# Photoelectron Angular Distribution and Phase in Two-Photon Single Ionization of H and He by a Femtosecond and Attosecond Extreme-Ultraviolet Pulse

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}[36].

_{ex}can be controlled by the pulse width, which manifests itself in a pulse width dependence of the PAD. This will eventually open a new avenue to the coherent control of the continuum wave packets. (In this connection, see [45] for the control of the resonant two-color two-photon excitation yield and [46] for the control of the photoelectron angular distributions of the nonperturbative resonant multi-photon ionization with ultrashort polarization-shaped pulses. See also a very recent review article for photoelectron angular distributions [47].)

_{ex}on pulse parameters in more detail. In our previous study [44], we have chosen He as a target atom for the following reasons: first, its single-electron excitation energies, e.g., 21.218 eV for 1s2p

^{1}P and 23.087 eV for 1s3p

^{1}P [48], coincide with the 13th and 15th harmonic photon energies of a Ti:Sapphire laser, respectively, and also with the typical wavelength range (50–61 nm) of EUV FELs such as the Spring-8 Compact SASE Source (SCSS) [18], the Free-electron LASer at Hamburg (FLASH) [17], and FERMI [49]. Second, its simple electronic structure allows for exact time-dependent numerical analysis [50–54], in great contrast to alkali atoms. Basically the same effect, however, is also expected for other atomic and molecular species. In the present study, in addition to He, we examine the case of a hydrogen atom, for which simulations are much less demanding and the scattering phase shift is precisely known as the Coulomb phase shift. We also investigate the case where the photon energy is close to the ionization threshold and show that the transition from 2PI involving Rydberg-manifold excitation to above-threshold 2PI is smooth. The Rydberg manifold behaves in a similar manner to the continuum, even when only a modest numbers of levels are excited by the pulse.

## 2. Gaussian Pulse

#### 2.1. Time-Dependent Perturbation Theory

_{f}of the final state after the pulse in the interaction picture can be written as,

_{mn}denotes the dipole transition matrix element between states m and n, ∆

_{m}= ω

_{m}− (ω

_{i}+ ω), ∆

_{f}= ω

_{f}− (ω

_{m}+ ω) with ω

_{m}being the energy eigen-value of state m, and the sum runs over all the intermediate bound and continuum states m. One can equivalently express Equation (1) by Equation (3) of [45] using the Fourier transform of f(t). Although a rectangular pulse is often assumed in previous work [2] and textbooks, we take, as a more realistic choice, a Gaussian profile $f(t)={E}_{0}{e}^{-{t}^{2}/2{T}^{2}}$, with E

_{0}and T being the field amplitude and the pulse width, respectively. More precisely, T is related to the full-width-at-half-maximum (FWHM) pulse width T

_{1/2}as ${T}_{1/2}=2\sqrt{\mathrm{ln}2}T$.

_{f}= ω

_{i}+ 2ω, in particular, one can perform the integrals in Equation (1) analytically to obtain a physically transparent expression:

_{m}= 0) participate in the second term, as expected for nonresonant paths. While either term dominates for a relatively long pulse (ps and ns), assuming that resonant excitation is not saturated, we can expect that the two terms are comparative for sufficiently short pulses and that their relative importance, which may be expressed as arg c

_{f}, varies with T. In such a situation, the amplitude ratio c

_{S}/c

_{D}between the final S and D continuum states is complex, since the branching ratio ${\mu}_{Sm}/{\mu}_{Dm}$ of the transitions from the intermediate P states m to each state depends on m. While the actual outgoing wave packets involve the contribution from the final states with ω

_{f}≠ ω

_{i}+ 2ω, it is instructive to write arg c

_{S}/c

_{D}using Equation (3) as follows:

_{m}T) ≈ (2∆

_{m}T)

^{−1}for all the other intermediate states. Hence, the competition between the resonant and nonresonant paths affects the interference between the outgoing S and D wave packets and manifests itself in the photoelectron angular distribution that depends on the pulse width.

