We demonstrate that monochromatic infrared laser pulses can generate polychromatic light in noncentrosymmetric crystals simultaneously covering the ultraviolet, visible, and infrared domains. The spatial shape of the beam and its energy can influence this multicolor frequency conversion, unveiling complex and interesting dynamics. We performed our experiments in a bulk crystal of periodically poled lithium niobate, working close to the optimal condition for second-harmonic generation. We used an input laser beam wide enough that, at very low intensities, the diffraction leaves its diameter unchanged along the propagation in the crystal. At high intensities instead, as we show in this work, such a spatially wide laser beam can be reshaped into a beam of much smaller diameter and guiding multispectral components. We also show how this outcome may permit exploitation of other parameters, like the crystal temperature, for tuning the spectrum of the generated multicolor light.
© 2015 Optical Society of America
Optical spatial solitary waves are self-trapped optical beams, in which the effect of diffraction is balanced by a self-induced and spatially varying nonlinear phase. Their behaviors and properties have been extensively studied both numerically and experimentally during the past several decades. Multicolor spatial soliton propagation mediated by the cascading of quadratic nonlinearities has been demonstrated in different types of crystals and in a variety of geometries. Spatial solitons are formed by the mutual trapping of waves parametrically interacting in a nonlinear medium; their interaction is ruled by the phase mismatch among the interacting waves, as will be discussed in the next sections. They can exist for a wide range of phase mismatches () including, at high intensities, also those values of at which one would expect to induce beam defocusing [1–8]. The observations of quadratic solitons have been limited to relatively narrowband spectra mainly around a fundamental wave (FH) and its second harmonic (SH). However, a guiding effect of a weak signal slightly detuned in wavelength from a fundamental beam was also demonstrated by means of coupling with a more powerful two-color quadratic spatial soliton .
Nonlinear quadratic crystals are to date mostly applied in industry for harmonic generation and parametric amplification [10,11]. The maximum conversion efficiency is achieved when the phase mismatch is zero (phase-matching) or zero on average in a periodic medium [quasi-phase-matching (QPM)]. Recent results have shown that nonlinear crystals are also excellent nonlinear media for broadband light generation, in particular when pumped by femtosecond pulses: their transparency window (from 200 nm to 4–5 μm) and their damage threshold are larger than those of glass fibers, which represent the most common solutions for supercontinuum (SC) sources. Moreover, nonlinear crystals provide a high nonlinear coefficient, which permits targeting frequency conversions in the ultraviolet (UV), visible (VIS), and infrared (IR) up to mid-infrared spectra in a few tens of millimeters of propagation [12–17]. Multioctave SC generation has been observed in gases and bulk transparent media with femtosecond pulses. These processes involved cascaded four-wave mixing among launched multiple harmonics through a filamentation process [18,19] or between two pumps in a noncollinear configuration .
Polychromatic solitary waves have been experimentally demonstrated in waveguide arrays (WAs). However, such multicolor self-trapping was either induced by the slow-time response of the photorefractive effect [21–23] or by using cubic nonlinearity in planar waveguides . In the first example, the input light launched into the WAs was already coming from a supercontinuum source, while in the second example only three beams, separated by a relatively small spectral detuning, were spatially self-guided.
Nonlinear parametric interactions in bulk media can also induce complex interplays that allow the generation of new types of spatiotemporal waveforms. Among others, recent works have investigated, for instance, the spatiotemporal reshaping induced by ionization in gases or the space–time nature of “light bullets” or rogue events in media with third-order nonlinearities [25–27]. Bulk quadratic crystals can also offer a great possibility to combine broadband generation and spatial beam shaping with power handling. Among numerous examples, the observations of colored conical emissions by means of spatiotemporal modulation instability or nonlinear X waves triggered by this mechanism have been reported in Refs. [1,28–30].
Spatial localization upon a large spectral domain can find potential applications in all-optical ultrafast signal processing, space–time resolved spectroscopy, or remote sensing. Further perspectives of complex multiplexing may be envisaged by considering interactions and collisions among self-trapped multicolor beams.
