Generation of intense subcycle optical pulses in a gas Yuichiro Kida1,* and Totaro Imasaka1,2 1
2
Department of Applied Chemistry, Graduate School of Engineering, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka 819-0395, Japan Division of Optoelectronics and Photonics, Center for Future Chemistry, Kyushu University, 744 Motooka, Nishiku, Fukuoka 819-0395, Japan * [email protected]
Abstract: The generation of intense subcycle laser pulses during the propagation of two-color femtosecond pulses in a gas medium is investigated theoretically and experimentally. Four-wave mixing induced by the laser pulses in a gas medium generates multi-octave laser radiation from the ultraviolet to the infrared, which forms stable subcycle laser pulses after a certain propagation distance in a gas medium with group-velocity dispersion. The intense subcycle laser pulses would allow the coherent control of the waveforms of soft-x-rays generated via high-harmonic generation. ©2015 Optical Society of America OCIS codes: (190.4380) Nonlinear optics, four-wave mixing; (320.5540) Pulse shaping; (320.7090) Ultrafast lasers; (320.7100) Ultrafast measurements; (320.7110) Ultrafast nonlinear optics.
References and links 1.
E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320(5883), 1614–1617 (2008). 2. K. Zhao, Q. Zhang, M. Chini, Y. Wu, X. Wang, and Z. Chang, “Tailoring a 67 attosecond pulse through advantageous phase-mismatch,” Opt. Lett. 37(18), 3891–3893 (2012). 3. M. Uiberacker, T. Uphues, M. Schultze, A. J. Verhoef, V. Yakovlev, M. F. Kling, J. Rauschenberger, N. M. Kabachnik, H. Schröder, M. Lezius, K. L. Kompa, H.-G. Muller, M. J. J. Vrakking, S. Hendel, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond real-time observation of electron tunnelling in atoms,” Nature 446(7136), 627–632 (2007). 4. E. Goulielmakis, Z.-H. Loh, A. Wirth, R. Santra, N. Rohringer, V. S. Yakovlev, S. Zherebtsov, T. Pfeifer, A. M. Azzeer, M. F. Kling, S. R. Leone, and F. Krausz, “Real-time observation of valence electron motion,” Nature 466(7307), 739–743 (2010). 5. M. Schultze, M. Fiess, N. Karpowicz, J. Gagnon, M. Korbman, M. Hofstetter, S. Neppl, A. L. Cavalieri, Y. Komninos, T. Mercouris, C. A. Nicolaides, R. Pazourek, S. Nagele, J. Feist, J. Burgdörfer, A. M. Azzeer, R. Ernstorfer, R. Kienberger, U. Kleineberg, E. Goulielmakis, F. Krausz, and V. S. Yakovlev, “Delay in photoemission,” Science 328(5986), 1658–1662 (2010). 6. A. N. Pfeiffer, C. Cirelli, M. Smolarski, R. Dörner, and U. Keller, “Timing the release in sequential double ionization,” Nat. Phys. 7(5), 428–433 (2011). 7. A. N. Pfeiffer, C. Cirelli, M. Smolarski, D. Dimitrovski, M. Abu-samha, L. B. Madsen, and U. Keller, “Attoclock reveals natural coordinates of the laser-induced tunnelling current flow in atoms,” Nat. Phys. 8(1), 76–80 (2011). 8. M. Fieß, B. Horvath, T. Wittmann, W. Helml, Y. Cheng, Y. Zeng, Z. Xu, A. Scrinzi, J. Gagnon, F. Krausz, R. Kienberger, “Attosecond control of tunneling ionization and electron trajectories,” New J. Phys. 13(3), 033031 (2011). 9. F. Krausz and M. I. Stockman, “Attosecond metrology: from electron capture to future signal processing,” Nat. Photonics 8(3), 205–213 (2014). 10. T. W. Hänsch, “A proposed sub-femtosecond pulse synthesizer using separate phase-locked laser oscillators,” Opt. Commun. 80(1), 71–75 (1990). 11. S. Yoshikawa and T. Imasaka, “A new approach for the generation of ultrashort optical pulses,” Opt. Commun. 96(1-3), 94–98 (1993). 12. A. E. Kaplan, “Subfemtosecond pulses in mode-locked 2 π solitons of the cascade stimulated Raman scattering,” Phys. Rev. Lett. 73(9), 1243–1246 (1994).
#234895 - $15.00 USD © 2015 OSA
Received 18 Feb 2015; revised 25 Apr 2015; accepted 28 Apr 2015; published 1 May 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.012373 | OPTICS EXPRESS 12373
13. M. Y. Shverdin, D. R. Walker, D. D. Yavuz, G. Y. Yin, and S. E. Harris, “Generation of a single-cycle optical pulse,” Phys. Rev. Lett. 94(3), 033904 (2005). 14. W. J. Chen, Z. M. Hsieh, S. W. Huang, H. Y. Su, C. J. Lai, T. T. Tang, C. H. Lin, C. K. Lee, R. P. Pan, C. L. Pan, and A. H. Kung, “Sub-single-cycle optical pulse train with constant carrier envelope phase,” Phys. Rev. Lett. 100(16), 163906 (2008). 15. S. Huang, G. Cirmi, J. Moses, K.-H. Hong, S. Bhardwaj, J. R. Birge, L. Chen, E. Li, B. J. Eggleton, G. Cerullo, and F. X. Kärtner, “High-energy pulse synthesis with sub-cycle waveform control for strong-field physics,” Nat. Photonics 5(8), 475–479 (2011). 16. A. Wirth, M. T. Hassan, I. Grguras, J. Gagnon, A. Moulet, T. T. Luu, S. Pabst, R. Santra, Z. A. Alahmed, A. M. Azzeer, V. S. Yakovlev, V. Pervak, F. Krausz, and E. Goulielmakis, “Synthesized light transients,” Science 334(6053), 195–200 (2011). 17. K. Yoshii, J. Kiran Anthony, and M. Katsuragawa, “The simplest route to generating a train of attosecond pulses,” Light: Sci. Appl. 2(3), e58 (2013). 18. T. Fuji and Y. Nomura, “Generation of phase-stable sub-cycle mid-infrared pulses from filamentation in nitrogen,” Appl. Sci. 3(1), 122–138 (2013). 19. C. Manzoni, S. W. Huang, G. Cirmi, P. Farinello, J. Moses, F. X. Kärtner, and G. Cerullo, “Coherent synthesis of ultra-broadband optical parametric amplifiers,” Opt. Lett. 37(11), 1880–1882 (2012). 20. E. J. Takahashi, P. Lan, O. D. Mücke, Y. Nabekawa, and K. Midorikawa, “Attosecond nonlinear optics using gigawatt-scale isolated attosecond pulses,” Nat. Commun. 4, 2691 (2013). 21. B. Kim, J. An, Y. Yu, Y. Cheng, Z. Xu, and D. E. Kim, “Optimization of multi-cycle two-color laser fields for the generation of an isolated attosecond pulse,” Opt. Express 16(14), 10331–10340 (2008). 22. C. Vozzi, F. Calegari, F. Frassetto, L. Poletto, G. Sansone, P. Villoresi, M. Nisoli, S. De Silvestri, and S. Stagira, “Coherent continuum generation above 100 eV driven by an ir parametric source in a two-color scheme,” Phys. Rev. A 79(3), 033842 (2009). 23. E. J. Takahashi, P. Lan, O. D. Mücke, Y. Nabekawa, and K. Midorikawa, “Infrared two-color multicycle laser field synthesis for generating an intense attosecond pulse,” Phys. Rev. Lett. 104(23), 233901 (2010). 