## Abstract

We have performed macroscopic calculations for the thin medium or low gas density regime and a central gas jet using microscopic numerical solutions of the time-dependent Schrödinger equation. In the case of a spatial phase distribution for broadband Gaussian pulses with a negative Porras factor, our theoretical results show an interference pattern in the angular distribution of below- and near-threshold harmonics, which is not present for the monochromatic Gouy phase distribution. The interference pattern is due to off-center contributions that are in-phase with those at the central points in the focus. The location of the maxima in the interference pattern can be estimated using the well-known double-slit formula with an effective slit separation.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

High-order harmonic generation (HHG) occurs due to the nonperturbative and highly nonlinear interaction of an atom, molecule, or solid with an intense laser pulse [1,2]. An intuitive physical picture of HHG is given by the three-step model [3–5]: The strong electric field of the laser suppresses the Coulomb barrier binding the electron to the residual ionic core, which permits the tunneling of an electron wave packet. The wave packet then propagates in the oscillating electric field and, depending on the time of release, it is steered back to the parent ion. With some probability, the wave packet recombines with the parent ion to the ground state and the energy absorbed by the electron during the propagation is emitted in the form of a high-energy photon.

Since nonlinear processes in ultrafast light-matter interaction, such as HHG, are induced at high laser peak intensities, they require the application of focused short-pulsed lasers with a broad frequency bandwidth. In contrast to a monochromatic Gaussian beam which experiences a $\pi$-phase shift across the focus, also commonly known as Gouy phase [6], broadband Gaussian pulses have a different spatial phase dependence. This more general phase distribution has been derived and studied by Porras and co-workers [7,8]. The exact distribution depends on the so-called Porras factor, which needs to be determined for a given laser system. It creates a spatially-dependent carrier-envelope phase (CEP), which is a parameter that controls many strong-field applications such as isolated attosecond pulse generation [9], laser driven electron dynamics in atoms, molecules, and nanostructures [10–13], or generation of relativistic electron beams [14]. The impact of the focal phase on electron backscattering at nanoscale metal tips [15] and photoelectron spectra [16] in few-cycle laser pulses has been demonstrated recently.

Since in high harmonic generation the electric fields, generated by atoms in the focus of the Gaussian laser pulse, are added coherently, one can expect to observe traces of the phase distribution in the radiation. Moreover, it can be expected that HHG spectra are more sensitive to the details of the phase distributions than the previously studied photoelectron spectra [16]. In particular, previous work has shown [17] that besides the often-considered phase matching along the propagation axis, phase matching in the transverse direction can also play an important role in HHG. The focal phase distribution, as derived by Porras and co-workers [7,8], gives the opportunity to further explore specific phase matching effects for a Gaussian beam. In this article we present results of simulations of macroscopic high harmonic generation, which indicate significant differences in the angular resolution of the emitted harmonics depending on the spatial phase distribution across the focus. Specifically, our results show a distinct interference pattern in the angular distribution for focal phase distributions with negative Porras factors.

The remainder of the paper is organized as follows: In section 2., we outline the numerical calculation of the microscopic HHG spectra and the macroscopic signals. For the single-atom response, ab-initio solutions of the time-dependent Schrödinger equation (TDSE) are obtained, which are then used in the macroscopic model simulations by applying interpolation of the TDSE results across intensity and carrier-envelope phase. Furthermore, we briefly discuss the spatial phase distribution for a broadband Gaussian laser pulse. In section 3., we first demonstrate the convergence of the macroscopic calculations for different spatial phase distribution. Then, we provide results for the angular distributions of the emitted radiation and discuss the interference pattern induced by the focal phase distribution. The paper ends with a brief summary.

## 2. Theory

In this section, we provide an overview of the theoretical and numerical methods used to obtain the microscopic and macroscopic HHG spectra. The differences in the spatial phase distribution for a monochromatic Gaussian (Gouy phase) and a broadband Gaussian laser beam (focal phase) are briefly discussed.

