Ultrabroadband supercontinuum and third-harmonic ...2013).pdf · third-harmonic generation in bulk...

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Ultrabroadband supercontinuum and third-harmonic generation in bulk solids with two optical-cycle carrier-envelope phase-stable pulses at 2 μ m Julius Darginaviˇ cius, 1 Donatas Majus, 1 Vytautas Jukna, 1,2 Nail Garejev, 1 Gintaras Valiulis, 1 Arnaud Couairon, 2 and Audrius Dubietis 1,1 Department of Quantum Electronics, Vilnius University, Saul˙ etekio Ave. 9, Building 3, LT-10222 Vilnius, Lithuania 2 Centre de Physique Th´ eorique, CNRS, Ecole Polytechnique, F-91128 Palaiseau, France [email protected] Abstract: We report on the generation of ultrabroadband supercontinuum (SC) by filamentation of two optical-cycle, carrier-envelope phase-stable pulses at 2 μ m in fused silica, sapphire, CaF 2 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 OCIS codes: (320.6629) Supercontinuum generation; (190.7110) Ultrafast nonlinear optics; (190.5940) Self-action effects; (190.3270) Kerr effect; (190.2620) Harmonic generation and mixing. References and links 1. The Supercontinuum Laser Source, 2nd ed., R. R. Alfano ed. (Springer NY, 2006). 2. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006). 3. A. Couairon and A. Mysyrowicz, “Femtoseconmd filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007). 4. A. Brodeur and S. L. Chin, “Ultrafast white-light continuum generation and self-focusing in transparent con- densed media,” J. Opt. Soc. Am. B 16, 637–650 (1999). 5. C. Nagura, A. Suda, H. Kawano, M. Obara, and K. Midorikawa, “Generation and characterization of ultrafst white-light continuum in condensed media,” Appl. Opt. 41, 3735–3742 (2002). 6. M. Bradler, P. Baum, and E. Riedle, “Femtosecond continuum generation in bulk laser host materials with sub-μ J pump pulses,” Appl. Phys. B 97, 561–574 (2009). 7. K. D. Moll and A. Gaeta, “Role of dispersion in multiple collapse dynamics,” Opt. Lett. 29, 995–997 (2004). 8. S. E. Schrauth, B. Shim, A. D. Slepkov, L. T. Vuong, A. L. Gaeta, N. Gavish, and G. Fibich, “Pulse splitting in the anomalous group-velocity-dispersion regime,” Opt. Express 19, 9309–9314 (2011). #192314 - $15.00 USD Received 17 Jun 2013; revised 11 Sep 2013; accepted 19 Sep 2013; published 15 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025210 | OPTICS EXPRESS 25210

Transcript of Ultrabroadband supercontinuum and third-harmonic ...2013).pdf · third-harmonic generation in bulk...

Ultrabroadband supercontinuum andthird-harmonic generation in bulk solidswith two optical-cycle carrier-envelope

phase-stable pulses at 2 μm

Julius Darginavicius,1 Donatas Majus,1 Vytautas Jukna,1,2 NailGarejev,1 Gintaras Valiulis,1 Arnaud Couairon,2 and Audrius

Dubietis1,∗1Department of Quantum Electronics, Vilnius University, Sauletekio Ave. 9, Building 3,

LT-10222 Vilnius, Lithuania2Centre de Physique Theorique, CNRS, Ecole Polytechnique, F-91128 Palaiseau, France

[email protected]

Abstract: We report on the generation of ultrabroadband supercontinuum(SC) by filamentation of two optical-cycle, carrier-envelope phase-stablepulses at 2 μm in fused silica, sapphire, CaF2 and YAG. The SC spectraextend from 450 nm to more than 2500 nm, and their particular shapesdepend on dispersive properties of the materials. Prior to spectral super-broadening, we observe third-harmonic generation, which occurs in thecondition of large phase and group velocity mismatch and consists of freeand driven components. A double-peaked third-harmonic structure coexistswith the SC pulse as demonstrated by the numerical simulations and verifiedexperimentally. The SC pulses have stable carrier envelope phase withshort-term rms fluctuations of ∼ 300 mrad, as simultaneously measured inYAG crystal by f-2f and f-3f interferometry, where the latter makes use ofintrinsic third-harmonic generation.

© 2013 Optical Society of America

OCIS codes: (320.6629) Supercontinuum generation; (190.7110) Ultrafast nonlinear optics;(190.5940) Self-action effects; (190.3270) Kerr effect; (190.2620) Harmonic generation andmixing.

