Avalanche Ionization (AI)

Avalanche ionization (AI), or cascade ionization, is an important photoionization mechanism that takes place during the interaction of high intensity laser pulses with condensed matter, or high-pressure gases. Y. R. Shen invoked this mechanism to explain the effects of ns laser pulses on materials. Kennedy et al. used an improved AI model to study the optical breakdown (OB) in water [1, 2, 17]. Chin et al. [3, 4] also took into consideration AI, along with MPI, in order to explain the self-focusing (SF) effect, and the spercontinuum generation in condensed media. Vogel et al. used a model similar to Chin's to describe what I call here the OB and B/OB ionization modes [5].

At the most simplistic level of explanation, AI is based on two important mechanisms: inverse bremsstrahlung, a three-body interaction, by which (quasi)-free electrons are accelerated, and impact ionization, a two-body collision by which the number of (quasi)-free electron population is multiplied. In condensed media, a quasi-free electron oscillates under the influence of the electromagnetic (EM) field generated by the laser light, and during its collisions with close-by atoms/molecules, it absorbs or looses energy. The case where energy is absorbed from the EM field is called inverse bremsstrahlung effect. At high intensities, the electron gains on average more energy than it loses during collisions. Accumulating energy over time, it might overpass the ionization threshold (or the threshold of valence-conduction band excitation) of the material, and during a fortunate collision it can create a second-generation quasi-free electron. This process is called impact ionization. The second-generation quasi-free electrons created in this manner can gain energy in the same way, and ionize other atoms or molecules. This process can be repeated as much as the EM field "seen" by the quasi-free electrons remains intense. AI is in fact a free electron amplifying process, which starts on a small initial population to generate high density plasma, up to 10²² electrons/cm³. The (quasi-)free electrons on which the avalanche ionization starts are called background electrons, and are normally created by thermo-ionic ionizations. They can also be induced by MPI or SPI, if the intensity, and/or the frequency of the laser pulse are appropriate. It is very common to see AI being considered along with MPI, as during the interaction of high intensity laser pulses with matter they both operate simultaneously, and influence each other, on a wide range of wavelengths and pulse durations. I prepared a separate section where I discuss AI and MPI in the context of laser pulses. Here we try to isolate some important characteristics that are specific to AI.

In order for AI to be initiated and to develop, a minimum density of matter is necessary. In other words, the effectiveness of this process requires that the collisions between (quasi-)free electrons and the surrounding atoms/molecules to be rather frequent. Early studies of laser-induced breakdown in gases [7, 8] show how this effect depends on gas pressure, and on the local intensity of the laser light. This, along with reference [9], represents probably the strongest empirical evidence of the avalanche process. AI is important in condensed phase.

Control parameters

In this section we discuss the impact of laser parameters on the avalanche ionization. Let's imagine a laser pulse traveling through a dielectric material. We assume an initial local population of (quasi-)free electrons, and we don't ask further questions about how they got there. These electrons oscillate in the EM field and can undergo the following processes:

• electron-phonon collision (e-ph), and inverse bremsstrahlung: if no EM field is present, the first interaction leads to energy loss, and works towards thermal equilibrium of the (quasi-)free electron population. In the presence of an EM field, there is a probability for a three-body collision, a photon being the third particle, where energy is absorbed by the (quasi-)free electron. This is the inverse bremsstrahlung effect, which leads to the acceleration of (quasi-)free electrons.

• electron-electron collision (e-e): it works toward thermal equilibrium of the (quasi-)free-electron gas.

• recombination: it makes a (quasi-)free electron "disappear"; the recombination rate becomes important only at very high ionization densities.

• impact ionization: it leads to energy loss, and to the creation of another (quasi-)free electron.

Figure 1 depicts the process of avalanche, taking into consideration only inverse bremsstrahlung, and impact ionization events.

In order to give a dynamic microscopic picture of AI, one must take into consideration all abovementioned events, including (see reference [6] for a theoretical treatment):

• their respective cross sections: for all interaction-events mentioned above.

