Effect of two-center interference on molecular ionization in multiphoton ionization regime

Shilin Hu; Jing Chen; Xiaolei Hao; Weidong Li; Li Guo; Shensheng Han

doi:10.1364/OE.25.023082

1. Introduction

Understanding of molecular ionization is the significant issue in strong-field molecular physics, since it is the first step that triggers a wealth of fascinating strong-field phenomena such as high-order harmonic generation [1, 2], nonsequential double ionization [3], and frustrated tunneling ionization [4,5]. Due to intensive investigations in the past two decades, most phenomena related to single ionization of atoms in laser fields have been well understood. However, investigations of molecular ionization show that it behaves very differently from that of atoms, due to the additional degrees of freedom in molecules. For example, charge-resonance-enhanced ionization of molecules takes place for larger internuclear separations [6–8]. Multiple ionization bursts appear for Hydrogen molecular ion due to the transient localization of the electron at one of the nuclei in the intense laser field [9]. Also, in the tunneling regime, it is natural to expect that different species show similar ionization behavior if they have analogous binding energy. However, it is found that the ionization strength of N₂ is analogous to that of Ar atom, while ionization suppression occurs for O₂ with respect to Xe atom [10–12], and the suppressed ionization of O₂ in intense mid-infrared laser fields is unambiguously ascribed to destructive interferences of contributions from different nuclei lately [13].

Thanks to the advance in molecular orientation and alignment techniques [14], investigation of alignment-dependence ionization for molecules subjected to intense fields has drawn great attentions recently [15, 16], which could be used as a tool for reconstructing the contour of the spatial electron distribution in the valence orbital [17]. However, a striking discrepancy occurs for the angular dependence of ionization for CO₂ between experiments and the theoretical simulation: the narrow ionization distribution of CO₂ and its maximum ionization at 45° between molecular axis and the laser direction are observed in experiments, while the peak ionization appears at alignment angle around 25° for the theoretical calculation [16]. Another case is the orientation dependence of ionization for OCS: the observation of the maximum ionization strength locates at alignment angle 90°, while the theory predicts at 45° [18]. Considerable efforts have been devoted to addressing the above issues, but the interpretation of the underlying mechanisms is still under debate [19–25].

So far, most theoretical simulations of molecular alignment-dependent ionization are performed by the strong-field approximation (SFA) [26–30] and molecular ADK theory (MO-ADK) [31, 32], because the computation is affordable, though the effect of coulomb potential and excited electronic structure are not taken into account. MO-ADK theory is only valid in the tunneling regime, and shows dependence on the field strength and molecular structure. For the SFA theory, the ionization rate of diatomic homonuclear molecule with bonding active orbital, e.g., H₂ and N₂, subjected to a linearly polarized laser field is relative to a trigonometric function cos²(P · R/2) [12], where P and R are the momentum of the emitted electron and the internuclear distance, respectively. The above trigonometric factor is not negligible for large internuclear distance, though most electrons are emitted with small momenta, which has been used to explain the oscillatory behavior of alignment-dependent ionization for model molecules with large internuclear separation [33]. Recently, SFA theory has been developed to account for coulomb correction, Stark shift, and the choice of gauges [12, 34–39]. Another approach used for calculation of molecular orientation-dependent ionization is numerical solution of the TDSE, which requires tremendous computational resources [33,40–43].

In experiments, most of alignment-dependent ionization yields were measured for ionization in the transition regime between the tunneling and multiphoton ionization, so theoretical study of the molecular ionization in multiphoton regime is also important for further understanding of the experimental findings. In addition, a complete description of the two-center interference effect on the molecular alignment-dependent ionization in the multiphoton ionization regime, given in a detailed manner based on a quantitative analysis, is still missing. In this paper, we present study on alignment-dependent ionization of $H_{2}^{+}$ based on a three-dimensional TDSE, MO-ADK theory and SFA theory, and find that two-center interference effect has an important impact on the angular-dependent ionization of $H_{2}^{+}$ . Atomic units (a.u.) are used unless otherwise indicated.

