47] and optical refrigeration^[ Reference Gragossian, Meng, Ghasemkhani, Albrecht and Sheik-Bahae^48]. In principle, the proposed geometries could be transferred to spectral broadening, but mode-matching requirements in such experiments reduce the attainable number of passes. We investigated slightly astigmatic MPCs, which generate Lissajous spot patterns on the mirrors^[ Reference Herriott and Schulte^41]. The subsequently larger beam coverage of their surface area results in a significant increase of the number of passes through an MPC. One way to introduce a slight astigmatism is to arrange several spherical mirrors in a circle, allowing the beam to propagate in the MPC in a star-like path with a non-zero angle of incidence $\varphi \ne 0$ on the mirrors (Figure 5(a)). The focal lengths of the spherical mirrors then differ slightly between planes, being ${f}_\mathrm{tan}\approx \frac{\mathrm{\mathcal{R}}\cos \left(\varphi \right)}{2}$ in the tangential (horizontal) plane and ${f}_\mathrm{sag}\approx \frac{\mathrm{\mathcal{R}}}{2\cos \left(\varphi \right)}$ in the sagittal (vertical) plane^[ Reference Jenkins and White^49].

Figure 5

(a) Simulated beam trajectory inside the 11-mirror MPC. (b) Calculated number of roundtrips of non-clipping patterns as a function of the MM-MPC radius ${r}_\mathrm{CM}$ defined as the distance between mirrors and the center of the cell, for a coupling hole diameter ${d}_\mathrm{h}$ =10 mm. (c) Tolerance on ${r}_\mathrm{CM}$ as a function of the hole diameter ${d}_\mathrm{h}$ for three selected patterns A–C with 374, 297 and 165 passes, respectively. (d) Simulated reflection pattern B with 297 passes on input mirror m = 0, where the ellipses represent the beam size at 1/e ${}^2$ , and showing the central coupling hole. (e) Experimental reflection pattern for 297 passes.

.

Figure 5 Long description

Panel a. A ray tracing diagram of an M M - M P C with a radius r sub C M of 495 mm. Red laser beams enter and exit through a coupling hole at m equals 0, reflecting between ten numbered mirrors arranged in a star-shaped radial pattern.

Panel b. A scatter plot showing the Number of passes on the y-axis from 0 to 400 versus the M M - M P C radius r sub C M in mm on the x-axis. Horizontal blue segments represent stable configurations. Three specific points are highlighted: A in green at approximately 380 passes, B in red at 300 passes, and C in gold at 180 passes.

Panel c. A line graph showing Tolerance on r sub C M in micrometers on the y-axis versus Coupling hole diameter d sub h in mm on the x-axis. Three curves correspond to the points in panel b. Curve C reaches the highest tolerance of 250 micrometers, followed by B at 100, and A at 50. All curves peak between 10 and 15 mm.

Panel d. A circular plot of beam spot positions on a mirror. The x and y axes represent dimensions in mm from negative 20 to 20. Red circles are distributed in concentric rings around a central hole.

Panel e. A green-tinted photograph showing the physical mirror surface with white laser spots corresponding to the pattern in panel d.

Such setups have recently been studied for absorption spectroscopy applications. To our knowledge, only experiments with up to five mirrors have been demonstrated^[ Reference Hudzikowski, Głuszek, Krzempek and Sotor^46, Reference Kong, Liu and Zhou^47]. However, to reduce aberrations introduced by off-axis reflections on the spherical mirrors and to increase further the number of reflections, we investigated the design of an 11-mirror MPC. For predicting the beam paths, Kong et al.^[ Reference Kong, Liu and Zhou^47] solved the ray propagation in three dimensions using the vector reflection principle and then applied a particle swarm optimization algorithm. We use here an analytical formula based on the ABCD matrix formalism that strongly reduces the computational cost. The formula yields the same results as the numerical propagation (Appendix A.4), since the paraxial approximation is valid in our regime with small angles on the mirrors.

We consider an MM-MPC with $M$ mirrors arranged in a circle. The laser beam is coupled in and out via a hole at the center of mirror $m=0$ . In the paraxial approximation, the $X-Y$ coordinates of the nth reflection relative to the center of the mirror $m$ are then as follows:

(6a) $$\begin{align}{X}_{m,n}={X}_{\mathrm{max}}\sin \left[\left( nM+m\right){\theta}_x+ nM\pi \right],\end{align}$$

