1.

Introduction

The cells of the retinal pigment epithelium naturally contain highly absorbing, micrometer-size melanosomes. Other biological and physical systems can be doped with highly absorbing micrometer- and nanometer-size absorbers, such as gold particles.^{1, 2} Their strong absorption relative to normal tissue makes these particles the likely site for the generation of light induced damage at threshold levels of illumination. In addition to the potential danger, the same biophysical mechanisms can be used for a variety of beneficial medical applications.

Much research has been carried out, both experimentally and theoretically, investigating the interaction of laser light with the visual system^{3, 4, 5, 6} and other biological tissues.⁷ Absorption of the laser energy generates various responses that include temperature rise^{8, 9, 10} (thermal), pressure^{11, 12, 13, 14, 15, 16, 17, 18} (acoustomechanical), and vaporization^{17, 19, 20} (phase transitions). In this paper, we present results for the modeling of the generation of dangerously high pressures and explosive vaporization utilizing a theoretical treatment that incorporates a full empirical equation of state of water. This is the most realistic treatment we have seen and enables us to predict thermomechanical effects, including nonlinear shockwaves and phase transitions that produce bubbles. This treatment is designed to predict the strength of shockwaves and the size of bubbles produced by laser pulses over a wide range of pulse duration and fluence that are absorbed by particles with known physical properties. However, the theoretical model can also be used in “reverse.” Experimental measurements of shockwave strength and bubble size can be used with this analysis to determine mechanical and thermal properties of absorbing particles, such as melanosomes, that are too small for conventional measurements of these properties.

Figure 1 is a schematic showing how the threshold laser fluence level $I_0$ for biological danger to the cell varies as a function of pulse duration. ${\tau }_L$ for each of the three damage mechanisms: temperature rise (thermal), explosive vaporization (bubble generation), and shockwaves. All three mechanisms can operate at all pulse durations. However, since their dependencies on ${\tau }_L$ are quite different, the mechanism responsible for threshold damage may change as a function of ${\tau }_L$ . For long ${\tau }_L$ , thermal damage requires the lowest $I_0$ and is the threshold damage mechanism. As ${\tau }_L$ shortens, thermal damage requires less fluence until ${\tau }_L$ decreases to a characteristic heat conduction time ${\tau }_H$ , that depends on absorber size and the spacing between absorbers, as explained in Ref. 9. For ${\tau }_L<{\tau }_H$ , the threshold fluence remains constant for thermal damage. The damage threshold fluence for explosive vaporization also decreases as ${\tau }_L$ shortens, but continues to decrease for ${\tau }_L<{\tau }_H$ . The characteristic vaporization bubble time ${\tau }_V$ is equal to the time scale of bubble expansion and for the system treated here is of the order of $100\mathrm{ns}$ . For pulse durations that are longer than this time, part of the laser energy will be absorbed after bubble growth has effectively ended, i.e., for a given fluence, the maximum bubble radius, and hence the damage threshold levels, will significantly depend on the pulse duration. Below ${\tau }_V$ , the fluence required for damage from explosive vaporization remains approximately constant. Since ${\tau }_V<{\tau }_H$ , for ${\tau }_L$ between ${\tau }_H$ and ${\tau }_V$ , the fluence required to cause damage from explosive vaporization may drop below the fluence needed to cause thermal damage, and explosive bubble generation may become the threshold damage mechanism.¹⁹ The fluence required for shockwave production decreases as ${\tau }_L$ decreases and becomes constant when ${\tau }_L<{\tau }_c$ , where ${\tau }_c$ is a characteristic time for shockwave production. Since ${\tau }_c<{\tau }_V$ , for ${\tau }_L$ between ${\tau }_V$ and ${\tau }_c$ , the fluence required to cause damage from shockwaves may drop below the fluence needed to cause bubble damage, and shockwave production becomes the threshold damage mechanism. Thus, the underlying biophysical mechanism for damage at threshold fluences may change as ${\tau }_L$ enters the regions between the characterisitic times ${\tau }_H$ , ${\tau }_V$ , and ${\tau }_c$ . In addition to different time scales, each of these mechanisms has different characteristic strengths and length scales. These differences can be important both for assessing danger, and in beneficial medical applications. In this paper, we show how to calculate ${\tau }_V$ and ${\tau }_c$ , as well as determining how the strength and extent of vaporization and shockwave effects depend on ${\tau }_L$ and $I_0$ .

