1. Introduction

Possibility of using periodically poled (PP) structures for non-linear light conversion has been outlined in the classical work on non-linear optics [1]. However only three decades after first efforts [2] and the development of crystal poling technology [3] the non-linear PP crystals have attracted attention of the practical applications [4–11] enabling a high efficiency at room temperatures, operation flexibility and easy temperature tuning at operated wavelengths, extended later on to terahertz wave generation by using frequency difference techniques [12–15]. Lithium tantalate (LT) is one of the best PP materials due to its high second order susceptibility and effective non-linearity, d_eff≈10 pm/V [11]. Previous reports provide a few experimental results and theoretical models related to continuous wave and ns pulsed operation with PP crystals aimed to SHG efficiency study, optical optimization, and towards revealing possible limitations associated with thermal dephasing, spectral bandwidth, damage generation mechanism due to two-photon absorption (TPA) followed by ionization and local defect generation [16–22], and thermal optimization of SHG by applying a temperature gradient along the crystal [23] used for high input power operation [24]. Other papers on non-linear conversion using LT extend significantly insight into the optical properties and related thermal effects leading to an optimized control in operation with this material [25–27].

In this communication we report our experimental data for quasi-phase-matched (QPM) temperature-controlled 18 ps pulsed high-frequency second-harmonic generation (SHG) by Mg-doped stoichiometric periodically poled LiTaO₃ (PPSLT). We also provide theoretical calculations revealing a complex mechanism involved in the inhibition of SHG efficiency, its saturation and stabilization for a wide range of the input intensity. Finally, our experimental data on temperature-controlled SHG operation are used for the estimation of TPA coefficient for generated 515 nm radiation.

2. Experimental results

The experimental setup schematically shown in Fig. 1 includes a PPSLT crystal mounted on the temperature controlled metal pedestal. An optical system including a femtosecond oscillator, stretcher, regenerative amplifier and a compressor with diffraction grating allows the generation of 18 ps pulses of fundamental wavelength with λ₁ = 1.03 μm with a spectral bandwidth of Δν≈300 GHz (Δλ≈1 nm) and pulse repetition rate ν_P = 10 - 100 kHz.

 

Fig. 1 Experimental scheme of temperature-controlled SHG.

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The experimental data are shown in Figs. 2(a) and 2(b). The crystals which have a poling period of Λ≈7.24 μm, are 2 mm in length (along beam propagation), 4 mm wide and 1 mm thick. QPM temperature is adjusted for PPSLT crystals under low intensity radiation input to ensure optimal SHG operation. In particular, Fig. 2(a) shows SHG efficiency vs the pulse peak input intensity for: (i) 20 kHz operation with the beam radius √2r₀ = 0.15 mm and QPMtemperature T_QPM = 37 °C for I = Q/2πτ_Pr₀² = 0.1-1 GW/cm², and (ii) 10 kHz operation with √2r₀ = 0.25 mm at T_QPM = 41 °C for I = 0.1-4 GW/cm². Both cases demonstrate similar behavior of SHG efficiency with increase in I. For the first case the efficiency achieves η≈0.33 at I≈1 GW/cm². For the second case SHG efficiency reaches this level at the same value of I, saturates at I≈3 GW/cm² and starts to decay for I>3 GW/cm².

 

Fig. 2 Experimental SHG efficiency vs peak input intensity for (a) 10 and 20 kHz operation with 0.5 mm beam diameter under T = T_QPM, and for (b) 10 kHz temperature-controlled operation with 0.3 mm beam diameter.

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Figure 2(b) shows SHG efficiency measured for a significantly larger range of I = 0.1-9.5 GW/cm² for 10 kHz operation and √2r₀ = 0.15 mm. First, SHG efficiency is measured for the QPM temperature maintained during the operation (T = T_QPM = 40 °C), and, second, for a series of the temperature-controlled SHG operations when the temperature of the metal pedestal was maintained below that of QPM, i.e. at T = 30, 32, 34 36 and 38 °C. In particular, SHG efficiency measured for T = T_QPM = 40 °C shows the saturation η≈0.35 at I = 4-5 GW/cm² followed by a monotonous decrease towards η≈0.23 at I = 9.5 GW/cm². However, SHG experiments made under lower values of T show that the related values of η reach their maxima under larger values of I. For example, in operation under T = 34 °C the efficiency reaches η≈0.35 at I = 8 GW/cm². Thus, the decrease of the operational temperature below that of QPM is compensated by the temperature increase induced by TPA.

Finally let us note, that our experimental data for the maximum of I = 9.5 GW/cm² show that the loss of η in operation at T = T_QPM = 40 °C can be completely recovered by decreasing the operational temperature on ΔT≈10 °C. Hence, the related temperature increase in PPSLT crystal operating at I = 9.5 GW/cm² is of order ΔT≈10 °C.

