Time-energy analysis of above-threshold ionization

L. Guo; S. S. Han; J. Chen

doi:10.1364/OE.18.001240

1. Introduction

When an atom is exposed to an intense laser field, the bound electron may be released through a so called ”Above-threshold ionization (ATI)” process, which produces highly structured electron spectra, consisting of a series of regularly spaced narrow energy peaks, each separated by one photon energy [1]. The low energy part of the photoelectron spectrum can be well described by the so-called Keldysh-Fasial-Reiss (KFR) theory [2, 3, 4]. On the other hand, after normally decreasing in magnitude in the low energy regime, the ATI spectra leave off in a plateau extending up to 10U_p (U_p denotes the ponderomotive energy), which can be successfully explained by a simple classical model, the ”simple-man theory” [5, 6, 7]. Recently, detailed measurements of ATI in argon and xenon have revealed that the plateau part of the spectrum is also dominated by narrow resonance-like structure appearing at specific intensities [8, 9]. Many theoretical works have been undertaken in an attempt to explain the feature of ATI spectra observed in these experiments [10, 11, 12, 13, 14, 15, 16]. More recently, the ATI spectra of noble gas atoms in ultrashort intense laser pulses at long wave-lengths show surprisingly pronounced hump structure in the very low-energy part of the photoelectron spectra and these humps are in contrast to the simple-man theory[17, 18]. So the new discoveries of ATI spectra always make us investigate the underlying mechanism.

The interference effect in the ATI spectrum was firstly studied by Reed and Burnett [19]. With a simple model, they attribute the subpeaks in each ATI peak, which is found by numerical solution of the one-dimensional time-dependent Schrödinger equation, to the interference between the ionization amplitude generated on the rising and falling edges of the laser pulse. Recently, this structure is reinvestigated by Wickenhauser et al. and is ascribed to contributions of different laser cycles to the final electron spectra [20]. In addition, interference between different trajectories released at different time is studied and is believed to be the origin of particular interference structures in photoelectron spectrum [21]. However, none of these works provide an explicit picture of the interference effect in the ATI process. To accomplish this and to understand the basic features and underlying mechanism further, it is essential to comprehend in full detail the time-energy characteristics of the electrons released by the intense time-varying electric field from atoms. To our knowledge, this kind of analysis of the time-energy characteristics have been never presented. As a comparison, in contrast, the time-frequency (energy) distribution of high harmonic generation (HHG) process has been investigated intensively in the past decade [22, 23, 24, 25, 26]. These studies significantly advance our comprehension of the HHG mechanism and play important roles in coherent control of the electron dynamics in the HHG process [27, 28].

The Wigner distribution, which has been proved powerful for investigation of quantum states in phase space [29, 30], has also been applied to time-frequency analysis of high harmonic generation process in atom-laser interaction [26]. In this paper, we propose a Wigner-distribution-like function to analyse the time-energy distributions of photoelectron emitted from an atom in laser fields with different angular frequencies. This method enable us to give a convincing and complete explanation for the interference effects in ATI peaks due to the finite pulse duration.

2. Theory

The first term of the S-matrix element[4] is

S_{fi} = - i \int_{- \infty}^{\infty} dt 〈 ψ_{Af} (p, t) ∣ V_{A} (t) ∣ φ_{0} 〉 e^{iI p^{t} .}

Here, V_A is the interaction potential between the applied laser field and the photoelectron ∣φ_i(t)⟩ = ∣φ₀⟩e^iIp^t is the atomic ground state where I_p is the ionization potential and ∣Φ_{A_f}(p,t)⟩ is the Volkov wavefunction with the final electron momentum p, which is in velocity gauge

∣ ψ_{A_{f}} (p, t) 〉 = \frac{1}{\sqrt{v}} \exp [i p·r - i \frac{p^{2}}{2} t - i \int_{- \infty}^{t} V_{A} (p, τ) dτ],

where v is normalization volume.

Introducing Eq. (2) into Eq. (1), we obtain

\begin{array}{l} S_{fi} & = \frac{- i}{\sqrt{v}} \int_{- \infty}^{\infty} dt ϕ_{i} (p) V_{A} (p, t) \\ \times \exp [i \int_{- \infty}^{t} V_{A} (p, τ) dτ + {iI}_{p} t + i \frac{p^{2}}{2} t] \\ = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} dtS' e^{i \frac{p^{2}}{2} t}, \end{array}

where S’ is given by

S' = \frac{- i \sqrt{2 π}}{\sqrt{v}} ϕ_{i} (p) V_{A} (p, t) \exp [i \int_{- \infty}^{t} V_{A} (p, τ) dτ + {iI}_{p} t]

It is noteworthy that the expression of Fourier Transform is

F (Ω) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} F' (t) e^{iΩt} dt .

