1. Introduction

The formation of the plasma and the cavitation bubble, followed by the subsequent propagation of the pressure front has been of interest in many laser-induced-breakdown (LIB) experiments performed in water [1–3]. The particular fields of interest are medical applications, especially, ophthalmic procedures such as posterior capsulotomy or iridotomy. The pulses from a Q-switched Nd:YAG laser are focused inside the eye, where the tissue is evaporated. In posterior capsulotomy the focus lies in the vitreous body just behind the intraocular lens, and in iridotomy the focus is on the iris. Even though the ocular microsurgery is performed by evaporating the tissue in the focal region of the laser beam, the cavitation bubble dynamics, the pressure-front propagation and its amplitude are of considerable interest as it is important to avoid adverse side effects on many delicate structures inside the eye that are located in close proximity to the focus. In iridotomy these may include the trabecular meshwork and the ciliary body, and in the case of capsulotomy the vitreous body and the implanted intraocular lens. The well-documented adverse effects associated with capsulotomy are an intraocular pressure increase, glaucoma, cystoid macular edema, retinal breaks or even retinal detachment [5,6] and intraocular lens damage [7]. The mechanism of retinal breaks and detachment after Nd:YAG posterior capsulotomy and the role of vitreous in this process is still not fully understood. However, the damage to the intraocular lens is known to be caused by focusing the Nd:YAG laser too close to the intraocular lens [7,8], and increasing the distance from the breakdown site to the intraocular lens was suggested as a way of minimizing the damage soon after the inception of the technique [8].

The methods previously used in the characterization of shock waves include hydrophone measurements [1] and different optical techniques, such as schlieren streak cameras and framing photography [2,3] or holographic recording [4]. Here we present an optodynamic method for the measurement of the cavitation bubble and the shock-wave front propagation. Two methods of measurement were employed, based on beam-deflection and interferometry in order to measure optodynamic signals. A beam-deflection probe beam was used to determine the shock-wave velocity at each particular position during two dimensional scanning. The shock wave locally changes the density, causing in turn a variation in the refractive index. The refractive-index gradient results in the deflection of the probe beam [9,10]. An arm-compensated interferometer was used to determine relative position of the LIB. The pulse durations under investigation in previous experiments spanned several orders of magnitude, from hundred-femtosecond-long pulses to hundred-nanosecond-long pulses. In this study we will limit ourselves to the region of commercially available photodisruptors where the pulse lengths lie in the nanosecond range.

2. Experimental

A schematic diagram of the experimental setup is shown in Fig. 1. It consists of an infrared, 1064-nm Q-switched Nd:YAG OptoYAG (Optotek d.o.o.) laser, designed for ocular photodisruption. Pulses with energy and length 13 mJ and 7 ns, respectively, were applied to the deionized water. The output focusing achromat lens was mounted into a holder whose position was controlled using a stepper motor. Vertical and horizontal scanning was realized by moving the lens and the beam-deflection probe, respectively, both in steps of 50 µm. The LIB was generated in deionized water.

The main principle is based on two-dimensional scanning of the beam-deflection probe and time-of-flight measurements. Two types of optical methods were used to determine the shock-wave velocity propagating from the LIB. First, close to the LIB the beam-deflection probe was applied to measure the shock-wave time-of-flight. Second, an arm-compensated Michelson interferometer was used to measure the relative position of the LIB during scanning. The propagation of the cavitation bubble boundary was measured simultaneously with the same probes. Therefore in order to obtain data representing the time-of-flight and determine their relative position in the two-dimensional mesh the optodynamic signal from beam-deflection probe and interferometer was used, respectively.

 

Fig. 1. Schematic diagram of the experimental setup. A Nd:YAG laser was used to induce the breakdown in water. A He-Ne laser and a quadrant diode with appropriate electronics represent the beam-deflection probe; another He-Ne laser together with control electronics was used in the compensated Michelson interferometer.

