1. Introduction

High-power continuous-wave (CW) lasers, which have average power ranging from several watts to kilowatts, are used in many industries, including laser processing, laser machining, and laser displays. LiNbO₃ and LiTaO₃ crystals are especially useful for second-harmonic generation (SHG) of CW light from the infrared to visible regions because of the high nonlinearity [1]. However, laser-induced breakdown and damage in nonlinear crystals limit output laser power in high-average-power lasers, where the breakdown threshold intensities of green light for LiTaO₃ are in the order of several tens of MW/cm² [2].

Since the 1990s, absorption due to the presence of long-lived excited states, like polarons, has been studied in LiNbO₃ and LiTaO₃ crystals [3]. The dynamics of polarons is investigated by various theoretical models [4–7]. The stretched-exponential decay behavior of polarons, however, is not yet completely clear.

We proposed that one of the causes of the catastrophic breakdown is a light-induced heating by the accumulated polarons [8]. Using a simple rate-equation model, we evaluated the absorption power density of polaron, $Q_p=\frac{1}{R_p}\frac{\beta {\sigma }_p}{2\overline{h}\omega }{I}^3$, where I and are the laser ω intensity and frequency, β is the two-photon absorption coefficient, σ_p is the absorption cross section of polarons, R_p is the decay rate of polarons. The absorption power density is proportional to the cube of the laser intensity. The light-induced heating causes the breakdown of LiNbO₃ crystal at relatively low intensities of the order of MW/cm².

In this paper, we investigate light-induced heating by absorption due to a long-lived polarons and thermal breakdown in LiNbO₃ crystals. In Sec. 2, we propose a modified rate-equation model for fundamental and second-harmonic laser lights. In Sec. 3, important parameters of the rate-equation are determined by reproducing experimental results and light induced heat is evaluated using the rate-equation model that could easily couple the thermal equation model to estimate heating. Some concluding remarks are given in Sec. 4.

2. Rate-equation model

The absorption mechanism for light-induced heating consists of the following three steps: (1) generation of conduction-band (CB) electrons and holes by two-photon absorption, (2) formation of electron and hole polarons [9] via the CB electrons and holes, and (3) light absorption by polarons.

In the previous rate-equation model [8], for simplicity, we treated polarons as one polaron without distinguishing between different types of polarons. In LiNbO₃ crystal, there are typically four kinds of polarons, which are the free electron polaron $({\text{Nb}}_{\text{Nb}}^{4+})$, bound electron polaron $({\text{Nb}}_{\text{Li}}^{4+})$, bound electron bipolaron $({\text{Nb}}_{\text{Li}}^{4+}:{\text{Nb}}_{\text{Nb}}^{4+})$, and bound hole polaron (O⁻). A kinetic model including iron impurities has been already proposed [10, 11]. Using the kinetic model, we propose a modified rate-equation that treats three kinds of polarons except the bipolaron, because ${\text{Nb}}_{\text{Nb}}^{4+}$, ${\text{Nb}}_{\text{Li}}^{4+}$, and O⁻ polarons are dominating in magnesium(Mg)-doped LiNbO₃ [12]. In the model, we neglect the iron impurities because impurities reduced crystals are used in SHG. A schematic of the model is shown in Fig. 1. In SHG of green light, there are infrared and green lights, hereafter referred to as IR and GR lights, respectively.

 

Fig. 1 Schematic of the rate-equation model. ${\text{Nb}}_{\text{Nb}}^{4+/5+}$, ${\text{Nb}}_{\text{Li}}^{4+/5+}$, O^−/2− are represented by 1, 2, and 3, respectively.

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The rate equations for the number densities of CB electrons, n_e, holes, n_h, and n_i (i = 1,2,3) show the number densities of the free electron polaron $({\text{Nb}}_{\text{Nb}}^{4+})$, bound electron polaron $({\text{Nb}}_{\text{Li}}^{4+})$, and bound hole polaron (O⁻), respectively, can be written as

3. Important parameters and light-induced heat

These rate equations have more than twenty parameters to characterize crystals. Most values, except absorption cross sections of polaron and two-photon absorption coefficients, however, have not been directly determined experimentally or theoretically. We use the absorption cross sections of polarons [12] and two-photon absorption coefficients β [13–15] already determined experimentally.

First, some important parameters are determined by the result that the generation time of polarons was reported to be less than 10⁻¹² s [16, 17], namely, CB electrons and holes generated by two-photon absorption are rapidly trapped before CB electrons or holes directly recombine. Because the generation time is less than 10⁻¹² s, we assume that the trapping coefficients T_e−1, T_e−2, and T_h−3 are 10⁻⁷ cm³/s and the recombination coefficients R_e−h, R_{1 −h}, R_2−h, and R_3−e would have would to be less than the trapping coefficients. Therefore, we assume that the recombination coefficients are much less than 10⁻⁷ cm³/s. Because there are almost no CB electron and hole when relaxation of n₁, n₂, and n₃ starts, the recombination coefficients (R_1−h,R_2−h,R_3−e) do not affect the polarons dynamics. Three densities, $n_i^0(i=1,2,3)$ and four coefficients R₁₋₃, R₂₋₃, γ₁₋₂, and γ₂₋₁ remain.

