Laser-Tissue Interactions Fundamentals and Applications - Markolf H. Niemz

128 3. Interaction Mechanisms

Fig. 3.55. Cavitation bubble within a human cornea induced by a single pulse from a Nd:YLF laser (pulse duration: 30ps, pulse energy: 1mJ, bar: 30μm)

shock wave formation even at the very threshold. Since adjacent tissue can be damaged by disruptive forces, the presence of these e ects is often an undesired but associated symptom. In contrast, picosecond or femtosecond pulses permit the generation of high peak intensities with considerably lower pulse energies. With these extremely short pulse durations, optical breakdown may still be achieved while significantly reducing plasma energy and, thus, disruptive e ects. Moreover, spatial confinement and predictability of the laser–tissue interaction is strongly enhanced.

Since both interaction mechanisms – plasma-induced ablation as well as photodisruption – rely on plasma generation, it is not always easy to distinguish between these two processes. Actually, in the 1970s and 1980s, all tissue e ects evoked by ultrashort laser pulses were attributed to photodisruption. It was only because of recent research that a di erentiation between ablations solely due to ionization and ablations owing to mechanical forces seems justified. For instance, it was found by Niemz (1994a) that, in the case of picosecond pulses, ablation without mechanical side e ects takes place at incident power densities of a few times the plasma threshold. Based on these findings and the theory describing the dependence of threshold parameters (see Figs. 3.45 and 3.46 in Sect. 3.4), approximate thresholds of both interaction types are listed in Table 3.14. The data represent estimated values for corneal tissue, assuming a mean ionization probability of about η = 10(J/cm2)−1 as obtained from Table 3.13.

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Table 3.14. Estimated parameters for the onset of plasma-induced ablation and photodisruption in corneal tissue

Pulse duration	Onset of plasma-induced ablation	Onset of photodisruption
	Energy density (J/cm2)	Energy density (J/cm2)
100fs	2.0	50
1ps	3.3	50
10ps	8.0	50
100ps	23	50
1ns	–	72
10ns	–	230
100ns	–	730
	Power density (W/cm2)	Power density (W/cm2)
100fs	2.0 × 1013	5.0 × 1014
1ps	3.3 × 1012	5.0 × 1013
10ps	8.0 × 1011	5.0 × 1012
100ps	2.3 × 1011	5.0	× 1011
1ns	–	7.2	× 1010
10ns	–	2.3	× 1010
100ns	–	7.3	× 109

The values listed in Table 3.14 are graphically presented in Fig. 3.56. Obviously, plasma-induced ablation is limited to a rather narrow range of pulse durations up to approximately 500ps. At longer pulse durations, the energy density necessary for achieving breakdown already induces significant mechanical side e ects.

In general, photodisruption may be regarded as a multi-cause mechanical e ect starting with optical breakdown. The primary mechanisms are shock wave generation and cavitation, completed by jet formation if cavitations collapse in fluids and near a solid boundary. In Fig. 3.57, a schematic sequence of these processes is illustrated indicating their relations to each other. Moreover, the distinction between photodisruption and plasma-induced ablation is emphasized.

The four e ects – plasma formation, shock wave generation, cavitation, and jet formation – all take place at a di erent time scale. This is schematically illustrated in Fig. 3.58. Plasma formation begins during the laser pulse and lasts for a few nanoseconds afterwards as already mentioned when discussing Fig. 3.49. This basically is the time needed by the free electrons to di use into the surrounding medium. Shock wave generation is associated with the expansion of the plasma and, thus, already starts during plasma formation. However, the shock wave then propagates into adjacent tissue and leaves the focal volume. Approximately 30–50ns later, it has slowed down to an ordinary acoustic wave. Cavitation, finally, is a macroscopic e ect start-

130 3. Interaction Mechanisms

Fig. 3.56. Distinction of plasma-induced ablation and photodisruption according to applied energy density

