mechanism of self-focusing of laser pulses on dynamic lenses in transparent media

ISSN 1063�7842, Technical Physics, 2012, Vol. 57, No. 5, pp. 649–656. © Pleiades Publishing, Ltd., 2012.Original Russian Text © S.V. Gribin, B.I. Spesivtsev, 2012, published in Zhurnal Tekhnicheskoi Fiziki, 2012, Vol. 82, No. 5, pp. 71–77.

649

It is well known that in most cases, the experimen�tally determined threshold of optical breakdown oftransparent materials turns out to be significantlylower than the threshold computed theoretically withregard to the processes of impact ionization and mul�tiphoton absorption [1–5].

In spite of the fact that an interrelation between theoptical breakdown threshold and physicochemicalproperties of a transparent medium must undoubtedlyexist, such a dependence has not been established sofar. This points to a nontrivial dependence of thebreakdown threshold on a set of characteristics of atransparent medium [1–3, 6, 7].

A considerable number of experimental works indi�cates that absorbing irregularities (AI) play an impor�tant role in the process of destruction of transparentmaterials by laser pulsed radiation [2, 3, 5, 8–12]. Inthis case, theoretical models of breakdown of opticalmaterials due to light absorption by AIs assume thatthe breakdown occurs either due to heating of the AIsup to the ionization temperature of the medium or dueto the formation of compression and shock wavesaround the AIs, which were observed in a wide range ofexperimental works and which exceed in amplitudethe ultimate strength of the material, or due to bothfactors simultaneously [2, 3, 6]. However, a number ofeffects (such as the displacement of breakdowntowards a light beam with a velocity higher than thesound velocity [8] and the absence of a unique depen�dence of the breakdown threshold of optical glass onthe absorption coefficient [13]) have not beenexplained convincingly in terms of these models.

Relevance of this work consists in theoreticaldescription of variations of characteristics of a laserpulse and an optical medium as functions of the pulseparameters (duration, power, shapes of leading andtrailing edges, and pulse repetition frequency) andphysicochemical properties of the medium.

The proposed mechanism [14] does not repudiatethe known self�focusing mechanisms [1] based onnonlinear polarization of the medium and electros�triction, especially important for the media with agreat value of the Kerr constant and laser pulses with ahigh power density. The main advantage of the pro�posed self�focusing mechanism is that it is based onthe generally recognized fact that it is impossible toobtain optical media, which do not contain micro�scopic absorbing impurities (graphite dust, metal con�tamination, and regions of segregation [2, 3, 15]). Thismechanism makes it possible to relate light resistanceof a wide class of traditional optical materials to theirphysicochemical properties and characteristics of alaser pulse. Understanding of these interrelationshipswill make it possible to purposefully work on the devel�opment of lasers generating pulses of a special shape,which are necessary for more effective destruction of agiven medium, as well as for free transmission throughit, and also for obtaining optical media, which are themost and the least transparent for a given laser pulse.

The proposed mechanism includes the followingbasic concepts.

(i) A transparent medium may contain the finest(0.01–10) μm AIs such as drops of metals, graphitedust, or microscopic regions of the medium contain�ing a higher�concentration absorbing impurity(including that at the molecular level).

(ii) If a powerful radiation pulse is supplied to anAI, the latter is quickly heated by several thousanddegrees (up to the plasma formation); this causes themotion of the heated region boundary with the forma�tion of a compression region in the surroundingmedium, which does not destroy the dielectric. Thisleads to a short�term change in the refractive index ofthe medium around the AIs. Such short�lived regionswith a modified refractive index will be called the

Mechanism of Self�Focusing of Laser Pulses on Dynamic Lensesin Transparent MediaS. V. Gribin and B. I. Spesivtsev*

St. Petersburg State Technological University of Plant Polymers, ul. Ivana Chernykh 4, St. Petersburg, 198095 Russia*e�mail: [email protected]

Received April 11, 2011; in final form, August 15, 2011

Abstract—We propose the physical and mathematical models for the mechanism explaining the reduction inexperimental breakdown thresholds of optical materials by self�focusing of radiation on dynamic irregulari�ties of the refractive index (dynamic lenses) as compared to avalanche ionization and multiphoton absorptionpredicted by theories.

