1. Introduction

Supercontinuum (SC) generation is one of the most spectacular nonlinear optical phenomena arising from nonlinear propagation of intense ultrashort light pulses in transparent dielectric media. The physical picture of SC generation is fairly well understood in the framework of femtosecond filamentation, which is a complex process involving an intricate coupling between spatial and temporal effects [1]. In the time domain, the complex interplay between self-focusing, multiphoton absorption, self-phase modulation and group velocity dispersion leads to self-steepening of the pulse, which ends up with either pulse splitting in normally dispersive medium or pulse compression in the range of anomalous group velocity dispersion. These dramatic transformations take place at the vicinity of the nonlinear focus, beyond which they produce an explosive spectral broadening, termed SC generation, as observed in a variety of wide bandgap solid state media and with input pulse wavelengths ranging from the ultraviolet to the mid-infrared, see e.g. [2–8]. The spectral extent of the SC is defined essentially by the laser wavelength and by linear or nonlinear properties of the medium, such as the medium dispersion, the bandgap and the nonlinear index of refraction [9, 10], which possess fundamental mutual relationships [11, 12].

The rapidly expanding field of applications calls for achieving broadband radiation with desired temporal and spectral properties, which in turn require setting an efficient control of the filamentation process. Since the relevant parameters of an isotropic medium are generally fixed for a given input wavelength, a certain control of SC generation was demonstrated by tailoring the shape [13, 14] and the chirp [15] of the input pulse, by varying the polarization state [16], phase [17, 18] and the numerical aperture [19] of the input beam. More sophisticated methods are based on performing two color filamentation in collinear [20] and noncollinear [21] geometries.

However, the achieved degree of control of the SC spectrum is rather limited, and most of these methods require more or less complex technical implementation. In that regard, the use of birefringent nonlinear media, which possesses both, quadratic and cubic nonlinearities, may provide a desired degree of freedom to perform the nonlinear interactions in a fully controlled fashion. The second-order cascading, which results from the phase-mismatched second harmonic (SH) generation [22], produces a cascaded-quadratic Kerr-like nonlinearity, which contributes to the nonlinear index of refraction. Thus the light wave sees a material with an effective nonlinear index of refraction, $n_2^{\text{eff}}=n_2^{\text{Kerr}}+n_2^{\text{casc}}$, whose cascaded counterpart, $n_2^{\text{casc}}\propto -d_{\text{eff}}^2/\mathrm{\Delta }k$, may be varied in sign and magnitude by setting an appropriate phase mismatch, Δk = k_2ω 2k_ω. Therefore the interplay between the cascaded-quadratic and intrinsic cubic (Kerr) nonlinearities may be readily exploited to either enhance or suppress the nonlinear effects [23, 24] and to access qualitatively different regimes of nonlinear propagation.

The self-defocusing regime $(n_2^{\text{eff}}<0)$ is achieved within a certain range of positive phase mismatch, leading to spectral broadening and SC generation via formation of temporal soliton in the range of normal group velocity dispersion [25–29]. In contrast, the self-focusing regime $(n_2^{\text{eff}}>0)$ is achieved within a wide range of either positive or negative phase mismatch, leading to SC generation via filametation [30–32]. More recent experiments demonstrate the feasibility of the self-focusing regime for spatial, spectral and temporal shaping of light within a wide input parameter space, ranging from picosecond [33] to few optical cycle pulses [34].

In this paper we demonstrate that the second-order cascading assisted filamentation of 120 fs, 800 nm laser pulses in beta barium borate (BBO) crystal in the operating conditions of prevailing self-focusing nonlinearity, leads to SC generation with fully controllable spectral extent, which is achieved by varying the phase mismatch parameter for SH generation. We propose a physical interpretation based on the competition between the self-steepening due to Kerr nonlinearity and the phase mismatch-dependent nonlinear effects which arise from the second-order cascading: self-phase-matched SH generation and quadratic self-steepening.

