1. Introduction

Second harmonic generation (SHG) has received much attention because it provides an efficient way for green and blue light generation [1–3], high-resolution imaging [4–6] and optical signal processing [7–9]. SHG occurs when a laser beam incidents upon a material with a nonzero second order susceptibility and has been demonstrated in various materials [10–12]. Among these materials, periodically poled 5 mol.% MgO doped LiNbO₃ (PPMgLN) is the most promising one for realizing highly efficient and compact blue and green lasers with moderate output power and good beam quality. It is of great interest because of the combined advantage of a large nonlinear coefficient and a good compatibility with waveguide fabrication process [13–15]. Two-dimensional spatial confinement provided by the waveguide significantly reduces the power required for highly efficient SHG [16]. However, it has been found that the SHG process using this material is inhibited for applications with relatively high power [14, 17].

The inhibition can be interpreted by the dephasing theory that absorptions induced inhomogeneous temperature rising and laser induced inhomogeneous refractive index changing result in considerable inhomogeneous phase mismatch in the irradiated zone. The phase mismatch will dramatically decrease the energy conversion from fundamental wave to SHG wave. For example, our past work reveals that inhibition of high power SHG in a PPMgLN waveguide is a result of significant phase mismatch induced by the photorefractive effect and the thermal effect along their radiated zone [18]. Great efforts have been made to minimize the inhibition of high power second harmonic generation in PPMgLN [19–26].

One way to minimize the inhibition is to broaden the bandwidth of second harmonic generation by chirped gratings. Basedon nonlinear coupled mode equations and undepleted pump approximation, Suhara et al. found that phase-matching bandwidth can be effectively broadened by appropriate design of chirped gratings [20].Since realizing a broadband device with a linearly chirped grating needs to ensure the smallness of change(~100 picometer) in the grating periods, Tehranchi et al. proposed a step-chirped grating which is engineered to highly broaden the bandwidth with larger step changes in the chirped period [21]. Although chirped gratings can increase the spectral or thermal bandwidth of the SHG, the conversion of fundamental wave into SHG wave becomes inefficient. Baranova et al. proposed an adiabatic SHG to resolve the bandwidth-efficiency tradeoff based on undepleted pump approximation [22]. Based on an analytical theory of SHG in a generic nonuniform second order nonlinear medium, Yaakobi et al. showed that by varying the properties of the medium gradually enough, the system can enter an autoresonant state in which the phases of the fundamental wave and of the SHG wave are locked [23].Subsequently, comprehensive physical models of adiabatic three-wave mixing were developed without using the undepleted pump approximation [24, 25]. The removal of undepleted pump restriction enables the exploration of adiabatic process in the fully nonlinear dynamic regime of nonlinear optics. Recently, Leshem et al. experimentally studied fully nonlinear adiabatic process by implementation in type I and type II SHG in two KTP nonlinear crystals [26].In an adiabatic SHG, the initial phase mismatch should be large enough so that nonlinear coupling is negligible and the rate of variation of the phase mismatch is sufficiently slow [25].To realize full conversion in adiabatic SHG, the signal intensities should be several orders of magnitude higher than the damage limits signal intensities under continuous-wave (CW) illumination, which makes the adiabatic CW SHG inefficient. Besides, these studies focused on a SHG with phase mismatch much smaller than the bandwidth. Although the bandwidth of the SHG can be very large, the efficiency is very low for CW operation due to the low damage limits signal intensity, which is ~2 MW/cm² at 532 nm [27]. However, little attention has been paid to the selection of an appropriate linearly chirped grating for minimizing the inhibition of CW SHG to improve the conversion efficiency.

The present paper presents a theoretical investigation on phase mismatch insensitive high power CW SHG by a coupled model in which phase mismatch from thermal dephasing and photorefractive effect are included. On basis of this model, it then describes the use of chirped gratings with linearly decreasing grating periods to reduce the sensitivity of PPMgLN waveguides to phase-matching conditions and maintain the SHG conversion efficiency. We found that the chirped gratings significantly reduced the sensitivity of the waveguide to phase mismatch in which the full width at half maximum (FWHM) bandwidth of the temperature tuning curve increased from 2.4 °C to 11.7 °C and the SHG output power increased from 0.64 W to 1.13 W.

