⇦Chinese Physics Letters, 2020, Vol. 37, No. 2, Article code 024201 Spider Structure of Photoelectron Momentum Distributions of Ionized Electrons from Hydrogen Atoms for Extraction of Carrier Envelope Phase of Few-Cycle Pulses * Jiu Tang (唐久)1,2, Guizhong Zhang (张贵忠)1,2**, Yufei He (何宇飞)1,2, Meng Li (李猛)1,2,3, Xin Ding (丁欣)1,2, Jianquan Yao (姚键铨)1,2 Affiliations 1College of Precision Instrument and Optoelectronics Engineering, Tianjin University, Tianjin 300072 2Key Lab of Optoelectronic Information Technology (Ministry of Education), Tianjin University, Tianjin 300072 3Key Laboratory of Operation Programming & Safety Technology of Air Traffic Management, Civil Aviation University of China, Tianjin 300300 Received 11 November 2019, online 18 January 2020 *Supported by the National Natural Science Foundation of China under Grant Nos. 11674243 and 11674242, and the Fundamental Research Funds for the Central Universities under Grant No. 3122016D014.
**Corresponding author. Email: johngzhang@tju.edu.cn Citation Text: Tang J, Zhang G Z, He Y F, Li M and Ding X et al 2020 Chin. Phys. Lett. 37 024201    Abstract The spider structure in the photoelectron momentum distributions (PMDs) of ionized electrons from the hydrogen atom is simulated by solving the time-dependent Schrödinger equation (TDSE). We find that the spider structure exhibits sensitive dependence on carrier envelope phase (CEP) of the few-cycle pulses. To elucidate the striking CEP dependence of the spider structure, we select three physical parameters $I_{\rm L}$, $I_{\rm R}$, and $I_{\rm R}/I_{\rm L}$ to quantitatively characterize the variations of the spider structure induced by altering the CEPs. $I_{\rm L}$ is the sum of the left half panel of the transverse cut curves (i.e., the sum of all the negative momenta along the laser polarization direction), $I_{\rm R}$ is the sum of the right half panel of the transverse cut curves (i.e., the sum of all the positive momenta along the laser polarization direction), and $I_{\rm R}/I_{\rm L}$ is the ratio between the two sums. These parameters are shown to have monotonic relation with the CEP value, which is exploited to extract the CEPs. We anticipate that our method will be useful for obtaining CEPs encoded in the spider structure of PMDs. DOI:10.1088/0256-307X/37/2/024201 PACS:42.25.Ja, 42.30.Rx, 32.80.-t © 2020 Chinese Physics Society Article Text Strong-field ionization is the foundation of understanding strong-field phenomena in atoms and molecules, and has been extensively studied experimentally and theoretically.[1–3] In tunnel ionization, electron wavepacket will be tunnel ionized from the parent ion. Tunneling electron wavepackets from different pathways will lead to interference in photoelectron momentum distributions (PMDs) when the final momenta of these wavepackets are equal. It has been demonstrated that PMDs have sensitive dependence upon the carrier envelope phase (CEP) in many experiments and theoretical calculations.[4–6] It is a formidable task to capture the CEP, which determines the actual time evolution of the electric field for few-cycle laser pulses and henceforth to control the relevant ionization processes.[7–10] Several methods have been proposed to determine CEPs such as the $f$-to-$2f$ interferometer,[11,12] attosecond streaking,[13,14] the left-right asymmetry of ionized electrons along the polarization axis[15,16] and the angular distributions of photons.[17] Some of them are based on tunneling ionization. In tunneling ionization of atoms, variation of the CEP is amplified to tunneling electron wavepackets. Consequently, any variation of the CEP of the laser pulses will induce prominent changes of the interference patterns in PMDs.[18,19] Therefore, the CEP dependence of PMDs may be used to measure the CEP values of the pulse. Over the past years, the so-called spider structure in the PMDs was found and extensively investigated.[20–23] The spider structure in the strong-field photoelectron holography (SFPH) spectra originates from the photoelectron wavepackets of direct (reference) and forwarding-scattering (signal) trajectories. In 2012, Hickstein et al.[20] intuitively demonstrated the formation of the prominent primary spider structure and the conspicuous inner spider structure. Recent experiments have shown that structural and dynamic information encoded in the spider structure can be extracted using certain methods.