• the energy distribution of all physical entities: instantaneous (quasi-)free electron spectrum [electrons-electron, electron-phonon, and photon-electron-phonon - or inverse bremsstrahlung - collisions "continuously" modify the (quasi-)free electron energy], instantaneous phonon occupancy distribution [energy is "continuously" transfered from the optoelectric field to the lattice by electron-phonon collisions and by direct photon absorption in the visible-IR spectral region], electronic structure of the material [band-gap, valance, vibrational states, etc].

• all relevant time-parameters: effective electron-atom/molecule collision-time [depends on average electron free path, field strength, and collision cross section], effective (impact)ionization-time [depends on the effective electron-atom/molecule collision-time, laser wavelength, and impact-ionization cross section], pulse duration.

• all relevant spatial characteristics: material density, material structure/lattice, spatial intensity distribution and laser polarization state, spatial plasma distribution and plasma shielding.

A complete formal description of AI is a very complex problem, and models can get very "ugly" very rapidly. Like in any other field of applied physics, the secret lies in our ability to recognize unimportant features, which can be safely neglected, given certain experimental conditions. It is also worth mentioning that the AI process is far from being an equilibrium process, and it is a big mistake to be treated as so. Energy is continuously transfered from the laser pulse into the material system through different channels, each having its own rate.

It is also important to note that in the case where MPI operates along with AI, the contribution of MPI to plasma generation must also be considered, in order to account for the final plasma density reached at the end of the laser pulse. The MPI input becomes part of the AI process, as it provides new seed electrons at any given moment during the time interval where the local intensity is higher then MPI's threshold.

In essence, for our own interests a good AI model must provide the evolution in time of the plasma density, and the free electron energy distribution, at all times during the laser pulse. Let's see now hot different laser parameters affect the AI process.

Average pulse intensity

If we try to describe AI in terms of intensity variations, we have to consider the local intensity "seen" by the atoms and molecules. The plasma build-up has a profound impact on the propagation of the laser pulse. The processes involved are scattering, reflexion, defocussing, and strong absorption (for plasma densities above the critical value). This means that during actual experiments, the input average intensity must always be scaled in order to obtain the local value. It turns out that the scaling factor depends on the pulse duration: in the sub-ps time regime, the local intensity it is much higher then in the ns time regime, for the same input. Moreover, it is observed that the intensity within the material reaches a saturation value. This means that no matter how much we increase the input intensity, the local intensity within the material will not increase further. This is true for F and OB photoionization modes. In the F mode, the saturation intensity (intensity clamping) is controlled by the equilibrium between the non-linear Kerr effect ,and the plasma effect on the propagation of the laser light. In the OB mode it is controlled by the plasma shielding effect. The B/OB and SP modes have a max intensity limit imposed by our categorization scheme: there is an intensity value for which we pass from the B/OB interaction regime to the OB, and another one for which we pass from SP to B/OB. In short, we have to consider the scaling factor between the input intensity and the local intensity, and the fact that the local intensity domain possesses an upper limit.

Having said that, the intensity levels can be so high in the sub-ps time regime, that the local physical properties of the material are drastically modified. Thus, the potential barrier necessary to promote an electron from its valance bond to the conduction band is reduced, or even eliminated. This is the reason why the effective ionization potential must be taken into account. In short, the whole dynamics of the AI is modified by the average intensity in the strong field regime, where the local band-gap of the material is greatly reduced. This perturbation can be expressed in terms of band-gap as follows:

where e is the electron charge, w and F are the frequency and amplitude of the EM field, and 1/m is the exciton reduced mass.

Moreover, for a given wavelength, the photon-electron-phonon collision depends on the local field strength. Consequently, the average local intensity is an important parameter, since it affects the ionization rate.

Pulse wavelength

The wavelength can affect the AI process for two fundamental reasons:

• photon-electron-phonon cross section: the probability of an inverse bremsstrahlung event varies with the laser frequency as (1/w)⁴, it is much higher in the visible-IR domain then in the UV.

