Synchrotron resonant radiation from nonlinear self-accelerating pulses

Solitons and nonlinear waves emit resonant radiation in the presence of perturbations. This effect is relevant for nonlinear fiber optics, supercontinuum generation, rogue waves, and complex nonlinear dynamics. However, resonant radiation is narrowband, and the challenge is finding novel ways to generate and tailor broadband spectra. We theoretically predict that nonlinear self-accelerated pulses emit a novel form of synchrotron radiation that is extremely broadband and controllable. We develop an analytic theory and confirm the results by numerical analysis. This new form of supercontinuum generation can be highly engineered by shaping the trajectory of the nonlinear self-accelerated pulses. Our results may find applications in novel highly efficient classical and quantum sources for spectroscopy, biophysics, security, and metrology. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement OCIS codes: (190.0190) Nonlinear optics; (190.4370) Nonlinear optics, fibers; (320.5540) Pulse shaping; (350.5610) Radiation. References and links 1. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). 2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007). 3. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979). 4. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008). 5. I. Kaminer, Y. Lumer, M. Segev, and D. N. Christodoulides, “Causality effects on accelerating light pulses,” Opt. Express 19(23), 23132–23139 (2011). 6. D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010). 7. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010). 8. C. Ament, P. Polynkin, and J. V. Moloney, “Supercontinuum generation with femtosecond self-healing Airy pulses,” Phys. Rev. Lett. 107(24), 243901 (2011). 9. C. Ament, M. Kolesik, J. V. Moloney, and P. Polynkin, “Self-focusing dynamics of ultraintense accelerating Airy waveforms in water,” Phys. Rev. A 86(4), 043842 (2012). 10. L. Zhang and H. Zhong, “Modulation instability of finite energy Airy pulse in optical fiber,” Opt. Express 22(14), 17107–17115 (2014). 11. L. Zhang, H. Zhong, Y. Li, and D. Fan, “Manipulation of Raman-induced frequency shift by use of asymmetric self-accelerating Airy pulse,” Opt. Express 22(19), 22598–22607 (2014). 12. A. Liu, G. Liu, J. Zhang, L. Zhang, and Y. Chen, “Dynamic propagation of finite-energy Airy pulses in the presence of higher-order effects,” J. Opt. Soc. Am. B 31(4), 889–897 (2014). 13. Y. Hu, A. Tehranchi, S. Wabnitz, R. Kashyap, Z. Chen, and R. Morandotti, “Improved intrapulse raman scattering control via asymmetric airy pulses,” Phys. Rev. Lett. 114(7), 073901 (2015). 14. S. Courvoisier, N. Götte, B. Zielinski, T. Winkler, C. Sarpe, A. Senftleben, L. Bonacina, J. P. Wolf, and T. Baumert, “Temporal Airy pulses control cell poration,” APL Photonics 1(4), 91–99 (2016). Vol. 26, No. 11 | 28 May 2018 | OPTICS EXPRESS 14710 #328191 https://doi.org/10.1364/OE.26.014710 Journal © 2018 Received 12 Apr 2018; accepted 16 May 2018; published 24 May 2018 15. N. Götte, T. Winkler, T. Meinl, T. Kusserow, B. Zielinski, C. Sarpe, A. Senftleben, H. Hillmer, and T. Baumert, “Temporal Airy pulses for controlled high aspect ratio nanomachining of dielectrics,” Optica 3(4), 389 (2016). 16. Y. Fattal, A. Rudnick, and D. M. Marom, “Soliton shedding from Airy pulses in Kerr media,” Opt. Express 19(18), 17298–17307 (2011). 17. J. A. Giannini and R. I. Joseph, “The role of the second Painlevé transcendent in nonlinear optics,” Phys. Lett. A 141(8), 417–419 (1989). 18. I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. 106(21), 213903 (2011). 19. A. Lotti, D. Faccio, A. Couairon, D. G. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, “Stationary nonlinear Airy beams,” Phys. Rev. A 84(2), 021807 (2011). 