## Radiative recombination in the presence of a few cycle laser pulse

Optics Express, Vol. 14, Issue 9, pp. 3715-3723 (2006)

http://dx.doi.org/10.1364/OE.14.003715

Acrobat PDF (265 KB)

### Abstract

We have investigated the laser-assisted radiative recombination in the presence of a few-cycle pulse with the aim of demonstrating means of controlling such process. Within the Coulomb-Volkov approach already employed to describe the radiative recombination assisted by a monochromatic laser field, we have found that the emitted photon spectrum is affected by both the cycle number *n*_{c}
and the carrier-envelope relative phase *φ*. In particular, it has been shown that the minimum and the maximum values of the emitted photon energy may be controlled by varying *n*_{c}
and *φ*. Finally, it has been found that the enhancement of radiative recombination occurring in the presence of a monochromatic field, takes place also by using a few-cycle laser pulse.

© 2006 Optical Society of America

## 1. Introduction

1. Y. Hahn, “Electron –ion recombination process – an overview,” Rep. Prog. Phys. **60**, 691–759 (1997). [CrossRef]

2. R. Neumann, H. Poth, A. Winnacker, and A. Wolf, “Laser-enhanced electron-ion capture and antihydrogen formation,” Z. Physik. A **313**, 253–262 (1983). [CrossRef]

10. M. Yu Kuchiev and V. N. Ostrovsky, “Multiphoton radiative recombination of electron assisted by a laser field,” Phys. Rev. A **61**, 033414 (2000). [CrossRef]

11. D. B. Milosevic and F. Ehlotzky, “Rescattering effects in soft-x-ray generation by laser-assisted electron-ion recombination,” Phys. Rev. A **65**, 042504 (2002). [CrossRef]

12. C. Leone, S. Bivona, R. Burlon, and G. Ferrante, “Strong-field and plasma aspects of multiphoton radiative recombination,” Phys. Rev. A **66**, 051403 (2002). [CrossRef]

- The LARR is enhanced when the average translation momentum is of the same order magnitude as or lower than the quiver momentum amplitude [12, 13];
12. C. Leone, S. Bivona, R. Burlon, and G. Ferrante, “Strong-field and plasma aspects of multiphoton radiative recombination,” Phys. Rev. A

**66**, 051403 (2002). [CrossRef] - The spectra exhibit large values at emitted photon energy,
*ω*_{X}, very higher than that possible in the field process [12**66**, 051403 (2002). [CrossRef] - The LARR may be described as a semiclassic model [17] in which the recombination is viewed as a two-step process. In the first step, the free electron propagates towards the ion and its motion is described classically with motion changes ascribed to the action of the laser field; in the second step, the free electron recombines with the ion instantaneously at a given value of the laser field phase. According to this picture, the spectrum of the emitted radiation exhibits large values in the range of the photon energy ,
17. S. Bivona, R. Burlon, G. Ferrante, and C. Leone, “Radiative recombination in a strong laser field. Low frequency approximation,” J. Opt. Soc. Am. B

**22**, 2076–2082 (2005). [CrossRef]*ω*_{X}, where the emission is classically allowed. Moreover, the maxima in the spectra occur when both the instantaneous value of the electric laser field and the instantaneous absolute value of the quiver velocity are, respectively, close to zero and to one of its maximum. Therefore, it turns out that the emission spectra are characterized by a maximum at the furthest edge, as in correspondence of the electron kinetic energy the oscillating electric field is zero.

*n*

_{c}and the carrier-envelope relative phase

*φ*. In particular, we will examine how the maximum value of the emitted photon energy may be controlled by varying

*n*

_{c}and

*φ*. We also will investigate whether the LARR enhancement found in the presence of a monochromatic laser field [12

**66**, 051403 (2002). [CrossRef]

## 2. Quantum theory of the laser assisted radiative recombination

*Z*

^{+}in the presence of a finite laser pulse, linearly polarized with a sin-square envelope, having the following electric field:

*E*

_{0}the electric field laser amplitude,

*the polarization vector and*ε ^

_{L}*φ*the carrier-envelope relative phase. In order to have an integer number of cycles we assume

*τ*=

*n*

_{c}

*T*with

*T*=2

*π/ω*the period of the carrier. We use atomic units throughout the paper. The electric field given by the Eq. (1) is zero before and after the pulse, and

*n*

_{c}will be taken equal or greater than 2. With this last choice the impulse imparted to the electron by the electric field of the laser pulse during its duration will be zero. Of course, the vector potential, taken as

*(*

**A***t*) = -

*c*

*dt*´

*E*(

*t*´), turns to be zero for

*t*≤ 0 and

*t*≤

*τ*.