24. P. Wei, J. Miao, Z. Zeng, C. Li, X. Ge, R. Li, and Z. Xu, “Selective enhancement of a single harmonic emission in a driving laser field with subcycle waveform control,” Phys. Rev. Lett. 110(23), 233903 (2013). 25. S. Haessler, T. Balciunas, G. Fan, G. Andriukaitis, A. Pugžlys, A. Baltuška, T. Witting, R. Squibb, A. Zaïr, J. W. G. Tisch, J. P. Marangos, and L. E. Chipperfield, “Optimization of quantum trajectories driven by strong-field waveforms,” Phys. Rev. X 4, 021028 (2014). 26. P. Wei, Z. Zeng, J. Jiang, J. Miao, Y. Zheng, X. Ge, C. Li, and R. Li, “Selective generation of an intense single harmonic from a long gas cell with loosely focusing optics based on a three-color laser field,” Appl. Phys. Lett. 104(15), 151101 (2014). 27. X. Wang, C. Jin, and C. D. Lin, “Coherent control of high-harmonic generation using waveform-synthesized chirped laser fields,” Phys. Rev. A 90(2), 023416 (2014). 28. A. V. Sokolov, D. D. Yavuz, and S. E. Harris, “Subfemtosecond pulse generation by rotational molecular modulation,” Opt. Lett. 24(8), 557–559 (1999). 29. T. Imasaka, S. Kawasaki, and N. Ishibashi, “Generation of more than 40 laser emission lines from the ultraviolet to the visible regions by two-color stimulated raman effect,” Appl. Phys. B 49(4), 389–392 (1989). 30. D. D. Yavuz, D. R. Walker, M. Y. Shverdin, G. Y. Yin, and S. E. Harris, “Quasiperiodic Raman technique for ultrashort pulse generation,” Phys. Rev. Lett. 91(23), 233602 (2003). 31. E. Sali, K. J. Mendham, J. W. G. Tisch, T. Halfmann, and J. P. Marangos, “High-order stimulated Raman scattering in a highly transient regime driven by a pair of ultrashort pulses,” Opt. Lett. 29(5), 495–497 (2004). 32. O. Shitamichi and T. Imasaka, “High-order Raman sidebands generated from the near-infrared to ultraviolet region by four-wave Raman mixing of hydrogen using an ultrashort two-color pump beam,” Opt. Express 20(25), 27959–27965 (2012). 33. K. Motoyoshi, Y. Kida, and T. Imasaka, “High-energy, multicolor femtosecond pulses from the deep ultraviolet to the near infrared generated in a hydrogen-filled gas cell and hollow fiber,” Appl. Sci. 4(3), 318–330 (2014). 34. J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962). 35. C. G. Durfee III, L. Misoguti, S. Backus, H. C. Kapteyn, and M. M. Murnane, “Phase matching in cascaded third-order processes,” J. Opt. Soc. Am. B 19(4), 822–831 (2002). 36. Y. Kida and T. Kobayashi, “Generation of sub-10 fs ultraviolet Gaussian pulses,” J. Opt. Soc. Am. B 28(1), 139– 148 (2011). 37. T. Fuji, T. Horio, and T. Suzuki, “Generation of 12 fs deep-ultraviolet pulses by four-wave mixing through filamentation in neon gas,” Opt. Lett. 32(17), 2481–2483 (2007). 38. T. Suzuki, M. Hirai, and M. Katsuragawa, “Octave-spanning Raman comb with carrier envelope offset control,” Phys. Rev. Lett. 101(24), 243602 (2008). 39. T. Fuji, T. Suzuki, E. E. Serebryannikov, and A. Zheltikov, “Experimental and theoretical investigation of a multicolor filament,” Phys. Rev. A 80(6), 063822 (2009). 40. P. Zuo, T. Fuji, T. Horio, S. Adachi, and T. Suzuki, “Simultaneous generation of ultrashort pulses at 158 and 198 nm in a single filamentation cell by cascaded four-wave mixing in Ar,” Appl. Phys. B 108(4), 815–819 (2012). 41. J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, 2006).
#234895 - $15.00 USD © 2015 OSA
Received 18 Feb 2015; revised 25 Apr 2015; accepted 28 Apr 2015; published 1 May 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.012373 | OPTICS EXPRESS 12374
42. C. G. Durfee 3rd, S. Backus, H. C. Kapteyn, and M. M. Murnane, “Intense 8-fs pulse generation in the deep ultraviolet,” Opt. Lett. 24(10), 697–699 (1999). 43. T. Fuji and T. Suzuki, “Generation of sub-two-cycle mid-infrared pulses by four-wave mixing through filamentation in air,” Opt. Lett. 32(22), 3330–3332 (2007). 44. Y. Kida, J. Liu, T. Teramoto, and T. Kobayashi, “Sub-10 fs deep-ultraviolet pulses generated by chirped-pulse four-wave mixing,” Opt. Lett. 35(11), 1807–1809 (2010). 45. M. Beutler, M. Ghotbi, and F. Noack, “Generation of intense sub-20-fs vacuum ultraviolet pulses compressed by material dispersion,” Opt. Lett. 36(19), 3726–3728 (2011). 46. E. R. Peck and K. Reeder, “Dispersion of air,” J. Opt. Soc. Am. 62(8), 958–962 (1972). 47. K. W. DeLong, R. Trebino, J. Hunter, and W. E. White, “Frequency-resolved optical gating with the use of second-harmonic generation,” J. Opt. Soc. Am. B 11, 2206–2215 (1994). 48. D. Keusters, H. Tan, P. O’Shea, E. Zeek, R. Trebino, and W. S. Warren, “Relative-phase ambiguities in measurements of ultrashort pulses with well-separated multiple frequency components,” J. Opt. Soc. Am. B 20(10), 2226–2237 (2003). 49. Y. Kida, Y. Nakano, K. Motoyoshi, and T. Imasaka, “Frequency-resolved optical gating with two nonlinear optical processes,” Opt. Lett. 39(10), 3006–3009 (2014). 50. F. Théberge, N. Aközbek, W. Liu, A. Becker, and S. L. Chin, “Tunable ultrashort laser pulses generated through filamentation in gases,” Phys. Rev. Lett. 97(2), 023904 (2006). 51. A. Houard, M. Franco, B. Prade, A. Durécu, L. Lombard, P. Bourdon, O. Vasseur, B. Fleury, C. Robert, V. Michau, A. Couairon, and A. Mysyrowicz, “Femtosecond filamentation in turbulent air,” Phys. Rev. A 78(3), 033804 (2008). 52. F. Théberge, M. Châteauneuf, G. Roy, P. Mathieu, and J. Dubois, “Generation of tunable and broadband farinfrared laser pulses during two-color filamentation,” Phys. Rev. A 81(3), 033821 (2010). 53. K. W. DeLong, D. N. Fittinghoff, R. Trebino, B. Kohler, and K. Wilson, “Pulse retrieval in frequency-resolved optical gating based on the method of generalized projections,” Opt. Lett. 19(24), 2152–2154 (1994). 54. S. Skupin, G. Stibenz, L. Bergé, F. Lederer, T. Sokollik, M. Schnürer, N. Zhavoronkov, and G. Steinmeyer, “Self-compression by femtosecond pulse filamentation: Experiments versus numerical simulations,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(5), 056604 (2006).
