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

108 3. Interaction Mechanisms

In particular, the important feature of optical breakdown is that it renders possible an energy deposition not only in pigmented tissue but also in nominally weakly absorbing media. This is due to the increased absorption coe cient of the induced plasma. Thereby, the field of potential medical laser applications is considerably widened. Especially in ophthalmology, transparent tissues – like cornea and lens – become potential targets of laser surgery. It is also worthwile noting that during the process there is no restriction of the photon energy involved. The electron may increase its kinetic energy by means of absorbing any amount of energy, thus leading to a very short risetime of the free electron density of the order of picoseconds. In order to achieve optical breakdown, though, the irradiance must be intense enough to cause rapid ionization so that losses do not quench the electron avalanche. According to Smith and Haught (1966), inelastic collisions and di usion of free electrons from the focal volume are the main loss mechanisms during avalanche ionization.

As already emphasized, the electric field strength E is the crucial parameter for the onset of plasma-induced ablation. In the following paragraphs, a model of plasma-induced ablation is developed which is based on fundamental relations of electrodynamics and plasma physics. It derives a general expression for the dielectric factor ε and culminates in a relation for the absorption coe cient αpl of the plasma. For visible and near infrared laser wavelengths, the original absorption coe cient α of most tissues is quite low compared to using UV light. However, calculations derived from the following model will prove that αpl >> α, thus leading to enhanced absorption and e cient ablation.

3.4.1 Model of Plasma-Induced Ablation

Our model is based on Maxwell’s equations of electrodynamics relating the electric and magnetic field strengths E and H to the electromagnetic inductions D and B, respectively. In order to account for free electrons and currents associated with plasma formation, the inhomogeneous version of Maxwell’s equations is considered. In their di erential form, these equations are given by

The laser beam itself is approximated by a plane electromagnetic wave

where k is the propagation vector of the electromagnetic wave. Assuming the relative magnetic permeability of the tissue to be μ = 1, the electromagnetic inductions can be expressed by

where μ0 and ε0 are the electromagnetic constants, and ε is the dielectric factor of the plasma. Inserting (3.28)–(3.31) into the first two of Maxwell’s equations leads to

k × E = ωμ0H ,

k × H = −ωεε0E + ijf = −ωεε0E + iσE ,

where σ is the electric conductivity of the plasma. Hence,

k × (k ×E) = −ω2εμ0ε0E + iωμ0σE ,

where c = 1/ε0μ0 is the speed of light. In electrodynamics, (3.32) is also referred to as the telegraph equation. Since electromagnetic waves are transversal, i.e. kE = 0, the amount k = |k| can be expressed by

with the complex dielectric factor

In the case of negligible absorption, (3.33) reduces to the simple relation of dispersion

where n is the index of refraction. However, since the plasma absorbs incident radiation, (3.35) is replaced by

110 3. Interaction Mechanisms

where α˜ is the index of absorption. Combining (3.33) and (3.36) leads to the general expression

The absorption coe cient α – which is not to be mistaken for α˜ – was defined in the previous section by (3.20). Together with (3.23) and (3.28), we thus obtain

2ω

αpl = − 2 Im(k) = − c α˜pl ,

which expresses the plasma absorption coe cient αpl and the index of plasma absorption α˜pl in terms of each other. We may separate (3.37) into

stating that the existence of an imaginary part of ε evokes absorption.

Next, we will consider the equation of motion of a plasma electron. It is

where me is the electron mass, ve is the electron velocity, e is the electron charge, and νei is the mean collision rate of free electrons and ions. Assuming that the plasma electrons are performing oscillations induced by the incident electromagnetic field at its frequency ω, we obtain

Together with the basic relation jf = −Neve ,

where N is the density of free electrons, (3.40) turns into

Therefore, plasma absorption is enhanced in the IR region of the spectrum. And, since ωpl2 N and νei N, we obtain the important relation

αpl N2 ,

stating that the absorption is a nonlinear function of the free electron density and, thus, of the absorbed energy itself.