References and links1. The Supercontinuum Laser Source, 2nd ed., R. R. Alfano ed. (Springer NY, 2006).2. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.

78, 1135–1184 (2006).3. A. Couairon and A. Mysyrowicz, “Femtoseconmd filamentation in transparent media,” Phys. Rep. 441, 47–189

(2007).4. A. Brodeur and S. L. Chin, “Ultrafast white-light continuum generation and self-focusing in transparent con-

densed media,” J. Opt. Soc. Am. B 16, 637–650 (1999).5. C. Nagura, A. Suda, H. Kawano, M. Obara, and K. Midorikawa, “Generation and characterization of ultrafst

white-light continuum in condensed media,” Appl. Opt. 41, 3735–3742 (2002).6. M. Bradler, P. Baum, and E. Riedle, “Femtosecond continuum generation in bulk laser host materials with sub-μJ

pump pulses,” Appl. Phys. B 97, 561–574 (2009).7. K. D. Moll and A. Gaeta, “Role of dispersion in multiple collapse dynamics,” Opt. Lett. 29, 995–997 (2004).8. S. E. Schrauth, B. Shim, A. D. Slepkov, L. T. Vuong, A. L. Gaeta, N. Gavish, and G. Fibich, “Pulse splitting in

the anomalous group-velocity-dispersion regime,” Opt. Express 19, 9309–9314 (2011).

#192314 - $15.00 USD Received 17 Jun 2013; revised 11 Sep 2013; accepted 19 Sep 2013; published 15 Oct 2013(C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025210 | OPTICS EXPRESS 25210

9. J. Liu, R. Li, and Z. Xu, “Few-cycle spatiotemporal soliton wave excited by filamentation of a femtosecond laserpulse in materials with anomalous dispersion,” Phys. Rev. A 74, 043801 (2006).

10. M. Durand, A. Jarnac, A. Houard, Y. Liu, S. Grabielle, N. Forget, A. Durecu, A. Couairon, and A. Mysyrowicz,“Self-guided propagation of ultrashort laser pulses in the anomalous dispersion region of transparent solids: anew regime of filamentation,” Phys. Rev. Lett. 110, 115003 (2013).

11. M. Kolesik, E. M. Wright, and J. V. Moloney, “Interpretation of the spectrally resolved far field of femtosecondpulses propagating in bulk nonlinear dispersive media,” Opt. Express 13, 10729–10741 (2005).

12. M. A. Porras, A. Dubietis, E. Kucinskas, F. Bragheri, V. Degiorgio, A. Couairon, D. Faccio, and P. Di Trapani,“From X- to O-shaped spatiotemporal spectra of light filaments in water,” Opt. Lett. 30, 3398–3400 (2005).

13. M. A. Porras, A. Dubietis, A. Matijosius, R. Piskarskas, F. Bragheri, A. Averchi, and P. Di Trapani, “Charac-terization of conical emission of light filaments in media with anomalous dispersion,” J. Opt. Soc. Am. B 24,581–584 (2007).

14. A. Saliminia, S. L. Chin, and R. Vallee, “Ultra-broad and coherent white light generation in silica glass by focusedfemtosecond pulses at 1.5 μm,” Opt. Express 13, 5731–5738 (2005).

15. M. L. Naudeau, R. J. Law, T. S. Luk, T. R. Nelson, and S. M. Cameron, “Observation of nonlinear opticalphenomena in air and fused silica using a 100 GW, 1.54 μm source,” Opt. Express 14, 6194–6200 (2006).

16. E. O. Smetanina, V. O. Kompanets, S. V. Chekalin, A. E. Dormidonov, and V. P. Kandidov, “Anti-Stokes wing offemtosecond laser filament supercontinuum in fused silica,” Opt. Lett. 38, 16–18 (2013).

17. M. Durand, K. Lim, V. Jukna, E. McKee, M. Baudelet, A. Houard, M. Richardson, A. Mysyrowicz, and A.Couairon, “Blueshifted continuum peaks from filamentation in the anomalous dispersion regime,” Phys. Rev. A87, 043820 (2013).

18. F. Silva, D. R. Austin, A. Thai, M. Baudisch, M. Hemmer, D. Faccio, A. Couairon, and J. Biegert, “Multi-octavesupercontinuum generation from mid-infrared filamentation in a bulk crystal,” Nat. Commun. 3, 807 (2012).

