Physical mechanisms controlling the generation of laser-induced stresses

The mechanisms responsible for the generation of stresses by pulsed-laser energy deposition in solids are elucidated with special attention given to laser-tissue interactions. These mechanisms include thermal expansion, subsurface cavity formation, ablative recoil and plasma formation and expansion. Scaling laws are presented for the magnitude of the stresses generated by each of these processes. The effect of laser parameters and material properties on the magnitude and temporal behavior of the stress transients is considered. The use of these scaling laws in conjunction with measurement of stress transients induced by pulsed laser sources may be a powerful tool in determining the physical processes which control the response of materials to pulsed energy deposition. In addition, the controlled generation and accurate measurement of acoustic transients may have important diagnostic and therapeutic applications.


Introduction
Pulsed lasers are frequently used as tools for the precise incision and excision of biological tissue.The stress transients generated within the tissue as a. result of pulsed laser heating and ablation are receiving increased attention for their potential therapeutic and destructive effects [3. 6].In addition, the measurement of the stresses resulting from pulsed heating of tissue is proving to be a valuable research tool in investigating the dynamics of laser ablation [5. 4].However, even with this interest in laser-induced stresses in tissue.no clear framework has been established to predict the nature of these stresses.
The objective of this paper is two fold.First, we will describe some of the l)hysiCal mechanisms responsible for the optical generation of stresses.Second.we will predict the magnitude and duration of the stress transients associated with these mechanisms.Emphasis is placed on the development of scaling laws which relate the characteristics of the stress transients to laser afl(l material parameters.Once these scaling laws are understood.experunents can he designed to discriminate between physical mechaiiisms that may be oJ)erative in a given laser-target interaction.

Thermoelastic Stress Generation
In most instances the thermoelastic effect is the dominant mechanism of stress generation in cases where energy depositioii results solely iii heating and thermal expansion of the medium.For a linearly elastic isotropic solid with constant thermophysical properties.the equation of motion in one-dimension takes the form [13]: The first.term in ecin.( 1 ) represents the spatial gradient of internal stresses within the medium, the second represents the spatial gradient of stresses produced by thermal expansion of the medium and the last represents the net particle acceleration produced by the stress gradients.
For the cases we consider.the temperature rise results from the absorption of laser energy by the tissue.Assuming that Beer's Iav governs the al)sorption of the laser radiation.eqn.(1 ) takes the form: where F = 3/3K'/pcL, 5 the Griineisen coefficient and "( x t ) = ( 1 -1? )q"( t' ) exp ( f'a X ) di the total radiant exposure deposited up to time t, q" being the incident irradiance.R the surface reflectivity and ia the optical absorption coefficient .If the target surface is free.stresses that.move towards the surface are reflected l)ack with a. change in sign and results in a bipolar stress transient.In this case the initia.land boundary COI1(litiOlI5 necessary to solve eqn.(2) are: on 1tI(yt__) = -::;--- = 0. ( -3/3K[TI(O+,t)-T] =o. ( Eqn. (3) requires that the medium be at rest and stress free as time approaches -c.Eqns.( 4) and (5) require that the particle displacement and stress be continuous across the free surface.The solution to eqns.( 2)-( 5) with these conditions can be found in [2].
To derive the appropriate scaling laws we consider the situation shown in figure 1 where laser radiation of irradiance q" aiul duration t is incident on the free surface of an absorbing medium with absorption coefficient .'We wish to characterize the stress transient measured by a stress measurement device located at a depth 5 >> jç1 .To do this.consider a layer of infinitesimal thickness dA located at a depth \ where d\/2 A << The tota.l energy absorbed by this layer at a given time / is f ftq"(t' ) exp ( \ ) dA dt'.
The absorption of the laser energy by the layer launches two compressive stress transients, one moving towards the free surface a;(t) and the other moving into the bulk of the solid a(t).Since the stress which travels towards the free surface gets reflected with a. change of sign, the measured stress is given by:  where c = [(h + 4p/3)/p]112 the )ropaga.tionvelocity of au acoustic wave.Alternatively. the measured stress can be expressed iii ternis of the laser the energy (leposition: J-.x, J-x \Ve define a. new variable = 1" -{i -[(5 -\)/c]} which allows eqn. (7)to be written more simply as: (.r = t) = J [P.a exp(-ita)jq" hi eqn.(8).we see that only the energy deposited ii the time interva.1 0 t 2\/c contributes to the measured stress.Since the characteristic length scale in this process is the absorption depth \ = jç we should consider those cases where the laser pulse duration.t is much smaller or much larger than the characteristic time 1/ji1c.Iii the limiting case where the laser pulse duration is much shorter than the acoustic transit time, i.e., ftaCtp << 1. sigiificant particle motion occurs only after the laser pulse.Accordingly.there is no relaxation of the stresses developed in the heated layer during the laser pulse.Thus the magnitude of the stresses scales with the volumetric energy deposition fLaE" and the Grüneisei coefficient F. In addition, the compressive and tensile contributions to o(x = 5t).a. (t -[( + )) /c]) and a. (1 -[(t5 - remaii separated in time.Thus the measured stress transient yields the distribution of the laser energy vithin the target since the time variable can be converted to a. spatial depth through the propagation velocity c.This clemonstiates the use of photoa.coustictecimiques to measure the (listributiOll of radiation vit1iiii a. medium: )rOViding a. means to determine its oJ)tica.lpropei'ties.This is shown in figure 2a.where tensile and compressive components of the stress transient.are exponentia.lswhose rise and decay times scale with the absorption coefficient In the limiting case where the laser l)UlSe duration is iimcli larger than the acoustic transit time.i.e., JLaCtp >> 1, significant particle motion occurs on the time scale of the laser pulse.This has the effect. of reducing the maximum unia.xia.lstress by a. factor proportional to [7].As a result.the magnitude of the laser induced stress is independent of the optical penetration depth and the scaling aa:x(x ) 0( FE" applies.Since 1//iaC S very small compared to tJ). the compressive and tensile contributions to a(x = The solid curve is the n.ea.sured stress transient for 1tct >> 1 for a. Gaussian laser pulse shape (dashed curve).are uo longer separated in time.Instead. the measured stress is the sum of a compressive stress generated at a given time plus a. tensile stress generated a. brief instant earlier.Since the magnitude of these stresses are J)ropOrtiollal to the laser irradiance.the magnitude of the stresses are proportional to the time derivative of the laser irradiance.Accordingly.the duration of the stress transient is simply the duration of the laser I)ulse.1/).This is illustrated in figure 2b for a Gaussiaii pulse shape.With the notable exception of Q-switched Nd:YAG.Q-switc.hedHo:YAG.and TEA CO2 laser pulses.most laser sources fall into this case when in the therinoelastic regime.