In Fig. 3.44, typical absorption coe cients αpl for various incident power densities have been collected. The plasma was induced in a cuvette filled with distilled water by picosecond pulses from a Nd:YLF laser at a wavelength of 1.053μm. The absorption coe cient has already gained two to three orders of magnitude if the incident energy density is twice the threshold value of plasma generation. In the example shown in Fig. 3.44, the absorption coe - cient increases from 0.1cm−1 at low fluences to 100cm−1 at approximately 23J/cm2. Therefore, the induced plasma serves as a shield for succeeding laser photons. This phenomenon is referred to as the plasma shielding e ect. At higher pulse energies, its significance even increases and will be discussed in Sect. 3.5. Primarily, the enhanced absorption of the plasma is caused by the high density of free electrons capable of absorbing laser photons. Thus, by means of plasma generation, a very e cient type of ablation is created.

112 3. Interaction Mechanisms

Fig. 3.44. Absorption of laser plasmas induced in distilled water by a Nd:YLF laser (wavelength: 1.053μm, pulse duration: 30ps). Unpublished data

The condition for plasma growth and sustainment is that losses – such as inelastic collisions and free electron di usion – do not quench the avalanche process. As the free electron density N increases, photon scattering is enhanced. Consequently, laser photons are not only absorbed but also scattered by the plasma. The critical density Ncrit at which a net amount of energy is not converted any further to plasma energy is obtained when the plasma frequency becomes equal to the frequency of the incident electromagnetic wave12. Mathematically, this condition is written as ωpl = ω and, when inserting (3.43),

For visible laser radiation, i.e. ω 1015 Hz, the electron density may thus reach values of up to a few 1020/cm3.

In the remaining part of this section, we will evaluate the threshold dependence of optical breakdown on pulse duration of the laser. In several studies

– as discussed below – a square root dependence of the threshold power density on pulse duration was observed when using picosecond or nanosecond laser pulses. This relationship, however, was only known as an empirical scaling law as stated by Koechner (1992). On the following pages, a novel model is developed which illuminates its theoretical background.

12Plasma oscillations are characterized by density-dependent eigenfrequencies. The highest amplitude occurs at resonance, i.e. at ωpl = ω. At ωpl > ω, the plasma oscillation is decelerated due to the phase shift of the external force.

The temporal behavior of the free electron density N(t) is described by

with the rate parameters β for avalanche ionization, γ for inelastic collision, and δ for electron di usion. Avalanche ionization is a two-step process. First, electrons gain energy in the electromagnetic field of the laser pulse, then further electrons are dissociated by subsequent collisions with either atoms or ions. According to Smith and Haught (1966), inelastic collisions and di usion of free electrons from the focal volume are the main loss mechanisms during avalanche ionization. They are accounted for in (3.47) by negative signs. The avalanche parameter β primarily depends on the incident intensity I(t). A reduced avalanche rate can be defined by

β = β −γ N(t) ,

which takes into consideration that some of the collisions are inelastic, i.e. no additional electrons are released but kinetic energy is transferred to the collision partners. This energy is lost for further avalanche processes and thus reduces the e ective rate parameter β . The rate parameter γ is multiplied by N(t), since the cross-section for the occurrence of inelastic collisions scales linearly with N(t). By substituting u(t) = 1/N(t), (3.47) turns into

∂u∂t + (β −δ)u(t) −γ = 0 .

This di erential equation has the following general solution

00

The initial condition is expressed by u(0) = 1/N0. Hence, it follows that c0 = 1/N0, where N0 is the initial electron density. After resubstituting N(t) in (3.48), the following general solution is derived

t

exp (β −δ)dt

1 + γ exp (β −δ)dt dt

N0

0 0

We will now assume a laser pulse with duration τ and constant intensity I0. According to Molchanov (1970), the parameter β can be approximated by

114 3. Interaction Mechanisms

where η is called the ionization probability and is expressed in units of (J/cm2)−1. In our model, multi-photon processes are not taken into account. It is rather assumed that the initial electrons for the avalanche are provided by thermal ionization as stated by Bloembergen (1974) who estimated an initial electron density of N0 108–1010 cm−3 and a breakdown density of

Nth 1018 cm−3.