19. S. V. Chekalin, V. O. Kompanets, E. O. Smetanina, and V. P. Kandidov, “Light bullets and supercontinuumspectrum during femtosecond pulse filamentation under conditions of anomalous group-velocity dispersion infused silica,” Quantum Electron. 43, 326–331 (2013).

20. M. Kolesik, E. M. Wright, A. Becker, and J. V. Moloney, “Simulation of third harmonic and supercontinuumgeneration for femtosecond pulses in air,” Appl. Phys. B 85, 531–538 (2006).

21. M. Kolesik, E. M. Wright, and J. V. Moloney, “Supercontinuum and third-harmonic generation accompanyingoptical filamentation as first order scattering processes,” Opt. Lett. 32, 2816–2818 (2007).

22. N. Akozbek, A. Iwasaki, A. Becker, M. Scalora, S. L. Chin, and C. M. Bowden, “Third-harmonic generation andself-channeling in air using high-power femtosecond laser pulses,” Phys. Rev. Lett. 89, 143901 (2002).

23. N. Akozbek, A. Becker, M. Scalora, S. L. Chin, and C. M. Bowden, “Continuum generation of the third-harmonicpulse generated by an intense femtosecond IR laser pulse in air,” Appl. Phys. B 77, 177–183 (2003).

24. F. Theberge, W. Liu, Q. Luo, and S. L. Chin, “Ultrabroadband continuum generated in air (down to 230 nm)using ultrashort and intense laser pulses,” Appl. Phys. B 80, 221–225 (2005).

25. N. Kortsalioudakis, M. Tatarakis, N. Vakakis, S. D. Moustaizis, M. Franco, B. Prade, A. Mysyrowicz, A. A.Papadogiannis, A. Couairon, and S. Tzortzakis,“Enhanced harmonic conversion efficiency in the self-guidedpropagation of femtosecond ultraviolet laser pulses in argon,” Appl. Phys. B 80, 211–214 (2005).

26. S. Suntsov, D. Abdollahpour, D. G. Papazoglou, and S. Tzortzakis, “Efficient third-harmonic generation throughtailored IR femtosecond laser pulse filamentation in air,” Opt. Express 17, 3190–3195 (2009).

27. X. Yang, J. Wu, Y. Peng, Y. Tong, S. Yuan, L. Ding, Z. Xu, and H. Zeng, “Noncollinear interaction of femtosecondfilaments with enhanced third harmonic generation in air,” Appl. Phys. Lett. 95, 111103 (2009).

28. S. Suntsov, D. Abdollahpour, D. G. Papazoglou, and S. Tzortzakis, “Filamentation-induced third-harmonic gen-eration in air via plasma-enhanced third-order susceptibility,” Phys. Rev. A 81, 033817 (2010).

29. Y. Liu, M. Durand, A. Houard, B. Forestier, A. Couairon, and A. Mysyrowicz, “Efficient generation of thirdharmonic radiation in air filaments: A revisit,” Opt. Commun. 284, 4706–4713 (2011).

30. E. Schulz, D. S. Steingrube, T. Vockerodt, T. Binhammer, U. Morgner, and M. Kovacev, “Gradient enhancedthird harmonic generation in a femtosecond filament,” Opt. Lett. 36, 4389–4391 (2011).

31. G. Mao, Y. Wu, and K. D. Singer, “Third harmonic generation in self-focused filaments in liquids,” Opt. Express15, 4857–4862 (2007).

32. J. Darginavicius, N. Garejev, and A. Dubietis, “Generation of carrier-envelope phase-stable two optical-cyclepulses at 2 μm from a noncollinear beta-barium borate optical parametric amplifier,” Opt. Lett. 37, 4805–4807(2012).

33. M. J. Weber, Handbook of Optical Materials (CRC Press NY, 2003).34. M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. MacDonald, V. Mahajan, and E. Van Stryland,

Handbook of Optics, Vol. 4, 3rd Ed. (McGraw-Hill, 2009).35. D. Faccio, A. Averchi, A. Couairon, A. Dubietis, R. Piskarskas, A. Matijosius, F. Bragheri, M. A. Porras, A.

Piskarskas, and P. Di Trapani, “Competition between phase-matching and stationarity in Kerr-driven opticalpulse filamentation,” Phys. Rev. E 74, 047603 (2006).