Cavity Formation
There is experimental evidence to indicate that as the radiant exposure increases, mechanical failure of the target may occur, resulting in cavity formation below the tissue surface [12.14].The conditions under which the cavity is formed are unclear although possible niechanisms include subsurface vapor formation 01' localized energy depositioii due to inhornogeneous absorption.In any event.once a cavity is formed the absorption of the laser radiatioii by the material within the cavity leads to an increase in cavity pressure which launches a. coiiipressive stress transient.If the 'aser pulse (Iliration is short.compared to the characteristic expailsioli time of the cavity.the laser-induced stress will reach a. maximum a.t the end of the laser I)ulse.After the en(l of the laser I)ulse mecha.nica.1 expa.nsioi of the cavity and therma.ldiffusion will the relax the interna.lpressure.Thus the stress transient will he longer than the laser pulse and have a decay time which scales a.s ttiecay 0( 1/cp, a being the thermal diffusivity of the medium surrounding the cavity.In addition, the peak stress will scale with the volumetric eiiergy deposition ItaE" regardless of the value of p.r, ct so long a.s the laser pulse duration is short compared to the characteristic thermal diffusion  time i.e.. ctti,, < O( I ).Stress generation characteristic of cavity formation has been observed to result from pulsed excirner laser irradiation of polyimide {15J.