Fig. 3.57. Scheme of the physical processes associated with optical breakdown. Percentages given are rough estimates of the approximate energy transferred to each e ect (incident pulse energy: 100%). Cavitation occurs in soft tissues and fluids only. In fluids, part of the cavitation energy might be converted to jet formation

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ing roughly 50–150ns after the laser pulse. The time delay is caused by the material during the process of vaporization. Usually, the cavitation bubble performs several oscillations of expansion and collapses within a period of a few hundred microseconds as will be shown below. Since the pressure inside the bubble again increases during collapse, each rebound of the cavitation bubble is accompanied by another shock wave. Furthermore, every collapse can induce a jet formation if the bubble is generated in the vicinity of a solid boundary. Each of the e ects contributing to photodisruption will be discussed in detail in the following paragraphs, because they all have their own physical significance.

Fig. 3.58. Approximate time scale for all processes contributing to photodisruption. Assumed is a 30ps laser pulse. The first and second occurrences of shock wave, cavitation and jet formation are indicated

3.5.1 Plasma Formation

The principles of laser-induced plasma formation have already been considered in Sect. 3.4. It should be emphasized, however, that the amount of energy absorbed during photodisruption is typically two or more orders of magnitude higher than during plasma-induced ablation. This is an immediate consequence of the di erent energy densities associated with either process as already emphasized in Fig. 3.1. Thus, the free electron density and the plasma temperature are also higher than for purely plasma-induced ablation. Therefore, in photodisruptive laser–tissue interactions, the following three e ects are enabled or become more significant:

1323. Interaction Mechanisms

•plasma shielding,

•Brillouin scattering,

•multiple plasma generation.

Once formed, the plasma absorbs and scatters further incident light. This property “shields” underlying structures which are also in the beam path. The importance of the plasma shielding e ect for medical laser applications was first recognized by Steinert et al. (1983) and Puliafito and Steinert (1984). In ophthalmology, the retina is considerably protected by this plasma shield during laser surgery of the lens or the vitrous. In Fig. 3.44, we have already encountered an increased absorption coe cient of the plasma. However, with the conditions of plasma-induced ablation, a significant amount of laser energy is still transmitted by the plasma. During photodisruptive interactions, thus at denser plasmas, the absorption coe cient is even enhanced, and the plasma serves as a very e ective shield.

In Brillouin scattering, incident light is scattered by thermally excited acoustic waves and shifted in frequency corresponding to potential phonon frequencies of the material. During the heating process of the plasma, acoustic waves are generated which lead to Brillouin scattering. When applying even higher irradiances, the laser light itself may create alterations in optical density by which, in turn, it is scattered. Accordingly, this e ect is called stimulated Brillouin scattering. It was described in detail by Ready (1971).

Finally, at the very high electric field strengths achieved during photodisruption, multiple plasma generation is enabled. Whereas close to the ablation threshold only one spark is induced at the very focus, several plasmas can be ignited at higher pulse energies. In the latter case, only the first section of the laser pulse will induce a plasma at the focal spot. Then, as the fluence increases during the pulse, succeeding radiation may also generate optical breakdown before reaching the smallest beam waist. Thus, a cascade of plasmas is initiated pointing from the focal spot into the direction of the laser source. This e ect is schematically illustrated in Fig. 3.59.

Fig. 3.59. Cascade of laser-induced plasmas. A Gaussian-shaped laser beam is incident from the left

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Plasma formation in distilled water has been extensively studied by Docchio et al. (1986), Zysset et al. (1989), and Vogel et al. (1994a). Some of their results were presented in Sect. 3.4 when discussing the threshold behavior of laser-induced optical breakdown. Moreover, Vogel et al. (1994a) have determined plasma sizes with time-resolved photography. They have observed that the length of plasmas is strongly related to the pulse duration. Their measurements are summarized in Fig. 3.60.

