Cleaning processes of encrusted marbles by Nd:YAG lasers operating in free-running and Q-switching regimes Salvatore Siano, Fabrizio Margheri, Roberto Pini, Piero Mazzinghi, and Renzo Salimbeni

The removal process of degraded superficial layers from marble samples by Nd:YAG lasers was studied while simulating operative conditions of stone artwork restoration. The effects of laser irradiation at 1064 nm with three different pulse durations of 6 ns, 20 ms, and 200 ms were investigated by time-resolved shadowgraphy and emission spectroscopy of the ejection plume to characterize the specific interaction regimes, with particular concern given to the occurrence of side effects, such as thermal and mechanical damages to the substrate, that could affect the laser cleaning procedure. © 1997 Optical Society of America Key words: Laser applications, laser cleaning, Nd:YAG, stone restoration.

1. Introduction

The removal of encrustations and degenerated layers from stone artworks by means of laser divestment was proposed by Asmus et al. in the 70’s,1 though the general idea of using pulsed laser radiation as a light eraser to remove a high absorbing layer from the surface of a lower absorption substrate goes back to the origins of laser technology.2 Probably owing to the large gap between the methodologies of laser technology and art restoration and to a natural caution on the part of restorers concerning new techniques that could potentially endanger the integrity of artifacts of high artistic and historic value, the effects of laser interaction in this specific field have not been studied as extensively as in other applications of laser ablation. Nevertheless, in recent years some important restoration interventions on marble and stone artworks have been performed with laser assistance, for example, on the portals of Amiens Cathedral and Notre Dame and on sculptures such as Donatello’s San Giovanni from Florence’s Opera del Duomo Museum and the statue of

The authors are with the Istituto di Elettronica Quantistica, Consiglio Nazionale delle Ricerche, Via Panciatichi 56y30, I-50127 Firenze, Italy. Received 13 August 1996; revised manuscript received 10 March 1997. 0003-6935y97y277073-07$10.00y0 © 1997 Optical Society of America

William Huskisson from the National Museum of Man Island. These operative tests of laser cleaning have opened a wide discussion in the art restoration community about the effectiveness and safety of this new technique, which is still far from a final validation. Recently, new interdisciplinary forums and meetings have been established to join together scientists involved in both art restoration and laser technology.3 Concerning the choice of the laser source for stone cleaning, the requirements of easy handling and portability, fiber-optic delivery, high removal efficiency and selectivity, and finally large commercial diffusion and relatively low cost have favored, so far, Nd:YAG laser devices. Two operative regimes have been employed up to now: the Q-switching mode, with pulse lengths of 5–10 ns and energy as much as 1 Jypulse, and the free-running mode, with pulse durations of 0.1–1 ms and energy as much as tens of joules. In both cases, important side effects have been evident during laser cleaning of stones. Longer pulses can induce undesired thermal damages such as melting and vitrification of the substrate.4 On the other hand, high power short pulses develop strong photomechanical effects that can lead to increased surface roughness or at least to local fragmentation of the substrate.5 This last feature is not surprising. In fact, even though in reports on laser restoration the cleaning process has often been described as a fast vaporization, general studies devoted to other applications of laser ablation have revealed more complex 20 September 1997 y Vol. 36, No. 27 y APPLIED OPTICS

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thermoacoustic features ~see, for example, Refs. 6 and 7!. An emerging application for laser cleaning that presents close relationships with artistic stone restoration is the removal of superficial decay due to atmospheric pollution or of unwanted paints and graffiti from the walls of modern buildings. Here efficiency and speed of the cleaning process, more than the achievement of high precision and control, represent the crucial factors since, usually, large areas are subject to laser treatment. In this respect, Q-switched devices appear to be the best choice.8 In art restoration, although it has already been shown that the laser cleaning technique can substantially reduce the processing time,9 it is more difficult to demonstrate that this procedure always provides remarkable improvements with respect to conventional techniques. For example, whether the laser treatment can improve the quality of the cleaned surfaces and permits the preservation of the original patina of ancient sculptures is still under discussion.10 This represents a key factor for both early and long-term success of laser restoration, considering that, in addition to artistic and historic implications, the degree of roughness can determine the future resistance of stone artworks to environmental attacks. Furthermore, comparative tests on a large variety of samples are needed to isolate the conditions under which laser cleaning will allow conservative restorations in those cases for which traditional methods have been proved to cause damage, as, for example, in the removal of encrustation from weak substrates such as sandstone and corroded marble.9,11 To provide the basis for a quantitative description of the laser cleaning process of encrusted marble, we performed a preliminary study on the interaction regimes induced by Nd:YAG laser pulses of different durations, with particular concern given to the prevention of undesired side effects. Time-resolved shadowgraphy provided information on the ejection dynamics and on the associated thermal and photoacoustic phenomena. Spectroscopic analysis of the ablation plume was also performed to assess the presence of a plasma phase.