In our work, we used a bulk crystal of periodically poled 5 mol% MgO-doped congruent lithium niobate (PPLN) pumped by a highly intense laser emitting 30 ps pulses at 1064 nm. Our input beam was relatively wide (100 μm) and propagated without influence of the diffraction and dispersion, which is not the configuration typically used to excite spatial solitons. We demonstrated experimentally and numerically the formation of a quadratic soliton-like filament involving a threefold broadband spectrum over two octaves. We showed how such polychromatic light could remain self-sustained for both signs of phase mismatch thanks to the effect of light intensity. Although the problem of beam filamentation is well known for third-order nonlinearity, the generation of a beam filament by second-order nonlinearity carrying such large multispectral domains has not yet been discussed so far, to the best of our knowledge.
2. EXPERIMENTAL SETUP
Our experimental setup is shown in Fig. 1. We used a periodically poled PPLN crystal fabricated by the HCP Photonics Corporation. The sample was 15 mm long, 1 mm thick, and 2 mm wide. The grating, designed for type-0 (eee) QPM second-harmonic generation (SHG) at 1064 nm, had a period () of 6.92 μm. The crystal was mounted in a temperature-controlled oven, and the phase-matching conditions for SHG were obtained for a temperature of 50°C. The experiments were carried out with a high-energy -switched, mode-locked Nd:YAG laser delivering 30 ps pulses at 1064 nm with a repetition rate of 20 Hz. The beam had a Gaussian transverse profile and was polarized parallel to the extraordinary axis of the PPLN crystal to access to the largest nonlinear coefficient. The pump beam was focused at the input face of the PPLN sample with a beam diameter of 100 μm measured at half of the maximum intensity (FWHMI). In our experiment, the diffraction length () of the input beam at 1064 nm was then approximately 3 times larger than the length of the crystal (L). This in practice means that, under a pure linear regime (that is, at very low intensities), the beam propagates through the crystal, keeping the diameter nearly unchanged. Note that this is not the usual condition for exciting spatial solitons, where instead . A polarizer (Pol) placed between two half-wave plates (HWPs) was used to adjust the energy and the polarization state of the pump beam. The spatial shape of the beam was monitored by a CCD camera recording a magnified image of the near field at the output face of the PPLN crystal. Different optical filters were alternatively introduced to select and analyze the various spectral components: 10 nm wide passband filters and a 3 nm wide interference filter at 1064 nm. We used two different spectrum analyzers to explore the UV–VIS and the IR parts of the spectrum. We set different values of phase mismatch for SHG and input energies to explore the different interplays between space and frequency.
3. NUMERICAL SIMULATIONS
Our numerical simulations were performed by solving the following unidirectional scalar pulse propagation equation (UPPE) and considering only second-order nonlinearities:1) stems from Maxwell equations under the hypothesis of forward propagation only . To simplify the computational effort, we limited our analysis to a single transverse coordinate to account for the spatial domain. Here, is the temporal coordinate, is the propagation coordinate, and is the corresponding temporal angular frequency. is the second-order nonlinear susceptibility, which is periodically varying in our experiments because of the periodic inversion of the ferroelectric domains of the PPLN crystal. The periodic function of was modeled by using a Fourier series expansion truncated at the fifth order. We considered . Moreover, we have limited our analysis to the case in which all interacting waves are polarized along the extraordinary axis of the PPLN crystal. and are the transverse and longitudinal components of the wave vector , and is the linear refractive index, which is a function of and of temperature . We have used the Sellmeier equation for lithium niobate to account for both of these variables . is the bidimensional Fourier transform linking the electric field to its Fourier transform , which is function of the spatial frequency and the angular frequency . Although this single equation comes with a series of approximations (see, also, for details and mathematical methods Refs. [33–35]) it permits us to model the simultaneous broadening of spectral and spatial frequencies, giving results in good qualitative agreement with the experiments, as will be discussed in Section 4. In our simulations, we neglected the material absorption. We considered pump pulses of 15 ps at 1064 nm. Moreover, following the experimental setup, we used an input beam diameter of 100 μm, crystal length of 15 mm, and phase-matching temperature for SHG of 50°C.
We verified the accuracy of 1D approximation of our UPPE model by comparing it to the coupled mode equations with 1D and 2D spatial components, and with a spectrum sufficiently large to account for the spectral broadening around FH and SH.