3. Calculations and discussion

In calculations and related discussion of our experiments we address the following points: (i) a relatively low efficiency of SHG operation under high intensity input pulse, (ii) a quasi-constant level of SHG efficiency observed for a wide range of input laser intensity and (iii) an estimation of TPA coefficient based on our experimental data of thermal control.

First, let us consider the wave vector mismatch given by [28]:

Second we consider the related dephasing factor given as follows:

In particular in Figs. 3(a) and 3(b) we show the values of (a) Δk_Q and (b) F as functions of temperature increase, ΔT, above T_QPM for the characteristic values of the spectral shift within the range of Δν = 300 GHz calculated by using Sellmeier equation for SLT [29]. Figure 3 shows that for the perfect temperature matching (ΔT = 0) the spectral shift ± Δν/4 = ± 75 GHz leads to ≈15% efficiency loss, whereas the shift ± Δν/2 = ± 150 GHz leads to ≈50% efficiency loss. Second, Fig. 3 suggests why SHG efficiency can remain quasi-constant within a wide range of input laser intensity, even under a significant temperature increase. That is, the Fig. 3 shows that for ν>ν₀ the initial wave vector mismatch is enhanced by the thermal effect, and the value of F decreases monotonically with increase in ΔT. In contrast, for ν<ν₀ the initial wave vector mismatch is inhibited (Δk_Q tends towards 0 with increase in ΔT) by the thermal effect when ΔT≤3.5 K and ΔT≤6.5 K, respectively for -Δν/4 and -Δν/2. Hence, for ν<ν₀ in this temperature range the value of F increases towards 1. Thus, Fig. 3 shows that for ΔT of few K the loss of SHG efficiency for ν>ν₀ can be compensated by its increase for ν<ν₀ .

 

Fig. 3 Dephasing vs temperature increase: (a) wave vector mismatch, Eq. (1), and (b) related dephasing factor, Eq. (2).

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However, the above effect cannot completely explain the low efficiency observed in our experiments. That is, let us consider the characteristic length of energy exchange between harmonics given (in Gaussian units) by [28]:

For instance, for I≈0.5 GWcm² this expression gives the value l = 0.85 mm suggesting that for the operation with the crystal length of L = 2 mm SHG efficiency must be close to its saturation. In contrast, experimental data given in Figs. 2(a) and 2(b) show that for I≈0.5 GW/cm² SHG efficiency is close to η≈0.2 and starts to saturate at I≈2 GW/cm² .

In order to clarify this point let us consider a simplified equation for SH wave amplitude neglecting all types of absorption and assuming that FH is not depleted along the beam being constant during the pulse duration (A₁ = const). In particular, we consider the following equation for SH wave amplitude distribution along the beam:

Let us obtain an estimate of the non-steady-state effect by neglecting the dephasing effect (Δk_Q = 0) and using a simplistic scaling approximation for the non-steady-state term as follows: $V_2^{-1}\partial A_2/\partial t\approx A_2/V_2{\tau }_P$. By integrating the value of A₂ from z = 0 (where A₂ = 0) along z we obtain for the output wave amplitude $A_2(L)\approx iG{V}_2{\tau }_P\left[1-\exp (-L/V_2{\tau }_P)\right]$, leading finally to the following expression for the output SH intensity:

These expressions show that SHG efficiency for short pulsed (ps) operation can be significantly lower as compared with the long pulse (ns) operation for the similar input intensity and crystal length. In particular, by using the values of V₂ = c/n₂ with n₂≈2.2 [29], τ_p = 18 ps and L = 2 mm we find: ζ = L/V₂τ_p = 0.815 and f(ζ)≈0.47. This estimate allows us to conclude that in our experimental operation the non-steady-state behavior of the wave amplitude can play a significant role in SHG operation leading to [1- f(ζ)]x100 = 53% loss of the maximal efficiency as compared with ns pulsed SHG operation where the efficiency can achieve the level of ≈0.6-0.7 (with ζ = L/V₂τ_p<<1 and f(ζ)≈1).

Hence, we can suggest that this effect combined with the dephasing effect of the spectral bandwidth of the input radiation is the reason of the low SHG efficiency (below η≈0.2) observed for I≈0.5 GW/cm² when the thermal dephasing is negligibly small. In particular, by using the values of (i) f(ζ)≈0.47, (ii) F = 0.5-0.85 from Fig. 3(b) and (iii) SHG efficiency η*≈0.6 achieved with a similar 2 mm long PPSLT crystal in ns pulsed operation under I = 0.36 GW/cm² [21] we find that for the present case the possible involvement of both effects leads to the value of η≈f(ζ)Fη*≈0.14-0.24 agreeing well with our data shown in Fig. 2.