Fig. 1

Schematic diagram for damage thresholds as a function of pulse duration. The dashed lines represent suprathreshold damage mechanisms, e.g., for long pulse durations, damage-producing bubbles can occur, however, they occur at fluences above the threshold for thermal damage.

2.

Model

3.

Shockwaves and Bubbles

In this section, we plot the results of exhaustive calculations for the size of shockwaves and bubbles expected for a wide range of laser pulse durations and fluences. These pulse durations and fluences are chosen because they are in the range for which pressure effects and vaporization may be important in determining threshold levels of retinal damage.

In all the calculations reported here we used the following parameters for the melanosome:^{12, 27} $a=1\mu \mathrm{m}$ , ${\rho }_0=1.35\mathrm{g}∕{\mathrm{cm}}^3$ , and $c_v=2.51\mathrm{J}∕\mathrm{gK}$ . We use an absorption coefficient of ${\alpha }_L=\mathrm{10,000}{\mathrm{cm}}^{-1}$ . The value of the absorption coefficient used represents an upper limit for any wavelength, hence it corresponds to a lower limit on threshold levels. As there is still some uncertainty with respect to the value of the absorption coefficient due to the difficulty in performing measurements on biological tissue,^{28, 29} later in this paper we present a table that enables a researcher to scale the fluence required to get the same shockwaves and bubbles for other values of ${\alpha }_L$ . Reliable numbers for the bulk modulus $B$ and bulk thermal coefficient of expansion $\alpha$ are also not available. Our results depend on the values used for these parameters, and independent measurements of them would be extremely valuable, as emphasized in Refs. 11, 12. To continue with the calculations, we adopt graphite as a substitute because of a chemical similarity to melanin,³⁰ and use graphite’s $B=39.4\mathrm{GPa}$ and $\alpha =2.98\times {10}^{-5}{\mathrm{K}}^{-1}$ . Later, we discuss how the results will change for other values of $B$ and $\alpha$ . The surrounding medium is aqueous with ${\rho }_0=1.0\mathrm{g}∕{\mathrm{cm}}^3$ , and thermal conductivity $\lambda =5.56\times {10}^{-3}\mathrm{J}∕\mathrm{cm}\mathrm{K}\mathrm{s}$ . Since for water we are using a tabulation relating $P$ , $v$ , $T$ , and $s$ , we do not use values for $c_v$ , $B$ , and $\alpha$ as is necessary when using the simple EOS, Eq. 5, for the absorbing melanosome.

In Fig. 2 and 3 we use a high fluence to present plots that are useful in interpreting the results that are presented in later figures. We use a laser pulse of duration ${\tau }_L=1\mathrm{ps}$ , $I_0=1.0\mathrm{J}∕{\mathrm{cm}}^2$ (suprathreshold for both shockwaves and bubble formation to elucidate the effects) with ${\alpha }_L=\mathrm{10,000}{\mathrm{cm}}^{-1}$ . The melanosome undergoes a $\Delta T=1595\mathrm{K}$ , and the shock front in the near field can be as large as $4\mathrm{kbar}$ , corresponding to a shock front speed of $U_s=2630\mathrm{m}∕\mathrm{s}$ ( $\approx 75\%$ above acoustic speed in water). Once the shock front leaves the near-field region, it decays to an acoustic type disturbance. Though not as dangerous as a shock front, a strong acoustic disturbance may cause biological damage, and this possibility remains to be investigated.

Fig. 2

Refer to text for laser and absorber properties. (a) Multiple snapshots of the pressure profile outside the melanosome in the aqueous medium at various times after the single laser pulse, including trailing shock fronts due to the remnant ringing of the absorber, (b) decreasing strength of the leading shock front as it propagates out into the medium, the decay is faster than the $1∕r$ dependence of an acoustic wave in three dimensions; and (c) log-log plot of (b) to emphasize the changing rate of $P$ decay as a function of $P$ .

