Theoretically, we can obtain the PMDs by solving the time-dependent Schr?dinger equation (TDSE); however, solving TDSE is difficult because of the microscopic process control to achieve the best nanoscopic physical picture. Therefore, the semiclassical theory becomes an alternative in numerical simulation. The distributed bouquet-like PMDs in momentum domains perpendicular to the laser polarization is relatively weak. However, this interference structure carries the characteristic information of the above-threshold ionization (ATI) and the inner-cycle interference (ICI) processes.

Results and Discussions We used the SFA and the saddle point approximation theory to simulate the bouquet-like PMDs of hydrogen atom ionization under different laser intensities. The light intensity range was 3.5×10¹³-7×10¹⁵ W/cm², as shown in Fig.1. The overall bouquet-shaped interference structure consists of various interference fringes: the ATI process-induced semicircle structure and the ICI process-induced partial circle structure. The interference of the electron wave packets generated by these two ionization processes results in the formation of bouquet-like PMDs. The coherent superposition between the ATI and ICI processes further turns these intersected circles into isolated island-shaped patterns.

The ATI interference fringes are concentric semicircles with centers at p_x=0 and p_y=0. As the perpendicular momentum p_y increases, these fringes become highly monotonically denser in the PMDs. When the PMDs are converted from the momentum space to the energy space, the ATI interference fringes become horizontal with equal spacing. This spacing is equivalent to the single photon energy (Fig.2). We derive a formula for the quantitative description of the ATI interference fringes valid over the laser intensity range of 3.5×10¹³-7×10¹⁵ W/cm².

Although the ICI interference fringes are also partial circles, these interference fringes center at the symmetric positions on the parallel momentum p_x axis. The ICI interference fringes are generated from the electrons ionized at the laser peak and valley times within the same optical cycle. The centers of the ICI interference fringes are located symmetrically in the positive and negative parts of the parallel momentum axis p_x. When the light intensity increases, the ICI interference fringes become highly denser. We also obtained a formula to describe the properties of the ICI interference fringes.

Additionally, we proposed an analog model of triple-slit interference in momentum space to better understand the formation of the bouquet-like PMDs. The wavefronts of electron waves that emanate from the middle slit form a set of concentric semicircles. These represent the ATI semicircles. The wavefronts of electron waves that ensue from the left and right slits form two symmetric concentric partial circles, representing the ICI partial circles. The intersections of these semicircles and partial circles are analogous to the interference maxima in the electronic bouquet-like PMDs (Fig. 4). Similarly, this intuitive model can help understand other interference processes in the strong field ionization of atoms and molecules.

In a stronger laser field, electrons move along different paths after ionization. This process involves two phases: first, the electron moves directly to the detector, and second, the ion scatters the other. Interference occurs when the electrons arrive at the detector with the same final momenta. Moreover, an interference structure in the photoelectron momentum distributions (PMDs) appeared. The interference structure encodes the dynamic information of electrons and the structural information of the parent ions, which are vital tools for studying the microscopic states of atoms and molecules in strong-field processes. Generally, these two kinds of ionized electrons are named reference electron: reaching the detector directly and being analogous to the reference light in laser holography, and the signal electron: returning to the parent ion and being scattered off by the ion and reaching the detector eventually, being analogous to the signal light in laser holography.

The study used the strong field approximation (SFA) theory and saddle point approximation method for investigations. Electrons are ionized at the peak of the laser field. In SFA, ionized electrons are affected only by the laser field, whereas the influence of parent ions is negligible. The final state of the ionized electron in the laser field was considered the Volkov state. It is simple to obtain the PMDs from the expression of the wave function amplitude of ionized electrons. We used the saddle point approximation method to calculate the bouquet-like PMDs. Moreover, we obtained the ionization time and rescattering time of the electrons and calculated the probability amplitudes by summing all the saddle points to obtain PMDs. Similarly, we carefully examined the bouquet-like PMDs produced by hydrogen atom ionization concerning laser intensity variations in the numerical investigation.

We performed an extensive numerical simulation of the bouquet-like PMDs induced by the stronger laser ionization of hydrogen atoms using SFA and saddle point approximation theories. The results show that the bouquet-like interference structure can be decomposed into semicircle-shaped interference fringes generated using the ATI process and partial-circle-shaped interference fringes obtained using the ICI process. We derived two analytical formulas using the classical action phase analysis to describe these two types of interference structures. These are valid over a relatively large laser intensity range. Furthermore, we proposed a physical picture of triple-slit interference in momentum space, which could help understand the bouquet-like PMDs, shedding light on studying electronic interference in strong field processes.