## Above-threshold ionization by few-cycle phase jump pulses |

Optics Express, Vol. 21, Issue 20, pp. 24309-24317 (2013)

http://dx.doi.org/10.1364/OE.21.024309

Acrobat PDF (1853 KB)

### Abstract

We theoretically investigate the above-threshold ionization of hydrogen atoms driven by few-cycle phase jump laser pulses. By numerically solving the three-dimensional time-dependent Schrödinger equation, we demonstrate that the phase jump plays an important role in the ionization process. The cutoff of the photoelectron energy spectrum can extend to a range of very high energy, and the yield of the photoelectrons can be dramatically enhanced by choosing proper phase jump times. Both the classical simulations and Fourier transform method are used to understand the spectra features found in our investigation.

© 2013 OSA

## 1. Introduction

1. P. Agostini, F. Fabre, G. Mainfray, G. Petite, and N. K. Rahman, “Free-free transitions following six-photon ionization of xenon atoms,” Phys. Rev. Lett. **42**, 1127–1130 (1979). [CrossRef]

2. P. Kruit, J. Kimman, H. G. Muller, and M. J. Van der Wiel, “Electron spectra from multiphoton ionization of xenon at 1064, 532, and 355 nm,” Phys. Rev. A **28**, 248–255 (1983). [CrossRef]

6. E. Mevel, P. Breger, R. Trainham, G. Petite, P. Agostini, A. Migus, J. P. Chambaret, and A. Antonetti, “Atoms in strong optical fields: evolution from multiphoton to tunnel ionization,” Phys. Rev. Lett. **70**, 406–409 (1993). [CrossRef] [PubMed]

7. F. Fabre, G. Petite, P. Agostini, and M. Clement, “Multiphoton above-threshold ionisation of xenon at 0.53 and 1.06μm,” J. Phys. B: At. Mol. Phys. **15**, 1353–1369 (1982). [CrossRef]

8. Y. Gontier and M. Trahin, “Energetic electron generation by multiphoton absorption,” J. Phys. B: At. Mol. Phys. **13**, 4383–4390 (1980). [CrossRef]

9. B. R. Yang, K. J. Schafer, B. Walker, K. C. Kulander, P. Agostini, and L. F. DiMauro, “Intensity-dependent scattering rings in high order above-threshold ionization,” Phys. Rev. Lett. **71**, 3770–3773 (1993). [CrossRef] [PubMed]

10. G. G. Paulus, W. Nicklich, and H. Walther, “Investigation of above-threshold ionization with femtosecond pulses: connection between plateau and angular distribution of the photoelectrons,” Europhys. Lett **27**, 267–272 (1994). [CrossRef]

14. P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. **71**, 1994–1997 (1993). [CrossRef] [PubMed]

*U*

_{p}and 10

*U*

_{p}(

*U*

_{p}is the ponderomotive energy).

15. L. Guo, S. S. Han, and J. Chen, “Time-energy analysis of above-threshold ionization,” Opt. Express **18**, 1240–1248 (2010). [CrossRef] [PubMed]

33. T. Shaaran, M. F. Ciappina, R. Guichard, J. A. Pérez-Hernández, L. Roso, M. Arnold, T. Siegel, A. Zaïr, and M. Lewenstein, “High-order-harmonic generation by enhanced plasmonic near-fields in metal nanoparticles,” Phys. Rev. A **87**, 041402 (2013). [CrossRef]

21. D. B. Miloševic, G. G. Paulus, and W. Becker, “High-order above-threshold ionization with few-cycle pulse: a meter of the absolute phase,” Opt. Express **11**, 1418–1429 (2003). [CrossRef]

22. T. Nakajima, “Above-threshold ionization by chirped laser pulses,” Phys. Rev. A **75**, 053409 (2007). [CrossRef]

23. Y. Xiang, Y. P. Niu, and S. Q. Gong, “Above-threshold ionization by few-cycle nonlinear chirped pulses,” Phys. Rev. A **80**, 023423 (2009). [CrossRef]