Fig. 3

For the same laser pulse as used in Fig. 2, the expansion of the bubble outward from the absorber’s surface at $r=a$ . The bubble is defined by the region of high specific volume $v$ . The time scale for bubble expansion is approximately 50 times larger than that for the pressure wave propagation displayed in Fig. 2. Note that the decreased specific volume in the proximity of the melanosome results from the increased pressure of the vapor due to heat transfer from the hot melanosome.

4.

Dependence of Shockwaves and Bubbles on Pulse Duration and Fluence

In Fig. 4 we plot the pressure and bubble results for different pulse durations and fluences. Figure 4(a) is a plot of the strength of the shock front at a location $1\mu \mathrm{m}$ outside the melanosome $(r=2a)$ . This position is chosen because the shockfront has traveled enough distance from the surface of the absorber to fully form, but a small enough distance to still have significant strength. We can see that the shock front amplitude as a function of $I_0$ is almost identical for ${\tau }_L=1\mathrm{ps}$ and $100\mathrm{ps}$ , but drops dramatically for ${\tau }_L=370\mathrm{ps}$ . This ${\tau }_L$ is also ${\tau }_c$ for the absorber, as already explained, and defines the stress confinement time. For ${\tau }_L<{\tau }_c$ , the pressure effects in the surrounding medium depend on $I_0$ , but are independent of ${\tau }_L$ . (As explained in Ref. 11, the concept of stress confinement is not valid in the core region of the absorber). For ${\tau }_L>{\tau }_c$ , the amplitude of the leading pressure front decreases inversely with ${\tau }_L$ . In addition to the decreasing amplitude, the pressure front smooths to a sinusoidal acoustic wave, with much smaller pressure gradients than the shockfronts generated by pulses with ${\tau }_L<{\tau }_c$ . Therefore, pulses with ${\tau }_L<{\tau }_c$ will cause much greater pressure induced effects in the surrounding medium than pulses with ${\tau }_L>{\tau }_c$ .

Fig. 4

Calculated results for pressure amplitude and maximum bubble radius for laser pulses of different fluence and pulse duration. Values used for properties of the absorbing melanosome are ${\alpha }_L=\mathrm{10,000}{\mathrm{cm}}^{-1}$ , $B=39.4\mathrm{GPa}$ , and $\alpha =2.98\times {10}^{-5}{\mathrm{K}}^{-1}$ . (a) The strength (pressure jump) of the expected shock front $1\mu \mathrm{m}$ outside the absorbing melanosome $(r=2a)$ , and (b) The maximum bubble size, as a function of fluence for different pulse durations.

Figure 4(b) shows the maximum bubble size $R_{\max }$ as a function of $I_0$ for different ${\tau }_L$ . The bubble size decreases by only 10% as the pulse duration increases from $1\mathrm{ps}$ to $1\mathrm{ns}$ , which is a manifestation of “cavitation confinement.”¹² For ${\tau }_L=100\mathrm{ns}$ , there is a significant decrease in the bubble size, and this defines the characteristic cavitation or vaporization confinement time ${\tau }_v$ , which is more than two orders of magnitude larger than ${\tau }_c$ . Vaporization resulting from laser absorption is mainly due to heat flow into the surrounding medium and is hot cavitation, though cold cavitation bubbles that are generated by a sudden decrease in pressure are incorporated in the EOS that we employ for the medium. Therefore, ${\tau }_v$ is determined by heat flow. In the next section, we show that the amplitude of both the shock fronts and bubbles depend on both the thermal and the mechanical properties of the absorber.

