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Study of the surface damage threshold and mechanism of KDP crystal under ultrashort laser irradiation

Yan Liu, Yujia Zhang, Xiaoqing Liu, Yang Liu, Jiezhao Lv, Changfeng Fang, Qingbo Li, and Xian Zhao

Open Access Open Access

Abstract

Here, we investigate the mechanism of surface damage threshold anisotropy induced by ultrashort laser in potassium dihydrogen phosphate (KDP) crystal. Carrier-lattice nonequilibrium interaction is simulated based on Brillouin light-scattering (BLS) spectroscopy and a complete self-consistent model to obtain the time evolution of carrier density and temperature as well as lattice temperature. The results indicate that the trend of the lattice temperature is consistent with the experimental phenomena. Meanwhile, the electron-phonon coupling effect, in addition to the electron density traditionally considered, is an important factor affecting damage and is a major contributor to the anisotropy of the damage threshold.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Potassium dihydrogen phosphate (KDP) is widely used in the fields of laser frequency conversion and electro-optical modulation due to its excellent properties, especially the ability to grow large-sized single crystals, making it an irreplaceable optical component in inertial confinement fusion (ICF) [1–3]. Unfortunately, its low laser-induced damage threshold (LIDT) has become a key factor in constraining system design and limiting system performance in high-power laser facilities [4,5]. Laser-induced damage to KDP crystal is categorized into body damage and surface damage based on the location of the damage [6]. In the past time, there have been many reports about the body damage of KDP crystal [7–11]. With the improvement of crystal material growth techniques, impurity particles inside the crystals have been gradually avoided and the body damage threshold of KDP crystal has been significantly improved.

In recent years, some research results have shown that laser-induced damage on the surface of KDP may be superior to the occurrence of bulk damage [12,13]. Moreover, the surface damage of KDP crystal will continue to grow under subsequent laser irradiation, which is more likely to cause catastrophic damage to crystal components. In addition, the advent of chirped-pulse amplification (CPA) technology has facilitated the construction of terawatt-scale laser facilities, which can produce ultrashort laser pulses [14]. Ultrashort pulses enable optical materials to withstand higher laser intensity than ever before. Therefore, the intrinsic damage mechanism of the surface damage of KDP crystal should be considered significantly.

Component changes and the relationship between electronic structure and laser-induced damage in KDP crystal irradiated by high fluence were analyzed by using micro-X-ray diffraction (µ-XRD) and X-ray absorption near-edge structure (XANES) spectroscopy [15,16]. For shorter pulses, damage is usually attributed to the generation of very large plasma concentration above the critical electron density (CED). Considering the synergistic effects of photoionization, collisional ionization and electron decay, a model dedicated to the prediction of LIDT has been developed using the evolution of the free electron density [17]. However, a single rate equation based on the CED only gives a single damage trend. Moreover, the CED mainly depends on the laser frequency, which is poorly dependent on the characteristic parameters of material [18]. In addition, laser-induced damage to KDP crystal is a complex phenomenon involving multiple coupled physical processes, resulting in incomplete understanding of the damage mechanism [19–23]. The surface damage of optical materials induced by ultrashort laser involves energy deposition and energy transfer. Currently, there are few reports on the latter in KDP. And electron-phonon coupling effect plays an important role in energy transfer. Currently, there are many reports on numerical models related to the interaction between ultrashort pulse and material, and the simulation results show good agreement with experimental phenomena [24–29]. Therefore, it is of high interest and practical significance to develop a theoretical model to understand the role of the electron-phonon coupling effect in the laser-induced surface damage mechanism of KDP crystal.

In this paper, we perform surface damage experiments on four common cutting types of KDP crystal. The experimental results show that the laser-induced surface damage threshold (LISDT) of KDP crystal changes with the direction of irradiation. Meanwhile, we simulate the carrier-lattice nonequilibrium interaction based on a complete self-consistent model in conjunction with Brillouin light-scattering (BLS) spectroscopy to obtain the time evolution of the carrier density and temperature as well as the lattice temperature. The simulation results indicate that the trend of lattice temperature calculated based on the carrier density criterion agrees well with the experimental phenomena. It is also found that in addition to the CED, which is traditionally considered, the electron-phonon coupling effect of the material is also a key factor affecting the damage and is a major contributor to the anisotropy of the LISDT.