Fig. 3.60. Plasma length as a function of incident pulse energy. Measured with a Nd:YAG laser (pulse duration: as labeled, focal spot diameter: 4μm) in distilled water. Data according to Vogel et al. (1994a)

Obviously, plasmas induced by 30ps pulses are approximately 2.5 times as long as plasmas induced by 6ns pulses of the same energy. Only at the respective thresholds of breakdown are the latter slightly longer. These di erent plasma lengths – and thus volumes – result in a considerably lower energy density of plasmas induced by picosecond pulses. Indeed, Vogel et al. (1994a) have observed a significant di erence in the corresponding intensities of visible plasma fluorescence. Moreover, the plasma volume determines the fraction of incident energy to be converted to shock waves or cavitations. If the plasma volume is larger – like in plasmas induced by picosecond pulses – more energy is required for ionization and vaporization of matter. Hence, this amount of energy can no longer contribute to the generation of potential shock waves or cavitations. Therefore, we can conclude that plasmas induced by picosecond pulses are less likely to cause mechanical tissue damage than plasmas from nanosecond pulses.

134 3. Interaction Mechanisms

The overall sequence of plasma formation is summarized in Table 3.15. In order to distinguish the physical parameters of plasma-induced ablation and photodisruption, two typical pulse durations of 10ps and 100ns are considered. These correspond to typical values of mode locked or Q-switched pulses, respectively. Energy densities and power densities at the threshold of plasma formation apply for corneal tissue and are taken from Table 3.14. Associated electric field strengths are calculated using (3.23). The critical electron density at the plasma threshold is derived from (3.46). It is not directly related to the pulse duration of the laser but does depend on its wavelength. In Sect. 3.4, we estimated typical densities of 1018/cm3 for visible laser radiation. For the process of photodisruption, higher electron densities were obtained by Boulnois (1986). Finally, the absorption coe cient of the plasma is given by (3.45). It is also wavelength-dependent and determines the extent of the plasma shielding e ect. In the case of plasma-induced ablation, some measured absorption coe cients are listed in Fig. 3.44. Although these data apply for water only, similar values can be assumed for corneal tissue because of its high water content. During photodisruption, higher absorption coe cients are accessible due to the increased electron density.

Table 3.15. Physical parameters of plasma formation in corneal tissue

Pulse duration	10ps		100ns
	8.0J/cm2		730J/cm2
Energy density
	8.0 × 1011	W/cm2	7.3 × 109	W/cm2
Power density
	2.5 × 107		2.3 × 106
Electric field strength		V/cm		V/cm
	1018–1019	/cm3	1018–1020	/cm3
Electron density of plasma
Nonlinear absorption of plasma	1–100/cm		1–10000/cm
(Plasma shielding)

The values listed in Table 3.15 provide a good estimate of the physical parameters associated with optical breakdown. In a first approximation, they apply for other targets, as well. In order to achieve a similar plasma electron density, roughly 100 times the energy density is needed when applying 100ns pulses rather than 10ps pulses. Therefore, provided the same focus size is chosen, plasmas induced by nanosecond pulses contain significantly more energy. This additional amount of energy must somehow dissipate into the surrounding medium. It is partly converted to the generation of shock waves, cavitation, and jet formation as will be discussed next.

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3.5.2 Shock Wave Generation

As discussed in Sect. 3.4, laser-induced optical breakdown is accompanied by a sudden adiabatic rise in plasma temperature to values of up to a few 10000K. Primarily, this temperature can be attributed to the kinetic energy of free electrons. Due to their high kinetic energy, the plasma electrons are not confined to the focal volume of the laser beam but rather di use into the surrounding medium. When the inert ions follow at a certain time delay, mass is moved which is the basic origin of shock wave generation. This shock wave soon separates from the boundary of the plasma. It initially moves at hypersonic speed and eventually slows down to the speed of sound. In Fig. 3.61, the geometry of shock wave generation is illustrated.