DOI: 10.1134/S1063784212050143

OPTICS, QUANTUM ELECTRONICS

650

TECHNICAL PHYSICS Vol. 57 No. 5 2012

GRIBIN, SPESIVTSEV

dynamic irregularities of the refractive index ordynamic compression lenses (DCLs).

(iii) The type of the dynamic lens (focusing ordiverging) formed in this case depends on the proper�ties of the medium (Fig. 1).

A correct estimation of the distribution of theenergy flux density in the region behind an AI requiresa solution to two independent problems. At the firststage (we call it conditionally the hydrodynamicstage), this is a solution of the problem on absorptionof optical radiation by AIs and the determination of aspace–time distribution of the refractive index aroundthe AIs. The second (optical) stage presumes the com�putation of deformation of the laser radiation field interms of the given refractive index distributionin accordance with the laws of optics of inhomoge�neous media [16, 17]. A physical model required forthe description of the process under consideration isthe model of an ideal compressible fluid containing asmall spherical volume heated uniformly due toabsorption of optical radiation [18]. As an example ofthe application of the model, we assume that the laserenergy is absorbed in the region of fluid with a charac�teristic size R0 ~ 1 μm, laser radiation flux is spatiallyhomogeneous within the absorption region, and theexternal medium is isotropic. The laser pulse durationis tp ~ 10 ns, the energy per pulse is Ei ~ 5–15 J, and thetemperature in this region is T ~ 104 K, which corre�sponds to the region of the primary ionization of the

fluid molecules. In this case, the target is thermallysmall [6], the absorbed laser energy is distributed uni�formly over the target, and the interaction with theexternal medium takes place not because of molecularheat conduction, but due to mechanical processes ofpressure and velocity leveling at the boundary of theregion, which rapidly expands into the surroundingmedium. We disregard the role of radiative heat con�duction, assuming that the main portion of reradiatedenergy is absorbed in a thin layer near the targetboundary and again takes part in the total energy bal�ance.

At the first stage, we consider the simplest case(i.e., a spherical shape of the volume being heated).Mathematically, such a model is described by the sys�tem of hydrodynamic equations

(1)

which includes the continuity equation, the equationof motion, and the energy balance equation, where r isthe radial coordinate, t is the time, ρ is the density, P isthe pressure, u is the mass velocity of the particles, a isthe velocity of sound in the medium, g is the thermalenergy source due to the absorption of optical radia�tion (g(t) ≠ 0 only within the ionization sphere, i.e., forr < Rg); ρt, ut, and Pt are the first time derivatives of thedensity, mass velocity, and pressure; and ρr, ur, and Pr

are the derivatives with respect to the radial coordi�nate. The initial parameters in the unperturbedmedium for t = 0 are as follows: P = P0, ρ = ρ0, anda = a0; in the ionized sphere, these parameters are P,ρ, and a; and the mass velocities of the particles areabsent. At the boundary of the expanding sphere, thenatural conditions for the balance of pressures andmass velocities are fulfilled for r = Rg. The system isclosed by the equations of state P = P(ρ, T) for fluidmedium and P = Pg(ρ, T) for ionized vapor, where T isthe absolute temperature. The form of function g(t) isdetermined under the assumption that the entireenergy falling on the cross section of the ionizedsphere is instantaneously converted into heat:

(2)

where W(t) is the flux density of the pulsed radiationenergy at the boundary of the energy release zone,PT is the derivative with respect to temperature, andCv

is the specific heat capacity at constant volume.Coefficient α describes the physical properties of

the medium; for example, according to the Kuznetsovequation of state [19], for water we can take

(3)The problem under consideration can be signifi�

cantly simplified on the assumption that the fluid flowis isentropic [20]. For water medium (even in the caseof flows with shock waves), the assumption of the isen�

ρt uρr ρ ur 2u/r+( )+ + 0,=

ut uur 1/ρPr+ + 0,=

Pt uPr a2 ρt uρr+( )+ + g t( ),=

g t( ) 3/4αW t( )/Rg, α 1/ ρ0Cv

( )PT,= =

α 0.1 MPa/s( ) cm( )3/W( ).=

dn/dp > 0, in bulk and on surface

Initial AI Compression Plasma Damage

dn/dp > 0, in bulk

1

2

3

Fig. 1. Diagram of a DCL inside a transparent medium(1 and 3) and on its surface (2). 1 and 2—media withdn/dp > 0, focusing DCL; 3—medium with dn/dp < 0,diverging DCL.


MECHANISM OF SELF�FOCUSING OF LASER PULSES 651

tropic flow is valid up to pressures on the order of1000–2000 MPa. In this case, system (1) can be writ�ten in the form

(4)

where t is the time, r is the distance from the center ofthe absorbing region, u is the velocity of the particles,and a is the local velocity of sound. The relation ofquantity a with pressure P and density ρ of themedium is given in the form a2 = dP/dρ. DependenceP(ρ) will be taken in the form of Theta function [21]:

(5)

where β, k are constants characterizing the medium.For water, B = 304.5 MPa and k = 7.15. P0 and ρ0 arethe initial pressure and density, respectively. Conse�quently,

(6)

The initial conditions for system (4) are a = a0 andu = 0, where a0 = 1500 m/s. The boundary conditionson a sphere of radius r = Rg(t) are a = a0 + Δag(t),where Δag(t) is the excessive local velocity of sound inthe fluid at the boundary Rg(t) with the expandinghigh�heated region for r ∞, a a0, and u 0.Applying the method of “nonlinearity introduction”[22] to the formulated problem, we obtain the follow�ing solution (in parametric form depending on param�eter τ):

(7)

where parameter τ is determined from the solution tothe transcendental equation

where

(8)

If we define on the boundary the function Δag(τ)(which characterizes the variation of the pressure inthe ionized sphere) monotonically increasing to a cer�tain maximal value Δag(τ)max, and then decreasing tozero, Eq. (8) will have one root during a certain timeinterval. Consequently, solution (7) will be a single�

2/ k 1–( )at uar aur 2au/r+ + + 0,=

ut uur 2/ k 1–( )aar+ + 0,=

P B+( )/ P0 B+( ) ρ/ρ0( )k,=

P B+( )/ P0 B+( ) a/a0( )2k/ k 1–( ).=

a a0 Δag τ( )Rg τ( )/r,+=

u = 2/ k 1–( ) Δag τ( )Rg τ( )/r a0Rg τ( )/r2 Δag τ( ) τd

0

τ

∫+ ,

a t τ–( )/a0 r/Rg τ( )=

– β r/Rg τ( ) β+( )2 γ–[ ]/ 1 β+( )2 γ–[ ]{ }ln

– γ β2+( ) Fr/Rg τ( ) F 1( )–[ ]/ γ ,

1/2 x β γ–+( )/ x β γ+ +( )[ ], xln 0,>

F x( ) x β+( )/ γ[ ], xarctan 0,<=

β 1/2 k 1+( )/ k 1–( )Δag τ( )/a0,=

γ β 2/ k 1–( )/Rg τ( ) Δag τ( ) τ.d

0

τ

∫–=

valued function of time and coordinate. If the pertur�bation in the medium propagates further, additionalroots of Eq. (8) can appear; i.e., solution (7) maybecome multivalued. Physically, this situation corre�sponds to the initiation of a shock wave. In this case,the multivaluedness of the solution from the point ofview of mathematics must be eliminated analytically byintroducing a discontinuity of the shock wave front,whose motion is governed by the following equation [22]:

(9)

with boundary conditions r = r0, tf = f0, where r0 and t0are the coordinate and the time of the shock wave frontinitiation, respectively. The plus and minus corre�spond to the parameters “behind” and “before” theshock wave front. The fact of the shock wave frontappearance in the flow corresponds mathematically tothe appearance of an envelope of family of curves (9),which is described by parameter τ. Therefore, inaccordance with the computational procedure for theenvelope [23], it is necessary to find the root of Eq. (8)and to determine the time of the front appearance, thedistance from the center of the explosion, and the pres�sure at the wave front using dependences (7) and (8).