2. Results and discussion

The experiments were performed using a commercial Ti:sapphire laser system (Spitfire-PRO, Newport-Spectra Physics) providing 120 fs, 800 nm laser pulses at a 500 Hz repetition rate. The input pulse energy was varied in the range of 50 nJ - 1 μJ, which corresponds to a peak power range of 0.37 – 7.4 P_cr, where ${\mathrm{P}}_{\text{cr}}=0.15{\lambda }_0^2/n_0{n}_2^{\text{kerr}}=1.12\text{MW}$ is the critical power for self-focusing in BBO, with n₀ = 1.66 being the linear and $n_2^{\text{Kerr}}=5.15\times {10}^{-16}{\text{cm}}^2/\mathrm{W}$ the Kerr nonlinear refractive index [35]. The ordinarily polarized laser beam was focused by a f = +125 mm lens into a 25 μm FWHM focal spot located at the front face of a 4.85-mm-long BBO crystal cut for type I phase matching. The crystal was placed on a mechanical rotation stage, which allowed varying the angle θ between the incident beam and the optical axis of the crystal in the phase matching plane for SH generation. Rotation of the crystal permitted the control of the phase mismatch parameter Δk in the positive and negative regions as shown in Fig. 1(a) as well as the effective nonlinear index of refraction, as illustrated in Fig. 1(b).

 

Fig. 1 (a) Calculated phase mismatch parameter Δk. (b) Nonlinear refractive indices: $n_2^{\text{kerr}}$ (blue dash-dotted line), $n_2^{\text{casc}}$ (red dotted curve) and $n_2^{\text{eff}}$ (green solid curve) versus the angle θ. Note the two available propagation regimes: self-defocusing, where $n_2^{\text{eff}}<0$ and self-focusing, where $n_2^{\text{eff}}>0$. The shaded areas mark the angle range where control of SC spectral extent was experimentally observed.

Download Full Size | PDF

Figure 2 presents the experimental data of spectral broadening versus the angle θ in the range of negative phase mismatch for several input pulse energies. The spectral ranges of ordinarily (around the fundamental wavelength) and extraordinarily (around the SH wavelength) polarized spectral components are labelled as o and e, respectively. The measured dynamics immediately reveal a number of interesting features, associated with spectral broadenings at ordinary and extraordinary polarizations.

 

Fig. 2 Experimentally measured spectra as functions of angle θ with various input pulse energies: (a) 160 nJ, (b) 190 nJ, (c) 280 nJ, (d) 410 nJ. Each plot is composed of 110 individual spectra. The logarithmic intensity scale is used to highlight fine spectral features. The solid and dashed curves depict the self-phase matching curve for an axial SH component and the corresponding wavelengths in the spectrally broadened fundamental pulse, respectively.

Download Full Size | PDF

First of all, at perfect phase matching for SH generation (θ = 29.2°), we capture a considerable spectral broadening around the SH (400 nm) and fundamental (800 nm) wavelengths. The nature of this phenomenon may be explained by filamentation of the SH beam itself. In fact, it is of no surprise considering efficient SH generation and bearing in mind the ${\lambda }_0^2$ dependence of the critical power for self-focusing. More precisely, with 45% energy conversion (see Fig. 3), the power of the SH pulse even with the input pulse energy of 160 nJ, well exceeds the critical power for self focusing, where P_cr = 0.23 MW for λ₀ = 400 nm, with an account of an increased nonlinear index of refraction, $n_2^{\text{Kerr}}=6.3\times {10}^{-16}{\text{cm}}^2/\mathrm{W}$ [35]. Hence filamentation of the SH pulse leads to generation of the extraordinarily polarized SC, which spans from 300 to 610 nm and induces a considerable spectral broadening (from 650 nm to 1 μm) around the fundamental wavelength due to cross-phase modulation.

 

Fig. 3 The total second harmonic energy conversion efficiency as a function of the angle θ with the input pulse energies of 160 nJ (red solid curve) and 280 nJ (blue solid curve). The inset shows a magnified fraction of the graph, which distinguishes between the self-phase-matched and phase mismatched (green dashed curve) SH generation.