1. Introduction In the past decades, ultrashort laser systems have been developed in various spectral ranges and applied for unveiling ultrafast photo-physical and chemical processes in atoms and molecules. Nearly single-cycle and a few-cycle laser pulses have enabled the generation of isolated attosecond pulses in the extreme ultraviolet (EUV) region [1,2] which have been utilized for the observation of electron motions in real time [3–9]. Even shorter optical pulses, sub-cycle optical pulses, allow detailed investigation of atomic-scale electron motion. Various approaches have been reported for the generation of subcycle laser pulses [10–19]. Among them, the optical shynthesizer generates an intense subcycle optical pulse with an energy of 0.3 mJ [16]. The active path-length stabilization generates subcycle laser pulses with stable waveforms for probing electron dynamics such as the instantaneous ac Stark shift in detail [16]. A train of intense subcycle pulses have important applications. Such a train of subcycle pulses is generated by synthesizing energetic multi-color femtosecond pulses. It has been recently reported that an energetic isolated attosecond pulse with a pulse energy exceeding one microjoule was generated using two-color multicycle laser fields forming optical beats [20]. Such a high-energy EUV pulse is a promising light source for nonlinear spectroscopy in the EUV. The waveforms of the harmonics strongly depend on the wavelengths of the twocolor driving laser fields [20–23] or the waveform of the synthesized electric field. It was reported that three-color multicycle laser pulses forming subcycle laser fields enhanced the intensities of harmonic emissions [24,25]. A waveform-controlled train of subcycle laser pulses with a high peak intensity is one of the desired light sources for the generation of energetic attosecond EUV pulses as well as the coherent control of high-harmonic generation (HHG) processes [20–27]. In this research, an approach to the generation of the intense train of subcycle optical pulses is reported. Two-color multicycle femtosecond laser fields are focused into a gas medium to generate multicolor laser fields composing an octave-spanning spectrum via degenerate four-wave mixing (FWM). During the subsequent propagation in a gas medium,
#234895 - $15.00 USD © 2015 OSA
Received 18 Feb 2015; revised 25 Apr 2015; accepted 28 Apr 2015; published 1 May 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.012373 | OPTICS EXPRESS 12375
the laser field synthesized by the multicolor laser fields drastically changes its shape, and forms a train of several intense subcycle optical pulses after a certain propagation distance (hereafter denoted as subcycle optical burst). Similar approach has been proposed by Harris et al. that employs group-velocity dispersion (GVD) for the synthesis of a train of pulses in a Raman-active gas with strongly driven molecular coherence by two-color nanosecond pulses [28]. In the present research, on the other hand, intense multicycle femtosecond pulses and nonresonant FWM are employed for the generation of the multi-color laser fields for the synthesis of subcycle laser pulses with a high peak intensity. The mutual coherence among the multicolor laser fields leads to a stable subcycle intensity profile without active feedback stabilization. The adjustment of the optical path length in the gas after the multicolor generation allows the control of the temporal intensity profile of the optical burst generated. 2. Theoretical description 2.1 Generation of intense subcycle optical pulses in a gas By focusing two-color laser pulses (P1, P2) with angular frequencies of ωP1 and ωP2 into a low-density gas medium, multicolor laser pulses are generated in a spectral range from the ultraviolet to the infrared via the degenerate FWM [13,14,29–33]. The phase mismatch and the influence of the GVD are small when the interaction length L is sufficiently short compared to the coherence length as in the experimental conditions described below. The real amplitude AS1(t), angular frequency ωS1, and phase φS1 of the first Stokes field generated by the FWM, in this case, are expressed as (Ln2/c)ωS1AP12(t)AP2(t), ωP2 - 2Ω, and 2φP1 - φP2 + π/2, respectively, under the assumption of the negligible pump depletion [34–36]. The real amplitude AAS1(t), angular frequency ωAS1, and phase φAS1 of the first anti-Stokes field generated by the FWM are, on the other hand, expressed as (Ln2/c)ωAS1AP1(t)AP22(t), ωP2 + Ω, and -φP1 + 2φP2 + π/2, respectively. The term Ω stands for the angular-frequency difference of ωP2 – ωP1. The parameters n2, c, AP1(t), AP2(t), φP1, and φP2 are the nonlinear refractive index of the medium, the speed of light in vacuum, the slowly-varying envelopes and phases of the two-color laser fields P1 and P2, respectively. The generated first anti-Stokes field then interacts with the two input laser fields to produce higher-order anti-Stokes fields via the cascaded interactions [13,14,29–33,35,37–40]. The mth-order anti-Stokes field has an amplitude of AASm (t ) = ( Ln2 / c ) ∏im=1 ω ASi AP1m (t ) AP 2 m +1 (t ) , an angular frequency of ωASm = m
ωP2 + mΩ, and phase φASm of –mφP1 + (m + 1)φP2 + mπ/2, respectively. After the generation, each laser field with an angular frequency ωi propagates with its own group velocity and propagation constant ki = niωi/c, where ni is the refractive index of the medium through which the laser field propagate. Under the retarded frame of reference (η, z) with the retarded time η of t – z/υg and υg being the group velocity of P2 [41], the phase of each field (i) evolves with the propagation distance z as ϕi (η , z ) = ϕi − ωi [η − z (ni / c − 1/ υ g )] . For any value of z, there exists a certain η (hereafter denoted as ηP) at which P1 and P2 are in φP1(ηP, z) = φP2(ηP, z) and phase such that η P = (ϕ P 2 − ϕ P1 ) / Ω + z[ωP 2 (nP 2 − nP1 ) / cΩ + (nP1 / c − 1/ υ g )] . Such a retarded time exits as long as the pulse durations of P1 and P2 are longer than the period of T = 2π/Ω. Besides, at a certain propagation distance, zl = 2π c (l − 1/ 4) / (nAS 1ω AS1 − 2nP 2ωP 2 + nP1ωP1 ) , the first antiStokes field is phase locked at ηP to the two laser fields P1 and P2 such that ΔφAS1(ηP, zl) = φAS1(ηP, zl) - φP2(ηP, zl) = 2πl, where l is an integer (l = 1, 2, 3, …). For such a zl to be present, the refractive index must be frequency dependent or GVD needs to be induced among the multicolor laser fields. The normal GVD induced in the medium changes the relative phases among the multicolor laser fields to phase lock the laser fields at the above propagation distance zl. The phase difference between S1 and P2 is ΔφS1(ηP, zl) = π/2 + (zl/c)(nS1ωS1 + nP2ωP2 – 2nP1ωP1) at zl and ηP, whereas that between the mth-order anti-Stokes field and P2 is
#234895 - $15.00 USD © 2015 OSA
Received 18 Feb 2015; revised 25 Apr 2015; accepted 28 Apr 2015; published 1 May 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.012373 | OPTICS EXPRESS 12376
