MPI and AI in the context of fs laser pulses

AI and MPI can operate simultaneously on a domain of intensity, wavelength, and pulse duration values. In the sections Multiphoton ionization (MPI), and Avalanche ionization (AI) I present each of these processes separately. Whenever they are active at the same time and at the same location, it becomes crucial to understand how they influence each other by sharing resources, and how their relative contribution to the plasma formation is influenced by the laser pulse parameters.

Whenever AI and MPI are active at the same time and location their relation is not a simple one. First, they are in competition, because the energy of a laser pulse is limited. Second, MPI feeds free electrons into AI, accelerating the avalanche. Third, the instantaneous ionization rate of each of these processes depends on the instantaneous local intensity, wavelength, and type of material. For a chirped pulse, or a pulse with a complex temporal intensity distribution, the instantaneous relative contribution to the plasma formation of AI and MPI varies over time. Although it becomes virtually impossible to treat formally real situations, experiments show that chirped multi-pick pulses can have interesting applications.

Usually simples models are used, where a non-chirped pulse with a Gaussian intensity temporal distribution (some times even a square function) is assumed. In this section we keep the problem simple, in order to better understand the interplay between AI and MPI in the context of pulsed laser applications.

There are two important laser parameters that we have to look at: the pulse duration and the wavelength. AI requires time to develop, whereas MPI responds in only a few fs. Experimental results show that in the ns time regime, AI is largely privileged over MPI. Moreover, AI is privileged at longer wavelengths, and MPI at shorter ones.

For intensities below OB threshold the evolution of free electron density is usually modeled using this simple rate equation:

The first two terms represent the MPI and AI contribution respectively, the last two terms account for electron losses, by diffusion out of the focal volume, and by recombination, respectively. The expressions for the first two terms are deduced form theoretical considerations, taking into account the conditions of a specific experiment, and introducing the appropriate approximations/simplifications. The theory of Keldish [1] is often used to account for the MPI ionization rate, which is rather adequate in the perturbation regime from UV to IR. Diffusion can be neglected in the fs time regime. Recombination can also be neglected in the the fs time regime, when plasma relaxation time is longer then the pulse duration, but it becomes important at very high plasma densities, usually generated at very high input power.

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Roughly, AI's ionization rate is proportional to the local intensity I, and the MPI ionization rate is proportional to I ^k (where k is the number of photons needed to be absorbed by the free electron, to reach the ionization threshold of the material).

The above approach is valid in situations where the plasma density doesn't pass the critical value. Once the OB threshold has been reached, the interaction between the plasma and the laser light becomes very important, and marks the transition into a different interaction regime. The energy stored in the remaining part of the laser pulse gets redistributed in a different manner over all active processes, and the interplay between AI and MPI is greatly affected.

The terms giving the MPI and AI ionization rates depend on the local and instantaneous intensity. Therefore, a temporal intensity distribution must be assumed, and plugged into equation (1). This simple approach gives the plasma density evolution during the pulse duration, which is an important parameter when it comes to the understanding of photolytic effects. But in order to get a more detailed descriptive picture one also needs to estimate the plasma energy density, and its variations in time and space. The nature of photolytic species depends on the ionization, excitation, and dissociation channels that are possible during, and just after the interaction. The nature and yield of the primary photolytic species can in principle be controlled by adjusting the wavelength, to select specific MPI processes, and the relative importance between AI and MPI; as well as the pulse duration, coupled with the chirp and the temporal intensity distribution (multi-pick pulse), to influence some long-term (above ps) occurring dissociation processes.

Photoionization plays a major part in material processing. But at the same time, there are other ionization/dissociation processes (for example: self ionization, dissociative ionization, electron attachment dissociation, photodissociation, etc.) that can also contribute to chemical and structural changes. How important are they? The observed optical effects induced to the ionizing laser pulse strongly depend on the charge density generated during the propagation through the media. The self-focusing effect is a good example: the optical Kerr effect and the accumulation of charge determine the propagation of a high power fs laser pulse in dielectric materials. As photoionization mechanisms, AI and MPI are sufficient to explain the most important optical features of these pulses during their non-linear propagation. In addition, they are sufficient to explain the spectrum of the supercontinuum generated [ref. S.L. Chin et al.] as well as the OB threshold [ref. S.L. Chin et al.]. The success of these predictions shows that our comprehension of AI and MPI is rather good, at least on the explored range of physical parameters. The other ionization/dissociation processes are probably expressing themselves at longer times (ps-ns), becoming more relevant at longer pulse durations.

Let's put all this in the context of photoionization modes. In the F mode, the intensity reached inside the medium attains a maximum value for a certain range of input intensities (the intensity is clamped). Filamentary ionization patterns are obtained, in which the plasma density is practically homogeneous, and remains at levels below OB threshold. Under these conditions, the relative contribution to the total ionization of the MPI and AI is fixed along a filament and for all filaments. In the case of B/OB mode the plasma density increases with the input intensity, and for a given wavelength and pulse duration, the relative contribution of AI and MPI might vary with the intensity. In the OB mode, the local intensity is limited by the effects of plasma on the propagation of the laser pulse (scattering, divergence, reflexion), by the high plasma absorption, and by plasma shielding.

Since MPI and AI don't have that same dependency on the pulse duration (or the time during which a certain volume of matter is shined by the laser ligth), whenever AI and MPI operate simultaneously, we can control their relative contribution to the total plasma density by changing the pulse duration.

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References

[1] Ionisation in the Field of a Strong Electromagnetic Wave; L. V. Keldish, Soviet Physics JETP Vol. 20, No. 5, May 1965

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