Comparison between Jacobi-Anger and saddle point methods to treat Above-threshold ionization

TL;DR

This work assesses two analytical tools—the Jacobi-Anger (JA) expansion and the saddle-point (SP) method—for modeling Above-Threshold Ionization (ATI) under the strong-field approximation (SFA). By applying these methods to few-cycle, elliptically polarized laser pulses, the authors quantify dipole and nondipole effects on photoelectron momentum distributions (PMD) and ATI spectra, providing a clear map of each method's regime of validity: JA captures full interference structures via a spectral decomposition, while SP isolates dominant ionization pathways for computational efficiency. The study shows that nondipole contributions, especially at long wavelengths, induce forward momentum shifts along the laser propagation direction that are more accurately captured by JA than by SP at lower intensities. Collectively, the results offer practical guidelines for selecting theoretical approaches when interpreting strong-field ionization spectra and momentum distributions in both dipole and nondipole regimes, with implications for mid-infrared and attosecond-scale experiments.

Abstract

We present a detailed comparison of theoretical approaches for modeling strong-field ionization by few-cycle laser pulses. The dipole approximation is shown to accurately capture interference structures in photoelectron spectra, while non-dipole effects introduce significant momentum shifts along the propagation direction. Two complementary analytical methods are used: the Jacobi-Anger expansion provides complete spectral decomposition of transition amplitudes, whereas the saddle-point method efficiently identifies dominant ionization pathways. Through this comparative study within the strong-field approximation framework, we establish validity conditions and practical advantages for each approach. Our results provide guidelines for selecting theoretical methods for advancing the interpretation of strong-field processes. These findings provide a roadmap for interpreting strong-field ionization spectra and momentum distributions, highlighting where non-dipole effects and method choice critically alter predictions.

Figures (10)

• Figure 1: Time-domain representation of the vector potential $\boldsymbol{A}(t)$ for a circularly polarized laser pulse (\ref{['eq:VectorPotentialCompact']}) is shown in the left panels, while in the right panel its corresponding frequency-domain representation, $[\mathcal{F}(\boldsymbol{A})](\omega)$. The plots include laser pulses with durations of two (red), four (blue), and eight (green) optical cycles, with a carrier-envelope phase of $\varphi_{\mathrm{cep}} = 0$ and a wavelength of 800 nm. The right panel’s vertical axis represents the absolute amplitude of the vector potential at a peak intensity of $5 \times 10^{14}$ W/cm$^2$.
• Figure 2: Visualization of the saddle points, Volkov phase, and contour integration for a four-cycle circularly polarized laser pulse. (Top) The complex-time saddle points (yellow dots) and the corresponding contour of integration (white curves) are overlaid on a density plot of the action’s imaginary part. (Middle) The real part of the Volkov phase, $e^{iS(\boldsymbol{p}, t)}$, is plotted as a function of real time, showing rapid oscillations. (Bottom) The Volkov phase evaluated along the integration contour or the steepest-descent path, $e^{iS(\boldsymbol{p}, t_s)}$, where $t_s$ represents the complex saddle-point time. The integration smooths the oscillatory behavior, highlighting the dominant contributions to the ionization amplitude.
• Figure 3: Photoelectron momentum distributions in the laser polarization plane for ionization of an argon atom by a circularly polarized laser pulse with a wavelength of 800 nm and peak intensity of $5 \times 10^{14}$ W/cm$^2$. The top row (JA) presents results obtained using the Jacobi-Anger expansion, while the bottom row (SP) corresponds to the saddle-point method. Each column represents different pulse durations: two-cycle (left), four-cycle (middle), and eight-cycle (right) pulses. The color scale represents the normalized probability amplitude.
• Figure 4: Normalized ATI spectra comparing the saddle-point method (SP) (red dashed lines) and Jacobi-Anger expansion (JA) (blue solid lines) for an argon atom ionized by a laser pulse with a wavelength of 800 nm and peak intensity of $5 \times 10^{14}$ W/cm$^2$. The spectra are presented as a function of scaled photoelectron energy $\varepsilon_p / \omega$, with different pulse durations: 2 cycles (left), 4 cycles (middle), and 8 cycles (right). The upper row corresponds to a carrier-envelope phase of $\varphi_{\mathrm{cep}} = 0$, while the lower row shows results for $\varphi_{\mathrm{cep}} = \pi$. The laser propagation direction is chosen such that the polar angle is $90^\circ$ and the azimuthal angle is $0^\circ$.
• Figure 5: Vector potential for a two-cycle laser pulse at a wavelength of 800 nm, with a peak intensity of $5 \times 10^{14} \, \mathrm{W/cm}^2$ and an ionization potential $I_p = 15.7596 \, \mathrm{eV}$. The left panel shows the vector potential for a $\phi_{\mathrm{CEP}}$ of 0, and the right panel for $\phi_{\mathrm{CEP}} = \pi$. The yellow dots indicate the positions of saddle-point solutions relevant to the ionization events.