[21–23] In this work, we propose and prove a method based on sensitive CEP dependence of PMDs for extracting the CEP values encoded in the spider structure by numerically solving the time-dependent Schrödinger equation (TDSE) for hydrogen atoms. Our numerical results show that the spider structure in PMDs is readily generated, from which the CEP values can be obtained. In addition, the present method reveals monotonic relation with the CEP value. Our proposed scheme can provide an alternative approach to determine the CEP of a few-cycle pulse. We numerically solve the two-dimensional time-dependent Schrödinger equation (TDSE) of the hydrogen atom using the atomic units. In the length gauge, the TDSE is written as (in atomic units) $$ i\frac{d\psi (r,t)}{dt}=\hat{H}\psi (r,t),~~ \tag {1} $$ where $\psi (r,t)$ is the wavefunction and $\hat{{H}}$ is the electron Hamiltonian expressed as $$ \hat H = - \frac{1}{2}{\nabla ^2} + V(r) + {\boldsymbol r}\cdot{\boldsymbol E}(t),~~ \tag {2} $$ where $r$ is the distance between the electron and the nucleus, and $V(r)$ is the Coulomb potential softened to avoid singularity and to match the ionization potential of hydrogen. In our simulation, $E(t)$ is taken as a sine function and is polarized along the $x$ axis. Then laser field can be written as $$ E(t)=E_{0} f(t)\sin (\omega t+\alpha),~~ \tag {3} $$ where $\omega$ is the carrier frequency of the laser pulse, $E_{0}$ is the field amplitude of the pulse, $\alpha$ is the CEP, and $f(t)$ is the envelope of the laser pulse. Our calculation of PMDs is based on the wavefunction-splitting technique, which allows us to reconstruct the external wavefunction in the momentum space and to calculate the photoelectron momentum spectra accurately. Chelkowski et al.[24] developed the wavefunction-splitting technique in 1998. According to the technique, the time evolution of the wavefunction in a laser field is implemented accurately over arbitrary time interval and the absorbed electron wavepacket can be reconstructed analytically. This splitting technique divides the total simulation space into three for overlapping with an absorbing potential to obtain the internal wavefunction $\psi_{\rm int}$ and the external wavefunction $\psi_{\rm ex}$ in the overlapping region. The absorbing potential is chosen to be (relative physical parameters have been elucidated in Ref. [24]) $$ V_{\rm abs} (r)=\Big[1+\exp \Big(\frac{r-r_{0} }{\Delta r}\Big)\Big]^{-1}.~~ \tag {4} $$ These wavefunctions are related by the formulas $$\begin{align} \psi (r,t)=\psi_{\rm in} (r,t)+\psi_{\rm ex} (r,t),\\ \psi_{\rm in} (r,t)=V_{\rm abs} (r)\psi (r,t), \\ \psi_{\rm ex} (r,t)=\left[ {1-V_{\rm abs} (r)} \right]\psi (r,t),~~ \tag {5} \end{align} $$ where the absorbing potential $V_{\rm abs} (r)$ relies on the distance between the electron and the nucleus. We use the following formula to calculate the external wavefunction, $$ \psi_{v} (r,t)=\exp \left[ {-i\Delta (t,0)r} \right]\psi_{\rm ex} (r,t).~~ \tag {6} $$ In the overlapping region, we use the following formula to calculate the Fourier transform $\phi_{v} (p,t)$ of the external wavefunction $\psi_{v} (r,t)$, $$ \phi_{v} (p,t)=(2\pi)^{-1/2}\int_{r_{\rm ex} }^{r_{\rm in} } {\exp (-ipr)\psi_{v} (r,t)} dr.~~ \tag {7} $$ We use the following formula to calculate the time evolution of the wavefunction in the momentum space, $$ \phi_{v} (p,t_{2})=U(t_{2},t_{1})\phi_{v} (p,t_{1}).~~ \tag {8} $$ Here $U$ is the exact field propagator, $$\begin{alignat}{1} U(t_{2},t_{1})=\,&\exp \Big\{-\frac{i}{2}\int_{t_{1} }^{t_{2} }\Big[ p^{2}+2\Delta (t,0)p\\ &+\Delta^{2}(t,0) \Big]dt \Big\}.~~ \tag {9} \end{alignat} $$ Using Eq. (8), we can implement the temporal propagation of the momentum-space wavefunction. Then, the accumulated momentum-space wavefunction $\phi_{v} (p,t_{f})$ is calculated with the assistance of the following formula: $$ \phi_{v}^{(k)} (p,t_{f})=\phi_{v}^{(k-1)} (p,t_{f})+\phi_{v} (p,t_{f}).~~ \tag {10} $$ Equation (10) means that the wavefunction $\phi_{v}^{(k)} (p,t_{f})$ obtained at the $k$th step is added to the wavefunction, $\phi_{v}^{(k-1)} (p,t_{f})$ obtained at the previous $(k-1)$th step, and $\phi_{v} (p,t_{f})$ stands for a sum of all the wavefunctions obtained before the $(k-1)$th step.[24] At each time instant $t_{k} =k\delta t$, we repeat the above steps until the end of the laser pulse, $t_{f} \geqslant t_{p}$. The final PMDs are obtained from the square of the momentum-space external wavefunction. The laser pulses used in our simulation have a sine-square envelope with a duration of four optical cycles (o.c.). The intensity and wavelength of the laser pulses are varied around $I=1.5\times 10^{14}$ W/cm$^{2}$ and $\lambda =1550$ nm, respectively.[22] The values of the CEPs are varied over a large range, we choose some of them to show a few values of $\alpha =0$, $\pi /4$, $\pi /2$, $3\pi /4$, $\pi$ in Fig. 1.
