1. Introduction

Compact and efficient high power laser sources in the blue-green spectral range have numerous applications such as laser spectroscopy, laser projection, medical instruments and communications [1–4]. Second-harmonic generation (SHG) based on periodically poled MgO doped congruent LiNbO₃ (PPMgLN) waveguide devices is attractive to construct compact and efficient blue-green lasers, because of the advantages of phase matching at arbitrary wavelength, a large nonlinear coefficient and strong spatial confinements of the interacting waves [5, 6].

Recently, 466 mW of continuous-wave (CW) 532 nm green-light generation has been demonstrated using a PPMgLN waveguide [7]. However, the inhibition of the SHG has been observed when the green light power exceeded 466 mW. Similar phenomenon has also been observed in the process of frequency doubling of diode laser in PPMgLN waveguide [8].

It has been suggested that the inhibition phenomenon was caused by significant temperature nonuniformities along the irradiated zone that is ultimately ascribed to the significant optical absorption by the device [8, 9]. With this assumption, in order to theoretically reproduce the experimental SHG power as a function of fundamental power, absorption coefficients about one order of magnitude higher than those known from the literatures have to be considered [8]. The large increases in the absorption coefficients suggest that the temperature nonuniformities may not always be the main reason for the inhibition of high power SHG.

It is well known that the LiNbO₃ (LN) device injected with a pump light of high power exhibits measurable photorefractive (PR) effect [10]. Since the PR effect is a source of beam distortion (PR damage), tremendous efforts have been made to reduce it in LN devices [11–14]. As is known, LN is usually periodically poled for the purpose of achieving quasi-phase matching (QPM). Meanwhile, it brings another benefit of reducing the PR effect [11, 12]. However, the PR effect in PPLN is still evident [13]. Another approach to reducing the PR damage is to dope MgO in LN at the concentration over threshold [14]. It was reported that the damage threshold of MgO doped LN is 8 × 10⁶ W/cm² while that of congruent LN is only 10³ W/cm². Although having considerably weaker PR effect than congruent LiNbO₃, PPMgLN still shows measurable photorefractively induced QPM wavelength shift even under milliwatt level light illumination [15]. Therefore, besides the optical absorptions, the photorefractive effect in the PPMgLN waveguide devices has to be taken into account in the investigation of the inhibition of high power SHG.

Since it is difficult to measure the actual temperature and refractive index distribution along the beam propagation direction inside the waveguide core [7], influences of the thermal effect and the photorefractive effect on the inhibition of the SHG can hardly be quantitatively verified via experiments. Thus, it is practical and achievable to investigate the problem theoretically [8, 9].

In this paper, we present a systematical numerical study of the SHG process in PPMgLN waveguides operating at near the room temperature based on the complete coupled mode equations in combination with both the photorefraction and the temperature nonuniformities. We clearly describe the influences of the photorefractive effect as well as the thermal effect on the inhibition of the high power SHG in the PPMgLN waveguide devices. Our study shows that in the PPMgLN waveguides, both of the two effects have some influences on the inhibition of the SHG conversion efficiency, but the role of each effect should be readdressed. The photorefractively induced phase mismatch is significant, which severely inhibits the SHG conversion efficiency. In contrast, the thermally induced phase mismatch is moderate, which plays a minor role on the inhibition of the high power SHG. In other words, it is the photorefractive effect instead of the thermal effect playing a determinative role on the inhibition of the high power SHG.

2. Theoretical model

We consider continuous-wave (CW) SHG in a 5 mol.% MgO doped congruent LiNbO₃ waveguide shown in Fig. 1. The 11.5 mm long ridge waveguide is located on the surface of a 5 mol.% MgO doped congruent LiNbO₃ substrate. The waveguide has been described in detail in [16]. A 1.064 μm CW laser was used as the fundamental wavelength laser. The poling period of the PPMgLN waveguide was Λ = 6.96 μm, which was chosen for 532 nm green-light generation at temperature T = 34 °C. The coordinate system in the theoretical model is set as follows: the x axis is parallel to the crystal axis Z which is in the lateral direction, the y axis is parallel to the crystal axis Y which is in the vertical direction, and the z axis is parallel to the crystal axis X which is in the light propagation direction. To explicitly investigate the inhibition of high power SHG process in PPMgLN waveguides, both the light absorptions and the photorefraction are taken into account in the coupled mode equations in this paper.

