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We report on generation of ultrabroadband, more than 4 octave spanning supercontinuum in thin CaF2 crystal, as pumped by intense mid-infrared laser pulses with central wavelength of 2.4 μm. The supercontinuum spectrum covers wavelength range from the ultraviolet to the mid-infrared and its short wavelength side is strongly enhanced by cascaded generation of third, fifth and seventh harmonics. Our results capture the transition from Kerr-dominated to plasma-dominated filamentation regime and uncover that in the latter the spectral superbroadening originates from dramatic plasma-induced compression of the driving pulse, which in turn induces broadening of the harmonics spectra due to cross-phase modulation effects. The experimental measurements are backed up by the numerical simulations based on a nonparaxial unidirectional propagation equation for the electric field of the pulse, which accounts for the cubic nonlinearity-induced effects, and which reproduce the experimental data in great detail.
© 2016 Optical Society of America
1. Introduction
Self-focusing of intense ultrashort laser pulses in dielectric media leads to a fascinating propagation regime, termed femtosecond filamentation, which was discovered in 1995 [1] and since then has been remaining an active research topic. Femtosecond filaments emerge from the interplay between a wealth of linear and nonlinear effects, while in the simplest approximation filamentary propagation is sustained by a dynamic competition between self-focusing, diffraction, and multiphoton absorption/ionization-induced free electron plasma [2].
In recent years a considerable effort was directed to study filamentation phenomena in solid-state dielectric media with ultrashort mid-infrared laser pulses, that give an access to the range of anomalous group velocity dispersion, where the interplay of self-action effects with anomalous group velocity dispersion facilitates generation of ultrabroadband supercontinuum [3–5], self-compression of the pulse down to few optical cycle duration [5–8] and eventually, formation of propagation invariant spatiotemporal light bullets [6, 9–13]. These studies also uncovered interesting features of the supercontinuum spectra, such as the occurrence of distinct blue peaks, whose spectral shifts vary form large to giant with increasing the input pulse wavelength [14–16]. In addition, they reported on occasional observations of third [3–5, 12] and fifth [17] harmonics, which were detected before the onset of supercontinuum generation.
So far, under typical experimental settings (loose focusing condition) for filamentation in solid-state dielectric media, which refer to Kerr-dominated filamentation regime, the harmonics spectra are overlaid by much more intense supercontinuum emission, and harmonics generation is regarded as an interesting, but generally irrelevant phenomenon. However, recent theoretical and numerical studies suggest that odd harmonics generation may produce a non-negligible impact on the filament propagation dynamics and contribute to spectral broadening [18–21]. In that regard, a supercontinuum in air spanning almost 5 octaves from the mid-infrared to the mid-ultraviolet regions accompanied by enhanced generation of odd-harmonics was recently reported to be induced by filamentation with 3.9 μm, 80 fs laser pulses from a multigigawatt OPCPA system [22, 23].
In this paper we show that odd harmonics generation brings a relevant contribution to spectral superbroadening in a solid-state dielectric medium as the plasma-dominated filamentation regime is accessed. More specifically, such regime is uncovered by self-focusing of 170 fs, 2.4 μm input pulses for short propagation lengths and few TW/cm2 input intensities in CaF2 crystal, in the absence of optical damage of the nonlinear medium. We demonstrate that plasma-induced compression of the driving pulse induces spectral superbroadening around the carrier wavelength as well as facilitates spectral broadening of third, fifth and seventh harmonics via cross phase modulation, giving rise to the generation of an extremely broadband supercontinuum spanning more than 4 octaves from the ultraviolet to the mid-infrared regions.
2. Experimental setup and numerical model
The experiment was performed using linearly polarized idler pulse with a central wavelength of 2.4 μm, duration of 170 fs and an energy up to 50 μJ from a commercial optical parametric amplifier, OPA (Topas-Prime, Light Conversion Ltd.) pumped by an amplified Ti:sapphire laser system (Spitfire-PRO, Newport-Spectra Physics). The idler beam from an OPA was spatially filtered, suitably attenuated and focused by an f = +100 mm lens into 55 μm FWHM spot located on the input face of CaF2 crystal.
