High-energy bow tie multi-pass cells for nonlinear spectral broadening applications

Multi-pass cells (MPCs) have emerged as very attractive tools for spectral broadening and post-compression applications. We discuss pulse energy limitations of standard MPCs considering basic geometrical scaling principles and introduce a novel energy scaling method using a MPC arranged in a bow tie geometry. Employing nonlinear pulse propagation simulations, we numerically demonstrate the compression of 125 mJ, 1 ps pulses to 50 fs using a compact 2 m long setup and outline routes to extend and broadening factors. Our analytical analysis shows, that highly efficient MPC-based pulse-post compression at large compression ratios can be extended to 100 mJ pulse energies and beyond using a table-top setup. More generally, our results extend the concept of multi-pass spectral broadening to non-standard MPCs demonstrating a viable route to greatly expand the parameter space of multi-pass post-compression.


Introduction
Ultrashort laser pulses play a crucial role in many fields ranging from time-and frequency domain spectroscopy and strong-field physics to surgery and welding [1]. The amplification of ultrashort pulses set constraints on their duration imposed by bandwidth limitations. The most common ultrafast high-power laser platforms are nowadays titanium:sapphire (Ti:Sa) and ytterbium (Yb)-based systems, providing amplified pulses with durations ranging from a few tens of fs (Ti:Sa) to hundreds of fs (Yb). Two different methods are commonly employed to overcome these pulse width limits: parametric amplification [2] and post-compression [3]. The latter method relies on nonlinear spectral broadening and either simultaneous or subsequent temporal compression. Various spectral broadening methods have been commonly used including nonlinear propagation within a single [4] or multiple plates [5], within solid-core [6] and photonic crystal fibers [7], hollow-core capillaries (HCCs) [8], filaments [9] or slab waveguides [10]. Recently, a new method based on nonlinear spectral broadening within multi-pass cells (MPC) was introduced [11,12]. The method offers high transmission [13,14], high-power handling [14,15], excellent beam quality [13,16] and large compression ratios [16,17] while being easily adaptable to a large variety of laser pulse parameters including durations ranging currently from 10 ps [18] to few-cycles [17,19,20] as well as pulse energies ranging from a few µJ [21] to more than 100 mJ [22].
A key challenge is the adaption of high-performance post-compression methods to high pulse energies. In particular, advancing post-compression methods into the Joule-class pulse energy regime could enable the application to strong-field physics with the potential to overcome peak power limitations of chirped-pulse amplification [23]. While approaches like HCC, MPC or filament-based methods can in principle be scaled to very high pulse energies [24], the required setup-sizes become impractically large. Alternative approaches have employed thin films as nonlinear media to spectrally broaden Joule-class pulses [23]. These methods suffer, however, from reduced beam quality compared to HCC and MPC-based methods and only small compression factors could be reached so far. Spatial and/or temporal pulse multiplexing methods have also been suggested for pulse energy scaling [25,26], increasing, however, system complexity especially at large multiplexing factors.
Here we outline basic geometrical energy scaling principles for MPCs and introduce a novel pulse energy scaling method. Our approach enables compact setups while maintaining advantageous properties of standard MPCs such as high throughput, high compression ratios and excellent beam quality. Using nonlinear pulse propagation simulations, we demonstrate compression of 125 mJ, 1 ps pulses at 1030 nm to 50 fs using a compact setup of 2 m length while maintaining a reasonable fluence at all mirrors. Moreover, via analytical energy scaling relations, we present routes to scale our approach to Joule-class pulses and present a first numerically tested example configuration at 1 J.