 

Fig. 1 Schematic diagram of second harmonic generation in a periodically poled MgO doped congruent LiNbO₃ waveguide.

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The electric field of the fundamental wave E_ω and the second harmonic wave E_2ω propagating along the positive z axis in the waveguide can be expressed as [4]:

The linear absorption is taken into account as α_i I_i, where α_i is the linear absorption coefficient and $I_i$is the light intensity. The nonlinear absorption is taken into account as$\beta I_{2\omega }^2$, where$\beta$ is the nonlinear absorption coefficient. The green-induced infrared absorption is taken into account as α_G I_ω, where α_G is the corresponding coefficient [17]. Thus, the coupled mode equations for the SHG process can be expressed as:

It is well known that the refractive index $n_i$ depends on the temperature T [20]. Thus, the optical field in the studied problem is coupled with the temperature distribution in the crystal. The equilibrium temperature distribution in the waveguide can be described by a 3D Poisson equation

3. Simulation results

The current investigation involves analyzing the influence of the PR effect and thermal effect on the inhibition of high power SHG in a periodically poled 5 mol.% MgO doped congruent LN waveguides. The finite difference method is used to numerically calculate the high power SHG in the waveguides and has been frequently used to calculate the coupled mode equations [9]. The method has been verified by studying the problem that neglecting the absorption losses and photorefraction as well as by comparing our simulation results with the experimental data extracted from [8]. The verification of our method can be found in the part of the detailed illustration of Fig. 2.

 

Fig. 2 SHG power (P_2ω) versus fundamental power (P_ω). The solid diamonds show the simulated ideal SHG power when the photorefraction and absorptions are neglected. The solid circles show the simulated SHG power when the absorption losses and the photorefraction are considered. The open circles show the measured SHG power versus fundamental power [8]. The solid line shows the corresponding tanh² fit.

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Two waveguides with different transverse dimensions were investigated to find out the performance of waveguides with different levels of damage sensitivities. The transverse mode profile e_i (x, y) of the waveguide can be expressed as a symmetrical Gaussian function in the lateral direction and a Hermite-Gaussian function in the vertical direction [22]. The 1/e² lateral and vertical diameters of the fundamental wave (the second harmonic wave) of waveguide I were selected as 4.8 μm (4.5 μm) and 4 μm (4 μm), respectively. Those of waveguide II were selected as 6.3 μm (5.5 μm) and 4.6 μm (4.6 μm), respectively.

At first, the high power SHG in waveguide I was simulated to reproduce the inhibition phenomenon and then analyzed to identify the contributions of each effect to the inhibition. The calculations were performed within an area of $50\times 40\times 11500$ μm³, discretized into 100 × 100 × 1000 point grid. The discretization parameters were adjusted to be small enough to get stable results. The normalized conversion efficiency was estimated to be η_eff = 400%/W⋅cm² [22]. The heat conductivity of MgO doped LN is $\kappa$ = 4 W/m⋅K [23]. The linear absorption coefficients of MgO doped LN are α_ω = 0.2 /m (λ = 1.064 μm) and α_2ω = 1 /m (λ = 532 nm) [24, 25]. The green induced infrared absorption coefficient of PPMgLN is α_G = β_GI_2ω + α₀ when the green light intensity is higher than 2 kW/cm², in which β_G = 0.0147 W/cm³ and α₀ = 0.0754 /cm [17].