It should be noted that the adequacy of the givenmodel (i.e., the validity of the assumption on the isen�tropic nature of the flow) must be controlled exactly atthe instant of the shock wave formation because it isthe shock wave front that is the only source of dissipa�tive loss. Functions Δag(τ) and Rg(τ) appearing inEqs. (7) must be determined from the solution of theproblem of the expansion of an ionized steam�gasbubble that absorbs laser energy. We write the system ofequations describing this problem in the Gilmoreapproximation [20]:

is the first law of thermodynamics and

(10)

is the equation of state of the ionized vapor, where Eg isthe internal energy of vapor in the bubble, Pg is thepressure in the bubble, and Vg is its volume, and

(11)

where Rs is the radius of the beam cross section at thebubble boundary, W is the density of the laser pulsepower in this zone, K is the coefficient of light absorp�tion by the target, and χ is the adiabatic exponent of avapor–gas mixture in the region of the primary ioniza�tion, χ = 1.25 [19]. Writing function g(t) in form (11),we take into account both the possibility to focus thelaser radiation on the target and irradiation of the tar�get by an unfocused beam. System (10) is closed by theequations at the vapor–water interface: dRg/dt = ug(t)and P(t) = Pg(t). Velocity ug(t) at the boundary of theexpanding bubble is determined on the basis of Eq. (7)for u = ug(τ) and r = Rg(τ), where

dtf/dr a+

a–

u+

u–

+ + +( )1–/2=

dEg/dt PgdVg/dt+ g t( )=

Eg PgVg/ χ 1–( )=

g t( ) 3/4KW t( )R*2

, R* min Rg Rs,( ),= =

652


GRIBIN, SPESIVTSEV

Thus, the problem of bubble expansion is reducedto the solution of the following system of differentialequations:

(12)

where Yg is the auxiliary function with the initial con�ditions for t = 0: Pg = 0, Rg = R0, and Yg = 0.

Let us now consider the radiation absorption on atarget of an arbitrary shape. The fluid flow appearingupon the expansion of the initial ionized region is notone�dimensional, but can be described approximatelywithin the limits of one�dimensional hydraulicapproximation

(13)

where r is the distance measured along a fixed fluidtube and S is the cross�sectional area of this tube,S ' = dS/dr. System of equations (13) is analogous tosystem (10) and is transformed into system (10) in thecase of spherical symmetry system (13). Simulta�neously, we make the following assumptions:

(a) ionized region expands uniformly in all direc�tions from the initial volume of the target, and

(b) the cross�sectional area of the lightpipe aver�aged over a solid angle is taken as the integral charac�teristic of geometrical divergence of the compressionwave.

Under these conditions, the problem becomes for�mally one�dimensional and has a solution of the fol�lowing systems of equations with the help of a com�puter: for compression waves in a fluid, we have

(14)

where Vg and Sg are the volume and cross�sectionalarea of the ionized region, respectively; Yg is the auxil�

iary function; and = dVg/dr. The initial conditionsfor T = 0 are Pg = P0, Rg = R0, and Yg = 0. The equa�tions for the ionized region expansion can be writtenin the form

