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Abstract
The technique of enhanced frequency chirping can be implemented in multi-pass cells to significantly improve the temporal contrast without a penalty in efficiency. In contrast to waveguides, multi-pass cells offer the unique opportunity to tailor the nonlinear interaction, as spectral broadening and dispersion can mostly be spatially separated. By including additional dispersion through the multi-pass cell mirrors, the pulse form can be controlled, which leads to a decrease of the spectral modulations. A pulse possessing this smoother spectrum can be compressed to reach higher peak powers and have more energy in the main feature compared to classical multi-pass cells.
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1. INTRODUCTION
Multi-pass cells (MPCs) are a tool for post-compression of ultrashort laser pulses based on spectral broadening using the nonlinear effect of self-phase modulation (SPM) [1]. In comparison with Ti:sapphire lasers, Yb-based laser systems have shown a power scalability in the kW regime [2], but are typically limited in pulse duration to values in the order of a couple hundred femtoseconds and therefore need additional post-compression to compete with Ti:sapphire lasers in regard to pulse duration. MPCs have successfully achieved compressions of average powers greater than 1 kW [3], compressions to pulse durations in the few-cycle regime [4,5], and broadening with compressibility of pulse energies greater than 100 mJ [6]. Ultrafast, high power lasers are desired for the generation of secondary sources such as X-ray [7,8] and THz radiation [9,10] or the acceleration of particles in a plasma [11].
The compressed laser pulses from SPM-based post-compressions have pre- and post-pulses due to the intrinsic mechanism of SPM. These not only take away energy from the main feature and thereby reduce the efficiency of the cell (since the total input energy is no longer used in the main pulse), but are also harmful for experiments like laser driven particle accelerators because the pre-pulse interaction with the matter changes the initial target state [11]. It is therefore desired to reduce the energy content of the side pulses. Here, we describe how using the technique of enhanced frequency chirping (EFC) in MPCs can improve the pulse contrast by reducing the energy in the side pulses while simultaneously increasing the peak power of the main pulse.
2. MANIPULATION OF PULSE FORM DURING BROADENING PROCESS
A. Temporal Contrast and Enhanced Frequency Chirping
For the characterization of temporal contrast, we consider two figures of merit: first, the power ratio ${P_{{\rm peak}}}/{P_{{\rm pre} - {\rm pulse}}}$, which is especially important for experiments in which the intensity of the pre-pulse is high enough to change the target properties before the main pulse arrives; second, the energy ratio, which is defined as the share of energy located between the two local minima enclosing the main temporal lobe of the pulse. The optimum for the energy ratio is therefore 100%.
A Gaussian laser pulse propagating through a medium in the normal dispersion regime will be broadened in time but not spectrally due to group velocity dispersion (GVD) and will be linearly chirped by the second order dispersion. An equivalent pulse exposed to SPM will be broadened spectrally but not in time and a nonlinear chirp is created. Exposure to adequate values of GVD and SPM at the same time leads to the effect of dispersive self-phase modulation (DSPM), which changes the pulse shape (and therefore the spectrum) and is the basis for enhanced frequency chirping [12].
The technique of EFC was introduced in 1982 by Grischkowsky and Balant [12] who described the potential of peak power boost improvement in fibers. The peak power boost is the ratio of the peak power of the compressed pulse after spectral broadening compared to the peak power of the input pulse. By choosing the fiber parameters for optimal DSPM, the peak power boost of a pulse can be increased, because the nonlinear form of the SPM chirp is pushed towards the pulse edges, leaving a longer time with close to linear chirp in the center of the DSPM-formed pulse and less pronounced spectral modulations [12].
Other contrast improvement methods that increase the power ratio such as nonlinear ellipse rotation [13], usage of saturable absorbers [14], or spectral filtering [15] all typically come with a reduction of the total energy [16,17]. Since the goal of post-compression is a maximization of the peak power, this additional loss of energy is unfortunate. While the influence of input chirp has been thoroughly investigated [16], DSPM in MPCs and its impact on the pulse contrast has not yet been considered. Because the energy previously deposited in side pulses is here located in the main pulse, EFC promises to be a good addition to the topic of temporal pulse contrast improvement. MPCs are especially suited for EFC since they offer unique possibilities to realize its implementation.
B. Application of EFC in MPCs
The method of EFC has been used in the fiber community since its discovery to manipulate the pulse shape and compression mainly by changing the fiber length [18]. We now show an adaption of the EFC technique that is employable in MPCs and discuss the resulting pulse contrast improvements in these systems.
