# Extremely Nonlinear Optics Using Shaped Pulses Spectrally Broadened in an Argon- or Sulfur Hexafluoride-Filled Hollow-Core Fiber

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## Abstract

**:**

_{6}) for demanding pulse-shaping experiments. The benefits of both gases for pulse-shaping are studied in the highly nonlinear process of high-harmonic generation. In this setup, temporally shaping the driving laser pulse leads to spectrally shaping of the output extreme ultraviolet (XUV) spectrum, where total yield and spectral selectivity in the XUV are the targets of the optimization approach. The effect of using sulfur hexafluoride for pulse-shaping the XUV yield can be doubled compared to pulse compression and pulse-shaping using argon and the spectral range for selective optimization of a single harmonic can be extended. The obtained results are of interest for extending the range of ultrafast science applications drawing on tailored XUV fields.

## 1. Introduction

_{P}, the laser peak intensity I and λ the wavelength of the driving laser.

**Figure 1.**Improvement approaches for application of gas-filled hollow-core fibers for spectral broadening.

_{6}) as the nonlinear medium. This molecule is a prominent example for ultrashort pulse generation via molecular phase modulation [13], dual-wavelength pumping [14,15], filamentation [16], generation of high-energy Raman-shifted Bessel beams [17] and spectral broadening and compression in hollow-core fibers [18].

_{6}as nonlinear medium is advantageous compared to argon because it results in higher XUV yield and improved spectral selectivity. The paper is structured as follows: In Section 2 we describe the experimental setup, discuss the spectral broadening in an HCF for argon and SF

_{6}and the general scheme of pulse-shaping for HHG. The results of the pulse-shaping experiments are presented in Section 3. Conclusions and possible applications follow in Section 4.

## 2. Experimental Section

#### 2.1. Experimental Setup

**Figure 2.**After spectral broadening in a gas-filled hollow-core fiber (HCF) laser pulses are sent through a prism-compressor with a computer-controlled deformable mirror for pulse shaping. Afterwards, the pulses are focused into an argon-filled capillary where they generate high harmonics. These are detected by an extreme ultraviolet (XUV) spectrometer and its signal is used as a feedback for an evolutionary algorithm.

^{2}membrane surface) and changes the optical path of individual wavelength components and thus changes the spectral phase. With this setup it is possible to shape the high-harmonic spectrum by implementing different optimization goals [19].

#### 2.2. Spectral Broadening in a Gas-Filled HCF

_{2}is the nonlinear refractive index, which is 1.6 ± 0.3 × 10

^{−19}cm

^{2}·W

^{−1}for SF

_{6}[20], being comparable to argon, but having a more complex behavior as is outlined as follows. SF

_{6}is a symmetric molecule and stretching vibrations have to be taken into account with particular importance on the A

_{1g}vibrational mode being the strongest Raman active one. The fundamental frequency Ω

_{Raman}of 775 cm

^{−1}corresponds to a temporal period of 43 fs. For the presented experiments, the laser pulse duration is longer than the Raman oscillation period but shorter than the dephasing time T

_{2}of the stretching vibration, which is referred as transient or non-stationary scattering [21] and is limited by

_{2}is the inverse of the scattering bandwidth (6 ps for SF

_{6}) [22]. Using fs pulses spectral broadening in SF

_{6}is governed by an interplay of SPM and stimulated Raman scattering (SRS), especially in an HCF where the intrinsic collinearity of the HCF avoids phase-matched SRS. SRS leads to sidebands in the spectrum at fixed positions (Figure 3b) in contrast to argon (Figure 3a), where only SPM is responsible for spectral broadening (experimental conditions as in Section 2.1.). Since achieving the same spectral broadening in SF

_{6}allows reduction of either the gas pressure or the input intensity compared to argon, it enables the transfer of this scheme to Yb based fiber lasers delivering rather long pulses [18].

**Figure 3.**A comparison of spectral broadening in argon-filled (

**a**) and SF

_{6}-filled (

**b**) HCF for different gas pressures shows possible benefits for pulse-shaping experiments. For SF

_{6}the spectral broadening is larger than for argon for otherwise comparable conditions. The generation of Raman sidebands is observed by vertical lines with higher amplitude at fixed positions which corresponds to the A

_{1g}vibrational mode. Both figures show the spectral intensity in a joint logarithmic color scale which is normalized to the output pulse energy of the HCF.