24. C. I. Blaga, F. Catoire, P. Colosimo, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. DiMauro, ”Strong-field photoionization revisited,” Nat. Phys **5**, 335 (2009). [CrossRef]

25. W. Quan, Z. Lin, M. Wu, H. Kang, H. Liu, X. Liu, J. Chen, J. Liu, X. T. He, S. G. Chen, H. Xiong, L. Guo, H. Xu, Y. Fu, Y. Cheng, and Z. Z. Xu, “Classical sspects in above-threshold ionization with a midinfrared strong laser field,” Phys. Rev. Lett. **103**, 093001 (2009). [CrossRef]

25. W. Quan, Z. Lin, M. Wu, H. Kang, H. Liu, X. Liu, J. Chen, J. Liu, X. T. He, S. G. Chen, H. Xiong, L. Guo, H. Xu, Y. Fu, Y. Cheng, and Z. Z. Xu, “Classical sspects in above-threshold ionization with a midinfrared strong laser field,” Phys. Rev. Lett. **103**, 093001 (2009). [CrossRef]

27. C. P. Liu and K. Z. Hatsagortsyan, “Coulomb focusing in above-threshold ionization in elliptically polarized midinfrared strong laser fields,” Phys. Rev. A **85**, 023413 (2012). [CrossRef]

28. S. Kim, J. Jin, Y.-J Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature (London) **453**, 757–760 (2008). [CrossRef]

34. M. F. Ciappina, Srdjan S. Aćimović, T. Shaaran, J. Biegert, R. Quidant, and M. Lewenstein, “Enhancement of high harmonic generation by confining electron motion in plasmonic nanostrutures,” Opt. Express **20**, 26261–26274 (2012). [CrossRef] [PubMed]

35. B. T. Torosov and N. V. Vitanov, “Coherent control of a quantum transition by a phase jump,” Phys. Rev. A **76**(5), 053404 (2007). [CrossRef]

37. P. K. Jha, H. Eleuch, and Y. V. Rostovtsev, “Coherent control of atomic excitation using off-resonant strong few-cycle pulses,” Phys. Rev. A **82**, 045805 (2010). [CrossRef]

*et al.*found that a phase-jump pulse may cause the breakdown of dipole blockade in strong Rydberg blockade regime [36

36. J. Qian, Y. Qian, M. Ke, X. L. Feng, C. H. Oh, and Y. Z. Wang, “Breakdown of the dipole blockade with a zero-area phase-jump pulse,” Phys. Rev. A **80**, 053413 (2009). [CrossRef]

*et al.*demonstrated that the center-of-mass motion of a Bose-Einstein condensate can be quenched by applying a carefully timed and sized jump in the phase of the rotating field [38

38. P. W. Cleary, T. W. Hijmans, and J. T. M. Walraven, “Manipulation of a Bose-Einstein condensate by a time-averaged orbiting potential using phase jumps of the rotating field,” Phys. Rev. A **82**, 063635 (2010). [CrossRef]

39. Y. Xiang, Y. P. Niu, H. M. Feng, Y. H. Qi, and S. Q. Gong, “Coherent control of high-order harmonic generation by phase jump pulses,” Opt. Express **20**(17),19289–19296 (2012). [CrossRef] [PubMed]

## 2. Theoretical model and method

**r**is the position vector of the electron,

*V*(

**r**) is the Coulomb potential, and

**E**(

*t*) =

*E*(

*t*)

*ẑ*is the electric field vector of the laser pulse with polarization direction along

*z*axis. For hydrogen atom, the Coulomb potential can be expressed as

*V*(

**r**) = 1/

*r*, where

*r*is the modulus of

**r**.

*F*,

*f*(

*t*),

*ω*, and

*φ*are the peak amplitude, field envelope, frequency and the carrier-envelope phase (CEP) of the laser pulse.