There are multiple differences between the propagation of a pulse in a fiber and in an MPC that prevent a simple transfer of the method. In a gas filled MPC, the effect of SPM is not distributed homogeneously throughout the propagation but is stronger in the focus region and weaker close to the mirrors. The GVD as the most dominant part of the total dispersion in a classical MPC comes from the interaction with the medium the beam travels through. An increase in gas pressure leads not only to an increased GVD but also a stronger nonlinearity. In addition to this, with advanced propagation, the spectrum is already broadened, and therefore, the impact of GVD increases since more frequencies are at hand.
An MPC, however, offers the possibility to use mirrors with a tailored dispersion via a coating with custom phase design, allowing to regulate the systems GVD almost arbitrarily. Since the dispersion is added in a single step, it is referred to as group delay dispersion (GDD) instead of GVD.
Due to the structure of an MPC, the common definitions of ${L_{\rm D}}$, ${L_{{\rm NL}}}$ and the resulting soliton number $N$ are not suitable and have to be adapted. To characterize the relative impact dispersive and nonlinear effects have on a pulse propagating in an MPC, we therefore define the dimensionless parameters ${Z_{\rm D}}$ and ${Z_{{\rm NL}}}$, characterizing half a roundtrip, as
where $\tau$ is the pulse duration, ${\beta _2}$ is the GVD parameter of the gas at the center wavelength, ${L_{{\rm MPC}}}$ is the length of the MPC, GDD is the GDD value of the dispersive mirror at the center wavelength, ${w_0}$ is the beam radius at the focus, ${n_2}$ is the nonlinear index of the gas, $\lambda$ is the wavelength, and $E$ is the pulse energy. Smaller values of ${Z_{\rm D}}$ and ${Z_{{\rm NL}}}$ represent a stronger impact of the effect. The new parameter $N = \frac{{{Z_{\rm D}}}}{{{Z_{{\rm NL}}}}}$ describes the ratio of the two effects. For $N \ll 1$, dispersion dominates and the pulse only broadens in time. For $N \gg 1$, the nonlinear effects dominate and new frequencies are generated without a significant change of the temporal form. If dispersion and SPM exercise a similar amount of impact, DSPM occurs.In MPCs with a solid medium, the ratio between dispersive and nonlinear effects can be changed by adjusting the position and thickness of the Kerr medium in the cell. The additional implementation of dispersive mirrors offers an alternative approach for these cases, if the possibility of changing the medium’s features is limited or more challenging. High energy gas-filled MPCs typically have values of $N\; \gt \;{10}$ [3–5]. All MPCs can profit from EFC, as the contrast improvement is also dependent on the properties of the compression, as discussed in Section 3.
In Fig. 1, the transformation of a pulse for different amounts of DSPM is depicted. In the temporal domain, the two most notable features are the reduction of the peak power due to the increase in pulse duration and the change of the pulse form from Gaussian to a more rectangular shape. For very strong DSPM ($N\; \lt \;{1}$ but not $\ll 1$), the pulse edges become so steep that wave breaking sets in, leading to oscillations on the pulse flanks [19]. The spectrum of each pulse is shown in Fig. 1(b). For increasing amounts of DSPM, the depth of the spectral modulations is reduced. The Fourier transform limit (FTL) of these smoothed out spectra will have less pronounced side pulses and therefore a higher contrast.
Fig. 1. (a) Temporal and (b) spectral form of laser pulses at the MPC output with decreasing $N$, showing the increasing impact of DSPM. While the pulses become longer and more rectangular, the spectral modulations become less pronounced.
3. NUMERICAL SIMULATIONS
A. Numerical Model
The numerical model solves the nonlinear envelope equation (NEE) in radial symmetry [20] similar to the model used in [21]. All orders of dispersion of the medium and the mirrors as well as SPM and self-steepening are considered in a way that effects due to DSPM can be investigated. The central wavelength is 1030 nm. The model also takes into account the transmission losses at the mirrors.
B. Exemplary Results
Since the goal of this MPC is the temporal compression via spectral broadening of a laser pulse, we compare two MPCs that achieve the same compression factor. The compression factor is defined as ${{\rm FWHM}_{{\rm in}}}/{{\rm FWHM}_{{\rm out}}}$ of the FTL of the pulse. Two MPCs with the same input can have the same compression factor but a different temporal (and spectral) form depending on the parameters of the MPC.
To assess the impact of EFC on an MPC, a classical MPC with typical parameters is used for comparison. For this reason, the MPC from Grebing et al. [3] is taken as reference. It compresses 1 mJ 200 fs long pulses to 31 fs in an argon-filled cell. The pulses propagate 13 roundtrips in the 1183 mm long cell, which operates close to the stability edge using mirrors with a radius of curvature of 600 mm. By exchanging the original mirrors (with a GDD close to zero) with mirrors that provide an additional phase in the form of positive GDD, the simulated energy ratio of the FTLs increases from 80% to 99%. The peak power boost increases by 20% from 5.5 to 6.6 for a mirror GDD value of $150\;{{\rm fs}^2}$. The GDD value for mirrors is given per single mirror bounce, and one MPC roundtrip consists of two mirror bounces.