2. Materials and methods

The current investigation involved analyzing the use of linearly chirped gratings to reduce the sensitivity of a PPMgLN waveguide to phase-matching and meanwhile increase the SHG output power. The PPMgLN waveguide has been widely used for highly efficient SHG [20, 28]. Frequency doubling of a 1064 nm CW laser with a PPMgLN waveguide was modeled and then calculated with a three dimensional coupled model equation. This method has been used for analyzing the inhibition phenomena occurred in PPMgLN waveguides [18], which can simulate the effect of photorefraction and heating upon the SHG. In this study, we chose a similar waveguide as the one we employed in the previous work. Thus, we can compare the results obtained in this study with that obtained in the previous one. As shown in Fig. 1(a), the waveguide is a periodically poled Y-cut 5% mol. MgO doped LiNbO₃ ridge waveguide [17, 18]. A polished PPMgLN crystal is bonded directly on the LN substrate. An airgap boundary, formed under the ridge structure, separates the PPMgLN waveguide from the LN substrate and performs strong confinement in the depth direction. The waveguide has a length l of 11.5 mm, a width of 5 μm and a height of 4 μm. It was chosen because notable inhibition of SHG has been observed for a fundamental power higher than 0.6 W [17]. The refractive index of the LN substrate can be found in [29].The normalized conversion efficiency used in this paper is 199%/W cm² which was determined through an experiment [17]. During the fabrication of the periodic structures, errors may be introduced in it [30]. Period errors that may affect a temperature acceptance bandwidth can be controlled very well with the rapid development of PPLN [14]. Therefore, we assumed that period errors were well controlled in this study. Fabrication errors that only change the conversion efficiency were taken into account in the normalized conversion efficiency in our calculation.

 

Fig. 1 (a) Schematic diagram of a chirped grating; (b) The dependence of SHG power P_2ω on the grating period Λ_E at the end facet of the waveguides at a fundamental power of 1.5 W.

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A waveguide with a linearly chirped grating was designed and then calculated for the SHG. As is known, the increase in SHG power along the z direction will result in an increase refractive index difference between SHG wave and fundamental wave [17, 18]. According to quasi-phase matching condition, the wave vector mismatch caused by the material dispersion should be compensated by the grating wave vector [30]. Hence, a larger grating wave vector is required to compensate the wave vector mismatch caused by material dispersion in the rear of the waveguide. The waveguide was designed to have linearly decreasing grating periods so that the grating has relatively larger wave vectors in the rear of the waveguide. This makes the grating could better compensate the wave vector mismatch of the whole waveguide. As shown in Fig. 1(a), the grating periods linearly vary from Λ_F at the front facet to Λ_E at the end facet. Since the wave vector mismatch is mainly caused by the second harmonic wave [18], the grating period Λ_F at the front facet is set to 6.96 μm aiming to satisfy the quasi-phase matching condition at a temperature of 34 °C. The grating period Λ_E at the end facet of the waveguide varied from 6.80 μm to 6.95 μm as shown in Fig. 1(b).The wavelength and temperature dependent Sellmeier equation for 5% MgO-doped CLN was taken from [31]. It is valid for0.5–4 μm spectral range and 20–200 °C temperature range.

3. Results and discussions

Data obtained in previous studies using models with no phase mismatch approximation indicates that sufficient intense signal are required to achieve high efficiency in broadband SHG using linear chirped poling periods [20–26]. According to Leshem et al., a pump intensity as high as ~486 MW/cm² was required to achieve a SHG conversion efficiency of 60% in a periodically poled KTP crystal with poling periods varied between 8.6 μm to 9.2 μm [26]. In our study, phase mismatch insensitive high power SHG was investigated by using a 3D thermal-optical model in which phase mismatch from thermal dephasing and photorefractive effect are taken into account [18]. This model has also been proven by comparing the simulation results with the experimental results in that paper. The calculated SHG powers were compared with that of the waveguide with uniform grating periods. Figure 1(b) shows the dependence of SHG power on the grating period Λ_E at a fundamental power of 1.5 W. Each of the SHG power shown in Fig. 1(b) was maximized by optimizing the waveguide temperature.

As can be seen in Fig. 1(b), the SHG power first increased and then decreased with the increasing Λ_E, and a peak SHG power which is around 1.13 W appeared at Λ_E = 6.95 μm. The corresponding SHG intensity was below the damage threshold (~2 MW/cm²) of 5 mol.% MgO doped LiNbO₃ at 532 nm [22]. As the Λ_E decreased to 6.94 μm, the SHG power declined to ~1.08 W. The SHG power decreased with the decreasing Λ_E. At Λ_E = 6.80μm, it declined to 0.22 W. To evaluate the effect of a chirped grating upon the SHG output power, we compared the SHG power versus fundamental power of the waveguide W_L with Λ_E = 6.95 μm with that of the waveguide W_U with uniform grating period as shown in Fig. 2(a). Since the damage limits signal intensity is ~2 MW/cm² [27], the maximum pump power was limited to 1.5 W. The curve of SHG power versus fundamental power for the waveguide W_U below a fundamental power of 1 W was in good agreement with experimental results [17, 18].