*t*=0, in a continuum state described by the wavefunction

*,*

**r***t*) , the transition amplitude of finding it in a bound state

**Ψ**

_{0}(

*,*

**r***t*), at the time

*t*, is given by

*the asymptotic electron momentum and*

**q***ω*

_{x}is the energy of the emitted x-ray photon during the recombination process,

*the polarization vector,*ε ^

_{X}*=*

**k**_{x}*(*◻

_{x}/

*c*)

*n̂*

_{X}, the momentum,

*n̂*

_{X}the propagation direction, and V the quantization volume of the high-frequency radiation. In our calculations

**Ψ**

_{0}(

*,*

**r***t*) is approximated by the unperturbed ground state of the hydrogenic ion with (

*Z-1*) charge and energy

*I*

_{0}= -0.5

*Z*

^{2}:

*,*

**r***t*), describes the incoming electron whose momentum, when the laser is off, is

*. It will be assumed to be approximated by the Coulomb-Volkov ansatz*

**q***,*

**r***t*) is the Volkov wavefunction

*(t)=*

**k**_{L}*(t)/*

**A***c*the momentum imparted to the electron by the radiation field and

*)*

**r***v*=

*Z*/

*q*, Γ(

*x*) and

_{1}

*F*

_{1}are, respectively, the gamma and the confluent hypergeometric functions. The ansatz Eq. (5) and slightly modified version of it [20–23

20. M. Jain and N. Tzoar, “Compton scattering in the presence of coherent electromagnetic radiation,” Phys. Rev. A **18**, 538–545 (1978). [CrossRef]

21. C. Leone, S. Bivona, R. Burlon, and G. Ferrante, “Two-frequency multiphoton ionization of hydrogen atoms,” Phys. Rev. A **38**, 5642–5651 (1988). [CrossRef] [PubMed]

24. A. Maquet, R. Taieb, and V. Veniard, “Two-color infrared-UV atomic photoionization,” in *Fundamental of laser-matter interaction*,
K.N. Drabovich and N. I. Koroteev, eds., Proc. SPIE **2796**, 31–38 (1995). [CrossRef]

24. A. Maquet, R. Taieb, and V. Veniard, “Two-color infrared-UV atomic photoionization,” in *Fundamental of laser-matter interaction*,
K.N. Drabovich and N. I. Koroteev, eds., Proc. SPIE **2796**, 31–38 (1995). [CrossRef]

24. A. Maquet, R. Taieb, and V. Veniard, “Two-color infrared-UV atomic photoionization,” in *Fundamental of laser-matter interaction*,
K.N. Drabovich and N. I. Koroteev, eds., Proc. SPIE **2796**, 31–38 (1995). [CrossRef]

*,*

**r***t*) and

**Ψ**

_{0}(

*,*

**r***t*) in Eq. (2) and proceeding in the usual way, the double differential probability of electron-ion recombination accompanied by the x-ray photon emission in the solid angle

*dω*

_{x}and having energy within the interval (

*ω*

_{x},

*ω*

_{x}+

*dω*

_{x}), for

*t*∈ [0, τ] , is equal to

*q*→-

*q*, while it is invariant when the contemporaneous transformations

*q*→-

*q*and

*φ*→-

*φ*.