• impact ionization energy vs photon energy: the number of absorbing collisions (inverse bremsstrahlung events) necessary for the (quasi-)free electron to achieve the impact ionization level depends on the energy of each photon absorbed, and on the effective band-gap of the material. Hence, in the case of liquid water (effective ionization potential = 7.3eV [5]) for a fs laser pulse at 800nm (1.55eV/photon) it takes eight inverse bremsstrahlung events to reach the kinetic energy required for impact ionization of 10.95eV [5].

In the case where MPI is coupled to AI (MPI feeding free-electrons into the avalanche process) the wavelength can become a very influential parameter, as MPI is very wavelength dependent. Moreover, for pulse durations where AI and MPI are in competition, AI is stronger at longer wavelengths, and MPI processes become dominant in the visible-UV domain. This behavior can be explained by taking into consideration the fact that the cross section of the inverse bremsstrahlung effect increases at longer wavelengths, as mentioned above.

If we consider real applications we also have to take into account the fact that the local intensity presents a saturation value, which is related to the plasma density. For a certain pulse durations domain, AI and MPI are in direct competition, and the comparison between their relative efficiency must be considered only on a finite intensity domain: from below the photoionization threshold, to the saturation value. In the UV domain, two-photon processes can be very efficient at intensity values lower that AI's threshold. In this case MPI's contribution to the plasma build-up can becomes dominant, the plasma density can be efficiently raised by MPI until the intensity increase is saturated, hence not giving the opportunity to AI to fully express itself. See section AI and MPI in the context of laser pulses.

Wavelength time distribution (chirp)

This parameter may be more important for pulses in the ns time regime. AI takes time to develop, and we know that it depends on the wavelength. If the pulse duration is longer, the chirp can have a detectable outcome on the effects generated by the avalanche. Perhaps, by controlling the chirp we can control the outcome of OB, keeping the other parameters of the laser pulse unchanged.

Pulse duration

AI requires time to develop: several inverse bremsstrahlung events are necessary for the electrons to increase their energy above the ionization threshold, and time is required for the population of free electrons to grow (see Figure 1). Studies suggest that for a laser pulse duration shorter than 25fs the contribution of AI can be neglected [6]. For example, in condensed phase the time between two collisions is in the order of 1fs [5]. For each inverse bremsstrahlung event the electron absorbs the energy of one photon. Therefore, a number of collisions equal to the integer part of E_ion/E_photon is required for the electrons to reach the ionization energy. In the case of liquid water (with effective ionization potential = 7.3eV [5]) for a fs laser pulse, at 800nm (1.55eV/photon), it takes eight fortunate collisions to reach the kinetic energy required for impact ionization at 10.95eV [5]. At 1fs/collision, the doubling time is 8fs.

Moreover, the energy spectrum of the (quasi-)free electron population is very dynamic. A few fs after the beginning of the laser pulse, the distribution is very heterogeneous, presenting picks at energy values equal to an integer multiple of the energy of one photon of light [6]. This is because (quasi-)free electrons are accelerated by the inverse bremsstrahlong process, by absorbing one photon at the time. As the time goes by, processes like electron-electron collision, and electron-phonon collision, tend to homogenize the energy spectrum. For example, for SiO₂ and an electric field of 150 MV/cm, it was proposed [6] that 10fs after the beginning of the laser pulse, the electron distribution becomes practically homogeneous. But this time is comparable with the doubling time, and we can argue that AI can be considered negligible. Even though in this case we cannot speak of an avalanche, because this would require more then one generation, nonetheless the fundamental mechanisms are at play. The energy spectrum of the (quasi-)free electrons population is relevant for the physico-chemical effects that follow the passage of the laser pulse, and ultimately it is related to the long therm effects in terms of chemical and structural transformations of the material.

Polarization

Experiments in the ns time regime, where AI's contribution to the plasma accumulation is greater, show that the OB intensity threshold in crystals depends strongly on the relative orientation between the polarization of the laser light and the lattice axes [10-16]. This is easily understood if we consider that for structured materials the mobility of (quasi-)free electrons becomes anisotropic.