2. Samples and experimental procedure

2.1 Samples and experimental conditions

The KDP crystal was grown from potassium dihydrogen phosphate solution by conventional temperature cooling method and treated by single point diamond turning technique. Laser-induced surface damage experiments were performed on four common cutting types of KDP crystal (z-cut, type-I, type-II, and x-cut) as shown in Fig. 1(a). The size of each sample is approximately 20 mm × 20 mm × 3 mm. The platform used to test the damage characteristic of KDP crystal is illustrated in Fig. 1(b). The laser used in the experiment is an amplified Ti: sapphire laser system with a standard output of 35 fs, 800 nm and repetition rate of 1 kHz. The number and power of pulses are adjusted by shutter and attenuator, respectively. The sample is moved using a 3-dimensional stepper-motor-controlled stage to change the position of the damage. To avoid the thermal effect, the distance between neighboring damage locations is 50 µm. The adjustment of the optical path as well as the damage process is monitored in real-time by a CMOS camera.

 figure: Fig. 1.

Fig. 1. (a) Cutting schematic diagram of KDP crystal sample. (b) Experimental setup for testing the multi-pulse LISDT of KDP crystal under the irradiation of pulse laser with 35 fs duration and 800 nm.

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2.2 Determination of surface damage threshold

Single-pulse damage sites are typically shallow ablation pits accompanied by a small amount of material removal (a few atomic layers) that are difficult to detect visually [30]. In contrast, multi-pulse damages not only magnify small damaged areas to a size that can be easily observed but also minimize statistical uncertainty in the testing process. Therefore, we adopt the multi-pulse LISDT (Fth(N)) and then derived the single-pulse LISDT (Fth(1)) based on the energy accumulation model shown below:

$${F_{th}}(N) = {F_{th}}(1){N^{S - 1}}$$
where N is the number of pulses and S is the energy accumulation factor. The model has been successfully applied to describe the laser-induced damage behavior of metal and dielectric under multi-pulse irradiation [31–33]. As shown below, the Fth(N) is calculated based on the relationship between damage area (s) and laser peak laser fluence (F0) [34,35].
$$\textrm{s}(N,{F_0}) = \pi \mathrm{\ \cdot }{\textrm{r}^2}\textrm{ = (}\pi \mathrm{\ \cdot }\textrm{w}_0^2\textrm{)}[{\ln {F_0} - \ln {F_{th}}(N)} ]$$
where r is the damage radius at the pulse number N and beam radius w0, and N = 15, 30, 50, 100, 500 and 1000 in Fth(N). The peak energy of a single pulse varies from 0.4 µJ to 4 µJ.

2.3 Brillouin light-scattering (BLS) spectroscopy measurements

BLS measurements were performed at room temperature using a single-mode solid-state laser with a wavelength of 532 nm in a backscattering geometry. The p-polarized incident laser was focused vertically onto the sample surface. The resolution of the BLS spectrum is 50 MHz. Each BLS spectrum is taken with a 50-GHz free-spectrum range distributed in 1024 channels and is accumulated over 30 min. Simultaneously, the scattered light was collected and analyzed in a six-pass tandem Fabry-Perot interferometer.

3. Results and discussions

3.1 Anisotropy of LISDT in KDP crystal

Figure 2(a) shows the damage morphology of x-cut KDP crystal at different numbers and energy of pulses. It indicates that multiple pulses irradiation can produce distinct sites of damage, which is more helpful in determining the occurrence of damage. The damage pattern is close to the circular shape produced by the Gaussian beam, and the damage becomes more severe with the increase of the number and energy of the pulses. Under high laser fluence, the detachment of fragments caused by shock waves occurs at the edge of the damage (This is not the focus of this work). As shown in the inset of Fig. 2(b), linear fitting is performed on the damage data under N = 50 based on Eq. (2). The fitting calculation shows that the beam radius w0 is approximately 4 µm. By extrapolating the fitting line to the damaged area s = 0, the multi-pulse damage threshold Fth(N) at N = 50 is approximately 0.94 J/cm2 (Fth(50) = 0.94 J/cm2). And fitting Fth(15), Fth(30), Fth(50), Fth(100), Fth(500) and Fth(1000) based on Eq. (1) yields a single pulse damage threshold Fth(1) of approximately 4.37 J/cm2 for x-cut KDP crystal. Similarly, the LISDT for z-cut, type-I and type-II KDP crystals are 2.18 J/cm2, 2.32 J/cm2 and 3.38 J/cm2 (Fig. 2(c) and Fig. S1), respectively.

 figure: Fig. 2.