(6b) $$\begin{align}\hspace{-28pt}{Y}_{m,n}={Y}_{\mathrm{max}}\sin \left[\left( nM+m\right){\theta}_y\right],\end{align}$$

where $m=0,\dots, M-1$ is the mirror index, ${X}_{\mathrm{max}}$ and ${Y}_{\mathrm{max}}$ are constants that only depend on the input angle and ${\theta}_x$ and ${\theta}_y$ are angular advances that are solutions of $\cos {\theta}_x=1-\frac{D_\mathrm{M}}{\mathrm{\mathcal{R}}\cos \varphi }$ and $\cos {\theta}_y=1-\frac{D_\mathrm{M}\cos \varphi }{\mathrm{\mathcal{R}}}$ , where ${D}_\mathrm{M}$ is the distance between the centers of two consecutive mirrors. The formalism is identical to an MPC with two astigmatic mirrors^[ Reference Herriott and Schulte^41] apart from the $nM\pi$ -term in Equation (6a). Only three parameters define the beam pattern: the distance ${r}_\mathrm{CM}$ from the center of the MM-MPC to each mirror, and the horizontal and vertical input angles. Beyond that, the diameters of the mirrors ${d}_m$ and of the coupling hole ${d}_\mathrm{h}$ determine feasible spot patterns since the beam must neither clip during the coupling in and out of the MM-MPC, nor clip on the hole while being reflected from the mirror $m=0$ .

In order to determine ${r}_\mathrm{CM}$ and the input angles that provide fully coupled out patterns, we evaluated all possible trajectories using Equations (6a) and (6b). Assuming the mode-matching condition for $\varphi \to 0$ , we computed the beam radius on the surface of the MM-MPC mirrors. Then, we selected only patterns in which all the beam spots remained at least one beam radius away from the in-/out-coupling hole, for a given mirror diameter ${d}_m$ and a hole diameter ${d}_\mathrm{h}$ (Figure 5(b)). For each valid pattern, we define a tolerance on ${r}_\mathrm{CM}$ as the range from the minimal to the maximal value of ${r}_\mathrm{CM}$ where the pattern can be achieved without clipping. This tolerance can be optimized for a given mirror diameter ${d}_m$ by appropriately choosing the hole diameter ${d}_\mathrm{h}$ , as illustrated in Figure 5(c). This optimization helps in designing the coupling hole in order to minimize the sensitivity of the pattern on ${r}_\mathrm{CM}$ . In a proof-of-concept experiment, we used 11 spherical mirrors with $\mathrm{\mathcal{R}}=500$ mm and ${d}_m=50.8$ mm (Figure 5(a)), and targeted a pattern with 27 reflections per mirror, resulting in a total of 297 passes (Figure 5(d)). We chose to conduct the experiments with 2-inch diameter mirrors that could be coated in a single ion beam sputtering run. While this was useful to find efficient alignment strategies and to test the setup stability, we compromised the ability to achieve a setting with the simulated 1300 passes owing to the insufficient ratio of the coupling hole and mirror diameters. However, our predictions show that it is possible to reach about four times more passes by implementing the studied 11-mirror configuration with 3-inch mirrors (Appendix A.5). Consequently, we expect that experimentally achieving more than 1000 passes does not require an increase of alignment degrees of freedom but merely an increase of the mirror surface area.

3.2 Experimental demonstration of an 11-mirror multi-pass cell

While the beam paths through the MM-MPCs can be readily computed, the experimental implementation requires the control over many degrees of freedom, including the steering of the input beam and the position and orientation of each MPC mirror. Whether such a complex system could be aligned in practice was uncertain, as previous experimental work had been limited to configurations with only up to five mirrors^[ Reference Hudzikowski, Głuszek, Krzempek and Sotor^46, Reference Kong, Liu and Zhou^47]. Fortunately, the degrees of freedom can be mostly decoupled. Firstly, the mirrors were placed using a precision rotation stage at the MM-MPC center connected to an adjustable rod. In order to set the angle of the mirrors, the input beam was centered on each mirror, that is, ${X}_{\mathrm{max}}={Y}_{\mathrm{max}}=0$ . This was achieved with high precision by generating first vertical ( ${X}_{\mathrm{max}}=0$ ) and second horizontal ( ${Y}_{\mathrm{max}}=0$ ) beam patterns by changing the input angles. By comparing the beam patterns for ${X}_{\mathrm{max}}\ne 0$ and ${Y}_{\mathrm{max}}\ne 0$ with the numerical results, an estimation of ${r}_\mathrm{CM}$ can be extracted. The length of the rod was then fine-tuned to the target position using a translation stage. The procedure was then repeated until the targeted pattern with 297 passes was obtained. Figures 5(d) and 5(e) show good agreement between the simulated and the experimentally obtained beam patterns. This confirms the feasibility of aligning a complex MM-MPC with 11 mirrors, despite the high number of degrees of freedom involved.