Fig. 1. Typical emission temporal profile of the SFR Nd:YAG laser.

with a system for laser beam diagnostics ~Big Sky 6.11! that for the two lasers revealed multimode distributions with no evidence of hot spots. A set of laboratory-produced samples simulating superficial degradation of marble artworks was prepared according to a standardized procedure, as per agreement with the restorers of the Opificio delle Pietre Dure in Florence, to obtain good repeatability of measurements. Each sample was composed of a substrate of white-gray Carrara marble of 3–5-mm thickness with a first deposit of a 100-mm-thick layer of calcium oxalate, representing the natural patina that should be safeguarded by the restoration intervention.5,10 A second layer of 500-mm-thick black gypsum ~96% gypsum, 3% carbon black, and 1% quartz powder! was deposited onto the first layer to simulate the degraded layer to be removed by laser ablation. After preparation, the samples were left to dry for several days prior to undergoing laser cleaning tests. Time-resolved images of the various phases of material removal were obtained with a pump-and-probe diagnostic setup,13 sketched in Fig. 2. The probe

2. Materials and Methods

Experiments have been carried out with two distinct Nd:YAG laser devices ~1064-nm emission wavelength! that provided three different pulse durations: ~1! a commercial laser ~Quanta Ray Model GCR-4! operating in two regimes: normal free-running ~NFR! mode with pulses of ;1 J and duration of 200 ms FWHM, and Q-switching ~QS! with a pulse energy of 500 mJ and duration of 6 ns FWHM; ~2! a homemade laser operating in a short pulse free-running mode ~SFR! especially designed for cleaning applications.12 This laser emits pulses of 1-J energy with 20-ms duration FWHM ~Fig. 1!, which is approximately the shortest value possible using all-solidstate flash-lamp drivers. The optical quality of laser emissions was checked 7074

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Fig. 2. Experimental set up for time-resolved shadowgraphy of the interaction volume during laser cleaning of encrusted marble.

beam from a nitrogen laser ~PRA Model LN 103, 337 nm, 0.5-ns pulse duration, 50-mJ pulse energy!, after spatial filtering and collimation, was sent tangentially to the sample surface to probe the air region where laser interaction and material removal occurred. The emissions of the probe laser and of the Nd:YAG laser were synchronized by a digital delay generator that permitted a temporal scan of the whole event with a time jitter of ;10 ns in the first few microseconds. The shadowgraphic patterns produced by local refractive-index perturbations and ejected particulates were detected by a CCD camera directly coupled to a frame grabber and stored in a personal computer for data analysis. A demagnification optic in front of the camera was sometimes used whenever a perturbation region larger than 5 mm in diameter was imaged. Plume spectroscopy was also performed to assess the occurrence of a plasma phase during laser-target interaction. The radiation emitted from the interaction volume was collected by an optical fiber and transmitted to the entrance slit of a 0.25-m spectrograph ~Jobin-Yvon Model M25!, equipped with a 300linesymm holographic grating. The spectra detection, recording, and analysis were performed by an optical multichannel analyzer ~EG&G PARC OMA III Model 1460!, with a 1024-channel intensified diode array detector ~Model 1420 G!. All the measurements were performed on the second shot of the laser at each location of the sample to avoid surface effects. The time evolution of the perturbation front was then measured for different shots and at different locations, moving the sample with a micromanipulator. Owing to the sample homogeneity, this was found to correlate well with the actual time evolution of the ablation process. 3. Results