Pulse energy limits of standard MPCs
We start by considering a standard Herriott-type MPC [27] with two identical concave mirrors with radius of curvature R placed at a distance L as discussed e.g. in [28]. A reentrant beam pattern with N round trips through the MPC can be reached by obeying the simple relation [27,29]: with k = 1, . . . , N − 1 denoting a variable integer which is related to the angular advance of consecutive beam spots on one MPC mirror ξ = 2πk/N [27]. In order to obtain a q-preserving MPC ensuring similar nonlinear pulse propagation characteristics for each round trip, the input beam needs to be mode-matched to the eigenmode of the MPC, which is identical to the mode of a corresponding cavity. Considering basic properties of a two-mirror cavity, the fundamental Gaussian eigenmode can be derived by solving the identity condition for the Gaussian beam parameter q = (M 1 q + M 2 )/(M 3 q + M 4 ) with M i denoting the elements of the ABCD matrix for a single round trip through the cavity. The corresponding focal spot radius w 0 of a linearly mode-matched beam (neglecting the impact of Kerr lensing on the MPC eigenmode) can be written as: expressed as a function of the integer variables k and N for a reentrant beam pattern or more generally as a function of the configuration parameter C = L/R, which can also be used for non-reentrant MPCs. Mode matching is possible for stable resonators, i.e. for 0 < C < 2. Equation (2) also provides an approximate solution for the typical operation condition close to the outer stability edge at k = N − 1 in the limit N → ∞.
Here and in the following, we use the simplified notation k/N → 1 to express this limit. In this limit, the focal spot size w 0 is minimized and the beam spot size at the mirrors w m reaches a maximum. Using simple Gaussian beam equations w m can be derived from equation (2) yielding: In order to spectrally broaden an input pulse, a nonlinear medium such as a gas or a glass plate can be inserted into the MPC. Considering self-phase modulation-based pulse propagation governed by the intensity-dependent refractive index change ∆n = n 2 I, the pulse energy E can be increased while maintaining key broadening characteristics by decreasing the nonlinearity. For gas-filled MPCs, the gas pressure p provides a simple way to tune the nonlinear refractive index n 2 ∝ p. For constant setup size, energy up-scaling is however limited by the laser-induced damage threshold (LIDT) F th of the MPC mirrors and by the focus peak intensity I 0 setting a limit via ionization at a threshold intensity I th . Peak fluence at the MPC mirrors and peak focus intensity of a linearly mode-matched beam can be calculated as: The maximum laser pulse energy thus reads as: Taking the example of an MPC with R = 1 m operated close to the stability edge with k/N = 14/15, λ = 1030 nm and pulse width τ = 1 ps, and considering a mirror damage threshold of F th = 500 mJ cm −2 , we obtain an energy limit of 24.5 mJ and a corresponding peak focus intensity of 4.6 · 10 13 W cm −2 at an MPC length of L = R · C = 1.978 m.
In order to increase the pulse energy beyond this limit, the inequalities in equation (6) indicates multiple tuning parameters: C (tunable via k/N), λ and R. For k/N → 1, F m decreases but I 0 increases. While the fluence-based pulse energy limit can thus be increased, ionization effects in the focus limit the maximum pulse energy for gas-filled MPCs. In addition, the MPC imaging properties at the stability edge k/N = 1 prevent homogenization of the spatial beam profile during the broadening process. Earlier works reporting the operation of MPCs at the stability edge (equivalent to a 4 f imaging geometry) yielded reduced spatial beam quality characteristics at rather low compression ratio [30]. Both limits defining E max scale linearly with λ, thus allowing larger pulse energies for longer wavelengths if F m can be kept constant.
The practically most effective pulse energy tuning option for a standard MPC is provided by the setup size, represented by R in equation (6), showing a straightforward linear scaling relation for the maximum pulse energy. Scaling the pulse energy linearly with setup length enables not only a constant intensity at the mirrors and in the focus, it further ensures fully scale-invariant spectral broadening characteristics provided that the pressure is reduced linearly with increasing pulse energy. The fundamental principle behind this energy scaling method is outlined in a more general context in [24] and can be motivated by basic scaling properties of the nonlinear wave equation, which can be used to describe the nonlinear pulse propagation process inside the MPC. In paraxial approximation and considering a Kerr-induced nonlinear polarization P NL , this equation can be written for the field envelopeÊ as [24]: where n, λ, ϵ 0 and ω, ρ denote refractive index, vacuum wavelength and permittivity as well as angular frequency and gas density, respectively. ∆ ⊥ denotes the transverse radial Laplace operator. Taking into account linear pressure dependencies (n − 1) ∝ ρ,P NL ∝ ρ and neglecting the weak pressure dependence of n in the denominator of the diffraction and the nonlinear terms, it can be shown that any solution of equation (7),Ê(rη, zη 2 , ρ/η 2 ) is fully scale-invariant if longitudinal coordinates z, radial coordinates r and the gas density are scaled appropriately with pulse energy E ∝ η 2 , using a general scaling parameter η.
