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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 24 — Nov. 23, 2009
  • pp: 21652–21665
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Laser-induced damage of KDP crystals by 1ω nanosecond pulses: influence of crystal orientation

Stéphane Reyné, Guillaume Duchateau, Jean-Yves Natoli, and Laurent Lamaignère  »View Author Affiliations


Optics Express, Vol. 17, Issue 24, pp. 21652-21665 (2009)
http://dx.doi.org/10.1364/OE.17.021652


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Abstract

We investigate the influence of THG-cut KDP crystal orientation on laser damage at 1064 nm under nanosecond pulses. Since laser damage is now assumed to initiate on precursor defects, this study makes a connection between these nanodefects (throughout a mesoscopic description) and the influence of their orientation on laser damage. Some investigations have already been carried out in various crystals and particularly for KDP, indicating propagation direction and polarization dependences. We performed experiments for two orthogonal positions of the crystal and results clearly indicate that KDP crystal laser damage depends on its orientation. We carried out further investigations on the effect of the polarization orientation, by rotating the crystal around the propagation axis. We then obtained the evolution of the damage probability as a function of the rotation angle. To account for these experimental results, we propose a laser damage model based on ellipsoid-shaped defects. This modeling is a refined implementation of the DMT model (Drude Mie Thermal) [Dyan et al., J. Opt. Soc. Am. B 25, 1087-1095 (2008)], by introducing absorption efficiency calculations for an ellipsoidal geometry. Modeling simulations are in good agreement with experimental results.

© 2009 Optical Society of America

1. Introduction

This paper deals with laser-induced damage (LID) of potassium dihydrogen phosphate (KH2PO4 mostly known as KDP), a uniaxial anisotropic crystal. It is a transparent dielectric material, preferentially used for its nonlinear optical and electro-optical properties. These characteristics are both required in specific facilities as LMJ (Laser MegaJoule, in France) or NIF (National Ignition Facility, in US). In such powerful facilities designed to reach ICF (Inertial Confinement Fusion), optical components are submitted to laser illuminations that may lead to damage and decrease their performances. Due to the dimensions required, KDP used in LMJ and NIF are specifically sliced from boules obtained by rapid growth techniques (a 50×50×50 cm3 crystal is obtained in two months [1

1. N. P. Zaitseva, J. J. De Yoreo, M. R. Dehaven, R. L. Vital, L. M. Carman, and H. R. Spears, “Rapid growth of large-scale (40–55 cm) KDP crystals,” J. Cryst. Growth 180, 255–262 ( 2001). [CrossRef]

]). Theses techniques induce new issues between conventionally- and rapidly- grown KDP, although it has been shown that there is no difference (in terms of material property [2

2. A. Ciapponi, S. Palmier, F. R. Wagner, J. Y. Natoli, H. Piombini, D. Damiani, and B. Bertussi, “Laser-induced fluorescence as a tool for the study of laser damage precursors in transparent materials,” Proc. SPIE 6998, 69981E-1 ( 2008). [CrossRef]

]). Among these issues, laser damage resistance and the understanding of damage physical processes are the main concerns to deal with. It is now conceded that laser damage initiates via the existence of precursor defects present or induced in the material. As it is difficult to get information on the nature of these initiators whose size is probably smaller than 100 nm [3

3. M. D. Feit and A. M. Rubenchik, “Implications of nanoabsorber initiators for damage probability curves, pulselength scaling, and laser conditioning,” Proc. SPIE 5273, 74 ( 2004). [CrossRef]

], quantifying their role in laser damage remains a topic of interest.

In order to understand the LID mechanisms and to obtain information on the nature of precursor defects, the literature on KDP mainly deals with chemicals analyses (spectra, structural and composition determination, etc) and non-destructive/destructive in-situ investigations (scattering, local absorption, fluorescence measurements, laser damage tests, etc) [4

4. M. Pommiès, D. Damiani, B. Bertussi, H. Piombini, H. Mathis, J. Capoulade, and J. Y. Natoli, “Detection and characterization of absorption heterogeneities in KH2PO4 crystals,” Opt. Commun. 267, 154–161 ( 2006). [CrossRef]

-8

8. H. Yoshida, T. Jitsuno, H. Fujita, M. Nakatsuka, M. Yoshimura, and T. Sasaki, “Investigation of bulk laser damage in KDP crystal as a function of laser irradiation direction, polarization, and wavelength,” Appl. Phys. 70, 195–201 ( 2000). [CrossRef]

]. Although we do not report here the overall list of KDP studies, the drastic increase of publications since 2000’s accounts for the progresses in the understanding of LID. As we interest more specifically in the nature of nanodefects, we remind here some examples of results to be considered as important in KDP studies. As KDP preferentially damages in bulk, mother-solution inclusions, impurities (metallic or not), lattice substitution, dislocations, etc have been naturally proposed [4

4. M. Pommiès, D. Damiani, B. Bertussi, H. Piombini, H. Mathis, J. Capoulade, and J. Y. Natoli, “Detection and characterization of absorption heterogeneities in KH2PO4 crystals,” Opt. Commun. 267, 154–161 ( 2006). [CrossRef]

]. But a clear correlation has not been highlighted between laser damage and absorption cartographies (or spectra), neither with impurities levels, nor the crystal sectors. In other concerns, surface contaminants have been revealed by fluorescence microscopy as potential precursors to laser damage [5

5. S. G. Demos and M. Staggs, “Application of fluorescence microscopy for noninvasive detection of surface contamination and precursors to laser-induced damage,” Appl. Opt. 41, 1977–1983 ( 2002). [CrossRef] [PubMed]

]. Other studies deal with the influence of polarization and propagation directions [6

6. A. K. Burnham, M. Runkel, M. D. Feit, A. M. Rubenchik, R. L. Floyd, T. A. Land, W. J. Siekhaus, and R. A. Hawley-Fedder, “Laser-induced damage in deuterated potassium dihydrogen phosphate,” Appl. Opt. 42, 5483–5495 ( 2003). [CrossRef] [PubMed]

-8

8. H. Yoshida, T. Jitsuno, H. Fujita, M. Nakatsuka, M. Yoshimura, and T. Sasaki, “Investigation of bulk laser damage in KDP crystal as a function of laser irradiation direction, polarization, and wavelength,” Appl. Phys. 70, 195–201 ( 2000). [CrossRef]

]. Burnham and co-workers report that laser-induced pinpoint bulk damage of (D)KDP (the partially deuterated form of KDP) at 351 nm depends on propagation direction relative to the crystallographic axes, not the polarization [6

6. A. K. Burnham, M. Runkel, M. D. Feit, A. M. Rubenchik, R. L. Floyd, T. A. Land, W. J. Siekhaus, and R. A. Hawley-Fedder, “Laser-induced damage in deuterated potassium dihydrogen phosphate,” Appl. Opt. 42, 5483–5495 ( 2003). [CrossRef] [PubMed]

]. Yoshida et al. have evaluated the dependence of damage threshold at 1064 nm. It seems to follow preferential axis direction, consistent with the molecular bonding structure in different direction of the crystal [8

8. H. Yoshida, T. Jitsuno, H. Fujita, M. Nakatsuka, M. Yoshimura, and T. Sasaki, “Investigation of bulk laser damage in KDP crystal as a function of laser irradiation direction, polarization, and wavelength,” Appl. Phys. 70, 195–201 ( 2000). [CrossRef]

].