#### 2.1 Microscopic HHG calculations

Calculations of the microscopic single-atom response were performed by solving the time-dependent Schrödinger equation with Hamiltonian given by (Hartree atomic units are used: $e = \hbar = m = 1$ a.u.):

where $\textbf {E}$(t) is the electric field. In all calculations, the initial state was set to be the ground state of the hydrogen atom potential. We have taken the laser field to be linearly polarized in the $\hat y$-direction. To ensure that the electric field integrates to zero, the vector potential was set to where $A_0 = \frac {c\sqrt {I}}{\omega }$, $\tau = \frac {2\pi N}{\omega }$, $I$ is the peak intensity, $\phi$ is the carrier-envelope phase and $N$ is the number of cycles in the pulse (full width). The electric field has been obtained asThe wavefunction has been expanded in 30 spherical harmonics and the radial part of the wavefunction and the potential have been discretized utilizing fourth order finite difference on a radial grid with spacing $dr = 0.2$ a.u and grid sizes up to $r_{max} = 100$ a.u. [18]. For the solution of the TDSE, we have used the Crank-Nicolson method to propagate the wavefunction starting from the initial state with time step $dt=0.1$. As absorbing boundary, we have used exterior complex scaling (ECS), where the edge of the grid (10%) is rotated into complex space by an angle $\eta =\pi /4$. To obtain the HHG spectra, the dipole acceleration $a(t)$ has been evaluated using the Ehrenfest theorem

Finally, the complex harmonic response $a(\omega )$ is then obtained by taking the Fourier transform of the dipole acceleration,

and the harmonic spectrum has been calculated as $P(\omega )=|a(\omega )|^2$.#### 2.2 Macroscopic HHG model simulations

### 2.2.1 Summation of fields from point emitters

To determine the macroscopic radiation signal at a far-field detector, we consider the thin medium or low gas density regimes that are free from longitudinal phase-matching effects. We define the arrangement and coordinates as follows: The driving laser propagates in the $\hat {z}$-direction, has linear polarization $\hat {y}$, and we consider only radiation polarized in this direction. The far-field detector is an arc in the $xz$-plane with its position defined by the angle $\theta$ from $z$ (propagation) to $x$. We then follow the approach used in [19], in which the macroscopic yield is obtained as the superposition of the fields generated at different points in the medium. This approximation to the full Maxwell solution relies on the dipole approximation and the assumption that generated radiation does not interact with the medium. The spectral distributions of the total radiation generated by a number of atoms, located at $\textbf {r}_j$ ($j = 1, 2, 3,\ldots$), is given by [19]:

In Eq. (6), $a_{j}(\omega )$ is the dipole acceleration in frequency domain of a single atom responding to the driving laser pulse. Since this is calculated without considering the delay of the driving laser reaching the radiator, $(\textbf {r}_j \cdot \hat {\textbf z})/c$, the corresponding additional phase has been added. Furthermore, it has been assumed that the relative location of the detector, denoted by $\textbf {r}_d$, is far away from the individual atoms, i.e. $|\textbf {r}_j| \ll |\textbf {r}_d|$ and, hence, $|\textbf {r}_d-\textbf {r}_j|\approx |\textbf {r}_d|-\textbf {r}_j \cdot \hat {\textbf r}_d$ and $\frac {1}{|\textbf {r}_d-\textbf {r}_j|}\approx \frac {1}{|\textbf {r}_d|}$. Since the prefactor in Eq. (6) universally scales the results, we have dropped it in the computations.

