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Abstract
Stable picosecond supercontinuum generated in long crystals is an excellent means of seeding broadband, high-energy CPA systems. The generated output energy and spectrum can be almost three times as stable as the pump for a wide range of input pulse parameters. In this work, we show this is an intrinsic property for crystals longer than the filament and for a range of input energy values. We present a description of the stability mechanism in both the visible and infrared regions together with experimental data that support the theoretical explanation.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Supercontinuum generation is an effective method of producing broadband coherent radiation which can be used as a seed for high-energy OPCPA systems. Because of the high intensities required from the driving laser, supercontinuum sources are most often generated with femtosecond pulses, but in many amplifier systems it is most convenient to drive the supercontinuum process with picosecond pulses.
The first experiments demonstrating super broadening of the spectrum of picosecond pulses focused into solid state optical media [1–3], liquids [4], and fibers [5] were published already in the 1970’s. These pioneering works gave rise to an intensive study of supercontinuum generation with picosecond pulses [6–8]. The broadband spectrum of the supercontinuum is generated in a medium where high peak intensity results in a strong self-focusing collapse, followed by temporal pulse splitting and self-steepening capable of reaching several $\mathrm {TW}/\mathrm {cm}^2$ and generating high plasma densities [9]. The main constraint in generating a supercontinuum experimentally is using a material that is able to withstand such plasma densities without permanent optical damage. Femtosecond pulses are often used as they can achieve high intensities with relatively low pulse energy, reducing the risk of damage to the medium [10].
With the development of picosecond pulse pumped OPCPA chains for the amplification of ultra-short pulses, picosecond supercontinuum generation has received renewed interest as a possible OPCPA seed source. The spectral extent of a supercontinuum originating from a narrowband picosecond pulse can be several octaves [11]. For these picosecond pulses, materials with a high damage threshold and a high nonlinear refractive index, such as sapphire [12] or YAG [13] are needed, as higher energy is required for self-focusing to occur in the crystal. In order to avoid damage to the material, loose focusing may also be necessary depending on the input pulse parameters, with shorter pump wavelengths being more prone to optical damage [14].
The output energy and spectral stability of the supercontinuum is crucial when seeding OPCPA systems. For femtosecond pulses, it was shown that certain spectral regions in the supercontinuum can be more stable than the pump in a YAG crystal a few millimeters long [15]. For picosecond pump pulses, the stability of supercontinuum was first studied in a 1.5 cm long YAG crystal and the spectral fluctuations were almost two times worse than the pump energy fluctuations [12]. Recently, however, it was experimentally observed that for input picosecond pulses at 1030 nm with energies sufficient to produce a single filament in a 13 cm long YAG crystal, the output energy in the spectral regions between 500-950 nm [16] and 1100-2400 nm [17] can be more than two times as stable as that of the pump.
We propose an explanation for this intrinsic stability in longer crystals based on both numerical and experimental results. In this paper, supercontinuum (SC) stability is always measured as the ratio of output energy fluctuations in a specific spectral region to the input energy fluctuations at the pump wavelength. The explanation of this stability comes from a combination of intensity clamping due to material ionization, chromatic dispersion and crystal length.
When a Gaussian pulse reaches a nonlinear focus due to self-focusing in a medium with normal dispersion, the temporal envelope splits into two separate sub-pulses [18]. Our simulations show that these sub-pulses propagate at different velocities and are responsible for the generation of new spectral components due to self-phase modulation and self-steepening. At first, the intensity of the pulses increases due to self-focusing until the self-focusing is halted by plasma defocusing. This process limits the maximum peak intensity which is defined by the ionization properties of the medium. This was already proposed as a stabilizing factor in 1975 [7]. The peak intensity subsequently decreases due to chromatic dispersion and spatial defocusing until the spectral broadening becomes insignificant. For a given numerical aperture and pulse duration, the processes that restrict the length and intensity of a single and complete filament, which governs the spectral broadening, are mainly defined by the material properties. Therefore, once the input energy is sufficient for filamentation to occur, and the whole filament is located in the medium, small fluctuations in that energy do not significantly influence the spectral broadening. For higher input energies, the process can repeat with additional nonlinear refocusing. But for input energies below the threshold for second refocusing, the input fluctuations do not translate significantly into output energy fluctuations in the newly generated spectral regions outside the initial narrowband pump spectrum.