*ϕ*is the jump phase introduced into the electric field at time

*t*

_{0}. The phase jump in the laser field (Eq. (2)) can be realized by modern femtosecond pulse-shaping technology, such as described in [35

35. B. T. Torosov and N. V. Vitanov, “Coherent control of a quantum transition by a phase jump,” Phys. Rev. A **76**(5), 053404 (2007). [CrossRef]

*F*is 0.0377 a.u. (corresponds to an intensity of

*I*= 5 × 10

^{13}

*W/cm*

^{2}), the frequency

*ω*is 0.057 a.u. (corresponds to a wavelength of

*λ*= 800 nm), and the CEP

*φ*= 0. We use a Gaussian shaped envelope

*f*(

*t*) = exp(−2ln2(t/

*τ*)

^{2}) with

*τ*= 5fs being full width at half maximum (FWHM) of the pulse.

40. A. Sanpera, P. Jönsson, J. B. Watson, and K. Burnett, “Harmonic generation beyond the saturation intensity in helium,” Phys. Rev. A **51**, 3148–3153 (1995). [CrossRef] [PubMed]

*ψ*(

**r**,

*t*) is obtained, the energy-resolved photoelectron spectra then can be calculated by using the window function technique developed by Schafer [42

42. K. J. Schafer and K. C. Kulander, “Energy analysis of time-dependent wave functions: application to above-threshold ionization,” Phys. Rev. A **42**, 5794–5797 (1990). [CrossRef] [PubMed]

43. K. J. Schafer, “The energy analysis of time-dependent numerical wave functions,” Comput. Phys. Commun **63**, 427–434 (1991). [CrossRef]

## 3. Results and discussions

*ϕ*and jump time

*t*

_{0}. As mentioned in our previous work [32

32. T. Shaaran, M. F. Ciappina, and M. Lewenstein, “Quantum-orbit analysis of high-order-harmonic generation by resonant plasmon field enhancement,” Phys. Rev. A **86**, 023408 (2012). [CrossRef]

*π*. In this paper two new cases are also considered which results in six phase jump cases, i.e.,

*t*

_{0}= ±0.75

*T*, ±0.50

*T*and ±0.25

*T*(where

*T*is the optical period of the laser pulse). As a reference, the case without phase jump (

*ϕ*= 0) is also considered.

*U*

_{p}and 10

*U*

_{p}cutoffs. But for the cases of

*t*

_{0}= ±0.75

*T*and

*t*

_{0}= ±0.25

*T*, the cutoff energy of the whole spectrum extends very much, and some oscillations appear. For the cases of

*t*

_{0}= ±0.50

*T*, the photoelectrons with higher energies far beyond the classical prediction 10

*U*

_{p}energy are present in the spectrum. Moreover, the yield of the photoelectrons is enhanced for all the cases with phase jump, especially for the cases of

*t*

_{0}= ±0.50

*T*in the low-energy regime.

44. C. P. Liu, T. Nakajima, T. Sakka, and H. Ohgaki, “Above-threshold ionization and high-order harmonic generation by mid-infrared and far-infrared laser pulses,” Phys. Rev. A **77**, 043411 (2008). [CrossRef]

*t*(ionization time, or born time) with zero velocity, and then they move under the influence of the laser field. For some ionization time, electrons may never return to the core (called the direct electrons). But for some other ionization time, electrons may return to the core at time

_{b}*t*and elastically backscatter (or rescatter) with the core, and subsequently they move in the laser field until the terminal time

_{r}*t*of the pulse (called the rescattered electrons). After the termination of the laser pulse, the kinetic energy of electron do not change anymore and the electron continues to fly to the detector. According to this theory, the final kinetic energy of electron can be calculated by solving the Newton equation of motion

_{f}*ẍ*= −

*E*(

*t*), ignoring the effect of the ionization potential. For the direct process, the kinetic energy of an electron born at time

*t*is while for the rescattering process, the kinetic energy of an electron is: Where

_{b}*A*is the vector potential of the laser field.

