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The scaling law of photoionization in few-cycle laser pulses is verified in this paper. By means of numerical solution of time-dependent Schrödinger equation, the photoionization and the asymmetry degree of photoionization of atoms with different binding potential irradiated by various laser pulses are studied. We find that the effect of increasing pulse intensity is compensated by deepening the atomic binding potential. In order to keep the asymmetric photoionization unchanged, if the central frequency of the pulse is enlarged by k times, the atomic binding potential should also be enlarged by k times, and the laser intensity should be enlarged by k3 times.
©2005 Optical Society of America
1. Introduction
Scaling is a powerful technique for analyzing many phenomena in intense laser fields. For photoionization in intense laser fields, a scaling law has been established which reveals a remarkably simple physical mechanism: the main characters of photoelectron angular distributions (PADs) in the nonrelativistic case, as well as many kinds of strong field phenomena, are determined by three dimensionless numbers: (1) the ponderomotive number up =Up/ħ ω, the ponderomotive energy Up in units of the laser photon energy ħω; (2) the binding number εb = Eb/ħω, the atomic binding energy Eb in units of ħω; and (3) the absorbed-photon number q [1]. The physical essences of processes with the same three parameters are just the same, even though their dynamic parameters look quite different.
The above scaling law holds for long pulses that can be uniquely determined by their frequency, intensity, and polarization. While for the laser pulses with duration as short as only few optical cycles, the temporal shape of the electric field varies dramatically with the carrier-envelope ( CE ) phase, i.e. the initial phase φ of the carrier wave with respect to the pulse envelope [2, 3]. All physical processes depending on the electric field in the few-cycle pulses are CE phase-dependent, such as the high-harmonic generation [4, 5], photoemission from metal surfaces [6], and dissociation in molecular systems [7]. Is the scaling law still applicable in the short-pulse case?
The CE phase-dependent phenomena of the photoionization of atoms and molecules in few-cycle laser pulses are characterized by the asymmetric ionization [8–13]. Very recently, the asymmetry in the ionization of Rydberg atoms by radio-frequency few-cycle pulses was demonstrated [14]. Similar results were also found in the photoionization of atoms by intense few-cycle laser pulses [15]. The similarity suggests a scaling relation for photoionization in various few-cycle pulses and a scaling law in few-cycle case is established [16]. The scaling law states that for given CE phases and cycle numbers, when the carrier frequency is enlarged by k times, in order to keep the physical process unchanged, the laser intensity should be enlarged by k3 times, and the binding energy of target atoms should be enlarged by k times.
The verifications from other sources provide strong support to the scaling law as well as to the theory itself. The scaling law in few-cycle case is established in the frame of a nonperturbative quantum scattering theory of Guo et al. [17]. Based on the analytical formula of photoionization in few-cycle case [11–13], we derived the scaling law by noting that the main characters of PADs in few-cycle pulses are determined by five-dimensionless parameters: the CE phase, the cycle number np, and the three parameters mentioned above [16]. As a theory with the potential to reveal the most basic parameters of strong-field physics, further theoretical or experimental proof using different methods is needed. Direct experimental tests to the scaling law are somewhat difficult in varying the laser frequency, the atomic binding energy, and the laser intensity in a large scale at one time.
In this paper, we verify the scaling law in few-cycle case using time-dependent Schrödinger equation (TDSE) method. The photoionization of atoms is obtained numerically, and the CE phase dependence of the asymmetric photoionization is shown. By comparing the photoionization, as well as the asymmetry, in different binding potentials and in various few-cycle pulses of different intensities and different frequencies, we will show that there exists a stable structure when the binding energy, the laser intensity, and the frequency satisfy the scaling law.
The paper is organized as follows: the numerical method is presented in Sec. 2, the dependence of photoionization on the laser intensity and on the binding energy is given in Sec. 3 and 4, respectively, the verification of the scaling law is presented in Sec. 5, and the conclusions and discussions are given in Sec. 6.