2. Theoretical methods

In the present paper, the simplest diatomic molecule $H_{2}^{+}$ in 800 nm laser field is chosen to investigate the angular dependence of molecular ionization, because it could be accurately treated by using a three-dimensional time-dependent Schrödinger equation in prolate spheroidal coordinates [33,41,42], and the results are benchmarks for the theoretical models (SFA and MO-ADK). Within the dipole approximation, the full-dimensional time-dependent Schrödinger equation of $H_{2}^{+}$ in the length gauge is written as

i \frac{\partial}{\partial t} Ψ (r, t) = [H_{1} (r) - r \cdot E (t)] Ψ (r, t) .

In prolate spheroidal coordinates, we define ξ = (r₁ + r₂)/R, η = (r₁ − r₂)/R, and the field-free Hamiltonian is

H_{1} = - \frac{2}{R^{2} (ξ^{2} - η^{2})} [\frac{\partial}{\partial ξ} (ξ^{2} - 1) \frac{\partial}{\partial ξ} + \frac{\partial}{\partial η} (1 - η^{2}) \frac{\partial}{\partial η} + \frac{ξ^{2} - η^{2}}{(ξ^{2} - 1) (1 - η^{2})} \frac{\partial^{2}}{\partial φ^{2}}] - \frac{2}{R} [\frac{1}{ξ + η} + \frac{1}{ξ - η}] .

Here R is the internuclear separation, and φ denotes the azimuthal angle. The laser-molecule interaction term is

H_{t} = - \frac{R}{2} (ξ η \cos β + \sqrt{(ξ^{2} - 1) (1 - η^{2})} \cos φ \sin β) E (t),

where time-dependent electric field E(t) is defined via

E (t) = - \frac{\partial A (t)}{\partial t}

. The vector potential is

A (t) = \frac{E_{0}}{ω} ε {sin}^{2} (π t / t_{m a x}) \cos (ω t + ϕ_{1})

with unit vector ε and the carrier envelope phase (CEP) of the pulse ϕ₁=0, 0 < t < t_max. E₀ is the maximum electric field, and t_max and ω are the pulse duration and the laser frequency, respectively. β is the alignment angle, which is defined as cos β = ε · R/|R|. The time-varying wave functions are expanded in terms of B-splines as

Ψ (ξ, η, φ, t) = \sum_{μ, ν, m} C_{μ, ν, m} (t) {(ξ^{2} - 1)}^{| m | / 2} B_{μ}^{k} (ξ) {(1 - η^{2})}^{| m | / 2} B_{ν}^{k} (η) e^{i m φ} / \sqrt{2 π},

where m is magnetic quantum number.

B_{μ}^{k} (ξ)

is the B-spline function, which is comprised of different polynomial pieces on adjacent subintervals of a fixed order k [44, 45], and μ and ν are the indexes of B splines in the ξ and η directions, respectively. The time-dependent wave functions are obtained by Crank-Nicolson method [33,44]. At the end of the pulse, the ionization probability is defined as P_ion = 1− Σ_n |〈ψ_n|Ψ(t_max)〉|², where ψ_n is the bound state calculated by diagonalization of field-free Hamiltonian matrix, and the sum over n includes all of the bound states. In the present calculation, the initial state is 1σ_g. We adopt 120 B-splines and 26 B-splines in ξ and η directions, respectively, and keep the magnetic quantum numbers from m=−7 to m=7. The parameter ξ is truncated at ξ_max=100 a.u. The diatomic molecule is exposed to a three-cycle laser pulse of frequency ω = 0.057 a.u, and the time step is 0.1 a.u. A cos^1/8 absorber function is used near the boundary to reduce the unphysical reflection.

