Rapid growth and ultra-precision machining of large-size KDP (KH2PO4) crystals with high laser damage resistance are tough challenges in the development of large laser systems. It is of high interest and practical significance to have theoretical models for scientists and manufacturers to determine the laser-induced damage threshold (LIDT) of actually prepared KDP optics. Here, we numerically and experimentally investigate the laser-induced damage on KDP crystals in ultra-short pulse laser regime. On basis of the rate equation for free electron generation, a model dedicated to predicting the LIDT is developed by considering the synergistic effect of photoionization, impact ionization and decay of electrons. Laser damage tests are performed to measure the single-pulse LIDT with several testing protocols. The testing results combined with previously reported experimental data agree well with those calculated by the model. By taking the light intensification into consideration, the model is successfully applied to quantitatively evaluate the effect of surface flaws inevitably introduced in the preparation processes on the laser damage resistance of KDP crystals. This work can not only contribute to further understanding of the laser damage mechanisms of optical materials, but also provide available models for evaluating the laser damage resistance of exquisitely prepared optical components used in high power laser systems.

During the past decades, the advent of chirped-pulse amplification (CPA) technique has prompted the construction of terawatt-class laser facilities, which are able to produce ultra-short laser pulses.1–3 To achieve such high laser peak power, a large number of optical components are required to temporally, spatially and spectrally control the laser beam. The wide application of ultra-short laser pulses enables the optical materials to be subjected to a higher laser intensity than ever before. Currently, the laser damage on optics is a main limitation in the development of laser systems with higher peak power,3 and the intense laser interaction with optical materials is an active field of research for both scientists and manufacturers.2,4–6 Among these optics, potassium dihydrogen phosphate (KH2PO4, KDP for short) crystals are remarkable nonlinear optical materials that are widely used as Pockel’s cells and frequency converter for its outstanding electro-optical properties.4,7 Due to the urgent demand of crystal parts in laser-driven fusion facilities, the research on laser interaction with KDP optics has been mostly focused on the bulk damage which arises from absorptive impurities in long-pulse duration (pico- and nano-second) regimes.8,9 However, in ultra-short pulse regime, the related research has been targeting metals,10 semiconductors11 and several specific kinds of dielectric materials (e.g., fused silica12,13 and dielectric films14), while for KDP crystals, it is still concentrated on the internal defects to explore the role of defect-induced interband states in the electronic photo-excitation and relaxation mechanisms during the damage process.5,15 In the actual process to prepare high-quality KDP surfaces, it is inevitable to introduce undesirable superficial flaws like scratches, micro-cracks and dents with micro/nano dimensions on finished surfaces due to the delicate mechanical and physical properties, such as high water solubility, low fracture toughness and thermal sensitivity.7,16–18 Surface morphology with these isolated flaws, even not so large in number, would greatly modulate the incident laser and correspondingly produce light intensification up to hundreds of times.18,19 It is well known that the electronic ionization rates, which determine the intrinsic laser damage mechanisms, are highly intensity-dependent.1,2,12,13,15 However, to our knowledge, current studies on ultra-short pulse laser-induced damage on KDP optics has rarely taken the surface flaws into consideration. In light of this, it is of high interest and practical significance to develop theoretical models for understanding the laser damage mechanisms and quantitatively estimating the effect of surface flaws on laser damage resistance of KDP crystal materials.

Though the actual fs laser-induced dielectric breakdown is a complicated process, it is generally accepted that the damage process can be well understood by the nonlinear excitation of electrons from valence band (VB) to conduction band (CB).1,2,12,20,21 When the electron density in CB reaches the critical density ncr, where the respective plasma waves are resonant with incident laser wave, the excited material would strongly absorb the laser energy via inverse bremsstrahlung, eventually resulting in permanent structural changes (regarded as damage). The generation of free-electron excited from VB to CB can be described by the following rate equation:2,20

n e ( t ) t = W PI ( I ( t ) ) + W II ( I ( t ) , n e ( t ) ) W rel ( n e ( t ) , t )
(1)