_{6}two observations should be pointed out: (i) the total spectral width is a few nanometers larger due to the enhanced redshifted part of the spectrum and (ii) the width of the central peak of the spectrum is larger due to the missing dip around 775 nm as for argon. Looking at the Fourier transform of both spectra there is not much difference in terms of pulse duration and shape. However, the presented pulse shaping scheme may benefit from the spectral characteristics of the SF

_{6}-broadened spectrum due to the more favorable shape: (i) as a minor contribution the increase in total bandwidth directly translates into a better illumination of the deformable mirror surface and (ii) as a major contribution the spectral amplitude distribution is pronounced between 760 nm and 825 nm.

_{6}is chosen such that the SRS sidebands are weak and cannot be observed in Figure 4a. In this case the spectral phase is still smooth and pulse compression by prisms is possible [18]. However, the cross-phase modulation products of the sidebands with the spectral distribution of the input pulse filling the spectral gaps compared to argon show clear evidence of Raman activity.

**Figure 4.**Spectrally broadened laser pulses after propagation through the hollow-core fiber (

**a**) and their corresponding Fourier transform assuming a flat spectral phase (

**b**) are compared for argon (black line) and SF

_{6}(blue line).

#### 2.3. Tailored High-Harmonic Generation in a Gas-Filled HCF

_{6}and argon is very demanding and can be best judged in a subsequent highly nonlinear interaction experiment where it should become clear whether the additional contribution from SRS is beneficial or not. The spectral shape of the high-harmonic spectrum from a hollow-core fiber (Figure 5a) depends strongly on the evolution of the electric field of the driving laser pulse which can be controlled by shaping the spectral phase and amplitude. In the presented experiment the spectral phase is controlled with the pulse-shaper in order to reach the predefined optimization goal.

_{p}) and the geometric contribution from the hollow-core fiber (inner radius a, l-th root of the Bessel function J

_{n}

_{−1}(z) u

_{nl}) for the m-th harmonic of the laser frequency ω

_{L}[23]. Since the description of the propagation of the generating IR laser pulse and the harmonic radiation is full of twists and turns [24,25,26], i.e., Equations (1) and (5) suggest a complex intensity and wavelength dependence, the implementation of an evolutionary algorithm for finding the best suited pulse-shape is an established technique.

**Figure 5.**Different optimization goals for pulse-shaping experiments in HHG experiments. (

**a**) is a typical HHG spectrum from a hollow-core fiber with a pulse that is compressed by translating the prisms for highest XUV yield and using a flat mirror surface; (

**b**) shows an optimization of the overall yield by adaptively shaping the deformable mirror. By temporally shaping the driving laser pulse, a specified region of the spectrum (blue area) can be enhanced (

**c**) or suppression of harmonics (blue arrows) can be achieved (

**d**).

## 3. Results and Discussion

_{6}spectrally broadened spectrum for generation of high-harmonics from a hollow-core fiber, two optimization scenarios are experimentally studied (pressure for both gases as in Figure 4): on the one hand, optimization of the total XUV yield and on the other hand selection of a single harmonic. For both experiments the conditions in the HHG-HCF are kept the same to ensure investigating only the influence of the slightly changed spectral shape for argon and SF

_{6}after broadening and the influence of the resulting temporal shaped laser pulses. For both experiments XUV spectra after optimization are shown as well as the development of the fitness parameter for the fittest individual of each generation. As parameters, 50 individuals per generation, a mutation rate of 0.5, a crossover rate of 0.4, a clone rate of 0.1 and 1500 ms acquisition time per individual were used.

#### 3.1. Optimization of the XUV Yield

_{6}. The 27th and 31st harmonics are slightly less pronounced, using SF

_{6}for broadening (Figure 6a). Inspecting the evolution of the fitness during the optimization as shown in the inset of Figure 6a it can be seen that the initial value for the fitness is much higher using SF

_{6}, but the gradient is smaller. These observations are a first hint that the temporal shape of the SF

_{6}-broadened pulse is slightly different. For 20 mbar backing pressure an increase for the 27th harmonic can be seen using SF

_{6}. Since the gas pressure and the properties of the HHG-HCF are the same, this effect can only be attributed to plasma dispersion and therefore a different temporal pulse shape. Also, other lower order harmonics are increased, as shown in Figure 6b, which significantly increases the fitness compared to the use of an argon broadened spectrum for pulse shaping. The same is true for 30 mbar (Figure 6c), here phase matching prevents low order harmonics in case of argon which can be compensated in case of SF

_{6}. For 40 mbar (Figure 6d) the absolute yield after optimization is similar for both gases, but the spectral distribution is different. In case of SF

_{6}lower order harmonics are more pronounced.