#### 2.2. Photoelectron Angular Distribution

_{2}and β

_{4}are the anisotropy parameters associated with the second- and fourth-order Legendre polynomials, respectively. The interference of the S and D wave packets produces a photoelectron angular distribution,

_{S}| and |c

_{D}|, respectively, and can be either positive or negative in principle, and δ

_{l}is the phase of the partial wave, or the apparent phase shift. The apparent phase shift difference,

_{sc}intrinsic to the continuum eigen wave-functions (scattering phase shift difference), which has previously been studied both theoretically [61–63] and experimentally [29], and the extra contribution δ

_{ex}= arg c

_{S}/c

_{D}(if ${\tilde{c}}_{S}/{\tilde{c}}_{D}>0$), π − arg c

_{S}/c

_{D}(if ${\tilde{c}}_{S}/{\tilde{c}}_{D}<0$) from the competition of the two paths. This situation presents a contrast to the case of the photo-ionization from photo-excited states [29], where the nonresonant path is absent and only δ

_{sc}is present (δ = δ

_{sc}).

_{2}and β

_{4}, one cannot distinguish between (W, δ) and (−W, π − δ). One cannot distinguish between δ and −δ, either. Hence, in what follows, let us take W as positive, i.e., W = |c

_{S}/c

_{D}| and define the value of δ within the range [0, π]. Then, δ

_{sc}in Equation (7) should be replaced by |δ

_{sc}| or π − |δ

_{sc}| for appropriate interpretation in some cases (see the discussion on Figure 1 below).

#### 2.3. Hydrogen Atom

_{sc,l}for an azimuthal quantum number l is at hand as,

_{S}and c

_{D}using the analytical expression for the dipole transition matrix elements in Equation (2), but it is not trivial to perform an integration over the continuum intermediate states, since the Dawson’s integral has a long tail (∼ 1/x). Instead, we take an approach in the time domain, namely, direct numerical solution of the time-dependent Schrödinger equation (TDSE) in the length gauge,

_{2}and β

_{4}by integrating the ionized part of |Φ(r)|

^{2}over r and ϕ, from which one obtains W and δ by solving Equation (8). We have monitored the convergence of the results by calculating the PAD at different times. The calculation has been done for a Gaussian pulse envelope with a peak intensity of 10

^{10}W/cm

^{2}, at which we have confirmed that the interaction is still in the perturbative regime.

_{1/2}varies from 21 fs to 1 fs. Thus, c

_{s}/c

_{d}is approximately real and negative in the short-pulse limit, which leads to the cancellation between Y

_{00}(θ, φ) and Y

_{20}(θ, φ) around $\theta =\frac{\pi}{2}$.

_{sc}| [thin black line in Figure 1(a)] or π − |δ

_{sc}| (thin dashed black line), depending on the sign of c

_{S}/c

_{D}. This situation corresponds to ħω ≲ 9.8 eV and 10.6 eV ≲ ħω ≲ 11.6 eV. The relative phase δ between the S and D partial waves does not depend on the pulse width [Figure 4(a)], except for the case of very short pulses where the broadened spectrum begins to be resonant with a 2p or 3p level, which belong to the second category. We can also see from Figure 1 that the sign of c

_{S}/c

_{D}changes, e.g., around ħω = 10.9 eV and ħω = 11.7 eV, where W sharply peaks or approaches to zero [Figure 1(b)]. The jump in δ is not completely step-function-like, since the contribution from the resonant path can be neglected no longer there.

_{sc}| or π −|δ

_{sc}|) and varies with pulse width, as we have already seen above [Figure 2(a)] and can also see in Figure 4(b). It tends to either |δ

_{sc}| or π − |δ

_{sc}| for the longer pulse width. Interestingly, at ħω = 11.7 eV [blue line in Figure 4(b)], δ first decreases with increasing pulse width, approaching |δ

_{sc}| = 1.40 (i.e., the resonant path is dominant), but then, for T

_{1/2}≳ 8 fs, increases again, tending to π − |δ

_{sc}| = 1.74 (i.e., the nonresonant paths are dominant). Similar features can be observed for 9.0 and 11.0 eV in Figure 4(a) as well.

_{1/2}≲ 4 fs and 7 fs for 12.75 and 13.0 eV, respectively, which we will discuss below.