To further investigate the spectral broadening concept, we used a burst-mode Yb:YAG laser^[ Reference Seidel, Balla, Li, Arisholm, Winkelmann, Hartl and Heyl^32], which was reconfigured to emit about 1 ps pulses with up to 1 mJ pulse energy at 1030 nm, and a Pharos laser (Light Conversion) emitting 210 fs pulses with up to 200 μJ pulse energy. A high-energy long-pulse laser was not available at the time of the experiments. The used lasers were located several meters away from the MM-MPC setup. For day-to-day operation, we stabilized the input beam position by two motor-controlled steering mirror mounts and simultaneous far- and near-field beam position measurements. The MPC mirrors were not actively controlled and did not have to be readjusted after their position was set. The transmission of the 297-pass MPC was about 88%, in good agreement with the specified reflectance of the used dielectric mirrors (Optoman, $R>99.95\%$ ). Firstly, we investigated if the introduced astigmatism deteriorates the beam quality of the transmitted beam. The M ${}^2$ factor was measured using a Spiricon M ${}^2$ -200s in the standard measurement mode. For the burst-mode laser operated at low power, no difference was observed between the input and output beam profiles (Figures 6(a) and 6(b)), nor in the measured M ${}^2$ values, amounting to 1.2 in the x-direction and 1.1 in the y-direction for both beams (Figures 6(c) and 6(d)). There is no additional astigmatism at the output, and the angle of incidence on the mirrors is small enough to avoid beam degradation from other aberrations such as coma. We attribute the preservation of the very good beam properties to meeting the so-called re-entrant condition^[ Reference Herriott, Kogelnik and Kompfner^50], that is, to coupling into and out of the MM-MPC at the same spot. As a consequence, the output beam was an image of the input beam, and thus the beam shape was preserved in the linear propagation case despite the astigmatism. During propagation through the MPC, however, the beam ellipticity oscillates, which originates from the inequality of ${f}_\mathrm{tan}$ and ${f}_\mathrm{sag}$ , and hence the different eigenmode beam sizes in the x- and y-directions.

Figure 6

Beam profiles at the first measurement position of the M ${}^2$ scan (‘far-field’) and at the focus of the (a) input and (b) output beams of the burst-mode laser at low input pulse energy. M ${}^2$ factor measurements of the (c) input and (d) output beams at low input pulse energy.

Figure 6 Long description

Top left panel labeled a shows two heatmaps for the input beam: far field on the left with an elliptical intensity profile and a 0.5 millimeter scale bar, and focus on the right with a circular profile and a 100 micrometer scale bar. Top right panel labeled b shows output beam heatmaps: far field on the left with a similar elliptical profile and focus on the right with a slightly elongated spot, both with the same scale bars as the input. A vertical color bar to the right indicates intensity from 0.0 to 1.0 in arbitrary units. Bottom left panel labeled c is a line graph with x axis labeled z in millimeters from negative 100 to positive 100, y axis labeled radius in millimeters from 0 to 1.2. Two curves are plotted: M squared sub x equals 1.22 in blue and M squared sub y equals 1.12 in red, both showing a V-shaped trend with a minimum at z equals 0. Bottom right panel labeled d is similar but for the output beam, with the same M squared values and a wider y axis up to 2 millimeters. The curves again show a V-shaped minimum at z equals 0.