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Since the lens was not in the water, the vertical displacement of the LIB site was not equal to the displacement of the lens due to refraction of the laser light at the air-water interface. Therefore, the position of the LIB site was determined by measuring the shock wave propagation time from the LIB site to the bottom of the water vessel, where the measuring mirror of the arm-compensated Michelson interferometer was fixed [11,12]. The output of the interferometer was fed to the photo detector. The high- and low-frequency parts of the signal were used to determine the leading edge of the shock-wave signal and to compensate the interferometer, respectively. The interferometer compensation was performed with a carefully designed electronic circuit with proportional-integral-derivative (PID) control. The signal from this circuit was fed to the piezo transducer, which compensated for the length of the reference arm. The interferometer was thus held in the range of its maximum sensitivity by compensating for any low-frequency mechanical disturbance from the environment. This relatively complex control was needed in order to operate the interferometer in an ordinary laboratory environment without any special requirements.

The other important part of the setup was the beam-deflection probe [9]. It consisted of a quadrant diode detector and a He-Ne probe beam that was focused close to the area of the LIB. When the pressure-shock front, generated by the LIB, crossed the path of the probe beam, it deflected, and consequently a signal was generated on the detector. The time-of-flight for the pressure front can be determined from the leading edge of the signal. A detailed description of the beam-deflection probe’s signal resulting from the shock wave spreading into the surrounding air or liquid is presented elsewhere [9,10].

3. Results and discussion

The signals from the beam-deflection probe and the Michelson interferometer are shown in Fig. 2. Not all the positions from a single, vertical scan close to the “incident beam” axis are shown; only every tenth signal trace was plotted in order to make the graph clearer. The vertical distance between consecutive curves corresponds to 500 µm. The top half of the curves corresponds to the probe beam being above the LIB site, while the bottom half of the curves corresponds to the probe being below the LIB site. After the shock waves and their reflection from the water boundary, lower-frequency signals can also be clearly seen (see Fig. 2(a)). They correspond to the expansion of the cavitation bubble. Although appearing simultaneously with the shock-wave front, the dynamics of this process is much slower. Fig. 2(b) shows the initial part of the same measurements. The curves on the graph represent the pressure front and the cavitation bubble at an early stage. The duration of the disturbance corresponding to the deflection of the probe beam due to the passage of the pressure-shock wave-front is relatively short. It is determined by the width of the probe beam and the width of the pressure front.

 

Fig. 2. Typical optodynamic responses of the beam-deflection probe (BDP) with different horizontal scales (a) and (b). They were used for determinations of the time-of-flight for the shock wave and the boundary of the cavitation bubble. Every tenth signal trace is shown, corresponding to a vertical displacement of 500 µm between the neighboring traces. The traces are approximately symmetrically distributed with respect to the LIB center. Signals from the compensated Michelson interferometer (MI), from which the relative positions of the LIB region are determined is also shown (c).

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Figure 3 shows the two-dimensional field of the time-of-flight measurements for the shock-wave pressure-front and the cavitation bubble, respectively. At an early stage of the spreading the pressure front is slightly conical (see Fig. 3(a)). This agrees with the shape of the laser-induced breakdown zone, reported elsewhere [2,3]. The basic mechanism of the appearance and the expansion of the breakdown zone can be explained as follows. After the start of the pulse the intensity of the laser beam first reaches the breakdown threshold in the focal plane. The plasma develops, and due to its strong absorption of the pulse the breakdown region expands toward the laser. The final breakdown zone, which represents the source of the shock waves and the cavitation bubble, has a conical shape in the initial stage of the process. However, during the further expansion the shock-wave front transforms to a spherical shape. Since the direction of the spreading is perpendicular to the iso-time lines, the gradients of the time-of-flight field are calculated in order to determine the velocity of the shock-wave front. It depends directly on the pressure, and is therefore important for our study because of its potential for having destructive effects. The pressure amplitude difference p_S between the pressure in the shock wave and in the surrounding water is calculated from the velocity-field data v_S using the following relation [3]:

where ρ ₀ is the mass density before compresion, c ₀ is the velocity of sound and c ₁ and c ₂ are empirical constants with values of 5190 m/s and 25306 m/s [13].