4. RESULTS AND DISCUSSION
We performed our experiments under various phase-mismatch values from to that we obtained with temperatures from 25°C to 160°C. We found the existence of two different regimes of nonlinear propagation, characterized by two different spatiospectral fingerprints and discriminated by an intensity threshold (). This threshold depended on the crystal temperature (i.e., on ) as shown in Fig. 2: was much lower and kept a nearly constant value () for all positive phase-mismatch values, as well as for some weakly negative values close to the exact phase matching (), where the efficiency of the parametric nonlinear coupling was still strong, whereas, increased rapidly, moving further to the negative side of phase mismatch.
Figure 3(d) shows how a FWHMI diameter of the fundamental beam, which was measured at the output of the crystal, varies upon the crystal temperature (i.e., upon ) when the pump intensity () was chosen below and above the threshold. In the first case, when , as demonstrated by a blue curve in Fig. 3(d), we observed a standard spatial self-defocusing/self-focusing of the FH with a corresponding increase/decrease of its output beam diameter for negative/positive . We indicate this scenario as Regime I of low conversion: the spatial beam shaping took place with narrowband optical spectra around the fundamental and SH wave [2,7,36–38]. Indeed the SHG cascading effect (i.e., when the pump and its second harmonic exchange periodically energy with low conversion) leads to the accumulation of a nonlinear spatial phase shift at the FH. However, when we increased the pump intensity above the threshold value , as shown by the red curve in Fig. 3(d), we observed a completely different and unusual dynamics of the nonlinear spatial reshaping. We refer to this second type of dynamics as Regime II of strong conversion: at those high intensities, for negative values of phase mismatch, the central part of the previously defocused beam started to decrease in diameter up to the formation of a parametric self-trapped wave. This effect was well reproduced by our numerical simulations presented in Figs. 4(a)–4(c): they demonstrate a spatial evolution of the fundamental beam upon input pump intensity at (i.e., 53°C). The efficiency of such process at the FH decreased when the phase mismatch grew larger in negative values [2,7,39]: experimental beam profiles at (i.e., 53°C) and (i.e., 60°C) are shown in Figs. 3(b) and 3(c), respectively. On the other hand, for positive phase mismatch, the self-focusing of Regime I was reinforced and the beam diameter remained 4 times smaller than that of the input one; a corresponding spatial beam shape, measured at (i.e., 40°C), is presented in Fig. 3(a). When Regime II was attained, we were able to keep trapped the beam disregarding the sign of phase mismatch thanks to a significant increase of the acceptance bandwidth of SHG occurring at high intensities, as was reported in Ref. . In our case, we can call this waveform a quadratic spatial filament, by extension of the known concept of filamentation in Kerr media and in contrast to the concept of a quadratic spatial soliton, mostly reported so far [2,7] in which the beam shape preserves its input diameter.
In standard spatial soliton experiments, the input excitation was commonly chosen in such a way that the diffraction length () was significantly shorter than the length of the nonlinear medium in order to demonstrate soliton propagation, balancing the diffraction by the nonlinear phase shift. In our experiment instead, at the input of the crystal, there was a marked disproportion between nonlinearity and diffraction: the diffraction length () was 3 times larger than the length of the PPLN sample. Therefore, in our case, at high intensities (Regime II) the self-trapped beam exhibited an output diameter much smaller than that at the input (from 100 μm down to a stable value of 25 μm). This fact can be seen as the equivalent of a spontaneous convergence toward a given quadratic spatial solitary filament, determined entirely by the nonlinear conversion dynamics, which is a situation rather different from the concept of spatial solitons. Moreover, it is important to emphasize that, different from what was reported in previous works, in our experiment we were able to significantly change the dynamics of the transition between the low and strong conversion regimes: in the regime of high intensities (Regime II), the spatial reshaping of the beam appeared strongly mixed with an atypical broadband spectral generation with the emergence of lateral sidebands around the FH and its second (SH) and third (TH) harmonic [40,41]. This is also in a good agreement with the numerical results shown in Fig. 4, where the spectral broadening [see Fig. 4(d), red curve] comes jointly with the formation of the spatial filament [see Fig. 4(c)], i.e., at high pump intensities.
