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When exposed to an intense laser field, free electrons undergo a series of movements before returning to their ground state, releasing higher harmonics in the process. As the order of these harmonics increases within a strong laser field, their intensity varies. Initially, the energy decreases rapidly with each successive order. Then, the curve levels off, maintaining a relatively constant intensity for a period. Eventually, the intensity peaks at a value of Ip + 3.17Up, where Ip represents the internal energy of the free electron, and 3.17Up is the maximum kinetic energy achieved by the electron in the laser field (with Up being the average kinetic energy gained by the electron). Following this peak, the intensity drops sharply. While the time-dependent Schrödinger equation provides a theoretical explanation for higher harmonics, the "three-step classical model" offers a simpler, more intuitive approach for studying them. This has made the exploration of higher harmonics one of the fastest-growing areas in the study of interactions between atomic physics and laser fields.
To better understand higher harmonics, it's essential to grasp how free electrons move in a strong laser field. Without a laser field, the potential of free electrons follows a specific curve. When a strong laser field is applied, the potential takes on a linear form, creating a potential barrier when combined with the original curve. Under the influence of the laser field, free electrons tunnel through this barrier, initially losing all velocity upon exiting. They then become free electrons, accelerating under the laser field’s influence until the field reverses direction, causing them to move back toward the nucleus. Upon reaching the nucleus again, the electrons recombine and return to their ground state, simultaneously emitting higher harmonics to shed excess kinetic energy.
For simplicity, we consider atoms moving in one-dimensional space. With a laser field varying as E0 sin(ωt), free electrons are driven by the Lorentz force and move away from the nucleus under the laser field’s influence. However, as the field reverses, the electrons accelerate back towards the nucleus. When they reunite with the nucleus, they release higher harmonics to dissipate excess energy and return to their initial positions. This movement can be viewed as a classic vibration.
The displacement of both free electrons and nuclei follows the classical Newtonian equation of motion: m(d²x/dt²) = -E₀ cos(ωt) + F_bind(x)/m + γ(dx/dt), where x is the position, v is the velocity, and γ accounts for damping effects. Assuming t₀ marks the time when electrons and nuclei recombine, we can determine velocity and kinetic energy at this point. Further simplifications allow us to transform the energy-time relationship into one involving energy differences and time intervals. By solving equations derived from these relationships, it becomes evident that the motion of free electrons resembles that of a simple pendulum, oscillating back and forth around the nucleus.
Calculations reveal that the maximum kinetic energy attained by free electrons in a laser field is approximately 3.17Up. Peaks in the energy-time graph correspond to different displacements and energy states, with higher harmonics carrying varying amounts of energy depending on displacement. These peaks often resemble one another, forming a plateau-like structure in the harmonic spectrum.
This analysis confirms that free electrons achieve a maximum kinetic energy of 3.17Up in a strong laser field, with the resulting harmonic energy peaking at Ip + 3.17Up. These findings align with results obtained through numerical simulations using classical Newtonian mechanics, validating the simplified model and enhancing our comprehension of the physical processes underlying higher harmonics emission.