### 2.2.2 Interpolation of single-atom TDSE results across intensity and CEP

Application of Eq. (6) to calculate the macroscopic response from atoms in a focused Gaussian laser beam requires single-atom TDSE solutions for a large number of intensities and CEPs (approximate methods such as the various forms of the strong-field approximation are not applicable to below-threshold HHG radiation). In the present simulations, $10^7$ single-atom results have been used to obtain each converged macroscopic HHG spectrum. In order to use ab-initio microscopic TDSE calculations, we have implemented an interpolation scheme. Exact TDSE results have been evaluated for a sample set over an intensity regime of two orders of magnitude. At intermediate values, the TDSE solutions have been approximated by cubic spline interpolation. In the present work, we have chosen to evaluate 100 exact TDSE results for a sample set of peak intensities that are equally spaced on a linear scale, though other spacing (e.g., logarithmic or Chebyshev, [20]) may be preferable in other cases. The success of the interpolation method can be seen from Fig. 1(a), in which we compare exact and interpolated spectra at a randomly-selected peak intensity of $9.1431 \times 10^{13}$ W/cm$^2$ driven by a 20-cycle laser pulse at 800 nm. The maximum absolute errors for these specific calculations are below $5 \times 10^{-7}$ for the harmonic yield. The effect of CEP $\phi$ on HHG spectral phase $\Phi (\omega )$ is well-approximated by [19]:

where $H(\omega )$ is the harmonic number rounded to the nearest odd integer. We have verified this approximation for the CEP via TDSE simulations in the regime of interest (see Fig. 1(b)).### 2.2.3 Geometry of laser focus and position of gas jet

Unless otherwise noted below, the spatial profile of the laser has been chosen to be a Gaussian beam with a beam waist of $W_0 = 30$ $\mu$m (Rayleigh length $z_R \approx 3500$ $\mu$m):

We have assumed a gas jet with a Gaussian density distribution along $z$, centered about $z_{\textrm {off}}$, a width of $\sigma _z = 800\; \mu$m $\approx 0.23 z_R$, and constant density along $x$ (the radial direction):

where the offset from the laser focus $z_{\textrm {off}}=0$ unless otherwise noted. We consider a low gas density such as $\rho _0=10^{18}$ cm$^{-3}$ – though our normalized results do not depend on the value of this overall scaling factor.#### 2.3 Focal phase distribution

The general form of the spatial phase distribution for broadband Gaussian laser pulses is given by [7,8]:

Figure 2 shows a comparison of the spatial phase profile for (a) $g_0 = 0$ (Gouy phase) and (b) $g_0 = -2$. We have focused on situations with negative Porras factor, which have been verified for the geometry of laser systems based on hollow-core fiber-compressors, but also used for Gaussian laser focus distributions before [16]. The specific value of $g_0 = -2$, chosen in the majority of our studies, has been reported in a recent measurement [15]. For negative values of $g_0$, the 2nd focal phase term has the same sign as the Gouy phase on-axis, but changes sign at certain radial distances, which depend on $g_0$ and $z$. This implies that for the total focal phase at each off-center position along the propagation distance, there are two points in the radial direction, symmetric over $r=0$, with $\Delta \Phi _F = 0$. Furthermore, these points are in-phase with all the central points at $z=0$ and for the location of the points $\lim _{z\to 0} r_{\textrm {zero}}(g_0, z) = W_0/\sqrt {2}$. Guided by previous work on the role of transverse phase matching [17], one may therefore expect that such a focal phase distribution imprints signatures, e.g., in the form of interference structures onto the angular distribution of the harmonic signals transverse to the propagation direction.

## 3. Results and discussions

In this section, we first compare on-axis HHG spectra for different focal phase distributions and demonstrate the convergence of the results as a function of the point emitters. We then test the expectation that the angular distributions of the harmonic radiation experiences interferences for the focal phase distribution.

