Synthesis of Two-Color Laser Pulses for the Harmonic Cutoff Extension

Transcription

1 Commun. Theor. Phys. 65 (2016) Vol. 65, No. 5, May 1, 2016 Synthesis of Two-Color Laser Pulses for the Harmonic Cutoff Extension Guo-Li Wang ( Á ), Li-Hua Zhou ( ÛÙ), Song-Feng Zhao ( Øô), and Xiao-Xin Zhou ( ²) Key Laboratory of Atomic and Molecular Physics and Functional Materials of Gansu Province, College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou , China (Received August 27, 2015; revised manuscript received March 9, 2016) Abstract Increasing simultaneously both the cutoff energy and efficiency is a big challenge to all applications of high-order harmonic generation (HHG). For this purpose, the shaping of the waveform of driving pulse is an alternative approach. Here, we show that the harmonic cutoff can be extended by about two times without reducing harmonic yield after considering macroscopic propagation effects, by adopting a practical way to synthesize two-color fields with fixed energy. Our results, combined with the experimental techniques, show the great potential of HHG as a tabletop light source. PACS numbers: Ky, Re, Qk Key words: harmonic cutoff extension, synthesized two-color laser field, propagation effects 1 Introduction The bright, spatially coherent radiation has many applications, such as spectroscopy, photochemistry, and so on. [1] From a practical point of view, different wavelengthes are often demanded for various applications. Besides some large-scale facilities, such as synchrotron radiation and free-electron laser, which can generate these radiations, tabletop devices are also desirable. When an intense laser field interacts with atoms or molecules, it drives nonlinear electron motion, producing high-order harmonics of the fundamental driving frequency. [2] This high-order harmonic generation (HHG) covers the wavelength spectral regions of extreme ultraviolet (XUV), soft X-ray and/or even hard X-ray. With the high photon flux, the HHG promises to be a tabletop light source. [3 4] HHG process produced by a single atom can be well understood by a semi-classical three-step model: [5] a bound electron first tunnel-ionizes into the continuum; then it is accelerated by the laser field; and finally it recombines with the parent ion on reversal of the laser field, emitting a high-energy photon. For a pure sinusoidal field the maximum electron recollision energy depends on the laser intensity, I, and wavelength, λ, scaling as Iλ 2. Although the increase of wavelength of the driving laser pulse provides a promising way to extend the highest harmonic energy (cutoff), it has a fundamental limitation that the harmonic yield drops quickly, which scaled roughly with λ 5 λ 7. [6 12] Hence, the search for ways of simultaneously increasing both the cut-off frequency and the HHG efficiency in the XUV spectral range is still among the most topical problems of nonlinear optics. [13] Although many relevant studies have been performed, [14 26] the work that is worth noting is Chipperfield et al. [27] They derived a general ideal waveform that could give the returning electron energy over 3 times higher than that of a pure sinusoidal wave. To approximate this ideal waveform, they used a genetic algorithm (GA) to find the optimal combination of fivecommensurate fields. We note that such optimizations are often on the single-atom level. The full description of experimental HHG spectra requires not only the theoretical treatment of the microscopic nonlinear laser-atom interaction but also the macroscopic propagation of the radiation through the nonlinear optical medium. [28 30] Our ultimate aim is to extend macroscopic harmonics. In this paper we first emphasize the role of longer wavelength, which is a necessary component in a synthesized waveform, by comparing the single-atom with macroscopic harmonic yields. We then develop a more practical way of optimizing waveform by only two-color wavelength, to increase the harmonic cutoff. We will see what factor can be achieved on the macroscopic level. 2 Theoretical Methods The driving field can be expressed as E(t) = i E i f i (t)cos(ω i t + φ i ), (1) where E i is the electric-field amplitude, ω i is the frequency of the laser pulses, φ i is the carrier-envelope phase (CEP), and f i (t) is the envelope. The parameters {E i, ω i, φ i } are determined by using GA. [31] In this work, we adopt micro-ga. The population size is chosen to be 10, the maximum number of genera- Supported by the National Natural Science Foundation of China under Grant Nos , , , the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant No , and the Basic Scientific Research Foundation for Institution of Higher Learning of Gansu Province wanggl@nwnu.edu.cn zhouxx@nwnu.edu.cn c 2016 Chinese Physical Society and IOP Publishing Ltd