2. TDSE method
Our research is based on the solution of the TDSE for an atom interacting with an intense, linearly polarized few-cycle laser pulse. In the length gauge, the Schrödinger equation can be written as [18] (atomic units are used throughout this paper)
where Ψ(r,t) is the time-dependent wave function of the electron and r denotes the position vector of electron with respect to its parent core. In the right side of the above equation, the first term in the large bracket is the kinetic energy term of electron, the second term is the Coulomb potential, and the last term is the dipole interaction in which E(t) is the electric field of the laser pulse. For linearly polarized laser pulses, the ionized electrons are mostly ejected out along the polarization vector [19]. To avoid the time-consumed calculations, we adopt the 1-dimension model to the considered problem. Then, the Schrödinger equation reads:
and the soft-Coulomb model potential is used
In the numerical simulation, chosen α as 0.775a.u., the binding energy of the model atom has the binding energy of 13.6 eV. With the implicit Crank-Nicholson scheme and the ‘trace-back’ algorithm for solving the tridiagonal equations, we numerically calculate the evolution of electron wave function. The laser pulse used in the calculation is defined by the vector potential
from which the electric field can be determined E(t) = -dA(t)/ dt. Here, np is the number of optical cycles in the pulse, φ denotes the CE phase, and E 0 is the peak value of the electric field.
The total ionization probability is defined as the absorption in the calculation boundaries and the absorption in one boundary is used to denote the partial ionization probability in the corresponding direction. The boundary is set to be far enough from the nucleus in order to avoid significant reflections from the boundary that could distort the dynamics of the process.
The ionization in few-cycle case is characterized by the asymmetric ionization in opposite directions, which is termed as inversion asymmetry [3, 11–13]. The asymmetry varies with the CE phase, the laser intensity, the pulse duration, as well as the atomic binding energy. A parameter defined by partial ionization probabilities is used to qualify the asymmetry degree [10, 14]
where P + and P - are photoelectron signal measured by two opposing detectors placed along the laser polarization vector. The influence of the laser intensity, the pulse duration and the atomic binding energy on the asymmetric ionization can be quantitatively reflected by this parameter. The scaling relation can also be reflected by the asymmetry parameter. For simplicity of description, we verify the scaling law by means of the asymmetric degree.
In the following sections, we first study the dependence of the photoionization and the asymmetry degree on the laser intensity and on the binding potential, then vary the two factors instantaneously and study the influence on the asymmetry degree, and finally we conclude the scaling law.
3. Dependence of photoionization on laser intensity
Fig. 1. Variation of photoionization probabilities with the CE phase at different laser intensities: (a) I=2.25×1014W/cm2; (b) I=3.51×1014W/cm2. Line with black squares is for P+ and that with red circles for P-. Curves in (c) show the variation of the asymmetry degree with CE phase. Line with real circles is for (a), and that with open circles is for (b). The atomic binding energy is set as 13.6eV, and the laser pulses are of five-cycle duration and of wavelength 800nm.
The laser intensity affects the photoionization in many aspects because the highly nonlinear essence of the multiphoton ionization. When the laser intensity is very low, the multiphoton ionization will not occur. Increasing laser intensity, the ionization becomes more and more probable. The ionization can not be significant until the laser intensity exceeds the ionization threshold. For further higher intensities, it is not always the case that a higher intensity corresponds to a higher ionization probability because the target materials may be depleted before the laser intensity reaches its maximum. In this section, we will study the dependence of the photoionization on the laser intensity in a range that the laser intensity is higher than the threshold and lower than the depletion value.
In few-cycle laser pulses, the photoionization and the asymmetry degree are both CE phase-dependent [10, 14]. For linearly polarized pulses, the maximal electric-field strength varies with the CE phase, which leads to the maximal ionization varying with the CE phase. The variation pattern depends on the laser intensity, as shown in Fig. 1.
Figure 1 shows the partial ionization probabilities and the corresponding asymmetry parameters varying with CE phase for a hydrogen atom irradiated by five-cycle laser pulses of different peak intensities. The laser pulses are of wavelength 800nm. The partial ionization probabilities at a relatively lower intensity are depicted in Fig. 1(a), and that at a relatively higher intensity is depicted in Fig. 1(b). In both cases, the partial ionization probabilities vary from their maximal value to their minimal value with the CE phase, respectively, and the partial ionization probabilities in opposite directions equal to each other for some special CE phases. The corresponding asymmetry degrees vary from their maxima to their minima. The influence of the laser intensity is two aspects. One is the maximal ionization probability at higher intensity is larger than that at lower intensity. The other is that the maximal/minimal partial ionization occurs at different CE phases, so does the CE phases for equally partial ionization probabilities. Correspondingly, there is a small shift of the asymmetric degree to right at higher intensities, as shown in Fig. 1(c). The shift becomes more obvious for even higher intensities.
Different from the results using strong field approximation (SFA) [8], our calculations shown in Fig. 1(b) exhibit that the photoionization is left-right asymmetric for φ = 0 or π and that zero asymmetry appears around φ = 0.7π . These results agree with the treatment by Chelkowski et al. using a 3-D TDSE model [10]. This difference probably arises from the neglecting of the Coulomb attraction to the ionized electrons in SFA [20]. It is also found that the asymmetry is more distinct at higher intensities than that at lower intensities when the pulse duration exceeds 4-cycles, and that the maximal asymmetry at lower intensities varies with the CE phase visibly, while at higher intensities not. The physical insight to these phenomena is presented elsewhere [15].