In the present work, we also calculate the angular-dependent ionization of $H_{2}^{+}$ based on length-gauge strong-field approximation (LGSFA) [46–48] and radiation-gauge strong-field approximation (RGSFA) [12,13]. The single ionization rate of a molecular ion subjected to the linearly polarized laser field with vector potential A(t)=E₀ε cos ωt is given by

W = 2 π N_{e} \sum_{n = n_{0}}^{\infty} \int | A_{p n} |^{2} δ (E_{p} + I_{p} - n ω) d P,

where N_e is the number of equivalent electrons in the active orbital, and n₀ denotes the minimum number of photons that the electron absorbs to be emitted. The transition amplitudes is A_pn = 1/T ∫ ψ^v^*(r, t)V_F (t)Φ₀(r, R)e^iI^p drdt, and ψ^v (r, t) denotes the Volkov wavefunction. I_p is the ionization potential, E_p=P²/2 +Up indicates the quasienergy, where P²/2=nω − U_p − I_p is the kinetic energy of the emitted electron after absorbing n photons, and

U_{p} (= E_{0}^{2} / 4 ω^{2})

denotes the ponderomotive energy. For the LGSFA, we adopt V_F (t) = r · E(t) and obtain

A_{p n} = \frac{1}{T} \int_{0}^{T} d t \exp [i (\frac{E_{0}^{2}}{8 ω^{3}} \sin 2 ω t + \frac{E_{0} ε \cdot P}{ω^{2}} \sin ω t)] \exp [i (\frac{P^{2}}{2} + I_{p} + U_{p}) t] E_{0} \sin ω t \tilde{F} (P + A (t)),

with

\tilde{F} (M) = \int \exp [- i M \cdot r] (ε \cdot r) Φ_{0} (r, R) d r .

The mechanical momentum is M=P + A(t), and ε is the unit vector. The initial state is represented as an appropriate superposition of scaled hydrogen 1s atomic orbital as

Φ_{0} (r, R) = [ϕ_{1 s} (r - \frac{R}{2}) + ϕ_{1 s} (r + \frac{R}{2})] / \sqrt{2 [1 + S_{1 s} (R)]}

, where S₁_s (R) denotes the atomic orbital overlap integral. The 1s orbital is written as

ϕ_{1 s} (r) = \frac{1}{\sqrt{μ}} κ^{3 / 2} e^{- κ r}

with

κ = \sqrt{2 I_{p}}

. Substituting the initial state Φ₀(r, R) into Eq. (7) and adopting undressed modified molecular SFA [46–48], we obtain the following expression

\tilde{F} (M) = \frac{2 \cos (M \cdot R / 2) \int \exp [- i M \cdot r] (ε \cdot r) ϕ_{1 s} (r) d r}{\sqrt{2 [1 + S_{1 s} (R)]}},

where the trigonometric function cos(M·R/2) is related to the interference effect of the electron wave packets detached from the two cores.

For the SFA theory in the radiation gauge, V_F (t) = A(t) · P + A²(t)/2 is employed in Eq. (5), and the total single ionization rate per molecule subjected to a linearly polarized laser field with a vector potential A(t)=E₀ε cos ωt can be written as [12,13]

W = 2 π N_{e} \sum_{n = n_{0}} P {(U_{p} - n ω)}^{2} \int d P J_{n}^{2} (P \cdot α, \frac{U_{p}}{2 ω}) | \int e^{- i r \cdot P} Φ_{0} (r, R) d r |^{2} .

Here J_n(x, y) is the generalized Bessel function with α=A₀ε/ω. The physical quantities of N_e, P, U_p, and Φ₀(r, R) have been mentioned above in the LGSFA. For

H_{2}^{+}

, the Fourier transform in Eq. (9) has the form

\tilde{ϕ} (P) = C Φ_{a t} (P) \cos (P \cdot R / 2)

with C being the normalization term. Here Φ_at (P) is the atomlike part, and cos(P · R/2) associated with the interference effect between the wave packets of the electrons emitted from the individual core. In this work, Eqs. (5) and (9) are calculated by numerical integration [13,33,49].