3.3 Spectral broadening in an air-filled 11-mirror MPC

We studied spectral broadening in air to benchmark our simulations and to compare the behavior of the implemented MM-MPC with standard Herriott-type MPCs. In recent MPC experiments with air as a nonlinear medium, spectral broadening factors of about 5 for 670 fs, 3 GW peak power pulses^[ Reference Omar, Vogel, Hoffmann and Saraceno^51]; of about 20 for 1.1 ps, 7 GW peak power pulses^[ Reference Schönberg, Rajhans, Escoto, Khodakovskiy, Hariton, Farace, Põder, Raab, Westerberg, Merdanov, Viotti, Arnold, Leemans, Hartl and Heyl^52]; and of about 10 for 185 fs, 740 MW peak power pulses^[ Reference Feng and Bromage^53] have been reported. We obtained a broadening factor of 15 with pulses of only about 260 MW peak power (Figure 7(a)). As expected from the spectral broadening limit considerations described in Section 1.3, this is far away from the simulated spectral broadening factors with 300 ps input pulses. Nevertheless, the experiments indicate the great potential of MM-MPCs to achieve ultrahigh B-integrals. For interpreting the results, the application of Equation (1) is most instructive, although this neglects saturation effects. Both ${I}_{\mathrm{eff}}$ and $1/{\overline{L}}_\mathrm{NL}$ are proportional to ${P}_\mathrm{p}$ . In addition, ${n}_2$ is similar in all experiments, because the build-up of the Raman response is faster than the durations of the pulses used in the experiments^[ Reference Feng and Bromage^53]. Consequently, evaluating the ratio between the broadening factor and the peak power constitutes a good measure to analyze the MPC geometry impact on the spectral broadening performance. It essentially represents the ratio of propagation length L to the effective pass-averaged beam area. With a spectral broadening factor of 58 GW⁻¹, we achieve clearly the highest ratio. The experiments in Ref. [Reference Feng and Bromage53] demonstrate a ratio of 13 GW⁻¹ because they minimized ${\overline{L}}_\mathrm{NL}$ by increasing the peak intensity up to the ionization limit. For ${P}_\mathrm{p}>740$ MW they observed increased transmission losses and a spectral blue shift, indicating the onset of spurious ionization. Although ${\overline{L}}_\mathrm{NL}$ was optimized, the physical propagation length $L$ through the two-mirror Herriott-type MPC was much lower than $L$ for the 11-mirror MPC demonstrated here. This highlights the shortcoming of standard MPCs, which MM-MPCs can overcome. The MPC with two concave and additional flat mirrors reported by Schönberg et al.^[ Reference Schönberg, Rajhans, Escoto, Khodakovskiy, Hariton, Farace, Põder, Raab, Westerberg, Merdanov, Viotti, Arnold, Leemans, Hartl and Heyl^52] also enables extremely large physical propagation lengths. However, the folded MPC geometry limits the peak intensity to the mirror damage threshold being much lower than the gas ionization threshold. Consequently, only a spectral broadening factor of 2.9 GW⁻¹ was reached. In the multi-mirror geometry presented here, the light is focused after each reflection, and thus much shorter nonlinear lengths are attainable. Eventually, Omar et al.^[ Reference Omar, Vogel, Hoffmann and Saraceno^51] set up a concave–convex MPC that gave weight to compactness rather than to high B-integrals. In this case $L$ was short because of the two-mirror geometry and ${\overline{L}}_\mathrm{NL}$ was long because of the intensity limit set by the convex mirror, and thus a lower spectral broadening factor of 1.7 GW⁻¹ was accomplished.

Figure 7

Spectral broadening with a 297-pass MPC exploiting the nonlinearity of air. Experiments were performed with (a) a 1 ps burst-mode Yb:YAG laser and (b) a 210 fs Pharos laser. The input energies are displayed in the legends. The spectra are offset by 7 dB for clarity.

Figure 7 Long description

The top panel labeled a shows spectral power in d B on the y-axis from minus 20 to 60 and wavelength in nanometers on the x-axis from 1000 to 1070. Eight colored lines represent input pulse energies of 2, 88, 106, 120, 140, 160, 180, 200, 240, and 280 microjoules, with colors progressing from yellow to purple. As input energy increases, the spectral width broadens, with the broadest and highest spectrum at 280 microjoules. The bottom panel labeled b shows spectral power in d B from minus 20 to 70 and wavelength from 960 to 1100 nanometers. Eleven colored lines represent input pulse energies of 2.6, 7, 10, 15, 20, 25, 30, 35, 40, 45, and 50 microjoules, again with colors from yellow to purple. The right edge is marked spectrometer cut-off. As input energy increases, the spectra broaden and the peak power increases, with the broadest spectrum at 50 microjoules. Both panels show that higher input energies result in greater spectral broadening.

Beyond the comparison of the obtained FT limits, we measured a pronounced peak around 1030 nm and a red-shifted asymmetric spectrum similar to the observations in Refs. [Reference Omar, Vogel, Hoffmann and Saraceno51,Reference Schönberg, Rajhans, Escoto, Khodakovskiy, Hariton, Farace, Põder, Raab, Westerberg, Merdanov, Viotti, Arnold, Leemans, Hartl and Heyl52], which is due to the Raman response of air. When using the 210 fs input pulses, the 1030 nm peak was not visible in the broadened spectra, probably due to better higher-order chirp management of the laser, in agreement with the results from Ref. [Reference Feng and Bromage53]. Already at 25 μJ input energy ( $\approx$ 110 MW peak power), a minimum FT limit of about 30 fs was reached (Figure 7(b)). At higher input pulse energies, spectral components extended beyond the reflection band of the mirrors, leading to losses and an increase in the ${\mathit{M}}^2$ factor. The Raman response of the molecular gas had an even more severe impact on the beam quality as we show in Appendix A.6, where spatio-spectral couplings that were observed in the experiment are reproduced in the simulations. By turning off the Raman scattering contribution in the simulation, the spatio-spectral couplings disappeared. We conclude that using an atomic gas as the nonlinear medium in the MPC is essential to avoid the beam degradation caused by Raman scattering.