ug t( ) 2/ k 1–( ) Δag t( ) a0/Rg t( ) Δag t( ) td

0

τ

∫+⎝ ⎠⎜ ⎟⎛ ⎞

.=

dRg/dt ug, dYg/dt Δag,= =

dPg/dt 3/4/π χ 1–( )g t( )/Rg3

3χPgug/Rg,–=

ug t( ) 2/ k 1–( ) Δag t( ) a0/Rg t( ) Δag t( ) td

0

τ

∫+⎝ ⎠⎜ ⎟⎛ ⎞

,=

Δag a0 P B+( )/ Pg B+( )[ ]2k/ k 1–( )1–{ },=

2/ k 1–( )at uar aur auS'/S+ + + 0,=

ut uur 2/ k 1–( )aar+ + 0,=

dRg/dt ug, dYg/dt Δag,= =

dPg/dt χ 1–( )g t( )/Vg χPgugVg'/Vg,–=

ug 2/ k 1–( )Δag a0RgSg'/Sg,+=

Δag a0 P B+( )/ Pg B+( )[ ]2k/ k 1–( )1–{ },=

Vg'

(15)

where τ is determined from the following equation:

(16)

Quantity g(t) is computed using Eq. (11). Theexpressions for Vg and Sg depend on the shape of theabsorption region. In the case of a cylindrical regionwith height Hc and radius Rc, the equations have theform

(17)

DISCUSSION OF THE RESULTS OF THE MODEL

The results of computations of the parametersinside the plasma region for quartz and water are givenin Fig. 2. The results of computations of the compres�sion wave parameters outside of the plasma region forquartz are given in Fig. 3. Figure 4 shows the ratio ofthe change in the refractive index (n – n0) in the regionof the compression wave to the value of the unper�turbed refractive index n0 at the atmospheric pressurefor instant t equal to half the laser pulse duration, tp/2.We computed the refractive index for quartz by the for�mula [24]

where n0q = 1.545, dn/dp = 0.00103/96.2 MPa (fromTable 1) and P is taken from Fig. 3 (curve 1); therefractive index for water was computed by the formula

where n0W = 1.334 and ρ and ρ0 are the densities ofwater in the compression wave and the initial density,respectively [19].

Deformation of the initial luminous flux on DCLwas computed on the basis of the ray theory [16].Figure 5 shows the ratio of laser energy density W onthe optical axis behind the dynamic lens and initialenergy density W0 (in front of the lens) depending onthe distance from the lens center for the instant equalto half the pulse duration.

The diffraction broadening angle was estimated interms of formula β = 0.61(λ/r) and turned out to besmaller than 0.0122 rad. Here, λ is the light wavelengthand r is the radius of the dynamic lens [17]. As can be

Δa Δag τ( ) Sg τ( )/S r( ),=

u 2/ k 1–( ) Sg τ( )/S r( ) Δag τ( )∫=

+ a0S ' r( )/S r( ) Δag τ( ) τd

0

τ

∫ ,

dr/dt a0 Δag u for t+ + τ, r Rg τ( ).= = =

Vg π Hc Rg Rc+( )22RgRc

22Rg

2Rc 4/3Rg

3+ + +[ ],=

Sg π 2Hc Rg Rc+( ) 2Rc2

4Rg2

2RgRc+ + +[ ].=

nq n0q dn/dpP,+=

nW n0W 0.334 ρ/ρ0 1–( ),+=



seen from Fig. 5, in the focal region of DCL, the laserenergy density per pulse increases by more than twoorders of magnitude.

TYPES OF DYNAMIC LENSES

In the general case, the dependence of the refrac�tive index of a transparent medium on the pressure isdescribed in the linear approximation by the followingequation [24]:

where the first term z∂ρ/∂p on the right�hand sidedescribes the change of the refractive index of themedium due to an increase in density, and the secondterm ρ∂z/∂p describes the change in the degree ofpolarizability upon an increase in pressure. Such acomplicated dependence of the refractive index on thepressure explains the fact that the attempts to deter�mine the correlation between the physicochemicalproperties of the medium and the value of the opticaldamage threshold fail. It can be seen from Table 1 thatin certain media (KBr, NaCl, quartz, and fused

dn/dp z∂ρ/∂p ρ∂z/∂p 24[ ],+=

0.2

0

15

25 50

P

R/R0

ti

3.0

0

10

8 16

P(1)