2. Experimental set-up and procedure

2.1 Facility set-up

This study has been performed on BLANCO facility (Banc LAser Nanoseconde pour Composants Optiques) at CEA/Cesta in France (see Fig. 1). This experimental system has been previously described in details in [15

15. S. Reyné, M. Loiseau, G. Duchateau, J.-Y. Natoli, and L. Lamaignère, “Towards a better understanding of multi-wavelength effect on KDP crystals,” Proc. SPIE 7361, 73610Z ( 2009). [CrossRef]

]. It is based on a Q-switched Nd:YAG laser supplying the 1064 nm fundamental wavelength (1ω). Laser injection seeding ensures a longitudinal monomode beam and a stable temporal profile. The laser delivers approximately 800 mJ at a nominal repetition rate of 10Hz. The laser beam is P-polarized and its polarization remains the same for the whole study (other configurations with circular or elliptical polarization have not been tested here). The laser beam is focused into the sample by a convex lens which focal length is f=4000 mm. It induces a depth of focus (DOF) higher than the sample thickness, ensuring the beam shape to be constant along the DOF.

Fig. 1. Scheme of BLANCO facility (see more details in [15])

The equivalent pulse duration τeq is 6.5 ns, captured by a phototube (with ≈70 ps rise/fall times, 1-σ standard deviation is about 2% of the average value). The beam equivalent surface Seq (i.e. defined as the surface given at 1/e for a Gaussian beam) is determined at the equivalent plane corresponding to the distance d between the focusing lens and the sample. At the focus point, the beam spot is millimetric Gaussian-shaped and diameter (at 1/e) is 700 ±27µm. The energy measurements are sampled by pyroelectric cells and systematically compared with full-beam calorimeter calibrations. Shot-to-shot laser fluence fluctuations (about 7%) are mainly due to fluctuations of Seq. This is the reason why fluence is determined for each shot. Hence the absolute fluence determination is driven by the whole measurement path so as fluence is given with an accuracy of as much as ±15%.

2.2 Procedure and metrology set-up

This study is divided into three parts. The first one concerns the acquisition of the spatial profiles of 1ω (transmitted) and 2ω (generated by Second Harmonic Generation (SHG)) beams at the exit of the KDP crystal. Secondly, we carried out energy measurements of these beams as a function of the rotation angle Ω. The third one is dedicated to the study of laser damage as a function of Ω.

In both of the two first metrologies, the energy and spatial determinations result from the mean of a set of 100 consecutive shots. We use this procedure to rid of shot-to-shot fluctuations, which are 6-7% for spatial measurements and <2% for energy measurements (at 1-σ standard deviation). Nevertheless, this procedure entails a conditioning effect. To achieve the first part of the tests, a specific system has been set-up. It consists in introducing a 1ω R max mirror (99.9% reflectivity, P-polarized with a P/S extinction ratio of 100:1) to separate 1ω and 2ω beams on two distinct paths. Optical densities (OD) are added to protect and filter (block) the detectors from residual beams. Either the CCD cameras or the pyroelectric cells are used to carry out the appropriate measurements. Spatial profiles are measured through CCD cameras placed in a plane where the beam magnification γ has been determined to 1.01±0.02. The pyroelectric cells are used to measure the energy of each shot. At the end, these two steps give us access to the energetic balance (in J/cm2, which is easy to understand when compared to damage tests), i.e. the 1ω transmission and the 2ω generation. At the same time, we are able to evaluate the losses induced by the crystal. A scheme of the set-up is given on Fig. 2.

Fig. 2. Scheme of the metrology set-up for the energy measurements of 1ω and 2ω beams. For spatial measurements, set-up is slightly modified. Image of the beam at the exit window is ensured through a 4-f set-up. A coupled system “waveplate + polarizer” adapted for each path is inserted in each path to adjust energy on the CCD cameras.

Concerning the third part of the experiments, laser damage tests yield to the Standard [16

16. ISO Standard No 11254-1:2000; ISO Standard No 11254-2:2001

]. We follow the 1-on-1 procedure and we use the specific data treatment for the crystal metrology (described in [17

17. L. Lamaignère, T. Donval, M. Loiseau, J. C. Poncetta, G. Razé, C. Meslin, B. Bertussi, and H. Bercegol, “Accurate measurements of laser-induced bulk damage density,” Meas. Sci. Technol. 20, 095701 ( 2009). [CrossRef]

]) to extract damage densities. About twenty sites are tested per fluence, on the basis of a ten of fluences set. We detect damage sites of few tens of microns. The laser damage pinpoints are illuminated by a He-Ne laser beam propagating collinearly to the test beam. Laser damage appears preferentially in the bulk of the crystal and is observed through a x20 objective mounted on a CCD camera. Note that surface damage appears at fluences higher than those obtained for bulk damage, so surface damage is not taken into account. For each laser shot, we determine if a damage site occurs or not, so as we extract a probability as function of the fluence. Each occurrence is measured to make results independent to the laser fluctuations. Since it is not possible to detect and visualize the precursor defects with the current techniques [4

4. M. Pommiès, D. Damiani, B. Bertussi, H. Piombini, H. Mathis, J. Capoulade, and J. Y. Natoli, “Detection and characterization of absorption heterogeneities in KH2PO4 crystals,” Opt. Commun. 267, 154–161 ( 2006). [CrossRef]

], we can not correlate the damage sites to the precursor defects. So, post-mortem observation of damage sites is not performed in this study, and no correlation between the defect shape and the crystal structure is done.