We present in Fig. 5 a paradigmatic example of such nonlinear multiple frequency conversion, generated at small negative phase-mismatch values (, i.e., 53°C). At high intensities (Regime II), we obtained a broadband generation between 950 and 1250 nm, between 460 and 600 nm, and between 330 and 380 nm. The IR and UV–VIS parts of the spectrum resulted from the concatenation of several parametric processes requiring simultaneity in time, as illustrated in Fig. 6. In particular, after initial conversion of the IR pump toward its SH and then its TH by sum frequency generation [SFG, see Fig. 6(a)], part of that energy was converted back toward the IR domain. As a consequence of nonzero for SHG, the SH did not get phase matched to the FH. Instead, a pair of sidebands around FH was induced by phase-matched difference frequency generation (DFG): their central wavelengths are indicated by and [see Fig. 6(b)]. The nonlinear process proceeded further by generating a pair of sidebands around the SH by SFG between the FH and IR sidebands, indicated as and [see Fig. 6(c)]. Finally, a similar pair of sidebands around TH was induced as and [see Fig. 6(d)]. Subscripts L and R on wavelengths identify the anti-Stokes and Stokes, respectively. As a consequence, a threefold spectral broadening located in the vicinity of the FH and its SH and TH was obtained. The measurements [see Figs. 5(a) and 5(b)] and the spatiotemporal numerical simulations (see Figs. 5(c) and 5(d)] are qualitatively in good agreement.
Similar behaviors were also obtained for larger negative and positive phase-mismatch values and pump intensities chosen accordingly. A series of experimental results are presented in Fig. 7. Figures 7(b) and 7(d) show that the higher is the PPLN temperature (more negative values of ), the narrower are the spectral sidebands. Their relative frequency detuning from the pump, SH, and TH also grows larger . Thus, by changing the temperature of the crystal, we could control the shape of the spectrum, moving from a flat continuum light to a discrete series of sharp and bright bands. When instead we decreased the PPLN temperature, i.e., for positive values of (i.e., at the self-focusing), we observed that the spectrum no longer exhibited any significant changes, disregarding the specific value of . The results are displayed in Figs. 7(a) and 7(c). The spectral shape thus became nearly insensitive to a change in the PPLN temperature. For , we observed experimentally the presence of additional sharp peaks in the spectrum, clearly visible in the IR domain far from phase matching (, i.e., 25°C), where the frequency conversion lost its efficiency.
To demonstrate the strong interplay between the spatial and spectral domains, we carried out a systematic study of the spatial distribution of the different spectral components for different values of and different pump intensities (Regime II). In addition to the FH (at 1064 nm) and its SH (at 532 nm) and TH (at 355 nm), we analyzed their related sidebands, selecting the wavelengths at 1000, 500, and 550 nm. Obtained results are presented in Fig. 8 for a pump intensity 2 times higher than the minimum intensity required to observe the spectral broadening . We can observe that the beam diameter for each of these colors was significantly smaller than that of the initial pump of 100 μm, with an average value of 25 μm (dotted line in Fig. 8). Note that the multispectral beam was self-confined despite the negative . For instance, at small negative phase mismatch (, i.e., 53°C), the output beam diameter measured at FWHM of its intensity became 28 μm at 355 nm, 24 μm at 500 nm, 25 μm at 532 nm, 27 μm at 550 nm, 19 μm at 1000 nm, and 40 μm at 1064 nm. Remarkably, the beam diameters of all these parametrically generated colors remained unchanged for input intensities up to 5 times higher than . In Fig. 9 we present the FWHMI output beam diameters of the analyzed colors as a function of the input pump intensity measured at (i.e., 53°C). Such a peculiar behavior remained valid over a wide range (more than ) of phase-mismatch values around the phase-matching point (see Fig. 8). The level of the spectral power density of the multicolor light was more than 20 dB below that of the pump at 1064 nm. Such a wideband self-guiding effect could be explained by the simultaneous parametric wave-mixing processes with a quadratic 1064 nm/532 nm spatial quasi-solitary filament taking place in a strong nonlinear conversion regime [8,9]. This quadratic filament transferred a nonlinear phase shift toward all generated wavelengths, when the parametric interactions among them reached a given efficiency. Note that, in our experiment, the SH component of the quasi-solitary filament had a 25 μm diameter (FWHMI), which resulted in decreasing the diffraction length for the analyzed wavelengths by more than 4 times, which is much below the length of the PPLN sample. Self-trapping of the generated light overcame the effect of diffraction and led to the formation of a wide-range multicolor spatial solitary filament.
