382 mW External-Cavity Frequency Doubling 461 nm Laser Based on Quasi-Phase Matching

Abstract

To rapidly improve strontium optical clocks, a high-power, high-efficiency, and high-beam-quality 461 nm laser is required. In blue lasers based on periodically poled KTiOPO₄ crystals, the optical absorption in the crystals can induce thermal effects, which must be considered in the design of high-efficiency external-cavity frequency doubling lasers. The interdependence between the absorption and the thermally induced quasi-phase mismatch was taken into account for the solution to the coupled wave equations. By incorporating multilayer crystal approximation, a theoretical model was developed to accurately determine the absorption of the frequency doubling laser. Based on experimental parameters, the temperature gradient in the crystal, the influence of the boundary temperature on the conversion efficiency, and the focal length of the thermal lens were simulated. Theoretical calculations were employed to optimize the parameters of the external-cavity frequency doubling experiment. In the experiment, in a bow-tie external cavity was demonstrated by pumping a 10 mm long periodically poled KTiOPO₄ crystal with a 922 nm laser, a 461 nm laser with a maximum output power of 382 mW. The conversion efficiency of the incident fundamental laser was 66.2%. The M² factor of the frequency doubling beam was approximately 1.4.

1. Introduction

With the rapid development of new nonlinear materials, blue lasers have a large number of applications in physics, biology, marine science, and other fields [1,2,3,4,5,6]. At present, the uncertainty of the optical clock has entered the order of 10⁻¹⁸. Rapidly developed optical clocks have had a significant impact on time–frequency science and application, and is expected to redefine “second” based on optical lattice clocks in the future [7].

Various optical clocks require blue light for trapping and detection, such as ytterbium atoms corresponding to 399 nm, calcium atoms corresponding to 422 nm, and strontium atoms corresponding to 461 nm. The strontium atomic optical clock is one of the most promising experimental platforms among many kinds of optical clocks [8,9,10,11]. Several institutions have begun the development of miniaturized optical clocks [12,13,14,15,16]. Consequently, there is a growing demand for high-power, high-efficiency, and high-beam-quality 461 nm single-frequency lasers.

Until now, the main methods to achieve 461 nm laser output have been external cavity diode lasers (ECDLs) and nonlinear frequency conversion technology.

The ECDL laser offers several benefits, including a straightforward design, high low-maintenance requirements, and excellent stability. At present, blue diodes have been used in commercial lasers, and a variety of tunable 461 nm ECDL lasers have appeared. The 461 nm external cavity diode laser (TLB 6802) developed by the New Focus company in the USA has an output power of 40 mW. However, due to its low power, a single laser setup is inadequate for fulfilling the demand of trapping and detecting strontium atoms. Therefore, a strontium atomic blue magneto-optical trap optical system requires an external cavity diode 461 nm seed laser to injection lock multiple slave lasers [17,18,19]. Although this scheme is widely used, the structure is complicated and the maintenance process is tedious. For strontium optical clocks, there is an urgent need for a watt-scale, narrow-linewidth, tunable 461 nm laser. Toptica and Moglab have commercialized tunable 461 nm ECDLs with output power levels of 200–250 mW. Such lasers can meet the experimental requirements of the compact ⁸⁸Sr optical clock physical system in [16] or a special designed physical system [20]. However, the reported portable ⁸⁷Sr optical clock physical systems have a power requirement of around 500 mW for the 461 nm lasers [13,14]. Due to the advancements in GaN blue diode technology, blue ECDLs with power outputs in the range of watts have been realized in laboratories [21,22,23]. It is believed that high-power 461 nm ECDLs will also appear in optical clock systems in the near future.

At present, only two kinds of nonlinear frequency conversion methods have been reported to generate blue lasers, i.e., sum frequency generation and frequency doubling (or called second harmonic generation, SHG). In 2003, Courtillot et al. used 813 nm and 1064 nm lasers to generate a 115 mW 461 nm laser in a KTiOPO₄ (KTP) sum frequency crystal [24]. Compared with sum frequency generation, the structure of the frequency doubling device is simpler. Based on the current frequency doubling technology, it has achieved efficient frequency doubling output with various blue wavelengths [25,26,27,28,29,30,31,32,33].