_{1/2}within the pulse width range investigated here [Figure 4(c),(d)]. These features, seen also for He [Figure 4(g)] and presumably general for all the atoms, can be understood as follows. Let us assume that the spectral width of the pulse contains a sufficient number of levels that the sum in Equation (1) can be approximately replaced by an integral,

_{m}) denotes the density of states multiplied with ${\mu}_{fm}{\mu}_{mi}$. By noting that F (∆T) ≈ (2∆T)

^{−1}, thus, F (∆T) ≈ F (∆)/T is a good approximation in most region of ∆T, Equation (12) can be rewritten as,

_{S}/c

_{D}, and thus δ and W, becomes roughly independent of T. Also, due to the continuity of the oscillator strength distribution across the ionization threshold [74], one can see that c

_{S}/c

_{D}also changes smoothly as a function of ħω. It should be noticed that, for the case of two-photon above-threshold ionization [37] and in the long-pulse limit, the first (resonant path) and second (nonresonant path) terms of Equation (12) tend to, e.g., the second and first terms of Equation (3.2) of [37], respectively.

_{n}= (n + δn)∆x, with ∆x(> 0) and δn being the increment of x and the offset of n, respectively. In Figure 5 we plot the ratio,

_{a}(u, q) denotes the theta function [60]. The approximation is surprisingly good even with ∆x = 1, though the FWHM of ${e}^{-{x}^{2}}$ is 2 ln 2 (= 1.66511 ⋯). This explains why ∆ varies smoothly even in the range of ħω where the level spacing is not negligibly small compared with the spectral width in Figure 1.

#### 2.4. Helium Atom

^{1}P and 1s3p

^{1}P states are 21.220 and 23.086 eV, respectively, in fair agreement with the experimental values (21.218 and 23.087 eV [48], respectively). Similarly to the case of a hydrogen atom, sufficiently long after the pulse has ended, we calculate β

_{2}and β

_{4}by integrating the ionized part of |Φ(r

_{1}, r

_{2})|

^{2}over r

_{1}, r

_{2}, θ

_{2}, ϕ

_{1}, ϕ

_{2}, from which one obtains W and δ by solving Equation (8). We use the values of δ

_{sc}from [63] to calculate δ

_{ex}= δ − δ

_{sc}. The calculation has been done for a Gaussian pulse envelope with a peak intensity of 10

^{11}W/cm

^{2}, at which we have confirmed that the interaction is in the perturbative regime. Our preliminary investigation indicates that the interaction begins to deviate slightly from this regime around 10

^{12}W/cm

^{2}if we increase the intensity, but the correction is still small below 10

^{13}W/cm

^{2}, which is the typical focal intensity of SCSS [40].

_{1/2}between 500 as and 21 fs. As expected from the discussions in Sections 2.1 and 2.2, and similarly to the case of a hydrogen atom (Subsection 2.3), both δ and W substantially change with pulse width, especially when the pulse is shorter than 10 fs. Accordingly, the PAD also varies as shown in Figure 3(b). One finds that the distribution to the direction perpendicular to the laser polarization again decreases as the pulse is shortened. As stated earlier, strictly speaking, Equation (4) is applicable only to ω

_{f}= ω

_{i}+ 2ω, and the actual PAD involves integration over ω

_{f}. Nevertheless, the results in Figure 2(b) can well be described by Equation (4) [44], except for δ in the ultrashort pulse regime T

_{1/2}≲ 1 fs, where the spectrum becomes broader than the level spacing. If we compare Figure 2(a) for H and (b) for He, both for the 2p resonance, the phase-shift difference δ has a similar pulse-width dependence. As for the amplitude ratio W = |c

_{s}/c

_{d}|, it is smaller than unity, i.e., |c

_{d}| > |c

_{s}| regardless of pulse width for H, while its variation is larger for He; |c

_{d}| > |c

_{s}| in the short-pulse limit, and |c

_{d}| < |c

_{s}| in the long-pulse limit. Accordingly, the variation in PAD is more prominent for He than for H (Figure 3).