The damage sensitivity (n*) and the nonlinear absorption coefficient (β) of PPMgLN at λ = 532 nm are not directly available. But those coefficients for 5 mol.% MgO doped congruent LN at λ = 514 nm can be found in [19], as well as those coefficients for congruent LN at λ = 514 nm and 532 nm [13, 26]. We assume that spectral dependence of the damage sensitivity (as well as the nonlinear absorption coefficient) of the 5 mol.% MgO doped congruent LN is similar to that of congruent LN. Note that the feature of spectral dependence has been used to estimate the nonlinear absorption coefficient [27]. From [13] and [26], we know that the damage sensitivity at λ = 532 nm should be 5.8 times smaller than that at λ = 514 nm. The damage sensitivity of 5 mol.% MgO doped congruent LN is reported to be n* = −1.26 cm²/GW at λ = 514 nm [19], thus the corresponding damage sensitivity at λ = 532 nm should be n* ≈-0.22 cm²/GW. We notice that the damage sensitivity of a 5 mol.% MgO congruent LN (n* = −1.26 cm²/GW) [19] was found dropping remarkably compared with that of congruent LN (n* = −1.5 × 10⁻¹ cm²/MW) [26]. That is attributed to the decrease of the saturated space charge field E_sc [28]. Similarly, based on the experiment data measured for congruent LN, it is known that the nonlinear absorption coefficient at λ = 532 nm is 1.8 times smaller than that at λ = 514 nm [13, 26]. The nonlinear absorption coefficient of 5 mol.% MgO congruent LN is measured to be β = 2.9 × 10³ cm/GW at 514 nm [19]; hence, the corresponding nonlinear absorption λ = 532 nm is estimated to be β ≈1.6 × 10³ cm/GW.

Both the damage sensitivity and the nonlinear absorption coefficient should be corrected due to periodically poling reduction [11, 27]. For PPMgLN, the magnitude of the damage sensitivity is suppressed compared with those in the homogeneous crystal (i.e., 5 mol.% MgO congruent LN in this study) by the square of the product of the poling-grating wave vectors (K_g) and the characteristic transverse dimension of the irradiance (w) [11]. For the specific SHG condition investigated in our paper, the magnitude of the suppression factor for damage sensitivity of PPMgLN at λ = 532 nm is calculated to be 16.5. And according to the report of Chen et. al [13], the sign of damage sensitivity is changed by periodically poling. At last, we obtain the damage sensitivity of our investigated periodical poled 5 mol.% MgO doped congruent LN waveguide (PPMgLN) at λ = 532 nm is n* ≈13.3 × 10⁻³ cm²/GW. Because the PR effect has a significant influence on the nonlinear absorption, periodically poling will also cause the reduction of nonlinear absorption. The suppression of nonlinear absorption caused by periodically poling is about 6 times larger than that of damage sensitivity [13]. Here, we notice that [13] bears a mistake for calculating the suppression factor of damage sensitivity from their experimental data; the accurate factor should be 50 instead of 20. Therefore, the suppression factor for the nonlinear absorption coefficient due to periodically poling should be 109. As we already know the nonlinear absorption of corresponding homogeneous crystal at λ = 532 nm which is β ≈1.6 × 10³ cm/GW, the nonlinear absorption coefficient of the investigated PPMgLN waveguide at λ = 532 nm is estimated to be β = 14.6 cm/GW. Here, the nonlinear absorption coefficient β is larger than the previously reported value (5.2 cm/GW) that measured by a femtosecond laser [29]. This is because the femtosecond laser pulses are too short for the marked PR process to be observed. Under femtosecond laser radiation, the photorefraction is largely suppressed and thus the contribution of third-order nonlinearity to the nonlinear absorption becomes dominating [27].

Simulations performed in previous studies considered the temperature nonuniformities as the main factor of the phase mismatch. According to the reports by Jedrzejczyk et al., to reproduce the experimental results, the linear absorption coefficients of the fundamental and SHG wave used in the simulation had to be increased from 0.2 /m to 5.2 /m and from 1 /m to 6/m if the PR effect was neglected [8]. It suggests that the temperature nonuniformities should not always be the main factor of the phase mismatch.

In our theoretical study, both the temperature nonuniformities and the photorefraction were considered in the simulation. Prior to modeling waveguide I based on the complete coupled-mode equations, we verified our method by simulating the corresponding case when the absorption losses and the photorefraction were neglected. For such a case, it is well known that the solution is

Then, the SHG was simulated using the complete coupled-mode equations in which both the absorption loss and the photorefraction were taken into account. In the simulation, the crystal temperature was optimized at each fundamental power to maximize the SHG power. The optimal operating temperature dropped slightly as the fundamental power increased. In Fig. 2, the solid circles show the simulation results of P_2ω versus P_ω according to the complete coupled mode equations. We note that the critical power of self-focusing is calculated to be 14 W based on the theory reported in [30], thereby the light power (0 ~1 W) considered in the simulation is not high enough for producing the self-focusing effect.