cpl-37-2-024201-fig1.png
Fig. 1. Photoelectron momentum distributions of hydrogen atoms induced by few-cycle laser pulses with different CEPs. The color lines in (f) encircle the areas for the transverse cuts, the right integrated momentum $\Delta I_{\rm R}$ is marked by the green lines, and the left integrated momentum $\Delta I_{\rm L}$ is marked by the red lines. These panels correspond to the CEP values of $\alpha =0$, $\pi /4$, $\pi /2$, $3\pi /4$ and $\pi$, respectively. The intensity of the laser pulses is $I=1.5\times 10^{14}$ W/cm$^{2}$, the laser wavelength is $\lambda =1550$ nm, and the laser pulse duration is four optical cycles.
In Fig. 1, we plot several photoelectron momentum distributions (PMDs) for different CEP values chosen from our simulation results. At first glance, the PMDs exhibit fairly perplex interference patterns, such as co-circular rings centered around the origin, thick interference arcs-curved to the left and right, and the famous spider legs. It has been established that the actual interference patterns are induced by many interfering pathways: for instance, inter-cycle interference,[25] which arises from the interference of electron wave packets ionized at different laser cycles, is manifested as co-circular rings; intra-cycle interference,[26] which arises from the interference of electron wave packets released within one laser cycle, is manifested as curved short arcs; and the direct and near-forward rescattering interference is reflected as the so-called spider legs. All these kinds of interference fringes are discerned in our simulation (see Fig. 1). The general feature for the spider structure in Fig. 1 is that the interference pattern is symmetric about the horizontal axis ($p_{y} =0$). More remarkably, one can observe that the PMDs are critically dependent on CEP, which is reflected in the asymmetry of the spider patterns between the left half and the right half of the PMDs (asymmetric about the vertical axis of $p_{x} =0$). For any CEP value, three types of interference patterns mentioned above are unambiguously observed in the right half panel of Fig. 1(a). Meanwhile, in the left half plane of Fig. 1(a), these three interference patterns are seen intermingled, resulting in an overall perplex structure. When the CEPs gradually increase, the interference patterns in the right half panel become more complicated while those in the left half panel become more visible. Eventually, when the CEP is set to be $\pi$ in Fig. 1(e), one can observe that the interference patterns are exactly a mirror image about the $p_{x} =0$ axis of the PMD patterns in Fig. 1(a). These observations imply that the CEPs are encoded and reflected in the variations of the interference patterns in PMDs, which is a remarkable effect to exploit. Theoretically, different CEPs will result in different electric-field forms of the pulses, and different electric-field forms will induce different dipole interactions in the time-dependent Schrödinger equation (TDSE), so that the spider-structured photoelectron momentum distributions (PMDs) will be different. Eventually, the left and right integrated momenta will be different. Therefore, we can utilize the properties and variations of the interference patterns in PMDs to extract CEP values. In the numerical simulation, we can obtain useful information from the perplex interference patterns by focusing on the changes of the interference patterns induced by different CEPs. In order to present the changes of the interference patterns in a clearer way, we use a window function[21] to average the PMDs over a narrow range of $p_{y}$. By choosing the cut position of $p_{y}$ and a suitable width $\Delta p_{y}$, we can optimize the window function $p_{y} \pm \Delta p_{y}$. Because the interference maxima of the spider structure lie in the symmetric axis $p_{y} =0$ and they are a key feature, we select $p_{y} =0$ a.u. as the cut position of $p_{y}$ for obtaining the transverse cut curves. To describe the striking CEP dependence of the PMDs, we select three physical parameters to characterize the variations of the transverse cut curves quantitatively. The first physical parameter, i.e., the left integrated momentum, $\Delta I_{\rm L}$, is the sum of the left half panel of the transverse cut curves (i.e., the sum of all the negative momenta along the laser polarization direction, red-encircled area, see Fig. 1(f)), the second one, i.e., the right integrated momentum, $\Delta I_{\rm R}$, is the sum of the right half panel of the transverse cut curves (i.e., the sum of all the positive momenta along the laser polarization direction, green-encircled area, see Fig. 1(f)), the third one, i.e., $\Delta I_{\rm R} /\Delta I_{\rm L}$, is the ratio between the two sums. It will be shown in the following that we can deploy the relative changes of these physical parameters to extract CEPs from the PMDs.