Finally, the KDP sample originates from a prismatic sector of a 2001 rapidly-grown boule using continuous filtration technique [18

18. N. Zaitseva, J. Atherton, R. Rozsa, L. Carman, I. Smolsky, M. Runkel, R. Ryon, and L. James, “Design and benefits of continuous filtration in rapid growth of large KDP and DKDP crystals,” J. Cryst. Growth 197, 911–920 ( 1999). [CrossRef]

,19

19. N. P. Zaitseva, I. L. Smolsky, L. Carman, R. Ryon, and Z. U. Rek, “Connection between filtration of solution during growth and defect structure of KDP group crystals,” 1998 SSRL Activity Report, Experimental progress Reports 7, 434–437 ( 1998).

]. The crystal is cut for type-I third harmonic generation (THG), with a cutting angle α=59°. It was produced and polished by Cleveland Crystals, Inc (CCI). The sample is a pristine plate polished on the sides, with dimensions of 100×100×10 mm3.

3. Results

We performed several sets of experiments at 1064 nm to highlight the influence of KDP crystal orientation. In anisotropic non linear crystals, several phenomena due to the electric susceptibility tensor χ (n) (with n the order of the tensor) are likely to happen assuming specific conditions and material properties. These processes may also induce an orientation dependence on LID. So we have to state whether or not the orientation influence on LID is due to the precursor defects. Also, we investigate the crystal inhomogeneity, self-focusing, walk-off and second harmonic generation (SHG) as potential candidates that might influence laser damage. Other effects may occur: either we do not have sufficient knowledge on them or we do not interest in since considered as weak. We report here (i) the main results extracted from preliminary studies entailed to evaluate the contribution of each candidate, and (ii) the results showing the influence of the KDP orientation, i.e. as a function of Ω.

3.1 Preliminary studies

Fig. 3 presents the evolution of the laser damage density as a function of 1ω fluence. Tests at 1ω have been performed for two orthogonal positions of the crystal, i.e. (a) the laser polarization along the ordinary axis (blue triangles), (b) the laser polarization along the extraordinary axis (red squares).

Fig. 3. Evolution of the laser damage density as a function of 1ω fluence. Blue triangles correspond to ordinary position and red squares to extraordinary position. Fitting curve (dotted points) is given by a power law. Inset: Representation of the two orthogonal test positions of the KDP crystal. In this configuration, the rotation angle Ω is 90° between these positions. o.a. is the optical axis of the crystal.

According to Fig. 3, results clearly appear different between the two positions as we can estimate a shift of about 10 J/cm2. This implies a factor 1.4–1.5 on the fluence at constant damage density. Many assumptions may be done to explain these observations. Crystal inhomogeneity was sometimes mentioned [18

18. N. Zaitseva, J. Atherton, R. Rozsa, L. Carman, I. Smolsky, M. Runkel, R. Ryon, and L. James, “Design and benefits of continuous filtration in rapid growth of large KDP and DKDP crystals,” J. Cryst. Growth 197, 911–920 ( 1999). [CrossRef]

-21

21. N. Zaitseva, L. Carman, I. Smolsky, R. Torres, and M. Yan, “The effect of impurities and supersaturation on the rapid growth of KDP crystals,” J. Cryst. Growth 204, 512–524 ( 1999). [CrossRef]

] since the beginning of studies on rapidly-grown crystals. Before improvements on crystal growth, it was tricky to determine damage threshold assuming the differences between crystal boule and even samples cut from the same sector. It was particularly stressed by the distinction between the prismatic and pyramidal sectors. So crystal inhomogeneity prevailed over damage threshold determination methods. But due to improvements in measurements repeatability [17

17. L. Lamaignère, T. Donval, M. Loiseau, J. C. Poncetta, G. Razé, C. Meslin, B. Bertussi, and H. Bercegol, “Accurate measurements of laser-induced bulk damage density,” Meas. Sci. Technol. 20, 095701 ( 2009). [CrossRef]

] and even considering error margins, results would not overlap. So we assert that the crystal inhomogeneity can not explain this shift between the two positions. In other concerns, self-focusing while well identified is not quantified with a good accuracy as the value of non linear index for KDP is very difficult to find in literature. Even if the impact of self-focusing is not clearly determined, we assess that self-focusing can be neglected considering the level of fluence for these tests (<50 J/cm2). As for walk-off, KDP crystal exhibit weak anisotropy, inducing a birefringence closer to 20 mrad. This is the reason why comparing the length of the sample with the length on which the energy of each beam is superposed, we can neglect the birefringence issue.

I(2ω,l)=2ω2deff2l2n2ωnωc3ε0(sin(Δk.l2)Δk.l2)2I2(ω).
(1)

One can see that Eq. (1) is maximized for the phase matched condition Δk=0. Else, conversion oscillates as sinc(Δk.l2).

Finally, even it is not the purpose of this paper, we briefly evaluated the main phenomena encountered in KDP crystal, whatever their linear or non-linear character. We ensure that these mechanisms are not the main contributors (even existing, participating or not) to this fluence shift. Our assessments are also in agreement with literature relative to (non)-linear effects in crystals, qualitatively considering the same range of operating conditions (ns pulses, beams of few hundreds of microns in size, intensity level below a hundred GW/cm2, etc).

3.2 Energy measurements at 1ω and 2ω̣

The first set of experiments deals with 1ω and 2ω measurements at the exit of the crystal. Fig. 4(a) shows the evolution of the ratios T 1ω and R 2ω as a function of the rotation angle Ω. T 1ω and R 2ω respectively represent the ratios between (a) 1ω-transmitted energy (E 1ω_out) and 1ω-inlet energy (E 1ω_in), and (b) 2ω-converted energy (E 2ω_out) and 1ω-inlet energy (E 1ω_in). These results are necessary as the initial step to determine if SHG may contribute to damage or not.

Fig. 4. (a) Correlation between 1ω and 2ω energy transmission at the exit of the crystal as a function of Ω, with fluence F 1ω=16.5 J/cm2 at the entrance. This correlation is highlighted by black arrows, corresponding to peaks of SHG. (b) Evolution of crystal losses (by reflections, intrinsic absorption, etc) as a function of Ω, for different 1ω-fluences.