36. D. Faccio, A. Averchi, A. Lotti, M. Kolesik, J. V. Moloney, A. Couairon, and P. Di Trapani, “Generation andcontrol of extreme blueshifted continuum peaks in optical Kerr media,” Phys. Rev. A 78, 033825 (2008).

#192314 - $15.00 USD Received 17 Jun 2013; revised 11 Sep 2013; accepted 19 Sep 2013; published 15 Oct 2013(C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025210 | OPTICS EXPRESS 25211

37. A. H. Chin, O. G. Calderon, and J. Kono, “Extreme midinfrared nonlinear optics in semiconductors,” Phys. Rev.Lett. 86, 3292–3295 (2001).

38. V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, C. M.Bowden, J. W. Haus, O. G. Kosareva, and V. P. Kandidov, “Role of phase matching in pulsed second-harmonicgeneration: Walk-off and phase-locked twin pulses in negative-index media,” Phys. Rev. A 76, 033829 (2007).

39. M. Mlejnek, E. M. Wright, J. V. Moloney, and N. Bloembergen, “Second harmonic generation of femtosecondpulses at the boundary of a nonlinear dielectric,” Phys. Rev. Lett. 83, 2934–2937 (1999).

40. G. Valiulis, V. Jukna, O. Jedrkiewicz, M. Clerici, E. Rubino, and P. Di Trapani, “Propagation dynamics andX-pulse formation in phase-mismatched second-harmonic generation,” Phys. Rev. A 83, 043834 (2011).

41. M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirec-tional equations,” Phys. Rev. E 70, 036604 (2004).

42. M. B. Gaarde and A. Couairon, “Intensity spikes in laser filamentation: diagnostics and application,” Phys. Rev.Lett. 103, 043901 (2009).

43. A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramırez-Gongora, and M. Kolesik, “Practitioner’s guideto laser pulse propagation models and simulation,” Eur. Phys. J. Special Topics 199, 5–76 (2011).

44. W. F. Krupke, M. D. Shinn, J. E. Marion, J. A. Caird, and S. E. Stokowski, “Spectroscopic, optical, and thermo-mechanical properties of neodymium- and chromium-doped gadolinium scandium gallium garnet,” J. Opt. Soc.Am. B 3, 102–114 (1986).

45. A. Couairon, L. Sudrie, M. Franco, B. Prade,and A. Mysyrowicz, “Filamentation and damage in fused silicainduced by tightly focused femtosecond laser pulses,” Phys. Rev. B 71, 125435 (2005).

46. L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Sov. Phys. JETP 20, 1307–1314 (1965).47. D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical emission, pulse

splitting, and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett.96, 193901 (2006).

1. Introduction

Supercontinuum (SC) generation is a well-established method for obtaining coherent broad-band radiation, spanning across ultraviolet, visible and infrared spectral range [1, 2]. In bulkdielectric media with Kerr nonlinearity, generation of ultrafast SC is tightly linked to fem-tosecond filamentation [3]; the broadband radiation emerges from a complex interplay betweenself-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 velocitydispersion (GVD), see e.g. [4–6]. On the other hand, it is well known that the material dis-persion plays an important role in self-focusing dynamics and filamentation of the ultrashortlaser pulses [3]. Experimental and numerical studies revealed that the interplay between theanomalous GVD and self-action effects results in extended filamentation length and conditionalpulse splitting [7,8], pulse compression rather than pulse splitting [9,10], different angular pat-tern of conical emission [11–13], ultrabroadband spectra [6, 14–18] and quasiperiodic collapseevents [19], hence giving rise to a new filamentation regime in general [10]. Another effectassociated with filamentation is the generation of third-harmonic (TH), whose occurrence ismediated by the same source of the nonlinear polarization [20, 21]. Indeed, TH generation is awell-known phenomenon, which accompanies filamentation and spectral broadening in gassesand in air in particular, see e.g. [22–30], however it is seldom observed [18, 31] and often ne-glected effect regarding filamentation and SC generation in condensed media.

In this paper we experimentally and numerically investigate SC generation in the regimeof anomalous GVD with two optical-cycle, carrier-envelope phase (CEP)-stable pulses at 2μm. We find that TH generation occurs prior to spectral broadening, and the TH pulse hasa specific double-peaked structure, which coexists with the SC also in the regime of spectralsuperbroadening. We also demonstrate that spectral beating between TH and ultrabroadbandSC produces the f-3f interference pattern, which could be readily applied for CEP stabilitymeasurements.