Fig. 2. (a) The damage morphology of x-cut KDP crystal at different numbers and energy of pulse. (b) Derived Fth(N) for N = 15, 30, 50, 100, 500, and 1000 in the case of x-cut KDP crystal. According to Eq. (1), Fth(1) is obtained by fitting the Fth(N) data points. The inserted figure, as an example, is obtained by linear fitting with Eq. (2) for N = 50. (c) Single-pulse LISDT of KDP crystal with different cutting types.

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As shown in Fig. 3, under the same condition, the damage profile of z-cut KDP crystal is more pronounced than that of x-cut KDP crystal, further indicating that z-cut KDP crystal has a lower damage threshold and is more prone to damage. Interestingly, the LISDT of KDP is anisotropic. To our knowledge, there are currently no relevant reports on this anisotropy. Therefore, the study of LISDT anisotropy in KDP crystal is very helpful in understanding the damage mechanism.

 figure: Fig. 3.

Fig. 3. Damage profile of x-cut and z-cut KDP crystals measured by atomic force microscopy (AFM) under the same condition: (a) N = 30, E = 0.5 µJ, (b) N = 30, E = 0.6 µJ.

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3.2 Photoionization—simply considering electron density

Femtosecond laser-induced damage to optical material is a process of photo-energy transfer to the material, involving nonlinear absorption of electrons and subsequent energy transfer. Two types of nonlinear excitation mechanisms play a role in this absorption: photoionization (PI) and impact ionization (II). Currently, the Keldysh photoionization theory is the most widely used and comprehensive explanation in the field of photoionization. The transition between multiphoton ionization (MPI) and tunnel ionization (TI) is represented by the Keldysh parameter γ [36]:

$$\mathrm{\gamma } = \frac{\mathrm{\omega }}{e}{\left[ {\frac{{m_e^\ast cn{\mathrm{\varepsilon }_0}{E_g}}}{{2I}}} \right]^{1/2}}$$
where ω and I are the laser frequency and laser intensity, $m_e^*$ and e are reduced mass and charge of the electron, n and Eg are refractive index and band-gap of material, c is velocity of light and ɛ0 is permittivity of free space.

Based on Eq. (4–9), Fig. 4(a) shows the ionization rate W and the Keldysh parameter γ as a function of the laser intensity for 800 nm in x-cut KDP crystal. When the γ is much greater (much less) than about 1.5, the photoionization is MPI (TI) process. When the γ is near 1.5, photoionization is a mixture of TI and MPI. The single rate equation determines the onset of damage by reaching a critical electron density (CED) [17]. Figure 4(b) and its localized enlargement Fig. 4(c) show that WPI-x-cut is the largest at the same energy density (WPI: x-cut > type-II > type-I > z-cut), which indicates that x-cut crystal is more likely to reach the CED first and be damaged, but this is contrary to the experimental results observed above. Therefore, simply considering the nonlinear absorption of electrons is incomplete, and transfer of energy also needs to be considered.