R/R0

ti

1.5

ti × a0/R0

P(2)

P/B R

/R0

P/B

R/R

0

Fig. 2. Dependence of dimensionless parameters of theplasma region on dimensionless time tpa0/R0. I—quartz.P is the curve describing dimensionless pressure in theplasma region; P/B, B = 1000 MPa; R is the radius of theplasma region; R0 is the radius of the absorbing irregularity,1 µm; a0 is the initial sound velocity in quartz, 5400 m/s.The laser pulse parameters: W0 is the energy density,15 J/cm2; tp is the pulse duration, 10 ns. II—water. B =306.5 MPa; a0 is the initial sound velocity in water, 1500 m/s;R0 is the radius of the absorbing irregularity, 1 µm; P(1) isthe curve describing dimensionless pressure in the plasmaregion; P/B, during irradiation by a laser pulse with energydensity W0 = 15 J/cm2; P(2), W0 = 5 J/cm2; tp is the pulseduration in both cases, 10 s.

Table 1. Dependence of refractive index on pressure

Material KBr NaCl LiF SiO2cryst.

SiO2melt. MgO Diamond

Refractive index at λ = 587 nm 1.560 1.546 1.392 1.545 1.458 1.738 2.418

1.554

Pressure P, MPa 96.61 96.36 96.32 95.76 96.02 96.08 96.00

96.15

Change in refractive index Δn × 105 227 115 0 103 83 –17 –11

107

dn/dp = Δn/P × 105 2.35 1.19 0 1.07 0.86 –0.18 –0.14

1.11

1

2

0.2

0.1

0 20 40 60r/R0

P(r)/B

Fig. 3. Dependence of dimensionless pressure P/B in thecompression wave around the plasma region in quartz ondimensionless distance r/R0 for various instants. 1—(+)for instant t = tp/2, 2—(*) for instant t = 1.5tp. W0, tp, B,and R0 are the same as in Fig. 2.

I

II

654


GRIBIN, SPESIVTSEV

quartz), the refractive index increases with pressure.The focusing DCLs appear in these media. In othermedia (MgO and diamond), the refractive indexdecreases with increasing pressure. Diverging lensesappear in such media. Information on the change inthe refractive index in polymers, solids, and fluidsupon the action of the compression waves is veryscarce in the literature; therefore, to estimate thechanges in the refractive index in this model, we usedthe data for the action of the steady�state pressure(Table 1) [24–26].

In the general case, function n(P) can have maximaand minima. Therefore, the kind of a DCL appearingin the given medium must be determined separatelyfor each medium. For longer laser pulses (1–100 μs),dependence n(T) of the refractive index around AIs onthe temperature can have a significant effect on thevariation of the refractive index. Thermal dynamiclenses (TDLs) appear. For the majority of media, coef�ficient dn/dT < 0, which is due to a decrease in thedensity of the substance during heating. Therefore, forlaser pulses of such a duration, a peculiar competitionappears between various mechanisms of variation of

the refractive index, which complicates still further thepattern of a transparent medium destruction.

SOME CONCLUSIONS AND ASSUMPTIONS CONCERNING THE MODEL

OF DYNAMIC LENSES

(i) The proposed model implies that the energydensity increases behind the DL both in the case offormation of focusing lenses (dn/dp > 0) and in thecase of formation of diverging lenses (dn/dp < 0).However, this increase in the density occurs in differ�ent regions. This must cause various forms of destruc�tion. In addition, the greatest ratio W/W0 of the energydensities is observed for media with dn/dp > 0. Thus, toobtain optical media with high optical resistance tolaser radiation, it is necessary either to produce mediawithout AIs (which is very difficult in practice) or toproduce media with dn/dp = 0 in the given range ofpressures (LiF, see Table 1).