The evolution of electron density ne(t) in CB depends on photoionization (PI) rate WPI, impact ionization (II) rate WII and a relaxation rate Wrel, which involves in the relaxation of excited electrons from CB to lower electronic state. The WPI is calculated by using the prevalent Keldysh’s PI theory,22,23 and the WII is obtained from four possible II models developed by Stuart,1,24–26 Sparks,27,28 Thornber26,29,30 and Drude.20,31 The experimentally measured relaxation times involving in self-trapped electron excitation is applied to describe the decay process of excited electrons.5,15,32 The detailed calculations of WPI, WII and Wrel for KDP crystal are presented in the Supplemental Material.33 Figure 1 shows the evolutions of ne(t) under the irradiation of a 68TW/cm2, 55fs pulse, which is calculated by solving Eq. (1). The Gaussian pulse intensity is included for reference and the respective contributions of PI, II and electron decay to the electron density are indicated. As presented, the ne(t) predicted by combined models of Keldysh’s PI, Drude II and electron relaxation can reach exactly at the critical density ncr, which implies that the threshold intensity for damage in KDP crystal predicted using this combined model is 68TW/cm2. While, for the combined models with Stuart, Sparks and Thornber II models, the estimated electron densities are much larger. However, the reported threshold intensity of KDP material is 68TW/cm2 under laser irradiation with the same parameters.5 This verifies the combined model of Keldysh’s PI, Drude II and electron relaxation as an optimal model to predict the LIDT of KDP materials. It is further inferred in Fig. 1 that the Keldysh’s PI predominantly contributes to the free electron yield that the PI alone produces the peak electron population at the peak of the pulse due to its strong dependent on light intensity. This result is consistent with the observation in Ref. 1. In addition, the PI-generated free electrons can serve as seed electrons for II process, and the II would then multiply the electron density by nearly two orders of magnitude to incur optical damage in the later part of the pulse. This means that the damage mechanism on KDP material is closely associated with both the PI and II ionizations. It is claimed in Ref. 20 that, more than 1019 to 1020 electrons/cm3 are typically necessary to initiate the significant II process. In Fig. 1, ∼9 × 1019 seed electrons/cm3 have been accumulated by PI before the II strongly works. We should keep in mind that in the theoretical models, the laser damage occurs at the location with the maximum light intensity, which should be in the center of the laser spot on the surface for a Gaussian pulse profile.

FIG. 1.

Calculated ne(t) based on Eq. (1) with the combined consideration of Keldysh’s PI, II models (Stuart, Sparks, Thornber and Drude), and electron relaxation for 55fs, 800nm pulse with peak intensity of 68 TW/cm2 for KDP materials. The red solid curve is the critical electron density n cr = ε 0 m e * ω 2 / e 2 , set as the damage criterion.

FIG. 1.

Calculated ne(t) based on Eq. (1) with the combined consideration of Keldysh’s PI, II models (Stuart, Sparks, Thornber and Drude), and electron relaxation for 55fs, 800nm pulse with peak intensity of 68 TW/cm2 for KDP materials. The red solid curve is the critical electron density n cr = ε 0 m e * ω 2 / e 2 , set as the damage criterion.