Fig. 4(a) presents the results obtained for a F 1ω=16.5 J/cm2. For clarity, Fig. 4(a) does not present the whole measurements obtained with five different fluences F 1ω. We would also indicate that at F 1ω=16.5 J/cm2, one would expect to observe damage site (according to Fig. 3 for position Ω=0°). But no damage has been observed. It is due to the measurements procedure, previously presented in subsection 2.2, that makes the crystal to get conditioned. This effect is usually studied in crystals to improve their laser damage resistance [13

13. G. Duchateau, “Simple models for laser-induced damage and conditioning of potassium dihydrogen phosphate crystals by nanosecond pulses,” Opt. Express 17, 10434–10456 ( 2009). [CrossRef] [PubMed]

,23

23. P. Demange, R. A. Negres, C. W. Carr, H. B. Radousky, and S. G. Demos, “Laser-induced defect reactions governing damage initiation in DKDP crystals,” Opt. Express 14, 5313–5328 ( 2006). [CrossRef] [PubMed]

,24

24. P. Demange, C. W. Carr, R. A. Negres, H. B. Radousky, and S. G. Demos, “Multiwavelength investigation of laser-damage performance in potassium dihydrogen phosphate after laser annealing,” Opt. Lett. 30, 221–223 ( 2005). [CrossRef] [PubMed]

]. Red and green curves represent T 1ω and R 2ω ratios respectively. Fig. 4(a) clearly shows the correlation between the variations of 1ω and 2ω energies. We also notice on the green curve the apparition of four SHG peaks (see arrows), three of them (for Ω=-55°, 45° and 65°) representing less than 5% of 1ω-inlet fluence whereas the fourth (for Ω=-30°) reaching about 20% of 1ω-inlet fluence. This value has led us to wonder whether the 2ω fluence is a key factor in laser damage. This will be discussed in section 3.3. But we do not perform complementary tests to eventually propose an interpretation of the role of a second harmonic when coupled to another during damage tests.

Fig. 4(b) represents the sum of T 1ω and R 2ω obtained for five different 1ω-inlet fluences, in order to evaluate the crystal losses (induced by reflection, intrinsic absorption, etc). Losses are estimated to be around 12 to 15 %, which is consistent with losses both induced by reflection (3.5% on each face, i.e. 7%) and absorption (it is commonly found 0.05 cm-1 to 0.07 cm-1 in literature for a KDP crystal). Moreover, the more E1ω_in is, the more E2ω_out and the less E1ω_out are. But in the end, the energetic balance (i.e. the sum of T 1ω and R 2ω) remains nearly constant whatever Ω and F 1ω. Thus, considering Fig. 4(a) and Fig. 4(b), it is possible to evaluate the level of 1ω and 2ω for each position Ω and for each fluence F 1ω_in. For a laser-induced damage fluence F 1ω_in, if now considering the positions Ω=0° and Ω=90° where T 1ω is comparable, one would expect not to observe differences on laser damage probability (or density). And yet, Fig. 3 shows that laser damage is different for these two positions. The analysis of Fig. 3, Fig. 4(a) and Fig. 4(b) suggests that SHG is not responsible for the laser damage orientation dependence at 1ω.

3.3 Laser damage probability vs. angle Ω at 1ω

The second set of experiments is dedicated to the study of the laser damage probability as a function of the rotation angle Ω. It is worth noting that rotating the crystal is equivalent to turning the beam polarization. We performed this test for two different fluences F 1ω (i.e. at 19 J/cm2 and 24.5 J/cm2) to investigate a potential effect due to fluence. Note that the choice of these F 1ω test fluences allows scanning damage probabilities in the whole range [0; 1]. Fig. 5 illustrates the damage probability as a function of Ω. Red squares and blue triangles respectively correspond to tests carried out at F 1ω=24.5 J/cm2 and at F 1ω=19 J/cm2. Curves in dashed points only remind the SHG signal (estimated from Fig. 4(a)) for these fluences.

Fig. 5 highlights two important information: (i) the influence of crystal orientation on LID, (ii) and how the SHG (i.e. mixing wavelengths) can influence the damage probability when its level becomes significant. To address the first point, let us interest in the variations of the damage probability as a function of Ω. In the range [-90°, 90°], apart from the points referenced by the black arrows (i.e. the peaks of SHG), we observe that globally laser damage probability increases and decreases in the ranges [-90°, 0°] and [0°, 90°] respectively. If we interest more precisely in the range [0°, 30°], the damage probability decreases progressively while the SHG level remains constant and weak. This confirms that SHG can definitely not explain this behavior. If considering now the points corresponding to the peaks of SHG locations (see black arrows), we notice that the laser damage probability is punctually altered. For these cases only, SHG tends to cooperate as soon as the 2ω level becomes upper than 1 J/cm2. In particular, for the maximum SHG peak at Ω=-30° (we would also include the point at Ω=-20°), damage probability saturates, meaning that 2ω fluence generated (about 6-7 J/cm2 for Ω=-30° considering F 1ω=24.5 J/cm2) is sufficient to enhance probability. Indeed, the more the F 1ω is, the more F 2ω is generated, and the more the probability is modified. So probability curves interpretation becomes delicate since damage probability is no more driven by only one wavelength but by two (and their mixing). It sets the glimpse of the coupling efficiency of two wavelengths. We are not able to explain yet the mechanisms due to wavelengths mixing in these punctual cases. But reader can refer to studies dealing with KDP laser damage under multi-wavelength illumination [15

15. S. Reyné, M. Loiseau, G. Duchateau, J.-Y. Natoli, and L. Lamaignère, “Towards a better understanding of multi-wavelength effect on KDP crystals,” Proc. SPIE 7361, 73610Z ( 2009). [CrossRef]

,24

24. P. Demange, C. W. Carr, R. A. Negres, H. B. Radousky, and S. G. Demos, “Multiwavelength investigation of laser-damage performance in potassium dihydrogen phosphate after laser annealing,” Opt. Lett. 30, 221–223 ( 2005). [CrossRef] [PubMed]

,25

25. P. Demange, R. A. Negres, A. M. Rubenchik, H. B. Radousky, M. D. Feit, and S. G. Demos, “The energy coupling efficiency of multiwavelength laser pulses to damage initiating defects in deuterated KH2PO4 non linear crystals,” J. Appl. Phys. 103, 083122 ( 2008). [CrossRef]

] and how it infers on damage.

Fig. 5. Evolution of laser damage probability as a function of Ω, for two different 1ω fluences. Blue triangles and red squares respectively correspond to F 1ω=19 J/cm2 and F 1ω=24.5 J/cm2, with scale on the vertical left axis. The vertical right axis represents the 2ω fluence level obtained by SHG as a function of Ω. The corresponding curves (blue and red dashes) are plotted to conveniently identify the correlation between peaks of SHG and damage probability enhancements (see black arrows).