#192314 - $15.00 USD Received 17 Jun 2013; revised 11 Sep 2013; accepted 19 Sep 2013; published 15 Oct 2013(C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025210 | OPTICS EXPRESS 25212

2. Supercontinuum generation

The experiment was performed using 15-fs (2.3 optical-cycle), CEP-stable pulses with centralwavelength of 2 μm from a home-built optical parametric amplifier [32]. Its output beam wassuitably attenuated and focused by an f = +100 mm lens into a 70 μm FWHM spot size lo-cated at the input face of the nonlinear medium. The output SC radiation was re-collimated andthen collected into a fiber tip of the spectrometer. The measurements were performed using twocalibrated 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. Wetested four different nonlinear wide bandgap materials, which are commonly used for SC gen-eration in the visible and near-infrared: sapphire (6 mm length), fused silica (5 mm), CaF2 (6mm), and YAG (6 mm), whose GVD curves are illustrated in Fig. 1.

Fig. 1. Group velocity dispersion of fused silica, sapphire, CaF2 [33] and YAG [34].

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 gener-ated in different nonlinear media, which demonstrate the importance of digression of the pumpwavelength from the zero GVD wavelength, that could be spotted from Fig. 1. The SC spectrumgenerated in fused silica [Fig. 2(a)] is very similar to those reported in a number of previousstudies [14–17] and has a deep extended minimum around 1 μm and distinct intense peak inthe visible (so-called blue peak), which then shows apparent red-shifted broadening of the bluepeak with increase of the input-pulse energy. The SC spectrum with very similar spectral fea-tures, 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. InCaF2, owing to its generally low dispersion, a broad and flat SC spectrum that spans from 450nm to > 2.5 μm was generated [Fig. 2(c)], with the longest wavelength being limited by ourdetection apparatus. And finally, in YAG, a smooth SC spectrum with elevated spectral intensityin the 600-1000 nm range [Fig. 2(d)] was generated using quite low input-pulse energy owingto higher nonlinearity of YAG as compared to other materials tested. Also note the differencesin visual appearance of characteristic coloring and angular divergence of the SC radiation in thesaturated regime of the spectral broadening (corresponding to higher values of the input-pulseenergy), as illustrated by the screen shots of the far field patterns taken at 15 cm distance fromthe output face of the nonlinear media. The angular spread in the visible wavelength range isregarded as conical emission, whose angles are set by dispersion-related phase matching condi-

#192314 - $15.00 USD Received 17 Jun 2013; revised 11 Sep 2013; accepted 19 Sep 2013; published 15 Oct 2013(C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025210 | OPTICS EXPRESS 25213

Fig. 2. SC spectra generated in (a) fused silica, (b) sapphire, (c) CaF2, (d) YAG. The dashedand solid curves represent SC spectra in the transient and saturation regimes of the spectralbroadening, respectively. Curve labels stand for the input-pulse energy. Images on the rightside show the corresponding far-field patterns of the SC emission in the visible range,recorded in the saturated regime of the spectral broadening.

tion [17], which coincides with the X-wave phase matching [35,36]. In other words, for a givenmaterial, the phase matching condition entirely determines the loci of the far-field that can bepopulated in priority, i.e., the dependence of angles as a function of wavelength in the conicalemission pattern. The effective scattering of a specific color at a given angle also depends on thenonlinear pulse-matter interaction ensuring that the phase matching condition is fulfilled over acertain propagation distance. Among the tested materials YAG has the highest nonlinear indexof refraction, leading to a strong laser-matter interaction, and the largest dispersion, thereforevisible 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 spectralbroadening a characteristic TH peak centered at 660 nm was observed. In the transient regimeof 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 strongand broad blue-shifted peak, which is not associated with TH generation [18].

3. Third-harmonic generation

The TH generation occurs at lower input pulse energy and is clearly observed before the onsetof spectral broadening and SC generation. For example, the TH radiation was detected with1.0 μJ input-pulse energy in fused silica and with 0.40 μJ input-pulse energy in YAG, andmeasured TH efficiency varied from 10−6 to 10−4, depending on the input-pulse energy. Acloser 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 testingshorter samples of the nonlinear media. An example of TH spectrum generated with 0.70 μJinput-pulse energy in YAG, before the onset of SC generation, is shown in Fig. 3(a). Interest-

#192314 - $15.00 USD Received 17 Jun 2013; revised 11 Sep 2013; accepted 19 Sep 2013; published 15 Oct 2013(C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025210 | OPTICS EXPRESS 25214

ingly, very similar modulation of the TH spectrum was observed in semiconductor materialsusing intense 3.5 μm pulses and was attributed to pulse splitting effect [37].

