## Abstract

Experimental results indicate that there is a distinct difference between the temperature tuning curves obtained with intracavity second harmonic generation (Intra-SHG) and extracavity SHG (Extra-SHG) configuration. By introducing the nonlinear SHG loss with the phase mismatch taken into consideration, the rate equations are established to describe the Intra-SHG laser performance. As for the widely used SHG crystal, KTP and LBO, theoretically calculated variation trends of the temperature insensitivity are in agreement with that obtained experimentally. The corresponding explanations have also been given in this paper. Our results may provide a new view on the temperature insensitivity of Intra-SHG configuration. The method employed in our study can be extended to other nonlinear crystals and intracavity nonlinear frequency conversions.

© 2012 OSA

## 1. Introduction

Intracavity second harmonic generation (Intra-SHG) achieved by placing the nonlinear crystal inside the fundamental laser cavity has become a mature technique for efficient frequency doubling the cw or high-repetition-rate lasers [1–5]. In comparison with the extracavity SHG (Extra-SHG), Intra-SHG can take advantage of the high power density inside the laser cavity. As for the nonlinear crystals, owing to the dependence of their refractive indices on angle, temperature and wavelength, any change of these parameters will cause phase mismatch. It has been well known that the generated second-harmonic power is strongly dependent on this phase mismatch. In Extra-SHG configuration, the phase-mismatch induced temperature (angle or wavelength) tuning curve is generally presented as a sinc^{2} function. However, an interesting question may be raised. For the given nonlinear crystal but employed as Intra-SHG converter, is the dependence of SHG power on the above-mentioned parameters same as that presented in Extra-SHG configuration?

According to our experimental results demonstrated in [6], the temperature tuning curve of the Intra-SHG KTP deviated from the expected sinc^{2} function. When the temperature tuning curve of the same KTP was measured with Extra-SHG configuration, it showed the typical shape. Due to the fact that the experimental condition was kept the same in the two measuring processes, this difference should be attributed to the use of Intra-SHG configuration. Study on the temperature insensitivity of Intra-SHG configuration may show some interests. To the best of our knowledge, there is no report on this issue up to now.

In this paper, by introducing the nonlinear SHG loss with the phase mismatch taken into consideration, the rate equations are established to describe the Intra-SHG laser performance. Based on the theoretical and experimental results for the widely used SHG crystals, it is found that the temperature insensitivity obtained with Intra-SHG configuration has an advantage over that obtained with Extra-SHG both for the type-II KTP and LBO. The difference in temperature insensitivity between the two configurations obtained with KTP is larger than that of LBO. The varying fundamental photon density and pulse width induced by the SHG conversion will play a positive role in improving the Intra-SHG temperature insensitivity.

## 2. Theoretical model

Two factors should be taken into account before. Firstly, when the laser crystal is selected, the fundamental laser wavelength (including the linewidth) is fixed. Wavelength insensitivity measurement is hard to make in the Intra-SHG configuration. Secondly, when the phase-matched crystal is inserted within the given cavity, angle tuning is likely to cause the damage of the nonlinear crystal due to the high power density inside cavity. However, temperature tuning of the Intra-SHG crystal is convenient to achieve. In addition, temperature has an important effect on the performance of the assembled nonlinear device. Therefore, this paper will focus on the temperature insensitivity of Intra-SHG crystals.

The phase mismatch$\Delta k$as a function of temperature can be written in the case of type-II SHG as follows:

*e*” and “

*o*” referring to the extraordinary and ordinary polarization, respectively.

For the Intra-SHG configuration, the SHG process is coupled into the dynamical behavior of fundamental laser. The discussions on the temperature insensitivity should turn to the model of rate equations. Furthermore, it can be anticipated that this dependence cannot be expressed analytically. To establish the rate equations, the SHG conversion can be considered as a nonlinear loss being equivalent to the output loss of the fundamental wave. Under the plane wave approximation and ignoring the turn-on time and turn-off time of the acoustic-optic (AO) Q-switch, the rate equations describing laser diode (LD) pumped AO Q-switched Intra-SHG laser can be presented as follows [7, 8]:

During the interval when the Q-switch is shut down, the initial inverted population density can be deduced from Eq. (2). Its value can be expressed by

## 3. Theoretical results and discussions

KTP is extensively used in the SHG conversion because of its large second-order susceptibility and phase-matching temperature bandwidths. Figure 1
shows the calculated dependences of normalized second-harmonic average output powers on$\Delta T$for a 1cm long type-II phase-matched KTP ($o+e\to e$) obtained with Extra-SHG and Intra-SHG configuration, respectively. $\Delta k$and$\Delta T$satisfy the relation of$\Delta k=\text{-}0.3\Delta {T}^{}{\text{cm}}^{-\text{1}},$ with the corresponding temperature derivatives of the refractive indices of KTP at o-1064 nm, e-1064 nm and e-532 nm selected as 3.4 × 10^{−5} °C^{−1}, 2.5 × 10^{−5} °C^{−1}and 3.2 × 10^{−5} °C^{−1}, respectively [9]. The other parameters used for the solution of rate equations are given in Table 1
. Nd:YAG is the laser crystal.