This is depicted in Fig. 2 where the reference MPC is shown in blue and the EFC-improved MPC is shown in red. The dashed black lines show the input pulse and spectrum. Figure 2(a) shows the pulses after propagation through the MPCs before compression. The EFC-MPC pulse shows the typically rectangular shape due to DSPM and is about three times as long as the one from the classical MPC (661 fs–234 fs). The impact of self-steepening is negligible. For the classical MPC, a gas pressure of 450 mbar is required, and for the EFC-MPC, 800 mbar. The pressure increase is necessary to achieve the same compression factor since the stronger stretched EFC-MPC pulse has a smaller peak power. This is compensated for by increasing the nonlinearity of the cell using a higher pressure.
Fig. 2. Simulation results for classical (blue) and EFC-MPC (red). (a) Temporal pulse profile at the input (black) and output. The pulse at the end of the EFC-MPC is significantly longer as well as rectangular shaped due to DSPM. (b) Spectrum at the input (black) and output for the two MPC types. The modulations are smoothed out in the EFC-MPC case. (c) Fourier transform limits of the MPC spectra from (b). Both have the same FWHM, but the EFC-MPC has an increased peak power and strongly reduced side pulses. The power is normalized to the peak power of the classical MPC.
The spectra shown in Fig. 2(b) both meet the expectation of strong modulations in the classical and a smoothed out form in the EFC case. The depth of the lobes is reduced from $ - 9.4\; {\rm dB}$ to $ - 0.5 \; {\rm dB}$, which results in a much cleaner FTL [Fig. 2(c)]. Although the spectra look different, the FWHM of their FTLs are identical. Comparing the FTLs in (c), the great potential of EFC in MPCs becomes visible, as the peak power is increased and most of the energy previously contained in the sidelobes is now part of the main feature.
The amount of pre-pulse reduction is shown in more detail in Fig. 3. Here the FTLs of different MPCs with increasing amount of mirror GDD (and gas pressure) reaching the same compression factor are compared. The power is normalized for each pulse so that the peak power is 1. Therefore, the $y$-axis gives the inverse power ratio. In the classical MPC case, the first pre-pulse (in terms of distance to the main pulse) almost reaches a value of 6%, and even the secondary pre-pulse exceeds 1%. The optimum energy ratio was reached in the case of a GDD value of $150\;{{\rm fs}^2}$ (green). The height of the first pre-pulse is 0.25% and the second is 0.02%. A further increase in GDD leads to the rising of additional oscillations at the edges of the pulse due to wave breaking and therefore to stronger side pulses.
Fig. 3. Power of the pre-pulses for the FTLs of MPCs with different amounts of mirror GDD. In the classical case of ${\rm GDD} = 0\;{{\rm fs}^2}$, the pre-pulse reaches a height of close to 6% of the main feature. This could be reduced to 0.25% in the optimized case with a GDD value of $150\;{{\rm fs}^2}$. A further increase leads to higher pre-pulses again due to the effect of wave breaking.
C. Limitations
When trying to find the optimal mirror GDD value for an EFC-MPC, a numerical simulation, testing different mirror GDDs, is necessary. This is because, depending on the MPC dimensions, energy, and gas, the optimal mirror GDD value can range from ${\lt}100\;{{\rm fs}^2}$ up to four digit values. Increasing the mirror GDD will lead to a higher energy ratio until the optimal value is reached. A further increase leads only to wave breaking and a reduction of the energy ratio.
Additional pulse contrast improvement could be achieved by combining EFC with other contrast enhancement methods, because the implementation of higher orders of dispersion in MPC mirrors constitutes a significantly higher order of complexity. The addition of GVD to the pulse before it enters the MPC is not effective since the effect of DSPM is based on the interplay of GVD and SPM.
Because of the comparatively reduced pulse duration in the EFC-MPC, the amount of B-integral per focal pass is not as uniform as in the classical case, but higher in the first passes and lower in the later ones. In this example, the peak B-integral was 1.4 rad in the first pass of the EFC-MPC compared to approximately 0.7 rad in the classical case. This amount is considered safe [22], and no beam quality degradation was observed.