_{sc}, and the PAD changes only slowly with T

_{1/2}[Figure 3(b)]. When the pulse is resonant (∆

_{r}T ≪ 1) and sufficiently long (T ≫ a

_{S}, a

_{D}) at the same time, assuming that the resonant excitation is not saturated, one can approximate the extra phase shift as

_{1/2}≲ 1 fs [Figures 3(b) and 6(b),(c)]. This is because the pulse becomes resonant with multiple levels; the spacing between the 1s2p and 1s3p is 1.9 eV. As we have shown in the previous subsection, when many neighboring states are resonantly excited by the pulse, the extra phase shift difference δ

_{ex}does not much depend on the pulse duration. Similarly to the case of the hydrogen atom, this especially applies when the photon energy lies in the Rydberg manifold, and exceeds the ionization potential (24.59 eV), i.e., in the case of above-threshold two-photon ionization.

_{ex}and the PAD, respectively, for different values of ħω, where we find trends similar to the case of H. While δ ≈ δ

_{sc}(δ

_{ex}≈ 0) and the PAD are nearly independent of T

_{1/2}for nonresonant pulses (ħω = 20.3 eV, T

_{1/2}≳ 3.5 fs), when the pulse is close to resonance with an excited level (ħω = 21.2, 21.3, and 23.0 eV), δ

_{ex}and PAD rapidly change with T

_{1/2}. On the contrary, δ

_{ex}is finite and nearly constant for ħω = 24.3, 24.6, and 25.0 eV [Figure 4(g)]; accordingly, the PAD hardly varies with pulse width. At T

_{1/2}≲ 1 fs, the spectrum is so broad that δ

_{ex}restarts to change slightly. One also sees that the transition across the ionization potential is smooth [Figure 4(g)], as has been seen for a hydrogen atom in Figures 1 and 4(c),(d). It should be pointed out that the extra phase shift difference due to free-free transitions plays a significant role in the recently observed time delay in photoemission by attosecond EUV pulses [75–79].

## 3. Pulse Shapes Typical of Experiments

#### 3.1. SASE-FEL Pulses

_{1/2}in the case of Gaussian pulses is already larger for the former, especially at T

_{1/2}< 10 fs [Figures 3(b) and 6]. This suggests that the controllability of PAD by pulse shape can be roughly estimated by the results for Gaussian pulses.

#### 3.2. High-Harmonic Pulses Containing Multiple Harmonic Orders

^{1}P state. We use the relative intensity of each harmonic component reported in Figure 1 of [82], neglecting inter- and intra-order chirps usually observed in experiments [83]. Each component is expressed as a Gaussian pulse with a common pulse width T

_{ov}, referred to as overall pulse width hereafter. We show the pulse shape and intensity spectrum for T

_{ov}= 7 fs in Figure 10. The whole pulse has a 7 fs envelope, while composed of many individual pulses with a pulse width ∼ 0.3 fs [Figure 10(a)]. The pulse spectrum, though composed of discrete harmonic peaks, is so broad as a whole that it spans from 13.8 eV (H9) to 26.1 (H17) [Figure 10(b)]. This latter manifests itself as a fact that each pulse in the pulse train [Figure 10(a)] has a sub-femtosecond duration, while the overall pulse width corresponds to the spectral width of each harmonic component.

_{ov}. Although δ changes with T

_{ov}especially at T

_{ov}< 4 fs, its variation is much smaller than for the case of a Gaussian pulse centered at 23.0 eV (squares in Figure 11). This is probably because the H9–H13 (nonresonant) and H17 (in the continuum) components would lead to a constant phase if alone, and that only resonant excitation by H15 contributes to the variation.