*U*

_{p}and 10

*U*

_{p}, respectively. This is well consistent with the quantum simulation as shown in Fig. 1. For the phase jump cases, we can observe the strong modifications of the electron energy distribution caused by phase jump. In particular for the cases of

*t*

_{0}= −0.75

*T*,

*t*

_{0}= −0.25

*T*,

*t*

_{0}= 0.25

*T*and

*t*

_{0}= 0.75

*T*, the maximum kinetic energies of the rescattered electrons extend to about 15

*U*

_{p}, 27

*U*

_{p}, 34

*U*

_{p}and 30

*U*

_{p}, respectively. These energies are approximately consistent with the quantum simulations as shown in Fig. 1.

45. Y. Xiang, Y. P. Niu, and S. Q. Gong, “Control of the high-order harmonics cutoff through the combination of a chirped laser and static electric field,” Phys. Rev. A **79**, 053419 (2009). [CrossRef]

*t*

_{0}= ±0.75

*T*and

*t*

_{0}= ±0.25

*T*, all the rescattered electrons which have the maximum kinetic energy (called the cutoff electrons) return to the core just at the time before half an optical period of the phase jump time

*t*

_{0}. At this moment, the laser field value is zero, and it will change the sign, and then the electrons backscatter at the same time as shown in Figs. 2(b), 2(c), 2(f) and 2(g). This coincidence in time exactly gives a positive force to the electrons to accelerate from this moment. For the laser pulse with phase jump, this acceleration process can last a whole optical period at most, while it can only last half an optical period at most for the pulse without phase jump. As a result, the electron could obtain a larger impulse from the laser pulse with phase jump.

*A*and the impulse with a reversed direction as

*B*in Fig. 2. In order to quantitatively show the significant effect of the laser field asymmetry, we calculate the cutoff electrons’ velocity at their backscattering time, the impulse obtained from the backscattering time to the terminal time and their final velocity at the terminal time. The results are shown in Table 1. We can see that, for all the phase jump cases, their cutoff electrons have a larger final velocity than that of the case without phase jump. A further view of these results shows us that the impulse obtained from backscattering time to the terminal time have different degrees of increase compared to that of the case without phase jump. This is because of the different degrees of asymmetry caused by the different phase jump times. For the cases of

*t*

_{0}= ±0.75

*T*and

*t*

_{0}= ±0.25

*T*, the asymmetry changes so much that the impulse obtained is much larger than that of the case without phase jump. Thus the electron’s final velocity is much larger even though its velocity at the backscattering time is smaller. However, for the cases of

*t*

_{0}= ±0.50

*T*, the symmetry doesn’t change so much and the impulse obtained is just a little bit larger than that of the case without phase jump. As a result, the final velocity is almost the same with that of the case without phase jump. In conclusion, the asymmetry of the laser field is the reason why the cutoff extends.

46. D. G. Arbó, E. Persson, and J. Burgdörfer, “Time double-slit interferences in strong-field tunneling ionization,” Phys. Rev. A **74**, 063407 (2006). [CrossRef]

*t*

_{0}= ±0.75

*T*and

*t*

_{0}= ±0.25

*T*. Take the case of

*t*

_{0}= −0.25

*T*for an example, we see from Fig. 2(f) that there exist many trajectories between 5

*U*

_{p}and 25

*U*

_{p}, and hence oscillations occur between 5

*U*

_{p}and 25

*U*

_{p}in the spectrum shown as the thin blue dash-dotted line in Fig. 1.

*ω*is weakened substantially, while the intensities of the lower and higher frequencies with a broad bandwidth are enhanced. Just as discussed in [47

47. C. P. Liu and T. Nakajima, “Anomalous ionization efficiency by few-cycle pulses in the multiphoton ionization regime,” Phys. Rev. A **76**, 023416 (2007). [CrossRef]

*n*-photon ionization becomes

*n′*(

*n′*<

*n*)-photon ionization, and then the ionization efficiency by few-cycle pulses will be larger than that by many-cycle pulses. Now, for all the phase jump pulses we used here, they have a much broader spectral bandwidth than the pulse without phase jump. Therefore, they have a larger ionization efficiency than that of the case without phase jump. Furthermore, we can see from Fig. 3 that, two distinct peaks appear astride the center frequency in 0.8

*ω*and 1.2

*ω*. And for the cases of

*t*

_{0}= ±0.50

*T*, the intensity of the high frequency peak 1.2

*ω*is stronger than that of all the other four phase jump cases. Then, the multiphoton ionization can be easier to take place for them than the other four cases. Thus the yield of the photoelectrons is much higher in the low-energy regime for

*t*

_{0}= ±0.50

*T*.