4. Influence of the binding potential
As mentioned in the previous section, there exists a threshold of laser intensity above which the photoionization becomes significant. The value of the threshold intensity varies with the laser frequency and the atomic binding potential. Thus, besides the intensity and the frequency of the laser field, the atomic binding potential also affects the photoionization process. In order to show the influence of the atomic binding potential on the photoionization, we also perform some calculations with varying atomic binding potentials.
2Figure 2(a) shows the variation of the partial ionization probabilities with the CE phase, for a hydrogen-like atom with binding energy of 18.2eV irradiated by a five-cycle laser pulse of peak intensity 3.51×1014 W/cm2. A comparison made with Fig. 1 (b) shows the influence of the atomic binding potential. Both ionizations are induced by laser pulses of equal pulse durations and equal peak intensities, but with different atomic binding energies (Fig. 1(b) is for 13.6eV). The influence of the atomic binding energy on the ionization can also be summarized as two aspects: one is that the ionization probability for smaller binding energy is far greater than that for larger binding energy, the other is the CE phases corresponding to the maximal ionization varies with the atomic binding energy. For H atom, the maximum of P+ appears for φ =5π/ 4, while for the H-like atom, the maximum of P+ appears for φ = π. Correspondingly, the asymmetry degree is subject to the similar effects, as shown in Fig. 2(b). The maximum of the asymmetry degree for larger atomic binding energy is a little larger than that for smaller atomic binding energy and there exists a small shift of the asymmetric degree to left for larger atomic binding energies. The shift will become more obvious for even larger binding energies.
Fig. 2. (a) Variation of photoionization probabilities with the CE phase at a deeper binding potential: Eb=18.2eV. Line with black squares is for P+, and that with red circles is for P-. A comparison can be made with Fig. 1(b); (b) Variation of the asymmetry degree with CE phase. Line with real circles is for (a), and that with open circles is for Fig. 1(b). The laser pulses are of five-cycle duration, wavelength 800nm, and peak intensity 3.51×1014W/cm2.
5. Verification of scaling law
We have shown in the previous two sections that the effect of increasing pulse intensity is compensated by deepening the atomic binding potential. In order to keep the ionization unchanged in different atoms, one should adopt a laser pulse with higher intensity for atoms with larger binding potential. A problem is to what extent the laser intensity should be increased?
In this section, the photoionization of atoms with different binding energies irradiated by different laser pulses is studied by varying the atomic binding energy and the pulse intensity simultaneously. The similarity in different laser intensities and different atomic binding energies proves the scaling law.
The photoionization, the partial ionization, and the asymmetry degree of the target atoms with binding energy 13.6eV irradiated by the laser pulses of wavelength 800nm and intensity I=3.51×1014W/cm2 are chosen as the benchmark. The asymmetry degree of two kinds of target atoms of different binding energies irradiated by laser pulses of different wavelengths and different peak intensities are calculated and compared with the benchmark. We notice that the calculations do not show any rule if the carrier frequency of the laser pulses does not change, then we perform the calculations with the frequency also changed correspondingly. According to the scaling law, when the laser frequency is enlarged by k times, the atomic binding potential should be enlarged by k times, and the laser intensity should be enlarged by is k3 times, then the photoionization process will be the same.
Fig. 3. The dependence on the CE phase of the asymmetric degree in five-cycle pulses. (O) is for Eb=13.6eV, λ=800nm, I=3.51×l014W/cm2; (A)-(D) are for Eb'=k Eb, λ'=λ/k, but for different intensities: (A) I1=kI; (B)I2=k2I; (C) I3=k3I; and (D) I4=k4I. In plot (a), λ'=616nm, thus k=1.3; in plot (b) λ'=400nm, thus k=2.0. The similarity in (O) and (C) verifies the scaling law.
In order to verify the scaling law, we enlarge the laser intensity by k, k2, k3, and k4 times, respectively. We plot in Fig. 3 the dependence of the asymmetry parameter on the CE phase in 5-cycle pulses. In order to keep the verifications more representative, two kinds of laser pulses are used, one is for the 5-cycle pulses of central wavelength 616nm ( k=1.3 ) shown in Fig. 3(a), the other is for the 5-cycle pulses of central wavelength 400nm ( k=2 ) shown in Fig. 3(b). From Fig. 3(a) we find that, among the five curves, the variation pattern of the asymmetry degree with the CE phase for 800nm (curve O) and that for 616nm with intensity of k3 I (curve C) are highly superposed, while distinctive differences appear in the comparison made between the benchmark curve to other curves. Similar phenomena can be also found in Fig. 3(b). The photoionizations for various atoms of different binding energies irradiated by various short pulses (with fixed np) of frequencies and intensities obeying the scaling relations exhibit the same dependence of the asymmetry on the CE phase, which confirms the scaling law.