 

Fig. 2 (a)The dependence of SHG power P_2ω on the fundamental power P_ω of the waveguide with Λ_E = 6.95 μm; (b) The temperature tuning curve of the SHG at a fundamental power of 0.2 W (solid diamonds) and 1.5 W (solid circles).

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In Fig. 2(a), it can be seen that the SHG power generated by the waveguide W_L with the linearly chirped grating was significantly higher than that by waveguide W_U using uniform grating for a fundamental power above 0.7 W. As the SHG process was inhibited by increasing phase mismatch in the W_U for fundamental above 0.3 W, the conversion of the energy from the fundamental wave to SHG wave became less efficient [18]. The SHG process of the W_L was almost not affected by the phase mismatch and the SHG power increased steadily with increasing fundamental power as compared with that of W_U. Therefore, the difference between the SHG power of W_L and W_U became greater with the increasing fundamental power. At a fundamental power of 1.5 W, the SHG power of W_L increased to 1.13 W which was ~77% higher than that of W_U.

The prior work done by Jedrzejczyk et al. indicates that the SHG using uniform grating is quite sensitive to temperature variation at high fundamental power [17]. For example, it showed that the FWHM width of the temperature tuning curve of SHG at P_ω = 0.747 W was ~45.8% narrower than that of SHG at P_ω = 0.162 W [17]. The narrowed FWHM width was a result of the photorefractive effect and the thermal effect induced imhomogeneous refractive index distribution [18]. In our study, the use of chirped grating could broaden the FWHM width of the temperature tuning curve by improving the phase mismatch tolerance at high pump powers. Figure 2(b) shows the temperature tuning curves obtained using the waveguide W_L.

As can be seen, the SHG at a fundamental power of 1.5 W had a much larger FWHM width which was ~4.5 times of that at a fundamental power of 0.2 W. Although the temperature tuning curve of the waveguide W_L deviated from the sinc² distribution of that in a waveguide with uniform grating, it had a nearly symmetric distribution. The FWHM width of temperature tuning curve of the waveguide W_L at P_ω = 0.2 W is ~2.6 °C which is higher than that (~2.4 °C) of the waveguide W_U at P_ω = 0.167 W [17]. However, at high pump powers, a striking difference between the temperature tuning curves of the waveguide W_L and W_U was noted when the width and the shape of the tuning curves were considered. The waveguide W_U has a narrowed FWHM width at a high pump power, whereas the waveguide W_L has a significantly broadened FWHM width at a high pump power. As shown in Fig. 2(b), the waveguide W_L has a temperature tuning curve with a FWHM width up to 11.7 °C which was ~4.5 times of that at a fundamental power of 0.2 W. As compared with that (~2.4 °C) of the waveguide W_U, the FWHM width of temperature tuning curve of the waveguide W_L is ~4.9 times. The result indicates that the chirped grating significantly improves the tolerance of the waveguide to phase mismatch. To find out how the chirped grating improved the SHG output power, we plotted the wave vector mismatch caused by material dispersion Δk_n of the waveguide W_L, the total wave vector mismatch Δk_T of the waveguide W_L and W_U and evolving of the SHG process along z axis of the waveguide W_L in Fig. 3.

 

Fig. 3 The wave vector mismatch caused by material dispersion Δk_nof the waveguide W_L in the transverse direction at z = l/2 (a) and z = l (b);(c) The total wave vector mismatch Δk_T of the waveguide W_U with a uniform grating period and the waveguide W_L with a linear tapered grating period;(d) The evolving of SHG power P_2ω along the z direction of the waveguide W_L at a fundamental power of 1.5 W.