Close modal

To further validate the feasibility of the LIDT model, single-pulse fs LIDTs are measured using three types of testing protocols based on direct measurement and derivation from multi-pulse LIDTs. A home-built Ti: sapphire liquid-nitrogen-cooled regenerative amplifier was employed to deliver the ultra-short pulse lasers with 773nm wavelength, 500Hz repetition rate. Using an external Pockels cell, single pulses are extracted from this amplifier system and then sent to the setup as shown in Fig. 2(a). Since the comparison of LIDTs between experiment and calculation for 800nm pulse has been done above, we make the comparison using the second harmonic. The output laser with 387nm-wavelength and 28fs-pulse duration is produced by a 100μm-thick nonlinear BBO crystal. A pair of symmetric transparent glasses with adjustable angles between them, combined with filter set, are used to adjust the applied laser energy. The pump laser is focused on the front sample surface with an effective Gaussian beam waist of 2ω0  = 23μm using an f = 100mm achromatic lens. Surface modification under each pulse shot is observed in-situ using a 10× infinity-conjugate microscope objective (Mitutoyo) and imaged onto a camera (Firefly by Point Grey). As opposed to long pulses, the laser damage in ultra-short pulse regime (τp < 10ps) shows an increasingly deterministic character, since the damage size in such short time is limited to a small region, where the laser intensity is sufficient to produce a plasma with no collateral damage.1,24,25 Therefore, the LIDTs in this work were measured following a deterministic R-on-1 protocol, in which the laser irradiates each test site with increasingly ramped energy until damage occurs.7,18,34 The damage fluence is defined as the lowest laser fluence at which damage occurs. A total of 10 sites are involved in each test and the LIDT is determined as the average damage fluence.

FIG. 2.

(a) Experimental setup used to test the single- and multi-pulse LIDTs on KDP materials under the irradiation of pulse lasers with 28fs duration and 387nm wavelength. (b) The measured LIDTs for pulse number of 10, 100 and 1000. The single-pulse LIDT is derived by fitting the data with energy-accumulation model of Eq. (2). (c) The morphology of tested sites after irradiation with 10, 100 and 1000 pulses.

FIG. 2.

(a) Experimental setup used to test the single- and multi-pulse LIDTs on KDP materials under the irradiation of pulse lasers with 28fs duration and 387nm wavelength. (b) The measured LIDTs for pulse number of 10, 100 and 1000. The single-pulse LIDT is derived by fitting the data with energy-accumulation model of Eq. (2). (c) The morphology of tested sites after irradiation with 10, 100 and 1000 pulses.

Close modal

We first measured the LIDT with single pulse by directly observing the surface modification after each laser shot. The damage is determined as any visible permanent surface change, which is both monitored on-line by the camera shown in Fig. 2(a) and postmortem inspection with optical microscope. The directly measured single-pulse threshold intensity is Ith = 127.10 ± 13.60TW/cm2, which is included in Fig. 3(b). In the actual direct test of single-pulse damage, the damage site is typically shallow ablation pit associated with such small amount of material removal (a few atomic layers)24 that it is very difficult to be detected visually. Further, the highly localized critical density plasma at the threshold implies faster cooling of the plasma for shorter pulse duration, resulting in increasing difficulty in judging the damage occurrence using plasma effects proposed by Du et al.30 In light of this, we tested the multi-pulse LIDTs first which is much easier to be observed, and then derived the single-pulse LIDT according to the energy-accumulation model shown below:35 

F th ( N ) = F th ( 1 ) N S 1
(2)

where the single-pulse LIDT Fth(1) can be derived from the LIDTs Fth(N) for N pulses. This energy-accumulation model has been successfully applied to describe the laser damage behaviors on metals and dielectrics irradiated with multiple pulses, in which the Fth(N) is lower than the Fth(1) due to the accumulation of damage induced by individual pulses (also termed incubation effect). In contrary to single-pulse case, multi-pulse damages can not only “amplify” the small damage site to an easily observable size, but also minimize the statistical uncertainty during the test process. The measured Fth(N) for N = 10, 100 and 1000 are given in Fig. 2(b). By fitting the data with Eq. (2), one can estimate the single-pulse Fth(1) = 2.92 ± 0.53J/cm2 (equivalent to threshold intensity of Ith = 104.11 ± 18.83TW/cm2). The derived LIDT is included in Fig. 3(b) as well for further comparison with the calculation results. The morphology of tested damage in Fig. 2(c) validate that irradiation with more number of pulses can generate larger damage sites, resulting in easier judgment of the damage occurrence.

FIG. 3.