4. Modeling and comparison to experiments

In this section, we first report a brief description of the DMT modeling and how computations have been performed by mixing the DDScat code (developed by Draine et al. in [14

14. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. B 11, 1491–1499 ( 1994). [CrossRef]

]) and the DMT model. We also present the improvements implied by the introduction of the ellipsoidal geometry. Then the numerical results are compared to experiments in order to state on the validity of this model.

4.1 DMT modeling description

Pre-existing DMT code developed under the framework of Dyan et al. [11

11. A. Dyan, F. Enguehard, S. Lallich, H. Piombini, and G. Duchateau, “Scaling laws in laser-induced potassium dihydrogen phosphate crystal damage by nanosecond pulses at 3ω,” J. Opt. Soc. Am. B 25, 1087–1095 ( 2008). [CrossRef]

] gives reliable laser damage modeling. This model considers a distribution of independent defects whose the temperature elevation is based on three physical mechanisms briefly explained just below: heat transfer [26

26. R. W. Hopper and D. R. Uhlmann, “Mechanisms of inclusion damage in laser glass,” J. Appl. Phys. 41, 4023–4037 ( 1970). [CrossRef]

,27

27. H. S. Carslaw and J. C. Jaeger, Conduction of heat in solids, (Oxford Science Publications, 2nd ed, 1959).

], Mie theory [28

28. H. C. Van de Hulst, Light scattering by small particles, (Dover publications, Inc., New York, 1981).

] and a Drude model. KDP is supposed to contain nanometer-scale defects that may initiate laser damage. Model solves the 3-D heating equation of these nanometer plasma balls (assimilated to spherical nanodefects actually) whose absorption efficiency is described through the Mie theory, introducing both the wavelength- and size-dependence. The plasma optical indices mandatory to Mie theory equations are then evaluated within the Drude model. Fourier’s equation resolution gives the sphere temperature evolution as a function of the precursor defect radius a. First assuming that a damage site appears as soon as the critical temperature Tc is reached, and then considering that any defect leads to a damage site, damage density is obtained from Eq. (2):

ρ(F)=a_(F)a+(F)Ddef(a).da
(2)

Where [a-(F), a +(F)] is the range of defects size activated at a given damage fluence level, Ddef(a) is the density size distribution of absorbers assumed to be (as expressed in [3

3. M. D. Feit and A. M. Rubenchik, “Implications of nanoabsorber initiators for damage probability curves, pulselength scaling, and laser conditioning,” Proc. SPIE 5273, 74 ( 2004). [CrossRef]

]):

Ddef(a)=Cdefap+1
(3)

Where Cdef and p are adjusting parameters. This distribution is consistent with the fact that the more numerous the precursors (even small and thus less absorbing), the higher the damage probability. The critical fluence Fc necessary to reach the critical temperature Tc for which a first damage site occurs can be written as [11

11. A. Dyan, F. Enguehard, S. Lallich, H. Piombini, and G. Duchateau, “Scaling laws in laser-induced potassium dihydrogen phosphate crystal damage by nanosecond pulses at 3ω,” J. Opt. Soc. Am. B 25, 1087–1095 ( 2008). [CrossRef]

]:

Fcγ.(TcT0)Qabs(ac).τx
(4)

The main parameters for running the DMT code are set as follow. Parameters can be divided into two categories: those that are fixed to describe the geometry of the defects (e.g. the aspect ratio) and those we adapt to fit to the experimental damage density curve for Ω=0° (Tc, n1, n2, Cdef, and p). The value of each parameter is reported in Table 1 and their choice is explained below. We assume a critical damage density level at 10-2 d/mm3 (it is consistent with experimental results in Fig. 3 that it would be possible to reach with a larger test area). This criterion corresponds to a critical fluence Fc=11 J/cm2 and a critical temperature Tc=6000 K. This latter value agrees qualitatively with experimental results obtained by Carr et al. [31

31. C. W. Carr, H. B. Radousky, A. M. Rubenchick, M. D. Feit, and S. G. Demos, “Localized dynamics during laser-induced damage in optical materials,” Phys. Rev. Lett. 92, 87401 ( 2004). [CrossRef]

], other value (e.g. around 10000 K) would not have significantly modified the results. Complex indices have been fixed to n1=0.3 and n2=0.11. Cdef and p necessary to define the defects size distribution are chosen to ensure that damage density must fit with experimentally observed probabilities (i.e. P=0.05 to P=1).

Table 1. Definition of the set of parameters for the DMT code at 1064 nm

table-icon
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It is worth noting that these parameters have been fixed for F 1ω=19 J/cm2, and remained unchanged for the calculations at F 1ω=24.5 J/cm2 (other experimental fluence used in this study). The only dependence is consequently given by Ω, through the determination of Qabs for each position. In other words, this model is expected to reproduce the experimental results for any fluence F 1ω tested on this crystal.

Lastly, as regards KDP crystals, lattice parameters a, b and c are such as a=bc conditions. We assume that the defects keep the symmetry of the crystal. So we consider that the defects are isotropic in the (a b) plane due to the multi-layered structure of KDP crystal. The principal axes of the defects match with the crystallographic axes. Assuming this, it is possible to encounter two geometries (either bc<1 or bc>1), the prolate (elongated) spheroid and the oblate (flattened) spheroid, represented on Fig. 6.

Fig. 6. Geometries proposed for modeling: (a) a sphere, which is the standard geometry used, (b) the oblate ellipsoid (flattened shape) and (c) the prolate ellipsoid (elongated shape). The value of the aspect ratio (between major and minor axis) is set to 2.

4.2 Numerical results and discussion

Through DDscat, we can finally extract the curve Qabs=f(Ω), which is re-injected in DMT code to reproduce the curve P |F=cste=f(Ω), i.e. the evolution of the laser damage probability as a function of Ω. Results are presented on Fig. 7.

Calculations have been performed by turns with the configurations presented in section 4.1. For each configuration, we consider that defects are all oriented in the same direction comparatively to the laser beam. For the prolate geometry, Qabs variations represented on the inset of Fig. 7 are clearly correlated to the variations of P(Ω). As regards the oblate one which has also been proposed, it has been immediately leaved out since variations introduced by the Qabs coefficient were anti-correlated to those obtained experimentally. Note that other geometries (not satisfying the condition a=b) have also been studied. Results (not presented here) show that either the variations of Qabs are anti-correlated or its variations are not large enough to reproduce experimental results whatever the 1ω fluence.