3. Results and discussions

In order to visualize orientation effect, the ionization probability is normalized to its maximum value unless otherwise indicated, and I₀ = 10¹⁴ W/cm² is used as the unit of laser intensity. Figure 1 displays the calculated ionization probabilities obtained by different theories. Figure 1(a) depicts the results obtained by the TDSE and the normalized ionization probability declines monotonically with alignment angle β from 0° to 90° for the laser intensity 3I₀ (the Keldysh parameter γ=0.92). The ratio of P_ion(90°)/P_ion(0°) is 0.496, which is nearly the same as the calculation (≈ 0.5) in [41] with the same laser intensity 3×10¹⁴ W/cm² (P_ion denotes the ionization probability). For laser intensity 0.8 I₀ (γ=1.78), the normalized ionization yield increases as a function of the alignment angle, which is not influenced by the CEP of few-cycle pulse, and this prominent feature also appears for $H_{2}^{+}$ subjected to the 20-optical laser pulse of a low intensity [43]. Because the TDSE under dipole approximation can give accurate enough results and is independent on the gauge chosen [50], the TDSE calculations are used to test the simulations of theoretical models such as SFA theory and ADK theory.

Fig. 1 (a): Normalized ionization probability as a function of alignment angle β obtained by TDSE with decreasing laser intensity for $H_{2}^{+}$ at R=2 a.u. (b) and (c): Same as (a), but calculated by MO-ADK and LGSFA, respectively (see text). I₀=10¹⁴W/cm².

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To gain more physical insights into the ionization dynamics of $H_{2}^{+}$ in different laser fields, MO-ADK model and LGSFA are used to investigate the above phenomena. Figure 1(b) shows the data calculated by the MO-ADK theory and the details could be found in [32, 33]. The asymptotic coefficients are C₀=4.451, C₂=0.604, and C₄=0.024, which are in good agreements with those in [43]. The ionization yield decreases as a function of alignment angle, and P_ion(90°)/P_ion(0°)=0.418 is independent on the laser intensity, which is in good accordance with that in [43], and the possible reason is that the angular dependence of the asymptotic behavior for the wavefunction does not change for different laser intensities. For 3I₀, the ratio P_ion(90°)/P_ion(0°) is less than 1, because the distribution of electron cloud for the initial state (1σ_g) is parallel to the molecular axis, which is in a reasonable agreement with the TDSE calculation. However, the data calculated by the MO-ADK theory shows disagreement with that of TDSE for lower laser intensity, since the MO-ADK theory is only valid in tunneling ionization regime. Figure 1(c) presents the results obtained by the LGSFA, which qualitatively reproduces the TDSE simulations. For 3I₀, the ionization yield declines with the increase of alignment angle, and P_ion(90°)/P_ion(0°) is 0.392 due to the asymptotic behavior of 1σ_g wave function. For the low intensity, the ionization probability ascends as a function of the alignment angle, though the electron cloud of the 1σ_g orbit distributes along the molecular axis. This indicates the interference effect maybe play an important role in molecular ionization at the low intensity. In [43], the ratios of P_ion(90°)/P_ion(0°) show no obvious dependence on laser intensities, which are calculated by LGSFA based on the saddle-point method.

Since LGSFA simulations are consistent with TDSE calculations, we turn to the physical mechanism behind the ionization dynamics of $H_{2}^{+}$ exposed to low laser intensity (0.8I₀) by LGSGA. For the LGSFA, the interference term is related to the mechanical momentum in Eq. (8), and we use a Wigner-distribution-like function, which enables us to obtain the time-energy distribution of a photoelectron emitted from an atom subjected to laser fields [51, 52], to extract the information of the mechanical momentum. We use a linearly polarized laser pulse with the vector potential A(t)=−E₀/ω sin² (ωt/n) cos(ωt)ε, and the time-varying electric field is defined as E(t)=−∂A(t)/∂t. E₀ and ω=0.057 a.u. are the peak field strength and the laser frequency respectively and the number of optical cycles is n/2=6. The Wigner-distribution-like function is obtained by the length-gauge SFA theory and the details of the calculations could be found in [51–54].

