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The phenomenon of free electrons emitting higher harmonics under the influence of a powerful laser field has been extensively studied through a three-step movement process. Initially, these electrons are propelled into an excited state by the laser field, and upon completing their trajectory, they return to the ground state, releasing higher harmonics in the process. These high-order harmonics exhibit varying intensities depending on their order. In the early stages, the energy decays rapidly with increasing order, followed by a plateau phase where the intensity remains relatively constant. Eventually, the intensity peaks at a value of Ip + 3.17Up, where Ip represents the internal energy of the free electron and Up denotes the average kinetic energy gained in the laser field. Following this peak, the intensity diminishes sharply. While the time-dependent Schrödinger equation provides a theoretical explanation for this phenomenon, the "three-step classic model" offers a simpler, more intuitive approach to studying high-order harmonics. As a result, research in this area has become one of the fastest-growing fields within the study of atomic interactions with laser fields.
To investigate these higher harmonics, it's essential to visualize the movement of free electrons in a strong laser field. Without a laser field, the potential energy of free electrons follows a certain curve. However, once a strong laser field is introduced, the potential takes on a linear form, creating a potential barrier when combined with the original curve. Under the influence of the laser field, electrons detach from the nucleus due to the tunneling effect, passing through the barrier—this marks the first step. Afterward, the electrons transition into a free state and accelerate under the influence of the laser field, reversing direction when the field opposes their motion—this constitutes the second step. Ultimately, the electrons reunite with the nucleus and return to their ground state—this is the third and final step. During this process, the electrons release higher harmonics to shed excess kinetic energy.
For simplicity, we consider the motion of electrons in one-dimensional space. Given a laser field with a variation described by E0 sin(ωt), the electrons are initially driven by the Lorentz force and move away from the nucleus under the influence of the internal laser field dynamics. However, as the field reverses its direction, the electrons are accelerated back toward the nucleus. Upon reuniting with the nucleus, they recombine and emit higher harmonics to expel surplus energy, returning to their initial positions. This movement can be modeled as a purely classical vibration.
The displacement of both the electrons and the nucleus follows the classical Newtonian equation of motion: m(d²x/dt²) = E0 sin(ωt) - V'(x) + γ(dx/dt), where x represents the position, v the velocity, and γ accounts for damping effects. By setting certain parameters to unity, the relationship between energy and time can be transformed into a relationship between energy and time difference. From this, we derive equations describing the motion of electrons under binding forces. Assuming that the free electron returns to the nucleus at time t, the displacement X becomes zero. This allows us to define the return time r = t - t₀. Using this framework, we establish the connection between the time at which higher harmonics are emitted and the return time.
Through analysis, it is evident that the motion of free electrons resembles that of a single pendulum, oscillating back and forth around the nucleus. The maximum kinetic energy attained by electrons in such conditions is approximately 3.17Up. Peaks in the energy-time relationship correspond to specific energies, each representing different displacement-energy combinations. Higher harmonics emitted by electrons carry varying amounts of energy depending on their displacement. These peaks align closely, forming a plateau-like structure in the spectrum of higher harmonics.
Our findings indicate that the maximum kinetic energy achieved by electrons in a strong laser field is 3.17Up, and the energy of the emitted higher harmonics equals Ip + 3.17Up. These conclusions derived from classical Newtonian mechanics align with those obtained via numerical simulations, validating the use of this simplified model and enhancing our understanding of the physical processes involved in higher harmonic generation.