We thus obtain fully scale-invariant spectral broadening characteristics which do not depend on the laser pulse energy if setup size and gas density are scaled appropriately with pulse energy [24]: Most importantly, there is no fundamental upper limit for the pulse energy via geometrical setup size scaling. In practice the limit is simply the setup size itself, as illustrated in figure 1. While setup lengths in the order of 1 m can be employed to compress 1 ps pulses with about 10 mJ, the setup length would increase to 20 m for 200 mJ, making further energy scaling impractical. Conversely, energy down-scaling is possible via a geometrical setup size reduction as indicated in figures 1(b) and (c) for 0.5 mJ, spectrally broadened in a 5 cm setup. Similar scaling principles can also be applied to bulk-based MPCs. As the density of the nonlinear medium cannot easily be changed, deviations from perfect scale-invariance can however be expected.

Bow tie multi-pass cells
We now introduce a pulse energy scaling route, which circumvents the above discussed setup size limitations. We employ known design principles for optical resonators, where large mode sizes can be achieved in a folded cavity design. Applying those principles and placing all folding mirrors in sections of large beam sizes, allows us to construct MPCs of compact size which support high pulse energies.
One of the most simple generalizations of a two-mirror resonator is a four-mirror resonator, often arranged in a so-called bow tie configuration. We here consider a corresponding bow tie MPC (BT-MPC) using two identical concave mirrors and two (or more) plane mirrors. This way, an MPC can be constructed for which the optical path L 1 from mirror 1 to mirror 2 has a different length compared to the returning path L 2 ̸ = L 1 , as illustrated in figure 2. A typical bow tie configuration would be reached when L 2 is folded by two plane mirrors. Generally, a BT-MPC can be arranged in different geometrical configurations. Figure 2 displays an example configuration utilizing a linear multipass pattern at the focusing mirrors in contrast to standard Herriot-type MPCs, which are usually arranged with circular pattern. As discussed by Herriott, elliptical or even linear patterns are easily possible depending on the input beam conditions [27].
Similar to equation (1), a general equation describing the solutions for a reentrant beam pattern in a BT-MPC with two identical concave mirrors of radii R and two or more planar folding mirrors can be derived. A reentrant MPC can be obtained if (M * ) N =Ĩ, where the ABCD matrixM * describes a single round trip through the MPC andĨ denotes the unity matrix with its elements A N = D N = 1. An equation describing the reentrant MPC can be found via the Gouy phase ξ accumulated during N round trips through the MPC [31]: With (M * ) N =Ĩ, equation (9) predicts the condition ξ N = 2πk with k denoting a variable integer. Assuming an optimized configuration where the beam acquires the same Gouy phase per pass through the MPC, we obtain for a single round trip: By solving equation (9) for N = 1 taking into account equation (10) and a suitable (M * ) describing the BT-MPC we obtain: with C 1,2 = L 1,2 /R defined analogously to equation (1). The solutions of equation (11) figure 3 (dashed black lines) using the example of N = 15 for k = 1, 2, . . . , 7. We consider in the following the parameter range C 1,2 > 1 with a tightly focused short beam path along L 1 and a weakly focused beam along L 2 , i.e. L 2 > L 1 . It can be shown that equation (11) describes lines with constant stability parameter G. Here, G is defined analogously to resonator optics as G = |(A + D)/2| = cos 2 (πk/N) with A and D denoting elements of an ABCD matrix describing a single round trip through the MPC. Figure 3 also displays the accumulated Gouy phase ξ after a single pass through the MPC. The white areas mark regions outside the stability range (G > 1). Equation (11) convergences towards equation (1) for L 1 = L 2 , representing the standard two-mirror Herriott cell. At the stability edge for k/N = 1, the BT-MPC converges towards a 4 f imaging configuration as explored in [30]. We now calculate basic properties of the BT-MPC assuming linear mode matching, i.e. neglecting Kerr lensing effects in the MPC. This simplification enables the derivation of simple analytical equations. Depending on the operation regime, Kerr lensing effects need to be taken into account to allow an accurate estimate of the MPC parameters [32,33], as shown in section 4. The