ΔφASm(ηP, zl) = mπ/2 + (zl/c)[nASmωASm – (m + 1)nP2ωP2 + mnP1ωP1], respectively. If the phase differences are close to integer multiples of 2π, a subcycle temporal intensity structure is formed at the phase-locking point ηP at the propagation distance zl by the octave-spanning multicolor laser fields. The temporal intensity structure of the laser field synthesized by the multicolor laser fields does not depend on the values of φP1 and φP2 except the phase locking point ηP. This is because the phase differences Δφi(ηP, zl) and zl are independent of φP1 and φP2. In other words, there is mutual phase coherence among the multicolor laser fields. The application of the above phase-locking scheme in the femtosecond regime allows one to generate an intense subcycle optical burst consisting of only several subcycle pulses. This indicates the generation of subcycle laser pulses with the several orders of magnitude higher intensity than those generated in the nanosecond regime [13,14,17]. In the femtosecond regime, nonresonant FWM can be employed for the generation of multicolor laser fields with an arbitrary frequency spacing [18,35,37,39,40,42–45]. This leads to a degree of freedom for the waveform control, which is useful for the coherent control of HHG [20–24,26,27]. The influence of the GVD needs to be carefully addressed in this regime since it tends to temporally separate the multicolor laser fields from each other, preventing the formation of a subcycle optical burst. Nevertheless, the above phase-locking scheme still works, provided that the pulse duration of each field forming the multicolor laser fields is longer than the period, T = 2π/ . For instance, the three laser fields P1, P2, and AS1 phase lock to each other after the propagation distance in air of zl of 0.36 m (l = 1), 0.84 m (l = 2), and 1.32 m (l = 3) without being separated from each other [Figs. 1(a) and 1(b)]. Here, the wavelength of P2 and 2πΩ/c are assumed to be 800 nm and 4155 cm−1, respectively, giving the multicolor laser fields, S1, P1, AS1, and AS2, with wavelengths of 2387, 1198, 600, and 481 nm, respectively, and the relative spectral intensities shown in Fig. 1(a). Each multicolor laser field is assumed to have a pulse duration of 50-fs and the refractive index of air is calculated by referring to the parameters reported elsewhere [46]. Interestingly, ΔΦAS 2 (η P ) and ΔΦS1 (η P ) are calculated to be 9.81 x 2π and 2.04 x 2π at z3 of 1.32 m, respectively,
indicating AS2 and S1 are also nearly phase locked to P1,
Fig. 1. (a) Simulated spectrum of the multicolor laser fields, (b) temporal intensity distribution of each multicolor laser field after the propagation distance of z = 1.31 m, and (c) intensity profile of the synthesized optical burst by the multicolor laser fields for φP1 - φP2 = 0 after the propagation distances z of 0, 1.31, and 1.50 m. The ordinate in (b,c) corresponds to the retarded time η (see text).
#234895 - $15.00 USD © 2015 OSA
Received 18 Feb 2015; revised 25 Apr 2015; accepted 28 Apr 2015; published 1 May 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.012373 | OPTICS EXPRESS 12377
P2, and AS1. A subcycle pulse is formed at the phase-locking point ηP as shown in Fig. 1(c). This is because of the second and/or higher-order dispersion in the refractive index of air with respect to frequency. Such subcycle pulses are not formed at z = 0 and at z = 1.50 m. 2.2 Probing variation in temporal intensity profile of optical burst Second-harmonic-generation frequency-resolved optical gating (SHG FROG) [47] allows to probe the evolution of the relative phases among the multicolor laser fields and thus the intensity structure of the optical burst with respect to the propagation distance. SHG-FROG traces for the optical bursts in Fig. 1(c) consist of several discrete spectral components with modulations with respect to delay with the period, T = 2π/Ω. Some modulations result from the interference among several sum-frequency wave-mixing processes, and have the information of the relative phases of the multicolor laser fields. Change in the relative phases leads to the change in the shape of the FROG trace, which is not the case for the test pulse consisting of only two-color laser fields [48]. In Fig. 2, the frequency components at 400 nm and 480 nm contained in the SHG-FROG traces are simulated for the optical bursts in Fig. 1(c). The signal at 480 nm is generated by the sum-frequency generations, ωS1 + ωAS1, and ωP1 + ωP2, while the signal at 400 nm is generated by the sum-frequency generations, ωS1 + ωAS2, ωP1 + ωAS1, and ωP2 + ωP2. For the phase-locked subcycle optical burst (z = 1.31 m), the two modulation signals are in phase [Fig. 2(a)], while for z = 1.50 m they are out of phase with respect to each other [Fig. 2(b)]. These features are experimentally observable only when there is mutual coherence among the multicolor laser fields, and allow probing the relative-phase evolution during the propagation.
Fig. 2. Simulated SHG-FROG signals at 400 nm (broken blue line) and 480 nm (solid red line) for the propagation distance z of (a) 1.31 m and (b) 1.50 m, respectively.
3. Experimental In the experiment, a Ti:sapphire regenerative amplifier (800 nm, 35 fs, 4 mJ, 1 kHz, Legend Elite-USP, Coherent Inc.) was combined with an optical parametric amplifier (OPA, OPerASolo, Coherent Inc.) to produce two-color femtosecond laser pulses emitting at 800 nm (P2) and 1198 nm (P1), respectively, similar to the previous works [32,33]. These two pulses had pulse durations of ca. 50 fs and slight frequency chirps according to the characterization with the FROG reported in [49]. As shown in Fig. 3, the two-color pulses were focused into air using an aluminum-coated concave mirror with a focal length of 500 mm for generating multicolor laser fields via the nonresonant, degenerate FWM. After being collimated with another concave mirror with the same focal length, the output beam was attenuated by reflecting on a fused-silica wedge. After being split into two beams with a two aluminumcoated mirrors one of which was mounted on a closed-loop piezoelectric actuator, which were acting as a beam splitter, they were sent to a translation stage for the adjustment of the optical path length (z) in air after the multicolor generation. The two beams were noncollinearly focused into a beta barium borate (BBO) crystal (5-μm-thick, φ = 90° and θ = 45°) to generate an SHG signal. By recording the spectrum of the SHG signal with a sensitivity-calibrated multi-channel spectrometer (Maya2000pro, Ocean Optics, Dunedin, FL, USA) while scanning the time delay of one of the two beams using the piezoelectric actuator, an SHGFROG trace was obtained. All the mirrors except the dichroic mirror were metal-coated for
#234895 - $15.00 USD © 2015 OSA
Received 18 Feb 2015; revised 25 Apr 2015; accepted 28 Apr 2015; published 1 May 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.012373 | OPTICS EXPRESS 12378
preventing additional GVD induced during the propagation of the multicolor laser fields except that in air.
Fig. 3. Experimental setup and the spectrum of the multicolor laser fields generated by the FWM: CM, concave mirrors; WG, fused-silica wedge; PZT; piezoelectric actuator; PM, parabolic mirror; BBO, BBO crystal.
