Self-Focusing

The self-focusing (SF) effect is the convergence of a high intensity laser beam during its propagation in a dielectric medium, caused by the change of the optical properties of this medium, under the influence of the intensity of the beam in question. It is responsible for a dramatic increase in the local intensity during propagation. SF was discovered by observing the damage induced to optical devices by very powerful laser pulses. It is better observed in materials with high Kerr non-linearity, and exhibits an intensity threshold dependent on the type of material.

Historically, the experimental study of SF was carried out using pulsed lasers, which offer a much higher power than the continuous lasers. Fundamentally, the phenomenon depends only on the intensity of the light and its spatial distribution, as well as on the non-linearity of the material. Al-Saidi observed SF of a continuous laser beam at 632.8nm, in CS₂ [10].

For a short historic review of the self-focusing effect see my theses part 1 and part 2 (written in French).

Explanation of the phenomenon

In the nonlinear regime, a high intensity laser beam influences its own propagation by modifying the optical properties of the medium through witch it propagates. The index of refraction n under these conditions can be written as:

n = no + Dn|E|²

Where no is the index of normal refraction, and Dn|E|² is called the nonlinear index of refraction (a parameter that depends on intensity, it represents the beam’s influence exerted on material).

This expression defines what is commonly called the optical Kerr effect. To explain this change of refraction index (Dn|E|²), we must invoke mechanisms that can operate locally at the time of the interaction between the beam of light and the material. The most important ones are: libration, reorientation and redistribution of molecules, electrostriction, deformation of the electronic cloud (or polarizability), and local heating. The dependence on intensity, and on wavelength of these mechanisms varies with the type of material. Moreover, each one of them possesses its own characteristic reaction time, which is also typical to the material. The reaction time becomes very important in the case of pulsed laser beams, when the optoelectrical field is applied during a limited time, roughly equal to the pulse duration. For a given set of values of wavelength, pulse duration, and intensity, only some of these mechanisms will be dominant at the time. For example, in the fs time regime the deformation of the electronic cloud, which has a response time of about 10^-16s, is by far the most significant for the optical Kerr effect.

Depending on the material, Dn can carry a negative or a positive sign. For the latter, the optical Kerr effect can cause SF, witch is accompanied by a dramatic increase in intensity, that can lead, in some circumstances, to ionization and dissociation (or photolysis) of atoms and molecules composing the medium. The spatial aspects of SF give the spatial distribution of the new species generated by photolysis.

Haw does an index of refraction having the form that appears in the equation above give an effect of convergence to the light beam? In essence, the radial distribution of the beam’s intensity modulates the radial distribution of the refraction index. This modulation of the refraction index translates further into a radial modulation of the speed of light, and ultimately into a radial phase modulation. For a Gaussian radial intensity distribution, the refraction index becomes smaller as we go from the center of the beam towards its edge. Consequently, the light on the edge of the beam travels faster that the light in the center, introducing a radial phase modulation. As a result, the beam converges after a certain distance of propagation within the material. The effect of SF forces the light beam to dimensions of only a few microns, generating a very high local intensity. In practice it is very difficult to achieve such tight convergence for low intensity beams, using conventional means such as lens, or parabolic mirrors.

Aspects of self-focusing and filamentary ionization

In reality, a laser beam has a global Gaussian space intensity distribution, but it can also manifest all sorts of irregularities. Let’s take for instance a laser pulse, and imagine that we look at a cross section (at a plan perpendicular to the propagation axes that cats through the laser pulse), as it travels through a dielectric medium. As the light propagates, the radial intensity irregularities are accentuated. The light converges towards the peaks with intensity above the SF threshold, thus forming small areas of high intensity levels. It seems that each peak with intensity higher then SF threshold undergoes its own SF, independent on the other peaks. Figure 1 shows the images of three plans perpendicular to the beam’s axis of propagation, at different positions on the trajectory. It illustrates very clearly how the radial distribution of the intensity "brakes into pieces", thus giving birth gradually to very well defined "small islands", where the intensity reaches, in certain cases, values beyond the threshold of photoionization. These areas of high intensity have a complex dynamic with reference to the coordinates attached to the laser pulse. This dynamic depends on the intensity, as well as on the ratio between the pulse duration and the reaction time of the Kerr mechanisms (mechanisms responsible for the Kerr effect).