$${W_{PI}} = \frac{{2\mathrm{\omega }}}{{9\pi }}{\left( {\frac{{\mathrm{\omega }m_e^\ast }}{{\mathrm{\hbar}\sqrt \Gamma }}} \right)^{3/2}}Q(\gamma ,x)\textrm{exp} \left( { - \pi \left\langle {x + 1} \right\rangle \frac{{K\left( {\sqrt \Gamma } \right) - E\left( {\sqrt \Gamma } \right)}}{{E(\zeta )}}} \right)$$
where ω is central frequency of laser, $m_e^*$ is the effective mass of electronic for KDP crystal, ħ is the reduced Planck constant, <x + 1 > denotes the integer part of the number x + 1, γ is the Keldysh parameter of KDP crystal, and Γ = γ2/(1+γ2), ζ = 1/(1+γ2). The other parameters of Eq. (4) are represented as follows:
$$Q(\gamma ,x) = \sqrt {\frac{\pi }{{2K\left( {\sqrt \zeta } \right)}}} \sum\limits_{n = 0}^\infty {\left\{ {\textrm{exp} \left( { - n\pi \frac{{K\left( {\sqrt \Gamma } \right) - E\left( {\sqrt \Gamma } \right)}}{{E\left( {\sqrt \zeta } \right)}}} \right)\Phi \left\{ {\pi \sqrt {\frac{{2\left\langle {x + 1} \right\rangle - 2x + n}}{{2K\left( {\sqrt \zeta } \right)E\left( {\sqrt \zeta } \right)}}} } \right\}} \right\}}$$
$$x = \frac{{2{E_g}}}{{\pi \mathrm{\hbar} \omega \sqrt \Gamma }}E\left( {\sqrt \zeta } \right)$$
$$\Phi (z) = \int\limits_0^z {\textrm{exp} ({y^2} - {z^2})dy}$$
where Eg is bandgap of KDP crystal, K and E are elliptic integrals, Ф(z) denotes Dawson integrals.
$$\begin{array}{l} {W_{MPI}} = \frac{{2\omega }}{{9\pi }}{\left( {\frac{{m_e^\ast \omega }}{\mathrm{\hbar}}} \right)^{3/2}} \times \Phi \left( {\sqrt {2\left\langle {\frac{\Delta }{{\mathrm{\hbar}\omega }} + 1} \right\rangle - \frac{{2\Delta }}{{\mathrm{\hbar}\omega }}} } \right) \times \textrm{exp} \left\{ {2\left\langle {\frac{\Delta }{{\mathrm{\hbar}\omega }} + 1} \right\rangle \left( {1 - \frac{1}{{4{\gamma^2}}}} \right)} \right\}{\left( {\frac{1}{{16{\gamma^2}}}} \right)^{\left\langle {\frac{\Delta }{{\mathrm{\hbar}\omega }} + 1} \right\rangle }}\\ \Delta = {E_g} + \frac{{{E_g}}}{{4{\gamma ^2}}} \end{array}$$
$${W_{TI}} = \frac{{2{E_g}}}{{9{\pi ^2}\mathrm{\hbar}}}{\left( {\frac{{m_e^\ast {E_g}}}{{{\mathrm{\hbar}^2}}}} \right)^{3/2}}{\left( {\frac{{\mathrm{\hbar}\omega }}{{{E_g}\gamma }}} \right)^{5/2}} \times \textrm{exp} \left( { - \frac{\pi }{2}\frac{{{E_g}\gamma }}{{\mathrm{\hbar}\omega }}\left( {1 - \frac{{{\gamma^2}}}{8}} \right)} \right)$$

 figure: Fig. 4.

Fig. 4. (a) Ionization rate and Keldysh parameter as a function of laser intensity for 800 nm light in x-cut KDP crystal. The blue solid line represents the photoionization rate (WPI) based on the full expression from Keldysh, while the purple dashed-dotted line represents the tunneling ionization rate (WTI), and the red dashed line represents the multiphoton ionization rate (WMPI). (b) WPI of four cutting types of KDP crystal. (c) Enlargement of the yellow area in Fig. 3(b).

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3.3 Self-consistent model—electron density + electron-phonon coupling effect

Given this, we construct the following complete self-consistent model to describe the transient changes of carrier concentration, carrier energy and lattice energy in the material during laser irradiation:

$$\frac{{\partial {n_e}(t)}}{{\partial t}} = {W_{PI}}(I(t)) + {W_{II}}(I(t),{n_e}(t)) - {W_{rel}}({n_e}(t),t)$$
$$\frac{{\partial ({n_e}{E_g} + {C_e}{T_e})}}{{\partial t}} = \nabla \mathrm{\ \cdot }({k_e}\nabla {T_e}) + {n_e}\mathrm{\theta }I + {W_{MPI}}\mathrm{\ \cdot }5\mathrm{\hbar}\mathrm{\omega } - G({T_e} - {T_l})$$
$${C_l}\frac{{\partial {T_l}}}{{\partial t}} = \nabla \mathrm{\ \cdot }({k_e}\mathrm{\ \cdot }{T_l}) + G({T_e} - {T_l})$$

Equations (10)–(12) describe the increase in carrier concentration, carrier energy and lattice energy, respectively. On the right-hand side of (10), photoionization, collision ionization and recombination of carrier are considered, respectively. On the right-hand side of (11), the energy diffusion, free carrier absorption and the increase of electron energy due to the photon energy exceeding the energy gap are indicated. The last term represents the energy exchange between the lattice and the plasma. On the right side of (12) the first term describes heat conduction through the lattice, and the second item describes energy exchange between the lattice and the plasma.