Fig. 3.61. Geometry of shock wave generation

Laser-induced shock waves in water were first investigated by Carome et al. (1966), Bell and Landt (1967), and Felix and Ellis (1971). Shock waves di er from sonic acoustic waves by their speed. Whereas the speed of sound in water, for instance, is 1483m/s at 37◦C, laser-induced shock waves typically reach speeds of up to 5000m/s at the very focus. Both hypersonic shock waves and sonic acoustic waves are referred to as acoustic transients. In order to derive a relation describing the pressure gradient at the shock front, let us consider a slab of tissue with a cross-section A0 which is passed through by a shock front at a speed us as seen in Fig. 3.62. During a time interval of dt, the shock front moves a distance of dxs, thus

us = ddxts .

The pressure inside the medium and its density are p0 and 0, respectively. The shock front induces a sudden increase in local pressure from p0 to p1 and of the density from 0 to 1. The conservation of mass demands compensation by other particles intruding from the left side in Fig. 3.62. These particles move at a particle speed up which is usually lower than us. During a time

136 3. Interaction Mechanisms

Fig. 3.62. Geometry of shock front moving through a slab of tissue

interval dt, a mass of ( 1 − 0)A0dxs must be provided. This is achieved at the particle speed up from a zone with a higher13 density 1:

up 1A0dt = ( 1 − 0)A0dxs .

	Hence,
	up =	1 − 0	us .	(3.61)
		1

Beside the conservation of mass, the conservation of momentum must also be fulfilled. A mass A0 1dxs begins to move at a speed up and thus receives a momentum A0 1updxs. This momentum is provided by two means:

–The mass A0 1updt intrudes at a speed up, thereby supplying a momentum of A0 1u2pdt.

–The shock front induces an increase in pressure from p0 to p1. This pressure gradient induces a mechanical force A0(p1 −p0) which generates a momentum A0(p1 −p0)dt during the time interval dt.

The conservation law of momentum thus asks for
A0 1updxs = A0 1up2dt + A0(p1 − p0)dt ,	(3.62)
or
p1 −p0 = 1upus − 1up2 .	(3.63)
Inserting (3.61) into (3.63) leads to a pressure increase
p1 −p0 = 0upus .	(3.64)

An empirical relationship between the shock speed us and particle speed up was first determined by Rice and Walsh (1957). For water at high pressures exceeding 20kbar, the following expression applies:

13In our model, the shock front has just passed the left area in Fig. 3.62, thus leaving a higher density 1 behind.

				3.5 Photodisruption	137
us = 1.483 + 25.306 log10	1 +		up	,	(3.65)
		5.19

where up and us must be inserted in units of km/s. For pressures lower than 20kbar, an approximation was given by Doukas et al. (1991), i.e.

us = a + bup ,

(3.66)

where a is the speed of sound and b is a dimensionless constant. In the case of water, b = 2.07 was estimated by Zweig and Deutsch (1992). Assuming a spherical shock wave with radius r, the conservation of momentum (3.62) leads to

4πr2 1upusΔt = c0 ,

where Δt is the risetime of the shock front, and c0 is a constant denoting the overall momentum. Replacing up within the last equation by an expression obtained from (3.66) yields

c1 us(us −a) = r2 ,

with

b

c1 = 4π 1Δt c0 ,

and the final solution

us(r) = 2		+					(3.67)
			4	+ r12 .
	a		a2		c

The parameter c1 can be empirically obtained. The particle speed is derived from (3.66)

up(r) = − 2b		+ b						(3.68)
				4	+ r2 .
	a	1		a2		c1

Finally, the resulting pressure is obtained by inserting the expressions for particle speed up and shock speed us into (3.64), i.e.

	0c1 1
p1(r) = p0(r) +	b r2 .	(3.69)

From (3.64) and (3.65), we can also derive two relationships of the form

us = us(p1) ,

up = up(p1) ,

respectively. In the case of water, they are graphically presented in Fig. 3.63. At p1 = 0kbar, the shock speed us approaches 1483km/s, whereas the particle speed up remains at 0km/s. However, these relationships can also be