^{1}P state. The corresponding 2PI photoelectron energy is 21.4 eV, while the other photoelectron peaks are separated by 3.07 eV from each other. Then, in order to highlight the contribution from H15, let us focus ourselves on the wave packet between 20 and 23 eV, for which the values of δ are plotted as circles in Figure 11. Only the results at T

_{ov}≥ 2 fs are plotted, since the pulse is so short at T

_{ov}< 2 fs that the photoelectron peaks are not well separated from each other anymore. As expected, we can see that δ now varies with T

_{ov}as rapidly as for the case of the Gaussian pulse, though its value is somewhat shifted due to an additional contribution from the H13+H17 and H17+H13 processes. The change in PAD corresponding to the wave packet between 20 and 23 eV, shown in Figure 12, is not so large as one might expect from the variation in δ, again probably due to the relatively small variation with pulse width below 10 fs even for the case of Gaussian pulses. Nevertheless, it is clearly different from what would be expected from the scattering phase shift difference [see the curve for 21 fs in Figure 6(c)], well indicating the presence of the competition between the resonant and nonresonant ionization paths.

## 4. Conclusions

_{sc}| or π−|δ

_{sc}| more precisely, which would be expected for single-photon ionization from an excited level [29] and nonresonant two-photon ionization, and rapidly changes with the pulse width when the pulse is resonant with an intermediate excited state and 2 fs ≲ T

_{1/2}≲ 10 fs. Accordingly, the photoelectron angular distribution varies with T

_{1/2}as well. Hence, the control of the competition between the resonant and nonresonant paths in H and He by pulse width is a unique feature of a-few-fs EUV pulses.

_{sc}| and π − |δ

_{sc}| but nearly constant independent of T

_{1/2}when the Rydberg manifold is excited and in the case of above-threshold two-photon ionization. The transition across the ionization threshold is smooth.

## Acknowledgments

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**Figure 1.**Photon-energy dependence of (

**a**) the relative phase (phase-shift difference) δ and (

**b**) the amplitude ratio W between the s and d wave packets for a pulse width of 7 fs (circles) and 14 fs (squares). The target atom is hydrogen. Thin sold and dashed lines in (

**a**): |δ

_{sc}| and π − |δ

_{sc}|, respectively. Vertical dotted lines: positions of 2p to 6p resonances.

**Figure 2.**Pulse-width dependence of the TDSE-derived apparent phase shift difference (relative phase) δ (left axis) and W = |c

_{s}/c

_{d}| (right axis). (a) The target atom is hydrogen, and ħω = 10.2 eV. The thin horizontal line denotes the value (= π/2)) of the Coulomb phase-shift difference; (b) The target atom is helium, and ħω = 21.2 eV. The thin horizontal line denotes the value (1.511 [63]) of the intrinsic scattering phase-shift difference δ

_{sc}for ħω = 21.2 eV.

**Figure 3.**Pulse-width dependence of the photoelectron angular distribution for (

**a**) H and ħω = 10.2 eV; and (

**b**) He and ħω = 21.2 eV.

**Figure 4.**Pulse-width dependence of the phase-shift difference δ for H [(

**a**)–(

**d**)] and He [(

**d**)–(

**g**)] for different values of photon energy. Panel (

**d**) is the close-up of the vertical-axis range between 2.5 and 3.0 radians of panel (

**c**). The pulse is nonresonant for (

**a**) and (

**e**), resonant with a single level for (

**b**) and (

**f**), and resonant with multiple levels for (

**c**), (

**d**), and (

**g**). The resonant excitation energies for He 1s2p

^{1}P, 1s3p

^{1}P, 1s4p

^{1}P, 1s5p

^{1}P levels are 21.218, 23.087, 23.742, and 24.046 eV [48].

**Figure 6.**Pulse-width dependence of the photoelectron angular distribution for (

**a**) ħω = 20.3 eV; (

**b**) ħω = 21.3 eV; (

**c**) 23.0 eV; (

**d**) 23.9 eV; (

**e**) 24.3 eV; and (

**f**) 25.0 eV. The target atom is helium.

**Figure 7.**Photon-energy dependence of (

**a**) the relative phase (phase-shift difference) δ and (

**b**) the amplitude ratio W between the S and D wave packets for a pulse width of 7 fs. The target atom is helium. Thin sold and dashed lines in (

**a**): |δ

_{sc}| and π − |δ

_{sc}|, respectively. Vertical dotted lines: positions of 1s2p

^{1}P to 1s6p

^{1}P resonances.

**Figure 8.**Typical pulse shapes generated by the partial-coherence method [81] for the case of 23.0 eV photon energy, 2 fs coherence time, and 7 fs mean pulse width.