By comparing Eq. (3) with Eq. (5), it can be found that S_fi is a Fourier-like transform of S’. Therefore we define a Wigner-distribution-like (WDL) function in analog with Ref. [26] to obtain the time-energy distribution of the photoionization process described by Eq. (1). The WDL function is defined as

f (t, \frac{p^{2}}{2}) = \frac{1}{π} \int_{- \infty}^{\infty} S' * (t + t') S' (t - t') e^{- 2 i \frac{p^{2}}{2} t'} dt' .

The reason why Eq. (6) is called the ”Wigner-distribution-like” function is that in Eq. (3), S’ is a function of p while in Eq. (5), F’(t) is not a function of Ω.

Integrating Eq. (6) over t, we obtain

\begin{array}{l} \int_{- \infty}^{\infty} f (t, \frac{p^{2}}{2}) dt & = \int_{- \infty}^{\infty} dt \frac{1}{π} \int_{- \infty}^{\infty} S' * (t + t') S' (t - t') e^{- 2 i \frac{p^{2}}{2} t'} dt' \\ = \frac{1}{2 π} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} S' * (t_{2}) S' (t_{1}) e^{- i \frac{p^{2}}{2} (t_{2} - t_{1})} d t_{1} {dt}_{2} \\ = S_{fi} S_{fi}^{*} = {∣ S_{fi} ∣}^{2} . \end{array}

In the process of above deduction, a liner transformation is made: t = (t ₂ + t ₁)/2,t’=(t ₂-t ₁)/2. We can see $f (t, \frac{p^{2}}{2})$ still satisfies the marginal relationship that

{∣ S_{fi} ∣}^{2} = \int f (t, \frac{p^{2}}{2}) dt .

In this paper, we use a linearly polarized laser pulse. The vector potential is given by

A (t) = \frac{E_{0}}{ω} \sin^{2} [\frac{ωt}{n}] \cos (ωt),

where E ₀ is the field peak strength, ω the laser frequency, and $\frac{n}{2}$ is the number of cycles. At the beginning of the pulse, both the A and E fields are zero.

Under these conditions, we have

f (t, \frac{p^{2}}{2}) = \frac{1}{π} \int_{0}^{T_{c}} S' * (t + t') S' (t - t') e^{- 2 i \frac{p^{2}}{2} t'} dt',

here, $T_{c} = \frac{πn}{ω}$ is the duration of the pulse.

3. Results and discussions

A clear insight into the fine detail of the time-energy characteristics of the ionized photoelec-trons can be gained from Fig. 1, which was obtained for an one-dimensional H atom exposed to a pulse with frequency ω=0.182 (atomic units are used throughout the paper), peak intensity of I=1×10¹⁴W/cm² and a duration of 60 optical cycles. It can be seen that many crescent structures appear in Fig. 1(a). The structure is symmetric about the peak electric field of the incident pulse. We also present the energy spectrum (Fig. 1(b)) by integrating Eq. (10) over t, namely Eq. (8). In Fig. 1(b), there are two main peaks at about $\frac{p^{2}}{2} = 0.006$ =0.006 and 0.208 and in addition to the main peaks, a series of subpeaks occur to the right of the main peaks.

Fig. 1. (a) Time-energy distribution of photoelectrons from H atoms irradiated by a laser pulse. The pulse duration is 60 cycles. The angular frequency ω0=0.182 a.u. and peak intensity I=1 × 10¹⁴W/cm². The lines denote the time-varying final kinetic energy obtained from Eq. (12), corresponding to the number of extra photons absorbed S = 0 (solid line) and S = 1 (dashed line), respectively. (b) Energy spectrum obtained by integrating Eq. (10) over t.

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According to the physical picture ATI, the positions of the main peaks in the energy spectrum can be determined by the formula:

\begin{array}{l} \frac{p^{2}}{2} & = (N_{0} + S) ω - I_{p} - U_{p} \\ = (N_{0} + S) ω - I_{p} - \frac{E_{0}^{2}}{4 ω^{2}} . \end{array}

Here N ₀ is the minimum number of photons necessary to overcome the ionization threshold and S is the number of excess photons absorbed in the continuum. $U_{p} = \frac{E_{0}^{2}}{{4 ω}^{2}}$ is the ponderomotive energy, E ₀ the peak electrical field, and I_p is the zero-field ionization energy of the atom. In Fig.1, the positions of the peaks at about $\frac{p^{2}}{2} = 0.006$ and 0.208 correspond to S = 0 and 1, respectively. Moreover, these subpeak structures can be attributed to interference between the photoelectron amplitudes produced at the same laser intensity on the rising and falling edge of the pulse [19].