cpl-37-2-024201-fig2.png
Fig. 2. Variation curves of the selected parameters of the transverse cuts versus CEP. The interference fringes are averaged over $p_{y}$ in an interval of $p_{y} =0\pm 0.2$. (a) Ratios of $\Delta I_{\rm R}$ to $\Delta I_{\rm L}$ as a function of CEP. The red dashed lines demonstrate an example of extracting a CEP of 0.2$\pi$ when the ratio curve reads 2.18. (b) The red curve represents for the transverse cuts of the left half panel, plotted as a function of CEP, and the green curve is for the similar cuts for the right half panel. The laser parameters are the same as those in Fig. 1. See text for details.
Choosing a width of $\Delta p_{y} =0.2$ a.u., we can use the extract procedure to obtain $\Delta I_{\rm L}$, $\Delta I_{\rm R}$ and $\Delta I_{\rm R} /\Delta I_{\rm L}$, as shown in Fig. 2. The variation curves in Fig. 2 clearly show that these three parameters manifest the monotonic relation with the CEP values. One can observe that, when CEP increases, $\Delta I_{\rm R}$ and $\Delta I_{\rm R} /\Delta I_{\rm L}$ monotonously decrease while $\Delta I_{\rm L}$ monotonously increases. The $\Delta I_{\rm R}$ curve and the $\Delta I_{\rm R} /\Delta I_{\rm L}$ curve in Fig. 2 reveal negative slopes, and the $\Delta I_{\rm L}$ curve reveals a positive slope, which are in accordance with the variation trends of the interference patterns in Fig. 1. Furthermore, the $\Delta I_{\rm R}$ values for CEP = 0 and $\pi$ are equal to $\Delta I_{\rm L}$ for CEP = $\pi$ and 0, respectively, which corroborates the mirror symmetry between Fig. 1(a) and Fig. 1(e). This unveils that the variations in the PMDs induced by the changes in the CEP are quantitatively reflected in the ratio $\Delta I_{\rm R} /\Delta I_{\rm L}$. All these monotonic curves prove that one can determine the CEP values from the relative changes of $\Delta I_{\rm L}$, $\Delta I_{\rm R}$ and $\Delta I_{\rm R} /\Delta I_{\rm L}$. The curve for $\Delta I_{\rm R} /\Delta I_{\rm L}$ serves as a calibration curve (Figs. 2(a) and 5(a)) for obtaining the CEP values. In experimental applications, one simply separately sums up the momenta in the negative and positive momentum directions along the laser polarization direction, take the ratio of the two sums, find the point of this ratio value on the ordinate (Fig. 2(a)), and read out the CEP value on the abscissa via the calibration curve. For example, when the ratio curve reads 2.18 (on the ordinate), the corresponding CEP reads 0.2$\pi$ (along the abscissa), as demonstrated in Fig. 2(a). To further demonstrate the feasibility of our proposed method, we also vary the laser pulse parameters in the simulation. Pertaining to laser intensity, we choose to present the resultant PMDs for three laser intensity values: $I=1.3\times 10^{14}$ W/cm$^{2}$, $1.4\times 10^{14}$ W/cm$^{2}$ and $1.6\times 10^{14}$ W/cm$^{2}$. For these PMDs, we use the same averaging and extracting procedures to obtain the parameters of $\Delta I_{\rm L}$, $\Delta I_{\rm R}$ and $\Delta I_{\rm R} /\Delta I_{\rm L}$ for the transverse cuts. The obtained curves are displayed in Fig. 3. It is obvious that all the curves show monotonic relations with the CEPs although the laser intensities are different. In addition, it appears that the curves are flatter when the CEPs are close to $\pi$, as shown in Figs. 3(a) and 3(b). Thus the accuracy of extracting CEP from the $\Delta I_{\rm R} /\Delta I_{\rm L}$ curves is deteriorated. However, the slopes of the $\Delta I_{\rm L}$ and $\Delta I_{\rm R}$ curves are steeper when CEPs are close to $\pi$, as shown in Figs. 3(d) and 3(e). Consequently, accurate CEP values can still be extracted from the $\Delta I_{\rm L}$ and $\Delta I_{\rm R}$ curves.