In Eq. (10), the photoionization term is described by the Keldysh ionization equation (Eq. (4)). The impact ionization term is described by the well-known Drude model [37]:

$$\begin{array}{l} {W_{II - Drude}} = \frac{\sigma }{{{E_g}}} \cdot I(t)\\ \sigma = \frac{{{e^2}}}{{c{\varepsilon _0}nm_e^\ast }} \cdot \frac{{{\tau _c}}}{{1 + {\omega ^2}\tau _c^2}}\;\;\;\;\;\;\;\;\;\;\;\;{\tau _c} = \frac{{16\pi {\varepsilon _0}\sqrt {m_e^\ast {{(0.1{E_g})}^3}} }}{{\sqrt 2 {e^4}{n_e}(t)}} \end{array}$$
where σ and τc represent the absorption cross section and the resulting collision time, respectively.

The relaxation term Wrel is given by:

$${W_{rel}} = \frac{{{n_e}(t)}}{{{\tau _r}}}$$
where the absorption relaxation time of five photons is 9000 fs [38].

The meanings of the symbols and associated values are listed in Table 1 [39]. Among them, some parameters refer to silicon material [40].

Tables Icon

Table 1. Model parameters

View Table

The electron-phonon coupling factor G can be simplified as the formula [41]:

$$G = \frac{{{\pi ^4}{{({n_e}{v_s}{k_B})}^2}}}{K}$$
where vs and kB represent sound velocity and Boltzmann constant, K represents the thermal conductivity of the material.

The electron-phonon coupling factor is a key parameter controlling the thermal relaxation rate between electron and the lattice [42], and its magnitude is related to the crystal structure. Figure 5 illustrates the structural frameworks and corresponding projection structures of z-cut and x-cut KDP crystals. Because we construct a one-dimensional model, we mainly analyze the longitudinal structure (laser incidence direction). It can be seen that the z-cut KDP crystal is composed of dense atomic stacking along the <001 > direction (laser incidence direction), containing K-O bond (239 kJ/mol) and strong P-O bond (597 kJ/mol). On the contrary, x-cut KDP crystal has loose atomic space along the <100 > direction (laser incidence direction) and weak hydrogen bonds between layers. This is consistent with the Vickers hardness data in KDP crystal, which are 183 ± 12 for the <001 > and 122 ± 17 for the <100 > direction [8]. It indicates that the elastic modulus (longitudinal sound velocity) of z-cut KDP crystal along the <001 > direction is larger than that of x-cut KDP crystal in the <100 > direction. Based on Eq. (15), the electron-phonon coupling factor of z-cut KDP crystal is greater than that of x-cut crystal. Because the crystal structure of type-II and type-I KDP crystals are located between z-cut and x-cut KDP crystals, the corresponding electron-phonon coupling factor is located between z-cut and x-cut KDP crystals. The above analysis indicates that under the same conditions, z-cut KDP crystal is more prone to energy exchange between electron and the lattice, making it easier for the lattice system to reach the damage temperature. Next, we conduct specific spectral analysis on the electron-phonon coupling effect of the four cutting types of KDP crystal.

 figure: Fig. 5.

Fig. 5. The structural frameworks and corresponding projection structures of z-cut KDP crystal (a) and x-cut KDP crystal (b).

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3.4 Brillouin light-scattering (BLS) spectroscopy

The Brillouin light-scattering (BLS) spectroscopy allows one to directly probe the acoustic phonon frequencies near the center of the Brillouin zone (BZ) and thus obtain the speed of sound [43]:

$${v_s} = {v_p} = \frac{{\mathrm{\lambda }f}}{{2n}}$$
where vp is the group velocity of the associated phonon mode, ƒ is the spectral position of the associated peak.