The asymmetric photoionization in few-cycle pulses and the variation of the asymmetry with the CE phase depend on the pulse duration. Is the verification of the scaling law affected by the pulse duration? To answer this problem, we perform calculations on several pulse durations, such as for n=3, 5, 7. We note that with the increasing pulse duration the CE phase-dependent asymmetry decreases, thus the proof of the scaling law by means of the CE phase-dependent asymmetry will be invalid for longer laser pulses where the asymmetry becomes negligible.
Fig. 4. Comparison of the asymmetry degree as a function of the CE phase in several pulse durations: (a) n=3; (b) n=5; (c) n=7.
In Fig. 4 we depict the variation of the asymmetry parameter with the CE phase for laser pulses of wavelength at 800nm, 616nm and 400nm with different durations. Similar to that in Fig. 3, the asymmetry for the atom with binding energy of 13.6eV irradiated by laser pulses of 800nm wavelength with intensity 3.51×1014W/cm2 is chosen as a benchmark. The intensities for the laser pulses of 616nm and 400nm wavelengths and the corresponding binding energies for the irradiated atoms are changed according to the scaling law. For extremely short pulses, such as n=3 shown in plot (a), small departure appears when the asymmetry degrees reach their extrema; and the departure becomes negligible for other two pulse durations, as shown in plots (b) for n=5 and (c) for n=7. In general, the scaling law still holds for the calculated pulse durations.
6. Conclusions and discussions
We have presented a numerical verification of the scaling law in few-cycle pulses by studying the asymmetry in photoionization of different atoms irradiated by various short pulses. We find that the effect of increasing pulse intensity is compensated by deepening atomic binding potential. In order to keep the ionization unchanged in different atoms and in different pulses, one should adopt a laser pulse with higher intensity for atoms with larger binding potential. Both scaling law and our calculations indicate that, in order to keep the asymmetric photoionization unchanged, the laser intensity should be enlarged by k3 times.
Compared to that in long-pulse cases, the scaling law in few-cycle case is determined by two additional dimensionless numbers deciding the distribution of the electric field in the pulse envelope, i.e. the CE phase φ and the cycle number np. Once the values of φ and np are given, the distribution of the electric field in the pulse envelope is fixed, and then the photoionization of atoms irradiated by the short pulse satisfy the scaling law is the same as that in long-pulse cases.
The scaling relation between atomic binding energy and photon energy is obvious, and the scaling of pulse duration is trivial in terms of the cycle number, but the scaling with intensity is not trivial yet. The essential of the scaling law is the cubic dependence of the laser intensity on the scaling ratio k, since
The scaling law in few-cycle case indicates that the ponderomotive number up is an essential parameter in intense laser fields.
Acknowledgments
We are thankful to the suggestive discussion with D.-S. Guo. To whom for materials: J. Zhang. This work is sponsored by Shanghai Rising-Star Program and supported by the Chinese National Natural Science Foundation under Grant No. 60408008 and Natural Science Key Foundation of Shanghai under Grant No. 04JC14036.