 

Fig. 3. Two-dimensional time-of-flight diagrams for shock wave (a) with contour spacing 30 ns, and cavitation bubble (b) with contour spacing 2 µs.

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The pressure amplitudes are shown in Fig 4(b). The decrease of the pressure with the distance from the LIB site can be theoretically predicted from the following relation for the shock-wave energy flux: j=$p_{\mathrm{S}}^2$(r)/(2ρ ₀ c ₀), where p_S was already introduced in Eq. (1), c ₀ is the velocity of sound and ρ ₀ is the mass density and r is distance from LIB center. The assumption that the energy of the wave is conserved, and therefore j∝1/r ², leads to the relation p∝1/r. It is clear from Fig 4(b) that in the region close to the breakdown (r <600 µm) the pressure decreases much faster, i.e., approximately as 1/r ^2.3. This discrepancy shows that the energy of the shock wave is not conserved and the potentially destructive flux of the mechanical energy is strongly attenuated in this region. When the distance increases from 150 µm to 600 µm the pressure amplitude drops by almost a factor of 14 (see Fig. 4(b)), indicating that the energy flux decreases almost 200 times. This is much faster than it should do for purely geometrical reasons, i.e., approximately 16 times. It is evident that the contribution due to the loss of the mechanical energy represents the important part of the flux drop, i.e., 12 times. It should be noted that in the above estimation we consider only the amplitude of the pressure; we neglect the effect of the shock-wave broadening during its propagation in water. The broadening is a non-linear process depending on the amplitude of the initial pressure. The measured and calculated values of the shock-wave broadening that can be found elsewhere [13] indicated the factor of 2 on much larger distance (from 0.1 mm 10 mm). Therefore we conclude that the broadening is not the decisive factor for the pressure amplitude decrease.

In the region where r>600 µm we find that the pressure is proportional to 1/r ^1.2. This result is close to the theoretical prediction for the case without energy loss (1/r), hence we called this area the non-attenuated region. From application point of view it is important that functional cell damage appears if the pressure of the shock waves exceeds 50 to 100 MPa [14]. These pressure values coincide with the boundary between the highly attenuated and non-attenuated region obtained with our measurements.

 

Fig. 4. Experimental shock-wave velocity (a) and pressure amplitude (b) in dependence on the distance from the LIB center (note the log scale). The velocity was obtained from of the time-of-flight measurements. The velocity data were smoothed by assuming that velocity decreases with distance from LIB site. Pressure was determined from velocity data using Eq.(1).

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It should be emphasized that the threshold for the LIB formations and the acoustic impedance of the water is close to that of the ocular fluids. Therefore the pressure amplitude (Fig. 4(b)) obtained from the experimental data can be directly applied to the real problem. For example, based on results from our experimental setup potential cell functional damage zone due to shock wave can be estimated during Nd:YAG procedures for given laser parameters.

4. Conclusion

We have mapped the velocity and pressure fields near the laser-induced breakdown site. The results of the optodynamic analysis show that two regions of the pressure front exist: a region with high attenuation of the shock wave, and a region where the shock wave spreads almost undisturbed. The pressure amplitude in the first region decreases much faster than the conservation of mechanical energy would predict, indicating a strong attenuation mechanism. Most of the energy is thus contained close to the breakdown site. The analysis of the measurement of the cavitation-bubble spreading shows a much slower dynamics.

The existence of the two attenuation regions is important for understanding the adverse effects of Nd:YAG procedures. One might speculate that the functional cell damage and the damage to the intraocular lens, lying inside the highly attenuated region, is governed by different processes than the liquefaction of the vitreous or its posterior detachment from the retina, where the distances involved are much larger – placing these effects in the non-attenuated region. It should be noted that plasma formation and the dynamics of the cavitation bubble also play an important role in damage. Increasing the distance from the LIB site to the intraocular lens, a feature known in commercially available photodisruptors as “offset”, will place the lens from the highly attenuated region into the non-attenuated region. This is in accordance with empirical clinical findings [7,8]. We believe that the experimental method presented here enables further research into the mechanisms of intraocular-lens damage.