The successive dynamics of harmonics generation and spectral broadening processes was investigated by fine tuning of the crystal length and the input pulse energy (intensity). For that purpose, a wedge-shaped CaF2 sample was mounted on a motorized translation stage and was moved across the input beam, allowing precise scanning of propagation length in the 300 μm–2 mm range. The input energy (intensity) was varied using a neutral metal-coated gradient filter (NDL-25C-2, Thorlabs Inc.) in the 5 – 35 μJ (0.8 – 5.5 TW/cm2) range, which corresponds to a peak power range of 0.7 – 5.2 Pcr, where Pcr = 0.15λ2/n0n2 = 40 MW is the critical power for self-focusing in CaF2, with n0 = 1.42 and n2 = 1.5 × 10−16cm2/W being linear and nonlinear refractive indexes, respectively. The energies of individual harmonics were measured by dispersing the harmonics beams in space using a fused silica prism with 60° apex angle and by using automated 16-bit digitized detectors: the TH energy was measured using a silicon photodiode SFH 291 (Siemens) with a sensitivity of 0.66 pJ/count, the FH energy was measured using a silicon photodiode BPW 34 B (Osram) with a sensitivity of 1.6 fJ/count, while the SH energy was measured using a photomultiplier tube FEU-84 with a sensitivity of 40 aJ/count. High dynamic range spectral measurements are described in the last section of the paper.
The experimental observations were backed up by the numerical simulations, which were performed using a model based on solving the nonparaxial unidirectional carrier-resolved propagation equation for the electric field E. It is written in the spectral domain and in the local pulse frame [24]:
where k(ω) = ωn(ω)/c is the dispersion relation and is the group velocity of the driving pulse. P̃ and J̃ are the nonlinear polarization and current, respectively, which are expressed as where is the third-order nonlinear optical susceptibility, and where ρnt = 2.1 × 1022 cm−3 is the neutral atom density, σ = 3.47 × 10−21 m2 is the cross section for inverse Bremsstrahlung, τc = 3 fs is the electron collision time. We used the Keldysh ionization rates, where W(I) is the ionization rate, calculated as a function of the intensity of the field I = ε0cn0|E|2/2, and Ui = 12 eV is the bandgap. The plasma density ρ was calculated from the equation: where τrec = 150 fs is the recombination time.3. Odd harmonics generation
First of all, under these operating conditions, third (TH), fifth (FH) and seventh (SH) harmonics with center wavelengths of 800 nm, 480 nm and 343 nm, respectively, were experimentally detected and successive dynamics of their energy versus the propagation length z are shown in Figs. 1(a)–1(c). Each data point in the plots represents an average of 40 consecutive laser shots, exposed to the same area of the crystal. The solid white curves mark the range of experimental parameter values (the input pulse intensity and propagation length) above which the optical damage on the output face of the crystal develops after that number of shots, as verified by monitoring an abrupt decrease of harmonics energies due to light scattering from the damage spot and by visual inspection of the crystal volume and output face under white-light illumination using a microscope objective with 10× magnification. We also verified that operation just slightly below that line, e.g. by decreasing the input pulse intensity by 7%, no optical damage of the crystal is observed after exposure of at least 1000 consecutive laser shots at 200 Hz repetition rate with pulse-to-pulse stability of 0.4% rms. Also, at these operating conditions, no apparent signatures that point to formation of color centers (change of crystal color or reduction of the blue shifted part of the spectrum) were detected.
Fig. 1 Experimentally measured (top row) and numerically simulated (bottom row) energies of (a,d) TH, (b,e) FH and (c,f) SH as functions of the driving pulse energy (intensity) and propagation length z. Dashed curves mark a virtual borderline between the Kerr-dominated and plasma-dominated filamentation regimes (see text for details). The black areas in the experimental graphs denote the region of the optical damage that occurs on the output face of the crystal, whose threshold is depicted by a solid curve.
Figures 1(d)–1(f) show the results of the numerical simulations, which reproduce the experimental data in great detail. The evolutions of TH, FH and SH energies versus the propagation distance z show a typical oscillatory behavior, as expected from strongly phase mismatched interactions. Indeed, the measured period of TH energy oscillations (85 μm) in the low intensity limit (with the input pulse intensity below 1 TW/cm2) well coincides with the calculated double coherent build-up length 2Lcoh = 2π/|Δk| = 91 μm, where Δk = k(3ω) − 3k(ω) = 69.2 mm−1 is the phase mismatch for TH generation in CaF2 with 2.4 μm input pulses. Interestingly, FH and SH energies oscillate with the same period as the TH energy, suggesting that FH and SH are generated by cascaded four-wave mixing processes based on the lowest, i.e. cubic nonlinearity [25]. Indeed, an extensive analysis of the FH oscillation periods performed in an earlier work [17] demonstrated that FH is generated via four-wave mixing between the TH and fundamental harmonics: 5ω = 3ω + ω + ω. Similar consideration applies to the SH generation; there are two possible four-wave mixing configurations, which involve either mixing between the fundamental and TH (7ω = 3ω + 3ω + ω) or mixing between the fundamental and FH frequencies (7ω = 5ω + ω + ω). However, the relative contributions of these two processes could not be unambiguously revealed in the present study.