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As can be seen in Fig. 3(a), the wave vector mismatch caused by material dispersion Δk_n of the waveguide W_Lis inhomogeneous in the xy plane. At the plane of z = l/2 (5.75 mm), the peak wave vector mismatch which was ~9034 cm⁻¹ appeared at x = 0.0 μm and y = 2.5 μm. In the xy plane, Δk_n decreases as it was far away from the peak point. At the edge, it reached the minimum value of ~9028 cm⁻¹. In a quasi-phase matched SHG, the efficiency is determined by total wave vector mismatch Δk_T which is equal to Δk_n minus wave vector of a grating k_G. According to the quasi-phase matching condition (Δk_T = 0), it is most desirable to achieve quasi-phase-matching by making the k_G close to the Δk_n. k_G of the waveguide W_L at z = l/2 was ~9030 cm⁻¹ which is close to the intermediate value of the Δk_n at that position. The peak wave vector mismatch of the waveguide W_L increased to 9036 cm⁻¹ at z = l as shown in Fig. 3(b). k_G of the waveguide W_L at z = l was ~9036 cm⁻¹ which was equal to peak wave vector mismatch at that position. As shown in Fig. 3(c), the offset of Δk_T of W_U and W_L has been optimized to obtain maximum outputs by tuning the temperature. However, total wave vector mismatch imhomogeneities cannot be reduced in the waveguide W_U. As can be seen, Δk_T of the waveguide W_U increased ~6.0 cm⁻¹ from z = 0.0 mm to z = 4.4 mm. The increased Δk_T inhibited the SHG process to a large extent. The Δk_T reach the peak at z = 4.4 mm and decreasing slightly from this point to end facet. Although, the variation in Δk_n was small (0.05%), the variation in the Δk_T was significant. As with the tapered grating, k_G also increased along the beam propagation direction. As can be seen, Δk_T of the waveguide W_L increased only ~1.3 cm⁻¹ from z = 0.0 mm to at z = 3.1 mm. Therefore, the total wave vector mismatch imhomogeneities is reduced to a large extent. The Δk_T reach the peak at z = 3.1 mm and decreasing slightly from this point to end facet. Although the SHG power of the waveguide W_L was ~76% higher than that of the waveguide W_U, the Δk_T of it was only slightly (< 27%) higher than that of the waveguide W_U in the second half of the waveguide. Therefore, the fact that k_G of the chirped grating grows in phase with Δk_n makes the SHG process in the W_L more efficient. There have ~1600 domains in the waveguide. The SHG output is a coherent superposition of the SHG wave generated at each domain. With reduced total wave vector mismatch inhomogeneities, the quasi-phase matching temperatures of the domains are close to each other at a high SHG power. Then, a wider temperature acceptance bandwidth can be obtained.

Data obtained in previous studies indicated that the strong inhibition of SHG began at the middle of the waveguide with uniform grating. According to our previous work, the SHG power reached the peak at the middle part of waveguide and declined in the rear of the waveguide [18]. In this study, the inhibition was suppressed by using the chirped grating. The evolving of P_2ω in the waveguide W_L is shown in Fig. 3(d).

As can be seen, the SHG process was almost not inhibited. A wider temperature acceptance bandwidth indicates that the waveguide W_L is insensitive to phase mismatch as compared with the W_U. It makes the W_L has a better performance than the W_U despite that the W_L has the higher phase mismatch. The SHG power grew steadily in the front part of the waveguide, which was consistent with that in the waveguide with uniform grating. However, in the rear of the waveguide, the evolving of the SHG power was not consistent with that with uniform grating. It can be observed that the SHG power kept growing in the rear of waveguide. Although the SHG power grew with a lower speed in middle part of the waveguide than that in the front part, the growing speed was recovered at the rear of the waveguide. Since the total wave vector mismatch did not increase exactly linearly along the waveguide, the chirped grating could not compensate the phase mismatching completely. There will be some residual total wave vector mismatch along the waveguide. Nevertheless, these results suggest that a chirped grating may provide a SHG waveguide that is insensitive to phase mismatch as compared with a waveguide with uniform grating, and thereby increasing the SHG output power.

4. Conclusions

Prior work has reported that the effectiveness of chirped grating in broadening the bandwidth of SHG. Leshem et al. [26], for example, demonstrated that thermal acceptance bandwidth can be increased to 100 °C with chirped poling periods. In the experiment, a pump intensity as high as 290 MW/cm² was used to achieve a conversion efficiency of 60%.However, the damage limits the pump intensity to several orders of magnitude lower than the intensity required for efficient conversion for CW lasers [27, 29].

In this study, we investigated phase mismatch insensitive high power SHG by using a 3D thermal-optical model in which phase mismatch from thermal dephasing and photorefractive effect are taken into account. We found that a chirped grating with linearly decreasing periods can reduce the sensitivity of SHG to phase-matching and increase SHG output power. By a tapered grating with decreasing periods from 6.96 μm to 6.95 μm, the FWHM width of the temperature tuning curve was broadened to 11.7 °C (the waveguide W_L) from 2.4 °C (the waveguide W_U) and the SHG output power was increased to ~1.13 W from ~0.64 W at a fundamental power of 1.5 W. These findings extend those of Leshem et al. [26], confirming that reduction of the inhibition of SHG can be achieved by engineering the phase-matching condition at a moderate signal intensity. Our result provides compelling evidence for phase-matching insensitive SHG and suggests that this approach appears to be effective in increasing SHG power that is required in many applications.