Recoil of Ablation Products
At large energy densities strong thermal and acoustic effects are generated within the medium and have the ability to achieve explosive removal of biological tissue.In this case.stress waves are generated by the recoil of the ablation products as well as by the thermoelastic response.However, due to the strong dependence of the recoil stresses on the absorbed radiant exposure.these stresses dominate the thermoelastic stresses at radiant exposures only nominally larger than the ablation threshold.
In order to derive the scaling law for the recoil pressure at the target surface, we will model the ablation process as one of steady vaporization as formulated by Landau and Lifshitz [11].We consider the onedimensional case where a layer of vapor at high temperature and pressure is created adjacent to the target surface.We assume the expansion velocity of the vapor u to be large compared to the sound velocity in the surrounding medium.This results in the radiation of a shock traveling at velocity u.This is shown schematically in figure 3 where we model the vaporization process as a piston moving at a. velocity u into the surrounding gas.In this case the equations of mass.momentum and energy conservation in a. reference frame moving with the shock wave are: In eqns.( 9) and ( 10). the subscripts and '2' refer to the regions downstream and upstream of the shock wave.Thus region 1 is the surrounding gas under ambient conditions and region 2 consists of a. mixture of ablation vapor and ambient gas which Iia.s undergone shock compression.ii.th".p. u and v refer to the specific entha.lpv.mass flux per unit area.. pressure, velocity and specific volume, respectively.
Our goal is to express the vapor flow velocity u in terms of the pre-aud post-shock pressures Pi and P2 By combining eqns . ( 9 )and ( 10 ) , we express u L as a function of the presures and specific volumes on either side of the shock: \Ve then combine eciis.( 9)-(1i) and use the relation ii = pt [/ ( -1)].where ', is the ratio of specific heats Cj)/CL to relate the ratio of specific volumes across the shock front with the pre-and post-shock pressures: -P1(71+l)+P2(v1 -1) (13) V1 ; (i -1) +P2 (i + 1)• Combiniig eqiis.( 12) aiid ( 13) we get the following quadratic equation for the compression ratio II = across the shock: where c1 = " is the adiabatic sound speed in region 1.Solving eqn.( 14) for II and simplifying for the case where u >> c1.we get: In order to find a we app1 conservation of energy globally to this process.We assume that the energy necessary to form the vapor is small compared to both the energy which heats the vapor and the work doiie l)y the expanding vapor on the surrounding air.In this case energy conservation can be written as: where TV is the work (lone by the expanding vapor on the surrounding air per unit surface area.aiid is the internal energy of the expandiig vapor per unit surface area.Letting : be the thickness of the vapor layer.T'V = P2 d and = ])2z/(-)2 _ i).Substituting these expressions into eqn.(16).recognizing that lit, = (dz/dt) and substituting eqn.( 15) for 1)2/Pt we get lit, = 2(2 1)cq" 1/3 Y2i1 hi + l)Pi Substituting eqn.( 17) into ( 15) and solving for p gives the result: , I fy1 (2'i + l)pi 1 (2 1)q I P2I . .
This result indicates that the recoil stress caii be reduced by decreasing the ambient pressure or replacing the ambient gas with one whose adiabatic sound speed is larger (i.e.. a. gas with lower molecula.rmass).For those interactions where a. steady state ablation process is not achieved due to the short duration of the laser pulse.two changes must.be made to the ai)ove expression.First.the irradiance q" should he replaced 1)V the radiant exposure (livided by the pulse duration ?'/1.Secondly.the amount of energy left ill the target.(i.e.. the threshold radiant exposure for ablation) should 1)e subtracted from the radiant exposure a.bsorl)ed by the target.This leads to the following scaiing law for the recoil stress    5 Stresses Generated by Plasma Formation and Expansion At higher energy densities.the irradiance of the laser radiation may be sufficient to ionize the ablation products and form a dense plasma adjacent to the target surface.Since the p1asma absorbs a. non-negligible portion of the incident laser radiation the gas-dynamics of the ablation flow is altered significantly.Thus the scaling laws derived in the previous section do not apply when plasma mediates tile laser-target interaction.
In modeling this situation we again assume that the velocity of the ablation products is much larger than tile sound velocity in the ambient gas thereby radiating a shock wave.However, since a portion of the laser energy is also absorbed by the plasma.. the shock wave is followed by a defla.grationwave which is HI turn followed by a. rarefaction resulting from tile plasma.expansion [9]. Figure 4 is a. pictorial representation of the physical situation we are considering.
We consider the case where tile process of free-free absorption or inverse Bremsstra.hlung is the mechallism responsible for i)la.slllaheating.This occurs when an electroll a.1)sOrhs a. photon thereby moving it from one free state to a. more energetic state in the field of an ion.Assuming that the plasma.collectively has 110 net charge, its absorption coefficient in tile limit where hu < kT is given by [1]: where 'I = M2 (M + 3) /2.Thus the pressure developed within the plasma due to 'aser heating is given by the scaling law: ($\1/8 (q/I\3/4 Since the recoil stress at the target surface differs only by a multiplicative factor dependent 011 the plasma characteristics, this scaling law also applies to the recoil stress a.t the target surface o. .This result shows that although the pla.sma mediates the interaction.the dependence of the recoil stress oi the optical thickness is very weak.This shows that our a.ssumj)tion that the optical thickness be Q( 1 ) is not critical to the final result.In addition, we find that although less energy is coupled into the target for plasma mediated interactions.the recoil stress is more strongly dependent on the incident irradiance.

Summary and Conclusions
In this paper we have considered the character of the stress transients generated by thermoelastic expansion, cavity formation, ablative recoil and plasma formation and expansion.The results indicate that the stresses generated by each of these mechanisms display unique characteristics with regards to their temporal behavior and with the scaling of their magnitudes with laser and material parameters.\Ve have shown that photoacoustic measurement techniques have the potential to determine the optical properties of a medium.We have also shown that the recoil stresses associated with ablation rise very sharply with increasing radiant exposure when close to the ablation threshold.rflus in the ablative regime.a reduction in the recoil stress caii be achieved with a lower pulse energy.Alternatively. the recoil stress can be reduced if the ambient pressure is reduced or if an ambient gas with lower molecular mass is used.Vihen ionization of the ablation products occurs, the scaling of the recoil stress with the incident radiant exposure is stronger than when plasma is iiot present.
As interest in the effect oflaser-induced stresses on biological systems increases.it is ofprime importance to characterize the stress transients to which these system will l)e exj)osed.To this end.it.is hoped that the fraiework presented here will aid in the (lesign of such studies.

Figure 3 :
Figure 3: Physical situation coiisidered whei determiniiig the recoil stress of the ablation products.The e1ocities i and it2 are given in a refereice frame moving with the shock wave.

Figure 4 :
Figure 4: Physical situation considered vhen determining the recoil effects produced by plasma formation and expansion.The velocities are given in the reference franie of the shock wave.See text for (letails.