In Fig. 5 , we compare $R_{\max }$ as a function of $I_0$ calculated from our numerical model with experimental measurements, as well as predictions from two other models. The curve labeled $1\mathrm{ps}$ employs the model of this paper, and is the same $1\mathrm{ps}$ curve plotted in Fig. 4(b). The curve labeled $10\mathrm{ns}$ is of the same model but uses a pulse duration of $10\mathrm{ns}$ . The curve labeled NB is the maximum bubble radius as a function of $I_0$ obtained by fitting the experimental data of Neumann and Brinkmann³³ for a pulse duration of $12\mathrm{ns}$ . We found that the curve $[R_{\max }\left(I_0\right)∕1\mu \mathrm{m}]={[0.3+44(I_0-0.12)]}^{0.5}$ , for $I_0>0.12$ in $\mathrm{J}∕{\mathrm{cm}}^2$ fits best the experimental data, and we use this function to extrapolate the experimental results of their Fig. 4 to the fluence scale of our Fig. 5. The curve labeled Gerstman is calculated from a model¹⁹ that makes simple assumptions about bubble growth but has the advantage of giving a straightforward analytical expression for $R_{\max }$ as a function of $I_0$

Fig. 5

Comparison of the maximum bubble radius obtained from model and experiment. The curves labeled $1\mathrm{ps}$ and $10\mathrm{ns}$ employ the numerical model of this paper for pulse durations of $1\mathrm{ps}$ and $10\mathrm{ns}$ , respectively. The curve labeled NB represents a best-fit curve to the experimental data of Neumann and Brinkmann.³³ The curve labeled Gerstman is calculated from Eq. 13. The curve labeled RP is calculated from Eq. 14 without the viscous and surface tension terms, and the curve labeled RPVS is calculated from Eq. 14 assuming ${\eta }_L={\rho }_L{\upsilon }_L=500\mu \mathrm{Pa}\mathrm{s}$ and $S=0.07\mathrm{N}∕\mathrm{m}$ .

We use the following conditions for the parameters in Eq. 12, their choice is more fully explained in Ref. 19. The parameter $q=2770\mathrm{J}∕\mathrm{g}$ is the heat needed to raise one gram of water initially at body temperature of $37°\mathrm{C}$ and $P=1\mathrm{atm}$ to water’s critical point temperature of $374°\mathrm{C}$ and pressure of $218\mathrm{atm}$ , ${\rho }_c=0.315\mathrm{g}∕{\mathrm{cm}}^3$ is the critical point density of water. Since under short pulse laser illumination the melanosome is expected to reach temperatures of thousands of degrees Celsius (e.g., at $1\mathrm{J}∕{\mathrm{cm}}^2$ retinal fluence), the initial temperature and radius of the bubble would be determined by the rate of energy transfer to the melanosome. For laser pulses such that the laser transfers all its energy on time scales shorter than the characteristic bubble formation time, which is the most damaging case, the water in contact with the melanosome will be heated up to the maximum allowed thermodynamic state of liquid-vapor coexistence, the critical point. Numerically we can estimate the bubble formation time by considering that it takes a time scale of a few hundred picoseconds for the vapor shell around the melanosome to double its specific volume.

As was done by Gerstman, we assume that the laser energy is used to raise the temperature of the melanosome to $374°\mathrm{C}$ , and to bring a certain volume of water to the critical point [cf. Eq. 11 in Gerstman ¹⁹]. The amount of water brought to the critical point is used to calculate the initial bubble radius. Because of the short time scales involved we further assume that the bubble will grow under adiabatic conditions and use the expression $P{V}^{\gamma }=\text{constant}$ , with $P$ and $V$ the pressure and volume, respectively, and $\gamma$ the ratio between the specific heat of the vapor at constant pressure and volume. Also in Eq. 12, $c_v=2.51\mathrm{J}∕\mathrm{g}\mathrm{K}$ is the specific heat of the melanosome, ${\rho }_0=1.35\mathrm{g}∕{\mathrm{cm}}^3$ is the static density of the melanosome, $\Delta T_c=374°\mathrm{C}-37°\mathrm{C}=337°\mathrm{C}$ is the temperature difference necessary to raise the melanosome to the critical point temperature of water, $P_0=218\mathrm{atm}$ is the critical point pressure, and $P_{\min }=1\mathrm{atm}$ is the ambient pressure, assumed to be the pressure inside the bubble at its maximum extension. The choice of $P_{\min }=1\mathrm{atm}$ was made by Gerstman ¹⁹ to balance two opposing effects: outward momentum of the liquid during bubble growth, which would cause an overshoot to $P_{\min }<1\mathrm{atm}$ inside the bubble and a larger $R_{\max }$ , versus viscous energy loss during expansion that would result in a smaller $R_{\max }$ . Using these values for the parameters, Eq. 12 for predicting bubble size as a function of fluence becomes