*t*

_{0}= ±0.50

*T*. This will make the multiphoton ionization be easier and even make the single-photon ionization take place. We consider that the photoelectrons with energy far beyond the classical prediction 10

*U*

_{p}cutoff mainly come from the multiphoton ionization or single-photon ionization of the very high energy photons.

*t*

_{0}= −0.75

*T*and

*t*

_{0}= −0.25

*T*, the laser field value at the born time is smaller than 0.6

*F*, which is shown in Figs. 2(b) and 2(f). According to [21

21. D. B. Miloševic, G. G. Paulus, and W. Becker, “High-order above-threshold ionization with few-cycle pulse: a meter of the absolute phase,” Opt. Express **11**, 1418–1429 (2003). [CrossRef]

*E*(

*t*)| ≥ 0.6

*F*) have considerable contribution to the electron spectrum. Based on this theory, the cutoffs of

*t*

_{0}= −0.75

*T*and

*t*

_{0}= −0.25

*T*are about 10

*U*

_{p}and 15

*U*

_{p}, which are obviously smaller than the quantum simulation values of 15

*U*

_{p}and 27

*U*

_{p}. In our calculations, since the Coulomb potential was used to characterize the level structure of hydrogen atom, there exists lots of excited bound states. As was reported by [44

44. C. P. Liu, T. Nakajima, T. Sakka, and H. Ohgaki, “Above-threshold ionization and high-order harmonic generation by mid-infrared and far-infrared laser pulses,” Phys. Rev. A **77**, 043411 (2008). [CrossRef]

*V*(

**r**) = −

*Z*·exp(−

*αr*) [48

48. J. L. Krause, K. J. Schafer, and K. C. Kulander, “Calculation of photoemission from atoms subject to intense laser fields,” Phys. Rev. A **45**, 4998–5010 (1992). [CrossRef] [PubMed]

*Z*and

*α*are chosen to be 2 and 1.1144, which results in only the existence of the ground state. Then we calculate the ionization probability by numerically solving three-dimensional time-dependent Schrödinger equation using Coulomb potential and the short-range potential. In Table 2 we list the ionization probabilities of the hydrogen atoms for

*t*

_{0}= −0.75 and

*t*

_{0}= −0.25

*T. P*and

_{coul}*P*are the ionization probability calculated for the Coulomb and short-range potential. We may approximate the contribution of the excited bound states by

_{sr}*P*−

_{coul}*P*. For the cases of

_{sr}*t*

_{0}= −0.75

*T*and

*t*

_{0}= −0.25, we can see that

*P*is much larger than

_{coul}*P*. This exactly demonstrates the important roles of the excited bound states in the ionization process. When the excited bound states are populated through multiphoton absorption, the ionization potential could be reduced that there does not need so strong laser field to tunneling ionize. Thus, tunneling process takes place even though the laser field is less than 0.6

_{sr}*F*. Therefore, the cutoff electrons have considerable yield for

*t*

_{0}= −0.75

*T*and

*t*

_{0}= −0.25

*T*, which results in the cutoffs reaching to the 15

*U*

_{p}and 27

*U*

_{p}.

## 4. Conclusions

*π*-phase jump. We found that the photoelectron energy spectrum is strongly modified by the phase jump time

*t*

_{0}. If proper

*t*

_{0}is selected, the cutoff of the spectrum could be dramatically extended, as well as the yield of the photoelectrons. We found that it is the asymmetry of the laser field that results in the cutoff extension. Since the phase jump makes the frequencies both lower and higher than center frequency

*ω*with a broad bandwidth appear, the yield of the photoelectrons is enhanced for all the cases with phase jump, especially for the cases of

*t*

_{0}= ±0.50

*T*in the low-energy regime. Moreover, we also demonstrated that the excited bound states play important roles in the ionization process.