The cascaded origin of harmonics generation is further confirmed by a rapid decrease of the harmonics energy with increasing their order. For instance, with a fixed parameter set (the input pulse intensity of 4.1 TW/cm2 and z = 0.31 mm), the experimentally measured energy conversion efficiencies of TH, FH and SH are 1.9 × 10−4, 8.7 × 10−8 and 1.4 × 10−11, respectively, which are the rather typical values for the cascaded harmonics generation process in a solid-state medium. These values agree very well with the numerically computed energy conversion efficiencies that yield 1.6 × 10−4, 9.7 × 10−8 and 7.8 × 10−11 for TH, TH and SH, respectively. Notice, that the numerical model takes into account the third-order nonlinearity only, and therefore by definition considers the cascading origin of harmonics generation.
More careful inspection of Fig. 1 reveals further interesting features of harmonics generation process. Firstly, TH, FH and SH energy oscillation periods are not constant; they slightly shrink with increasing the input pulse intensity, as evident from slightly tilted harmonics energy oscillation patterns with respect to the vertical axes. The reduction of the oscillation periods is attributed to self- and cross-phase modulation effects [17], which contribute to the changes in the wavevector length of the interacting waves and so the net reduction of the coherent build-up length.
Secondly, there occurs an abrupt change of the tilt angle of harmonics energy oscillation patterns, which coincides with a notable increase of the harmonics (FH and SH, in particular) energies. With increasing the input pulse intensity, a virtual borderline that marks the change of the character of harmonics energy oscillations, shifts closer to the input face of the crystal as highlighted by dashed curves in Fig. 1, which serve as guides for the eye. Figure 2 shows the experimentally measured and numerically computed harmonics energy oscillations at a fixed input pulse intensity of 4.8 TW/cm2 in the z range around that virtual borderline in more detail. Figure 2(a) depicts the TH energy oscillations vs z, where an apparent change of the TH oscillation character takes place at around z = 1 mm, and thereafter the energy of the TH starts to increase. Similar features are observed in FH and SH energy oscillations, as shown in Figs. 2(b) and 2(c); here however, the growth of FH and SH energies is more dramatic.
Fig. 2 Energy oscillations of (a) TH, (b) FH and (c) SH as functions of the propagation distance z. Solid and dashed curves represent the experimental and numerical data, respectively. The input pulse intensity is 4.8 TW/cm2.
4. Spectral broadening and supercontinuum generation
In order to explain the above observations, we refer to the numerically simulated temporal and spectral dynamics presented in Fig. 3. Figure 3(a) shows the evolution of the on-axis intensity profile of the driving pulse at the fundamental frequency. In the early stage of the self-focusing (in the z range below 1 mm) there is just a slight increase of the peak intensity of the driving pulse, without an apparent change of its temporal profile. Thereafter (for z > 1 mm) the driving pulse experiences a remarkable (8.5-fold) self-compression from 170 fs to 20 fs, which is induced by rapidly increasing density of the free electron plasma, as shown in Fig. 3(b). The self-compression takes place in the most intense part of the beam, and the self-compressed pulse contains ∼ 10% of the input energy.
Fig. 3 Numerically simulated dynamics of (a) on-axis intensity profile of the driving pulse at 2.4 μm, (b) pulse duration (red curve) and free electron plasma density (blue curve), and (c) successive evolution of the spectrum. The input pulse intensity is 4.8 TW/cm2.
Figure 3(c) illustrates the successive evolution of the spectrum, which exhibits a dramatic broadening around the carrier wavelength for z > 1 mm, where plasma-induced pulse compression takes place. More interestingly, at the same time the spectra of the individual harmonics experience considerable broadenings as well. The spectral broadening of the individual harmonics may be attributed to the cross-phase modulation induced by an intense driving pulse. Eventually, for longer propagation lengths, z > 1.4 mm, the spectral broadenings around the carrier wavelength and the individual harmonics overlap, covering an extremely wide spectral range from the ultraviolet to the mid-infrared.