4. Results and discussion The beam diameters of the two input beams at the focus were ca. 200 μm, by which the effective interaction length for the FWM (the confocal parameters) was estimated to be 60 mm. The length was shorter than the coherence length of 480 mm and 200 mm calculated for the FWM that generates AS1 at 600 nm and AS3 at 401 nm, respectively. The input pulse energies of P1 and P2 were 250 μJ and 160 μJ, respectively. No indication of filamentation [18,37,39,40,43,45,50–52], such as plasma column and change in the beam divergence, was observed in air. There was no detectable energy loss during the FWM process (less than the energy fluctuation of the input P1 and P2,
2
Department of Applied Chemistry, Graduate School of Engineering, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka 819-0395, Japan Division of Optoelectronics and Photonics, Center for Future Chemistry, Kyushu University, 744 Motooka, Nishiku, Fukuoka 819-0395, Japan * [email protected]
Abstract: The generation of intense subcycle laser pulses during the propagation of two-color femtosecond pulses in a gas medium is investigated theoretically and experimentally. Four-wave mixing induced by the laser pulses in a gas medium generates multi-octave laser radiation from the ultraviolet to the infrared, which forms stable subcycle laser pulses after a certain propagation distance in a gas medium with group-velocity dispersion. The intense subcycle laser pulses would allow the coherent control of the waveforms of soft-x-rays generated via high-harmonic generation. ©2015 Optical Society of America OCIS codes: (190.4380) Nonlinear optics, four-wave mixing; (320.5540) Pulse shaping; (320.7090) Ultrafast lasers; (320.7100) Ultrafast measurements; (320.7110) Ultrafast nonlinear optics.
References and links 1.
E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320(5883), 1614–1617 (2008). 2. K. Zhao, Q. Zhang, M. Chini, Y. Wu, X. Wang, and Z. Chang, “Tailoring a 67 attosecond pulse through advantageous phase-mismatch,” Opt. Lett. 37(18), 3891–3893 (2012). 3. M. Uiberacker, T. Uphues, M. Schultze, A. J. Verhoef, V. Yakovlev, M. F. Kling, J. Rauschenberger, N. M. Kabachnik, H. Schröder, M. Lezius, K. L. Kompa, H.-G. Muller, M. J. J. Vrakking, S. Hendel, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond real-time observation of electron tunnelling in atoms,” Nature 446(7136), 627–632 (2007). 4. E. Goulielmakis, Z.-H. Loh, A. Wirth, R. Santra, N. Rohringer, V. S. Yakovlev, S. Zherebtsov, T. Pfeifer, A. M. Azzeer, M. F. Kling, S. R. Leone, and F. Krausz, “Real-time observation of valence electron motion,” Nature 466(7307), 739–743 (2010). 5. M. Schultze, M. Fiess, N. Karpowicz, J. Gagnon, M. Korbman, M. Hofstetter, S. Neppl, A. L. Cavalieri, Y. Komninos, T. Mercouris, C. A. Nicolaides, R. Pazourek, S. Nagele, J. Feist, J. Burgdörfer, A. M. Azzeer, R. Ernstorfer, R. Kienberger, U. Kleineberg, E. Goulielmakis, F. Krausz, and V. S. Yakovlev, “Delay in photoemission,” Science 328(5986), 1658–1662 (2010). 6. A. N. Pfeiffer, C. Cirelli, M. Smolarski, R. Dörner, and U. Keller, “Timing the release in sequential double ionization,” Nat. Phys. 7(5), 428–433 (2011). 7. A. N. Pfeiffer, C. Cirelli, M. Smolarski, D. Dimitrovski, M. Abu-samha, L. B. Madsen, and U. Keller, “Attoclock reveals natural coordinates of the laser-induced tunnelling current flow in atoms,” Nat. Phys. 8(1), 76–80 (2011). 8. M. Fieß, B. Horvath, T. Wittmann, W. Helml, Y. Cheng, Y. Zeng, Z. Xu, A. Scrinzi, J. Gagnon, F. Krausz, R. Kienberger, “Attosecond control of tunneling ionization and electron trajectories,” New J. Phys. 13(3), 033031 (2011). 9. F. Krausz and M. I. Stockman, “Attosecond metrology: from electron capture to future signal processing,” Nat. Photonics 8(3), 205–213 (2014). 10. T. W. Hänsch, “A proposed sub-femtosecond pulse synthesizer using separate phase-locked laser oscillators,” Opt. Commun. 80(1), 71–75 (1990). 11. S. Yoshikawa and T. Imasaka, “A new approach for the generation of ultrashort optical pulses,” Opt. Commun. 96(1-3), 94–98 (1993). 12. A. E. Kaplan, “Subfemtosecond pulses in mode-locked 2 π solitons of the cascade stimulated Raman scattering,” Phys. Rev. Lett. 73(9), 1243–1246 (1994).
#234895 - $15.00 USD © 2015 OSA
Received 18 Feb 2015; revised 25 Apr 2015; accepted 28 Apr 2015; published 1 May 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.012373 | OPTICS EXPRESS 12373
13. M. Y. Shverdin, D. R. Walker, D. D. Yavuz, G. Y. Yin, and S. E. Harris, “Generation of a single-cycle optical pulse,” Phys. Rev. Lett. 94(3), 033904 (2005). 14. W. J. Chen, Z. M. Hsieh, S. W. Huang, H. Y. Su, C. J. Lai, T. T. Tang, C. H. Lin, C. K. Lee, R. P. Pan, C. L. Pan, and A. H. Kung, “Sub-single-cycle optical pulse train with constant carrier envelope phase,” Phys. Rev. Lett. 100(16), 163906 (2008). 15. S. Huang, G. Cirmi, J. Moses, K.-H. Hong, S. Bhardwaj, J. R. Birge, L. Chen, E. Li, B. J. Eggleton, G. Cerullo, and F. X. Kärtner, “High-energy pulse synthesis with sub-cycle waveform control for strong-field physics,” Nat. Photonics 5(8), 475–479 (2011). 16. A. Wirth, M. T. Hassan, I. Grguras, J. Gagnon, A. Moulet, T. T. Luu, S. Pabst, R. Santra, Z. A. Alahmed, A. M. Azzeer, V. S. Yakovlev, V. Pervak, F. Krausz, and E. Goulielmakis, “Synthesized light transients,” Science 334(6053), 195–200 (2011). 17. K. Yoshii, J. Kiran Anthony, and M. Katsuragawa, “The simplest route to generating a train of attosecond pulses,” Light: Sci. Appl. 2(3), e58 (2013). 18. T. Fuji and Y. Nomura, “Generation of phase-stable sub-cycle mid-infrared pulses from filamentation in nitrogen,” Appl. Sci. 3(1), 122–138 (2013). 19. C. Manzoni, S. W. Huang, G. Cirmi, P. Farinello, J. Moses, F. X. Kärtner, and G. Cerullo, “Coherent synthesis of ultra-broadband optical parametric amplifiers,” Opt. Lett. 37(11), 1880–1882 (2012). 20. E. J. Takahashi, P. Lan, O. D. Mücke, Y. Nabekawa, and K. Midorikawa, “Attosecond nonlinear optics using gigawatt-scale isolated attosecond pulses,” Nat. Commun. 4, 2691 (2013). 21. B. Kim, J. An, Y. Yu, Y. Cheng, Z. Xu, and D. E. Kim, “Optimization of multi-cycle two-color laser fields for the generation of an isolated attosecond pulse,” Opt. Express 16(14), 10331–10340 (2008). 