focal spot radii w 01,2 of a beam linearly mode-matched to the fundamental Gaussian eigenmode of a BT-MPC can be calculated as: where w 01,2 denote the two foci with w 02 > w 01 for L 2 > L 1 . The above equations display approximated solutions for k = (N − 1)/2 in the limit N → ∞ considering an odd integer N. Again, we use a simplified notation to express this limit: k/N → 1/2 with N odd. The limit denotes the operation condition for which the spot size ratio w 02 /w 01 and the beam sizes at all mirrors are maximized. Note that the mirror spot size for a standard MPC is maximized for k/N → 1 instead. For large w 02 , the beam along L 2 appears approximately collimated, i.e. w m ≈ w 02 with increasing beam diameter as L 2 increases. A collimated beam along L 2 brings along an important advantage of the BT-MPC: the beam path can be folded without increasing the maximum mirror fluence, providing great opportunities for the construction of compact MPC systems as the system length is only determined by L 1 ≈ R. Assuming linear mode-matching, the fluence at the focusing mirrors can be estimated as: The corresponding maximum focus peak intensity occurring at the focus along L 1 reads as: While the beam spot sizes at the mirror surfaces increases with L 2 causing a decreasing fluence, the focus peak intensity at the tighter focus intersected by L 1 increases. Ionization at the tight focus of the BT-MPC can be avoided by operation inside a vacuum chamber at a low residual gas pressure. For spectral broadening one or multiple glass plates can be placed within a section of the MPC with large beam diameter. Thus, the fluence on the mirrors is the only pulse energy limit: With increasing L 2 , E * max increases reaching values above 1 J for L 2 = 100 m and L 1 ≈ 1 m as illustrated in figure 4. In figure 4(c), the fluence-limited E * max is displayed as a function of C 2 for different system lengths L 1 together with the corresponding peak power reached assuming for simplicity a Gaussian pulse shape and compression to 50 fs. At large asymmetries L 2 /L 1 , the collimated beam path along L 2 has to be folded many times in order to keep the system compact. This is possible using today's multi-layer mirror technology supporting pulse durations of <30 fs with losses at the few-parts per million level, thus enabling system transmissions above 90% even for a large number of mirror reflections.
For increasing asymmetry L 2 /L 1 , the stability range of a BT-MPC decreases, as visible in figure 4. This implies increasing stability demands on the setup, in particular with respect to the distance L 1 . Small changes of L 1 can translate into mode-matching issues. Depending on the exact geometry, variations of L 1 from round trip to round trip may occur, in particular if geometries using linear multi-pass patterns, as illustrated in figure 2, are used. In addition, it is advantageous to design the BT-MPC such that all angles of incidence at the focusing mirrors are kept small in order to minimize astigmatism. While these constraints will need to be considered for a experimental setup design, they are not setting a fundamental limit for the proposed scaling approach using BT-MPCs. Possible solutions to circumvent these constraints may include the division of the focusing and/or folding mirrors into multiple small mirrors, providing increased flexibility for beam angle control, enabling e.g. to place all mirrors in a cylindrically symmetric arrangement.

Numerical verification of the BT-MPC scheme
In order to test the applicability of BT-MPCs to high peak power spectral broadening, we conducted two-dimensional (2D+1) nonlinear pulse propagation simulations with the SISYFOS code assuming radial symmetry and considering the Kerr effect (Raman and ionization are not included as these effects are not relevant for the parameters considered). Example BT-MPC configurations were tested with a full three-dimensional (3D+1) version of the code, confirming good agreement with the radially symmetric code. The code was previously applied to bulk and MPC broadening simulations [32,34]. We selected thin silica plates as nonlinear media and consider a MPC placed inside a vacuum chamber to avoid ionization at the focus. The fluence at the Kerr plates as well as the Kerr-lensing effect were minimized by placing the silica  plates close to the curved mirrors in the long arm of the BT-MPC. For the simulations, a BT-MPC equivalent cavity was considered which is shown in figure 5. Virtual plane mirrors are introduced at the symmetry planes.