Figure 10 shows the spatial profile of the different spectral components at the output of the 15 mm long PPLN crystal in the strong conversion regime (Regime II). In particular, we present the measured 2D output intensity images, as well as one of their radial sections (vertical profiles); for reference, we also show the input shape of the beam, which remained substantially unchanged at the output at low intensity since . The experimental results were obtained at 53°C (i.e., ) and (that is, at ). The output spatial profile of the FH is compared to the initial beam profile [Fig. 10(a), black dashed curve]. We clearly observe experimentally a strong spatial reshaping in the core of the beam due to the local intensity [see also Fig. 3(b)], as is also numerically demonstrated in Fig. 4(c). Let us remind the reader that, when grew larger in negative values, this effect became less effective and only a small portion of the central part of the beam had a contraction in diameter [see Fig. 3(c)], which was, however, enough to trap the other generated colors. Note that this fact can explain the larger, and growing with negative , FWHMI diameter value of the fundamental component of the observed self-confined multispectral filament [red curve in Fig. 3(d), bright blue asterisks in Figs. 8 and 9]. In fact, for , the beam at 1064 nm exhibited a consistent narrowing seen above the half-maximum level. The measurements reported in Fig. 10(a) are in good agreement with our simulations presented in Fig. 10(b). In both cases we perceived such intriguing behavior of spatial self-trapping on a large spectral domain.
In this work we studied the spatial–spectral nonlinear dynamics in PPLN crystals in the regime of high pump intensities. We reported the observation of a strong nonlinear coupling self-induced in the crystal between the spatial profile of the beam and its broadband optical spectrum. We showed what we believe is the first experimental demonstration of a broadband multicolor two-dimensional spatial solitary filament over a wide range of 1000 nm by means of second-order nonlinearity in PPLN crystal. We highlighted that, in our experiment, the formation of such polychromatic quadratic filament was possible even at negative values of phase mismatch. We measured the temperature-dependent intensity threshold, above which complex nonlinear spatial reshaping appears combined with broadband conversion in the IR, VIS, and UV domains based on temporal quadratic modulation instability by only using a single monochromatic pump at 1064 nm. We demonstrated that the crystal temperature could control the shape of such a threefold broadband spectrum, while this polychromatic beam remained focused. Numerical simulations qualitatively confirmed our experimental results.
Such an intrinsic coupling between spectral broadening and spatial reshaping might be used to develop polychromatic laser sources in the UV–VIS domain. To generate a high-energy broadband spectrum, it may be more advantageous to work with a small negative mismatch, which provides relatively small threshold for frequency conversion (), permitting an increase in pump intensity: the residual self-defocusing observed in the tails of the beam might prevent the beam from reaching the damage threshold of the crystal. Such a spatial self-defocusing may be then potentially used as a mechanism of self-protection of the nonlinear crystal when generating a broadband spectrum. The opposite situation of positive phase mismatch is also interesting for applications: one could potentially exploit the resulting enhanced spatial self-confinement offered by a positive mismatch to minimize the input energy required for large-band frequency conversions.
Army Research Office (ARO) (W911NF-11-0297); BpiFrance; Ministero dell’Istruzione, dell’Università e della Ricerca (MIUR) (2012BFNWZ2).