22. C. Vozzi, F. Calegari, F. Frassetto, L. Poletto, G. Sansone, P. Villoresi, M. Nisoli, S. De Silvestri, and S. Stagira, “Coherent continuum generation above 100 eV driven by an ir parametric source in a two-color scheme,” Phys. Rev. A 79(3), 033842 (2009). 23. E. J. Takahashi, P. Lan, O. D. Mücke, Y. Nabekawa, and K. Midorikawa, “Infrared two-color multicycle laser field synthesis for generating an intense attosecond pulse,” Phys. Rev. Lett. 104(23), 233901 (2010). 24. P. Wei, J. Miao, Z. Zeng, C. Li, X. Ge, R. Li, and Z. Xu, “Selective enhancement of a single harmonic emission in a driving laser field with subcycle waveform control,” Phys. Rev. Lett. 110(23), 233903 (2013). 25. S. Haessler, T. Balciunas, G. Fan, G. Andriukaitis, A. Pugžlys, A. Baltuška, T. Witting, R. Squibb, A. Zaïr, J. W. G. Tisch, J. P. Marangos, and L. E. Chipperfield, “Optimization of quantum trajectories driven by strong-field waveforms,” Phys. Rev. X 4, 021028 (2014). 26. P. Wei, Z. Zeng, J. Jiang, J. Miao, Y. Zheng, X. Ge, C. Li, and R. Li, “Selective generation of an intense single harmonic from a long gas cell with loosely focusing optics based on a three-color laser field,” Appl. Phys. Lett. 104(15), 151101 (2014). 27. X. Wang, C. Jin, and C. D. Lin, “Coherent control of high-harmonic generation using waveform-synthesized chirped laser fields,” Phys. Rev. A 90(2), 023416 (2014). 28. A. V. Sokolov, D. D. Yavuz, and S. E. Harris, “Subfemtosecond pulse generation by rotational molecular modulation,” Opt. Lett. 24(8), 557–559 (1999). 29. T. Imasaka, S. Kawasaki, and N. Ishibashi, “Generation of more than 40 laser emission lines from the ultraviolet to the visible regions by two-color stimulated raman effect,” Appl. Phys. B 49(4), 389–392 (1989). 30. D. D. Yavuz, D. R. Walker, M. Y. Shverdin, G. Y. Yin, and S. E. Harris, “Quasiperiodic Raman technique for ultrashort pulse generation,” Phys. Rev. Lett. 91(23), 233602 (2003). 31. E. Sali, K. J. Mendham, J. W. G. Tisch, T. Halfmann, and J. P. Marangos, “High-order stimulated Raman scattering in a highly transient regime driven by a pair of ultrashort pulses,” Opt. Lett. 29(5), 495–497 (2004). 32. O. Shitamichi and T. Imasaka, “High-order Raman sidebands generated from the near-infrared to ultraviolet region by four-wave Raman mixing of hydrogen using an ultrashort two-color pump beam,” Opt. Express 20(25), 27959–27965 (2012). 33. K. Motoyoshi, Y. Kida, and T. Imasaka, “High-energy, multicolor femtosecond pulses from the deep ultraviolet to the near infrared generated in a hydrogen-filled gas cell and hollow fiber,” Appl. Sci. 4(3), 318–330 (2014). 34. J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962). 35. C. G. Durfee III, L. Misoguti, S. Backus, H. C. Kapteyn, and M. M. Murnane, “Phase matching in cascaded third-order processes,” J. Opt. Soc. Am. B 19(4), 822–831 (2002). 36. Y. Kida and T. Kobayashi, “Generation of sub-10 fs ultraviolet Gaussian pulses,” J. Opt. Soc. Am. B 28(1), 139– 148 (2011). 37. T. Fuji, T. Horio, and T. Suzuki, “Generation of 12 fs deep-ultraviolet pulses by four-wave mixing through filamentation in neon gas,” Opt. Lett. 32(17), 2481–2483 (2007). 38. T. Suzuki, M. Hirai, and M. Katsuragawa, “Octave-spanning Raman comb with carrier envelope offset control,” Phys. Rev. Lett. 101(24), 243602 (2008). 39. T. Fuji, T. Suzuki, E. E. Serebryannikov, and A. Zheltikov, “Experimental and theoretical investigation of a multicolor filament,” Phys. Rev. A 80(6), 063822 (2009). 40. P. Zuo, T. Fuji, T. Horio, S. Adachi, and T. Suzuki, “Simultaneous generation of ultrashort pulses at 158 and 198 nm in a single filamentation cell by cascaded four-wave mixing in Ar,” Appl. Phys. B 108(4), 815–819 (2012). 41. J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, 2006).
#234895 - $15.00 USD © 2015 OSA
Received 18 Feb 2015; revised 25 Apr 2015; accepted 28 Apr 2015; published 1 May 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.012373 | OPTICS EXPRESS 12374
42. C. G. Durfee 3rd, S. Backus, H. C. Kapteyn, and M. M. Murnane, “Intense 8-fs pulse generation in the deep ultraviolet,” Opt. Lett. 24(10), 697–699 (1999). 43. T. Fuji and T. Suzuki, “Generation of sub-two-cycle mid-infrared pulses by four-wave mixing through filamentation in air,” Opt. Lett. 32(22), 3330–3332 (2007). 44. Y. Kida, J. Liu, T. Teramoto, and T. Kobayashi, “Sub-10 fs deep-ultraviolet pulses generated by chirped-pulse four-wave mixing,” Opt. Lett. 35(11), 1807–1809 (2010). 45. M. Beutler, M. Ghotbi, and F. Noack, “Generation of intense sub-20-fs vacuum ultraviolet pulses compressed by material dispersion,” Opt. Lett. 36(19), 3726–3728 (2011). 46. E. R. Peck and K. Reeder, “Dispersion of air,” J. Opt. Soc. Am. 62(8), 958–962 (1972). 47. K. W. DeLong, R. Trebino, J. Hunter, and W. E. White, “Frequency-resolved optical gating with the use of second-harmonic generation,” J. Opt. Soc. Am. B 11, 2206–2215 (1994). 48. D. Keusters, H. Tan, P. O’Shea, E. Zeek, R. Trebino, and W. S. Warren, “Relative-phase ambiguities in measurements of ultrashort pulses with well-separated multiple frequency components,” J. Opt. Soc. Am. B 20(10), 2226–2237 (2003). 49. Y. Kida, Y. Nakano, K. Motoyoshi, and T. Imasaka, “Frequency-resolved optical gating with two nonlinear optical processes,” Opt. Lett. 39(10), 3006–3009 (2014). 50. F. Théberge, N. Aközbek, W. Liu, A. Becker, and S. L. Chin, “Tunable ultrashort laser pulses generated through filamentation in gases,” Phys. Rev. Lett. 97(2), 023904 (2006). 51. A. Houard, M. Franco, B. Prade, A. Durécu, L. Lombard, P. Bourdon, O. Vasseur, B. Fleury, C. Robert, V. Michau, A. Couairon, and A. Mysyrowicz, “Femtosecond filamentation in turbulent air,” Phys. Rev. A 78(3), 033804 (2008). 52. F. Théberge, M. Châteauneuf, G. Roy, P. Mathieu, and J. Dubois, “Generation of tunable and broadband farinfrared laser pulses during two-color filamentation,” Phys. Rev. A 81(3), 033821 (2010). 53. K. W. DeLong, D. N. Fittinghoff, R. Trebino, B. Kohler, and K. Wilson, “Pulse retrieval in frequency-resolved optical gating based on the method of generalized projections,” Opt. Lett. 19(24), 2152–2154 (1994). 54. S. Skupin, G. Stibenz, L. Bergé, F. Lederer, T. Sokollik, M. Schnürer, N. Zhavoronkov, and G. Steinmeyer, “Self-compression by femtosecond pulse filamentation: Experiments versus numerical simulations,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(5), 056604 (2006).