Table 1

As a function of αLa ; Percent of incident energy absorbed C(αLa) , and scaling factor S(αLa)=C(1.0)∕C(αLa) for fluence to produce an identical shock front and bubble as for any of the results presented in this study if αLa is changed from 1.0 ( αL=10,000cm−1 , a=1μm ).

The curve labeled RPVS is calculated from the well-known Rayleigh-Plesset³⁴ equation including viscosity and surface tension:

The difference between the Gerstman prediction and the RP line is due to the difference in the dynamics of the bubble growth. We see from Fig. 5 that the present numerical model and the Gerstman model both do a good job in explaining the experimental data, with the present numerical model better at lower fluences and the Gerstman model better at higher fluences. The RP line greatly overestimates the size of the bubble. The RPVS line does a better job, but also overestimates the bubble size. This overestimation by the RPVS model may be evidence that the growth of large bubbles is thermally controlled³⁴ by heat loss, which the RPVS model does not take into account. The better agreement between the Gerstman model and the experiments is possibly due to not including inertial effects, which compensates for not allowing heat loss during bubble growth. The present numerical model explicitly includes heat loss. Another possible explanation for the overestimation of maximum bubble size in the Gerstman and RPVS models is that bubble nucleation occurs at temperatures and pressures below the critical point¹⁹ for pulses longer than the characteristic bubble formation time. A lower nucleation temperature would lead to a lower initial vapor pressure and subsequently to smaller bubble sizes.

5.

Dependence of Shockwaves and Bubbles on Melanosome Properties

As discussed at the beginning of the previous section, important physical properties of melanosomes are not well known. To obtain the results presented in the previous sections, we chose specific values representing these properties. Since once they are measured, the values that we chose may turn out not to be those of melanosome, we also investigated how our results will change if the values of some of these parameters are changed. These results are also useful for investigations of interesting microparticles and nanoparticles other than melanosomes.

5.2.

Dependence on $B$

Since it does not occur as a simple factor as does ${\alpha }_L$ in $C\left({\alpha }_{L}a\right)$ , the effect of $B$ is not as simple to determine. For a specific absorber, such as the melanosomes in the retina, the prediction of the shockfront strength requires knowledge of the value of $B$ . Alternatively, if absorbing particles are inserted into a cell to generate shockwaves for biomedical applications, the $B$ of the particle material will affect the size of the shockfront as well as other characteristics of the pressure response such as ${\tau }_c$ . Thus, understanding the dependencies on $B$ is important for guiding the choice of particle composition. The shock front strength depends on $B$ in a nonlinear fashion, as seen in Fig. 6 . For six different pulse durations, ${\tau }_L=1\mathrm{ps}$ , $10\mathrm{ps}$ , $100\mathrm{ps}$ , $250\mathrm{ps}$ , $1\mathrm{ns}$ , and $10\mathrm{ns}$ , we calculate how the pressure amplitude at $u=2a$ depends on $B$ for $I_0=100\mathrm{mJ}∕{\mathrm{cm}}^2$ . We see that for a given ${\tau }_L$ , the value of $B$ of the absorbing particle controls the strength of the pressure wave that is generated. There are several competing effects that depend on $B$ . For ${\tau }_L<{\tau }_c$ , the shockwave pressure amplitude in the medium increases with $B$ approximately¹¹ as