As can be seen from Fig. 1, the temperature tuning curves show distinct differences between the two configurations. For Intra-SHG, greatly broadened central band and heightened side lobes are impressive. The FWHM (full width at half maximum) of central band is up to 35 °C, greatly larger than that of 20 °C displayed in Extra-SHG configuration. When the temperature greatly deviates from the exact phase-matched temperature, the Intra-SHG laser still output considerable power. The peak value of the secondary and thirdly side lobe can reach 0.63 (Δ*T* = 29 °C) and 0.37 (Δ*T* = 50 °C), respectively.

LBO is also an efficient SHG crystal with high threshold for optical damage. However, its temperature insensitivity is exceptionally low for type-II SHG conversion at 1064 nm in comparison with KTP. What about its Intra-SHG temperature insensitivity? Figure 2
shows the calculated temperature tuning curves of 1cm long LBO ($o+e\to o$) obtained with the two configurations. $\Delta k$and$\Delta T$satisfy the relation of$\Delta k=\text{-}0.76\Delta {T}^{}{\text{cm}}^{-\text{1}},$ with the corresponding temperature derivatives of the refractive indices of LBO at o-1064 nm, e-1064 nm and o-532 nm selected as −1.9×10^{−6} °C^{−1}, −13×10^{−6} °C^{−1}and −0.9×10^{−6} °C^{−1}, respectively [10]. The other parameters can also be found in Table 1. As can be seen from Fig. 2, the differences between the two temperature tuning curves become less. However, the calculated temperature FWHM of Intra-SHG configuration (8 °C) is also larger than that of Extra-SHG configuration (6.2 °C).

The difference between the Intra-SHG and Extra-SHG temperature insensitivity can be understood as follows. The SHG powers in both configurations depend on the fundamental photon density, pulse width and the sinc^{2} function. However these parameters do not change in both configurations in the same way. In Extra-SHG configuration, the fundamental beam is hardly depleted under the small-signal approximation. So the fundamental photon density and pulse width will remain unchanged. Therefore, the generated second-harmonic power only depends on the sinc^{2} function for the Extra-SHG configuration. However, the Intra-SHG output power depends on three variables, the maximum fundamental photon density${\varphi}_{m},$ the value of sinc^{2} function and pulse width$\tau ,$ as expressed in Eq. (5). Moreover, due to the fact that the SHG conversion can be considered as a nonlinear loss to fundamental laser cavity, ${\varphi}_{m}$and$\tau $will change accordingly with the temperature of Intra-SHG crystal. Figure 3
and 4
display the dependence of calculated${\varphi}_{m}$and$\tau $on the temperature of Intra-SHG crystal for KTP and LBO, respectively. The corresponding Intra-SHG temperature tuning curve and sinc^{2} function are also inserted within Fig. 3 and Fig. 4 for the convenience of discussion.

When the temperature deviates from the exact phase-matched temperature for KTP, the value of sinc^{2} function will decrease. This will make the equivalent output coupling of the fundamental wave decrease. Then, the corresponding intracavity circulating fundamental power (or${\varphi}_{m}$) will grow rapidly, as displayed in Fig. 3. Rapid growing${\varphi}_{m}$will dominate the performance of Intra-SHG laser and make the green average output power increase. However, the increase of Intra-SHG average output power will in turn lead to the decrease of fundamental photon lifetime, resulting in slight decrease of pulse width$\tau $. When$\Delta T=\pm {10}^{\circ},$ the declining sinc^{2} function, narrowing pulses and growing${\varphi}_{m}$will reach a balance. Maximum Intra-SHG output power is achieved accordingly. When the temperature is further increased (or decreased), the contribution from the declining sinc^{2} function will dominate the Intra-SHG performance, resulting in the drop of SHG output power. When$\Delta T=\pm {20}^{\circ}$in Fig. 3, the second-harmonic average output power drops to minimum as the sinc^{2} function approaches the first zero. However, both${\varphi}_{m}$and$\tau $reach the maximum and start to vary with the sinc^{2} function henceforth. It is found that${\varphi}_{m}$and$\tau $will all maintain higher values than that in central band, which can account for the considerable output power achieved in the side lobes. Therefore, the varying fundamental photon density and pulse width induced by the SHG conversion will play a positive role in improving the Intra-SHG temperature insensitivity.