1. Introduction In the past decades, ultrashort laser systems have been developed in various spectral ranges and applied for unveiling ultrafast photo-physical and chemical processes in atoms and molecules. Nearly single-cycle and a few-cycle laser pulses have enabled the generation of isolated attosecond pulses in the extreme ultraviolet (EUV) region [1,2] which have been utilized for the observation of electron motions in real time [3–9]. Even shorter optical pulses, sub-cycle optical pulses, allow detailed investigation of atomic-scale electron motion. Various approaches have been reported for the generation of subcycle laser pulses [10–19]. Among them, the optical shynthesizer generates an intense subcycle optical pulse with an energy of 0.3 mJ [16]. The active path-length stabilization generates subcycle laser pulses with stable waveforms for probing electron dynamics such as the instantaneous ac Stark shift in detail [16]. A train of intense subcycle pulses have important applications. Such a train of subcycle pulses is generated by synthesizing energetic multi-color femtosecond pulses. It has been recently reported that an energetic isolated attosecond pulse with a pulse energy exceeding one microjoule was generated using two-color multicycle laser fields forming optical beats [20]. Such a high-energy EUV pulse is a promising light source for nonlinear spectroscopy in the EUV. The waveforms of the harmonics strongly depend on the wavelengths of the twocolor driving laser fields [20–23] or the waveform of the synthesized electric field. It was reported that three-color multicycle laser pulses forming subcycle laser fields enhanced the intensities of harmonic emissions [24,25]. A waveform-controlled train of subcycle laser pulses with a high peak intensity is one of the desired light sources for the generation of energetic attosecond EUV pulses as well as the coherent control of high-harmonic generation (HHG) processes [20–27]. In this research, an approach to the generation of the intense train of subcycle optical pulses is reported. Two-color multicycle femtosecond laser fields are focused into a gas medium to generate multicolor laser fields composing an octave-spanning spectrum via degenerate four-wave mixing (FWM). During the subsequent propagation in a gas medium,
#234895 - $15.00 USD © 2015 OSA
Received 18 Feb 2015; revised 25 Apr 2015; accepted 28 Apr 2015; published 1 May 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.012373 | OPTICS EXPRESS 12375
the laser field synthesized by the multicolor laser fields drastically changes its shape, and forms a train of several intense subcycle optical pulses after a certain propagation distance (hereafter denoted as subcycle optical burst). Similar approach has been proposed by Harris et al. that employs group-velocity dispersion (GVD) for the synthesis of a train of pulses in a Raman-active gas with strongly driven molecular coherence by two-color nanosecond pulses [28]. In the present research, on the other hand, intense multicycle femtosecond pulses and nonresonant FWM are employed for the generation of the multi-color laser fields for the synthesis of subcycle laser pulses with a high peak intensity. The mutual coherence among the multicolor laser fields leads to a stable subcycle intensity profile without active feedback stabilization. The adjustment of the optical path length in the gas after the multicolor generation allows the control of the temporal intensity profile of the optical burst generated. 2. Theoretical description 2.1 Generation of intense subcycle optical pulses in a gas By focusing two-color laser pulses (P1, P2) with angular frequencies of ωP1 and ωP2 into a low-density gas medium, multicolor laser pulses are generated in a spectral range from the ultraviolet to the infrared via the degenerate FWM [13,14,29–33]. The phase mismatch and the influence of the GVD are small when the interaction length L is sufficiently short compared to the coherence length as in the experimental conditions described below. The real amplitude AS1(t), angular frequency ωS1, and phase φS1 of the first Stokes field generated by the FWM, in this case, are expressed as (Ln2/c)ωS1AP12(t)AP2(t), ωP2 - 2Ω, and 2φP1 - φP2 + π/2, respectively, under the assumption of the negligible pump depletion [34–36]. The real amplitude AAS1(t), angular frequency ωAS1, and phase φAS1 of the first anti-Stokes field generated by the FWM are, on the other hand, expressed as (Ln2/c)ωAS1AP1(t)AP22(t), ωP2 + Ω, and -φP1 + 2φP2 + π/2, respectively. The term Ω stands for the angular-frequency difference of ωP2 – ωP1. The parameters n2, c, AP1(t), AP2(t), φP1, and φP2 are the nonlinear refractive index of the medium, the speed of light in vacuum, the slowly-varying envelopes and phases of the two-color laser fields P1 and P2, respectively. The generated first anti-Stokes field then interacts with the two input laser fields to produce higher-order anti-Stokes fields via the cascaded interactions [13,14,29–33,35,37–40]. The mth-order anti-Stokes field has an amplitude of AASm (t ) = ( Ln2 / c ) ∏im=1 ω ASi AP1m (t ) AP 2 m +1 (t ) , an angular frequency of ωASm = m
ωP2 + mΩ, and phase φASm of –mφP1 + (m + 1)φP2 + mπ/2, respectively. After the generation, each laser field with an angular frequency ωi propagates with its own group velocity and propagation constant ki = niωi/c, where ni is the refractive index of the medium through which the laser field propagate. Under the retarded frame of reference (η, z) with the retarded time η of t – z/υg and υg being the group velocity of P2 [41], the phase of each field (i) evolves with the propagation distance z as ϕi (η , z ) = ϕi − ωi [η − z (ni / c − 1/ υ g )] . For any value of z, there exists a certain η (hereafter denoted as ηP) at which P1 and P2 are in φP1(ηP, z) = φP2(ηP, z) and phase such that η P = (ϕ P 2 − ϕ P1 ) / Ω + z[ωP 2 (nP 2 − nP1 ) / cΩ + (nP1 / c − 1/ υ g )] . Such a retarded time exits as long as the pulse durations of P1 and P2 are longer than the period of T = 2π/Ω. Besides, at a certain propagation distance, zl = 2π c (l − 1/ 4) / (nAS 1ω AS1 − 2nP 2ωP 2 + nP1ωP1 ) , the first antiStokes field is phase locked at ηP to the two laser fields P1 and P2 such that ΔφAS1(ηP, zl) = φAS1(ηP, zl) - φP2(ηP, zl) = 2πl, where l is an integer (l = 1, 2, 3, …). For such a zl to be present, the refractive index must be frequency dependent or GVD needs to be induced among the multicolor laser fields. The normal GVD induced in the medium changes the relative phases among the multicolor laser fields to phase lock the laser fields at the above propagation distance zl. The phase difference between S1 and P2 is ΔφS1(ηP, zl) = π/2 + (zl/c)(nS1ωS1 + nP2ωP2 – 2nP1ωP1) at zl and ηP, whereas that between the mth-order anti-Stokes field and P2 is
#234895 - $15.00 USD © 2015 OSA
Received 18 Feb 2015; revised 25 Apr 2015; accepted 28 Apr 2015; published 1 May 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.012373 | OPTICS EXPRESS 12376