2 602 Communications in Theoretical Physics Vol. 65 tions is set to be The fitness function is the classical maximum electron recollision energy. The optimization is carried out under the constraint that the field fluence is fixed. Once the GA searching is finished, harmonic spectrum can then be calculated for given laser parameters. In present simulations, the single-atom response is solved by the strong field approximation (SFA) method. [32] For the Ne target used in the present work, the induced dipole is calculated by using the model potential given by Ref. [33]. The macroscopic propagation of the fundamental and harmonic fields in an ionizing medium are obtained by solving Maxwell equations. [28 29] For the fundamental field we include the dispersion, absorption, Kerr and plasma effects. For the harmonics, the dispersion and absorption due to the neutral atoms are included. 3 Results and Discussions To test the role of longer-wavelength component of optimized wave, we first optimize the waveform for generating maximum return electron energy by using a five-color field: a fundamental pulse with optical period T, its first three harmonics, and a field at half the fundamental frequency, similar to Chipperfield et al., [27] but we remove the restriction of ratio of relative fluence, 2.0, between fundamental and half the fundamental frequency. The optimization returned the fractions of total fluence which are 0.672, 0.247, 0.042, 0.023, and 0.017, for five frequency components (from lower to higher), and 0.91π, 0.0, 1.50π, 0.77π, 0.12π for five phases. Figure 1(a) shows the optimized waveform for one optical cycle, 2T, together with the optimization of Chipperfield et al. [27] The main difference for two waveforms is that there are more field centered near plus maximum amplitude in our optimization. These extra field energy will generate higher recollision electron energy. In order to interpret the cutoff extension, we compare the maximum-recollision-energy electron trajectories that are generated in a fundamental and optimized filed in Fig. 1(b). Note that in both cases the fluence within the interval 0 < t < 2T is the same. The trajectory duration in a pure sinusoidal field is 0.65T. To obtain high recollision energy, the time the electron spends in the continuum increases to 0.92T in the optimized field, which will also reduce harmonic yield, as a result of extra electron wave packet spreading. Fortunately, this reduction can be compensated by increasing tunneling probability, which is determined by the laser s electric field at ionization time. In addition, from the point view of energy, higher recollision electron energy can be obtained in the optimized field due to the use of more field energy (shadow area) to accelerate the electron back into the core. [27] We then calculate the single-atom high harmonic spectra for Ne for a number of waveforms. The same pulse duration of 16 fs full width at half maximum (FWHM) is used for all waveforms, and the total laser energy is fixed equal to that of a pure sinusoidal pulse with a peak intensity of W/cm 2. We take a flat-top envelope with Gaussian ramps of 0.45 times of total duration. Figure 1(c) compares the simulated HHG yields, for clarity we also show the smoothed spectra. It is shown that the harmonic cutoffs increase from 79 ev for the fundamental 800 nm pulse to 237 ev for the optimized waveform, with the same level of harmonic yield. Compared with the result of Chipperfield et al., a more dramatic increasing of cutoff of a factor of 3 is achieved, due to a more dominant long-wavelength component, which is very close to the 3.11 for the perfect waveform. A 1550 nm pulse can also generate the same harmonic cutoff with the optimized waveform, but the yield drops about more than 100 times, corresponding to a wavelength scaling law of λ 7, as most results reported before. We note that most of previous theoretical studies of wavelength scaling are on single-atom level. In Figs. 1(d) and 1(e) we show the time-frequency analysis [34] of the harmonic spectra generated by 800 nm and optimized wave shown in Fig. 1(a). In addition to the number of bursts of harmonics is reduced in the optimized field, an obvious feature is that the contribution of short and long trajectories are different for two fields. Clearly, at the single-atom level the optimized pulse has much stronger contributions from short-trajectory electrons, however, for the case of single-color, the longtrajectory electrons dominate the harmonic emission. It is well known that the harmonics from long-trajectory electrons