Figure 2(a) presents the time-varying vector potential and electric field, and Figs. 2(b) and 2(c) show the time-momentum distribution of H atom in different laser intensities, respectively, which are obtained by the Wigner-distribution-like function based on the LGSFA theory. In general, the ionization mainly occurs near t=2.75T and t=3.25T (T=2π/ω) in Figs. 2(b) and 2(c), which are peaks of the electric field. A closer inspection reveals that the distribution shows positive and negative values with increasing P at t=3T, which can be attributed to the interference between the electrons produced at times next to t=2.75T and t=3.25T [52]. A closer comparison between the simulation in Fig. 2(b) and the data in Fig. 2(c) shows that the stripes become wider with decreasing laser intensity around t=2.75T and t=3.25T. According to the time-momentum distribution in Figs. 2(b), 2(c) and M=P+A(t) (the direction of P is assumed to be parallel to that of A(t)), we could obtain the distribution of mechanical momentum f (M) at a specific time. In Figs. 2(d) and 2(e), we plot the distribution of mechanical momentum at t = 2.75T, and it is found that the distribution of mechanical momentum for 3I₀ is considerably narrower comparing with that for 0.8I₀. With f (M) and the interference term, the ionization probability of $H_{2}^{+}$ for different alignment angle can be written as P_ion = ∫ cos²(M·R/2) f (M)dM. Because electrons ejected around t=2.75T and t=3.25T contribute dominatingly to the ionization, and the electrons emitted at other moments contribute negligibly, the ionization probability is obtained by the integral around the time at t=2.75T and t=3.25T, which is present in Fig. 2(f). It can be seen that the ionization probabilities ascend with the alignment angle, while the ratio of P_ion(0°)/P_ion(90°) is much smaller for 0.8I₀ compared with 3I₀, due to the wider distribution of the mechanical momentum for 0.8I₀ as shown in Fig. 2(e). For 3I₀, the simulation in Fig. 2(f) is inconsistent with those obtained by the TDSE and LGSFA in Fig. 1. This can be attributed to that the distribution of the 1σ_g orbital with respect to molecular axis is not taken into account, which plays an important role in tunneling ionization. For 0.8I₀, the result is in qualitative accordance with that calculated by the TDSE and LGSFA in Fig. 1, so the increase of the ionization probability from β=0° to 90° can be attributed to the interference effect which becomes evident at low laser intensities due to the wide distribution of mechanical momentum at ionization moment. This mechanism is different from the competition between multiphoton ionization and tunneling ionization, which is used to explain the change of P_ion(0°)/P_ion(90°) for different laser intensities in [43].

Fig. 2 (a): Vector potential and electric field of a six-cycle pulse (T=2π/ω). (b) and (c): Time-momentum distribution of H atom calculated by LGSFA in laser fields with intensities 3I₀ and 0.8I₀, respectively (the black line denotes -A(t)). (d) and (e): Normalized distribution of mechanical momentum for H atom in laser fields with intensities 3I₀ and 0.8I₀ at t=2.75T, respectively. (f): Normalized ionization probabilities of $H_{2}^{+}$ obtained by the model P_ion =∫ cos²(M · R/2) f (M)dM near the peak electric field. For more details see the text. I₀=10¹⁴W/cm².

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We also investigate the alignment-dependent ionization probability of $H_{2}^{+}$ at R=2 a.u. by RGSFA, which is present in Fig. 3(a), and the ionization probabilities of 3I₀ and 0.8I₀ increase monotonically with alignment angle from 0° to 90°. However, the ratio of P_ion (90°)/P_ion (0°) is larger for 3I₀ in comparison with 0.8I₀, which is not in agreement with that of the TDSE. To clearly understand the results in Fig. 3(a), we show the momentum distributions of photoelectrons from $H_{2}^{+}$ calculated without the trigonometric function and the trigonometric term in Fig. 3(b) for the RGSFA. As the intensity decreases, the contribution from small momentum becomes more significant, and this leads to the increase of the interference effect cos(P · R/2). So the ratio of P_ion (90°)/P_ion (0°) for 0.8I₀ is smaller than that for 3I₀, which is consistent with calculation in [13]. The two-center interference effect on molecular ionization has been studied extensively based on RGSFA [12, 13, 55–57], and the interference effect depends on the canonical momentum P, while in LGSFA the interference term is related to mechanical momentum M, which depends on the ionization moment and is hard to be extracted. For the first time, the effect of the mechanical momentum distribution at the ionization moment on molecular ionization has been studied by LGSFA, and the information of mechanical momentum is extracted by Wigner-distribution-like function. Moreover, the LGSFA simulation is in better agreement with the TDSE calculation than the RGSFA, which can be attributed to that the initial state Φ₀(r, R) exp(iI_pt) is the eigenstate of the Hamiltonian H₁ for the LGSFA but is not the eigenstate of the Hamiltonian H₁ for the RGSFA [58].