20. R. Bekenstein and M. Segev, “Self-accelerating optical beams in highly nonlocal nonlinear media,” Opt. Express 19(24), 23706–23715 (2011). 21. I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental observation of self-accelerating beams in quadratic nonlinear media,” Phys. Rev. Lett. 108(11), 113903 (2012). 22. Y. Hu, S. Huang, P. Zhang, C. Lou, J. Xu, and Z. Chen, “Persistence and breakdown of Airy beams driven by an initial nonlinearity,” Opt. Lett. 35(23), 3952–3954 (2010). 23. P. K. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers,” Opt. Lett. 11(7), 464–466 (1986). 24. N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51(3), 2602–2607 (1995). 25. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation is photonic crystal fiber,” Rev. Mod. Phys. 78(4), 1135–1184 (2006). 26. V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. H. Pfeiffer, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip-based optical frequency comb using soliton Cherenkov radiation,” Science 351(6271), 357–360 (2016). 27. N. Y. Joly, J. Nold, W. Chang, P. Hölzer, A. Nazarkin, G. K. Wong, F. Biancalana, and P. S. Russell, “Bright spatially coherent wavelength-tunable deep-UV laser source using an Ar-filled photonic crystal fiber,” Phys. Rev. Lett. 106(20), 203901 (2011). 28. K. F. Mak, J. C. Travers, P. Hölzer, N. Y. Joly, and P. S. Russell, “Tunable vacuum-UV to visible ultrafast pulse source based on gas-filled Kagome-PCF,” Opt. Express 21(9), 10942–10953 (2013). 29. G. Chang, L. J. Chen, and F. X. Kärtner, “Highly efficient Cherenkov radiation in photonic crystal fibers for broadband visible wavelength generation,” Opt. Lett. 35(14), 2361–2363 (2010). 30. D. Novoa, M. Cassataro, J. C. Travers, and P. S. Russell, “Photoionization-induced emission of tunable fewcycle midinfrared dispersive waves in gas-filled hollow-core photonic crystal fibers,” Phys. Rev. Lett. 115(3), 033901 (2015). 31. I. Babushkin, A. Tajalli, H. Sayinc, U. Morgner, G. Steinmeyer, and A. Demircan, “Simple route toward efficient frequency conversion for generation of fully coherent supercontinum in the mid-IR and UV range,” Light Sci. Appl. 6(2), e16218 (2017). 32. E. Rubino, J. McLenaghan, S. C. Kehr, F. Belgiorno, D. Townsend, S. Rohr, C. E. Kuklewicz, U. Leonhardt, F. König, and D. Faccio, “Negative-frequency resonant radiation,” Phys. Rev. Lett. 108(25), 253901 (2012). 33. T. X. Tran and F. Biancalana, “Diffractive resonant radiation emitted by spatial solitons in waveguide arrays,” Phys. Rev. Lett. 110(11), 113903 (2013). 34. D. V. Skryabin, Y. V. Kartashov, O. A. Egorov, M. Sich, J. K. Chana, L. E. Tapia Rodriguez, P. M. Walker, E. Clarke, B. Royall, M. S. Skolnick, and D. N. Krizhanovskii, “Backward Cherenkov radiation emitted by polariton solitons in a microcavity wire,” Nat. Commun. 8(1), 1554 (2017). 35. C. R. Lourés, T. Roger, D. Faccio, and F. Biancalana, “Super-resonant radiation stimulated by high-harmonics,” Phys. Rev. Lett. 118(4), 043902 (2017). 36. L. G. Wright, S. Wabnitz, D. N. Christodoulides, and F. W. Wise, “Ultrabroadband dispersive radiation by spatiotemporal oscillation of multimode waves,” Phys. Rev. Lett. 115(22), 223902 (2015). 37. A. Bendahmane, F. Braud, M. Conforti, B. Barviau, A. Mussot, and A. Kudlinski, “Dynamics of cascaded resonant radiations in a dispersion-varying optical fiber,” Optica 1(4), 243–249 (2014). 38. C. Brée, I. Babushkin, U. Morgner, and A. Demircan, “Regularizing aperiodic cycles of resonant radiation in filament light bullets,” Phys. Rev. Lett. 118(16), 163901 (2017). 39. Y. Hu, Z. Li, B. Wetzel, R. Morandotti, Z. Chen, and J. Xu, “Cherenkov Radiation control via self-accelerating wave-packets,” Sci. Rep. 7(1), 8695 (2017). 