#### 3.1 Dependence of macroscopic HHG on spatial phase distribution

In Fig. 3, we show how the higher harmonic radiation on-axis with a gas jet positioned at the center of the focus (i.e., $z_{\textrm {off}}=0$) depends on the macroscopic phase. The macroscopic result for a (non-physical) laser with no spatial phase variation (purple line) strongly suppresses most off-harmonic radiation, which is often of similar strength as the harmonic radiation in the single-atom result (black dashed line). Including the Gouy phase of a monochromatic Gaussian laser ($g_0 = 0$, red line), radiation is suppressed proportional to the harmonic order, but the structure of the off-harmonic radiation does not greatly differ. When the additional focal phase is included ($g_0 = -2$, green line), on-axis radiation is further suppressed for higher harmonics.

The results in Fig. 4 demonstrate the increased convergence requirements when including the focal phase ($g_0 = -2$, panel (b)) as compared to calculations with only the Gouy phase term ($g_0 = 0$, panel (a)). In case of the focal phase, orders of magnitude more emitters, especially for harmonics above the ionization threshold, are required. We attribute this to a combination of lower radiation intensity – requiring more emitters to bring the noise floor below regions of interest – and more rapid phase variation within the gas jet – requiring more emitters to well-sample the phase structure.

#### 3.2 Interferences in angular distributions

In Fig. 5, we present simulated angle-resolved macroscopic high harmonic spectra generated by a driving spatial Gaussian laser profile with a central wavelength of 800 nm ($\omega _0 = 0.057$ a.u.), a peak intensity of $7 \times 10^{13}$ W/cm$^2$, and a $\sin ^2$-envelope of 20 cycles full width. The gas jet was positioned at $z_{\textrm {off}}=0$. The results in the upper row of Fig. 5 have been obtained assuming focal phase distributions with (a) $g_0 = 0$ (Gouy phase distribution) and (b) $g_0 = -2$.

The comparison reveals a significant difference in the angular distribution of the below-threshold harmonic lines for the two phase distributions. The spectrum obtained with negative Porras factor clearly exhibits the expected interference pattern in the harmonic lines, while there is no such signature in the spectrum calculated for the Gouy phase distribution. The exact interference pattern depends on the details of the focal phase distributions. However, it turns out that the position of the side maxima in the interference pattern can be roughly approximated by the double-slit formula:

where $n$ is an integer, $L$ denotes an effective separation of the slits perpendicular to the laser propagation direction and $\lambda _{em}$ is the wavelength of the emitted radiation. The white lines in Fig. 5 show predictions for the side maxima, fitted to match the numerical data using $L = 0.925 L_0$. This effective slit separation is slightly smaller than the distance between the two radial positions $L_0 = \sqrt {2}W_0$ for $\Delta \Phi _F = 0$ near $z=0$ (see dashed lines in Fig. 2). Since the divergence angle of the emitted radiation and, for the focal phase distribution, the overall HHG signal (c.f., Fig. 3) quickly decreases with an increase of the harmonic number, the interference pattern can be best seen in the below- and near-threshold harmonics of the spectra. We further note that the off-harmonic radiation at energies in between the harmonics does not show an interference pattern. This is due to the incoherent nature of this radiation which results from fluorescence due to resonant excitation of the hydrogen atom [21].In the second row of Fig. 5, we present results of model calculations in which we deliberately removed the intrinsic HHG phase. Specifically, in Eq. (6), we replaced $a_j(\omega )\rightarrow \vert {a_j(\omega )}\vert e^{\phi H(\omega )}$. This retains the phase differences due to driving laser CEP and spatial phase distribution, while removing the intensity dependence of the spectral phase. We note that, in our numeric TDSE simulations, it is not possible to separately zero out the phase for different trajectories. In panel (c), HHG spectra from the Gouy phase distribution show significant differences compared to the results including intrinsic phase (panel (a)). However, we note that with respect to the aspect studied in this work, the spectra continue to lack angular nodes. For the focal phase distribution (panel (d)), angular nodes still appear in agreement with predictions. However, excluding the intrinsic phase has the effect of also producing nodes in the off-harmonic radiation. These results demonstrate that interference effects can be explained without consideration of intrinsic HHG phase.