Next, we use our experimental data for a simplistic estimation of TPA coefficient for 515 nm radiation via few analytical expressions. In particular, assuming that for the related range of input intensity βη²I²>>α₁(1-η)I + α₂ηI (where α₁≈0.002 cm⁻¹ and α₂≈0.025 cm⁻¹) we can estimate the temperature increase of a single pulse (SP) by neglecting the heat diffusion as follows:

In contrast with SP the high frequency operation involves the effect of heat diffusion. The average temperature increase can be estimated by using the thermal linear superposition of pulse repetition by taking into account the heat diffusion effect as follows [20,21]:

By combining Eqs. (8) and (9) and using the experimental values of the temperature shift ΔT_AV = ΔT_exp corresponding to the recovery of SHG efficiency found in Fig. 2 b for related values of η and I, we obtain the following range for the TPA coefficient:

By using I = 9.5 GW/cm² and η = 0.35 we find for the single pulse: ΔT_SP ≈0.17 K for β≈2.7 cm/GW, and ΔT_SP ≈0.08 K for β≈1.2 cm/GW. By using the calculated value of S_N = 111.5 we find ΔT_AV ≈19.7 K for β≈2.7 cm/GW, and ΔT_AV ≈8.8 K for β≈1.2 cm/GW. These values agree with the experimental data in Fig. 2(b). These estimates show that the average temperature increase is about 2 orders of magnitude higher than that of SP. Moreover, Fig. 2(b) shows that the decrease in η starts at I≈5 GW/cm². The estimation of the temperature for I≈5 GW/cm² and β≈2.7 cm/GW gives ΔT_AV ≈5.4 K agreeing well with the value of ΔT under which the dephasing factor F is found to reach the maximum in Fig. 3(b).

We have to note that the above estimated TPA coefficient agrees well with the experimental value β≈1.2 cm/GW for 532 nm radiation [31] and with the theoretical estimate of TPA coefficient in LT for SH (515 nm): $\beta \approx 2{\sigma }_{ng}^{(2)}{N}^{*}/\hslash {\omega }_2$≈2.6 cm/GW, where ${\sigma }_{ng}^{(2)}$≈2.5x10⁻⁵⁰ cm⁴ s/photon² is the TPA cross section [28] and N*≈1.9x10²² cm⁻³ is the molecular density of LT.

Let us now consider the possible contribution of green induced infrared absorption (GRIIRA) by estimating the density of free electrons from $d{n}_e/dt\approx \beta {(\eta I)}^2/E_g-{\gamma }_R{n}_e^2$, by setting in which dn_e/dt = 0 we can find the maximal value:

By making the estimation for ε_e = 10 eV (V_e = 1.9x10⁸ cm/s and γ_R≈3.8x10⁻⁶ cm³/s), I = 9.5 GW/cm², η = 0.35 and β≈1.2-2.6 cm/GW we find the following range for n_e-max≈(0.7-1)x10¹⁷ cm⁻³. Additionally we can estimate the free electron density decay between the pulses from$d{n}_e/dt\approx -{\gamma }_R{n}_e^2$. By integrating this equation and assuming that the decay starts from the value given by Eq. (11) we find: $n_e(t)\approx n_{e-\max }/(1+n_{e-\max }{\gamma }_{R}t)$. By using n_e-max≈10¹⁷ cm⁻³, γ_R≈3.8x10⁻⁶ cm³/s one finds that for the moment when a new pulse starts, i.e. t = ν⁻¹ = 10⁻⁴ s, the electron density decays towards n_e = 2.9x10⁹ cm⁻³. We can also assume that after pulse the free electrons equalize their energy with the lattice within few ps. Under T_e = 300 K the values of V_e≈10⁷ cm/s and γ_R≈0.2x10⁻⁶ cm³/s give n_e≈5x10¹⁰ cm⁻³ for t = 10⁻⁴ s.

This estimate shows that in contrast with the temperature pulse superposition effect the electron density does not accumulate under 10-20 KHz pulse repetition, and by using Eq. (11) for the value of n_e we can estimate the absorption by the free electrons as follows:

In particular, for the maximal value of n_e≈10¹⁷ cm⁻³ we find that Eq. (12) gives α_1-e≈0.13 cm⁻¹ (FH) as compared with α₁≈0.002 cm⁻¹, and α_2-e≈0.032 cm⁻¹ (SH) as compared with α₂≈0.025 cm⁻¹.