Compared the time-energy distribution in Fig. 1(a) with the energy spectrum in Fig. 1(b), it is found that the positions of crescent structures except that at about $\frac{p^{2}}{2} = 0.12$ in Fig. 1(a) correspond to those of the main peaks of the energy spectrum in Fig. 1(b). The crescent structure at about $\frac{p^{2}}{2} = 0.006$ lying between the main peaks at about $\frac{p^{2}}{2} = 0.006$ and 0.208, is artifact pertaining to the Wigner distribution [ 26, 31]. When the energy spectrum is calculated by integrating Eq. (10) over t, the negative values offset the positive values in the distribution, resulting in the absence of the peak at about $\frac{p^{2}}{2} = 0.12$ in Fig. 1(b). The appearance of the first crescent pattern of Fig. 1(a) can be understood as following: when the electric field reaches the peak, the ponderomotive energy is maximum. According to Eq. (11), the final kinetic energy of the ionized photoelectron is minimum. With decreasing intensity (from the center of the pulse to the edge) the ponderomotive energy decreases, which leads to increasing kinetic energy. So when the time from the center of the pulse increases, the energy of the electrons ionized by laser field also increases. The patterns of these smaller crescent structures which occur to the right of the main crescent structures are similar to these of the main structures in the time-energy spectrum and these smaller crescent structures in Fig. 1(a) correspond to the subpeaks of the energy spectrum in Fig. 1(b).

Fig. 2. The energy spectrum and the time-energy distribution in the regime of the first ATI peak of Fig. 1. The opened circles denote the points from which the electron is emitted to produce the interference minima (see text).

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For a comparison of the crescent structures in Fig. 1(a) with above explanation based on Eq. (11), we also calculate the positions of the time-varying finial kinetic energy. When the electric-field envelope is approximately constant over one oscillation of the laser field, we can take an envelope of the electric field approximately as $E (t) = E_{0} \sin^{2} [\frac{ωt}{n}]$ So the finial kinetic energy of the electrons ionized by the laser field can be written as a simple energy conservation equation

\begin{array}{l} \frac{p^{2}}{2} & = (N_{0} + S) ω - I_{p} - U_{p} (t) \\ = (N_{0} + S) ω - I_{p} - \frac{E_{0}^{2} \sin^{4} [\frac{ωt}{n}]}{{4 ω}^{2}} . \end{array}

The lines in Fig. 1(a) are the results based on Eq. (12) with S = 0 (solid line) and 1 (dashed line), respectively. Clearly, these results are in good agreement with the time-energy distribution as shown in Fig. 1(a).

We further calculate the time-energy positions of the interference minima as shown in Fig. 2, together with the energy spectrum for comparison. The interference minima in the energy spectrum were predicted in Ref.[19]:

E = - I_{p} + Nω - \frac{E_{0}^{2}}{{4 ω}^{2}} + {[\frac{3 π}{8}]}^{\frac{2}{3}} {[\frac{E_{0}^{2}}{{2 n}^{2}}]}^{\frac{1}{3}} {(4 q - 1)}^{2 / 3},

where E is the interference minimal energy; N is the number of the absorbed photons and the parameter q=1, 2, 3,…

Fig. 3. Time-energy distribution of photoelectrons from H atoms. The parameters are the same as those in Fig. 1, except the angular frequency ω =0.1 a.u. The solid line is the energy spectrum. The dash dotted line is calculated by Eq. (12) (S = 0). The opened circles denote the points from which the electron is emitted to produce the interference minima (see text).

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To obtain Eq. (13), the following approximation is used:

U_{p} (t) = - \frac{E_{0}^{2}}{{4 ω}^{2}} \sin^{4} [\frac{ωt}{n}] \approx \frac{E_{0}^{2}}{{4 ω}^{2}} + \frac{E_{0}^{2}}{{2 n}^{2}} {[t - \frac{T_{c}}{2}]}^{2} .

From Fig. 2, it can be seen that the opened circles represent the predicted minima of the interference patterns and the positions are well consistent with the energy spectrum, especially for the first pair of points. The other points slightly shift to higher energy than the minimums in the energy spectrum. This is due to the approximation of Eq. (14), which becomes less valid as the time from the center of the pulse increases.

Fig. 3 shows the all kinds of calculations mentioned above with the same parameters as Fig. 1 except that ω = 0.1. Comparing with that of ω = 0.182, when the frequency decreases, there are more subpeaks and more substructures in the energy spectrum (solid line) and the time-energy distribution, respectively. It can be clearly seen that the centers of the crescent structures correspond to the peaks in the energy spectrum. The positions of the time-varying final kinetic energy calculated by using Eq. (12) (dash dotted line) also agree well with the main crescent structure of the time-energy distribution. The opened circles (representing positions of the interference minima) are in good agreement with the minimums in the energy spectrum when the time is close to the peak of the pulse. The higher ATI peaks (not shown here) are similar.