Figure 6 shows the BLS data for four cutting types of KDP crystal on the Stokes side. It can be seen in Fig. 6(a) that three peaks corresponding to one longitudinal (LA) and two transverse (TA1 and TA2) acoustic phonon polarization branches are resolved. The two transverse phonon frequencies of the z-cut KDP sample are the same, which is related to the fourfold rotation symmetry shown in Fig. 6(b). Since we construct a one-dimensional model, we mainly consider the LA phonon frequency. Based on Eq. (15) and (16), the LA phonon frequency gradually increases from x-cut to z-cut (30.19 GHz < 30.59 GHz < 30.81 GHz < 33.57 GHz), which implies that z-cut KDP crystal has larger sound velocity and electron-phonon coupling factor (G: x-cut < type-II < type-I < z-cut). This further indicates that under the same conditions, more energy is exchanged between the lattice and the plasma of the z-cut KDP sample compared to other cut types, resulting in a higher lattice temperature and greater susceptibility to damage, which is consistent with the experimental results.

 figure: Fig. 6.

Fig. 6. (a) BLS spectra of four cutting types of KDP crystal on the Stokes side. (b) Structure of unit cell of KDP crystal. (top) 2-D structure on acb plane, (bottom) 3-D structure.

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3.5 Damage mechanism—simulation calculation based on self-consistent model

Finally, we conduct simulation calculations based on the above analysis. A one-dimensional finite difference method is used to solve the self-consistent model in the laser propagation direction (z), and the thickness in the z-direction is d = 10 µm. The room temperature is Troom = 300 K = Te(z,0) = Tl(z,0). The laser wavelength is 800 nm, and laser pulse shape is considered as a Gaussian pulse with a duration of tp = 35 fs. The maximum laser intensity reaches the material surface after t0 = 100 fs. According to Beer-Lambert law, the laser intensity can be expressed as follows:

$$I = \sqrt {\frac{{4\ln 2}}{\pi }} (1 - R){I_0}\textrm{exp} ( - 4\ln 2\frac{{{{(t - {t_p})}^2}}}{{t_0^2}})\textrm{exp} ( - \alpha z)$$
where R is KDP surface reflectivity (R = 0.04) and $\sqrt {\textrm{4ln2}\textrm{ / }\mathrm{\pi }} {\textrm{I}_\textrm{0}}$ is the maximum laser intensity.

Because we investigate the mechanism of LISDT (damage area s→0) anisotropy in KDP, we mainly analyze the time evolution of carrier density, carrier temperature and lattice temperature at z = 0. The initial carrier density is n0 = 1 × 1012 cm−3.

The single rate equation determines the onset of damage by reaching the CED. For KDP crystal, the calculation of the CED can be expressed as [37]:

$${n_{cr}} = \frac{{{\mathrm{\varepsilon }_0}m_e^\ast {\mathrm{\omega }^2}}}{{{e^2}}} = 5.21 \times {10^{21}}\textrm{ c}{\textrm{m}^{\textrm{ - 3}}}$$

Figure 7(a) shows the time evolution of carrier temperature, carrier density and lattice temperature in the x-cut KDP crystal when the laser energy density reaches 3.5 J/cm2. At this fluence, the carrier density is close to the CED. Note that there are two peaks in carrier temperature, which may be due to the following reasons. Early in the laser irradiation, due to the initial carrier concentration inside the crystal and the extremely small number of electron-hole pairs generated by photoionization, the carrier heat capacity is small, leading to a rapid and significant increase in carrier temperature. As time increases, the concentration of carrier increases, resulting in a larger heat capacity. Meanwhile, the heat transfer from carrier to the lattice leads to the loss of carrier energy. The above reasons result in a decrease in carrier temperature after the first peak. Subsequently, due to impact ionization, the carrier density increases dramatically, which results in a large absorption of laser energy, causing the carrier temperature to rise again. Finally, after the end of pulse irradiation, thermal equilibrium is established by the electron-phonon coupling effect of the carrier-lattice, and the two subsystems will reach the same temperature. These results indicate that our simulations are consistent with expected physical processes.

 figure: Fig. 7.

Fig. 7. (a) Time evolution of carrier density (ne), carrier temperature (Te), and lattice temperature (Tl) at the front surface (z = 0) of x-cut KDP crystal heated by a 35-fs laser pulse at fluence F = 3.5 J/cm2. Time evolution of carrier density (ne) (b), and lattice temperature (Tl) (c) of four cutting types of KDP crystal heated by a 35-fs laser pulse at fluence F = 3.5 J/cm2.