Fig. 7. Evolution of the laser damage probability as a function of Ω for two fluences: 19 J/cm2 and 24.5 J/cm2. The according modeling curves are plotted in dashed lines. Inset: Evolution of the absorption efficiency coefficient Qabs as a function of Ω for aeff=100 nm, which is the equivalent size of an ellipsoid associated with the volume of a sphere whose radius a is 100 nm. Results are obtained from DDScat 7.0 code for the prolate configuration.

5. Conclusion

We studied the effect of THG-cut KDP crystal orientation on the laser damage probability at 1064 nm. We first showed the evolution of the damage density as a function of fluence for two extreme positions of the crystal. For any level of damage density, an important shift in fluence has been observed. This led us to analyze the evolution of the laser damage probability as a function of the rotation angle around the propagation axis. In the range [0°, 90°], we concluded that the laser damage probability is not influenced by SHG and follows a monotone curve. These experiments have then been compared to modeling. We insist on the fact that SHG is not implemented in the model as it is not the prime contributor for laser damage in the range [0°, 90°]. In the model, we have proposed a new description of the nanodefects based on an ellipsoidal geometry. This model integrates calculations of absorption efficiency coefficient via the DDScat code. Thus, this model accounts for 1ω laser damage density curves as a function of the orientation of KDP crystal. This approach allows giving an explanation for orientation dependence on laser damage whereas modeling based on spherical geometry failed. Besides, several geometries do not qualitatively match. According to the size, the distribution and the shape (i.e. few tens of nanometers, few ppm and elongated shape), we give additional information that allow discriminating the potential defects listed in literature. This study is a promising way to give a reliable model of the physical mechanisms implied in laser damage. By a better description (or knowledge) of defects, solutions would be proposed to perform the crystal growth and to increase the crystal laser-damage resistance.

In addition, in the range [-90°, 0°], due to the high level of SHG, crystal is submitted to simultaneous illumination of 1ω and 2ω beams in the bulk. We observed experimentally that the damage probability is altered in the scope of this scenario. The model is no more reliable in this range since we do not take into account wavelength combination. Nevertheless, in literature few studies provide explanations on the mechanisms that occur in the configuration of wavelength mixing. Given the range of 2ω fluences, there is no doubt that a coupling efficiency establishes between the two wavelengths. If taken separately, the level of 2ω fluences can never induce damage sites (or at least not detectable). But once mixed with 1ω fluences, the damage probability is not merely added to the damage probability induced by 1ω, it is noticeably enhanced. A coupling effect between 3ω and 2ω has also been observed by Demange [25

25. P. Demange, R. A. Negres, A. M. Rubenchik, H. B. Radousky, M. D. Feit, and S. G. Demos, “The energy coupling efficiency of multiwavelength laser pulses to damage initiating defects in deuterated KH2PO4 non linear crystals,” J. Appl. Phys. 103, 083122 ( 2008). [CrossRef]

] when mixing these wavelengths, consistent with a precursor defects discrimination. Wavelength mixing is clearly an interesting way to approach laser damage and its understanding, by the discrimination of the physical mechanisms implied in these processes. Multi-wavelengths mixing studies are in progress, so as at 2ω and 3ω. Results at 2ω and 3ω would also be an additional source of information and would give some details as the same as those obtained at 1ω.

Acknowledgments

This work was performed under the auspices of the CEA-Cesta (the French Atomic Energy Commission) in Power Laser Department. The authors also want to greatly thank C. Maunier for his fruitful assistance and advice on KDP crystals.

References and links

1.

N. P. Zaitseva, J. J. De Yoreo, M. R. Dehaven, R. L. Vital, L. M. Carman, and H. R. Spears, “Rapid growth of large-scale (40–55 cm) KDP crystals,” J. Cryst. Growth 180, 255–262 ( 2001). [CrossRef]

2.

A. Ciapponi, S. Palmier, F. R. Wagner, J. Y. Natoli, H. Piombini, D. Damiani, and B. Bertussi, “Laser-induced fluorescence as a tool for the study of laser damage precursors in transparent materials,” Proc. SPIE 6998, 69981E-1 ( 2008). [CrossRef]

3.

M. D. Feit and A. M. Rubenchik, “Implications of nanoabsorber initiators for damage probability curves, pulselength scaling, and laser conditioning,” Proc. SPIE 5273, 74 ( 2004). [CrossRef]

4.

M. Pommiès, D. Damiani, B. Bertussi, H. Piombini, H. Mathis, J. Capoulade, and J. Y. Natoli, “Detection and characterization of absorption heterogeneities in KH2PO4 crystals,” Opt. Commun. 267, 154–161 ( 2006). [CrossRef]

5.

S. G. Demos and M. Staggs, “Application of fluorescence microscopy for noninvasive detection of surface contamination and precursors to laser-induced damage,” Appl. Opt. 41, 1977–1983 ( 2002). [CrossRef] [PubMed]

6.

A. K. Burnham, M. Runkel, M. D. Feit, A. M. Rubenchik, R. L. Floyd, T. A. Land, W. J. Siekhaus, and R. A. Hawley-Fedder, “Laser-induced damage in deuterated potassium dihydrogen phosphate,” Appl. Opt. 42, 5483–5495 ( 2003). [CrossRef] [PubMed]

7.

F. R. Wagner, A. Hildenbrand, J. Y. Natoli, M. Commandré, F. Théodore, and H. Albrecht, “Laser damage resistance of RbTiOPO4: evidence of polarization dependent anisotropy,” Opt. Express 15, 13849–13857 ( 2007). [CrossRef] [PubMed]

8.

H. Yoshida, T. Jitsuno, H. Fujita, M. Nakatsuka, M. Yoshimura, and T. Sasaki, “Investigation of bulk laser damage in KDP crystal as a function of laser irradiation direction, polarization, and wavelength,” Appl. Phys. 70, 195–201 ( 2000). [CrossRef]

9.

J. B. Trenholme, M. D. Feit, and A. Rubenchik, “Size-selection initiation model extended to include shape and random factors,” Proc. SPIE 5991, 59910X ( 2006). [CrossRef]

10.

C. W. Carr, H. B. Radousky, and S. G. Demos, “Wavelength dependence of laser-induced damage: determining the damage initiation mechanisms,” Phys. Rev. Lett. 91, 127402 ( 2003). [CrossRef] [PubMed]

11.

A. Dyan, F. Enguehard, S. Lallich, H. Piombini, and G. Duchateau, “Scaling laws in laser-induced potassium dihydrogen phosphate crystal damage by nanosecond pulses at 3ω,” J. Opt. Soc. Am. B 25, 1087–1095 ( 2008). [CrossRef]

12.