cpl-37-2-024201-fig3.png
Fig. 3. Top three panels show the ratio $\Delta I_{\rm L} /\Delta I_{\rm R}$ of the transverse cuts, and the bottom three panels show the variations for the left integrated momentum (red curve) and right integrated momentum (green curve) of the transverse cuts. All the cuts are taken for $p_{y} =0\pm 0.2$ and for different laser intensities. The panels in the three columns correspond to laser intensities of $I=1.3\times 10^{14}$ W/cm$^{2}$(a, d), $1.4\times 10^{14}$ W/cm$^{2}$(b, e) and $1.6\times 10^{14}$ W/cm$^{2}$(c, f), respectively. The other laser parameters are the same as those in Fig. 1.
cpl-37-2-024201-fig4.png
Fig. 4. The three top panels show the ratio $\Delta I_{\rm L} /\Delta I_{\rm R}$ of the transverse cuts. Bottom panels show the variations for the left integrated momentum (red curve) and the right integrated momentum (green curve) of the transverse cuts. The first column is for the transverse cuts averaged over $p_{y}$ in the interval of $p_{y} =0\pm 0.2$ for $\lambda =1460$ nm. The second column is for the transverse cuts averaged over $p_{y}$ in the interval of $p_{y} =0\pm 0.2$ for $\lambda =1655$ nm. The third column is for the transverse cuts averaged over $p_{y}$ but in the interval of $p_{y} =0\pm 0.3$ for $\lambda =1655$ nm. The other laser parameters are the same as those in Fig. 1.
We also perform simulations of the PMDs for varying laser wavelengths. We just focus on the two cases that the transverse cuts are implemented for laser wavelengths of $\lambda =1460$ nm and 1655 nm, respectively. The curves in Figs. 4(a)–4(d) show the similar trends of variation as those in Fig. 3. In Figs. 4(b) and 4(e), we note that the curves are flatter for CEPs between $7\pi /8$ and$\pi$. By optimizing the width as $\Delta p_{y} =0.3$ a.u. for a better signal-to-noise ratio, the $\Delta I_{\rm L}$ curve becomes steeper again for the same CEP range, thus accurate CEP values can still be extracted. Therefore, we show that the proposed scheme is robust against laser wavelength changes.
cpl-37-2-024201-fig5.png
Fig. 5. Variations of the momenta and momentum ratio with CEPs for the Gaussian pulse envelope of $\exp (-t^{2}/\tau^{2})$, where $\tau =2\pi /\omega$. The curves are for the transverse cuts averaged over $p_{y}$ in the interval $p_{y} =0\pm 0.2$. (a) Variations of the ratio of $\Delta I_{\rm R}$ to $\Delta I_{\rm L}$ as a function of CEP. (b) Red and green curves show the variations for left integrated momentum and right integrated momentum of the transverse cuts, respectively. (c) Plots of all the previous $\Delta I_{\rm L} /\Delta I_{\rm R}$ curves for sine-square envelope with different laser parameters: the ratio curves in Fig. 2(a), Figs. 3(a)–3(c) and Figs. 4(a)–4(b) are displayed by the blue, black, red, green, yellow and purple colors, respectively. The other laser parameters are the same as those in Fig. 1.