1. A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002). [CrossRef]
2. W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995). [CrossRef]
3. P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001). [CrossRef]
4. B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B 7, R53–R72 (2005). [CrossRef]
5. F. Baronio, C. De Angelis, P. Pioger, V. Couderc, and A. Barthélémy, “Reflection of quadratic solitons at the boundary of nonlinear media,” Opt. Lett. 29, 986–988 (2004). [CrossRef]
6. V. Couderc, E. Lopez Lago, C. Simos, and A. Barthélémy, “Experiments in quadratic spatial soliton generation and steering in a noncollinear geometry,” Opt. Lett. 26, 905–907 (2001). [CrossRef]
7. B. Bourliaguet, V. Couderc, A. Barthélémy, G. W. Ross, P. G. R. Smith, D. C. Hanna, and C. De Angelis, “Observation of quadratic spatial solitons in periodically poled lithium niobate,” Opt. Lett. 24, 1410–1412 (1999). [CrossRef]
8. P. Pioger, V. Couderc, L. Lefort, A. Barthélémy, F. Baronio, C. De Angelis, Y. Min, V. Quiring, and W. Sohler, “Spatial trapping of short pulses in Ti-indiffused LiNbO3 waveguides,” Opt. Lett. 27, 2182–2184 (2002). [CrossRef]
9. V. Couderc, E. Lopez Lago, A. Barthélémy, C. De Angelis, and F. Gringoli, “Trapping of a weak probe through coupling with a two-color quadratic spatial soliton,” Opt. Commun. 203, 421–425 (2002). [CrossRef]
10. M. L. Bortz, M. Fujimura, and M. M. Fejer, “Increased acceptance bandwidth for quasi-phasematched second harmonic generation in LiNbO3 waveguides,” Electron. Lett. 30, 34–35 (1994). [CrossRef]
11. G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74, 1–18 (2003). [CrossRef]
12. M. Zhi and A. V. Sokolov, “Broadband coherent light generation in a Raman active crystal driven by two-color femtosecond laser pulses,” Opt. Lett. 32, 2251–2253 (2007). [CrossRef]
13. N. K. M. Naga Srinivas, S. Sree Harsha, and D. Narayana Rao, “Femtosecond supercontinuum generation in a quadratic nonlinear medium (KDP),” Opt. Express 13, 3224–3229 (2005). [CrossRef]
14. C. Langrock, M. M. Fejer, I. Hartl, and M. E. Fermann, “Generation of octave-spanning spectra inside reverse-proton-exchanged periodically poled lithium niobate waveguides,” Opt. Lett. 32, 2478–2480 (2007). [CrossRef]
15. B. Q. Chen, M. L. Ren, R. J. Liu, C. Zhang, Y. Sheng, B. Q. Ma, and Z. Y. Li, “Simultaneous broadband generation of second and third harmonics from chirped nonlinear photonic crystals,” Light Sci. Appl. 3, e189 (2014).
16. B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, “Ultrafast and octave-spanning optical nonlinearities from strongly phase-mismatched quadratic interactions,” Phys. Rev. Lett. 109, 043902 (2012). [CrossRef]
17. B. B. Zhou, H. R. Guo, and M. Bache, “Energetic mid-IR femtosecond pulse generation by self-defocusing soliton-induced dispersive waves in a bulk quadratic nonlinear crystal,” Opt. Express 23, 6924–6936 (2015). [CrossRef]
18. P. B. Petersen and A. Tokmakoff, “Source for ultrafast continuum infrared and terahertz radiation,” Opt. Lett. 35, 1962–1964 (2010). [CrossRef]
19. T. Fuji and Y. Nomura, “Generation of phase-stable sub-cycle mid-infrared pulses from filamentation in nitrogen,” Appl. Sci. 3, 122–138 (2013). [CrossRef]
20. H. Crespo and J. T. Mendonça, “Cascaded highly nondegenerate four-wave-mixing phenomenon in transparent isotropic condensed media,” Opt. Lett. 25, 829–831 (2000). [CrossRef]
21. D. N. Neshev, A. A. Sukhorukov, A. Dreischuh, R. Fischer, S. Ha, J. Bolger, L. Bui, W. Krolikowski, B. J. Eggleton, A. Mitchell, M. W. Austin, and Y. S. Kivshar, “Nonlinear spectral-spatial control and localization of supercontinuum radiation,” Phys. Rev. Lett. 99, 123901 (2007). [CrossRef]
22. X. Qi, I. L. Garanovich, A. A. Sukhorukov, W. Krolikowski, A. Mitchell, G. Zhang, D. N. Neshev, and Y. S. Kivshar, “Polychromatic solitons and symmetry breaking in curved waveguide arrays,” Opt. Lett. 35, 1371–1373 (2010). [CrossRef]