(ii) The results of [2, 3, 27] are in a good agreementwith the presented model. Thus, proceeding from thevalue of change of refractive index dn/dp for variousoptical materials (Table 1), we can expect that thematerials with a higher value of dn/dp have lower laserdestruction thresholds. In addition, the self�focusingeffect is larger (the destruction threshold is lower) thesmaller the wavelength of laser radiation since the dif�fraction divergence of the rays decreases in this case.Experimental thresholds of laser destruction at differ�ent wavelengths for a number optical dielectrics, forwhich the value of dn/dp is known, are given in Table 2borrowed from [2].

Comparing the data from the tables, we see that achange in the laser destruction threshold correlatesespecially well with the change in the value of dn/dp atwavelengths of 0.69 and 1.06 μm for wideband crystals

0.003

0.002

0.001

(n − n0)/n0

0 10 20 r/R0

Fig. 4. Dependence of the ratio of the change in the refractiveindex (n – n0) in quartz to initial refractive index n0 of quartz inthe compression wave around the plasma region on dimen�sionless distance r/R0 for instant t = tp/2. The values of W0, tp,B, and R0 are the same as in Fig. 2. nq = n0, q + dn/dp P; n0, q =

1.545; dn/dp = 0.00103/96.76 (MPa)–1 (see Table 1), P istaken from Fig. 3 (1).

200

100

0 50 100 L/R0

W/W0

Fig. 5. Ratio of laser energy density W on the optical axisbehind the DCL in quartz to energy density W0 in front ofthe dynamic lens as a function of dimensionless distanceL/R0 from the lens center for instant t = tp/2. The values ofW0, tp, B, and R0 are the same as in Fig. 2.

Table 2. Destruction thresholds for transparent dielectricsat different wavelengths, GW/cm2 [2]

Material KBr NaCl LiF SiO2cryst.

SiO2melt.

Wavelength, μm

1.06 50 120 360 230 400

0.69 58 150 360 280 600

0.266 50 45 240 70 40



(KBr, NaCl, LiF, and SiO2). Destruction of these crys�tals is associated in [2] with the presence of absorbingimpurities.

A higher value of the destruction threshold of fusedquartz can apparently be explained by a considerablysmaller concentration of the absorbing defects [2] orthe size effect [3]. In spite of the fact that the depen�dence of dn/dp on pressure at various wavelengths forthese materials is not given in the literature, a changein the destruction thresholds decreases upon adecrease in the laser radiation wavelength for all mate�rials upon a transition to λ = 0.266 μm, as follows fromthe allowance for diffraction divergence in the focusof DCL.

(iii) For the materials with a weak dependence ofdn/dp on pressure (LiF) and superpure materials, thelaser destruction threshold is apparently determinedby an electronic–thermal mechanism of defect gener�ation [28].

(iv) The model shows that in the media with a smallnonlinearity of the sound velocity (quartz), shockwaves have no time to form. However, in media with arelatively high nonlinearity (water), the shock front isformed at a distance of several tens of micrometersfrom AIs.

(v) The model predicts [29] that when the com�pression wave escapes from a solid optical mediuminto a gas, the kinetic energy of the compression waveis converted into the thermal energy of a thin near�sur�face gas layer. This leads to gas heating by several hun�dred degrees and melting of the solid surface to a char�acteristic arc shape. The traces of such melting werefound in [27] during irradiation of the optical materialby a nanosecond laser pulse.

(vi) During irradiation of the materials with AIslocated on a surface, a plasma torch is formed in airand a cavity is formed in the material.

(vii) Self�focusing of picosecond pulses on a DCLis obviously impossible because the lens size over thepulse duration would be too small (for fused quartz, r ~5 × 10–9 m). However, for the frequency regime of laseroperation, such a focusing mechanism is also possible.

APPLICATIONS OF THE MODEL

The model was applied for explaining the size effectin glass [30] and in the development of new technolo�gies such as disinfection of a fluid by laser microexplo�sions [31, 32] and a laser adhesiometer [33, 34].

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Translated by N. Wadhwa

mechanism of self-focusing of laser pulses on dynamic lenses in transparent media

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