## 3. A semiclassical model

25. N. M. Kroll and K. M. Watson, “Charged-particle scattering in the presence of a strong electromagnetic wave,” Phys. Rev. A **8**, 804–809 (1973). [CrossRef]

*, when the laser pulse off, by solving the Newton’s equation its kinetic momentum at some time*

**q***t*∈ [0,

*τ*] is given by

*P*the double differential probability that the recombination takes places in the whole pulse duration

*τ*, the double differential recombination probability in the time interval

*dt*reads

*P*/

*τ*equal to the LARR rate in the low-frequency approximation by using a quantum mechanical treatment (see Eq. (12) of Ref [17

17. S. Bivona, R. Burlon, G. Ferrante, and C. Leone, “Radiative recombination in a strong laser field. Low frequency approximation,” J. Opt. Soc. Am. B **22**, 2076–2082 (2005). [CrossRef]

*t*, we find the counterpart of Eq. (6) obtained by full quantum mechanical treatment

*l*numbers the real solution of equation

*ω*

_{x}(

*t*) =0 in the interval [0,

*τ*]. We remark that, as discussed extensively in Ref. [17

**22**, 2076–2082 (2005). [CrossRef]

*(*

**E***t*

_{l})=0, so that the semiclassical approaches cannot be used when the photon energy

*ω*

_{x}≈

*ω*

_{x}(

*t*

_{0}), Eq. (10), where

*t*

_{0}is a time instant in which the electric laser field instantaneous is close to zero. However, the above discussion allows predicting the position of the maxima in the double differential probability.

## 4. Calculations

*=*

**q***q*

*;*

**ẑ***q*>0), the propagation direction of the emitted radiation will be chosen in the x-direction, the X-ray and the laser polarization vectors will be taken along the z-axis (

*=*ε ^

_{X}*=*ε ^

_{L}*). Further we will choose a Ti: sapphire laser pulse of intensity*z ^

*I*

_{L}=6×10

^{14}W cm

^{-2}(

*ω*=1.5498 eV) and the hydrogen as model atom (

*Z*=1).

**DDP**) at the end of the laser pulse as a function of the emitted photon energy,

*ω*

_{x}, for three different values of the carrier envelope relative phase (

*φ*= -90°,0°, 90°), and the energy of the incoming electron

*ε*

_{q}=90 eV. The red line corresponds to DP (

*τ*), Eq. (6), calculated by the quantum theory. The blue line corresponds to CDP (

*τ*), Eq. (13). The left and the right panels show, respectively, the probabilities for

*n*

_{c}=2 and

*n*

_{c}=3.

**22**, 2076–2082 (2005). [CrossRef]

*ω*

_{X}(

*t*), Eq. (10), corresponding to the time instant where the electric field is zero, that is when the instantaneous kinetic momentum

*q*(

*t*), Eq. (9), exhibit a minimum or a maximum value. For instance the two maxima at the edges of the DDP correspond, respectively, to the absolute maximum and minimum values of

*q*(

*t*). We note that the DDP spectra are practically confined in the range of

*ω*

_{x}where the emission is classically allowed, while the DDP is strongly reduced elsewhere. The shape of the reported spectra is in agreement with one reported in Ref [18

18. S. X. Hu and L. A. Collins, “Phase control of the inverse above-threshold-ionization processes with few-cycle pulses,” Phys. Rev A **70**, 013407 (2004). [CrossRef]

*q*(

*t*) depend on

*φ*, we may conclude that the DDP spectra may be controlled by varying

*φ*.

*n*

_{c}their shape depends on the value of

*φ*. However this dependence is reduced when

*n*

_{c}increases, as confirmed also by calculations here not reported.

*n*

_{c}=2 and

*ϕ*= -90

*°*.

*ω*

_{X}=437.5 eV, the blue curve to

*ω*

_{X}=10.4 eV and the green curve to

*ω*

_{X}=62.4 eV. In the insert we report the time dependence of the emitted photon energy

*ω*

_{X}(

*t*), Eq. (10), evaluated classically. By inspection of the curves shown in Fig. 2, we may conclude that a given X-ray frequency is emitted within particular short time intervals that are a small fraction of the laser period. These short intervals are located around the instant in which the electric field is zero, as predicted by the semiclassical model.