2. Numerical model and experimental set-up
The numerical model used to analyze the supercontinuum stability is based on a scalar, unidirectional, paraxial nonlinear envelope equation [Eq. (1)] directly derived from the wave equation in the pulse frame under the minimal approximation as it is derived in [19] in the following form
The nonlinear polarization $P_{\mathrm {nl}}(r,z,t)$ describes the effects stemming from the nonlinear response of the medium. Our model contains information about the Kerr-effect and also about the energy loss due to plasma generation arising from optical field ionization (OFI) and avalanche ionization (AV) as $P_{\mathrm {nl}}=P_{\mathrm {kerr}}+P_{\mathrm {ofi}}+P_{\mathrm {av}}$. The nonlinear polarization driven by the Kerr effect can be calculated as
where $n_2=6.5\cdot 10^{-20}\,\mathrm {m}^2/\mathrm {W}$ is taken to be the nonlinear refractive index for YAG [12], and $I$ and $E$ are the intensity and amplitude, respectively, of the optical beam in time domain. $\mathcal {F}_t[\cdot ]$ represents the Fourier transform from time domain to frequency domain.The term $P_{\mathrm {ofi}}$ takes into account both the multi-photon ionization and the tunneling ionization in the following form
Finally, the term containing information about the avalanche ionization can be written as
The free-electron density can be calculated from the Drude model by the following rate equation
The implemented numerical model was then used to analyze the propagation dynamics of filamention pumped by 1.5 ps pulses at 1030 nm in a 13 cm long YAG crystal. Experimentally, the Gaussian pulses in space and time were focused by a 400 mm lens and the crystal was placed at a distance of 370 mm after the lens for the optimal stability (see Fig. 1). In the numerical model, we assume that the geometrical focal spot is located approximately 3 cm in the crystal. This assumption neglects the effects of the air-YAG interface and the self-focusing happening before the focal plane. Therefore, we start the simulation 3 cm from the front side of the 13 cm YAG crystal in order to decrease the simulation runtime. We have justified this by doing simulations for the full length of the YAG crystal which did not show any significant differences in the resulting spectrum or energy. For this reason, the presented numerical results contain a propagation axis from 0 to 100 mm (instead of 0 to 130 mm). It was also experimentally observed that the stability of the output supercontinuum is optimal when the input pump beam with 3 mm diameter at $1/e^2$ is cropped by a 2.6 mm diameter iris, which was also incorporated into our model. The scheme of the experimental set-up can be seen in Fig. 1.
Fig. 1. Experimental set-up used to analyze the stability of a picosecond pulse driven supercontinuum in YAG. The spectrum and energy at the output of the system is measured after a 950 nm low-pass filter or a long-pass 1050 nm filter for the visible or IR measurements respectively.
3. Energy stability
For picosecond supercontinuum pumped by stable sources with a good beam quality, it seems to be always possible to find a stable regime of picosecond supercontinuum generation when using a longer YAG crystal. The stability can be optimized by, for instance, the focusing geometry or by cropping the input pump beam with an iris [24]. For the setup presented in Fig. 1, we have measured the output SC energy in the visible domain, by using the low-pass 950 nm filter, as a function of the pump energy and many numerical simulations were performed for different input energies in order to analyze the stability in the different regimes. The comparison between the experimental and numerical data can be seen in Figs. 2(a) and (b) respectively.
Fig. 2. SC energy in the 500-950 nm domain as a function of pump energy: (a) measured; (b) simulated.
The overall shape of the function in both graphs (a) and (b) agrees with the theoretical description of the stability mechanism as will be explained in the Section 5. The stable regime is between the threshold energy and the second refocusing, observable as a plateau in the energy curve. The main difference between the simulation and measurement is in the length of the plateau region defined by the necessary input energy for refocusing. Experimentally, the energy required for second refocusing is around $9.5\,\mathrm{\mu} \mathrm {J}$ while in the simulation it is only $6\,\mathrm{\mu} \mathrm {J}$. Therefore, the measured stable plateau is approximately two times longer. This might be caused by slightly different experimental conditions such as imperfect pulse contrast or beam profile, or by an imperfect model of the plasma interaction.
In order to address the dependence of the stability with respect to the length of the YAG rod, the stability was measured for different positions of the crystal. This is not strictly equivalent to measuring stability for different crystal lengths due to the added effect of self-focusing; nevertheless, it demonstrates the limitation in using shorter crystals. The best stability was achieved when the front side of the crystal was located 370 mm from the lens, where the stability of the SC was more than two times better than that of the pump. However, the stability was very similar for different positions of the crystal ranging from 320 mm to 370 mm. By moving the crystal closer towards the lens, the filament is generated further in the YAG rod and for positions closer than 310 mm the stability worsens significantly as can be seen in Fig. 3. When the front-side of the crystal is moved closer than 290 mm, the filament is generated in the last few centimeters of the crystal and the output energy is less stable than the pump.