Download Full Size | PDF

Secondly, we measure a complex spectrum of the extraordinarily polarized SH, which consists of two distinct peaks. The narrow spectral peak centered at exactly 400 nm is attributed to the phase mismatched SH, whose intensity rapidly drops as the crystal is rotated out of phase matching. The second peak is broader and much more intense and is tunable in the 300–400 nm range. Its occurrence is attributed to the so-called self-phase matching (solid curves in Fig. 2), which is inherent to SH generation by narrow light beams carrying broadband pulses [36], and which is accessed by a certain range of spectral components (dashed curves in Fig. 2) present in a spectrally broadened fundamental pulse. Figure 3 shows the energy conversion efficiency to SH as a function of the angle θ for the input pulse energies of 160 nJ and 280 nJ. Here the total energy was estimated by the integration of the spectra presented in Figs. 2(a) and 2(c) over the entire 300 – 1100 nm range, while the energy contained in the SH was estimated by the integration in the 300 – 450 nm range. At the vicinity of phase matching, the SH energy was estimated by the integration over wider, 300 – 610 nm spectral range, due to SC generation by the SH pulse. For θ > 32°, we may readily distinguish between the energy conversions to self-phase-matched SH (tunable in wavelength) and phase-mismatched SH (at 400 nm), as shown in the inset.

Thirdly, and most importantly, at the ordinary polarization, the spectral broadening around the carrier wavelength shows a qualitatively different behavior in the θ range above and below 42.5°. In the range of very large negative phase mismatch (θ > 42.5°) and for relatively low input pulse energies the spectral broadening is almost absent [Fig. 2(a)] or unstable, as seen from ragged blue-shifted edge of the SC spectrum in Fig. 2(b). At higher input pulse energies, a stable SC, which spans from 410 nm to 1.1 μm is generated, whose spectral extent is independent on the phase mismatch [Figs. 2(c) and 2(d)]. These results are rather obvious, since in this angle range the contribution of the cascaded second-order nonlinearity to the effective nonlinear index of refraction quickly vanishes, as seen from Fig. 1(b). Therefore for the input pulse energy of 160 nJ, its power just very slightly exceeds P_cr and the beam is not able to self-focus within the crystal. For the input pulse energy of 190 nJ, the nonlinear focus is located very close to the output face of the crystal, resulting in unstable spectral broadening due to small fluctuations of laser pulse energy. Only with the input energies of 280 and 410 nJ a stable filamention regime is achieved.

However, the most striking behavior of the spectral broadening is observed in the θ range of 30° – 42.5°. Here the extension of the blue-shifted part of the SC spectrum shows a very strong dependence on the angle θ, while the extent of the red-shifted part is almost constant over the entire angle range, especially with higher input pulse energies, as seen in Figs. 2(c) and 2(d). The most fascinating feature of the SC spectrum is that its blue-shifted cut-off gradually extends to shorter wavelengths while varying angle θ from 30° to 42.5°. Moreover, the cutoff wavelength is remarkably stable for a given crystal orientation and does not change with the input pulse energy, suggesting a complete control of the blue-shifted spectral extent of the SC via phase mismatch. This is illustrated in more detail in Fig. 4, which shows the cut-off wavelength (defined at the 10⁻⁴ intensity level) as a function of the input pulse energy for several values of angle θ.

 

Fig. 4 The blue-shifted cut-off of the SC spectrum as a function of the input pulse energy and angle θ.

Download Full Size | PDF

In order to get more insight into the underlying phenomena, we consider the effects of pulse splitting and self-steepening, which play the key role in producing the spectral broadening in the filamentation regime [37, 38]. For that purpose we measured the axial cross-correlation functions of the SC pulses generated at several crystal orientations, which yield the SC spectra with different blue-shifted broadenings. The cross-correlation measurements were performed by sampling the output pulse with a short, 25-fs gating pulse at 700 nm from a noncollinear optical amplifier (Topas-White, Light Conversion Ltd) via sum-frequency generation in a thin (20 μm) BBO crystal.

The results presented in Fig. 5 confirm, that the pulse splitting is clearly detected over the entire range of angle θ, where the spectral broadening takes place. For a fixed input pulse energy and different values of θ, we capture the split sub-pulses at various stages of their evolution, in line with the whole evolution cycle of femtosecond filaments in the range of normal group velocity dispersion [39]. More precisely, as the pulse splitting event takes place at the nonlinear focus, a large part of the energy is expelled out of propagation axis, and thereafter with further propagation rebuilds the axial pulse at the center (at the zero delay). If its power exceeds the critical power for self-focusing, the reconstructed pulse undergoes another self-focusing cycle with subsequent secondary splitting event.