Eq. 15

$$P_{\mathrm{liq}}\left(u\right)=\frac{\alpha B}{c_v}\left(\frac{v_{\mathrm{liq}}}{v_{\mathrm{liq}}+v_{\mathrm{abs}}}\right){\left(\frac{a}{u}\right)}^{\beta }C\left({\alpha }_{L}a\right)I_0,$$

where $v_{\mathrm{liq}}$ and $v_{\mathrm{abs}}$ are the acoustic speed of pressure waves in the liquid medium and absorber, respectively. However, as $B$ increases, ${\tau }_c$ decreases, as displayed in Eq. 11. For large enough $B$ , ${\tau }_c$ drops below ${\tau }_L$ and the shockfront strength then decreases with increasing $B$ , and softens into an acoustic wave. These two competing effects lead to a peak in the shockfront amplitude as a function of $B$ for the constant ${\tau }_L$ curves in Fig. 6. The graphs in Fig. 6 use $I_0=100\mathrm{mJ}∕{\mathrm{cm}}^2$ , ${\alpha }_L=\mathrm{10,000}{\mathrm{cm}}^{-1}$ , and two different value for $\alpha$ : Fig. 6(a) uses $\alpha =2.98\times {10}^{-5}{\mathrm{K}}^{-1}$ , and Fig. 6(b) uses $\alpha =2.4\times {10}^{-4}{\mathrm{K}}^{-1}$ .

Fig. 6

Dependence of the pressure response on the mechanical parameter $B$ . Maximum pressure at $r=2\mu \mathrm{m}$ for (a) $\alpha =2.98\times {10}^{-5}{\mathrm{K}}^{-1}$ and (b) $\alpha =2.4\times {10}^{-4}{\mathrm{K}}^{-1}$ . For large $\alpha$ and short pulse durations the curves peak around $B\approx 80\mathrm{GPa}$ . Values used for laser and absorber properties: $I_0=100\mathrm{mJ}∕{\mathrm{cm}}^2$ ; ${\alpha }_L=\mathrm{10,000}{\mathrm{cm}}^{-1}$ . The labels for the curves in (b) are the same as for the curves (a).

We found that the strength of the predicted shock front at $r=2a$ will be largest for an absorber with $B=80\mathrm{GPa}$ if $\alpha =2.4\times {10}^{-4}{\mathrm{K}}^{-1}$ . For a $10\text{-}\mathrm{ps}$ pulse, an absorber with $B=80\mathrm{GPa}$ will produce a shock front that is more than 40% stronger than an absorber with $B=20\mathrm{GPa}$ . As $B$ is increased above $80\mathrm{GPa}$ , the dropoff in the size of the expected shock front as a function of $B$ depends on the length of the laser pulse. We also see that the value of the absorber’s $\alpha$ plays an important role, which we investigate in the next section.

5.3.

Dependence on the Thermal Expansion Coefficient

The strength of the shock front and the size of the bubble both depend on the value used for $\alpha$ . We therefore investigated the effect of using an absorber with a different coefficient of thermal expansion $\alpha$ . Figure 7(a) shows the dependence of the shock font strength on $\alpha$ , and Fig. 7(b) shows the dependence of the bubble size on $\alpha$ . The values for other parameters used in the calculation for Fig. 7 were $I_0=880\mathrm{mJ}∕{\mathrm{cm}}^2$ , ${\alpha }_L=\mathrm{10,000}{\mathrm{cm}}^{-1}$ , and $B=39.4\mathrm{GPa}$ . The dependence of both shockwaves and bubbles on $\alpha$ is notably stronger¹¹ for ${\tau }_L<{\tau }_c$ .

Fig. 7

Dependence of the response on the thermal expansion coefficient $\alpha$ for (a) maximum pressure at $r=2\mu \mathrm{m}$ and (b) maximum bubble radius as a function of $\alpha$ for different pulse durations. Values used for laser and absorber properties: $I_0=0.88\mathrm{J}∕{\mathrm{cm}}^2$ , ${\alpha }_L=\mathrm{10,000}{\mathrm{cm}}^{-1}$ , and $B=39.4\mathrm{GPa}$ .

