Margarita Khokhlova: Analytical theory of resonant high-order harmonic generation

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Analytical theory of resonant high­order harmonic generation

M. A. Khokhlova1,2, V. V. Strelkov1

1. Theoretical Department, General Physics Institute of Russian Academy of Sciences,119991, 38 Vaviliva str., Moscow, Russia

2. Department of Physics, M. V. Lomonosov Moscow State University, 119991, Leninskie gory str., Moscow, Russia

Properties of resonant high­order harmonics generated in intense laser field are actively studied both 

experimentally [1­3] and theoretically [4­7]. Very efficient generation of the harmonic resonant with the 

transition from the bound to the autoionizing state was demonstrated in the experiments using plasma plumes 

[1,2] and Xe jet [3].

We suggested analytical quantum­mechanical theory describing the effect of quasi­stationary state on HHG 

[7]. We start with the time­dependent Schrodinger equation for an atom or ion in an external laser field. The 

wave function is presented as a sum of the ground state, unperturbed continuum and the quasi­stationary state. 

To solve the Schrödinger equation we derived the perturbation method in which the solution obtained in the 

absence of the quasi­stationary state by Lewenstein et al. in Ref. [5] is taken as an unperturbed solution. 

Assuming that (i) the ionization rate is low, (ii) the quasi­stationary state population is low, (iii) the quasi­

stationary state is not affected with the laser field, and (iv) the quasi­stationary state width * is much less than 

the quasi­stationary state energy, we find the following equation for the spectral complex amplitude of the 

microscopic response at the frequency Z

 (atomic units are used):

 P ( Z ) P nr ( Z ) F (Z )

a

 * / 2 o

F (Z )

 « 1 Q »

¬

 'Z i* / 2 1⁄4

where is the 

 P

 nr

 (Z

 )

 spectrum of the non­resonant contribution, * is the resonance width, Q is a complex 

parameter 

 defined by the properties of the generating atom or ion, but not 

depending on the laser field. So the resonant harmonic line is presented as a product of the F (Z ) Fano­like 

[8] factor and the harmonic line which would be emitted in the absence of the AIS. 

Our theory allows calculating not only the resonant harmonic intensity, but also its phase. We show that there 

is a rapid variation of the phase in the vicinity of the resonance. Our calculations reasonably agree with recent 

harmonic phase measurements [9]. 

The other direction of our research is the study of the phase properties of the cut­off harmonics. We study the 

harmonic phase dependence on the laser intensity both analytically and numerically. Moreover, we investigate 

the dephasing between adjacent harmonics to study the emission time of the attosecond pulses. Thus, the 

optimum conditions for the shortest attosecond pulse generation using the cut­off harmonics are suggested.

References

>1@ R. A. Ganeev, "High­order harmonic generation in a laser plasma: a review of recent achievements", J. Phys. B: At. Mol. Opt. Phys. 40, 

R213 (2007).

[2] R. A. Ganeev, V. V. Strelkov, C. Hutchison, A. Zaïr, D. Kilbane, M. A. Khokhlova, and J. P. Marangos, "Experimental and theoretical 

studies of two­color­pump resonance­induced enhancement of odd and even harmonics from a tin plasma", Phys. Rev. A 85, 023832 (2012).

[3] A. D. Shiner, B. E. Schmidt, C. Trallero­Herrero, H. J. Wörner, S. Patchkovskii, P. B. Corkum, J­C. Kieffer, F. Légaré & D. M. 

Villeneuve, "Probing collective multi­electron dynamics in xenon with high­harmonic spectroscopy", Nature Physics 7, 464 (2011).

[4] V. Strelkov, "Role of Autoionizing State in Resonant High­Order Harmonic Generation and Attosecond Pulse Production", Phys. Rev. 

Lett. 104, 123901 (2010).

[5] M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, Anne L’Huillier, and P. B. Corkum, "Theory of high­harmonic generation by low­frequency 

laser fields", Phys. Rev. A 49, 2117­2132 (1994).

[6] Jan Rothhardt, Steffen Hädrich, Stefan Demmler, Manuel Krebs, Stephan Fritzsche, Jens Limpert, and Andreas Tünnermann, "Enhancing 

the Macroscopic Yield of Narrow­Band High­Order Harmonic Generation by Fano Resonances", Phys. Rev. Lett. 112, 233002 (2014).

[7] V. V. Strelkov, M. A. Khokhlova, and N. Yu Shubin, "High­order harmonic generation and Fano resonances", Phys. Rev. A 89, 053833 

(2014).

[8] U. Fano, "Effects of Configuration Interaction on Intensities and Phase Shifts", Phys. Rev. 124, 1866­1878 (1961).

[9] S Haessler, V Strelkov, L B Elouga Bom, M Khokhlova, O Gobert, J­F Hergott, F Lepetit, M Perdrix, T Ozaki and P Salières, "Phase 

distortions of attosecond pulses produced by resonance­enhanced high harmonic generation", New J. Phys. 15, 013051 (2013).

[10] J. Seres, E. Seres, D. Hochhaus, B. Ecker, D. Zimmer, V. Bagnoud, T. Kuehl and C. Spielmann, "Laser­driven amplification of soft X­

rays by parametric stimulated emission in neutral gases", Nature Physics 6, 455–461 (2010)