**Figure 6.**Optimized high-harmonic spectra from an argon-filled HCF are shown for temporally shaped pulses after spectral broadening in argon (black curve) and SF

_{6}(red curve). The optimization was performed to achieve the maximum yield between 23 and 43 nm and the fitness of the fittest individual of each generation is shown in the inset. The optimizations were performed at (

**a**) 10 mbar; (

**b**) 20 mbar; (

**c**) 30 mbar and (

**d**) 40 mbar argon backing pressure in the high-harmonic generation (HHG)-HCF.

_{6}broadened and subsequently shaped pulse show a larger amount of lower order harmonics, which increases the total bandwidth of the XUV spectrum. In a certain parameter range, the XUV yield can be doubled compared to the use of argon for spectral broadening.

#### 3.2. Selection of Single Harmonics

**Figure 7.**Optimization of single harmonics for (

**a**) argon broadened and (

**c**) SF

_{6}broadened shaped pulses. The colored areas show the chosen spectral area for the optimization and the respective optimization result is shown as a line plot. The accompanying development of the fitness is shown in (

**b**) and (

**d**) in the same color as the optimized spectra. In case of using SF

_{6}the spectral fraction of 25th to 29th harmonic can be optimized over 50%.

_{6}for spectral broadening only two different evolutions of the optimization can be observed as shown in Figure 7c,d. For high energy harmonics (blue, red and black marked regions) a single harmonic can be increased above 50% of the spectrum. For lower order harmonics (green and purple marked regions) even after optimization the spectrum contains all harmonics and the spectral fraction of a single harmonic is below 30%. This different behavior using SF

_{6}instead of argon for spectral broadening can be attributed to a slightly changed phase profile due to the SRS contribution, i.e., will enhance the dynamic range for the phase only pulse-shaping, because spectral components, which are not needed for putting together the ideal pulse in time domain, can be suppressed by stretching them in the time domain. On the other hand, the role of the missing spectral parts in case of argon can be hardly enhanced by a phase shaper.

_{6}for spectral broadening the spectral window for selective optimization is wider and allows us to enhance also 29th and 25th harmonic substantially and the corresponding development of the fitness is very similar.

## 4. Conclusions and Outlook

_{6}for spectral broadening. Furthermore, it was shown that the characteristic of the XUV spectrum can be easily influenced by simply changing the gas which is used for spectral broadening without any further change of a single optical element. Thus, nonlinear fiber optics with molecular media offer large potential in ultrafast science. This can be of great interest when building up XUV beamlines for spectroscopy. Since using an HCF for spectral broadening is a standard technique in ultrafast science and nowadays a large variation of pulse-shaping devices becomes increasingly common to be implemented in laser systems, the general idea of switching to a molecular gas can be easily realized and offers, as in case of HHG, a lot of potential.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Hoffmann, A.; Zürch, M.; Spielmann, C. Extremely Nonlinear Optics Using Shaped Pulses Spectrally Broadened in an Argon- or Sulfur Hexafluoride-Filled Hollow-Core Fiber. *Appl. Sci.* **2015**, *5*, 1310-1322.
https://doi.org/10.3390/app5041310

**AMA Style**

Hoffmann A, Zürch M, Spielmann C. Extremely Nonlinear Optics Using Shaped Pulses Spectrally Broadened in an Argon- or Sulfur Hexafluoride-Filled Hollow-Core Fiber. *Applied Sciences*. 2015; 5(4):1310-1322.
https://doi.org/10.3390/app5041310

**Chicago/Turabian Style**

Hoffmann, Andreas, Michael Zürch, and Christian Spielmann. 2015. "Extremely Nonlinear Optics Using Shaped Pulses Spectrally Broadened in an Argon- or Sulfur Hexafluoride-Filled Hollow-Core Fiber" *Applied Sciences* 5, no. 4: 1310-1322.
https://doi.org/10.3390/app5041310