Lastly, we verify that our method is valid for other pulse shapes. We now consider a few-cycle pulse expressed as $E(t)=E_{0} f(t)\sin (\omega t+\alpha)$, where the pulse envelope is changed to a Gaussian: $\exp (-t^{2}/\tau^{2})$ with $\tau =2\pi /\omega$. The other laser parameters are the intensity $I=1.5\times 10^{14}$ W/cm$^{2}$, laser pulse wavelength $\lambda =1550$ nm. In Figs. 5(a) and 5(b), we give the simulation results using our proposed algorithm. Compared to the envelope function used previously, the ratio curves in Fig. 5(a) are seen to have a larger curvature, and flatter for CEPs close to $\pi$. In Fig. 5(b), however, the $\Delta I_{\rm L}$ and $\Delta I_{\rm R}$ curves all show monotonic relations with the CEP values, following the similar trends with the previous situations: when the CEPs increase gradually from zero to $\pi$, the $\Delta I_{\rm L}$ curve and the $\Delta I_{\rm R}$ curve vary monotonically, implying that the properties of the transverse cuts can be a tool for extracting CEPs from the spider structures in PMDs. In addition, we plot all our previous $\Delta I_{\rm L} /\Delta I_{\rm R}$ curves for sine-square envelope with different laser parameters in Fig. 5(c). It is obvious that the ratio curves remain approximately a similar trend even though the laser parameters are varied. In practical experiment, hydrogen gas should be first ionized for further atomic ionization. The photoelectron momentum distributions are acquired for atomic hydrogen. Gas jet position and laser intensity variation over the Rayleigh range can be rendered with an averaged intensity. In summary, by solving the TDSE for hydrogen atoms, we have simulated the PMDs of hydrogen atoms. It is found that the spider structure in PMDs exhibits sensitive dependence on CEPs of the few-cycle pulses. We employ three parameters to quantitatively characterize the variations of the spider features in the PMDs induced by altering the CEPs. In addition, we find that the momentum ratio curve shows a monotonic relation with the CEP value, which can be used to extract the CEP values. It is demonstrated that our proposed method is robust against changes in the laser pulse parameters. Our CEP characterization is doable in the visible and infrared spectral ranges for interrogating strong field dynamics.[27,28] We anticipate that this novel method may serve as an alternative tool for extracting CEP information encoded in the spider structure of PMDs. References Angular-dependent asymmetries of above-threshold ionization in a two-color laser fieldClassical-Quantum Correspondence for Above-Threshold IonizationDirect photodetachment of F − by mid-infrared few-cycle femtosecond laser pulsesPhotodetachment of F - by a few-cycle circularly polarized laser fieldAngularly resolved electron spectra of H − by few-cycle laser pulsesAbove-threshold ionization by few-cycle pulsesDependence of dynamic Lorentz frequency shift on carrier-envelope phase and including local field effectsCarrier-Envelope Phase Effects in Plasma-Based Electron Acceleration with Few-Cycle Laser PulsesWavelength scaling of high-order-harmonic-generation efficiency by few-cycle laser pulses: Influence of carrier-envelope phaseOptical Frequency Synthesizer for Precision SpectroscopySingle-shot measurement of carrier-envelope phase changes by spectral interferometryAtomic transient recorderDirect Measurement of Light WavesAbsolute-phase phenomena in photoionization with few-cycle laser pulsesMeasurement of the Phase of Few-Cycle Laser PulsesDetermining the Carrier-Envelope Phase of Intense Few-Cycle Laser PulsesCarrier-Envelope-Phase Characterization for an Isolated Attosecond Pulse by Angular StreakingCharacterizing carrier envelope phase of an isolated attosecond pulse with annular photoionization momentum spectraDirect Visualization of Laser-Driven Electron Multiple Scattering and Tunneling Distance in Strong-Field IonizationNear-Forward Rescattering Photoelectron Holography in Strong-Field Ionization: Extraction of the Phase of the Scattering AmplitudeDetermination of the Ionization Time Using Attosecond Photoelectron InterferometryQuantum Interference of Glory Rescattering in Strong-Field Atomic IonizationElectron-nuclear dynamics of multiphoton H 2 + dissociative ionization in intense laser fieldsIntracycle and intercycle interferences in above-threshold ionization: The time gratingTime double-slit interferences in strong-field tunneling ionizationSelectivity of Electronic Coherence and Attosecond Ionization Delays in Strong-Field Double IonizationAttosecond Ionization and Tunneling Delay Time Measurements in Helium
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