will vanish after propagation, as a result, whether the macroscopic harmonic yield of optimized wave is high than that of 800 nm wave? [14] Figure 1(d) shows the simulated macroscopic spectra. In the simulation, the lasers with beam waist of 40 µm are focused 1.5 mm before a 1 mm long gas jet. The gas jet has a Lorentz density distribution with an FWHM of 0.5 mm and a peak density of pressure 3 Torr. The amplitudes and phases of the fields are such that on axis at the center of the gas jet they formed optimized waveform with a total fluence equal to that of a sinusoidal 800 nm field with the same pulse duration and a peak intensity of W/cm 2. We can see that harmonic yield generated by optimized waveform drops more than one order compared to that generated by sinusoidal 800 nm field, even though these two waves have comparable single-atom harmonic yields and different contributing trajectories, as discussed above. To understand this unpredictable drops, we also simulate the macroscopic spectrum of single-color 1550 nm field. The comparison shows that after propagation the decrease of the yield becomes more obvious, i.e., more than 3 orders, and this gives a scaling rule of about λ 11, which is in accordance with previous theoretical and experimental investigations (see Ref. [29] and references therein). Therefore, combined with the simulation of Chipperfield et al., we can conclude that the role of longer-wavelength component in the optimized field is twofold. On the single-atom level, longerwavelength component can increase effectively maximum electron recollision energy. On the other hand, the HHG yields for long-wavelength driving lasers under the same experimental conditions appear quite unfavorable. In the optimization we must balance the pros and cons of the longer-wavelength component.

3 No. 5 Communications in Theoretical Physics 603 Fig. 1 (a) Optimized waveforms of present and Chipperfield et al. [27] (b) Comparison of the electric fields of a pure sinusoidal and optimized field, and electron trajectories with maximum recollision energy are generated in two fields. In both cases the fluence within the interval 0 < t < 2T is the same. (c) Single-atom harmonic yields of Ne generated by optimized field and single-color 800 nm and 1550 nm fields. The total field energy is the same for three laser fields. (d) (e) The time-frequency analysis of the harmonic spectra generated by 800 nm and optimized wave. (f) The macroscopic harmonic spectra that spectra are shown in (c) after propagated in a 1 mm gas jet with a gas pressure of 3 Torr. The other parameters are given in the text. Next, we will focus on a more practical way to increase cutoff energy, i.e., find an optimum waveform by synthesizing only two-color laser fields. To fix the total field energy, unlike the case of commensurate components that the optimization is performed in a given optical cycle, we optimize wave for a whole pulse. Here, we set each fields has the same pulse duration, 6-cycle FWHM for fundamental one. In Fig. 2(a) we show three waves, one is fundamental single color with wavelength λ, second is optimized one by synthesizing wavelengths λ + 1.3λ, with intensity ratio of 1:1.46 and CEPs of 1.56π, 1.78π, the third one is also an optimized one, but with wavelengths λ+1.16λ, intensity ratio of 1:1 and CEPs of 1.63π, 0.13π. We can infer from above discussion that the first optimized wave will generate higher cutoff with lower yield compared to second optimum. In Fig. 2(b) we show that this is indeed the case. We compare the harmonic spectra of Ne generated by waves shown in Fig. 2(a) for λ = 800 nm, total laser intensity = W/cm 2, and pulse duration =16 fs. By combining a strong longer-wavelength 1041 nm component, the harmonic cutoff can be extended from 79 ev to 166 ev, without losing the harmonic yield. A comparable 930 nm wavelength increases cutoff energy to 153 ev, and harmonic intensity is also enhanced. Figure 2(c) shows the macroscopic harmonic spectra for three waveforms are generated under the same conditions as in Fig. 1(f). Propagation effects decrease harmonic yields generated by optimized waves somewhat due to a long-wavelength component. Nevertheless, a factor of 1.9 for the extension of cutoff energy with the same harmonic yield can be accomplished. This achievement can be maintained even for a higher gas pressure, 30 Torr, as shown in Fig. 2(d). Increasing the pressure from 3 to 30 Torr enhances the harmonic yields by a factor of about 100, which is close to the quadratic dependence on the pressure, indicating that good phase matching is maintained in this pressure range, because of a low ionization level (6.7% in the two-color field. [35] ).