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Figure 7(b) shows that the carrier density of the four cutting types of KDP crystal approaches the CED when the laser energy density reaches 3.5 J/cm2. As analyzed in section 3.2 above, the x-cut KDP has a high carrier density due to its high photoionization rate and subsequent collision ionization rate (carrier density: x-cut > type-II > type-I > z-cut). Considering the single rate equation, x-cut KDP crystal is more likely to reach the CED, form plasma and undergo damage under the same conditions. However, experiments show that the LISDT is exactly opposite to the trend of carrier density (LISDT: x-cut > type-II > type-I > z-cut). Therefore, it is not enough to only consider the traditional CED for the anisotropy of LISDT of KDP, and the subsequent energy transfer process still needs to be taken into account. Corresponding damage occurs when the lattice temperature reaches the melting point, boiling point, or critical temperature. As shown in Fig. 7(c), the lattice temperature is the lowest due to the relatively small electron-phonon coupling factor of the x-cut KDP, while the z-cut KDP is exactly the opposite (Tl: x-cut < type-II < type-I < z-cut). It indicates that the z-cut KDP is more likely to reach the lattice damage temperature and undergo damage under the same conditions, while the opposite is true for the x-cut type. Furthermore, the trend of the lattice temperature perfectly matches the experimental results. Therefore, the anisotropy of LISDT of KDP crystal is not only related to the CED, but also involves electron-phonon coupling effect. Moreover, the electron-phonon coupling effect is the main factor leading to the anisotropy of the LISDT in KDP crystal. Since some parameters of the model are referenced silicon material, 3.5 J/cm2 and other calculated values may deviate from the reality of KDP crystal. However, it does not affect the analysis of the anisotropy of the damage thresholds for KDP crystal in four cutting types, as these parameters have a fair influence on them.

4. Conclusion

Based on BLS spectroscopy and a self-consistent model, the damage mechanism of the anisotropy of LISDT for KDP crystals is investigated. The finite difference method is used to simulate the carrier-lattice nonequilibrium interaction of KDP crystal irradiated by ultrashort pulse to obtain the temporal evolution of carrier density and temperature as well as the lattice temperature. The results indicate that the trend of lattice temperature and LISDT of the KDP cut-type are very consistent, indicating that the electron-phonon coupling effect of the material is an important factor affecting the LISDT, in addition to the traditionally considered CED. Moreover, the electron-phonon coupling effect is the main factor leading to the anisotropy of the LISDT in KDP crystal. Accurate temperature-dependent material properties are required for improved damage prediction, and these issues still need to be further investigated.

Funding

Key Technologies Research and Development Program (GG20210301-1); Fundamental Research Fund of Shandong University (62350079614137).

Acknowledgments

The authors express their appreciation to Lisong Zhang for his assistance in sample polishing.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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Supplementary Material (1)

NameDescription
Supplement 1       Illustrates the single-pulse damage thresholds of type-?, type-?, and z-cut KDP for 3.38 J/cm2, 2.32 J/cm2 and 2.18 J/cm2, respectively.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (7)
Fig. 1.
Fig. 1. (a) Cutting schematic diagram of KDP crystal sample. (b) Experimental setup for testing the multi-pulse LISDT of KDP crystal under the irradiation of pulse laser with 35 fs duration and 800 nm.

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Fig. 2.
Fig. 2. (a) The damage morphology of x-cut KDP crystal at different numbers and energy of pulse. (b) Derived Fth(N) for N = 15, 30, 50, 100, 500, and 1000 in the case of x-cut KDP crystal. According to Eq. (1), Fth(1) is obtained by fitting the Fth(N) data points. The inserted figure, as an example, is obtained by linear fitting with Eq. (2) for N = 50. (c) Single-pulse LISDT of KDP crystal with different cutting types.

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Fig. 3.
Fig. 3. Damage profile of x-cut and z-cut KDP crystals measured by atomic force microscopy (AFM) under the same condition: (a) N = 30, E = 0.5 µJ, (b) N = 30, E = 0.6 µJ.