*n*

_{c}=2 and three different values of

*φ*(blue curve

*φ*=-90°, red curve

*φ*=0°, green curve

*φ*=90°. The lower panel show the corresponding data for

*n*

_{c}=6

*n*

_{c}=2 considerable modification occurs in the single differential probability by varying

*φ*, while the dependence on

*φ*is strongly reduced for

*n*

_{c}=6. Calculations here not reported show that the dependence on

*φ*may be neglected for very large values of

*n*

_{c}, as expected on the basis of the results in the presence of a monochromatic laser field.

## 5. Concluding remarks

*n*

_{c}and the carrier envelope relative phase φ only for very small values of

*n*

_{c}. The mean features of the reported curves may be reproduced by the semiclassical model, in which the recombination occurs instantaneously at a given value of the kinetic momentum, obtained by solving the Newton’s equation. According to this pictures the double differential probability exhibits two maxima at the emission spectrum edges, corresponding to the absolute minimum and maximum values of the kinetic momentum

*q*(

*t*). Therefore, as the time evolution of

*q*(

*t*) depends on

*φ*, the emitted photon energy spectrum may be controlled by varying the carrier- envelope relative phase.

## Acknowledgments

## References and links

1. | Y. Hahn, “Electron –ion recombination process – an overview,” Rep. Prog. Phys. |

2. | R. Neumann, H. Poth, A. Winnacker, and A. Wolf, “Laser-enhanced electron-ion capture and antihydrogen formation,” Z. Physik. A |

3. | U. Schramm, J. Berger, M. Grieser, D. Habs, E. Jaeschke, G. Kilgus, D. Schwalm, A. Wolf, R. Neumann, and R. Schuch, “Observation of laser-induced recombination in merged electron and proton-beams,” Phys. Rev. Lett. |

4. | F. B. Yousif, P. Vanderdonk, Z. Kucherovsky, J. Reis, E. Brannen, J. B. A. Mitchell, and T. J. Morgan, “Experimental-observation of laser-stimulated radiative recombination,” Phys. Rev. Lett. |

5. | S. Borneis, F. Bosch, T. Engel, M. Jung, I. Klaft, O. Klepper, T. Kuhl, D. Marx, R. Moshammer, R. Neumann, S. Schroder, P. Seelig, and L. Volker, “Laser-stimulated two-step recombination of highly charged ions and electrons in a storage ring,” Phys. Rev. Lett. |

6. | U. Schramm, T. Schussler, D. Habs, D. Schwalm, and A. Wolf, ”Laser-induced recombination studies with the adiabatically expanded electron beam of the Heidelberg TSR,” Hyperfine Interact. |

7. | S. Pastuszka, U. Schramm, M. Grieser, C. Broude, R. Grimm, D. Habs, J. Kenntner, H. J. Miesner, T. Schussler, D. Schwalm, and A. Wolf, “Electron cooling and recombination experiments with an adiabatically expanded electron beam,” Nucl. Instrum. Methods Phys. Res. A |

8. | S. Asp, S. R. Schuch, D. R. DeWitt, C. Biedermann, H. Gao, W. Zong, G. Andler, and E. Ustiniano, ”Laser-induced recombination of D |

9. | M. L. Rogelstad, F. B. Yousif, T. J. Morgan, and J. B. A. Mitchell, “Stimulated radiative recombination of H |

10. | M. Yu Kuchiev and V. N. Ostrovsky, “Multiphoton radiative recombination of electron assisted by a laser field,” Phys. Rev. A |

11. | D. B. Milosevic and F. Ehlotzky, “Rescattering effects in soft-x-ray generation by laser-assisted electron-ion recombination,” Phys. Rev. A |

12. | C. Leone, S. Bivona, R. Burlon, and G. Ferrante, “Strong-field and plasma aspects of multiphoton radiative recombination,” Phys. Rev. A |

13. | S. Bivona, R. Burlon, G. Ferrante, and C. Leone, “Strong field effects of multiphoton radiative recombination,” Laser Phys. |

14. | S. Bivona, R. Burlon, G. Ferrante, and C. Leone, “Influence of a plasma medium on laser assisted radiative recombination,” Laser Phys. Lett. |

15. | S. Bivona, R. Burlon, G. Ferrante, and C. Leone, “Control of multiphoton radiative recombination through the action of two-frequency fields,” Laser Phys. Lett. |