4 604 Communications in Theoretical Physics Vol. 65 Fig. 2 (a) Waves for a single color with wavelength of λ, synthesized fields by λ + 1.3λ and λ λ. Each field has the same total energy. (b) The smoothed single-atom harmonic spectra generated by waves shown in Fig. 2(a) for the λ = 800 nm. (c) The macroscopic harmonic spectra generated by waves shown in Fig. 2(a) in a 1 mm gas jet with pressure of 3 Torr. (d) Same as (c), but for a pressure of 30 Torr. The other parameters are given in the text. Fig. 3 Single-atom (a) and macroscopic harmonic spectra of Ne generated from 1 mm gas jet positioned after (b) 1.5 mm, (c) 1.0 mm, (d) 2.0 mm of laser focus, by waveforms that shown in Fig. 2(a) with a fundamental wavelength of 1200 nm. Other conditions are the same as those used in Figs. 2(b) and 2(c).

5 No. 5 Communications in Theoretical Physics 605 We also check the wavelength dependence on optimized waveform. We calculate harmonic spectra of Ne generated by a single-color 1200 nm and two-color synthesized from 1200 nm and 1395 nm fields, and the results are shown in Figs. 3(a) and 3(b), other conditions are the same as those used in Figs. 2(b) and 2(c). Figures show that the maximum harmonic energy extends from about 150 ev for sinusoidal field to 315 ev for optimized wave, i.e., the extension is about a factor of 2.1, which is in accordance with the case for fundamental wavelength of 800 nm. In view of the experiment, we last discuss the influence of gas-jet position on the optimized waveforms. Compared with the HHG generated from gas jet placed 1.5 mm after laser focus (Fig. 3(b)), Figs. 3(c) and 3(d) show the spectra generated also from 1 mm gas gets, but with different positions, i.e., 1.0 mm and 2.0 mm downstream from the laser focus, respectively. It reveals a same factor of cutoff extension for all jet positions. But, it is worth noting that the more higher laser intensity and gas pressure would reduce the extension, because the electron ionization probability generated in optimized field is bigger greatly than that in the single-color field. 4 Conclusion To summarize, we show that by optimally synthesizing two-color laser fields, the harmonic cut-off can be extended by about 2 times without losing the harmonic yield. Our results, combined with the techniques of waveform synthesizing, [36 38] optical parametric amplification (OPA) and optical parametric chirped-pulse amplification (OPCPA), show the HHG might be a tabletop light source. References [1] J.M. Dudley and G. Genty, Phys. Today 66 (2013) 29. [2] M.C. Kohler, T. Pfeifer, K.Z. Hatsagortsyan, and