ISSN 10637761, Journal of Experimental and Theoretical Physics, 2009, Vol. 108, No. 6, pp. 947–962. © Pleiades Publishing, Inc., 2009. Original Russian Text © S.B. Popruzhenko, V.D. Mur, V.S. Popov, D. Bauer, 2009, published in Zhurnal Éksperimental’noі i Teoreticheskoі Fiziki, 2009, Vol. 135, No. 6, pp. 1092–1108. ATOMS, MOLECULES, OPTICS Multiphoton Ionization of Atoms and Ions by HighIntensity XRay Lasers S. B. Popruzhenkoa, b, V. D. Mura, V. S. Popovc, and D. Bauerb a Moscow Institute of Engineering Physics, Kashirskoe sh. 31, Moscow, 115409 Russia b MaxPlank Institut für Kernphysik, Postfach 103980, 69029, Heidelberg, Germany c Institute for Theoretical and Experimental Physics, ul. Bol’shaya Cheremushkinskaya 25, Moscow, 117218 Russia email: [email protected] Received December 31, 2008 Abstract—Coulomb corrections to the action function and rate of multiphoton ionization of atoms and ions in a strong linearly polarized electromagnetic field are calculated for high values of the Keldysh adiabaticity parameter. The Coulomb corrections significantly increase the ionization rate for atoms (by several orders of magnitude). An interpolation formula proposed for ionization rate is valid for arbitrary values of the adiaba ticity parameter. The high accuracy of the formula is confirmed by comparison with the results of numerical calculations. The general case of elliptic polarization of laser radiation is also considered. PACS numbers: 32.80.Rm, 32.80.Fb, 03.65.Sq DOI: 10.1134/S1063776109060053 1. INTRODUCTION been mainly associated with experiments using high power optical and IR lasers with a wavelength on the Considerable advances have been made in recent 1 years in the physics of the interaction of highintensity order of a micrometer or longer. Accordingly, theo laser fields with matter in connection with the devel retical investigations were also associated mainly with opment and commissioning of powerful sources of low frequencies of the laser field. It should be recalled coherent radiation in the UV and Xray wavelength that electromagnetic field in the theory of multipho ranges. Such sources are based on freeelectron lasers. ton ionization is assumed to be a lowfrequency field if A unique device of this type (FLASH) operates at parameter γ, introduced by Keldysh [15], is much present at the DESY laboratory (Hamburg, Germany) smaller than unity. Ionization in this case is known as [1, 2]. In 2002, the FLASH device generated electro tunneltype ionization. The opposite limit (γ ӷ 1) is magnetic radiation pulses at a wavelength of about usually referred to as multiphoton ionization. When 100 nm (photon energy បω ≈ 13 eV) with a duration on atoms and positive ions are ionized by the field of a 13 2 the order of 100 fs and an intensity of up to 10 W/cm highintensity infrared optical laser, the Keldysh [3]. Since 2007, experiments with radiation pulses of a parameter is usually on the order of unity or smaller. wavelength of 13 nm (បω ≈ 93 eV), a duration of about 16 2 The development of highpower Xray lasers extends 10 fs, and an intensity of up to 10 W/cm have been the experimental potential to the range of high fre carried out [4]. In the nearest future, transition to a wavelength of 6 nm is planned and use will be made of quencies and small wavelengths. Typical values of the Keldysh parameter in experiments [4, 7] amounted to the third and fifth harmonics of the fundamental fre γ ≈ quency (the latter harmonic corresponds to a wave 30–100, which corresponds to the multiphoton length of 1.2 nm) [2]. Numerous experiments on the ionization mode. Until recently, much less attention interaction of highintensity coherent rf radiation with has been given to the development of the theory in this atoms, molecules, multiply charged ions, nanostruc range of parameters. In this study, we analyze mul tures, and solids were performed at the DESY labora tiphoton ionization of atoms and positive ions in a tory in 2002–2007 using freeelectron lasers (see [4–9] strong highfrequency (rf) field (i.e., for γ ӷ 1) and and the literature cited therein). derive analytic expressions for the total ionization probability per unit time. The ionization rate calcu One of the fundamental nonlinear processes lated using these formulas is compared to the result of induced by a highintensity laser field is multiphoton ionization of atoms, which was observed for the first exact numerical calculations. time in 1965 [10] (on the modern state of the art, see 1 Among powerful lasers, the most widely used are titanium–sap reviews [11–14]). Until recently, the physics of inter phire, neodymium, and CO2 lasers with wavelengths of 0.79– action of highintensity laser fields with matter has 0.80, 1.06, and 10.6 µm, respectively. 