40. X. Liu, A. S. Svane, J. Lægsgaard, H. Tu, S. A. Boppart, and D. Turchinovich, “Progress in Cherenkov femtosecond fiber lasers,” J. Phys. D Appl. Phys. 49(2), 023001 (2016). 41. K. C. Li, L. L. Huang, J. H. Liang, and M. C. Chan, “Simple approach to three-color two-photon microscopy by a fiber-optic wavelength convertor,” Biomed. Opt. Express 7(11), 4803–4815 (2016). 42. P. Hannaford, Femtosecond Laser Spectroscopy (Springer US, 2005). 43. M. R. Zaghloul and A. N. Ali, “Algorithm 916: Computing the Faddeyeva and Voigt Functions,” Acm. Tran. Math. Software 38(2), 1–22 (2012). Vol. 26, No. 11 | 28 May 2018 | OPTICS EXPRESS 14711


Introduction
Finite energy Airy pulses are non-spreading electromagnetic expressed as a truncated Airy function [1]. They are the temporal counterpart of finite energy Airy beam [1,2], firstly introduced in the context of quantum mechanics [3]. Similar to Airy beams, Airy pulses also display truly remarkable properties such as quasi-nondispersive evolution, self-reconstruction and self-acceleration [1,2,4]. Compared to the self-bending trajectory of Airy beams, Airy pulses exhibit self-accelerating or self-decelerating dynamics [5], and have asymmetric temporal profile with rapidly oscillating tails due to the cubic phase modulation. Various authors have shown that Airy pulses have many exciting applications, including linear light bullets generation [6,7], supercontinuum generation [8], self-focusing dynamics [9,10], manipulation of Raman-induced frequency effects [11][12][13], optimizing laser-cell membrane interactions [14], laser processing [15], and more. Previous investigations on RRs considered pulses with symmetric profiles as Gaussian and hyperbolic secant pulses. More recently, RRs emitted from a self-accelerating wavepackets have been also reported [39]. However, the investigation is restricted to the linear Airy pulse. An open question concerns the dynamics of RRs from the nonlinear selfaccelerating solitons. At a first analysis, one can expect that this asymmetric Airy-like pulse may lead to fairly non-trivial emission of RRs, but, to the best of our knowledge, RRs process from the nonlinear self-accelerating solitons has not been reported before.
Here we show that RRs emission from Airy pulses has a very broadband structure, due to the curved energy path of the nonlinear self-accelerating solitons. Due to the time varying velocity of these pulses, the resonant emission can be fairly more complex than for standard solitons, and the results are extremely relevant for novel broadband emission sources. Moreover, exploring these pulses in combination with RRs may improve existing application as wavelength conversion [40], biophotonics [41] and supercontinuum generation [8,25], also including quantum control [42].

Theoretical model and nonlinear self-accelerating pulses
We consider the generalized nonlinear Schrödinger equation ( Figure 1(a) shows the nonlinear self-accelerating solution with varying amplitude A (see Ref. 18). Their maximum amplitude and corresponding position first increase exponentially and then become saturated with an increasing A , as shown in Fig. 1(b). It should be pointed out that such nonlinear self-accelerating solitons are obtained numerically and have infinite energy. We use the expression of linear truncated Airy pulse as an input pulse for analytical analysis. Based on these, we fix 2 4 W t z = − , and find a perturbative solution to Eq. (1) by considering a modified finite energy Airy pulse a slowly varying term, with 1 C = . The energy of the Airy pulse is  Figure 1(d) plots the nonlinear phase shift as a function of the parameter a and the energy E of the Airy pulse. We verified that the Airy pulse is robust with respect to self-phase modulation by numerical simulation of Eq. (1).