C.H. Keitel, Adv. Atom. Mol. Opt. Phys. 61 (2012) 159. [3] K. Bourzac, Nature (London) 486 (2012) 172. [4] T. Popmintchev, et al., Science 336 (2012) [5] P.B. Corkum, Phys. Rev. Lett. 71 (1993) [6] J. Tate, T. Auguste, H.G. Muller, P. Salières, P. Agostini, and L.F. DiMauro, Phys. Rev. Lett. 98 (2007) [7] K. Schiessl, K.L. Ishikawa, E. Persson, and J. Burgdörfer, Phys. Rev. Lett. 99 (2007) [8] A.D. Shiner, et al., Phys. Rev. Lett. 103 (2009) [9] M.V. Frolov, N.L. Manakov, and A.F. Starace, Phys. Rev. Lett. 100 (2008) [10] P. Colosimo, et al., Nature Phys. 4 (2008) 386. [11] C.J. Lai, et al., Phys. Rev. Lett. 111 (2013) [12] C.Z. Cheng, X.X. Zhou, and P.C. Li, Acta Phys. Sin. 60 (2011) [13] A.V. Andreev, S.Y. Stremoukhov, and O.A. Shoutova, Springer Proceedings in Physics 147 (2014) 7. [14] C. Jin, G.L. Wang, H. Wei, A.T. Le, and C.D. Lin, Nat. Commun. 5 (2014) [15] C. Jin, G.L. Wang, A.T. Le, and C.D. Lin, Sci. Rep. 4 (2014) [16] S. Haessler, T. Balčiunas, G. Fan, G. Andriukaitis, A. Pugžlys, and A. Baltuška, Phys. Rev. X 4 (2014) [17] J. Solanpää, J.A. Budagosky, N.L. Shvetsov-Shilovski, A. Castro, A. Rubio, and E. Räsänen, Phys. Rev. A 90 (2014) [18] P.F. Wei, J. Miao, Z.N. Zeng, C. Li, X.C. Ge, R.X. Li, and Z.Z. Xu, Phys. Rev. Lett. 110 (2013) [19] C. Altucci, J.W.G. Tisch, and R. Velotta, J. Mod. Optic. 58 (2011) 1. [20] Eiji J. Takahashi, et al., Phys. Rev. Lett. 104 (2010) [21] H.C. Bandulet, et al., Phys. Rev. A 81 (2010) [22] F. Calegari, C. Vozzi, M. Negro, et al., Opt. Lett. 34 (2009) [23] T. Siegel, et al., Opt. Express 18 (2010) [24] M. Negro, et al., AIP Conf. Proc (2012) 45. [25] M. Negro, et al., Laser Phys. Lett. 8 (2011) 875. [26] M. Klaiber, M.C. Kohler, K.Z. Hatsagortsyan, and C.H. Keitel, Phys. Rev. A 85 (2012) [27] L.E. Chipperfield, J.S. Robinson, J.W.G. Tisch, and J.P. Marangos, Phys. Rev. Lett. 102 (2009) [28] M.B. Gaarde, J.L. Tate, and K.J. Schafer, J. Phys. B 41 (2008) [29] C. Jin, A.T. Le, and C.D. Lin, Phys. Rev. A 83 (2011) [30] G.L. Wang, C. Jin, A.T. Le, and C.D. Lin, Phys. Rev. A 84 (2011) [31] D.L. Carroll, FORTRAN genetic algorithm driver, http: // GADriverFreeVersion.aspx (accessed in August 2015) [32] M. Lewenstein, P. Balcou, M.Y. Ivanov, A. L Huillier, and P.B. Corkum, Phys. Rev. A 49 (1994) [33] X.M. Tong and C.D. Lin, J. Phys. B 38 (2005) [34] X.M. Tong and Shih-I. Chu, Phys. Rev. A 61 (2000) (R). [35] X.M. Tong and C.D. Lin, J. Phys. B 38 (2005) [36] S.W. Huang, et al., Nat. Photonics 5 (2011) 475. [37] H.S. Chan, Z.M. Hsieh, W.H. Liang, et al., Science 331 (2011) [38] W.J. Chen, H.Z. Wang, R.Y. Lin, C.K. Lee, and C.L. Pan, Laser Phys. Lett. 9 (2012) 212.