16. | S. Bivona, R. Burlon, G. Ferrante, and C. Leone, “Control of radiative recombination by a strong laser field,” Appl. Phys. B |

17. | S. Bivona, R. Burlon, G. Ferrante, and C. Leone, “Radiative recombination in a strong laser field. Low frequency approximation,” J. Opt. Soc. Am. B |

18. | S. X. Hu and L. A. Collins, “Phase control of the inverse above-threshold-ionization processes with few-cycle pulses,” Phys. Rev A |

19. | J. Z. Kaminski and F. Ehlotzky, “Time-frequency analysis of x-ray generation by recombination in short laser pulses,” Phys. Rev. A |

20. | M. Jain and N. Tzoar, “Compton scattering in the presence of coherent electromagnetic radiation,” Phys. Rev. A |

21. | C. Leone, S. Bivona, R. Burlon, and G. Ferrante, “Two-frequency multiphoton ionization of hydrogen atoms,” Phys. Rev. A |

22. | F. Ehlotzky, A. Jaron, and J. Z. Kaminsky, “Electron-atom collisions in a laser field,” Phys. Rep. Rev. Section of Phys. Lett. |

23. | D. B. Milosevic and F. Ehlotzky, “Scattering and reaction processes in powerful laser fields,” Adv. At. Mol., Opt. Phys. |

24. | A. Maquet, R. Taieb, and V. Veniard, “Two-color infrared-UV atomic photoionization,” in |

25. | N. M. Kroll and K. M. Watson, “Charged-particle scattering in the presence of a strong electromagnetic wave,” Phys. Rev. A |

**OCIS Codes**

(020.4180) Atomic and molecular physics : Multiphoton processes

(270.6620) Quantum optics : Strong-field processes

(320.2250) Ultrafast optics : Femtosecond phenomena

**ToC Category:**

Atomic and Molecular Physics

**History**

Original Manuscript: January 20, 2006

Revised Manuscript: April 18, 2006

Manuscript Accepted: April 18, 2006

Published: May 1, 2006

**Citation**

Saverio Bivona, Riccardo Burlon, Gaetano Ferrante, and Claudio Leone, "Radiative recombination in the presence of a few cycle laser pulse," Opt. Express **14**, 3715-3723 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-9-3715