In preliminary trials we measured the laser ablation thresholds of the black gypsum layer with the three available pulse lengths of 6 ns ~QS!, 20 ms ~SFR!, and 200 ms ~NFR! that were ;1, 5, and 14 Jycm2, respectively. Further measurements were performed at laser fluence in the ranges of 1–22, 10 –32, and 20 –130 Jycm2, respectively, for the three pulse durations reported above. Visual and microscopic observations revealed that at fluence levels of 1–3 Jycm2 ~QS!, 10 –20 Jycm2 ~SFR!, and 30 –50 Jycm2 ~NFR!, which have been assumed as operative ranges for stone cleaning, the calcium oxalate patina underlying the black gypsum deposit was not attacked by either of the longer laser pulses, though such an attack was observed to occur under QS laser irradiation. This result suggests that for the encrusted marble model we used in our tests, the QS cleaning process could not provide a sufficient degree of selectivity between the deposited layer and the substrate to be safeguarded, even at a fluence value just above the ablation threshold of the deposited layer. Some examples of the shadowgraphic images displaying material removal, as well as laser-induced

Fig. 3. Sequences of shadowgraphic images showing the evolution of the ablation plume and of the associated acoustic and thermal phenomena observed for three different sample irradiation conditions. ~a! QS pulse, 1.7 Jycm2; ~b! SFR pulse, 17 Jycm2; ~c! NFR pulse, 44 Jycm2. Real sizes of the image frames are ~a! 10.2 3 14.4 mm2, and @~b! and ~c!# 5 3 5 mm2.

acoustic and thermal effects developing in the air region before the target surface, are shown in Figs. 3~a!, 3~b!, and 3~c! for the 6-ns, 20-ms, and 200-ms pulse durations, respectively. The QS laser pulse generates an intense plasma, as confirmed by the spectroscopic analysis described below. The front of this plasma region expands at supersonic speed from the target surface and drives the development of a shock wave. The dark shadowgraphic ring corresponding to the shock front is typically observable at ;1 ms after laser irradiation. A dark cloud expanding at lower speed from the target surface is clearly indicative of the presence of ejected material and suggests that material removal involves mostly particulates opaque to the probe laser beam. These observations point out the relevant role of the photomechanical effect for the 6-ns pulse-induced ablation process, as is quantitatively demonstrated in Section 4. The complete detachment from the target surface of the ablated material cloud when irradiating with QS pulses occurs after ;200 ms @see Fig. 4~a!#. Conversely, time-resolved imaging of the effects of longer laser pulses does not show significant photomechanical effects. Only weak acoustic waves departing from the focal region are barely visible during the first microseconds of laser irradiation @see, for example, the first frame of Fig. 3~b!#; they are related to the first intense spikes of the pulse shape. For both 20- and 200-ms laser pulses, the sequence of frames shows the formation of a perturbed hot region, whose front expands at constant subsonic speed, as reported in Fig. 5 for the SFR pulse. Then the front 20 September 1997 y Vol. 36, No. 27 y APPLIED OPTICS

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Fig. 4. Last phase of the plume evolution for ~a! QS, ~b! SFR, ~c! NFR laser pulses.

of the hot region degenerates in turbulent motions. This occurs soon after the end of laser irradiation for the SFR pulse and during irradiation for the NFR pulse. The turbulence region produces a shadow that remains visible for as long as 1 ms after irradiation @Figs. 4~b! and 4~c!#. Time-integrated images of the plume emission for the three different irradiation regimes are reported in Fig. 6 to give an indication of the size of the whole heated region in each case. The spectra obtained from these ablation plumes points out the presence of plasma for both QS and SFR pulses but not for the NFR pulse. Typical QS and SFR spectra around 400 nm shows intense 393.3–396.8-nm lines of Ca II over a continuum @Fig. 7~a!#. Also, self-absorption is observable for these ionized lines and for the 422.7-nm Ca I neutral line, revealing a denser plasma in the region close to the target with respect to the outer shell. On the other hand, for NFR pulses, ionized calcium lines are detectable only at a fluence higher than 100 Jycm2 @Fig. 7~b!#.

Fig. 6. Time-integrated images of the plume detected by a standard CCD camera ~50-ms exposure time!. ~a! QS, ~b! SFR, ~c! NFR pulses.

gimes of laser-target interaction, depending on pulse duration: ~a! blast wave expansion induced by an optical detonation for QS pulses, ~b! laser-sustained combustion for SFR, and ~c! heating and vaporization for NFR. These behaviors are analyzed in some detail in the following subsections.