 

Fig. 5 The axial cross-correlation functions measured with two input pulse energies: 190 nJ (top row) and 280 nJ (bottom row) and at various angles θ: (a),(d) 32.4°, (b),(e) 37.5°, (c),(f) 42.6°.

Download Full Size | PDF

Indeed, the measured cross correlation functions confirm the above scenario in great detail, taking into account the inverse relationship between $n_2^{\text{eff}}$ and θ as shown in Fig. 1(b). As for the high values of $n_2^{\text{eff}}$, the nonlinear focus is located closer to the input face of the crystal, there remains a sufficient propagation distance to rebuild the axial pulse after the pulse splitting event, as shown in Fig. 5(a). In contrast, for the lower values of $n_2^{\text{eff}}$, only the split sub-pulses are measured, as the pulse splitting takes place closer to the output face of the crystal [Figs. 5(b) and 5(c)].

Similar considerations apply to the results shown in Figs. 5(d) and 5(e), here however, due to higher input pulse energy, the nonlinear focus and so the pulse splitting event is shifted even closer to the input face of the nonlinear crystal, therefore the reconstructed pulse is present in all measurements. For the highest value of $n_2^{\text{eff}}$, we capture the splitting of the reconstructed pulse after the secondary nonlinear focus, as shown in Fig. 5(d). The secondary splitting produces periodic modulation in the SC spectrum due to temporally separated pairs of the sub-pulses at the leading and trailing edges. This is barely seen in Fig. 2(c) but clearly emerges in Fig. 2(d).

Therefore the cross-correlation measurements confirm the fundamental relationship between pulse splitting and SC generation. On the other hand, the spectral superbroadening is a direct consequence of self-steepening, and the spectral blue shift is associated with the self-steepening of the trailing front of the trailing sub-pulse [37]. The earlier studies of spectral broadening in the absence of filamentation demonstrated that quadratic self-steepening due to second-order cascading has a nearly identical effect as cubic self-steepening, with the additional property of being inversely proportional to the phase mismatch parameter Δk, and hence controllable in sign and magnitude [40, 41]. More precisely, in the range of relatively small negative phase mismatch (note the opposite sign in the definition of Δk in [40]), the quadratic self-steepening is expected to amplify the effect of cubic self-steepening, giving rise to a considerable blue-shifted spectral broadening. However, our measurements presented in Fig. 2 show directly the opposite. Therefore, at this point we refer to the results of Fig. 3 and its inset in particular, which demonstrate that even at large phase mismatch, a still reasonable amount of the input pulse energy is converted into the self-phase-matched SH. More importantly, the blue-shift of the self-phase-matched SH wavelength suggests that it is generated only by the trailing sub-pulse, whose carrier frequency is blue-shifted [42].

In order to estimate the energy fraction carried by the trailing sub-pulse, we performed the numerical simulation using a standard filamentation model in media with cubic nonlinearity, which is based on solving the paraxial unidirectional propagation equation with cylindrical symmetry for the nonlinear pulse envelope coupled with an evolution equation for the electron density generated by the high-intensity pulse [43]. The model simulated filamention as if the second-order nonlinear effects were absent, corresponding to the experimental situation far away from the phase matching. The numerical data show that a light filament with a 5 μm FWHM diameter emerges after 1.5 mm of propagation, and is immediately followed by an almost symmetric pulse splitting and SC generation. The energy balance is as follows: 18% of the input energy is lost due to five photon absorption and absorption by the free electron plasma, and the central core contains 14% of the remaining energy, so the trailing sub-pulse carries just 6% of the input pulse energy. Assuming the total energy conversion to the self-phase-matched SH of, e.g. 2.8% at θ = 32° (as shown in the inset of Fig. 3), almost a half of the trailing sub-pulse energy is converted into the self-phase-matched SH.