As for LBO displayed in Fig. 4, both${\varphi}_{m}$and$\tau $will grow first when the temperature deviates from the exact phase-matched temperature. This will also result in the improvement of Intra-SHG temperature insensitivity. Then they will continue to maintain a relatively high level and slightly vary with the temperature. However, the difference between the Intra-SHG and Extra-SHG temperature tuning curves for LBO is less compared with that of KTP. This phenomenon can be understood as follows. KTP has higher effective nonlinear coefficient and larger FWHM of sinc^{2} function than that of LBO. Its SHG nonlinear loss will induce sufficient modulation of${\varphi}_{m}$and$\tau $, resulting in large modulation bandwidth accordingly. This induced bandwidth of${\varphi}_{m}$and$\tau $band will consequently determine the large broadening of Intra-SHG KTP temperature tuning curve as well as heightened side lobes.

## 4. Experimental results and discussions

We conducted an experiment to validate the above theoretical model. The experimental arrangement was shown schematically in Fig. 5 . The pump source was a LD side-pumped Nd:YAG module. In the Intra-SHG configuration, the fundamental cavity was made up of an input mirror M1, the Nd:YAG module, an AO Q-switch and an end mirror M3. M1 was a concave mirror with 800 mm radius of curvature and high-reflection coated at 1064 and 532 nm. The AO Q-switch had antireflection coatings at 1064 nm on both surfaces. It was driven at a 24 MHz center frequency with the radio frequency power of 50 W. The 532 nm cavity consisted of an internal flat mirror M2, nonlinear crystal and a flat output-coupling mirror M3. M2 had the antireflection coatings at 1064 nm on one surface, with high-transmission coatings at 1064 nm and high-reflection coatings at 532 nm on the other surface. The output-coupler M3 was high-reflection coated at 1064 nm and high-transmission coated at 532 nm on one surface. Its other surface was antireflection coated at 532 nm. The dichroic mirror M2 was employed to make the generated second-harmonic immune from the loss induced by the absorption and thermally induced depolarization of laser crystal. As for the Extra-SHG configuration, M3 was replaced with a 1064 nm output coupler. The nonlinear crystal was removed from the cavity and placed at the position behind a lens that had the same focused beam size as that in Intra-SHG setup. The generated second-harmonic and residual fundamental wave would be separated by a filter that was high-reflection coated at 1064 nm and high-transmission coated at 532 nm. An oven with temperature instability of ± 0.1 °C was used to change the temperature of nonlinear crystal. A power meter (FieldMaxII-TO, Coherent Inc.) was used to measure the second-harmonic power.

It should also be noticed that temperature change of the nonlinear crystal can be the result of not only the ambient temperature variations (such as the used oven), but also its absorption of the fundamental or harmonic beam. According to the data presented in [6], the linear absorption loss of 1 cm long common KTP (an average between the data at 1064 nm and 532 nm) is 1.2%, greatly larger than that of 1 cm long GTR-KTP (0.1%). Firstly, we theoretically investigate the effect of linear absorption loss on the Intra-SHG temperature insensitivity. A linear absorption loss can be additionally introduced into the first differential equation in Eq. (2). Figure 6
shows the calculated results for common KTP and GTR-KTP. The presented information can be understood as follows. The introduction of linear absorption loss will not change the shape of sinc^{2} function. Moreover, due to the fact that 1.2% linear absorption loss is greatly smaller than the SHG nonlinear loss, the modulation of${\varphi}_{m}$and$\tau $induced by the linear absorption loss can be ignored. Therefore, the central band of temperature tuning curve for common KTP shows little difference with the normal situation (GTR-KTP). However, higher linear absorption loss can decrease the fundamental photon density. This will result in the drop of second-harmonic output power, as shown in Fig. 6.