D. Necessity for Higher Order Dispersion Compensation
The different temporal pulse form in the EFC-MPC has an additional effect on the compressibility of the broadened pulse, originating in the different instantaneous frequency. The instantaneous frequency is the frequency of the pulse at a certain temporal position and is given by ${\omega _i}(t) = {\omega _0} + \frac{{\partial \phi}}{{\partial t}}$, where $\phi$ is the time domain phase of the pulse envelope. Because of the fast change of the intensity appearing in steep DSPM pulses, the instantaneous frequency change is also higher. This leads to a stronger presence of higher order dispersion terms to be compensated to compress the pulse. Due to the symmetry in the case of DSPM, the fourth order dispersion (FOD) has a stronger impact than the third order dispersion (TOD).
While in the classical MPC design the compression after the MPC is usually only done with negative GDD, this is not enough in the EFC-MPC case. In the classical approach, the application of higher order dispersion for compression improves the peak power and contrast only slightly, and orders greater than the FOD are necessary to achieve 99% of the FTL peak power. In the case of the EFC-MPC, the total amount of phase is much higher since the pulse propagates through gas at a higher pressure and GDD mirrors. Removing the GDD alone, however, is not enough since higher order phase terms developed due to the steeper pulse edges. Compensating only for GDD leads to emerging side pulses, defeating the main goal of this concept. Since the main component of the additional phase is FOD, it is possible to remove it using mirrors with a custom phase coating. The introduction of this additional parameter, reduces the flexibility of the MPC slightly, as for different MPC settings, not only the GDD, but also the TOD and FOD in the compression stage need to be adjusted.
In Fig. 4, the phases and compression possibilities of a classical MPC (top) and an EFC-MPC (bottom) are compared. The classical MPC FTL in black, which would equal a spectral phase of zero, is reached neither by the compression with GDD (blue, ${-}1299\;{{\rm fs}^2}$) nor the compression with GDD, TOD, and FOD (orange, ${-}1358\;{{\rm fs}^2}$, ${+}6217\;{{\rm fs}^3}$, ${+}3330\;{{\rm fs}^4}$, respectively). In the spectrum, the spectral phase of the uncompressed pulse (solid line) and the two compressions (GDD compression: dashed; GDD, TOD, FOD compression: dotted) is depicted. In both cases, the function of the phase still shows a strong deviation from the FTL phase. The total amount of accumulated phase is higher in the EFC-MPC case, but it does not exhibit strong modulations in the central region. A compression with only GDD (${-}4707\;{{\rm fs}^2}$), however, leads to a negative phase at both ends of the spectrum due to the amount of FOD present. This leads to the emergence of side pulses in the time domain [Fig. 4(c), blue line]. The compression with GDD, TOD, and FOD (${-}5303\;{{\rm fs}^2}$, ${+}903\;{{\rm fs}^3}$, ${+}776840\;{{\rm fs}^4}$, respectively), however, results in a flat and close to zero spectral phase, which means the compressed pulse is very close to the FTL.
Fig. 4. Time and spectral information of classical (top) and EFC-MPC (bottom). In the time domain, the graphs show the FTL (black), the compressed pulse using only GDD (blue), and the compressed pulse using GDD, TOD, and FOD (orange). In the spectral domain, the change of the phase (solid line) after compression with only GDD (dashed line) and after compression with GDD, TOD, and FOD (dotted line) is shown. The necessity for using higher orders of dispersion for compression is much higher in the case of the EFC-MPC.
In the case of few-cycle MPCs, the compensation of higher orders of dispersion has to be considered because the difference between the generated frequency and the center frequency is so high that these terms are no longer negligible. There, typically, the TOD is the most relevant after GDD, whereas in the EFC-MPC case, FOD has the greater impact. The design and manufacturing of mirrors with a fitting phase compensating these orders of dispersion is possible according to coating suppliers. The peak power boost between a pulse from the classical MPC compressed with only GDD and a pulse from the EFC-MPC compressed with GDD, TOD, and FOD is 21.6% in this example.
4. CONCLUSION
We have theoretically shown that the technique of enhanced frequency chirping can be easily implemented in the design of multi-pass cells to significantly improve the temporal contrast. Using a numerical simulation computing DSPM behavior in MPCs, we were able to show an increase of energy ratios from 80% up to 99% and a reduction of pre-pulse power from 6% to 0.25% of the peak power compared to classical MPCs. This could enhance MPC technology to even higher peak powers and ensure operation in sensitive experiments without a reduction of energy or additional optical components. In upcoming experiments, we plan to realize the described comparison of a classical and an EFC-MPC using GDD mirrors in the MPC and compression mirrors suitable for higher order dispersion compensation. The predicted results of EFC technology denote not only a big step in combating the issue of SPM-induced pre-pulses, but also an increase in efficiency in pulse compression using MPCs.
Funding
Freistaat Thüringen (MULTIPASS); European Research Council (835306, SALT); Fraunhofer-Gesellschaft (CAPS).
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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