As the zones of high intensity advance through the material, part of the energy stored in the light pulse is dissipated through the various mechanisms mentioned above. The dissipative effects can reach equilibrium with the continuous energy flow, from the edges towards the center, maintained by SF. As the energy is progressively depleted, the intensity decreases below the SF threshold, the propagation becomes linear, and the light diverges. Experiments show that the core of maximum intensity persists a certain distance (Figure 2), which depends on the total energy stored around it, and on the efficiency of the dissipative mechanisms. This is how the length of light filaments, and equivalently the length of fossil filaments, is explained. They begin when the core is formed, that is when the maximum intensity value is reached, and they end where the local surrounding energy is depleted, and cannot sustain SF anymore.

The diameter of these filaments appears to be independent of the entry intensity of the laser pulse, but it depends on the characteristics of the medium. Experiments carried out in a mixture of two different liquids showed that by continuously changing the proportions of the mixture, from one extreme to the other, the diameter of the filaments changes in a continuous way, from the characteristic value of one liquid to that of the other [8, 15].

Can diffraction limit the range of SF? It is well known that a beam of finite size undergoes diffraction, which has a defocusing effect, contrary to SF. The effect of diffraction is inversely proportional to the square of the beam’s ray (µ1/r2). In alkalis vapors, researchers observed filamentary traces of light having dimensions limited by diffraction [11]. On the other hand, in water, in the fs time regime, and at 800nm, the two effects should counterbalance each other at a size of approximately 2 microns. Experiments show that the minimum diameter of filaments in water created by an 800nm beam is around 10 microns [12-14]. In conclusion, for high-pressure gases and condensed materials, the SF is stopped before the diffraction can have an important influence.

For extremely high intensity levels the Kerr effect is stronger, and the dissipating mechanisms might not be able to balance SF. The beam is collapsed even more, the intensity can reach higher values, and OB can occur. For powers beyond the OB threshold nothing interesting happens, other then the increase of the area occupied by the hot plasma generated, as well as an increase in local temperature, and of the amplitude of the shockwave produced by the Coulombian explosion (read the section on optical breakdown mode OB). Let’s see in more detail what happens at powers above SF threshold and below OB threshold, for various time regimes, and various types of materials.

We mentioned above that some type of intensity dependent processes stop the SF effect, at a level witch depends on the type of material, and on the optical characteristics of the laser beam. Experiments made in atomic gases, with powers below OB threshold, show that in certain cases the saturation of the index of refraction is the cause of the filament diameter limitation [reference]. On the other hand, in the case of molecular gases, the experiments show that SF stops before the saturation of the index of refraction [reference]. The cause invoked in this case is the ionization of molecules under the action of the intense optoelectric field. The accumulation of free charge modifies the value of the refraction index in a manner contrary to the Kerr effect. For a given intensity level, the plasma becomes sufficiently dense and counterbalances the Kerr effect, thus stopping the beam’s convergence. This was first proposed in 1972 by Eli Yablonovith and N. Bloembergen [17, 18].

In condensed phase, SF stops well before the saturation of the index of refraction. In this case, the molecules interact more strongly with one another, introducing even more degrees of freedom into their interaction with the laser pulse. In the ns time regime, experiments carried out in liquid CS₂ show that the mechanisms that stop SF are the stimulated Raman and Brillouin emissions [8]. It is worth mentioning that in this time regime processes that require atomic nucleus motion can occur.

In the sub-picosecond regime, for powers below OB threshold, Bluembergen [17] proposed photonization (avalanche ionization - AI) as SF stopping mechanism in condensed media. Later, experiments in water and air (for pressures around 1 atm), in the fs time regime at 800nm, made by S.L. Chin reinforced the claims that photoionization is at the origin of SF stopping. He invokes MPI and AI as ionization mechanisms. His conclusions are supported by a combination of experimental results and simulations.

For a better understanding of the mechanism behind the stabilization of SF, we must look at the two parameters that play the most important part: duration and intensity of the laser pulse. The duration of the pulse is important because any physical process takes a finite time to respond to a stimulation, to develop, and then to relax. With pulsed lasers, we can produce light pulses of duration comparable to the ultra-fast characteristic reaction time of the physical mechanisms responsible for optical effects. We cannot consider the response time of the material as being instantaneous anymore. Thus, the ratio between the pulse duration and the medium’s characteristic response time becomes an important parameter in the interpretation of experimental results. Moreover, for a given time regime, the intensity of the light will influence different processes in a different way.