Such an efficient conversion is due to the fact, that generation of self-phase-matched SH is group velocity matched with the pump as well, as the SH radiation takes the spatiotemporal shape of an X-wave, as demonstrated in [36]. As a result, the trailing sub-pulse undergoes a considerable energy loss and distortion, which counteracts the effect of both, quadratic and cubic self-steepenings, preventing the blue-shifted spectral broadening of the SC. As the energy conversion to self-phase-matched SH gradually drops with increasing θ, the role of self-steepenings increases, resulting in larger blue-shift. Eventually, at very large Δk (θ > 42.5°), the generation of self-phase-matched SH ceases completely, as well as the effect of quadratic self-steepening. Here the blue-shifted spectral broadening is entirely controlled by cubic self-steepening, yielding a constant blue-shifted extension of the SC. The above considerations are also supported by the fact that the red-shifted leading sub-pulse has no self-phase matching, so the red-shifted extent of the SC is not affected and remains fairly constant over the entire Δk range.

A different situation is then expected at positive phase mismatch, where self-phase matching is still possible [44], but within a reduced θ range, wherein the spectral components of the fundamental pulse satisfying the self-phase matching condition are absent and so no self-phase matched SH is generated. Figure 6 illustrates the spectral measurements in the θ range of 10 – 28°, which demonstrate markedly different dynamics of spectral broadening, which were measured with elevated input pulse energies of 0.6 μJ and 1 μJ as due to reduced values of $n_2^{\text{eff}}$, resulting from a negative contribution of $n_2^{\text{casc}}$ [see Fig. 1(b)]. More specifically, we observe the control of both, blue-shifted and red-shifted spectral broadenings in the θ angle range of 17 – 25°. This result may be explained by the competition between quadratic and cubic self-steepenings, which in the present case are of opposite signs [40], resulting in reduction of the overall self-steepening, acting on both, leading and trailing fronts of the sub-pulses. As the phase mismatch increases (θ decreases), the contribution of quadratic self-steepening gradually ceases leading to increased blue and red shifts of the SC spectrum. Eventually, for θ < 17°, the overall spectral extent of the SC becomes fairly constant, defined by the cubic self-steepening only. Finally, an interesting, yet unexplained spectral behavior accompanied by a considerable spectral broadening of a weak phase-mismatched SH is captured within a narrow θ range of around θ = 25°, which is close to the region of self-defocusing.

 

Fig. 6 Experimentally measured spectra in the range of positive phase mismatch with the input pulse energies of (a) 0.6 μJ and (b) 1 μJ. The curve designations are the same as in Fig. 2. A weak signal centered at 400 nm is the phase-mismatched SH.

Download Full Size | PDF

3. Conclusion

In conclusion, we demonstrated fully controllable SC generation with 120 fs, 800 nm laser pulses in BBO crystal in the regime of second-order cascading-assisted filamentation. We uncover that the control of the SC spectral extent stems from the complex interplay between the self-phase-matched second harmonic generation and self-steepening processes which arise from cubic and cascaded-quadratic nonlinearities, and which bring remarkably different contributions to the spectral broadening in the range of negative and positive phase mismatch. In particular, we show that in the range of negative phase mismatch, the blue-shifted spectral broadening and the short wavelength cut-off of the SC spectrum is fully controlled in the 410 – 700 nm range by varying the phase mismatch parameter. The achieved spectral control is very robust in terms of the input pulse energy and is attributed to efficient generation of the self-phase-matched SH, which introduces a considerable energy loss and distortion of the trailing sub-pulse shape, counteracting the joint effect of cascaded-quadratic and cubic self-steepenings. In contrast, in the range of positive phase mismatch, where self-phase matched SH is absent and where the effective self-focusing nonlinearity is still prevailing, the control of the entire SC spectrum is demonstrated and is explained by the competition between the cubic and phase-mismatch-dependent cascaded-quadratic self-steepenings.

We also uncover a number of other interesting findings regarding the SH generation. In particular, we observe filamentation of the SH pulses at perfect phase matching, which leads to extraordinarily polarized SC, which spans from 300 nm to 610 nm and induces a considerable spectral broadening around the fundamental wavelength, ranging from 650 nm to 1 μm. In the range of negative phase mismatch, we also capture the self-phase-matched SH generation, which yields tunable ultraviolet radiation in the 300 – 400 nm range.

We expect that the discovered mechanism of the spectral control is universal and could be applied to any nonlinear crystal possessing both quadratic and cubic nonlinearities, opening the route to perform ultrafast nonlinear interactions over a wide spectral range in a simple and fully controlled fashion.