It is observed that, initially, the SHG simulated by the complete coupled mode equation is in good agreement with the tanh² fit. Then, the SHG power starts to deviate from the tanh² distribution when P_ω reaches 0.3 W. After that, the difference of SHG power between that calculated from the complete coupled mode equation and that predicted by tanh² function becomes larger and larger after the fundamental power exceeds 0.3 W. This is the inhibition of SHG and similar phenomena have also been observed in experiments [8]. For the purpose of comparison, the problem simulated was set the same to that illustrated in [8] and the corresponding experimental results extracted from [8] are also plotted in Fig. 2, indicated by the open circles. By comparison, one sees an excellent agreement between the experimental results and our simulation results, which can be another evidence to support that the theoretical model we have developed could be reasonable. It is to emphasize that we have theoretically reproduced the inhibition of high power SHG by taking the photorefraction into account. In comparison with the theoretical study in [8], our work is more reliable because we have not increased the absorption coefficients of the fundamental nor SHG wave.

It is known that the inhibition of SHG is mainly due to the introduction of phase mismatch Δk. The phase mismatch can arise from both the temperature nonuniformities and the photorefraction. For high power SHG, the absorptions can lead to temperature nonuniformities which are one source of the phase mismatch. Meanwhile, the increase of green light power can also lead to the rising of photorefraction which could be the other factor of the phase mismatch. However, comparison of the contributions from these two factors on the phase mismatch has not been reported yet.

Next, we calculated the refractive index changes induced by the two factors, respectively, to identify the contribution from each of them. Figure 3(a) shows the photorefractively induced refractive index change (Δn_p) (solid line) and the thermally induced refractive index change (Δn_T) (dashed line) along the beam propagation direction at the pump power of 1 W which is sufficiently high to lead to the inhibition of SHG. As can be seen in Fig. 3(a), both the photorefraction and temperature nonuniformities can cause notable refractive index change in the high power SHG, but obviously the photorefraction induced refractive index changes are an order of magnitude larger than those induced by temperature nonuniformities. Such calculated results reflect evidently that the photorefractive effect is the dominating factor influencing the phase mismatch, which is completely different from the conclusion reported in [8] and [9]. Therefore, the inhibition is mainly induced by the PR effect for PPMgLN waveguides operating at near the room temperature. As we mentioned before, the PR effect can be reduced significantly for PPMgLN waveguides operating at sufficiently high temperature [15]. In that case, the inhibition should be dominated by the temperature nonuniformities.

 

Fig. 3 (a) Photorefractively induced refractive index change Δn_P (solid line) and thermally induced refractive index change Δn_T (dashed line) along the beam propagation direction at a pump power of 1W. (b) The corresponding distributions of the fundamental power P_ω (z) (solid line) and the SHG power P_2ω (z) (dashed line) along the beam propagation direction at a pump power of 1W.

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Figure 3(b) displays the corresponding distributions of the fundamental power P_ω (z) (solid line) and SHG power P_2ω (z) (dashed line) along the beam propagation direction at a pump power of 1W. One observes that the curve of photorefractively induced refractive index change suffers similar variation trend to that of the SHG Power, i.e., increases firstly and decreases later. That is not hard to understand. As we mentioned in Section 2, because the PR effect from the fundamental wave is too small to take into account, the photorefractively induced refractive index change mainly comes from the SHG wave.

In addition, Fig. 3(b) tells us that the fundamental power cannot be fully converted into SHG power and the inhibition of SHG happens during the wave propagation. The physical insight of this phenomenon can be explained as follows. At the beginning of light propagation, the refractive index change gradually increases when the SHG power grows; see Fig. 3(a) and 3(b). After light propagating a certain length, the refractive index change becomes significant, and then generates apparent phase mismatch as illustrated by Eq. (5), which in turn inhibits the growing of SHG power. From Fig. 3, we see that the inhibition of SHG starts to appear after light passing the plane of z ~4 mm. Due to larger phase mismatch generated by the photorefractively induced refractive index change, the SHG power increases more and more slowly and then it reaches saturation at the central part of the waveguide (z ~7 mm). After light passing the plane of z ~7 mm, the phase slip between these two waves caused by their different phase velocities leads to an alternation of the direction of power flow [31]. Thus, the SHG power begins to decrease and the fundamental power begins to increase after this plane.