In a condensed medium, in the sub-picosecond time regime, and for powers below OB threshold, the photoionisation plays a crucial role in the stabilization of SF. Looking at the intensity irregularities on a transverse cross section of a light pulse, above SF threshold peaks individually undertake SF. The Kerr effect is sustained by the very fast polarizability of the medium, and this mechanism conveys SF a given "strength". Slow responding balancing mechanisms such as Raman or Brilouin are not active in this time regime. Other short-time responding mechanisms are induced by the increase in intensity, and their balancing effects increase as the intensity becomes higher. It turns out that photoionization is the dominant one in these conditions. The density of charge generated modifies the refraction index contrary to the Kerr effect, eventually establishing equilibrium, and stopping farther SF. The maximum convergence of the light, and equivalently the maximum intensity level, is reached whenever these two processes counterbalance each other. At this point the core is formed. Following along this SF zone, the energy dissipated by ionization in the core is replaced by the energy influx maintained by SF. Whenever the energy stored in the surroundings of the core is depleted the core disappears. The volume in space traced by the core forms the light filament and its associated fossil filament. Viewed only a few fs after the passage of the light pulse, the fossil filament is, in this case, a partially ionized region. As time passes, other physico-chemical relaxation processes take place, and the situation migrates towards a new equilibrium. Some transformations can be permanent. In water, studies conducted by A. Brodeur and S.L. Chin [19] show that the density of free electrons should be about 10¹⁷ electrons/cm^-3 in order to balance the SF effect. They propose MPI and AI as mechanisms of ionization responsible for the accumulation of the free charge. Models proposed later by S.L. Chin et al. describe very well the SF process, taking into account not only the contribution of the plasma to the refraction index, but also the diffraction, and the group velocity dispersion (the latter being important in the sub-ps time regime) [reference]. It predicts with a very good empirical adequacy: the evolution of the space-time intensity distribution during propagation (formation of filaments, intensity maximum value, diameter and length of filaments, pulse time-spliting, self steepning), nonlinear optical effects (continuum generation), and the OB intensity threshold. The experimental results conducted by myself demonstrate for the first time the existence of ionization fossil filaments, having the characteristics predicted by Chin’s models on SF.

Non-linear optical processes

In the ns time regime, the equilibrium intensity is below AI and MPI threshold, and SF is counterbalanced mainly by the stimulated Raman and Brillouin emission. It is also important to recognize that these mechanisms have a limited dissipative efficiency. By gradually increasing the entry intensity of the ns laser pulse, the capacity of energy dissipation of Raman and Brillouin conversions can be exceeded. In this case, SF becomes stronger at higher intensities, and surpasses the counter-effect of Raman and Brillouin conversion, causing the further collapse of the beam. Eventually, the AI intensity threshold is reached. OB is observed at intensities lower then the threshold of other nonlinear optical processes. Thus, in the ns regime it is not possible to observe self-phase modulation, nor 2 to 4 photons mixing, for their intensity threshold is higher than OB threshold. It is therefore impossible to observe continuum generation.

As pulse duration is diminished, the contribution of AI to the plasma generation becomes less important. This raises the OB intensity threshold, making it more stable and more predictable. As OB intensity threshold becomes higher in the fs time regime, other optical non-linear effects can be observed, such as: self-phase modulation, multiphoton absorption, and 2-4 photons wave mixing. On the other hand, stimulated Raman emission decreases for powers just above the SF threshold, and becomes important only for very high powers. The Brillouin stimulated emission, disappears. The generation of continuum, which develops inside the core, is observed with an intensity threshold practically equal to that of SF. Being generated in the core, the continuum becomes more intense by increasing the intensity, as the filaments become longer, and as their number increases. Group velocity dispersion also becomes very important in the sub-ps time regime.

The nonlinear propagation also affects the temporal intensity distribution, inducing effects like self-steepening [20, 23, 24] and pulse splitting [21, 22].

Self-focusing, filamentation, and continuum generation

We spoke about continuum generation in the ps and fs time regime. A. Embroiderer et al. explored the link between continuum generation and SF. Their experimental results show that the power threshold for continuum generation and SF practically coincide. The continuum generation is easily detectable with the naked eye. It presents therefore a practical interest, as it becomes a sign for SF and filamentation.

The continuum is generated within filaments. As mentioned above, the intensity of the continuum increases with the length and the number of the filaments.

The spectrum of the continuum generation is also related to the maximum intensity reached inside the core; it can extend from the near infrared to the UV. The spatial color distribution of the continuum gives information on the core’s intensity distribution and diameter. The sampling and measurement of the continuum’s spectrum constitutes a convenient way to estimate and monitor these other important parameters.