(a) Derived Fth(N)s for N = 5, 10, 20, 50, 100, 500 and 1000 from the damage crater area using lasers with “above-threshold” fluences. The Fth(N) data are fitted by Eq. (2) to determine Fth(1). The inset is an example of crater area variation with respect to fluence and the linear fitting with Eq. (4) for N = 20. (b) Comparison of Fth(1) determined by theoretical calculation, direct measurement, crater area extrapolation and Fth(N) derivation.

FIG. 3.

(a) Derived Fth(N)s for N = 5, 10, 20, 50, 100, 500 and 1000 from the damage crater area using lasers with “above-threshold” fluences. The Fth(N) data are fitted by Eq. (2) to determine Fth(1). The inset is an example of crater area variation with respect to fluence and the linear fitting with Eq. (4) for N = 20. (b) Comparison of Fth(1) determined by theoretical calculation, direct measurement, crater area extrapolation and Fth(N) derivation.

Close modal

Beside of visual acquisition for judging the occurrence of laser damage, measuring the damage sizes with laser fluences above the damage threshold to deduce the damage threshold is another effective method to precisely determine the LIDT. This “above-threshold” measurement permits the determination of LIDT not to be affected by the instrument resolution and sensitivity.21,25 For a Gaussian laser beam, the spatial distribution of fluence reads:

F ( r ) = F 0 e 2 r 2 / ω 0 2
(3)

where F0 is the peak fluence and r is the distance from the beam spot center. In this pulse fluence profile, the damage crater is generally formed inside the limited area, where the fluence exceeds the damage threshold Fth. By rearranging Eq. (3), the damage crater area (A), peak laser fluence (F0) and number of pulse (N) satisfy the following relations:

A ( N , F 0 ) = π ( D / 2 ) 2 = ( π ω 0 2 / 2 ) In F 0 In F th ( N )
(4)

where D is the crater diameter under the irradiation of pulse number N and peak fluence F0. Hence, for various pulse number, by irradiating the KDP surface with “above-threshold” fluences, we can obtain the multi-pulse LIDTs based on Eq. (4) by means of measuring the corresponding crater areas. Then the single-pulse LIDT can be derived by fitting Eq. (2) as stated above. As presented in Fig. 3(a), the Fth(N)s are first determined using Eq. (4) by extrapolating the fitted curves of crater areas to A = 0 (damage area→0, taking N = 20 for example as shown in the inset). By fitting the extrapolated Fth(N)s with Eq. (2), the single-pulse LIDT can be then estimated: Fth(1) = 3.55 ± 0.36J/cm2 (equivalent to the intensity of 126.96 ± 12.96TW/cm2). Using the developed LIDT model with the combined considerations of PI, II and electron relaxation, the single-pulse LIDT is calculated to be Ith = 132TW/cm2 for the same laser parameters used in the experiments. It is worth noting that the relaxation time is τr = 300fs for 387nm wavelength, because it is involved in 3-photon absorption for KDP materials (7.7eV band-gap15).

Figure 3(b) shows the comparison of Ith(1) determined by numerical calculation and experimental measurements with three testing protocols. Considering the relative error of laser intensity of ±10%, the calculated Ith(1) based on the developed Keldysh-Drude-Relaxation model is reasonably consistent with the experimentally measured results. The discrepancy in the experiment results arises from the minimum adjustable angle between the pair of glasses used to adjust the laser fluence as shown in Fig. 2(a). The controllable laser fluence range between damage and no damage would directly determine the relative error in the experiments. Besides, the calculated Ith(1) in Fig. 3(b) is slightly higher than that of the experiment results. This is for the reason that the theoretical calculation is based on intrinsic damage mechanisms without the consideration of practical factors like surface contaminations or flaws introduced in the sample preparation process, which would lower the damage thresholds. What’s more, the determined threshold intensity for 28fs pulses is in the magnitude of 1014 W/cm2, indicating that the PI works in the regime of Keldysh parameter γ ≈ 1 as depicted in Fig. S1(a) (see the Supplemental Material33). This means that both multi-photon and tunneling ionizations should be considered to fully describe the PI process during the nonlinear excitation of KDP materials.

