We report on the generation of ultrabroadband supercontinuum (SC) by filamentation of two optical-cycle, carrier-envelope phase-stable pulses at 2 μm in fused silica, sapphire, CaF2 and YAG. The SC spectra extend from 450 nm to more than 2500 nm, and their particular shapes depend on dispersive properties of the materials. Prior to spectral super-broadening, we observe third-harmonic generation, which occurs in the condition of large phase and group velocity mismatch and consists of free and driven components. A double-peaked third-harmonic structure coexists with the SC pulse as demonstrated by the numerical simulations and verified experimentally. The SC pulses have stable carrier envelope phase with short-term rms fluctuations of ∼ 300 mrad, as simultaneously measured in YAG crystal by f-2f and f-3f interferometry, where the latter makes use of intrinsic third-harmonic generation.
© 2013 Optical Society of America
Supercontinuum (SC) generation is a well-established method for obtaining coherent broadband radiation, spanning across ultraviolet, visible and infrared spectral range [1, 2]. In bulk dielectric media with Kerr nonlinearity, generation of ultrafast SC is tightly linked to femtosecond filamentation ; the broadband radiation emerges from a complex interplay between self-focusing, self-phase modulation, four-wave mixing, pulse-front steepening, pulse splitting, generation of optical shocks, multiphoton absorption and generation of free electron plasma. To date, SC generation was extensively studied in wide bandgap dielectrics with ultraviolet, visible and near infrared femtosecond laser pulses, under conditions of normal group velocity dispersion (GVD), see e.g. [4–6]. On the other hand, it is well known that the material dispersion plays an important role in self-focusing dynamics and filamentation of the ultrashort laser pulses . Experimental and numerical studies revealed that the interplay between the anomalous GVD and self-action effects results in extended filamentation length and conditional pulse splitting [7,8], pulse compression rather than pulse splitting [9,10], different angular pattern of conical emission [11–13], ultrabroadband spectra [6, 14–18] and quasiperiodic collapse events , hence giving rise to a new filamentation regime in general . Another effect associated with filamentation is the generation of third-harmonic (TH), whose occurrence is mediated by the same source of the nonlinear polarization [20, 21]. Indeed, TH generation is a well-known phenomenon, which accompanies filamentation and spectral broadening in gasses and in air in particular, see e.g. [22–30], however it is seldom observed [18, 31] and often neglected effect regarding filamentation and SC generation in condensed media.
In this paper we experimentally and numerically investigate SC generation in the regime of anomalous GVD with two optical-cycle, carrier-envelope phase (CEP)-stable pulses at 2 μm. We find that TH generation occurs prior to spectral broadening, and the TH pulse has a specific double-peaked structure, which coexists with the SC also in the regime of spectral superbroadening. We also demonstrate that spectral beating between TH and ultrabroadband SC produces the f-3f interference pattern, which could be readily applied for CEP stability measurements.
2. Supercontinuum generation
The experiment was performed using 15-fs (2.3 optical-cycle), CEP-stable pulses with central wavelength of 2 μm from a home-built optical parametric amplifier . Its output beam was suitably attenuated and focused by an f = +100 mm lens into a 70 μm FWHM spot size located at the input face of the nonlinear medium. The output SC radiation was re-collimated and then collected into a fiber tip of the spectrometer. The measurements were performed using two calibrated fiber spectrometers AvaSpec-2048 and AvaSpec-NIR256-2.5 (both from Avantes), that covered effective wavelength ranges of 400 – 1100 nm and 1 – 2.5 μm, respectively. We tested four different nonlinear wide bandgap materials, which are commonly used for SC generation in the visible and near-infrared: sapphire (6 mm length), fused silica (5 mm), CaF2 (6 mm), and YAG (6 mm), whose GVD curves are illustrated in Fig. 1.
