Efficient High-Power Ultrashort Pulse Compression in Self-Defocusing Bulk Media

Peak and average power scalability is the key feature of advancing femtosecond laser technology. Today, near-infrared light sources are capable of providing hundreds of Watts of average power. These sources, however, scarcely deliver pulses shorter than 100 fs which are, for instance, highly beneficial for frequency conversion to the extreme ultraviolet or to the mid- infrared. Therefore, the development of power scalable pulse compression schemes is still an ongoing quest. This article presents the compression of 90 W average power, 190 fs pulses to 70 W, 30 fs. An increase in peak power from 18 MW to 60 MW is achieved. The compression scheme is based on cascaded phase-mismatched quadratic nonlinearities in BBO crystals. In addition to the experimental results, simulations are presented which compare spatially resolved spectra of pulses spectrally broadened in self-focusing and self-defocusing media, respectively. It is demonstrated that balancing self- defocusing and Gaussian beam convergence results in an efficient, power-scalable spectral broadening mechanism in bulk material.


Supplement 1: Magnitude and dispersion of the effective nonlinear refractive index
Fig. S1 displays magnitude and dispersion of the effective nonlinear refractive index for phase-mismatched quadratic nonlinearities near the type I second harmonic resonance. Fig. S1(a), which is calculated from the Sellmeier equations in ref. S1 , shows where this resonance occurs, i.e. for which tuning angle θ the SHG process is phase-matched, and indicates the regions of positive and negative n (cas) 2 . The curves in Figs. S1(b) and (c) have been calculated from Eq. (2) of the main text and a constant n (Kerr) 2 = 4 · 10 −16 cm 2 /W. Fig. S1(b) demonstrates that BBO crystals must be tuned near resonance to achieve a defocusing nonlinearity at 1030 nm which implies dispersion of the n 2 as Fig. S1(c) illustrates.

Supplement 2: FROG measurements
Fig. S2 shows the measured and the retrieved FROG traces after the third broadening stage. The retrieval grid was 512 × 512, the delay spacing 2 fs and the wavelength spacing 0.26 nm. The FROG error of the retrieval was 0.3 %. The homogeneity of the beam is evidenced by the excellent agreement between the FROG and OSA spectrum. Since the FROG contains an only 10 µm thick BBO crystal for SHG, it behaves like a spatial filter S2 . Moreover,Fig. S2(d) shows that additional control of the phase of the output pulses would lead to a further increase of peak power from 60 MW to 75 MW. The simulation utilized the FROG retrieval shown in Fig. S2(b) and added the GDD of the tailored mirrors used in Kerr-effect-based bulk broadening experiments presented in ref. S3

Supplement 3: Ray model for self-focusing and self-defocusing nonlinearities
In order to visualize the homogenization effect upon combining self-defocusing nonlinearities with beam convergence, the simple model proposed by R.Y. Chiao et al. S4,S5 is adapted. The initial model was used to find a threshold for self-trapping of optical beams. It treated the laser beam as a step-index fibre-like waveguide with a cladding of refractive index n and a core of refractive index n + n 2 I where I is the laser intensity. This led to a good approximation of a medium's critical power. Here, the laser beam is treated as a graded-index waveguide with the refractive index profile where I 0 is the peak intensity, r the distance to the beam centre and w the beam radius at the front facet of the nonlinear crystal. Typically, ray tracing in graded index optical elements is performed by applying Fermat's principle. For cylindrical symmetry, this leads to the equation where z is the propagation length inside the crystal. The differential equation was solved numerically with a Runge-Kutta method. It is to note that no z-dependence was assigned to I 0 and w which is a strong approximation since the beam size is changing inside the crystal. Ray packet equivalents (consiting of 100 rays each) of three Gaussian beams S6 were launched. Their propagation through a 10 mm long nonlinear crystal was plotted in Fig. S3. Fig. S3(a) presents the modelling of the self-focusing example discussed in the main text (cf. Fig. 3(e)). The laser beam is focused to a waist of 20 µm. In analogy to the simulation presented in the main text, the BBO crystal with n 2 = +8 · 10 −16 cm 2 /W is placed at z min = 4z R . The rays which are initially in the beam centre (red rays) remain in this most intense region due to the self-focusing effect. Therefore, they undergo strong SPM while the rays which are initially in the wing of the beam (black rays) hardly notice the self-focusing effect, diverge strongly, and thus do not undergo SPM. This behaviour is in good agreement with the simulation result shown in Fig. 3(e) of the main text. The central part of the beam is spectrally broadened while the spectrum of the wings is not affected by the nonlinearity of the crystal. The plot of Fig. S3(b) corresponds to the simulation result shown in Fig. 3(c) of the main text. The n 2 is set to −8 · 10 −16 cm 2 /W and z min = −7z R . The rays behave fundamentally different from the self-focusing example. The initially central rays are strongly diffracted and cross the rays from the wings within the defocusing medium, i.e. they undergo SPM only at the beginning of the crystal while the rays from the wings will spectrally broaden towards the end of the crystal. This leads to the homogenization effect in the case of combining self-defocusing and beam convergence. Finally, the case of self-defocusing without divergence is studied in Fig. S3(c). The peak intensity is set to 200 GW/cm 2 and n 2 = −8 · 10 −16 cm 2 /W like before. Fewer rays cross propagation length (cm) Figure S3. Model for 100 rays describing a Gaussian beam. The darker the rays the longer the initial distance to the optical axis. (a) Self-focusing nonlinearity and beam divergence: The central rays stay close to the optical axis while the outer rays diverge strongly. (b) Self-defocusing nonlinearity and beam convergence: The inner rays are diffracted out of the inner beam region while the outer rays move towards the beam centre. (c) Self-defocusing nonlinearity, no divergence: the mixing of outer and inner rays is clearly reduced in comparison to (b). For all cases, an initial peak intensity of 200 GW/cm 2 was assumed.
within the crystal. Basically, only the initially centred rays undergo SPM in the beginning of the nonlinear medium. The rays from the wings diffract slower but they do not propagate towards the beam centre due to the missing convergence. In conclusion, the simple ray model provides an illustrative picture of the homogenization effect which was quantitatively investigated in the main text.

