The free electrons under the action of a strong laser field return to the ground state after three steps of movement, emitting higher harmonics. For high-order harmonics in a strong laser field, the intensity changes with the order. The energy decayed rapidly with the order in the previous stages, then the curve is basically a platform that has the same intensity, and then the curve reaches the maximum value, which is Ip + 3.17Up where / p is the internal energy of the free electron, 3. 17UP It is the maximum kinetic energy obtained by free electrons in the laser field (Up is the average kinetic energy obtained by free electrons in the laser field). Finally, the intensity decays rapidly with the order. Although theoretically, higher harmonics can be explained by the time-dependent Schrödinger equation, the "three-step classic model" proposes a simple and intuitive new method for studying higher harmonics. Now, the study of higher harmonics has become the fastest growing subject in the field of the interaction of physical atoms and laser fields.
In order to study the higher harmonics, it is necessary to understand the moving image of free electrons in a strong laser field (eg) If no laser field is added, the potential of free electrons is as in the curve 1. If a strong laser field is added, as shown in the figure At this moment, the potential of the strengthened laser field is like a straight line 2 and the curve 1 and the straight line 2 are superimposed, and the obtained potential like the curve 3 forms a potential barrier. At this time, the electrons are separated from the nucleus movement under the action of a strong laser field. Due to the tunneling effect, free electrons penetrate the barrier, which is the first step (such as line 4), and the velocity is zero when the barrier is penetrated. It then becomes a free electron and accelerates the cloud force under the action of a strong laser field until the direction of the laser field is opposite, and the free electron moves toward the nucleus again (such as curve 5), which is the second step. Finally, the free electron moves to the top of the nucleus, rejoins the nucleus, and returns to the ground state (such as line 6). At this time, the free electron will emit higher harmonics to take away excess kinetic energy.
To simplify the model, we discuss atoms moving in one-dimensional space. If the change of the laser field is E0) sw0f, the free electrons are driven by the Lorentz force and leave the nucleus under the control of the internal driving of the laser field at a general intensity. But after a while, the direction of the laser field is reversed, and free electrons are accelerated back to the nucleus. When free electrons meet the nucleus again, they can recombine and emit higher harmonics to take away excess energy and return to the internal position. We can regard the movement of free electrons as a completely classic vibration.
The displacement of the free electron and the nucleus follow the classic Newtonian equation of motion: = E0Cswt) t / + V / t ~ + XiVi and Xi are the internal velocity and position.
Suppose that at t, what we need is velocity and kinetic energy (kinetic energy when electrons and nuclei recombine). This is easy to determine. Below we discuss the movement of electrons in the X direction under the action of binding forces. In order to further simplify the model, we can make 0 = 1, so that the relationship between energy E and time t is transformed into the relationship between energy E0 and time difference t. Equation (1) can be obtained from equation (4) and equation (5). If the free electron returns to the nucleus at time t, then the displacement X is 0. From this condition, r = t-t0 is the return time. Equation (10) gives the relationship between the time t at which higher harmonics are emitted and the return time t.
From equation (9), we can get from equation (10), which can also be solved. It can be proved that the motion of free electrons is similar to the vibration of a single pendulum. Free electrons vibrate back and forth around the nucleus. When the displacement is zero, the velocity reaches The maximum value.
Through calculation, it is found that the maximum kinetic energy of free electrons in the laser field is about 3.17Up. In the relationship between energy and time, we can see that there is a peak with the largest energy, the corresponding energy is Ip + 3.17Up and the other peaks represent Different energies correspond to different displacement and energy possibilities. The higher harmonics emitted by free electrons all carry energy, the displacement is different, and the energy is not necessarily the same. The peaks are similar (similar), corresponding to a section of the platform on the higher harmonics.
Through the above derivation, it is found that the maximum kinetic energy obtained by free electrons moving in a strong laser field is 3.17Up. The higher harmonic energy released is Ip + 3.17Up. This is the conclusion drawn by the classical Newtonian equation of motion The results obtained by numerical simulation in the method are consistent. This verifies the correctness of using this simplified model, and thus deepens the understanding of the physical image that emits higher harmonics.
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