Figure 2 shows the angle-integrated SC spectra, as recorded using two different input-pulse energies, which roughly represent transient (lower energy) and saturated (higher energy) regimes of the spectral broadening. Note the distinct differences of SC spectral shapes, as generated in different nonlinear media, which demonstrate the importance of digression of the pump wavelength from the zero GVD wavelength, that could be spotted from Fig. 1. The SC spectrum generated in fused silica [Fig. 2(a)] is very similar to those reported in a number of previous studies [14–17] and has a deep extended minimum around 1 μm and distinct intense peak in the visible (so-called blue peak), which then shows apparent red-shifted broadening of the blue peak with increase of the input-pulse energy. The SC spectrum with very similar spectral features, just with slight shift of the blue peak to the green is generated in sapphire [Fig. 2(b)], which exhibits essentially similar dispersive characteristics and nonlinearity as fused silica. In CaF2, owing to its generally low dispersion, a broad and flat SC spectrum that spans from 450 nm to > 2.5 μm was generated [Fig. 2(c)], with the longest wavelength being limited by our detection apparatus. And finally, in YAG, a smooth SC spectrum with elevated spectral intensity in the 600–1000 nm range [Fig. 2(d)] was generated using quite low input-pulse energy owing to higher nonlinearity of YAG as compared to other materials tested. Also note the differences in visual appearance of characteristic coloring and angular divergence of the SC radiation in the saturated regime of the spectral broadening (corresponding to higher values of the input-pulse energy), as illustrated by the screen shots of the far field patterns taken at 15 cm distance from the output face of the nonlinear media. The angular spread in the visible wavelength range is regarded as conical emission, whose angles are set by dispersion-related phase matching condition , which coincides with the X-wave phase matching [35,36]. In other words, for a given material, the phase matching condition entirely determines the loci of the far-field that can be populated in priority, i.e., the dependence of angles as a function of wavelength in the conical emission pattern. The effective scattering of a specific color at a given angle also depends on the nonlinear pulse-matter interaction ensuring that the phase matching condition is fulfilled over a certain propagation distance. Among the tested materials YAG has the highest nonlinear index of refraction, leading to a strong laser-matter interaction, and the largest dispersion, therefore visible extension of SC in YAG has the largest angular spread.
In all examined cases, prior to SC generation as well as in the transient regime of spectral broadening a characteristic TH peak centered at 660 nm was observed. In the transient regime of spectral broadening, the TH peak is best visible in CaF2 and YAG, see Fig. 2(c) and (d), whereas in fused silica and sapphire it becomes rapidly masked by the occurrence of a strong and broad blue-shifted peak, which is not associated with TH generation .
3. Third-harmonic generation
The TH generation occurs at lower input pulse energy and is clearly observed before the onset of spectral broadening and SC generation. For example, the TH radiation was detected with 1.0 μJ input-pulse energy in fused silica and with 0.40 μJ input-pulse energy in YAG, and measured TH efficiency varied from 10−6 to 10−4, depending on the input-pulse energy. A closer inspection of stand-alone TH spectra revealed a remarkable fast periodic modulation, whose frequency changed with the nonlinear material and its length, as verified by testing shorter samples of the nonlinear media. An example of TH spectrum generated with 0.70 μJ input-pulse energy in YAG, before the onset of SC generation, is shown in Fig. 3(a). Interestingly, very similar modulation of the TH spectrum was observed in semiconductor materials using intense 3.5 μm pulses and was attributed to pulse splitting effect .
In what follows, we show that distinct spectral modulation is a signature of a double-peaked TH pulse, which occurs naturally, without the splitting of the input pulse. We interpret our results in the framework of phase and group-velocity mismatched TH generation . Such an operating condition imposes that TH radiation consists of two pulses, representing so-called free and driven waves, which, in analogy with phase and group velocity-mismatched second harmonic generation, are solutions of the homogenous and the inhomogenous wave equations, respectively, see [38–40] for more details. The first pulse, i.e. the free wave travels with the group velocity uf, as set by the material dispersion, and walks-off from the pump pulse, i.e. the input-pulse at fundamental frequency. The second pulse, the driven wave, travels with the velocity ud of the nonlinear polarization, i.e., with the velocity of the intensity peak of the fundamental frequency pulse, which we evaluate in first approximation as its group velocity. Consequently, TH radiation at the output of the nonlinear medium consists of two pulses separated in time by the amount τ = |νfd|z, where νfd = 1/uf − 1/ud is the group velocity mismatch and z is the medium length, which produce beating in TH spectrum. Inserting the relevant values of YAG: z = 6 mm and νfd = 115 fs/mm, the estimated temporal separation between the free and driven TH pulses thus is 690 fs, which is very close to that of 670 fs, as retrieved from the fringe pattern shown in Fig. 3(a). In the spectral domain, the TH spectral intensity could be expressed as , where Δϕ = ϕf − ϕd denotes the instant phase difference between the free and driven TH pulses. The fringe pattern shown in Fig. 3(a) is thus regarded as 3f-3f interferogram, whose variation over time yields the phase jitter between free and driven TH pulses, regardless on CEP fluctuations of the pump pulse. Figure 3(b) shows a series of 1000 3f-3f interferograms, whose fringes are remarkably stable in time, yielding the root-mean-square (rms) phase jitter of 48 mrad [Fig. 3(c,d)], which originates from intensity-dependent phase matching condition due to nonlinear refractive index and intensity fluctuations of the pump pulse.