947 948 POPRUZHENKO et al. In analyzing multiphoton ionization of atoms by It should be emphasized that ionization potential I highintensity laser radiation, the Keldysh theory [15], and asymptotic coefficient Cκ in expression (2) corre also known as the strong field approximation [16], is spond to the atomic state for which the ionization widely used. In this approximation, the electromagnetic probability is calculated rather than to the state in a field of a wave is taken into account exactly, and the Cou shortrange potential. Finally, the Keldysh parameter lomb interaction of an emitted electron with the atomic is defined as core is disregarded, which makes it possible to derive 2mIω 1 convenient analytic formulas for the ionization probabil γ == , (4) ity and momentum spectra of photoelectrons (see [17, eᏱ 2K0F 18] and the literature in [12]). For ionization of negative បω ions, the strong field approximation leads to good quan where K0 = I/ is the multiquantum parameter of the titative agreement with the results of exact numerical cal process. In the Keldysh theory, it was assumed that culations and experimental data (see, for example, [19] K 1, F 1, (5) and the literature therein). At the same time, in the case 0 of neutral atoms and positive ions, the Coulomb interac here, parameter γ can assume arbitrary values. Hence tion suppresses the potential barrier through which an forth, we will use atomic units: ប = e = m = 1. electron tunnels, which considerably increases the ion Expression (1) for the Coulomb correction to the ization probability. For example, in a constant electric ionization rate is applicable not only in the case of a field of strength Ᏹ, ionization rate w for the s state of an constant field, but also in the tunnel mode (γ < 1) for a atom differs from analogous quantity wsr for an energy plane electromagnetic wave with an arbitrary polariza level in the shortrange well [12, 20, 21] (with the same tion. Usually, this correction increases the tunnel ion ionization potential I) by a factor of [22] ization probability by several orders of magnitude (this ()Ᏹ effect was reliably established in experiments; see, for ()Ᏹ, ω ≡ ()Ᏹ w QC = 0 Q0 = example, [24]). At present, the expressions for tunnel wsr()Ᏹ (1) ionization rate taking into account the Coulomb cor ⎛⎞2ប2κ3 2n* ⎛⎞2 2n* rection (and also known as the Ammosov–Delone– 1, = ⎝⎠Ᏹ = ⎝⎠ Krainov formulas [25]; see [26]) are widely used for me F calibrating the laser pulse intensity. where In this study, we obtain the Coulomb correction Ᏹ ω κ2ប Ᏹ Ᏹ QC( , ) in the opposite limit of high frequency, ()Ᏹ 2 ⎛⎞2 ch γ ӷ 1, in the simplest case of ionization of the s state wsr = Cκ exp⎝⎠– . (2) m Ᏹch 3Ᏹ of an atom or a positive ion by a linearly polarized field (ρ = 0). In this case, the formulas for tunnel ប Ᏹ ប2κ3 Here, k = 2mI/ and ch = /me are the character ionization probability are completely inapplicable istic momentum and the electric field of the atom (for the even for obtaining qualitative estimates. For exam Ᏹ ≡ Ᏹ 2 5 ប4 16 2 ground state of the hydrogen atom, ch a = m e / = ple, for an intensity on the order of 10 V/cm used 9 2 2 –1/2 5.14 × 10 V/cm), n* = ᐆe m/ប k = ᐆ(I/IH) is the in experiment [4], successive tunnel ionization effective principal quantum number of the level (I = leads to the formation of ions up to Xe8+–Xe9+ H 21+ me4/2ប2 = 13.6 eV is the ionization potential of the instead of observed Xe ions, and the tunnel ion ization probability for the Xe10+ ion for the given hydrogen atom), m and –e are the electron mass and –24 charge, and ᐆ is the charge of the atomic core (ᐆ = 1 for intensity amounts to a value on the order of 10 for a pulse with a duration of about 10 fs. In our calcu neutral atoms and ᐆ = 0 for negative ions H–, Na–, etc.). Ᏹ Ᏹ lations, we use the imaginary time method (ITM), Ratio F = / ch is often referred to as the reduced field, in which subbarrier trajectories satisfying the classi ᏶ ⎛⎞I –3/2 cal equation of motion, but in imaginary time, are F = 0.169 , (3) 2⎝⎠ considered for describing quantum tunneling 1 + ρ IH through a barrier [27, 28]. In the derivation of the where ρ is the ellipticity of radiation (–1 ≤ ρ ≤ 1; ρ = 0 Coulomb correction, we will use the Kapitza and ±1 corresponds to the linear and circular polariza method [29, 30] for describing the averaged motion tions of the wave, respectively), and is the intensity of a particle in a rapidly oscillating field. The appli measured in units of 1015 W/cm2. Finally, C is the cation of the ITM for analyzing Coulomb effects in κ the spectral–angular distributions of photoelec dimensionless asymptotic coefficient of the atomic 2 trons in a highintensity laser field is considered in wavefunction at distances r ӷ ប/κ from the nucleus.