Fig. 3. Measured RMS of the fluctuations of the input pump and the visible part of SC in the stable regimes for different positions of the 13 cm long YAG rod relative to the focusing lens with a focal distance of 400 mm.
4. Spectral stability
In our application, we aim to use the SC as a seed for a CPA system as a replacement for a Ti:Sapphire oscillator. For this reason, we are interested in the spectral region of 500-950 nm. As described in the previous section, the stable regime of SC generation in the 500-950 nm region is between the threshold energy and the energy that gives rise to the second nonlinear focusing of the filament. This can be experimentally observed by increasing the input energy and observing the visible spectrum as illustrated in Fig. 4 which shows the simulated and measured spectra in the different regimes of stability. At the threshold energy, a sudden change in the spectrum can be observed as new spectral components are generated between 550 nm and 700 nm.When the energy is increased further, spectral components >700 nm are generated until a stable regime where the increasing energy ceases to influence the newly generated spectrum. In this regime, the energy fluctuations can be observed only in the spectral components close to the 1030 nm pump. For even higher energies, the filament refocuses and spectral modulations can be observed which nicely matches our numerical model.
Fig. 4. Simulated and measured SC spectra in the visible domain after the propagation through 13 cm of YAG for input pump energy just above the threshold, in the stable regime and in the unstable regime of second focusing.
A similar effect can be also observed in the IR domain as shown in Fig. 5. Just above the threshold energy for SC generation, the spectrum quickly broadens and generates new frequency components around $2\,\mathrm{\mu} \mathrm {m}$. In the region between $1.6\,\mathrm{\mu} \mathrm {m}$ and $2.3\,\mathrm{\mu} \mathrm {m}$, the energy of the SC remains stable until the second focusing where significant modulation propagates from 1030 nm to longer wavelengths with increasing energy. This was also recently observed in YAG for picosecond pulses at 1030 nm [17] for very similar experimental conditions.
Fig. 5. Simulated and measured SC spectra in the IR domain after the propagation through 13 cm of YAG for input pump energy just above the threshold, in the stable regime and in the unstable regime of second focusing.
5. Description of stability mechanism
There are two main processes affecting the stability of supercontinuum. Firstly, the ionization properties of the medium ensure that the maximum intensity reached in the crystal changes very little with respect to the input energy. This process is called intensity clamping and is dependent on the order of multiphoton absorption, and, therefore on the band-gap of the material [23]. Secondly, chromatic dispersion together with beam divergence by diffraction and plasma defocusing decrease the peak intensity after the nonlinear focus which limits the length of the filament and thus the distance during which the self-phase modulation (SPM) broadens the spectrum. A more thorough analysis of the effect of chromatic dispersion on the spectral broadening can be found in [25]. Before the formation of the filament, the combined effect of SPM, self-focusing and normal chromatic dispersion alter the initial temporal Gaussian envelope, which becomes more Lorentzian and then it splits into two separate sub-pulses [18]. Then, both sub-pulses experience considerable self-shortening due to the plasma effects and self-steepening. The trailing pulse possesses a steep trailing edge, therefore it is responsible for the spectral broadening in the visible part of the spectrum, while the leading pulse broadens the visible part with its steep rising edge. This is in agreement with the description of this effect in [14]. The intensity of both pulses decreases at distinct rates due to the different chromatic dispersion at different wavelengths. The leading pulse contains longer wavelengths, so it is closer to the zero-GVD point and experiences smaller chromatic dispersion while the trailing pulses are temporally stretched much faster. Consequently, the distance during which the energy couples from 1030 nm to new spectral components is limited to about a centimeter in the visible region and to a few centimeters in the infrared region for this specific focusing geometry and it is, in general, not highly dependent on the input pump energy. In this regime, pump energy fluctuations only affect the strength of the self-focusing that shifts the filament position in the crystal, but the intensity clamping and chromatic dispersion ensure a constant length and intensity of the filament.