4. Analysis and Discussion

The time-resolved shadowgraphy together with the plume spectroscopy evidenced three different re-

Fig. 5. Experimental data and linear fittings of the expansion behavior of the front of the hot region induced by SFR pulses of increasing fluences. Front speeds of 106, 177, 211, and 276 mys result at laser fluences of 11, 17, 28, and 32 Jycm2, respectively. 7076

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Fig. 7. Spectra of the plume emission. ~a! QS, 1.7 Jycm2 fluence, showing emission lines of Ca I and Ca II; ~b! NFR at fluences in the range 40 –130 Jycm2. In this case, ionized Ca lines appear only at fluences higher than 100 Jycm2.

Fig. 8. Time evolution of the front of the shock wave induced by QS pulses of increasing fluences. Fittings combining planar and spherical expansion geometry have been calculated according to a model of optical detonation ~see Subsection 4.A!.

A.

QS Interaction

Laser pulses in the nanosecond range can give rise to a so-called optical detonation when the laser intensity is high enough to rapidly induce on the target surface a plasma that directly absorbs the radiation in a thin layer ~light-absorption wave!. The highly overcompressed and overheated gas forces a supersonic expansion of the plasma front, producing an extension of the ionized region ~hydrodynamic regime!. An approximate law of motion for the shock front evolution can be obtained starting from the expression of the detonation speed for a laser pulse with top hat spatial and temporal shapes14:

F

2~g2 2 1!ID cD 5 r0

G

1y3

,

(1)

where ID is the detonation intensity ~representing in this case a fraction of laser intensity IL!, r0 ' 1.2 kg m23 is the density of unperturbed air, g ' 1.2 is the adiabatic coefficient for air that can be assumed for temperatures of thousands of Kelvins.15 The jump of the gas density on the shock front is ryr0 5 ~g 1 1!yg ' 11. The detonation phase will be followed by planar and then spherical decays of the shock. Thus the complete time evolution of the front speed is described by the following approximate expressions:

5SD cD

1y3

tD cf ~t! 5 cD t cDtD1y3t14y15t23y5

t0 , t # tD tD , t # t1 ,

Fig. 9. ~a! Calculated front speed and ~b! pressure acting on the target surface for QS pulses with increasing fluences of 1.7, 3, and 22 Jycm2, respectively ~from bottom to top!.

(2)

t . t1

where tD 5 tL 2 t0, tL being the laser pulse duration, t0 the starting time of the normal detonation regime, and t1 the time of switching from planar to spherical decay.16 If laser irradiation is provided at a level that is much higher than the breakdown threshold of the target, one can reasonably assume that tD ' tL,

whereas in general the real duration of the detonation phase should be taken into account. The relationship between the front speed cf ~t! and the pressure acting on the target surface at any time is given by16 ps 5

S D g11 2g

2gy~g21!

r0 cf2. g11

(3)

Figure 8 reports the experimental data of the displacement of the shock front versus time for three fluence values of 1.7, 3, and 22 Jycm2 and the corresponding fittings obtained from integration of Eq. ~2!. Figures 9~a! and 9~b! report the front speed cf and the pressure at the target surface ps for the same laser fluence. This analysis shows that at fluences of 1–3 Jycm2 that are typical values for cleaning applications, the initial pressure is ;10 – 60 bar, whereas at 22 Jycm2 a peak pressure value as high as 410 bar can be produced on the target surface. The duration of pressure pulses is ;20 ns in all cases. It is worth noting that these pressure peaks can be considered as the lower limit of the real ones, since the above de20 September 1997 y Vol. 36, No. 27 y APPLIED OPTICS

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scription does not include the recoil of material ejection that is due to the plasma-target energy transfer. B.

SFR Interaction

The behavior observed with the SFR pulse suggests a description of the process in terms of a lasersustained combustion, where a plasma is initially generated and then is pumped by a laser intensity much lower than that required for optical breakdown.17 The plasma onset is favored by the high absorption of the dark layer and by some initial spikes of the laser pulse shape of high instantaneous power ~see Fig. 1!. In such a condition, only a plasma of low electron density can develop, characterized by a low absorption of the incident radiation and subsonic expansion of the front. The temperature inside this plasma region can be very high ~of the order of 104 K!,14 but the mass density is indeed low. Considering that the laser energy in this case increases the enthalpy of the gas H 5 ε 1 pyr ~where ε is the specific internal energy! rather than the internal energy and assuming an unperturbed value of H .. H0 from the energy conservation law applied to the front of the ionization zone, the following expression for the density jump can be obtained: gnf p r < , r0 ~g 2 1!IL