– Case I: (0 ≤ t ≤ τ). By making the simplifying assumptions that the parameters η and δ are not time-dependent, (3.49) reduces to

e(β−δ)t

N(t) = N0 1 + βγN−0δ e(β−δ)t −1 .

According to Bloembergen (1974), a certain threshold ratio s can be defined for the occurrence of optical breakdown

It is now supposed that optical breakdown occurs near the end of the laser pulse, i.e. at t = τ, and that the free electron density saturates at this threshold condition. Hence,

Therefore, (3.51) reduces to

Because of (3.52), the logarithm in (3.53) is evaluated in a series at the

The rate parameters for inelastic collision and di usion are defined by γ = (Nττc)−1 and δ = τ−d 1, where τc and τd are the respective mean time constants. Then, together with (3.50), the following two relations are obtained for the threshold parameters of plasma generation

where Eth and Ith are the threshold values of incident energy density and intensity, respectively. The term (ln2 − 0.5) has been neglected, because Bloembergen (1974) estimated a typical ratio of s = ln(Nth/N0) 18, where Nth is the electron density at the threshold of optical breakdown.

– Case II: (t > τ). The integration in (3.49) is performed by means of

1 + γN0 e(β−δ)t dt + γN0 e(β−δ)τ−δ(t −τ) dt

0τ

with the final solution

eβτ−δt

N(t) = N0 1 + βγN−0δ e(β−δ)τ −1 − γNδ 0 eβτ(e−δt −e−δτ) . (3.58)

The last equation generally describes the decrease in electron density after the laser pulse due to collision and di usion processes.

In Figs. 3.45 and 3.46, the threshold dependences of incident energy density and intensity on pulse duration are illustrated, respectively, assuming τc = 1fs according to Bloembergen (1974) and τd = 500ps according to Zysset et al. (1989). It is supposed that the electron di usion time is of the order of plasma lifetimes achieved with picosecond pulses. With these values, numerical evaluation of (3.54) and (3.55) yields that Eth τ x and Ith τ x−1 with

Primarily, the two limits within the three time domains are determined by the time constants of inelastic collision and di usion modified by the empirical threshold parameter s. There remains, of course, an uncertainty in the absolute value of these time constants. This uncertainty is thus carried forward to the domain limits, i.e. insertion of longer time constants would shift the two limits to longer pulse durations, as well.

Indeed, the results of a series of experiments indicate a relationship of

√

the form Eth τ for pulse durations in the picosecond and nanosecond range. These studies have been made with a variety of anorganic target materials such as NaCl and SiO2 as performed by Fradin et al. (1973a), Bloembergen (1974), Vaidyanthan et al. (1980), van Stryland et al. (1981), Taylor et al. (1987), and Du et al. (1994). A similar threshold dependence was obtained for various biological tissues as investigated by Stern et al. (1989), Niemz et al. (1993a), and Vogel et al. (1994b).

In Table 3.12, the exponent in the relationship Eth τ x is listed for various targets and pulse durations. They are calculated from threshold data

which were published by di erent authors. Obviously, most of the values fit

√

very well to the postulated relationship Eth τ, i.e. x 0.5, except the data obtained by Zysset et al. (1989). The latter discrepancy might be caused by performing the experiments at extremely small focal spot sizes and the associated errors in threshold determination.

Table 3.12. Exponents in the relationship Eth τ x as calculated from published threshold data. Pulse durations shorter than 1ps were not considered to remain in the time domain of the square root dependence

In Figs. 3.47a–b, the threshold dependence of the incident energy density is illustrated in corneal tissue and teeth. The data were obtained with two di erent laser systems, a Nd:YLF laser and a Rhodamine 590 dye laser, respectively. The dye laser was necessary to provide pulse durations in the femtosecond range. From the slope of each curve, the corresponding exponents x = 0.52 and x = 0.46 are calculated, respectively. At the shortest pulse durations, both curves tend to decrease in slope.