References and links
1 . D.-S. Guo , Jingtao Zhang , Zhizhan Xu , Xiaofeng Li , Panming Fu , and R. R. Freeman , “ Practical scaling law for photoionization angular distributions ,” Phys. Rev. A 68 , 043404 (1-5) ( 2003 ). [CrossRef]
2 . A. Baltuška , Th. Udem , M. Uiberacker , M. Hentschel , E. Goulielmakis , Ch. Gohle , R. Holzwarth , V.S. Yakovlev , A. Scrinzi , T. W. Hänsch , and F. Krausz , “ Attosecond control of electronic processes by intense light fields ,” Nature 421 , 611 – 615 ( 2003 ). [CrossRef] [PubMed]
3 . G. G. Paulus , F. Grasbon , H. Walther , P. Villoresi , M. Nisoli , S. Stagira , E. Priori , and S. De Silvestri , “ Absolute-phase phenomena in photoionization with few-cycle laser pulses ,” Nature 414 , 182 – 185 ( 2001 ). [CrossRef] [PubMed]
4 . A. de Bohan , Ph. Antonie , D. B. Milošević , and B. Piraux , “ Phase-dependent harmonic emission with ultrashort laser pulses ,” Phys. Rev. Lett. 81 , 1837 – 1840 ( 1998 ). [CrossRef]
5 . Zhinan Zeng , Ruxin Li , Wei Yu , and ZhiZhan Xu , “ Effect of the carrier-envelope phase of driving laser field on the high-order harmonic attosecond pulse ,” Phys. Rev. A 67 , 013815 (1-6) ( 2003 ). [CrossRef]
6 . C. Lemell , X. M. Tong , F. Krausz , and J. Burgdörfer , “ Electron emission from metal surfaces by ultrashort pulses: determination of carrier-envelope phase ,” Phys. Rev. Lett. 90 , 076403 (1-4) ( 2003 ). [CrossRef] [PubMed]
7 . A. Brown and W. J. Meath , “ On the effects of absolute laser phase on the interaction of a pulsed laser with polar versus nonpolar molecules ,” J. Chem. Phys. 109 , 9351 – 9365 ( 1998 ). [CrossRef]
8 . D. B. Milošević , G. G. Paulus , and W. Becker , “ Phase-dependent effects of a few-cycle laser pulse ,” Phys. Rev. Lett. 89 , 153001 (1-4) ( 2002 ). [CrossRef] [PubMed]
9 . G. G. Paulus , F. Lindner , H. Walther , A. Baltuška , E. Goulielmakis , M. Lezius , and F. Krausz , “ Measurement of the phase of few-cycle laser pulses ,” Phys. Rev. Lett. 91 , 253004 (1-4) ( 2003 ). [CrossRef]
10 . S. Chelkowski , A. D. Bandrauk , and A. Apolonski , “ Phase-dependent asymmetries in strong-field photoionization by few-cycle laser pulses ,” Phys. Rev. A 70 , 013815 (1-9) ( 2004 ). [CrossRef]
11 . Jingtao Zhang and Zhizhan Xu , “ Above-threshold ionization of Kr atoms in an infinite sequence of circular polarized few-cycle pulses ,” Phys. Rev. A 68 , 013402 (1-6) ( 2003 ). [CrossRef]
12 . Jingtao Zhang , Xunli Feng , D.-S. Guo , and Zhizhan Xu , “ Phase-dependent angular distributions of photoelectrons in an infinite sequence of linearly polarized few-cycle pulses ,” Phys. Rev. A 69 , 043409 (1-6) ( 2004 ). [CrossRef]
13 . Jingtao Zhang , Shaohui Li , and Zhizhan Xu , “ Above-threshold ionization of xenon atoms in a bichromatic phase-controlled laser field of linear and circular polarizations ,” Phys. Rev. A 69 , 053410 (1-6) ( 2004 ). [CrossRef]
14 . A. Gürtler , E. Robicheaux , W. J. van der Zande , and L. D. Noordam , “ Asymmetry in the strong-field ionization of Rydberg atoms by few-cycle pulses ,” Phys. Rev. Lett. 92 , 033002 (1-4) ( 2004 ). [CrossRef] [PubMed]
15 . Xiaoming Zhang , State Key Laboratory for Mesoscopic Physics, Department of Physics, Peking University, Beijing 100087, China and Jingtao Zhang are preparing a manuscript to be called “ Photoionization of H atoms in few-cycle laser pulses .”
16 . Jingtao Zhang , State Key Laboratory of High Field Laser Physics, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China is preparing a manuscript to be called “ A scaling law of photoionization in few-cycle regime .”
17 . D.-S. Guo , T. Åberg , and B. Crasemann , “ Scattering theory of multiphoton ionization in strong fields ,” Phys. Rev. A 40 , 4997 – 5005 ( 1989 ). [CrossRef] [PubMed]
18 . Q. Su and J. H. Eberly , “ Suppression of ionization and atomic electron localization by short intense laser pulses ,” Phys. Rev. A 43 , 2474 – 2479 ( 1991 ). [CrossRef] [PubMed]
19 . P. H. Bucksbaum , M. Bashkansky , and D. W. Schumacher , “ Above-threshold ionization in helium ,” Phys. Rev. A 37 , 3615 – 3618 ( 1988 ). [CrossRef] [PubMed]
20 . S. Chelkowski and A. D. Bandrauk , “ Asymmetries in strong-field photoionization by few-cycle laser pulses: Kinetic-energy spectra and semiclassical explanation of the asymmetries of fast and slow electrons ,” Phys. Rev. A 71 , 053815 ( 2005 ). [CrossRef]