Second, three densities must be determined by the composition of crystals. The model includes anti-site defects (Nb_Li), which number density $n_2^0$ is an important parameter. We consider that the number density $(n_2^0)$ of anti-site defects expresses the difference of the Mg-doped, congurent, and stoichiometric composition. We are interested in Mg-doped LiNbO₃, which is popular in SHG. The density of the unit cell is 3.1 × 10²¹ cm⁻³. Then, we assumed that $n_1^0=3.1\times {10}^{21}{\text{cm}}^{-3}$, $n_2^0=3.1\times {10}^{18}{\text{cm}}^{-3}$, and $n_3^0=3.3\times {10}^{19}{\text{cm}}^{-3}$, which represent ${\text{Nb}}_{\text{Nb}}^{5+}$, ${\text{Nb}}_{\text{Li}}^{5+}$, and O²⁻ near the Li vacancy, respectively.

Third, by fitting the experimentally obtained stretched exponential curve for Mg-doped LiNbO₃ [14] which was fitted by exp[−(t/τ)^β ], where τ = 40ns, β = 0.37, the remaining four parameters are determined to be R₁₋₃ = 4 × 10⁻¹¹ cm³/s, R₂₋₃ = 0.1 × 10⁻¹¹ cm³/s, γ₁₋₂ = 4 × 10⁻¹¹ cm³/s, and γ₂₋₁ = 4 × 10⁻cm³/s. The ratio γ₁₋₂/γ₂₋₁ strongly depends on the temperature. Figure 2 shows the temporal evolution of the light-induced absorption of the model and experimental results. The results of the model calculation using these parameters reproduce the stretched-exponential decay well. Figure 3 shows the temporal evolution of the number density of polarons. The apparent stretched-exponential decay is reproduced by the absorption contribution varying from the free electron polaron $({\text{Nb}}_{\text{Nb}}^{4+})$ to the bound electron polaron $({\text{Nb}}_{\text{Li}}^{4+})$. The bound hole polaron (O⁻), which absorption cross section of 780 nm is small compared with electron polarons, do not almost contribute the absorption.

 

Fig. 2 Temporal evolution of the light-induced absorption. Solid red and black lines show the calculation of this model and experimental results [14], respectively. Dotted and dashed black lines show the exponential decay function of decay constants τ = 40 and 167 ns, respectively. The stretched exponential curve in the experiment is fitted by exp[−(t/τ)^β], where τ = 40ns, β = 0.37. The average decay time $<\tau >=\frac{\tau }{\beta }\mathrm{\Gamma }[1/\beta ]$, where Γ is the gamma function, is 167 ns. Solid red line almost overlaps with solid black one.

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Fig. 3 Temporal evolution of the number densities. Solid, dotted and dashed lines show the number densities of n₁, n₂, and n₃, respectively.

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Light-induced heat in SHG is evaluated using the rate equation with these parameters. All parameters for simulation are shown in Table 1. Using the number densities of polarons, we can obtain the absorption power density of polarons, Q_p ≃ α₁I₁ + α₂I₂, where α_1,2 = Σ_i=1,3σ_i(ω_1,2)n_i. Figure 4 shows the absorption power density as a function of the second harmonic and fundamental laser intensities.

Table 1. All parameters for simulation. ACS, RC, and TC show absorption cross section, recombination coefficient, and transition coefficient, respectively.

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Fig. 4 Absorption power density as a function of intensity of second harmonic (SH) laser. Circle, down-pointing triangle, triangle, diamond, and square indicate fundamental laser intensities 0, 20, 40, 60, and 100 MW/cm², respectively.

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We evaluate temperature increase by the same conditions as previous study [8]. Absorption power density Q = 22 MW/cm³ is the critical power density required for reaching melting temperature of 1530 K. The critical intensity of second harmonic laser $I_2^{\text{cr}}=34\text{MW}/{\text{cm}}^2$ without fundamental laser. When intensity of the fundamental laser remains 48 MW/cm² and 130 MW/cm², $I_2^{\text{cr}}=20\text{MW}/{\text{cm}}^2$ and 10 MW/cm², respectively. If a critical temperature exists before reaching melting temperature, the critical intensity could be less than $I_2^{\text{cr}}=10\text{MW}/{\text{cm}}^2$.

4. Concluding remarks

We propose a modified rate-equation based on the polaron kinetics previously proposed [10, 11]. Important parameters of the rate-equation are determined by reproducing the experimental results [14]. We show that the kinetic model which includes the relaxation between free and bound polarons reproduces the experimental results well. However, it is possible that there are different sets of parameters that produce similar curve. The kinetic model is non-dimensional model; however, because hopping charge transport is also important [4, 5], it will be necessary to decide these coefficients more carefully in future studies.

We evaluate the temperature increase and estimate the critical intensity of breakdown using the rate equation. The laser intensities close to experimental findings and light-induced heat was estimated as a trigger of catastrophic breakdown.