The temporal and spectral dynamics therefore suitably explain the change in the character of harmonics energy oscillations. A rapid increase of the driving pulse intensity due to plasma-induced compression results in an abrupt reduction of harmonics oscillation periods, as well as in an increase of the harmonics energies. From a more general viewpoint, these findings manifest the onset of the plasma-dominated filamentation regime, where self-compression of the driving pulse is due to plasma defocusing that pushes the rear part of the pulse out of propagation axis. The plasma-induced pulse compression mechanism is essentially similar to that reported in normally dispersive gaseous media in femtosecond [26, 27] and picosecond [28, 29] filamentation regimes, as well as in normally dispersive solid-state media with picosecond laser pulses [30], where plasma generation results in a catastrophic blow-up of the trailing part of the pulse. Our present findings suggest that, in the first approximation, this compression mechanism is regardless of the sign of group velocity dispersion and is in much contrast to the Kerr-dominated filamentation regime in the range of anomalous group velocity dispersion, where pulse compression is achieved due to opposite effects of self-phase modulation and anomalous group velocity dispersion, see e.g. [6, 9, 10]. In order to justify the negligible role of dispersion, we calculate the nonlinear and dispersive lengths, which are expressed as Ln = 1/(n2k0I0) and , respectively, where k0 = ωn0/c, I0 is the pulse intensity, τFWHM is the pulsewidth and k″ = d2k/dω2. For the input pulse with τFWHM = 170 fs and I0 = 4.8 TW/cm2 we get Ln = 0.36 mm and Ld = 212 mm. Even if the compressed pulse and the increase of the peak intensity along propagation are taken into account, (e.g. τFWHM = 25 fs and I0 = 50 TW/cm2 at z = 1.6 mm), Ld = 0.46 mm, which is still larger than the nonlinear length Ln=0.04 mm. Therefore a virtual borderline in Fig. 1, which marks the change in the character of harmonics energy oscillations, in fact indicates the transition from the Kerr-dominated to the plasma-dominated filamentation regimes.
Figure 4 presents the experimentally measured and numerically computed spectra highlighting the relevant spectral features associated with Kerr-dominated and plasma-dominated filamentation regimes, which were accessed for lower input pulse intensity and shorter propagation length (3.4 TW/cm2, z = 0.48 mm) and for higher intensity and longer propagation length (4.1 TW/cm2, z = 1.94 mm), respectively. In the experiment, high dynamic range spectral measurements were performed using a home-built scanning spectrometer with Si, Ge and PbSe detectors, operating in the 0.2–1.1 μm, 0.7–1.9 μm and 1.5–3.6 μm spectral ranges, respectively, as schematically illustrated on the top of Fig. 4(a). The measured spectra where corrected to sensitivity functions of the detectors and transmission of the bandpass optical filters used in the measurement. Finally, spectra from each detector were slightly scaled to achieve consistency in the overlap regions.
Fig. 4 The output spectra which correspond to Kerr-dominated (black curves) and plasma-dominated (red curves) filamentation regimes: (a) experimental data; here the horizontal bars on the top indicate the spectral regions of the detectors, (b) numerical simulation.
In the Kerr-dominated filamentation regime, the output spectrum consists of a series of isolated narrow-band spectral peaks corresponding to the driving pulse and its odd harmonics. In the plasma-dominated filamentation regime, the individual spectral peaks show considerable broadenings, which overlap and merge into an ultrabroadband, multi-octave supercontinuum radiation, which covers the wavelength range from 250 nm in the ultraviolet to more than 7 μm in the mid-infrared (at the 10−12 intensity level), approaching the infrared absorption edge of CaF2 crystal. Note that even a broad spectral peak around the ninth harmonics at 267 nm appears in the numerical simulation, as shown in Fig. 4(b); however its energy was beyond the experimental detection range. A good agreement between the experimentally measured and numerically computed spectra is achieved although the experimental spectra were recorded within a reduced dynamic (10−8) and wavelength (from 200 nm to 3.6 μm) ranges, as due to sensitivity limitations of our detection apparatus.
5. Conclusion
In conclusion, we performed experimental and numerical investigation of odd-harmonics and supercontinuum generation with intense mid-infrared femtosecond laser pulses in thin CaF2 crystal within a wide range of input pulse intensities and propagation lengths. Such a broad operating parameter space allowed us to capture in detail the dynamics of odd harmonics generation and spectral broadening processes, leading to a clear distinction between two filamentation regimes. The Kerr-dominated filamentation regime is detected for low input pulse intensities and short propagation lengths and is characterized by phase-mismatched generation of third, fifth and seventh harmonics with characteristic energy oscillations, which also demonstrate that fifth and seventh harmonics are generated by cascaded four-wave mixing via cubic nonlinearity. In contrast, the plasma-dominated filamentation regime leads to plasma-induced compression of the driving pulse, which in turn induces spectral superbroadening around the carrier wavelength as well as facilitates large scale spectral broadening of third, fifth and seventh harmonics via cross phase modulation, eventually giving rise to the generation of a supercontinuum spanning more than 4 octaves from the ultraviolet to the mid-infrared spectral regions. We believe that the uncovered nonlinear propagation regime is of importance for better understanding the light-matter interaction processes in solid state dielectric media and revealing a debated role of odd-harmonics generation, in particular.
Funding
This research was funded by a grant No. APP-8/2016 from the Research Council of Lithuania.
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