A typical KDP surface finished by single point diamond turning is presented in Fig. 4. Though the machined KDP surface meets the stringent specification of high-quality optical components,17 some isolated scratches are still introduced on the surface due to abnormal cutting of diamond tool in the machining process. The surface roughness Ra can reach 4.3nm, which is almost the highest surface quality that can be currently machined for brittle KDP materials. The scratches are typically several-micron wide and sub-micron deep. Surface features on optics with this magnitude of size are reported to strongly modulate incident lasers, resulting in great light intensification.19 With the measured geometric information of scratch shown in Fig. 4, we modeled the scratch-induced light intensification using finite difference time domain (FDTD) method.18 The light intensity distribution exhibited in Fig. 4(e) presents that the uniform incident laser field is highly distorted by the adjacent scratches, with the intensified interference ripples located among the central region (as denoted by blue arrows). The light intensity is enhanced to 1.29 times as large as that for an ideal flat KDP surface without any flaws. This light intensification can be taken into consideration in the proposed Keldysh-Drude-Relaxation model. And one can then quantitatively estimate the role of surface scratches in decreasing the LIDT of KDP optics by using the dependence of electronic ionization rates on light intensity as shown in Eqs. (S1) and (S6) (see Supplemental Material33). With the presence of 2.5μm-width and 0.1μm-depth scratches, it is calculated that the damage threshold would descend from 132TW/cm2 to 102TW/cm2, which can provide directive guidance for lasers and optics manufacturers to evaluate the surface quality of their KDP elements. This method is also applicable to other optics for quantitatively evaluating the effect of surface flaws introduced in the engineering fabrication processes on their laser damage resistance. Further, instead of complex pump-probe measurements, the electron relaxation times τr in different laser conditions are possible to be achieved by parameter sweeping of the developed LIDT model and comparing with the experimentally tested LIDT.

FIG. 4.

Characterization of scratches on diamond-finished high-quality KDP surface and its induced light intensification. The isolated scratches are introduced during the machining process and characterized by stereoscopic optical (a), interferometric (b) and atomic force (c) microscopes. The cross section profile (d) of the surface with scratches shows that the scratches are generally several-micron wide and sub-micron deep. (e) The simulated light intensification caused by scratches with 2.5.μm-width and 0.1.μm-depth.

FIG. 4.

Characterization of scratches on diamond-finished high-quality KDP surface and its induced light intensification. The isolated scratches are introduced during the machining process and characterized by stereoscopic optical (a), interferometric (b) and atomic force (c) microscopes. The cross section profile (d) of the surface with scratches shows that the scratches are generally several-micron wide and sub-micron deep. (e) The simulated light intensification caused by scratches with 2.5.μm-width and 0.1.μm-depth.

Close modal

In summary, a theoretical model taking the combined considerations of PI, II and electron relaxation has been optimized for predicting the LIDTs of KDP crystal in the ultra-short pulse regime. The laser damage tests were carried out to experimentally determine the single-pulse LIDTs using several types of testing methods. The testing results, combined with previously reported experimental data, were well consistent with those calculated from the model, which validates its feasibility in determining the LID threshold of KDP materials. According to the developed model with the light intensification considered, the quantitative role of surface scratches in decreasing the LIDT was analyzed, that the threshold intensity would descend from 132TW/cm2 to 102TW/cm2 with the presence of 2.5μm-width and 0.1μm-depth surface scratches. This model possesses further scalability in quantitatively evaluating the laser damage resistance of other exquisitely prepared optical components used in the high power laser systems.

This work was supported by the National Natural Science Foundation, China Grant No. 51275113 and the scholarship from China Scholarship Council. The Chowdhury group acknowledges support from the Air Force Office of Scientific Research, USA under Grant No. FA9550-12-1-0454.

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