G. Duchateau and A. Dyan, “Coupling statistics and heat transfer to study laser-induced crystal damage by nanosecond pulses,” Opt. Express 15, 4557–4576 ( 2007). [CrossRef] [PubMed]

13.

G. Duchateau, “Simple models for laser-induced damage and conditioning of potassium dihydrogen phosphate crystals by nanosecond pulses,” Opt. Express 17, 10434–10456 ( 2009). [CrossRef] [PubMed]

14.

B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. B 11, 1491–1499 ( 1994). [CrossRef]

15.

S. Reyné, M. Loiseau, G. Duchateau, J.-Y. Natoli, and L. Lamaignère, “Towards a better understanding of multi-wavelength effect on KDP crystals,” Proc. SPIE 7361, 73610Z ( 2009). [CrossRef]

16.

ISO Standard No 11254-1:2000; ISO Standard No 11254-2:2001

17.

L. Lamaignère, T. Donval, M. Loiseau, J. C. Poncetta, G. Razé, C. Meslin, B. Bertussi, and H. Bercegol, “Accurate measurements of laser-induced bulk damage density,” Meas. Sci. Technol. 20, 095701 ( 2009). [CrossRef]

18.

N. Zaitseva, J. Atherton, R. Rozsa, L. Carman, I. Smolsky, M. Runkel, R. Ryon, and L. James, “Design and benefits of continuous filtration in rapid growth of large KDP and DKDP crystals,” J. Cryst. Growth 197, 911–920 ( 1999). [CrossRef]

19.

N. P. Zaitseva, I. L. Smolsky, L. Carman, R. Ryon, and Z. U. Rek, “Connection between filtration of solution during growth and defect structure of KDP group crystals,” 1998 SSRL Activity Report, Experimental progress Reports 7, 434–437 ( 1998).

20.

S. G. Demos, M. Staggs, and H. B. Radousky, “Bulk defect formations in KH2PO4 crystals investigated using fluorescence microscopy,” Phys. Rev. B 67, 224102 ( 2003). [CrossRef]

21.

N. Zaitseva, L. Carman, I. Smolsky, R. Torres, and M. Yan, “The effect of impurities and supersaturation on the rapid growth of KDP crystals,” J. Cryst. Growth 204, 512–524 ( 1999). [CrossRef]

22.

N. Bloembergen, Nonlinear Optics, (World Scientific, 4th ed, 1965).

23.

P. Demange, R. A. Negres, C. W. Carr, H. B. Radousky, and S. G. Demos, “Laser-induced defect reactions governing damage initiation in DKDP crystals,” Opt. Express 14, 5313–5328 ( 2006). [CrossRef] [PubMed]

24.

P. Demange, C. W. Carr, R. A. Negres, H. B. Radousky, and S. G. Demos, “Multiwavelength investigation of laser-damage performance in potassium dihydrogen phosphate after laser annealing,” Opt. Lett. 30, 221–223 ( 2005). [CrossRef] [PubMed]

25.

P. Demange, R. A. Negres, A. M. Rubenchik, H. B. Radousky, M. D. Feit, and S. G. Demos, “The energy coupling efficiency of multiwavelength laser pulses to damage initiating defects in deuterated KH2PO4 non linear crystals,” J. Appl. Phys. 103, 083122 ( 2008). [CrossRef]

26.

R. W. Hopper and D. R. Uhlmann, “Mechanisms of inclusion damage in laser glass,” J. Appl. Phys. 41, 4023–4037 ( 1970). [CrossRef]

27.

H. S. Carslaw and J. C. Jaeger, Conduction of heat in solids, (Oxford Science Publications, 2nd ed, 1959).

28.

H. C. Van de Hulst, Light scattering by small particles, (Dover publications, Inc., New York, 1981).

29.

B. T. Draine and P. J. Flatau, “User guide for the discrete dipole approximation code DDScat 7.0”, http://arxiv.org/abs/0809.0337.

30.

B. T. Draine and P. J. Flatau, “The discrete dipole approximation for periodic targets: theory and tests,” J. Opt. Soc. Am. A 25, 2693–2703 ( 2008). [CrossRef]

31.

C. W. Carr, H. B. Radousky, A. M. Rubenchick, M. D. Feit, and S. G. Demos, “Localized dynamics during laser-induced damage in optical materials,” Phys. Rev. Lett. 92, 87401 ( 2004). [CrossRef]

32.

J. Y. Natoli, J. Capoulade, H. Piombini, and B. Bertussi, “Influence of laser beam size and wavelength in the determination of LIDT and associated laser damage precursor densities in KH2PO4,” Proc. SPIE 6720, 672016 ( 2007). [CrossRef]

OCIS Codes
(140.3330) Lasers and laser optics : Laser damage
(160.0160) Materials : Materials

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: August 18, 2009
Revised Manuscript: October 2, 2009
Manuscript Accepted: October 2, 2009
Published: November 11, 2009

Citation
Stéphane Reyné, Guillaume Duchateau, Jean-Yves Natoli, and Laurent Lamaignère, "Laser-induced damage of KDP crystals by 1ω nanosecond pulses: influence of crystal orientation," Opt. Express 17, 21652-21665 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-24-21652