Fig. 3. (a) TH spectrum in YAG (shown in linear intensity scale) as averaged over 1000laser shots and (b) variation of interference fringes over time. (c) and (d) show statistics ofthe retrieved phase jitter between free and driven TH pulses.

In what follows, we show that distinct spectral modulation is a signature of a double-peakedTH pulse, which occurs naturally, without the splitting of the input pulse. We interpret our re-sults in the framework of phase and group-velocity mismatched TH generation [38]. Such anoperating condition imposes that TH radiation consists of two pulses, representing so-calledfree and driven waves, which, in analogy with phase and group velocity-mismatched secondharmonic 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 thegroup velocity u f , 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 thevelocity ud of the nonlinear polarization, i.e., with the velocity of the intensity peak of thefundamental 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 sepa-rated in time by the amount τ = |νfd|z, where νfd = 1/u f −1/ud is the group velocity mismatchand z is the medium length, which produce beating in TH spectrum. Inserting the relevant valuesof YAG: z = 6 mm and νfd = 115 fs/mm, the estimated temporal separation between the freeand driven TH pulses thus is 690 fs, which is very close to that of 670 fs, as retrieved from thefringe pattern shown in Fig. 3(a). In the spectral domain, the TH spectral intensity could be ex-pressed as I(3ω) = I f (3ω)+ Id(3ω)+2

√Id(3ω)I f (3ω)cos(3ωτ +Δφ), where Δφ = φ f −φd

denotes the instant phase difference between the free and driven TH pulses. The fringe patternshown in Fig. 3(a) is thus regarded as 3f-3f interferogram, whose variation over time yields thephase jitter between free and driven TH pulses, regardless on CEP fluctuations of the pumppulse. Figure 3(b) shows a series of 1000 3f-3f interferograms, whose fringes are remarkablystable in time, yielding the root-mean-square (rms) phase jitter of 48 mrad [Fig. 3(c,d)], whichoriginates from intensity-dependent phase matching condition due to nonlinear refractive indexand intensity fluctuations of the pump pulse.

4. Numerical simulations

In order to verify our interpretation, we performed numerical simulations by using a radiallysymmetric form of the generic unidirectional carrier resolving models (forward Maxwell equa-tion) [41–43] for propagating the frequency components of the infrared pulse:

#192314 - $15.00 USD Received 17 Jun 2013; revised 11 Sep 2013; accepted 19 Sep 2013; published 15 Oct 2013(C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025210 | OPTICS EXPRESS 25215

∂ E(z,ω,r)∂ z

= i[k(ω)−ω/vg]E +i

2k(ω)∇2⊥E +

12ε0cn(ω)

[iωP(z,ω,r)− J(z,ω,r)], (1)

where E(z,ω,r) = F [E(z, t,r)] denote the Fourier components of the electric field E(z, t,r),z is the propagation coordinate, k(ω) and n(ω) denote the frequency dependent wavenumberand refraction index of the medium and account for the chromatic dispersion via a Sellmeierrelation [34]. Diffraction is described by the second term on the right hand side of Eq. (1). Thenonlinear 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 thefrequency dependent coefficients in front of the nonlinear polarization. The third order sus-ceptibility χ(3) = (4/3)ε0cn2

0n2 is obtained from the nonlinear index coefficient for the opticalKerr effect n2 = 7× 10−16 cm2/W [44]. Plasma effects (nonlinear absorption, plasma gener-ation, plasma defocusing and absorption) are calculated in the temporal domain by resolvinga coupled system of equations for the density ρ(z, t,r) of the electron-hole and electron-ionplasma generated by optical field ionization and avalanche [45], and for the current source termJ(z, t,r)≡ Je + Ja:

∂ρ∂ t

=W (E)(1− ρρb

)+σUi

ρE2, (2)

∂Je

∂ t+

Je

τc=

q2e

meρE(z, t,r), (3)

Ja = ε0cn0W (E)