**Figure 9.**Photoelectron angular distribution by a chaotic pulse for a photon energy, coherence time (CT), and mean pulse width (MPW) indicated in each panel. The target atom is helium. The red and blue curves are for Gaussian pulses whose widths are given by the CT and MPW, respectively of the chaotic pulse. For example, in panel (

**b**), the photon energy, CT, and MPW, are 21.2 eV, 3.5 fs, and 7 fs, respectively.

**Figure 10.**(

**a**) Pulse shape and (

**b**) intensity spectrum of a 7 fs high-harmonic pulse (attosecond pulse train) composed of multiple harmonic orders used in the present study (see text).

**Figure 11.**Pulse-width dependence of the phase shift difference δ for high-harmonic pulses composed of multiple harmonic orders (see text). The target atom is helium. Black triangles: derived from the whole 2PI photoelectron wave packets. Red circles: derived from the 2PI photoelectron wave packets within the 20–23 eV energy range. Green squares: for Gaussian pulses with a photon energy of 23 eV. The thin horizontal line denotes the value (1.410 [63]) of the intrinsic scattering phase-shift difference δ

_{sc}for ħω = 23.0 eV.

**Figure 12.**Pulse-width dependence of the photoelectron angular distribution for high-harmonic pulses. The target atom is helium.

**Table 1.**W and δ calculated for chaotic pulses generated by the partial-coherence method for several pairs of coherence time (CT) and mean pulse width (MPW). The target atom is helium. The average values and standard deviation errors of W and δ by typically 48 runs are listed. The rows with the same CT and MPW values are for fully coherent Gaussian pulses.

ħω (eV) | CT (fs) | MPW (fs) | W | δ |
---|---|---|---|---|

21.2 | 2 | 5 | 1.31 ± 0.08 | 2.03 ± 0.04 |

2 | 7 | 1.42 ± 0.09 | 1.91 ± 0.04 | |

3.5 | 7 | 1.49 ± 0.09 | 1.87 ± 0.03 | |

2 | 2 | 1.05 | 2.26 | |

3.5 | 3.5 | 1.41 | 2.05 | |

5 | 5 | 1.57 | 1.92 | |

7 | 7 | 1.67 | 1.81 | |

21.3 | 2 | 7 | 2.21 ± 0.17 | 2.26 ± 0.06 |

7 | 21 | 1.92 ± 0.08 | 1.62 ± 0.009 | |

2 | 2 | 1.13 | 2.34 | |

7 | 7 | 2.04 | 1.87 | |

21 | 21 | 2.31 | 1.59 | |

23.0 | 2 | 7 | 0.645 ± 0.061 | 2.14 ± 0.08 |

3.5 | 7 | 0.890 ± 0.058 | 2.15 ± 0.06 | |

7 | 21 | 1.50 ± 0.154 | 1.79 ± 0.033 | |

2 | 2 | 0.464 | 2.71 | |

3.5 | 3.5 | 0.679 | 2.45 | |

7 | 7 | 1.08 | 2.10 | |

21 | 21 | 1.31 | 1.58 |

© 2013 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Ishikawa, K.L.; Ueda, K. Photoelectron Angular Distribution and Phase in Two-Photon Single Ionization of H and He by a Femtosecond and Attosecond Extreme-Ultraviolet Pulse. *Appl. Sci.* **2013**, *3*, 189-213.
https://doi.org/10.3390/app3010189

**AMA Style**

Ishikawa KL, Ueda K. Photoelectron Angular Distribution and Phase in Two-Photon Single Ionization of H and He by a Femtosecond and Attosecond Extreme-Ultraviolet Pulse. *Applied Sciences*. 2013; 3(1):189-213.
https://doi.org/10.3390/app3010189

**Chicago/Turabian Style**

Ishikawa, Kenichi L., and Kiyoshi Ueda. 2013. "Photoelectron Angular Distribution and Phase in Two-Photon Single Ionization of H and He by a Femtosecond and Attosecond Extreme-Ultraviolet Pulse" *Applied Sciences* 3, no. 1: 189-213.
https://doi.org/10.3390/app3010189