Supplement 4: Spectral broadening in dependence on tuning angle and distance to the focal point
In order to investigate the sensitivity of the pulse compression scheme on the alignment of the BBO crystals, simulations with different phase-matching angles and varied crystal positions have been conducted. The simulations refer to the first broadening stage. The results are shown in Fig. S4. In Fig. S4(a) the tuning angle is fixed to θ = 21.5 • , i.e. the effective n 2 is approximately −8 · 10 −16 cm 2 /W. The crystal length is 10 mm which corresponds to about 8 Rayleigh lengths. If the n 2 was 0, the maximal peak intensity inside the crystal would be about 3 TW/cm 2 for z min between -8 and 0 z R . This would lead to crystal damage. It is to note that the critical intensity  Figure S4. (a) Dependence of the Fourier limit and the maximal peak intensity inside the 10 mm long crystal on z min . The Rayleigh length z R is 1.22 mm for a beam waist of 20 µm. The tuning angle θ is fixed to 21.5 • . (b) Dependence of the Fourier limit, the maximal peak intensity inside the crystal and the second harmonic (SH) losses on the tuning angle θ for a fixed z min = −7 z R .
has not been measured for the oscillator parameters. For BBO, it is specified to be 40 GW/cm 2 for tens of ps pulse duration S7 while tens of TW/cm 2 did not damage the crystal when 4 fs pulses were used S8 . For the utilized high-power fs system, a damage threshold of several hundreds of GW/cm 2 is estimated. Crystal damage was observed upon detuning of the crystal angle by a few degrees. The self-defocusing effect strongly reduces the maximal peak intensity inside the crystal to only about 6 % of the free-space value if z min = -7 z R . This, however, depends on the tuning angle as Fig. S4(b) shows. The slope of the maximal peak intensity in dependence on the tuning angle is about -75 GW/cm 2 / • in the vicinity of θ = 21.5 • , i.e. the angle has to be set carefully before power is sent into the broadening crystal in order to avoid its damage. The generated second harmonic provides a good measure for the tuning angle. For the 10 mm crystal about 2 % (1.8 W) of the 90 W are converted to green light. The Fourier limit of the broadened pulses is also very sensitive to the tuning angle as Fig. S4(b) indicates. In the vicinity of θ = 21.5 • , the slope is -30 fs/ • . It is remarkable that the peak intensity grows with the Fourier limit. This is due to the tuning angle dependence of n 2 . An increase in θ leads to higher absolute value of the nonlinearity, subsequently to a lower Fourier limit as well as to stronger self-defocusing, and thus to a reduced maximal peak intensity inside the crystal. Peak intensity and the broadening factor are also strongly dependent on the crystal position with respect to the focal point. In the graph shown in Fig. S4(a), crystal damage is expected if z min is in the interval between ≈ ±4 Rayleigh lengths. If the crystal is positioned behind the focus, the broadening factor is relatively low. In the example, the best positions for operation are between -8 and -4 z R where a Fourier limit of sub-100 fs is reached and the peak intensity is kept below the damage threshold. As described in the main text, z min ≈ −10 mm was chosen which enabled a relatively clean pulse compression by compensating for first order chirp only. The initial experimental setup required a careful choice of nonlinearity and divergence as described in this section. The presented simulations helped finding an appropriate setting. After having established the reported configuration, the setup has been operated for several months on a day-to-day basis without the need of major changes.