A key factor for the occurrence of the interference effect is that for nonzero values of $g_0$, the phase shift equals zero at radial positions in the focus with $z \ne 0$. This raises the question of how the interference pattern depends on parameters of the set-up. Results of those studies are presented in Fig. 6. In the plots, we show how the positions of the first minimum (circles) and the first maximum (squares) in the interference pattern of the 7th harmonic depend on (a) the width of the central gas jet $\sigma _z$, (b) the Porras factor $g_0$ and (c) the beam waist $W_0$. Other parameters are kept the same as in Fig. 5(b).

The positions of the first two extrema remain almost unchanged for variations of the central gas jet width and the Porras factor. We note that an increase of the gas jet width leads to a decrease in the contrast between the maxima and minima in the interference pattern until for even larger jet widths, i.e. $\sigma _z > 0.5 z_R$, the pattern visually disappears. This is likely due to the enhanced impact of out-of-phase contributions from locations in between the nodal points which tend to destructively interfere with the contributions from points with $\Delta \Phi _F = 0$. Within the limits for the jet widths established in Fig. 6(a), the position of the nodal points does not change significantly along the propagation axis. This explains the stability of the extrema over the range of jet widths. The Porras factor $g_0$ controls the convexity (concavity) of the curve along which the nodal points appear as a function of $z$. However, near the center of the focus the change is not significant enough to impact the position of the extrema in the interference patters for $\sigma _z \approx 0.23 z_R$ (c.f., Fig. 6(b)). On the other hand for a variation of the laser beam width $W_0$ (Fig. 6(c)), the locations of the interference maxima approximately scale with $W_0^{-1}$ – as expected for the simplified interpretation based on the double-slit interference pattern. Small variations from the trend are seen at the smallest values of $W_0$. Since $z_R\propto W_0^2$, this can be understood again as likely due out-of-phase contributions from the relatively wide gas jet.

Finally in Fig. 7, we present results of HHG spectra for the focal phase distribution if the gas jet is positioned before the laser. For the position closer to the laser focus (Fig. 7(a), $z_{\textrm {off}}=-0.5z_R$), some variation is present in the angular distribution of the 9th and 11th harmonic with slight signatures of the first side maxima observed previously (see Fig. 5(b)). At $z_{\textrm {off}}=-1.0z_R$ (Fig. 7(b)), we do not observe any signature of the interference anymore. This shows the significance of the central points in the focus, which are in-phase with the two symmetric points in the radial directions at $z \ne 0$. We further note that the harmonic radiation at the off-center gas jet positions gets stronger, which likely indicates that best overall phase matching can be achieved at these positions. While it is not the focus of the present work, it may be interesting in future to exploit the phase matching conditions along the propagation axis for the focal phase distribution.

## 4. Summary

In summary, we have identified interference patterns in the angular distributions of below- and near-threshold harmonics using macroscopic numerical simulations for the thin medium or low gas density regime and a position of the gas jet at the center of the laser focus. The conclusions are based on numerical results which are obtained from ab-initio solutions of the time-dependent Schrödinger equation at the single-atom level and macroscopic model simulations assuming point-like field emitters. Interpolation across intensity and carrier-envelope phase is used to obtain the large number of microscopic HHG spectra to simulate the macroscopic response. The interference effects are found in the case of a spatial phase distribution for broadband Gaussian pulses with a negative Porras factor, while they are not present for the monochromatic Gouy phase distribution. Results of model calculations have shown that the intrinsic phase of the harmonics does not impact the appearance of the angular nodes in the interference pattern. Our analysis indicates that the interferences are due to off-center contributions which are in-phase with those at the central points in the focus.

## Funding

Air Force Office of Scientific Research (FA9550-16-1-0121); National Science Foundation (PHY-1734006).

## Acknowledgments

We thank Carlos Hernandez-Garcia for discussions concerning the implementation of the macroscopic HHG simulations.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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