The same calculations as those depicted in Fig. 3 are performed for H atoms irradiated by a 60 cycles pulse with frequency ω = 0.05691 (λ = 800 nm) and a peak intensity of I =1 × 10¹⁴W/cm². The results are shown in Fig. 4. Comparing with higher frequencies (see Fig. 1 and Fig. 3), though the main peaks are still visible, many more complex structures due to interference appear in Fig. 4. This is due to the fact that when the frequency is low, for each value of a given final kinetic energy, there are more than two ionization times satisfying Eq. (12) during the laser pulse. The reason can be understood as following: in low frequency laser field, the range of the ponderomotive energy U_p(t) becomes lager than the photon energy. As the time goes from the center of the pulse to the edge, U_p(t) decreases and when its decreased magnitude exceeds the photon energy, the minimum photon number N ₀ required for ionization decreases by 1 and another pair of curves satisfying Eq. (12) appears in the time-energy distribution as shown in Fig. 4. Clearly, this additional emission of electron wavepacket will also contribute to the photoelectron energy spectrum if its amplitude is not negligible. In addition, the interference between the photoelectron amplitudes produced at these times makes the energy spectrum in Fig. 4 more complex than those in Fig. 1 and Fig. 3. Moreover, it is noteworthy that the interference minima points obtained from Eq. (13), except the few points near the main peak, are not consistent with the minimums in the energy spectrum. The reason is that the method used to calculate the interference minima here is under the condition that only two saddle points are taken into account. So it is expected that the interference minima points obtained from Eq. (13) become invalid when the number of the saddle points dominated the ionized wave amplitude are more than two. However, since the emission of photoelectron close to the peak intensity of the pulse (center of the pulse) plays dominating role, the interference minima points which occur close to the main peak in the spectrum still well consist with the prediction of Eq. (13) as shown in Fig. 4.

Fig. 4. (a)Time-energy distribution of photoelectrons from H atoms. The parameters are the same as those in Fig. 1, except the angular frequency ω = 0.057 a.u. The green dash dotted line and green dashed line are calculated by Eq. (12) corresponding to S = 0 and S = 1, respectively. The circles denote the points from which the electron is emitted to produce the interference minima (opened circles and filled circles correspond to S = 0 and S = 1, respectively)(see text); (b) The energy spectrum.

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4. Conclusions

In conclusion, we propose a Wigner-distribution-like function to investigate the time-energy distribution of the photoelectrons emitted from an atom in intense laser fields. The time-energy distribution shows many crescent structures. The positions of the crescent structures are consistent with the final kinetic energy of ATI photoelectron acquired in the laser field given by the well-known simple energy conservation equation with a time-dependent ponderomotive energy U_p. In addition, these crescent structures give rise to interference pattern in the photoelectron energy spectrum which shows a main peak with several side peaks for each ATI peak. When the laser frequency is high, the interference minimums can be well predicted by a formula derived from the S-matrix theory using the saddle-point approximation. Nevertheless, when the frequency decreases, i. e., the ponderomotive energy increases, more ATI channels are involved and make the interference pattern more complicate. The proposed WDL function enable us to give a convincing and complete explanation for the interference effects in ATI peaks due to the finite pulse duration.

In this paper, we mainly focus on the analysis of ATI in long pulses. It is noteworthy that the proposed method, by which the relationship between the energy of the photoelectron and its ionization moment can be explicitly shown, can be applied to explicitly study the intriguing carrier-envelop phase effect in the interference effect of ATI in few-cycle pulses[32]. In addition, applying WDL function to the second term in the S-matrix expansion [4, 33]may provide further physical insight of the interaction between the ion and the photoelectron, i. e., the rescattering process, in ATI. All this work mentioned above is undergoing and will be reported in the future.

Acknowledgements

JC acknowledge Dr. Jung Hoon Kim for simulated discussions. This project is supported by the National Science Foundations of China under Grant No. 10705042, the National Basic Research Program of China Grant No. 2006CB806000, the Shanghai Excellent Academic Leaders Project Grant No. 07XD14007, the CAEP Foundation No. 2006z0202 and No. 2008B0102007 and the Open Fund of the State Key Laboratory of High Field Laser Physics ( Shanghai Institute of Optics and Fine Mechanics).

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Time-energy analysis of above-threshold ionization

Abstract

1. Introduction

2. Theory

3. Results and discussions

4. Conclusions

Acknowledgements

References and links

Cited By

Figures (4)

Equations (14)

Optics Express