Fig. 3 (a): Normalized ionization probability as a function of alignment angle β calculated by RGSFA with decreasing laser intensity for $H_{2}^{+}$ at R=2 a.u. (b): Corresponding normalized momentum distributions of photoelectrons from $H_{2}^{+}$ calculated without the trigonometric function at intensities of 3I₀ and 0.8I₀ for RGSFA. The green full triangles denote the trigonometric part cos²(PR/2) as a function of momentum (see text). I₀ =10¹⁴ W/cm².

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In Figs. 4(a) and 4(b), we study the angular dependence of ionization probability of $H_{2}^{+}$ at R=4 a.u. by TDSE and LGSFA for laser intensities 2.2I₀ and 0.58I₀, respectively, which have the same Keldysh parameters as those in Fig. 1 for 3I₀ and 0.8I₀. The ionization yields decrease as a function of alignment angles for TDSE and LGSFA calculations irrespective of laser intensity, which are different from those in Fig. 1, since the wavefunction of the initial state 1σ_g is more concentrated along the molecular axis for R=4 a.u. than that of R=2 a.u. as shown in Fig. 5.

Fig. 4 (a) and (b): Normalized alignment-dependent ionization probability of $H_{2}^{+}$ obtained by TDSE and LGSFA with decreasing laser intensity for R=4 a.u. and I₀ = 10¹⁴W/cm² (see text).

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Fig. 5 Two-dimensional distribution of the 1σ_g wavefunction of $H_{2}^{+}$ at different internuclear distances: (a): R=2 a.u. (b): R=4 a.u.

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4. Conclusions

In conclusion, we have studied the alignment dependence of ionization of $H_{2}^{+}$ with internuclear distance R=2 a.u. subjected to different laser fields using the three-dimensional TDSE. Our results show that the peak ionization probability occurs when the polarization direction of the laser field is parallel to the molecular axis in the tunneling ionization regime, while the ionization yield increases as a function of alignment angle in the multiphoton ionization regime. The results obtained by the RGSFA show disagreement with the TDSE calculations. Based on the calculations of the MO-ADK and LGSFA theories, the decrease of ionization probability as a function of alignment angle in tunneling ionization regime is ascribed to the parallel distribution of the electron cloud with respect to the molecular axis for the 1σ_g orbital. As the laser intensity decreases, the two-center inference effect becomes significant due to the wide distribution of the initial mechanical momentum of photoelectron at ionization moment according to the LGSFA, which gives rise to the increasing ionization with ascending orientation angle in the multiphoton ionization regime, and the information of mechanical momentum is extracted by Wigner-distribution-like function. As the internuclear distance increases to R=4 a.u., the ionization yields decrease as a function of the alignment angles irrespective of laser intensity for both TDSE and LGSFA calculations, which is different from that of R=2 a.u. due to that the 1σ_g wavefunction concentrates more in the molecular axis when R becomes large. Our work demonstrates that two-center interference effect plays a pivotal role in the orientation ionization of $H_{2}^{+}$ at R=2 a.u. and provides a further understanding of molecular alignment ionization in multiphoton ionization regime.

Funding

National Key program for Science and Technology Research and Development (No. 2016YFA0401100); National Basic Research Program of China (Grant No. 2013CB922201); National Natural Science Foundation of China (Grants No. 11304329, No. 11334009, No. 11425414, No. 11374197, No. 11447114, and No. 11504215).

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Effect of two-center interference on molecular ionization in multiphoton ionization regime

Abstract

1. Introduction

2. Theoretical methods

3. Results and discussions

4. Conclusions

Funding

References and links

Cited By

Figures (5)

Equations (9)

Optics Express