Another way of showing the strong dependence on $\alpha$ is presented in Fig. 8 , where the fluence required to produce a shock front of $1\mathrm{kbar}$ at $r=2a$ is plotted as a function of $\alpha$ . For Fig. 8 we set ${\tau }_L=0.1\mathrm{ns}$ , ${\alpha }_L=\mathrm{10,000}{\mathrm{cm}}^{-1}$ , and $B=80\mathrm{GPa}$ . The curve in Fig. 8 displays a $1∕\alpha$ dependence, as expected from the relationship between $P$ , $\alpha$ , and $I_0$ , shown in Eq. 15.

Fig. 8

Critical fluence $I_c$ required to obtain a shock front of $1\mathrm{kbar}$ at a position of $1\mu \mathrm{m}$ outside the surface of the melanosome $(r=2\mu \mathrm{m})$ as a function of the thermal expansion coefficient of the melanosome $\alpha$ . Values used for other parameters: ${\alpha }_L=\mathrm{10,000}{\mathrm{cm}}^{-1}$ , $B=80\mathrm{GPa}$ , and ${\tau }_L=0.1\mathrm{ns}$ . For reference we also give a best fit curve showing the $1∕\alpha$ dependence of the critical fluence.

Because of the uncertainty in the values of the physical properties of a melanosome, we determined a set of values that would produce the strongest shock front for a given fluence. We also required the values of the melanosome parameters to be reasonable based on existing evidence. We found that those values are $B=80\mathrm{GPa}$ and $\alpha =2.4\times {10}^{-4}{\mathrm{K}}^{-1}$ . Using these values for $B$ and $\alpha$ , we determined the fluence that would produce a shock front of $1000\text{bar}$ , and found that to be $I_0=120\mathrm{mJ}∕{\mathrm{cm}}^2$ if ${\alpha }_L=\mathrm{10,000}{\mathrm{cm}}^{-1}$ . This value of $120\mathrm{mJ}∕{\mathrm{cm}}^2$ decreases slightly for ${\tau }_L<100\mathrm{ps}$ .

6.

Summary

We have presented detailed results from calculations that show the expected size of shockwaves and bubbles generated by lasers pulses of various durations and fluences incident on a spherically absorbing melanosome immersed in an aqueous medium. The nonlinearity of the process requires careful attention to the numerical algorithm employed. The strength of the shock front drops rapidly as it expands outward, and eventualy decays to acoustic strength and speed. Whether or not these strong acoustic waves can still cause damage was not investigated.

We showed that stress confinement is valid for shockwaves in the surrounding medium when pulse durations are shortened below a characteristic absorber time. For melanosomes, this ${\tau }_c$ is likely to have an order of $100\mathrm{ps}$ . We also showed that cavitation confinement for bubbles also occurs, and that bubbles become relevant for pulse durations shorter than $100\mathrm{ns}$ . This difference in confinement time scales is a reflection of the difference in the dynamics; vaporization propagates outwards at speeds that are approximately $1∕50$ that of pressure waves.

A table was presented that gave a factor by which the fluence should be scaled to get the same pressure and bubble results for absorbers with different coefficients of absorption. We also showed that the size of the shock front that is generated depends on the physical properties of the absorbing melanosome, such as its bulk modulus and thermal coefficient of expansion. The importance of these properties makes it crucial for them to be measured independently.

The results of this paper can be used to guide biomedical procedures involving the retina or other tissues with strongly absorbing particles, and experiments that investigate shockwave and bubble formation from spherical absorbers in biological tissues or other materials. The results can also be used as a basis for determining the biological molecules or structures in a cell that are most important when an incident laser pulse causes damage.

Finally, note that the previous discussion bears relevance to understanding the transition between bulk and nonbulk properties in materials. From Eq. 11 we see that for a constant bulk modulus, the critical acoustic time depends linearly on the radius of the absorber. However, as the size of the absorber is reduced, we expect to eventually reach a regime where finite size effects will influence the bulk modulus of the material. Hence, as the size of the absorber is made smaller, and as one approaches the nonbulk regime, where finite size effects become important, one would observe a deviation from the linear relationship expressed in Eq. 11, when measuring the oscillation time of the absorber.