ΔφASm(ηP, zl) = mπ/2 + (zl/c)[nASmωASm – (m + 1)nP2ωP2 + mnP1ωP1], respectively. If the phase differences are close to integer multiples of 2π, a subcycle temporal intensity structure is formed at the phase-locking point ηP at the propagation distance zl by the octave-spanning multicolor laser fields. The temporal intensity structure of the laser field synthesized by the multicolor laser fields does not depend on the values of φP1 and φP2 except the phase locking point ηP. This is because the phase differences Δφi(ηP, zl) and zl are independent of φP1 and φP2. In other words, there is mutual phase coherence among the multicolor laser fields. The application of the above phase-locking scheme in the femtosecond regime allows one to generate an intense subcycle optical burst consisting of only several subcycle pulses. This indicates the generation of subcycle laser pulses with the several orders of magnitude higher intensity than those generated in the nanosecond regime [13,14,17]. In the femtosecond regime, nonresonant FWM can be employed for the generation of multicolor laser fields with an arbitrary frequency spacing [18,35,37,39,40,42–45]. This leads to a degree of freedom for the waveform control, which is useful for the coherent control of HHG [20–24,26,27]. The influence of the GVD needs to be carefully addressed in this regime since it tends to temporally separate the multicolor laser fields from each other, preventing the formation of a subcycle optical burst. Nevertheless, the above phase-locking scheme still works, provided that the pulse duration of each field forming the multicolor laser fields is longer than the period, T = 2π/ . For instance, the three laser fields P1, P2, and AS1 phase lock to each other after the propagation distance in air of zl of 0.36 m (l = 1), 0.84 m (l = 2), and 1.32 m (l = 3) without being separated from each other [Figs. 1(a) and 1(b)]. Here, the wavelength of P2 and 2πΩ/c are assumed to be 800 nm and 4155 cm−1, respectively, giving the multicolor laser fields, S1, P1, AS1, and AS2, with wavelengths of 2387, 1198, 600, and 481 nm, respectively, and the relative spectral intensities shown in Fig. 1(a). Each multicolor laser field is assumed to have a pulse duration of 50-fs and the refractive index of air is calculated by referring to the parameters reported elsewhere [46]. Interestingly, ΔΦAS 2 (η P ) and ΔΦS1 (η P ) are calculated to be 9.81 x 2π and 2.04 x 2π at z3 of 1.32 m, respectively,
indicating AS2 and S1 are also nearly phase locked to P1,
Fig. 1. (a) Simulated spectrum of the multicolor laser fields, (b) temporal intensity distribution of each multicolor laser field after the propagation distance of z = 1.31 m, and (c) intensity profile of the synthesized optical burst by the multicolor laser fields for φP1 - φP2 = 0 after the propagation distances z of 0, 1.31, and 1.50 m. The ordinate in (b,c) corresponds to the retarded time η (see text).
#234895 - $15.00 USD © 2015 OSA
Received 18 Feb 2015; revised 25 Apr 2015; accepted 28 Apr 2015; published 1 May 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.012373 | OPTICS EXPRESS 12377
P2, and AS1. A subcycle pulse is formed at the phase-locking point ηP as shown in Fig. 1(c). This is because of the second and/or higher-order dispersion in the refractive index of air with respect to frequency. Such subcycle pulses are not formed at z = 0 and at z = 1.50 m. 2.2 Probing variation in temporal intensity profile of optical burst Second-harmonic-generation frequency-resolved optical gating (SHG FROG) [47] allows to probe the evolution of the relative phases among the multicolor laser fields and thus the intensity structure of the optical burst with respect to the propagation distance. SHG-FROG traces for the optical bursts in Fig. 1(c) consist of several discrete spectral components with modulations with respect to delay with the period, T = 2π/Ω. Some modulations result from the interference among several sum-frequency wave-mixing processes, and have the information of the relative phases of the multicolor laser fields. Change in the relative phases leads to the change in the shape of the FROG trace, which is not the case for the test pulse consisting of only two-color laser fields [48]. In Fig. 2, the frequency components at 400 nm and 480 nm contained in the SHG-FROG traces are simulated for the optical bursts in Fig. 1(c). The signal at 480 nm is generated by the sum-frequency generations, ωS1 + ωAS1, and ωP1 + ωP2, while the signal at 400 nm is generated by the sum-frequency generations, ωS1 + ωAS2, ωP1 + ωAS1, and ωP2 + ωP2. For the phase-locked subcycle optical burst (z = 1.31 m), the two modulation signals are in phase [Fig. 2(a)], while for z = 1.50 m they are out of phase with respect to each other [Fig. 2(b)]. These features are experimentally observable only when there is mutual coherence among the multicolor laser fields, and allow probing the relative-phase evolution during the propagation.
Fig. 2. Simulated SHG-FROG signals at 400 nm (broken blue line) and 480 nm (solid red line) for the propagation distance z of (a) 1.31 m and (b) 1.50 m, respectively.
3. Experimental In the experiment, a Ti:sapphire regenerative amplifier (800 nm, 35 fs, 4 mJ, 1 kHz, Legend Elite-USP, Coherent Inc.) was combined with an optical parametric amplifier (OPA, OPerASolo, Coherent Inc.) to produce two-color femtosecond laser pulses emitting at 800 nm (P2) and 1198 nm (P1), respectively, similar to the previous works [32,33]. These two pulses had pulse durations of ca. 50 fs and slight frequency chirps according to the characterization with the FROG reported in [49]. As shown in Fig. 3, the two-color pulses were focused into air using an aluminum-coated concave mirror with a focal length of 500 mm for generating multicolor laser fields via the nonresonant, degenerate FWM. After being collimated with another concave mirror with the same focal length, the output beam was attenuated by reflecting on a fused-silica wedge. After being split into two beams with a two aluminumcoated mirrors one of which was mounted on a closed-loop piezoelectric actuator, which were acting as a beam splitter, they were sent to a translation stage for the adjustment of the optical path length (z) in air after the multicolor generation. The two beams were noncollinearly focused into a beta barium borate (BBO) crystal (5-μm-thick, φ = 90° and θ = 45°) to generate an SHG signal. By recording the spectrum of the SHG signal with a sensitivity-calibrated multi-channel spectrometer (Maya2000pro, Ocean Optics, Dunedin, FL, USA) while scanning the time delay of one of the two beams using the piezoelectric actuator, an SHGFROG trace was obtained. All the mirrors except the dichroic mirror were metal-coated for
#234895 - $15.00 USD © 2015 OSA
Received 18 Feb 2015; revised 25 Apr 2015; accepted 28 Apr 2015; published 1 May 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.012373 | OPTICS EXPRESS 12378
preventing additional GVD induced during the propagation of the multicolor laser fields except that in air.
Fig. 3. Experimental setup and the spectrum of the multicolor laser fields generated by the FWM: CM, concave mirrors; WG, fused-silica wedge; PZT; piezoelectric actuator; PM, parabolic mirror; BBO, BBO crystal.
4. Results and discussion The beam diameters of the two input beams at the focus were ca. 200 μm, by which the effective interaction length for the FWM (the confocal parameters) was estimated to be 60 mm. The length was shorter than the coherence length of 480 mm and 200 mm calculated for the FWM that generates AS1 at 600 nm and AS3 at 401 nm, respectively. The input pulse energies of P1 and P2 were 250 μJ and 160 μJ, respectively. No indication of filamentation [18,37,39,40,43,45,50–52], such as plasma column and change in the beam divergence, was observed in air. There was no detectable energy loss during the FWM process (less than the energy fluctuation of the input P1 and P2,