Moreover, whenever MPI and AI operate in the same time, and at the same location, MPI can feed AI with free electrons. But MPI is also directly influenced by the polarization of the laser pulse, and in turn AI will be affected. See sections on MPI and AI and MPI in the context of laser pulses for more details.

Material impurities

Impurities, or defects in the case of crystals, can be seen as good electron donors, which could feed AI. A small difference in the initial concentration of free electrons can make a big difference in terms of plasma density at the end of a laser pulse.

Other important mechanisms

Usually AI competes for energy with other optical effects like:

• Brillouin conversion: important in the ns time regime

• Raman conversion: important in the ns time regime, as well as in the sub-ps time regime at very high intensities.

• Wave mixing, higher harmonics generation: more important in the sub-ps time regime

• Group velocity dispersion: important in the sub-ps time regime.

Competition also comes from other photoionization processes like SPI and MPI, and photodissociation processes (the latter being relevant in the ns time regime).

A simple model

When treating ionizing effects of laser pulse a simple rate equation is used to account for the avalanche ionization rate by electron, where some simplifications are made based on the local intensity level, pulse duration compared to relaxation and recombination times, and wavelength [17, 18]:

The first term is related to the energy gain of the electrons from the electric field, whereas the second term describes the energy transfer from the electrons to the heavy molecules during elastic collisions. DE is the band gap of the material, t is the free time between two consecutive collisions, w is the frequency of the laser, c is the vacuum speed of light, n is the refractive index, M is the mass of molecules composing the material, and I is the laser local intensity.

We also have to take into consideration that electron recombination also takes place in real situations. The probability of this event grows as the square of the plasma density. An dynamic equilibrium can be attained between AI and recombination, where plasma growth is saturated.

To be completed !

References

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[2] A First-Order Model for Computation of Laser-Induced Breakdown Treshold in Ocular and Aqueous Media : Part II-Comparaison to Experiment; Paul K. Kennedy, S. A. Boppart, Daniel X. Hammer, Benjamin A. Rockwell, Gary D. Noojin, and W. P. Roach, IEEE Journal of Quantum Electronics, Vol. 31, No. 12, 2250, December 1995

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[4] Band Gap Dependence of the ultrafast white-light continuum; A. Brodeur and S.L. Chin; Phys rev lett, Vol 80, No 20, 4406, 18 may 1998

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[12] Laser-induced damage in beta-barium metaborate; H. Nakatani, W. R. Bosenberg, L. K. Cheng, and C. L. Tang; Appl. Phys. Lett. 53 (26. 26 December 1988)

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[15] Bulk Laser Damage in CsLiB₆O₁₀ Crystal and Its dependence on Crystal Structure, M. Yoshimura, T. Kamimura, K. Murase, Y. Mori, H. Yoshidaand , M. Nakatsuka, T. Sasaki; Jpn. J. Appl. Phys. Vol. 38 (1999) pp. L129-L131, Part 2, No.2A, 1 February 1999

[16] Laser-induced damage in deuterated potassium dihydrogen phosphate; Alan K. Burnham, Michael Runkel, Michael D. Feit, Alexander M. Rubenchik, Randy L. Floyd, Teresa A. Land, Wigbert J. Siekhaus, and Ruth A. Hawley-Fedder; Applied Optics, 20 September 2003, Vol. 42, No. 27

[17] Theory and simulation on the threshold of water breakdown induced by focused ultra short laser pulses; Q. Feng, J.V. Moloney, A.C. Newell, E.M. Wright, K. Cook, P.K. Kennedy, D. X. Hammer, B.A. Rockwell, C. R. Thompson; IEEE, Journal of Quantum Electronics, Vol. 33, No. 2, February 1997.

[18] Laser-Induced Plasma Formation in Water at Nanosecond to Femtosecond Time Scales: Calculation of Thresholds, Absorption Coefficients, and Energy Density; Joachim Noack, Alfred Vogel; IEEE Journal of Quantum Electronics, Vol. 35, No. 8, August 1999