Finally, we carried out the comparison of SHG in waveguides of different transverse dimensions. Due to the increased transverse dimension of waveguide II, the normalized conversion efficiency was reduced to η_eff = 225%/W⋅cm² [22]. Similar to the method used for estimating the coefficients of waveguide I, the damage sensitivity and the nonlinear absorption coefficient of the PPMgLN of waveguide II were estimated to be n* ~9 × 10⁻³ cm²/GW and β ~10 cm/GW (λ = 532 nm), respectively.

Figure 4(a) displays the relation between the output SHG conversion efficiency (η = P_2ω /P_ω) and the input fundamental power (P_ω) for waveguide I and waveguide II by the solid circles and open circles, respectively. As can be seen in Fig. 4(a), the SHG conversion efficiency of waveguide I reaches its maximum at P_ω ~0.45 W, but for waveguide II, its peak appears when P_ω is ~0.8 W. In other words, given the same fundamental power (e.g., P_ω = 0.5 W), SHG in waveguide I is inhibited more seriously by the refractive index change than that in waveguide II. The physic origin of this phenomenon is obvious as larger transverse dimension means lower light intensity and thus leads to smaller phase mismatch due to refractive index change. Furthermore, the damage sensitivity n* of PPMgLN caused by an irradiance of lager transverse dimension is smaller as a result of periodically poling reduction. Hence, saturation of SHG for waveguide II is postponed to take place under a pump light of higher power compared with that for waveguide I.

 

Fig. 4 (a) SHG conversion efficiency η versus the input fundamental power P_ω in waveguide I (solid circles) and waveguide II (open circles). (b) SHG conversion efficiency η versus the input fundamental light intensity I_ω in waveguide I (solid circles) and waveguide II (open circles).

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To check if the two efficiency maxima correspond to the same beam intensity, we plot the conversion efficiency versus the input fundamental light intensity I_ω in waveguide I (solid circles) and waveguide II (open circles) in Fig. 4(b). One sees that the maximum conversion efficiencies of the two waveguides are indeed achieved at the same fundamental light intensity I_ω ~3 MW/cm². This is an evidence to demonstrate that the transverse dimension of the irradiance is one of major factors that impacting the inhibition of SHG. Note that although the maximum conversion efficiencies of the two waveguides are achieved at nearly the same fundamental light intensity, Waveguide II suffering relatively weaker PR effect performs much better than Waveguide I in high power SHG applications as larger transverse dimension of irradiance means higher light power. Supressing of the PR effect remains a crucial issue for SHG in PPMgLN and specially tailored QPM gratings [31] or segmented crystal heating [32] could also be applied in waveguide SHG devices in order to enable output of even higher visible power.

4. Conclusions

Prior works have suggested that the inhibition of high power SHG in PPMgLN waveguide is mainly caused by temperature nonuniformities. However, these studies have to increase the value of the reported absorption coefficients to reproduce the experimental results. In this study, we have considered both the thermal effect and the photorefractive effect in investigating the inhibition of high power SHG in PPMgLN waveguides operating at near the room temperature. We have found that not only the temperature nonuniformities but also the photorefractive effect are responsible for inducing the refractive index nonuniformities that destroy the phase matching condition. In addition, the photorefractive effect is found to be the major factor influencing the refractive index change. In this study, all the coefficients are consistent with the experimentally measured data. These findings clarify the mechanism of inhibition of high power SHG in PPMgLN operating at near the room temperature. Most notably, this is the first study to investigate the influence of photorefractive effect on the inhibition of high power SHG. By comparing SHG in waveguides with different transverse dimensions, we have found that the waveguide with weaker PR effect is advantageous in high power SHG due to relatively smaller photorefractively induced phase mismatch. The findings of this work provide some guidance on designing higher power SHG waveguides devices in future.