23. D. N. Neshev, A. Dreischuh, V. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Observation of polychromatic vortex solitons,” Opt. Lett. 33, 1851–1853 (2008). [CrossRef]
24. G. Fanjoux, J. Michaud, M. Delqué, H. Maillotte, and T. Sylvestre, “Generation of multicolor vector Kerr solitons by cross-phase modulation, four-wave mixing, and stimulated Raman scattering,” Opt. Lett. 31, 3480–3482 (2006). [CrossRef]
25. Z. H. He, J. A. Nees, B. Hou, K. Krushelnick, and A. G. R. Thomas, “Ionization-induced self-compression of tightly focused femtosecond laser pulses,” Phys. Rev. Lett. 113, 263904 (2014). [CrossRef]
26. D. Majus, G. Tamosauskas, I. Grazuleviciute, N. Garejev, A. Lotti, A. Couairon, D. Faccio, and A. Dubietis, “Nature of spatiotemporal light bullets in bulk Kerr media,” Phys. Rev. Lett. 112, 193901 (2014). [CrossRef]
27. S. Birkholz, E. T. J. Nibbering, C. Bree, S. Skupin, A. Demircan, G. Genty, and G. Steinmeyer, “Spatiotemporal rogue events in optical multiple filamentation,” Phys. Rev. Lett. 111, 243903 (2013). [CrossRef]
28. S. Trillo, C. Conti, P. Di Trapani, O. Jedrkiewicz, J. Trull, G. Valiulis, and G. Bellanca, “Colored conical emission by means of second-harmonic generation,” Opt. Lett. 27, 1451–1453 (2002). [CrossRef]
29. H. Zeng, J. Wu, H. Xu, K. Wu, and E. Wu, “Colored conical emission by means of second-harmonic generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 92, 143903 (2004). [CrossRef]
30. C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90, 170406 (2003). [CrossRef]
31. M. Conforti, F. Baronio, M. Levenius, and K. Gallo, “Broadband parametric processes in χ(2) nonlinear photonic crystals,” Opt. Lett. 39, 3457–3460 (2014). [CrossRef]
32. D. N. Nikogosyan, Nonlinear Optical Crystals: A Complete Survey (Springer, 2005).
33. M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A 81, 053841 (2010). [CrossRef]
34. S. Wabnitz and V. V. Kozlov, “Harmonic and supercontinuum generation in quadratic and cubic nonlinear optical media,” J. Opt. Soc. Am. B 27, 1707–1711 (2010). [CrossRef]
35. M. Conforti, F. Baronio, and C. De Angelis, “Ultra-broadband optical phenomena in quadratic nonlinear media,” IEEE Photon. J. 2, 600–610 (2010). [CrossRef]
36. K. Krupa, F. Baronio, M. Conforti, S. Trillo, A. Tonello, and V. Couderc, “Zero focusing via competing nonlinearities in beta-barium-borate crystals,” Opt. Lett. 39, 925–928 (2014). [CrossRef]
37. F. Wise and J. Moses, “Self-focusing and self-defocusing of femtosecond pulses with cascaded quadratic nonlinearities,” in Self-Focusing: Past and Present, R. W. Boyd, S. L. Lukishova, and Y. R. Shen, eds. (Springer, 2009), pp. 481–506.
38. A. V. Buryak, Y. S. Kivshar, and S. Trillo, “Parametric spatial solitary waves due to type II second-harmonic generation,” J. Opt. Soc. Am. B 14, 3110–3118 (1997). [CrossRef]
39. K. Krupa, R. Fona, A. Tonello, A. Labruyère, B. M. Shalaby, S. Wabnitz, and V. Couderc, “Self-increased acceptance bandwidth of second harmonic generation for high-energy light sources,” in International Workshop on Spatiotemporal Complexity in Nonlinear Optics (SCNO), Como, Italy, August 31, 2015, paper 3843079.
40. K. Krupa, A. Labruyère, B. M. Shalaby, A. Tonello, F. Baronio, and V. Couderc, “Broadband light generation in PPLN crystal and its interplay with spatial nonlinear effects,” in Nonlinear Photonics, OSA Technical Digest (Optical Society of America, 2014), paper NTu2A.2.
41. K. Krupa, A. Labruyère, B. M. Shalaby, A. Tonello, F. Baronio, and V. Couderc, “Complex spatial and spectral evolutions in cascaded second-order nonlinear process,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2014), paper JTh2A.107.
42. S. Trillo and P. Ferro, “Modulational instability in second-harmonic generation,” Opt. Lett. 20, 4438–4440 (1995). [CrossRef]