To illustrate these mechanisms, several numerical simulations were carried out for different input pump energies under the same experimental conditions. Figure 6 shows the numerical results for three different pump energies lying in three distinct regimes of stability. The top row shows the on-axis temporal envelope as the filament propagates in the YAG crystal where the temporal splitting can be observed. The bottom row shows the spatially averaged spectrum for the same energy. In Figs. 6(a) and (b), the input pulse has just enough energy for the nonlinear focus to form before the rear side of the crystal where the peak intensity is high enough for significant spectral broadening. We call this the threshold energy. Around the threshold, slight fluctuations of the input energy will induce high output fluctuations as the filament is being cut-off by the rear side of the crystal. Figures 6(c) and (d) represents a situation where the energy is higher, so the nonlinear focus occurs sooner in the crystal (around 67 mm) and the intensity of both sub-pulses decreases below the limit for spectral broadening before the filament reaches the edge of the crystal. Around this energy, the SC generation is the most stable. Finally, Figs. 6(e) and (f) illustrates the case where the energy is high enough to overcome the divergence of the filament and refocuses it in the crystal, thus inducing a second pulse-splitting in time. In this regime, spectral modulation can be observed as the sub-pulses interfere with each other as is also shown in Fig. 4 and Fig. 5. This is in good agreement to what we empirically observed in YAG for ps-pulses in [16] and what was observed by other groups in other media for femtosecond pulses in [26] and [27]. The first two leading pulses interfere in the longer-wavelength domain while the last two pulses in the visible domain which makes the SC generation less stable than in Figs. 6(c) and (d). This is further demonstrated by the spectrogram, shown in Fig. 7, that was evaluated around the position of second pulse splitting at 63 mm in Fig. 6(e). It illustrates the separation of spectral content of the four sub-pulses, where the first two leading pulses contain mainly infrared wavelengths while the two trailing pulses are responsible for the spectral broadening in the visible domain.
Fig. 6. Evolution of the on-axis temporal envelope (top row) and the averaged spectrum (bottom row) during filamentation in YAG for three different energies. (a) and (b) correspond to 4 $\mathrm{\mu}$J, which is just above the threshold for supercontinuum generation; (c) and (d) to 5 $\mathrm{\mu}$J, which represents the stable regime of supercontinuum generation; (e) and (f) to 6.5 $\mathrm{\mu}$J, which is high enough for the beam to refocus in the crystal resulting in a second pulse splitting that modulates the spectrum both in the IR and visible domains and make the generation more unstable.
In summary, the picosecond pump pulses and the loose focus geometry nake the filament longer than what is common in femtosecond supercontinuum. The filament length depends on the focusing conditions and on the pulse duration and for our experimental set-up in Fig. 1 it was 1-2 cm long. For stable supercontinuum generation, it is important that the filament is not being cut-off by the rear side of the rod before the nonlinear frequency generation has ceased. If this condition is not met, the output SC becomes less stable as was illustrated in Fig. 3 by moving the crystal closer to the focusing lens up to the point where the filament was generated 1-2 cm from the rear side of the crystal. For this reason, in the configuration we consider here, a 5-6 cm long crystal would be advisable which would accommodate the 1-2 cm filament and the self-focusing before filamentation.
6. Conclusion
The explanation for the intrinsic stability of supercontinuum generation was presented. Numerical simulations for SC in YAG pumped by picosecond pulses were carried out and compared to experimental results in order to analyze the propagation dynamics of Gaussian pulses undergoing the filamentation process. From these simulations, the combined effects of intensity clamping, plasma defocusing and chromatic dispersion was seen to limit the length of the filament which is little dependent on the input energy fluctuations. Therefore, in the stable regime, the generation of new spectral components is comparable for a broad range of input energies. For picosecond pulses, longer focus geometry is required to avoid damage to the material, which increases the length of the filament. The length of the filament was, in our case, approximately 1 cm in the visible domain and 2 cm in the infrared domain. In order to accommodate the full length of the filament and the distance required for damage-free self focusing, a crystal of at least 5-6 cm would be required in our case. From our simulation it appears that the filament length changes proportionally to the initial pump duration with shorter pulses producing shorter filaments that in turn require shorter crystals, but the optimum length of the crystal for stable SC generation depends also on the focusing geometry and material parameters. In general, when considering the optimum crystal length for a given experimental setup, this stable regime can be achieved when the crystal is long enough for the filament to fully evolve.
Funding
Ministerstvo Školství, Mládeže a Tělovýchovy (CZ.1.07/2.3.00/20.0091, LQ1606, RVO 68407700); European Regional Development Fund (ADONIS CZ.02.1.01/0.0/0.0/16 019/0000789).
Disclosures
The authors declare no conflicts of interest.
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