Results and discussions
We then consider the effect of higher-order dispersion on the nonlinear propagation of Airy pulse. Following the approach of Akhmediev and Karlsson [24], we write the solution as u f + where f is a small perturbation that satisfies the following equation ω ω is the nonlinear phase-mismatch including the nonlinear wavevector NL k . We note that for a large range of values of a . From Eq. (4) we obtain a number of theoretical predictions, which can be written analytically by approximating because of the exponential factor in the integral that rapidly decay with respect to z . These approximations are validated below by the comparison with the numerical simulation of Eq. (1). We first consider the case of a very long propagation L → ∞ and we study the amount of generated energy at the phase-matched frequency 0 k Δ = (we also neglect the quadratic and cubic term in the phase). We find Note that because of the exponential function, the generated content does not always increase with L but saturates at a maximum value after the pump Airy pulse has spread upon evolution. We then consider the normalized spectrum of the generated frequencies Neglecting the cubic term z , Eq. (7) is written in closed form as the Faddeyeva function, commonly employed in plasma physics [43], In Fig. 2 we show the plot of the generated frequencies when varying the truncated coefficient of the Airy pulse. At strong truncation ( 10 a = ) the spectrum shows distinctive peaks which correspond to the phase-matched resonant radiation as in the case of standard solitons in the case of third-order dispersion (TOD) [ Fig. 2(a)] and fourth-order dispersion (FOD) [ Fig.  2(b)]. When the truncated coefficient decreases ( 1 a = ) satellites peak arises, which ultimately broadens in a large spectral emission mimicking the typical feature of the broadband synchrotron emission ( 0.1 a = ). This behavior is found in the presence of third order dispersion, when the spectrum is asymmetrical ( 3 =0.03 δ , 4 0 δ = ), and for fourth order dispersion, when the spectrum is symmetrical as far as the resonant radiation is not dominant It should be pointed out that, in our simulations, we use a truncated nonlinear Airy pulse because nonlinear self-accelerating pulses is obtained numerically and have infinite energy. The temporal truncated positon c t is introduced to keep ( ) 0 c u t t < = . The energy of truncated nonlinear Airy pulse is finite and increases with an increasing c t . We test our theoretical model by solving Eq. (1) for various parameters and including TOD and FOD; as shown in Fig. 3 we find a good agreement of the generated spectrum with Eq. (8). As predicted, when the truncated coefficient of the Airy pulse decreases we observe a very broadband emission due to synchrotron emission.   In our theory, it is possible to detail the dynamics of the generation of frequencies versus the propagation distance z . We adopt the stationary phase method in Eq. (7) and consider the phase term in the integral

Conclusion
In conclusion, we have theoretically predicted and numerically verified that the curved trajectory of an Airy beam in the time domain allows for generating very broadband emission. This effect strongly resembles synchrotron radiation, which is the broadband emission of electromagnetic radiation by charged particles accelerated on curved trajectories. In simple terms, the spectral content can be explained by resonant radiation emission, because the solitonic dynamics of the Airy pulse emit resonant frequencies, as it happens for standard solitons propagating at a constant speed. However, as the velocity of the Airy pulse changes with time, also the emitted frequencies change, resulting in a very broad band emission. We reported an analytic theory for the generated spectrum and its dynamics.
As the curved trajectory of the pulses can be optimized by a proper design of the fiber parameters, eventually varying with respect to propagation distance by dispersion management, one can produce oscillating pulses with periodic velocities or time dependent accelerations. The generated spectrum will depend on the specific trajectory and can be predicted by the theory here reported. We expect that this specific approach may lead to novel broadband sources and supercontinuum emission with engineered spectral for applications in spectroscopy, metrology and quantum sources.