First, we simulate spectral broadening of Gaussian pulses with 1 ps duration and 125 mJ energy considering an initially Gaussian spatial beam shape. In the absence of nonlinearity, the BT-MPC consists of focusing mirrors with a radius of curvature of 2 m and an unfolded arm length of L 2 = 10 m. In order to reach a large spot size on the mirrors we consider k ≈ N/2, i.e. L 1 = R + δ with δ = 2 mm. The corresponding peak intensity in the long arm was 360 GW cm −2 (360 mJ cm −2 ). To achieve a minimum fluence level at the mirror when Kerr nonlinearity is introduced, we apply nonlinear mode-matching methods [32,33]. Nonlinear mode-matching can be reached by adjusting the input beam parameters e.g. via adjusting the input spot size at the virtual plane mirror M3. For the BT-MPC geometry considered here, this simple approach is not sufficient to reach a fluence level at the mirrors below damage threshold. Therefore, additional cavity length adjustments are required. Extending the long arm length to L 2 = 35 m results in spatially homogeneous broadening and only moderate fluence enhancements compatible with the damage thresholds of the optics. The MPC length lies in the range of demonstrated solid-state laser oscillators with a repetition rate of 8.6 MHz [35]. The implementation of such a BT-MPC therefore appears feasible. Folding of the long arm can lead to a physical length of the cavity of only about 2 m. After 34 round trips in the cavity, a broadening factor of 20 is reached with two 1-mm thick Kerr media placed symmetrically close to the focusing mirrors. A large round trip number is chosen to reduce the B-integral per pass required for the targeted compression factor, as a reduced B-integral simplifies nonlinear mode-matching. The resulting broadened spectrum is shown in figure 6(a). We test the compressibility of the pulses by simulating a dispersive mirror compressor. A flat group delay dispersion (GDD) of −400 fs 2 per bounce is added to the full electric field obtained from the MPC simulation until the peak power was maximized. After 25 bounces a near transform-limited pulse with a duration of 50 fs and a peak power of 1.3 TW is reached (b), assuming an MPC transmission of 87% and a lossless compressor. Figures 6(c) and (d) show spatially resolved spectrum, beam profile and homogeneity parameter parameter (defined as in [21]), indicating excellent spatial output beam quality. The numerically calculated M 2 value of the output beam is M 2 = 1.15. Direct calculation of M 2 is very sensitive to numerical noise in the winds of the spatial beam profile, we therefore use a method mimicking a realistic experimental scenario as implemented by commercially available M 2 measurement tools, i.e. considering propagation of the output beam through a focus while applying spatial apertures with a diameter equal to twice the knife edge beam width at a 10/90 intensity fraction.
In a second study, we increase the pulse energy from 125 mJ to 1 J. In order to keep the fluence on the mirrors comparable to the previous example, a long arm length of L 2 = 75 m is required in the linear propagation regime if δ is reduced to 0.1 mm. Nonlinear mode-matching for a 1 mm thick substrate per half-cavity results in a long arm length of 276 m, i.e. a relatively large extension comparable to the 125 mJ case. While this optical path length reaches the limits of earlier demonstrated oscillator lengths [35], it should be noted that an MPC has reduced stability requirements compared to oscillators as no longitudinal modes are formed. Figure 6(e) shows the spectral broadening of the 1 ps pulses after 14 round trips. Compression is simulated taking into account dispersive mirrors with a GDD of −1000 fs 2 . A pulse duration of 99 fs (96 fs Fourier transform limit) was achieved by adding a total GDD of −15 000 fs 2 . The resulting peak power is 5.6 TW under the assumption of 94.5% total transmission (f). Again, an excellent spatial beam quality is achieved at the output (g,h) corresponding to M 2 = 1.08.

Conclusion
We have discussed general pulse energy scaling principles and limitations for MPCs employed for spectral broadening applications and introduced a new MPC type. The new MPC design enables large beam spot sizes at all mirror surfaces and thus operation at high pulse energies while keeping the setup size compact. In contrast to other guided-wave spectral broadening approaches, our method supports very high spatial beam quality despite the large pulse energies and broadening factors. Our analytical analysis shows, that highly efficient MPC-based pulse-post compression at large compression ratios can be extended to 100 mJ pulse energies and beyond using a table-top setup. More generally, our results extend the concept of multi-pass spectral broadening to non-standard MPCs demonstrating a viable route to greatly expand the parameter space of multi-pass post-compression.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.