E2 Ui(1−ρ/ρb)E, (4)

where the two components for the current source terms Je and Ja account for plasma inducedeffects and for the absorption of energy required for optical field ionization, respectively. Inparticular, 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 absorp-tion. The quantity τc = 3 fs denotes the typical collision time in transparent solids, Ui = 6.5eV denotes the gap between valence and conduction bands for YAG. The intensity dependentphotoionization rate W (E) follows Keldysh’s formulation for condensed media in the multi-photon limit [46]: 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 ratefollows 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 (wave-

length 2 μm, pulse duration 15 fs, beam width at the entrance face of the YAG plate 70 μmFWHM, pulse energy in the range 0.6− 0.8 μJ). Note that in the simulation we used slightlyreduced input-pulse energy so as to account for low contrast of the pulse used in the experiment,see Fig. 3(b) of [32]. Details on the numerical schemes and resolution methods are describedin [43].

Figure 4 compares the experimentally measured [taken from Fig. 2(d)] and numerically sim-ulated 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 spectralbroadening, respectively. A good agreement between the numerical simulations and experi-mental data was obtained; note, how experimentally observed spectral features, such as distinctTH peak, elevated spectral intensity in the 600-1000 nm range and a slight decrease of spectralintensity in the 1200-1400 nm range were reproduced by the numerics.

#192314 - $15.00 USD Received 17 Jun 2013; revised 11 Sep 2013; accepted 19 Sep 2013; published 15 Oct 2013(C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025210 | OPTICS EXPRESS 25216

Fig. 4. Comparison of experimentally measured (blue curves) and numerically simulated(red curves) angle-integrated spectra in YAG in (a) transient and (b) saturated regimes ofspectral broadening.

Fig. 5. Angularly resolved spectra in YAG in the transient (a) and saturated (c) regimesof spectral broadening. Insets show enlarged portions of the spectra, so as to highlightmodulation around TH spectral range. (b) and (d) show the respective normalized temporalprofiles of the pump (black curves) and TH (red curves) pulses after spectral filtering.Additional arrows in (d) indicate driven (on the left) and free (on the right) TH pulsesin the regime of spectral superbroadening.

#192314 - $15.00 USD Received 17 Jun 2013; revised 11 Sep 2013; accepted 19 Sep 2013; published 15 Oct 2013(C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025210 | OPTICS EXPRESS 25217

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 broad-ening, respectively. The insets of the figures show the magnified portions of the spectra in the550-750 nm range, highlighting the spectral beatings in the TH spectrum in Fig. 5(a), and thespectral 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 THwavelength (red curves), as retrieved using a super-gaussian filter with FWHM width of 150nm and centered at 660 nm.

In the transient regime of spectral broadening, the retrieved TH temporal profile consists oftwo distinct peaks separated by 700 fs [Fig. 5(b)], which are attributed to free and driven THpulses, in fair agreement with analytical interpretation and experimental results presented inthe previous section. Note also that the driven TH pulse remains short as being locked underthe 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 experiencesconsiderable temporal broadening due to normal GVD.

In the saturation regime of spectral broadening, a broad SC is generated, with its spectrumspanning from ∼ 500 nm to 4 μm (only a part up to 2.5 μm is shown) and that is accompaniedby strong conical emission, ending-up with a pronounced fish-tail in the green-red spectralrange. The SC radiation overlaps the TH spectral peak [Fig. 5(c)]; these produce a distinctspectral beating, which is confined to the beam axis and to the TH spectral range, as shownmagnified in the inset. The temporal profiles were retrieved by the same filtering procedureshowing that TH radiation yet consists of isolated free and driven pulses, which coexist withthe broadband main pulse that develops a certain sub-structure and an extended tail, as shownin Fig. 5(d). Filtering recovers also an intense peak in between the TH pulses, located at 170fs with respect to the intensity peak of the main pulse, and which could be attributed to SCspectral components in the 600-700 nm range that populate the main pulse tail. However, moredetailed numerical investigation by varying the input-pulse energy and propagation distancedisclosed that the intense peak is also contributed by generation of a secondary free TH pulseas a result of pulse-front steepening-induced spontaneous formation of an X wave in the normalGVD region [47]. The propagation distance at which the X wave is generated decreases withincreasing the input-pulse energy (not shown here), however, in general, in the time domain thecontributions of the X wave and the secondary free TH pulse are impossible to distinguish asthey 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 ob-served spectral beating between SC and TH directly produces the f-3f interferogram, whosetime series readily provide the statistics of CEP fluctuations. This could be done by proper fil-tering the conical components of the SC, so as to select the axial portion of the spectrum, whereclear spectral beating between SC and TH is observed according to the results of numericalsimulation presented in Fig. 5(c). Moreover, we have verified experimentally that the spectralbeating is more or less well-detectable in all investigated media, see examples of spectra infused silica and YAG shown in Fig. 6.