References

[1] Interference of transverse rings in multifilamentation of powerful femtosecond laser pulses in air; S.L. Chin, S. Petit, W. Liu, A. Iwasaki, M.-C. Nadeau, V.P. Kandidov, O.G. Kosareva, K.Yu. Andrianov; Optics Communications 210 329–341 ; 15 September 2002

[2] Intensity clamping of a femtosecond laser pulse in condensed matter; W. Liu, S. Petit, A. Becker, N. Akozbek, C.M. Bowden, S.L. Chin; Optics Communications 202 (2002) 189–197

[3] Visualization of the evolution of multiple filaments in methanol; H. Schroeder, S.L. Chin; Optics Communications 234 (2004) 399–406

[4] Self-focusing and stimulated Raman and Brillouin scattering; Y. R. Shen, Y. J. Shaham; Phys. Rev., Vol. 163, No. 2, 10 November 1967

[5] Self-Focusing of laser light and interaction with stimulated scattering processes; M. Maier, G. Wendl, and W. Kaiser; Phys. Rev. Lett., Vol. 24, No. 8, 23 February 1970

[6] Explanation of Limiting Diameters of the Self-Focusing of Light; O. Rahn and M. Maier; Phys. Rev. Lett., Vol. 29, No. 9, 28 August 1972

[7] Backward stimulated light scattering and the limiting diameters of self-focused light beams; P. L. Kelley; Phys. Rev. A, Vol. 8, No. 1, July 1973

[8] Raman-limited beam diameters in self-focusing of laser light; O. Rahn and M. Maier; Phys. Rev. A, Vol. 9, No. 3, March 1974

[9] Avalanche Ionisation and the Limiting Diameter of Filaments Induced by Light Pulses in Transparent Media; Eli Yablonovith and N. Bloembergen; Phys. Rev. Lett., Vol. 29, No. 14, 907, 2 October 1972

[10] Laser-beam profiles in liquid CS₂: direct evidence for self-focusing of a laser beam in CS₂; I A Al-Saidi; J. Phys. D: Appl. Phys. 32 (1999) 874–875

[11] Self-focusing of light by potassium vapor; D. Grischkowsky; Phys. Rev. Lett., Vol. 24, pp. 866-869, 1970

[12] Beam filamentation and the white light continuum divergence; A. Bordeur, F. A. Ilkov, S. L. Chin; Optics Communications, Vol. 129, pp. 193-198, 1996

[13] Femtosecond laser pulse filamentation versus optical breakdown in H₂O; W. Liu, O. Kosareva, I.S. Golubtsov, A. Iwasaki, A. Becker, V.P. Kandidov, S.L. Chin; Appl. Phys. B, Vol. 76, pp. 215-229, 2003

[14] Ultrafast wihte-light continuum generation end self focusing in transparent condensed media; A. Brodeur, S.L. Chin; Jurnal of Optical Society of America B, Vol. 16, No. 4, pp. 637-650, 1999

[15] Self-Focusing: Experimental; Y. R. Shen; Progress in Quantum Electronics, Vol. 4, pp. 1-34, 1975

[17] Avalanche Ionisation and the Limiting Diameter of Filaments Induced by Light Pulses in Transparent Media; Eli Yablonovith, N. Bloembergen; Phys. Rev. Lett., Vol. 29, No. 14, 2 October 1972

[18] The influence of electron plasma formation on superbroadening in light filaments; N. Bloembergen; Optics Communications, Vol. 8, No. 4, August 1973

[19] Ultrafast wihte-light continuum generation end self focusing in transparent condensed media; A. Brodeur, S.L. Chin; J. Opt. Soc. Am. B 637, Vol. 16, No. 4, April 1999;

[20] Self-transformation of a powerful femtosecond laser pulse into a white-light laser pulse in bulk optical media (or supercontinuum generation); V.P. kandidov, O.G. Kosareva, I.S. Golubtsov, W. Liu, A. Becker, N. Akozbek, C.M. Bowden, S.L. Chin; Appl. Phys. B 77, 149–165 (2003)

[21] Observation of Pulse Splitting in Nonlinear Dispersive Media; Jinendra K. Ranka, Robert W. Schirmer, Alexander L. Gaeta; Phys. Rev. Lett., Vol. 77, No. 18, 28 October 1996

[22] Self-Guided Propagation of Ultrashort IR Laser Pulses in Fused Silica; S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz; A. Couairon, L. Bergé; Phys. Rev. Lett., Vol. 87, No. 21, 19 November 2001

[23] Catastrophic Collapse of Ultrashort Pulses; Alexander L. Gaeta; Phys. Rev. Lett., Vol. 84, No. 16, 17 April 2000

[24] Propagation Dynamics of Intense Femtosecond Pulses: Multiple Splittings, Coalescence, and Continuum Generation; Alex A. Zozulya, Scott A. Diddams, Amelia G. Van Engen, Tracy S. Clement; Phys. Rrev. Lett., Vol. 82, No. 7, 15 February 1999