## Acknowledgments

## References and links

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32. | T. Shaaran, M. F. Ciappina, and M. Lewenstein, “Quantum-orbit analysis of high-order-harmonic generation by resonant plasmon field enhancement,” Phys. Rev. A |

33. | T. Shaaran, M. F. Ciappina, R. Guichard, J. A. Pérez-Hernández, L. Roso, M. Arnold, T. Siegel, A. Zaïr, and M. Lewenstein, “High-order-harmonic generation by enhanced plasmonic near-fields in metal nanoparticles,” Phys. Rev. A |

34. | M. F. Ciappina, Srdjan S. Aćimović, T. Shaaran, J. Biegert, R. Quidant, and M. Lewenstein, “Enhancement of high harmonic generation by confining electron motion in plasmonic nanostrutures,” Opt. Express |

35. | B. T. Torosov and N. V. Vitanov, “Coherent control of a quantum transition by a phase jump,” Phys. Rev. A |

36. | J. Qian, Y. Qian, M. Ke, X. L. Feng, C. H. Oh, and Y. Z. Wang, “Breakdown of the dipole blockade with a zero-area phase-jump pulse,” Phys. Rev. A |

37. | P. K. Jha, H. Eleuch, and Y. V. Rostovtsev, “Coherent control of atomic excitation using off-resonant strong few-cycle pulses,” Phys. Rev. A |

38. | P. W. Cleary, T. W. Hijmans, and J. T. M. Walraven, “Manipulation of a Bose-Einstein condensate by a time-averaged orbiting potential using phase jumps of the rotating field,” Phys. Rev. A |

39. | Y. Xiang, Y. P. Niu, H. M. Feng, Y. H. Qi, and S. Q. Gong, “Coherent control of high-order harmonic generation by phase jump pulses,” Opt. Express |

40. | A. Sanpera, P. Jönsson, J. B. Watson, and K. Burnett, “Harmonic generation beyond the saturation intensity in helium,” Phys. Rev. A |

41. | M. Gavrila, |

42. | K. J. Schafer and K. C. Kulander, “Energy analysis of time-dependent wave functions: application to above-threshold ionization,” Phys. Rev. A |

43. | K. J. Schafer, “The energy analysis of time-dependent numerical wave functions,” Comput. Phys. Commun |

44. | C. P. Liu, T. Nakajima, T. Sakka, and H. Ohgaki, “Above-threshold ionization and high-order harmonic generation by mid-infrared and far-infrared laser pulses,” Phys. Rev. A |

45. | Y. Xiang, Y. P. Niu, and S. Q. Gong, “Control of the high-order harmonics cutoff through the combination of a chirped laser and static electric field,” Phys. Rev. A |

46. | D. G. Arbó, E. Persson, and J. Burgdörfer, “Time double-slit interferences in strong-field tunneling ionization,” Phys. Rev. A |

47. | C. P. Liu and T. Nakajima, “Anomalous ionization efficiency by few-cycle pulses in the multiphoton ionization regime,” Phys. Rev. A |

48. | J. L. Krause, K. J. Schafer, and K. C. Kulander, “Calculation of photoemission from atoms subject to intense laser fields,” Phys. Rev. A |

**OCIS Codes**

(190.4180) Nonlinear optics : Multiphoton processes

(270.6620) Quantum optics : Strong-field processes

(320.7110) Ultrafast optics : Ultrafast nonlinear optics

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: August 7, 2013

Revised Manuscript: September 18, 2013

Manuscript Accepted: September 18, 2013

Published: October 3, 2013

**Citation**

Pidong Hu, Yueping Niu, Yang Xiang, and Shangqing Gong, "Above-threshold ionization by few-cycle phase jump pulses," Opt. Express **21**, 24309-24317 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-20-24309

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