Let us also consider the absorption mechanism which can be attributed to the defects and electrons shallow traps generated by high energy collisions of free electrons. By using the value of Eq. (11) we can estimate the number of high energy collisions able to produce defects as ≈n_eR_ionδτ≈10¹⁶-10¹⁷ cm⁻³, where R_ion≈10¹⁰-10¹¹ s⁻¹ is the rate of high energy electron collisions [21] and δτ ≈10 ps is the characteristic time for onset of n_e≈10¹⁷ cm⁻³. By using now the typical value for the absorption cross section of entrapped electrons σ_a≈10⁻¹⁸ cm² we can estimate the maximal limits (assuming that all collisions produce defects) for the related absorption coefficients as ≈σ_a n_eR_ionδτ≈0.01-0.1 cm⁻¹.

The above estimates show that these GRIIRA mechanisms can seriously increase FH absorption. However, for the related heat release and induced temperature increase this effect remains negligibly small. In particular, by taking into account all related values we can find α_1Σ = α₁ + α_1-e + σ_an_eR_ionδτ≈0.002 + 0.13 + 0.1≈0.23 cm⁻¹ and α_2Σ = α₂ + α_2-e + σ_an_eR_ionδτ≈0.025 + 0.032 + 0.1≈0.16 cm⁻¹. That is, for η≈0.35 and I = 9.5 GW/cm² the value of the absorbed energy given by α_1Σ(1-η)I + α_2ΣηI≈1.9 GW/cm³ is much lower than that by TPA βη²I²≈30 GW/cm³. Hence, the contribution of GRIIRA into the heat release remains within few % compared with that of TPA. Additionally, let us note that for the estimation of the absorption by the free electrons and shallow traps electrons we have used the maximal electron density and collision rate giving us an upper limit of this effect during the pulse.

Finally, let us estimate the possible effect of TPA on the SHG efficiency by integrating dI₂/dz = -βI₂² with an initial value of I_2-in . Obtaining I₂(δz) = I_2-in(1 + βI_2-in δz) ⁻¹ one finds that for the maximal values of η≈0.35, I = 9.5 GW/cm² and I_2-in = η I the factor of (1 + βI_2-in δz)⁻¹≈0.7 for δz = 0.5 mm meaning that TPA absorption can strongly contribute to the observed saturation of SHG efficiency. However, for η≈0.3-0.35 and I = 2-3 GW/cm² this effect is not strong and TPA does not contribute to the onset of saturation. Hence, the role of TPA in saturation of SHG efficiency is two-fold. First, by heat generation and temperature increase (Eqs. (7)–(9)) TPA leads to the thermal dephasing affecting SHG efficiency in line with Eqs. (1) and (2). Additionally, due to TPA the generated second harmonic is prone to the strong attenuation with the distance of order ≈I₂⁻¹ dI₂/dz≈-βηI≈-(0.8-8) cm⁻¹ for I = 1-9.5 GW/cm².

4. Conclusions

We have performed the experimental study of high repetition pulsed ps SHG for input radiation at 1.03 μm with a spectral bandwidth 300 GHz by using PPSLT for the input peak intensity I = 0.1-9.5 GW/cm² and under repetition rate 10-20 kHz. For I>3 GW/cm² SHG efficiency achieves the saturation level of η≈0.35 which can be maintained within a wide range of I = 3-9.5 GW/cm². The loss of SHG efficiency observed for I>5 GW/cm² can be recovered to the level of η≈0.35 by using temperature-controlled operation. By combining our experimental data and explicit analytic expressions for the temperature increase under the effect of TPA and heat diffusion involved in high frequency operation we estimated the TPA coefficient for 515 nm radiation to be within the range of β≈1.1-2.7 cm/GW, agreeing well with the theoretical estimate of β≈2.6 cm/GW. Our analysis suggests that SHG efficiency observed in our experiments is defined by the complex interaction of several effects: (i) the strong contribution of the spectral bandwidth of the input radiation to the wave vector mismatch and dephasing, (ii) the non-steady-state effect which decreases SHG efficiency in ps pulsed operation as compared with ns pulsed operation for the similar input intensity and crystal length, (iii) complex negative and positive contribution of the thermal effect associated with TPA to dephasing depending on the spectral range, and (iv) strong attenuation of the generated SH by TPA. The complex action of thermal dephasing is due to the specific dependence of the related wave vector mismatch on the input frequency and temperature increase. The calculations using Sellmeier equation for SLT suggest that for the spectrum range with ν>ν₀ the initial mismatch is enhanced by the increase in T. In contrast, for ν<ν₀ the initial mismatch is inhibited by the increase in T until ΔT does not overcome the level of few K (3.5 – 6.5 K). Thus, with the temperature increase of few K the decrease of SHG efficiency for ν>ν₀ can be compensated by SHG efficiency increase for ν<ν₀, resulting in the efficiency stabilization over the wide range of input intensity as observed in our experiments.