(4)

where nf is the speed of the hot front. The observed evolution of the front was found to be linear during laser irradiation ~Fig. 5! with estimated speeds in the range 100 –300 mys for laser fluences of 10 –30 Jycm2. Considering that the subsonic speed of the front can be associated with an almost negligible overpressure in the hot region with respect to the outer pressure, one can reasonably assume p ' 1 bar. Then Eq. ~4! gives ryr0 ' 0.7–2 3 1022 and the electron density has to be accordingly low. Such a plasma is quite transparent to the 1064-nm laser radiation, whereas some shielding effects take place in close proximity to the target surface, mostly because of the presence of ejected particles. In summary, it can be argued that plasma formation induced by SFR pulses does not play a crucial role in laser-target interaction, as was found to occur in the previous QS case. C.

NFR Interaction

The observed behavior for the NFR pulse is characterized by the presence of turbulent motions from the beginning of irradiation @Fig. 3~c!#. This case is expected to be similar to the regime induced by cw irradiation. In fact, shadowgraphs of the interaction region did not show any ejection of solid material or acoustic waves expanding from the target surface. This feature confirms that the material removal takes place by continuous vaporization from the target along the laser-pulse duration. The model that is often applied to describe the interaction process in similar irradiation conditions is the thermal model of a semi-infinite wall that pro7078

vides the following expression for the temperature rise at the target surface9,18:

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T~0, t! 5

Î

2AIL kt L , K p

(5)

where A is the target absorbance, IL is the laser intensity, K is the thermal conductivity, and k is the thermal diffusivity. The penetrating depth of heat is related to the pulse duration through the equation19: L 5 Î4kt L.

(6)

For NFR pulses one obtains L ' 20–30 mm, to be compared with L ' 5–10 mm for SFR pulses, which suggests a larger extension of the thermally damaged zone in the former case, as we typically observed. 5. Conclusions

This experimental study was intended to provide the restorers and scientists involved in conservation of marble and stone artworks with quantitative evaluations of the laser cleaning process, which could complete the picture usually furnished by subjective examinations and postprocess analysis of treated samples, such as those that have usually been reported to date. The study also helps clarify the mechanism of laser-material interaction and suggests some criteria to identify the optimum irradiation conditions for this specific application. In particular, we focused our attention on the dependence of ablation dynamics induced by Nd:YAG lasers on pulse duration in order to achieve preliminary indications of the optimum temporal range suitable for stone artwork cleaning. The analysis was conducted on the basis of time-resolved shadowgraphy and emission spectroscopy diagnostic techniques. Strong plasma-mediated photomechanical effects induced by QS pulses at typical operative fluence have been evident. It is reasonable to conclude that they are responsible for the roughness of cleaned surfaces, as reported by various authors. On the other hand, thermal side effects similar to those caused by cw irradiation are associated with long NFR pulses. A good compromise seems to be represented by SFR irradiation that was observed to induce a lasersustained combustion regime characterized by a weaker plasma formation that did not cause substantial shielding effects to the incoming laser beam and developed lower pressure values compared with QS operation, with no observable mechanical damage. Moreover, SFR pulsed ablation showed a clear selectivity with respect to QS irradiation in the removal of the absorbing layer by preserving the underlying patina. These observations lead us to conclude that, for optimum irradiation conditions, the pulse duration should be short enough to induce a pressure transient that is necessary for fast and efficient material removal, but not so short as to induce mechanical damage to the substrate. In this respect the SFR laser