Figure 7(a)
and 7(b) shows the normalized average output powers obtained with the two configurations for GTR-KTP and common KTP, respectively. The GTR-KTP and common KTP crystals were all with the dimensions of 4×4×10 mm^{3} and antireflection coated at 1064 and 532 nm on both surfaces. They were cut with the type-II phase-matching configuration. The initial testing temperature is set to be 15 °C due to the fact that it is the dew point temperature in our laboratory. Under the LD pump power of 80 W and AO repetition rate of 10 kHz, the Intra-SHG GTR-KTP laser output the maximum average power of 10 W (9 W for common KTP). For the Extra-SHG configuration, the maximum green average output power of 4.8 W (4.2 W for common KTP) was obtained under the same LD pump power and AO repetition rate as that in Intra-SHG configuration. As can be seen from Fig. 7(a), the two curves present the similar variation trend with that displayed in Fig. 1. The Intra-SHG temperature tuning curve is also characterized by the broad central band (15-58 °C) and high side lobes (such as 62-87 °C and 87-111 °C). Furthermore, the side lobes (A1-A2 and A2-A3) also present periodical variation, agreeing with the calculated trend. The normalized peak output power of the side lobes A1-A2 and A2-A3 are up to 0.52 and 0.32, respectively. As can be seen from Fig. 7(b), the Intra-SHG temperature tuning curve of common KTP is similar with that of GTR-KTP, agreeing with the theoretical results displayed in Fig. 6.

As can be seen from Figs. 7(a) and 7(b), the Extra-SHG temperature tuning curves fail to present the sinc^{2} shape. This phenomenon can be understood as follows. Actually, when the temperature was increased above 60 °C in Extra-SHG configuration, it was found that the measured power was nearly unchanged with the temperature of KTP. The peak value of first side lobes of sinc^{2} function is 0.045. This should support the output power of about 200 mW according to the maximum output power of 4.8 W obtained with sinc^{2} function approaching unity. It was also found that the power of residual fundamental wave leaking through the filter was on the order of 200 mW. When the temperature was increased above 60 °C, the generated second-harmonic power should increase with sinc^{2} function, but the power of residual fundamental wave would be decreased due to the SHG conversion. Therefore, the total measured power was nearly unchanged with the temperature. However, this will not affect the comparison between the temperature insensitivity obtained with the Intra-SHG and Extra-SHG configuration.

One issue should be pointed out here. For the Intra-SHG laser with 10 W average output power, the fundamental power leaking through the mirror M1 and M3 was measured to be 500 mW. The intracavity circulating fundamental power could be calculated to be about 500 W considering the 99.9% reflectivity of M1 and M3 at 1064 nm. The actual conversion efficiency from the intracavity circulating fundamental power to green power is very low (~2%), which is well within the small signal approximation limit. Furthermore, it is experimentally found that the SHG intensity still follows a square-law dependence on the fundamental intensity when the actual conversion efficiency is less than 5%.

Figure 8
shows the normalized average output powers obtained with the two configurations for LBO. The 4×4×10 mm^{3} LBO was antireflection coated at 1064 and 532 nm on both surfaces. It were cut along the direction ($\theta ={21.1}^{\circ},\varphi ={90}^{\circ}$), satisfying the type-II phase-matching configuration. Under the LD pump power of 100 W and AO repetition rate of 10 kHz, the Intra-SHG LBO laser output the maximum average power of 6.4 W. For the Extra-SHG configuration, the maximum green average output power of 3.5 W was obtained under the same LD pump power and AO repetition rate as that in Intra-SHG configuration. As can be seen from Fig. 8, the FWHM of Intra-SHG temperature tuning curve is 8 °C, slightly larger than that of 7 °C in Extra-SHG configuration. This result is also in good agreement with the theoretical analysis.

Based on the theoretical and experimental results mentioned above, it is found that the experiences obtained from Extra-SHG configuration are hard to characterize the Intra-SHG temperature insensitivity. For example, when the length of LBO is increased from 1 cm to 2 cm, the corresponding temperature bandwidth will be reduced by half for Extra-SHG configuration. However, the Intra-SHG temperature bandwidth will be decreased from 8 °C to 4.8 °C according to theoretical calculation. Therefore, the phase-matching properties of nonlinear crystal used as intracavity frequency converter should refer to the model of rate equations with modified nonlinear loss introduced.

## 5. Conclusions

The rate equations describing the Intra-SHG laser performance are established by introducing the nonlinear SHG loss with the phase mismatch considered. Based on the theoretical and experimental results, it is found that the temperature insensitivity obtained with Intra-SHG configuration has an advantage over that obtained with Extra-SHG both for the type-II KTP and LBO. The difference in temperature insensitivity between the two configurations obtained with KTP is larger than that of LBO. The varying fundamental photon density and pulse width induced by the SHG conversion will play a positive role in improving the Intra-SHG temperature insensitivity. Our results may provide a new view on the temperature insensitivity of Intra-SHG configuration. The method employed in our study can be extended to other nonlinear crystals and intracavity nonlinear frequency conversions.

## Acknowledgments

The authors would like to thank Prof. Deyuan Shen for fruitful discussions. This work was supported by A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).

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