† Corresponding author. E-mail:
We ab initio investigate the interaction between the hydrogen atom and the inhomogeneous field which is induced by resonant plasmons within a metal nanostructure. Same as normal laser pulse (homogeneous field), only odd-harmonic generation occurs when the bow-tie nanostructure is utilized. For the single nanotip case, the even-harmonic generation can be distinctly found in the harmonic emission spectrum. By investigating the symmetry and trajectories of different inhomogeneous fields, we demonstrate that the breaking symmetry of system can enable even high harmonic generations.
Atoms and molecules driven by intense laser pulses radiate high-order harmonics generation (HHG) whose frequencies are integer multiples of the incident laser frequency. HHG is a promising method to produce a unique table-top source of coherent x-ray and attosecond pulses for ultrafast spectroscopy.[1] The physical mechanism behind HHG has been clearly declared by the classical three-step model.[2] Step one: the electron is liberated by tunneling from the potential barrier that built by the collaboration of atomic potential and the electric potential of laser field. Then step two, the electron oscillates in the laser field, and gains large amount of kinetic energy (from 1 to hundreds of photons) from the laser field. Finally, in the step three, the electron recombines with the parent ion and releases the gained energy by radiating a harmonic photon.
Symmetry plays an important role in the HHG of atomic and molecular systems. If the medium (atom or molecule) is inversion-symmetric, such as an atomic or non-oriented molecular gas, the suppression of even-harmonics has been certified,[3] only odd harmonics can be observed in experiments.[4–6] The absence of even-harmonics can be explained by the concept of inversion symmetry. In the quantum system with a center of inversion symmetry, the parity of the systemʼs eigenstates must be +1 or −1. In the view point of perturbation theory, the cause and mechanism of the n-th order HHG can be treated as follows: absorbing n photons of laser frequency and then releasing a harmonic photon. Since the transition between two states must be different parity, therefore, an even number of photons could not be absorbed by the system. This is why the atoms could not produce even-harmonics, i.e., even-harmonic generation is absolutely forbidden in atoms. More details by studying the symmetry properties to obtain the selection rules for even and odd harmonics can be found in Refs. [3] and [7]. In an ensemble of oriented diatomic molecules, except the homonuclear molecules, the inversion symmetry is broken. It has been predicted that the even-harmonic generation in the asymmetry molecules shined by laser field.[8–11] There are plenty of investigations on the selection rules[12,13] and applications[10,14,15] on the even harmonics of asymmetric molecules.
In recent years, people focus on HHG in a spatially inhomogeneous field[16–18] produced by a metal nanostructure when it is illuminated by a short laser pulse. By using such an inhomogeneous field, relying on the strong confinement of plasmonic hot spots, HHGs in nanostructures show a lot of novel characteristics, such as extending the harmonic cutoff energy[17,19–22] and increasing the harmonic yield.[23,24] Moreover, the isolated attosecond pulse has been proposed to generate by selecting the quantum path with inhomogeneous fields. Those characteristics just fit one of the most major branches of HHG studies. On the other hand, Ciappina et al. observed the generation of even-harmonics within inhomogeneous field of single-tip nanostructure.[25] To understand how the induced inhomogeneous field plays its role in even-harmonics generations, in this work, we performed a theoretically numerical analysis to investigate the HHG from a hydrogen atom in two different types of spatially inhomogeneous field. By comparing HHG spectra in two kinds of inhomogeneous field which generated by bow-tie and single-tip nanostructures respectively, we will discuss how the inhomogeneous field produces modifications in the harmonic periodicity, and reveal the mechanism of even-harmonics generations by laser-induced inhomogeneous field. By analyzing the HHG process using time-frequency and semiclassical calculations, we present strong evidence about the atomic even-harmonic generation due to symmetry-breaking effects induced by the spatially inhomogeneous field. Unless specifically stated otherwise, the atomic units (a.u.) are used throughout this paper.
To fully describe the HHG process in quantum mechanics, it is required to solve the three-dimensional (3D) time-dependent Schrödinger equation (TDSE). However, the previous investigation on simulations of one-dimensional (1D) and 3D TDSEs shows that the harmonic spectra and the positions of the cutoff are remarkably similar.[26] Therefore in this work, we will perform the simulation of HHG process from a hydrogen atom in a spatially inhomogeneous field by solving the 1D TDSE in length-gauge
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In this section, the effect of electric field inhomogeneity on the spectral profile of the high-order harmonic spectrum will be discussed. We have calculated the HHG spectra of hydrogen atoms emitted by two kinds of inhomogeneous field which generated by the bow-tie [Fig.
Figure
For both kinds of inhomogeneous field, it can be seen that an apparent peak–valley structure appears in the harmonic spectrum. To clarify the peak–valley structure and illustrate the dynamics from the harmonic platform in two inhomogeneous fields, we employed the wavelet transform of time-dependent induced dipole acceleration. First, for comparison, in Figs.
![]() | Fig. 2. Wavelet time-frequency spectra (absolute value) of the dipole acceleration and classical kinetic simulation. |
To illustrate the possible half-wave symmetry breaking, we have calculated the time-dependent probability to find the electron and analyzed the profiles of the classical trajectories in both cases. Figure
The HHG processes of hydrogen atoms under the interaction with two kinds of spatially inhomogeneous field were investigated by numerically solving the TDSE. These spatially inhomogeneous fields lie in a vicinity of the metal nanostructure when they are illuminated by ultrashort laser pulses. Due to the effect of inhomogeneity of induced field, the inhomogeneous field produced modifications in the harmonic periodicity. The odd and even-harmonics were both observed in the single nanotip case. The bow-tie nanostructure could produce only odd harmonics, it still follows the same symmetry rules as that of homogeneous field with atomic targets. By employing the classical equation of motion for an electron moving in a linearly polarized inhomogeneous electric field and time–frequency analysis, we demonstrated the origin of even HHG from the symmetry-breaking of the total atom–laser system.
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