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References

  1. N. P. Zaitseva, J. J. De Yoreo, M. R. Dehaven, R. L. Vital, L. M. Carman, and H. R. Spears, "Rapid growth of large-scale (40-55 cm) KDP crystals," J. Cryst. Growth 180, 255-262 (2001). [CrossRef]
  2. A. Ciapponi, S. Palmier, F. R. Wagner, J. Y. Natoli, H. Piombini, D. Damiani, and B. Bertussi, "Laser-induced fluorescence as a tool for the study of laser damage precursors in transparent materials," Proc. SPIE 6998, 69981E-1 (2008). [CrossRef]
  3. M. D. Feit and A. M. Rubenchik, "Implications of nanoabsorber initiators for damage probability curves, pulselength scaling, and laser conditioning," Proc. SPIE 5273, 74 (2004). [CrossRef]
  4. M. Pommiès, D. Damiani, B. Bertussi, H. Piombini, H. Mathis, J. Capoulade, and J. Y. Natoli, "Detection and characterization of absorption heterogeneities in KH2PO4 crystals," Opt. Commun. 267, 154-161 (2006). [CrossRef]
  5. S. G. Demos and M. Staggs, "Application of fluorescence microscopy for noninvasive detection of surface contamination and precursors to laser-induced damage," Appl. Opt. 41, 1977-1983 (2002). [CrossRef] [PubMed]
  6. A. K. Burnham, M. Runkel, M. D. Feit, A. M. Rubenchik, R. L. Floyd, T. A. Land, W. J. Siekhaus, and R. A. Hawley-Fedder, "Laser-induced damage in deuterated potassium dihydrogen phosphate," Appl. Opt. 42, 5483-5495 (2003). [CrossRef] [PubMed]
  7. F. R. Wagner, A. Hildenbrand, J. Y. Natoli, M. Commandré, F. Théodore, and H. Albrecht, "Laser damage resistance of RbTiOPO4: evidence of polarization dependent anisotropy," Opt. Express 15, 13849-13857 (2007). [CrossRef] [PubMed]
  8. H. Yoshida, T. Jitsuno, H. Fujita, M. Nakatsuka, M. Yoshimura, and T. Sasaki, "Investigation of bulk laser damage in KDP crystal as a function of laser irradiation direction, polarization, and wavelength," Appl. Phys. 70, 195-201 (2000). [CrossRef]
  9. J. B. Trenholme, M. D. Feit, and A. Rubenchik, "Size-selection initiation model extended to include shape and random factors," Proc. SPIE 5991, 59910X (2006). [CrossRef]
  10. C. W. Carr, H. B. Radousky, and S. G. Demos, "Wavelength dependence of laser-induced damage: determining the damage initiation mechanisms," Phys. Rev. Lett. 91, 127402 (2003). [CrossRef] [PubMed]
  11. A. Dyan, F. Enguehard, S. Lallich, H. Piombini, and G. Duchateau, "Scaling laws in laser-induced potassium dihydrogen phosphate crystal damage by nanosecond pulses at 3ω," J. Opt. Soc. Am. B 25, 1087-1095 (2008). [CrossRef]
  12. G. Duchateau and A. Dyan, "Coupling statistics and heat transfer to study laser-induced crystal damage by nanosecond pulses," Opt. Express 15, 4557-4576 (2007). [CrossRef] [PubMed]
  13. G. Duchateau, "Simple models for laser-induced damage and conditioning of potassium dihydrogen phosphate crystals by nanosecond pulses," Opt. Express 17, 10434-10456 (2009). [CrossRef] [PubMed]
  14. B. T. Draine and P. J. Flatau, "Discrete-dipole approximation for scattering calculations," J. Opt. Soc. Am. B 11, 1491-1499 (1994). [CrossRef]
  15. S. Reyné, M. Loiseau, G. Duchateau, J.-Y. Natoli, and L. Lamaignère, "Towards a better understanding of multi-wavelength effect on KDP crystals," Proc. SPIE 7361, 73610Z (2009). [CrossRef]
  16. ISO Standard No 11254-1:2000; ISO Standard No 11254-2:2001
  17. L. Lamaignère, T. Donval, M. Loiseau, J. C. Poncetta, G. Razé, C. Meslin, B. Bertussi, and H. Bercegol, "Accurate measurements of laser-induced bulk damage density," Meas. Sci. Technol. 20, 095701 (2009). [CrossRef]
  18. N. Zaitseva, J. Atherton, R. Rozsa, L. Carman, I. Smolsky, M. Runkel, R. Ryon, and L. James, "Design and benefits of continuous filtration in rapid growth of large KDP and DKDP crystals," J. Cryst. Growth 197, 911-920 (1999). [CrossRef]
  19. 1. N. P. Zaitseva, I. L. Smolsky, L. Carman, R. Ryon, and Z. U. Rek, "Connection between filtration of solution during growth and defect structure of KDP group crystals," 1998 SSRL Activity Report, Experimental progress Reports 7, 434-437 (1998).
  20. S. G. Demos, M. Staggs, and H. B. Radousky, "Bulk defect formations in KH2PO4 crystals investigated using fluorescence microscopy," Phys. Rev. B 67, 224102 (2003). [CrossRef]
  21. N. Zaitseva, L. Carman, I. Smolsky, R. Torres, and M. Yan, "The effect of impurities and supersaturation on the rapid growth of KDP crystals," J. Cryst. Growth 204, 512-524 (1999). [CrossRef]
  22. N. Bloembergen, Nonlinear Optics (World Scientific, 4th ed, 1965).
  23. P. Demange, R. A. Negres, C. W. Carr, H. B. Radousky, and S. G. Demos, "Laser-induced defect reactions governing damage initiation in DKDP crystals," Opt. Express 14, 5313-5328 (2006). [CrossRef] [PubMed]
  24. P. Demange, C. W. Carr, R. A. Negres, H. B. Radousky, and S. G. Demos, "Multiwavelength investigation of laser-damage performance in potassium dihydrogen phosphate after laser annealing," Opt. Lett. 30, 221-223 (2005). [CrossRef] [PubMed]
  25. P. Demange, R. A. Negres, A. M. Rubenchik, H. B. Radousky, M. D. Feit, and S. G. Demos, "The energy coupling efficiency of multiwavelength laser pulses to damage initiating defects in deuterated KH2PO4 non linear crystals," J. Appl. Phys. 103, 083122 (2008). [CrossRef]
  26. R. W. Hopper and D. R. Uhlmann, "Mechanisms of inclusion damage in laser glass," J. Appl. Phys. 41, 4023-4037 (1970). [CrossRef]
  27. H. S. Carslaw and J. C. Jaeger, Conduction of heat in solids (Oxford Science Publications, 2nd ed, 1959).
  28. H. C. Van de Hulst, Light scattering by small particles (Dover publications, Inc., New York, 1981).
  29. B. T. Draine and P. J. Flatau, "User guide for the discrete dipole approximation code DDScat 7.0," http://arxiv.org/abs/0809.0337.
  30. B. T. Draine and P. J. Flatau, "The discrete dipole approximation for periodic targets: theory and tests," J. Opt. Soc. Am. A 25, 2693-2703 (2008). [CrossRef]
  31. C. W. Carr, H. B. Radousky, A. M. Rubenchick, M. D. Feit, and S. G. Demos, "Localized dynamics during laser-induced damage in optical materials," Phys. Rev. Lett. 92, 87401 (2004). [CrossRef]
  32. J. Y. Natoli, J. Capoulade, H. Piombini, and B. Bertussi, "Influence of laser beam size and wavelength in the determination of LIDT and associated laser damage precursor densities in KH2PO4," Proc. SPIE 6720, 672016 (2007). [CrossRef]

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