source we developed and utilized for this study provided a preliminary validation of the possible advantages furnished by laser cleaning in the microsecond regime. The authors thank M. Mazzoni of Istituto di Elettronica Quantistica for the use of her laboratory facilities and M. Matteini from the Opificio delle Pietre Dure in Florence for helpful information about preparation of the experimental samples. This research has been supported by the Tuscany Regional Government and by the Special Project Cultural Heritage of the Italian National Research Council. References 1. J. F. Asmus, C. G. Murphy, and W. H. Munk, “Studies on the interaction of laser radiation with art artifacts,” in Developments in Laser Technology II, R. F. Wuerker, ed., Proc. SPIE 41, 19 –30 ~1973!. 2. A. L. Shawlow, “Lasers,” Science 149, 13–22 ~1965!. 3. See, for example, Proceedings of the Seventh International Congress on Deterioration and Conservation of Stone, J. Delgado Rodriguez, F. Menriques, and F. Telmo Jeremias, eds. ~Laboratorio Nacional de Engenhari Civil, Lisboa, Portugal, 1992!; Proceedings of Third International Symposium on the Conservation of Monuments in the Mediterranean Basin, V. Fassina, H. Ott, and F. Zerra, eds. ~Soprintendenza Beni artistici e storici di Venezia, Venice, Italy, 1994!; Workshop on Lasers in the Conservation of Artworks ~LACONA! ~Heraklion, Greece, 1995!; Proceedings of First International Congress on Science and Technology for the Safeguard of Cultural Heritage in the Mediterranean Basin ~to be published!. 4. M. I. Cooper, D. C. Emmony, and J. H. Larson, “A comparative study of the laser cleaning of limestone,” in Proceedings of the Seventh International Congress on Deterioration and Conservation of Stone, J. Delgado Rodriguez, F. Menriques, and F. Telmo Jeremias, eds. ~Laboratorio Nacional de Engenhari Civil, Lisboa, Portugal, 1992!, pp. 1307–1311. 5. M. S. D’Urbano, C. Giovannone, P. Governale, A. Pandolfi, U. Santamaria, “A standardized methodology to check the effects of laser cleaning of stone surfaces,” in Proceedings of Third International Symposium on the Conservation of Monuments

6.

7.

8. 9. 10.

11.

12.

13.

14. 15.

16. 17.

18.

19.

in the Mediterranean Basin, V. Fassina, H. Ott, and F. Zezza, eds. ~Soprintendenza Beni artistici e storici di Venezia, Venice, Italy, 1994!, pp. 955–962. K. Do¨rschel and G. Mu¨ller, “Photoablation,” in Future Trends in Biomedical Applications of Lasers, L. A. Svaasand, ed., Proc. SPIE 1525, 253–279 ~1991!. R. O. Esenaliev, A. A. Oraevsky, V. S. Letokhov, A. A. Karabutov, and T. V. Malinsky, “Studies of acoustical and shock waves in the pulsed laser ablation of biotissue,” Lasers Surg. Med. 13, 470 – 484 ~1993!. K. Liu and E. Garmire, “Paint removal using lasers,” Appl. Opt. 34, 4409 – 4415 ~1995!. J. F. Asmus, “More light for art conservation,” IEEE Circuits Devices Mag. 6 –14 ~March 1986!. M. Matteini and A. Moles, “Le patine di ossalato di calcio sui manufatti di marmo,” in Quaderni dell’Opificio delle Pietre Dure e Laboratori di Restauro di Firenze, Marble Restoration Issue ~Opus libri, Firenze, Italy, 1986!, pp. 38 – 45. M. I. Cooper and J. H. Larson, “Laser cleaning of marble sculpture,” in Abstracts of Lasers in the Conservation of Artworks ~Heraklion, Greece, 1995!, p. 6. F. Margheri and P. Mazzinghi, “A short pulse, free running Nd:YAG laser for stone artwork restoration,” Opt. Commun. ~to be published!. S. Siano, R. Pini, R. Salimbeni, and M. Vannini, “A diagnostic set-up for time-resolved imaging of laser-induced ablation,” Opt. Laser Eng. 25, 1–12 ~1996!. Yu. P. Raizer, Laser-Induced Discharge Phenomena ~Consultants Bureau, Plenum, New York, 1977!. Ya. B. Zel’dovich and Yu. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena ~Academic, New York, 1967!, Vol. 1. A. N. Pirri, “Theory for momentum transfer to a surface with a high-power laser,” Phys. Fluids 16, 1435–1440 ~1973!. Yu. P. Raizer, “Subsonic propagation of a light spark and threshold conditions for the maintenance of plasma by radiation,” Sov. Phys. JETP 31, 1148 –1154 ~1970!. M. I. Cooper, D. C. Emmony, and J. H. Larson, “The evaluation of cleaning of stone sculpture,” in Proceedings of the Bath Meeting, C. A. Brebbia and R. J. B. Frewer, eds. ~Computational Mechanics, Southampton, U.K., 1993!, pp. 259 –266. H. S. Carslaw, Conduction of Heat in Solids, 2nd ed. ~Clarendon Press, Oxford, 1989!, p. 75.

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