4. Numerical simulations
In order to verify our interpretation, we performed numerical simulations by using a radially symmetric form of the generic unidirectional carrier resolving models (forward Maxwell equation) [41–43] for propagating the frequency components of the infrared pulse:34]. Diffraction is described by the second term on the right hand side of Eq. (1). The nonlinear polarization P(z, t, r) = ε0χ(3)E3(z, t, r) describes self-phase modulation and third-harmonic generation. Self-steepening and nonlinear chromatic dispersion are described by the frequency dependent coefficients in front of the nonlinear polarization. The third order susceptibility is obtained from the nonlinear index coefficient for the optical Kerr effect n2 = 7 × 10−16 cm2/W . Plasma effects (nonlinear absorption, plasma generation, plasma defocusing and absorption) are calculated in the temporal domain by resolving a coupled system of equations for the density ρ(z, t, r) of the electron-hole and electron-ion plasma generated by optical field ionization and avalanche , and for the current source term J(z, t, r) ≡ Je + Ja: Eq. (3) is the phenomenological model from which Drude-like models are derived. The corresponding current Je includes the effects of plasma defocusing and plasma absorption. The quantity τc = 3 fs denotes the typical collision time in transparent solids, Ui = 6.5 eV denotes the gap between valence and conduction bands for YAG. The intensity dependent photoionization rate W(E) follows Keldysh’s formulation for condensed media in the multi-photon limit : W(E) = σKE2Kρb, where K = 11 photons, σ11 = 2 × 10−137 cm22W−11s−1 and ρb = 7 × 1022 cm−3 denotes the density of background neutral atoms. The avalanche rate follows the Drude model, leading to an inverse Bremsstrahlung cross section σ = 2 × 10−21 cm2.
The parameters for the input optical pulse corresponded to the experimental values (wavelength 2 μm, pulse duration 15 fs, beam width at the entrance face of the YAG plate 70 μm FWHM, pulse energy in the range 0.6 – 0.8 μJ). Note that in the simulation we used slightly reduced input-pulse energy so as to account for low contrast of the pulse used in the experiment, see Fig. 3(b) of . Details on the numerical schemes and resolution methods are described in .
Figure 4 compares the experimentally measured [taken from Fig. 2(d)] and numerically simulated angle-integrated SC spectra, where the numerical spectra were computed using the input-pulse energies of 0.6 and 0.8 μJ, representing the transient and saturated regimes of spectral broadening, respectively. A good agreement between the numerical simulations and experimental data was obtained; note, how experimentally observed spectral features, such as distinct TH peak, elevated spectral intensity in the 600–1000 nm range and a slight decrease of spectral intensity in the 1200–1400 nm range were reproduced by the numerics.
Figure 5 illustrates the results of numerical simulations in more detail. Figures 5(a) and (c) show the angularly resolved spectra in the transient and saturated regimes of spectral broadening, respectively. The insets of the figures show the magnified portions of the spectra in the 550–750 nm range, highlighting the spectral beatings in the TH spectrum in Fig. 5(a), and the spectral beatings produced by overlap of TH and SC spectra in Fig. 5(c). Figures 5(b) and (d) show the respective temporal profiles of the main (pump) pulse (black curves) and pulses at TH wavelength (red curves), as retrieved using a super-gaussian filter with FWHM width of 150 nm and centered at 660 nm.
In the transient regime of spectral broadening, the retrieved TH temporal profile consists of two distinct peaks separated by 700 fs [Fig. 5(b)], which are attributed to free and driven TH pulses, in fair agreement with analytical interpretation and experimental results presented in the previous section. Note also that the driven TH pulse remains short as being locked under the envelope the main (pump) pulse, which does not broaden much due to the interplay of self-phase modulation and anomalous GVD, while the free TH pulse moves away and experiences considerable temporal broadening due to normal GVD.
In the saturation regime of spectral broadening, a broad SC is generated, with its spectrum spanning from ∼ 500 nm to 4 μm (only a part up to 2.5 μm is shown) and that is accompanied by strong conical emission, ending-up with a pronounced fish-tail in the green-red spectral range. The SC radiation overlaps the TH spectral peak [Fig. 5(c)]; these produce a distinct spectral beating, which is confined to the beam axis and to the TH spectral range, as shown magnified in the inset. The temporal profiles were retrieved by the same filtering procedure showing that TH radiation yet consists of isolated free and driven pulses, which coexist with the broadband main pulse that develops a certain sub-structure and an extended tail, as shown in Fig. 5(d). Filtering recovers also an intense peak in between the TH pulses, located at 170 fs with respect to the intensity peak of the main pulse, and which could be attributed to SC spectral components in the 600–700 nm range that populate the main pulse tail. However, more detailed numerical investigation by varying the input-pulse energy and propagation distance disclosed that the intense peak is also contributed by generation of a secondary free TH pulse as a result of pulse-front steepening-induced spontaneous formation of an X wave in the normal GVD region . The propagation distance at which the X wave is generated decreases with increasing the input-pulse energy (not shown here), however, in general, in the time domain the contributions of the X wave and the secondary free TH pulse are impossible to distinguish as they are generated at the same moment and in the same spectral region.