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### References

- Y. Hahn, "Electron -ion recombination process - an overview," Rep. Prog. Phys. 60, 691-759 (1997). [CrossRef]
- R. Neumann, H. Poth, A. Winnacker, and A. Wolf, "Laser-enhanced electron-ion capture and antihydrogen formation," Z. Physik. A 313, 253-262 (1983). [CrossRef]
- U. Schramm, J. Berger, M. Grieser, D. Habs, E. Jaeschke, G. Kilgus, D. Schwalm, A. Wolf, R. Neumann, and R. Schuch, "Observation of laser-induced recombination in merged electron and proton-beams," Phys. Rev. Lett. 67, 22-25 (1991). [CrossRef] [PubMed]
- F. B. Yousif, P. Vanderdonk, Z. Kucherovsky, J. Reis, E. Brannen, J. B. A. Mitchell, and T. J. Morgan, "Experimental-observation of laser-stimulated radiative recombination," Phys. Rev. Lett. 67, 26-29 (1991). [CrossRef] [PubMed]
- S. Borneis, F. Bosch, T. Engel, M. Jung, I. Klaft, O. Klepper, T. Kuhl, D. Marx, R. Moshammer, R. Neumann, S. Schroder, P. Seelig, and L. Volker, "Laser-stimulated two-step recombination of highly charged ions and electrons in a storage ring," Phys. Rev. Lett. 72, 207-209 (1994). [CrossRef] [PubMed]
- U. Schramm, T. Schussler, D. Habs, D. Schwalm, and A. Wolf, "Laser-induced recombination studies with the adiabatically expanded electron beam of the Heidelberg TSR," Hyperfine Interact. 99,309-316 (1996). [CrossRef]
- S. Pastuszka, U. Schramm, M. Grieser, C. Broude, R. Grimm, D. Habs, J. Kenntner, H. J. Miesner, T. Schussler, D. Schwalm and A. Wolf, "Electron cooling and recombination experiments with an adiabatically expanded electron beam, " Nucl. Instrum. Methods Phys. Res. A 369, 11-22 (1996). [CrossRef]
- S. Asp, S. R. Schuch, D. R. DeWitt, C. Biedermann, H. Gao, W. Zong, G. Andler, and E. Ustiniano, "Laser-induced recombination of D+," Nucl. Instrum. Methods: Phys. Res. B 117, 31-37 (1996). [CrossRef]
- M. L. Rogelstad, F. B. Yousif, T. J. Morgan, and J. B. A. Mitchell, "Stimulated radiative recombination of H+ and He+," J. Phys. B 30, 3913-3931 (1997). [CrossRef]
- M. Yu Kuchiev and V. N. Ostrovsky, "Multiphoton radiative recombination of electron assisted by a laser field," Phys. Rev. A 61, 033414 (2000). [CrossRef]
- D. B. Milosevic and F. Ehlotzky, "Rescattering effects in soft-x-ray generation by laser-assisted electron-ion recombination," Phys. Rev. A 65, 042504 (2002). [CrossRef]
- C. Leone, S. Bivona, R. Burlon, and G. Ferrante, "Strong-field and plasma aspects of multiphoton radiative recombination," Phys. Rev. A 66, 051403 (2002). [CrossRef]
- S. Bivona, R. Burlon, G. Ferrante, and C. Leone, "Strong field effects of multiphoton radiative recombination," Laser Phys. 13, 1077-1082 (2003).
- S. Bivona, R. Burlon, G. Ferrante, and C. Leone, "Influence of a plasma medium on laser assisted radiative recombination," Laser Phys. Lett. 1, 86-92 (2004). [CrossRef]
- S. Bivona, R. Burlon, G. Ferrante, and C. Leone, "Control of multiphoton radiative recombination through the action of two-frequency fields," Laser Phys. Lett. 1, 118-123 (2004). [CrossRef]
- S. Bivona, R. Burlon, G. Ferrante, and C. Leone, "Control of radiative recombination by a strong laser field," Appl. Phys. B 78, 809-812 (2004). [CrossRef]
- S. Bivona, R. Burlon, G. Ferrante, and C. Leone, "Radiative recombination in a strong laser field. Low frequency approximation," J. Opt. Soc. Am. B 22, 2076-2082 (2005). [CrossRef]
- S. X. Hu and L. A. Collins, "Phase control of the inverse above-threshold-ionization processes with few-cycle pulses," Phys. Rev A 70, 013407 (2004). [CrossRef]
- J. Z. Kaminski and F. Ehlotzky, "Time-frequency analysis of x-ray generation by recombination in short laser pulses," Phys. Rev. A 71, 043402 (2005). [CrossRef]
- M. Jain and N. Tzoar, "Compton scattering in the presence of coherent electromagnetic radiation," Phys. Rev. A 18, 538-545 (1978). [CrossRef]
- C. Leone, S. Bivona, R. Burlon, and G. Ferrante, "Two-frequency multiphoton ionization of hydrogen atoms," Phys. Rev. A 38,5642-5651 (1988). [CrossRef] [PubMed]
- F. Ehlotzky, A. Jaron, and J. Z. Kaminsky, "Electron-atom collisions in a laser field," Phys. Rep. Rev. Section of Phys. Lett. 297, 64-153 (1998)
- D. B. Milosevic and F. Ehlotzky, "Scattering and reaction processes in powerful laser fields," Adv. At. Mol., Opt. Phys. 49, 373-532 (2003). [CrossRef]
- A. Maquet, R. Taieb, and V. Veniard, "Two-color infrared-UV atomic photoionization," in Fundamental of laser-matter interaction, K.N. Drabovich and N. I. Koroteev, eds., Proc. SPIE 2796, 31-38 (1995). [CrossRef]
- N. M. Kroll and K. M. Watson, "Charged-particle scattering in the presence of a strong electromagnetic wave," Phys. Rev. A 8, 804-809 (1973). [CrossRef]

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