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Fig. 4.
Fig. 4. (a) Ionization rate and Keldysh parameter as a function of laser intensity for 800 nm light in x-cut KDP crystal. The blue solid line represents the photoionization rate (WPI) based on the full expression from Keldysh, while the purple dashed-dotted line represents the tunneling ionization rate (WTI), and the red dashed line represents the multiphoton ionization rate (WMPI). (b) WPI of four cutting types of KDP crystal. (c) Enlargement of the yellow area in Fig. 3(b).

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Fig. 5.
Fig. 5. The structural frameworks and corresponding projection structures of z-cut KDP crystal (a) and x-cut KDP crystal (b).

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Fig. 6.
Fig. 6. (a) BLS spectra of four cutting types of KDP crystal on the Stokes side. (b) Structure of unit cell of KDP crystal. (top) 2-D structure on acb plane, (bottom) 3-D structure.

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Fig. 7.
Fig. 7. (a) Time evolution of carrier density (ne), carrier temperature (Te), and lattice temperature (Tl) at the front surface (z = 0) of x-cut KDP crystal heated by a 35-fs laser pulse at fluence F = 3.5 J/cm2. Time evolution of carrier density (ne) (b), and lattice temperature (Tl) (c) of four cutting types of KDP crystal heated by a 35-fs laser pulse at fluence F = 3.5 J/cm2.

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Tables (1)

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Table 1. Model parameters

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Equations (18)

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F t h ( N ) = F t h ( 1 ) N S − 1
s ( N , F 0 ) = π   ⋅ r 2  = ( π   ⋅ w 0 2 ) [ ln ⁡ F 0 − ln ⁡ F t h ( N ) ]
γ = ω e [ m e ∗ c n ε 0 E g 2 I ] 1 / 2
W P I = 2 ω 9 π ( ω m e ∗ ℏ Γ ) 3 / 2 Q ( γ , x ) exp ( − π ⟨ x + 1 ⟩ K ( Γ ) − E ( Γ ) E ( ζ ) )
Q ( γ , x ) = π 2 K ( ζ ) ∑ n = 0 ∞ { exp ( − n π K ( Γ ) − E ( Γ ) E ( ζ ) ) Φ { π 2 ⟨ x + 1 ⟩ − 2 x + n 2 K ( ζ ) E ( ζ ) } }
x = 2 E g π ℏ ω Γ E ( ζ )
Φ ( z ) = ∫ 0 z exp ( y 2 − z 2 ) d y
W M P I = 2 ω 9 π ( m e ∗ ω ℏ ) 3 / 2 × Φ ( 2 ⟨ Δ ℏ ω + 1 ⟩ − 2 Δ ℏ ω ) × exp { 2 ⟨ Δ ℏ ω + 1 ⟩ ( 1 − 1 4 γ 2 ) } ( 1 16 γ 2 ) ⟨ Δ ℏ ω + 1 ⟩ Δ = E g + E g 4 γ 2
W T I = 2 E g 9 π 2 ℏ ( m e ∗ E g ℏ 2 ) 3 / 2 ( ℏ ω E g γ ) 5 / 2 × exp ( − π 2 E g γ ℏ ω ( 1 − γ 2 8 ) )
∂ n e ( t ) ∂ t = W P I ( I ( t ) ) + W I I ( I ( t ) , n e ( t ) ) − W r e l ( n e ( t ) , t )
∂ ( n e E g + C e T e ) ∂ t = ∇   ⋅ ( k e ∇ T e ) + n e θ I + W M P I   ⋅ 5 ℏ ω − G ( T e − T l )
C l ∂ T l ∂ t = ∇   ⋅ ( k e   ⋅ T l ) + G ( T e − T l )
W I I − D r u d e = σ E g ⋅ I ( t ) σ = e 2 c ε 0 n m e ∗ ⋅ τ c 1 + ω 2 τ c 2 τ c = 16 π ε 0 m e ∗ ( 0.1 E g ) 3 2 e 4 n e ( t )
W r e l = n e ( t ) τ r
G = π 4 ( n e v s k B ) 2 K
v s = v p = λ f 2 n
I = 4 ln ⁡ 2 π ( 1 − R ) I 0 exp ( − 4 ln ⁡ 2 ( t − t p ) 2 t 0 2 ) exp ( − α z )
n c r = ε 0 m e ∗ ω 2 e 2 = 5.21 × 10 21  c m  - 3
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