5. Measurements of CEP fluctuations from the beating between SC and TH spectra
Finally, we experimentally demonstrate the practical use of intrinsic TH generation, as the observed spectral beating between SC and TH directly produces the f-3f interferogram, whose time series readily provide the statistics of CEP fluctuations. This could be done by proper filtering the conical components of the SC, so as to select the axial portion of the spectrum, where clear spectral beating between SC and TH is observed according to the results of numerical simulation presented in Fig. 5(c). Moreover, we have verified experimentally that the spectral beating is more or less well-detectable in all investigated media, see examples of spectra in fused silica and YAG shown in Fig. 6.
Taking the SC generation in YAG as an example, we devised a setup shown in Fig. 7, which simultaneously measured f-3f and, for a comparison, a more conventional f-2f interferogram. Specifically, the f-3f interferogram was recorded after careful filtering of the conical part of the SC by placing 1-mm iris aperture at 5 cm distance from the output face of YAG crystal. A conventional f-2f interferometer consisted of a collimating lens, bulk dispersive medium (DM, 25-mm-long YAG slab), which introduced the necessary delay between the spectral components of the SC (namely, between those located at 2 μm and 1 μm), the second-harmonic (SH) generator (SHG, 0.5-mm-thick BBO crystal cut for type I phase matching) and a polarizer. The resulting interference pattern is shown in Fig. 8(a), where blue-shaded areas mark the interference fringes around 1 μm and 660 nm, generated by sampling the SC pulse with the SH and TH pulses, respectively. Figure 8(b) shows variation of the interference pattern over 10 s period after recording 1000 single-shot spectrograms. The f-2f interference pattern yields SC pulse CEP rms fluctuations of 300 mrad. The f-3f interference pattern is more complex and has beatings at several frequencies, as shown enlarged in the inset of Fig. 8(c), and by means of the inverse Fourier transform could be decomposed into seven peaks, as schematically illustrated in Fig. 8(c). Application of arbitrary amplitude filters on the particular peaks (schematically shown as blue-shaded areas) delivers information on phase fluctuations. The farthest peaks at ±640 fs (labelled as 3f-3f) yield phase jitter of 55 mrad, which is attributed to the phase fluctuations between free and driven TH pulses. The remaining peaks at ±170 fs and ±470 fs (labelled as f-3f) provide CEP fluctuations of 315 mrad and 275 mrad, which are attributed to phase jitters between the SC and driven and free TH pulses, respectively.
The recovered peak positions in time fairly agree with those retrieved from the numerical simulation, as shown in Fig. 5(d).
In conclusion, we demonstrated ultrabroadband SC generation by filamentation of two optical-cycle, CEP-stable pulses at 2 μm in wide-bandgap solids: sapphire, fused silica, CaF2, and YAG, in the regime of anomalous GVD. The measured SC spectra span from 450 nm to more than 2.5 μm, and their particular shapes crucially depend on digression of the pump wavelength from the the zero GVD wavelength. In that regard, CaF2 and YAG provide the SC radiation with the smoothest spectral coverage across the entire detected spectral range. We also detect TH generation, which occurs prior to spectral supebroadening. Periodic modulation of the TH spectrum reveals a double-peaked temporal structure of the TH pulse, consisting of free and driven components, which are generated in the regime of large phase and group-velocity mismatch. We find that double-peaked TH structure persists also in the regime of spectral superbroadening and coexists with strong SC emission, as verified experimentally and by the numerical simulations. We also devised an experimental setup, which simultaneously measures the CEP stability of the SC pulses by means of f-2f and f-3f interferometry. Given a good agreement between the results obtained by f-2f and f-3f interferometry, the f-3f interferometry based on intrinsic TH generation, suggests a simple and straightforward method to measure CEP fluctuations, despite rather complex temporal structure of the TH pulse.
This research was funded by Grant No. VP1-3.1-ŠMM-07-K-03-001 from the Lithuanian Science Council.
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