Taking the SC generation in YAG as an example, we devised a setup shown in Fig. 7, whichsimultaneously 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 ofthe SC by placing 1-mm iris aperture at 5 cm distance from the output face of YAG crystal. Aconventional 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

#192314 - $15.00 USD Received 17 Jun 2013; revised 11 Sep 2013; accepted 19 Sep 2013; published 15 Oct 2013(C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025210 | OPTICS EXPRESS 25218

Fig. 6. Visible and near-infrared part of the SC generated in (a) fused silica and (b) YAG.In (a) red and blue curves show the stand-alone TH and filtered part of the SC blue peak,respectively. In (b) black and red curves denote angle-integrated and filtered SC spectra,respectively. Note linear and logarithmic scales used for data in fused silica and YAG,respectively, and emerging modulation in the filtered SC spectra as due to spectral beatingbetween SC and TH in the regime of spectral superbroadening.

Fig. 7. Setup for simultaneous measurement of f-3f and f-2f interferograms. See text fordetails.

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 theinterference fringes around 1 μm and 660 nm, generated by sampling the SC pulse with the SHand TH pulses, respectively. Figure 8(b) shows variation of the interference pattern over 10 speriod after recording 1000 single-shot spectrograms. The f-2f interference pattern yields SCpulse CEP rms fluctuations of 300 mrad. The f-3f interference pattern is more complex and hasbeatings at several frequencies, as shown enlarged in the inset of Fig. 8(c), and by means of theinverse Fourier transform could be decomposed into seven peaks, as schematically illustrated inFig. 8(c). Application of arbitrary amplitude filters on the particular peaks (schematically shownas blue-shaded areas) delivers information on phase fluctuations. The farthest peaks at ±640fs (labelled as 3f-3f) yield phase jitter of 55 mrad, which is attributed to the phase fluctuationsbetween free and driven TH pulses. The remaining peaks at ±170 fs and ±470 fs (labelled asf-3f) provide CEP fluctuations of 315 mrad and 275 mrad, which are attributed to phase jittersbetween the SC and driven and free TH pulses, respectively.

The recovered peak positions in time fairly agree with those retrieved from the numericalsimulation, as shown in Fig. 5(d).

#192314 - $15.00 USD Received 17 Jun 2013; revised 11 Sep 2013; accepted 19 Sep 2013; published 15 Oct 2013(C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025210 | OPTICS EXPRESS 25219

Fig. 8. Measurement of CEP fluctuations of the SC pulse: (a) spectrogram averaged from1000 single-shot spectra, (b) variation of f-2f and f-3f interference patterns in time, (c)Power spectrum of the spectrogram around 660 nm, which is shown enlarged in the inset.

6. Conclusion

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, andYAG, in the regime of anomalous GVD. The measured SC spectra span from 450 nm to morethan 2.5 μm, and their particular shapes crucially depend on digression of the pump wave-length from the the zero GVD wavelength. In that regard, CaF2 and YAG provide the SC ra-diation with the smoothest spectral coverage across the entire detected spectral range. We alsodetect TH generation, which occurs prior to spectral supebroadening. Periodic modulation ofthe TH spectrum reveals a double-peaked temporal structure of the TH pulse, consisting of freeand driven components, which are generated in the regime of large phase and group-velocitymismatch. We find that double-peaked TH structure persists also in the regime of spectral su-perbroadening and coexists with strong SC emission, as verified experimentally and by thenumerical simulations. We also devised an experimental setup, which simultaneously measuresthe CEP stability of the SC pulses by means of f-2f and f-3f interferometry. Given a goodagreement between the results obtained by f-2f and f-3f interferometry, the f-3f interferometrybased on intrinsic TH generation, suggests a simple and straightforward method to measureCEP fluctuations, despite rather complex temporal structure of the TH pulse.

Acknowledgment

This research was funded by Grant No. VP1-3.1-SMM-07-K-03-001 from the Lithuanian Sci-ence Council.

#192314 - $15.00 USD Received 17 Jun 2013; revised 11 Sep 2013; accepted 19 Sep 2013; published 15 Oct 2013(C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025210 | OPTICS EXPRESS 25220