75
Self-Focusing and Nonlinear Effects
Chapter 4
4. Self-Focusing and
Nonlinear Effects
Vitaly Gruzdev
4.1 Introduction
Scaling of rate of laser-induced eects with laser irradiance* (i.e., surface density of power;
W/cm
2
) is one of the fundamental features that control dynamics of laser–material interac-
tions. At low irradiance, the scaling is linear for most interactions: (1) zero irradiance pro-
duces zero rate, and (2) increase in irradiance by factor of K leads to increase in rate by factor
of K (Figure 4.1). ose properties of the linear scaling are mathematically expressed as follows:
RS
I
=
, (4.1)
where
R is the rate of an eect
I is the laser irradiance
Coecient S does not depend on irradiance (while it might depend on other laser
parameters, e.g., wavelength)
*
Many laser–material interactions, for example, laser damage, are characterized in terms of uence, that is, surface den-
sity of energy measured in J/cm
2
. Peak irradiance is related to peak uence via a coecient that depends on pulse dura-
tion only. erefore, linear or nonlinear scaling with irradiance means correspondingly linear or nonlinear scaling with
uence. Due to this fact, we discuss the nonlinear eects in terms of scaling with irradiance. For eects in bulk materi-
als, irradiance is also referred to as intensity following the recent trend in scientic publications.
Laser-Induced Damage in Optical Materials. Edited by Detlev Ristau © 2015 CRC Press/Taylor & Francis
Group, LLC. ISBN: 978-1-4398-7216-1.
4.1 Introduction ........................................................ 75
4.2 Overview of Elementary Laser-Induced Processes ...................... 77
4.3 Nonlinear Absorption ............................................... 84
4.3.1 Photo-Ionization and Multiphoton Absorption ...................... 85
4.3.2 Experimental Verications of the Photo-Ionization Models ..........102
4.3.3 Photoexcitation of Defects ......................................106
4.3.4 Nonlinear Effects due to Collision-Driven Electron Dynamics ........108
4.4 Nonlinear Propagation Effects ........................................116
4.4.1 Nonlinear Refraction and Self-Focusing ..........................116
4.4.2 Self-Induced Modications of Laser Spectrum ....................121
4.5 Summary .........................................................123
References .............................................................124
76
Laser-Induced Damage in Optical Materials
For example, the rate of light absorption scales according to Equation 4.1 at low irra-
diance and is therefore referred to as linear absorption (Ashcro and Mermin 1976,
Ridley1993).
Strong linear absorption is believed to be a major driving mechanism of laser-induced
damage (LID) of absorbing materials (e.g., metal surfaces of mirrors) (Anisimov etal.
1971, Ready 1971, Wood 2003). For those materials, LID occurs at surface at relatively
low laser irradiance that is not favorable to induce signicant nonlinear eects capable
of initiating and driving LID. In contrary, laser irradiance required to produce LID
of transparent materials (i.e., materials with negligible linear absorption at laser wave-
length) is high enough to induce various eects that nonlinearly scale with irradiance.
Among them, two major groups are processes of nonlinear absorption and eects of
nonlinear propagation (that are of special importance for bulk damage). eir rates
grow with irradiance faster than in linear case given by Equation 4.1 due to dependence
of coecient S on irradiance. For example, N-photon absorption scales as N-th power of
irradiance I
N
with N ≥ 2 (Figure 4.1), and rate of absorption due to tunneling ionization
exponentially scales with the square root of irradiance (Keldysh 1965).
1.41.2
Irradiance I
1.00.80.60.40.2
0
2
4
6
R
R
8
10
12
14
16
1.6 1.8
(a)
(b)
2.0
0.01
10
1
10
0
10
–1
10
–2
10
–3
10
–4
0.05 0.1 0.5
Irradiance I
1
FIGURE 4.1 Linear (solid line), power I
4
(dashed line), and threshold-type (dash-dotted line) scaling of
rate R of a laser-induced process with laser irradiance I. (a) depicts the scaling in regular axes while (b)
depicts the same scaling in log–log axes. Note that slope of the linear scaling is smaller than that of non-
linear power scaling in the log–log system shown in (b), while both are given by straight lines.
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