In the current state of developed laser diode technology, blue light diodes lag behind infrared diode products in terms of power and lifetime. The drawback of low power can be compensated for by employing a master oscillator power amplifier (MOPA) laser system, while the issue of limited lifetime can be counteracted through redundant power design and modular replacement design. However, these approaches increase the complexity of the optical clock. Compared with ECDL, although the structure of the frequency doubling laser is more complex, the technology of external-cavity frequency doubler lasers is more mature. Additionally, frequency doubler crystals have become widely used optical materials. The primary issues impeding the application of external-cavity frequency doubling lasers in miniaturized optical clocks are cavity stability, power degradation over time, and the lifespan of the crystal. The linear F-P cavity or semi-monolithic cavity technology can be used to shorten the cavity length [34,35] and improve the long-term alignment of the laser cavity. The relationship between the laser waist spot and the cavity length can also be kept in a stable interval by optimizing the cavity parameters. Enhancing the mechanical and thermal properties of the cavity can be achieved through engineering methods, including glue fixation, non-adjustable mirror frames, active temperature control of the cavity, and filling the cavity with inert gas. Avoiding using crystals with lower damage thresholds or easy deliquescence as SHG crystals can effectively improve the lifetime of the SHG cavity. Finally, the power consumption of the system was reduced by optimizing the parameters of the SHG laser to achieve high-efficiency and high-beam-quality blue light output. Five 461 nm lasers with different frequencies are used in a strontium atomic optical clock, so an optical system is often needed to realize laser beam splitting and frequency shifting. Optical systems driven by a single high-power 461 nm laser offer notable advantages in terms of long-term stability and maintenance challenges compared to laser sources using MOPA structures.

Previously, potassium niobate (KNbO₃) crystals were considered to be the best choice for efficient frequency doubling 461 nm lasers. Although this crystal has a high nonlinear coefficient (d_eff = 18 pm/V), it has two obvious disadvantages. Firstly, its matching temperature is about 150 °C, hindering the long-term temperature control stability of the crystal. Secondly, its strong blue light absorption makes it unsuitable for high-power 461 nm lasers [26,35]. In research on high-power blue light lasers, LiB₃O₅ (LBO) crystals or β-BaB₂O₄ (BBO) crystals are also used, both of which have high transmittance to blue light and high damage thresholds. In 2021, Tinsley et al. from the University of Florence reported an external-cavity SHG 461 nm laser with 1 W output power, utilizing LBO as the nonlinear crystal. The SHG cavity was pumped by a VECSEL laser, and the conversion efficiency was nearly 57% [36]. Feng and Vidal et al. from the University of Bordeaux in France used a 922 nm laser to generate a 461 nm laser with more than 1 W of power in the LBO in 2021, and the conversion efficiency reached 87% [30]. Up to now, this has been the highest conversion efficiency reported for a frequency doubling 461 nm laser. However, the LBO crystals and BBO crystals have low effective nonlinear coefficients. Due to deviation effects, the output laser spot degrades into an ellipsoid shape accompanied by a higher-order transverse mode output [33,37]. Toptica Photonics developed a frequency doubling 461 nm laser with an output power of watts (TA-SHG pro). However, they have not shared the material making up the SHG crystal. With the development of quasi-phase matching technology, periodically poled KTiOPO₄ (PPKTP) crystals have been widely used in blue light frequency doubling technology due to its advantages of high nonlinear coefficient and high damage threshold [38,39,40,41,42]. In 2003, Schwedes et al. of the Max Planck Institute of Quantum Optics in Germany, using a 20 mm long PPKTP crystal, created a 461 nm laser with an output power of 205 mW and a conversion efficiency of 40% [43]. In 2005, Le Targart et al. of the BNM-SYRTE research group in France achieved a 234 mW 461 nm laser output with a conversion efficiency of 75% by optimizing the focusing parameter of the fundamental laser and using a PPKTP crystal 20 mm in length [44]. In 2009, Zhao Yang et al. from the National Institute of Metrology reported a 461 nm frequency doubling laser based on a PPKTP crystal 20 mm in length, with a maximum output power of 208 mW and a maximum coupling conversion efficiency of 73.3% [45]. Since there is no departure angle of the quasi-phase matched crystal, the laser spot is generally in TEM₀₀ mode. Therefore, the blue light can be directly coupled to the strontium atomic physics system through a free-space optical path, thus greatly improving the laser utilization rate.

Although the PPKTP crystal can output high-power, high-efficiency, and high-beam-quality frequency doubling blue light, it has a strong absorption at 461 nm laser. When the crystal outputs a high-power 461 nm laser, the temperature gradient inside the crystal is induced. As a result, quasi-phase mismatching occurs between the fundamental and the frequency doubling lights, and the conversion efficiency decreases. The thermal effect also produces thermal lens effect in the crystal, which reduces the mode matching degree between the fundamental laser and the intrinsic mode of the frequency doubling external cavity. Theoretical studies on thermal lenses have been proceeding for years [30,31,32,46], but quantitative calculations on thermal quasi-phase mismatch and the thermal lens effects induced by frequency doubling blue light absorption in PPKTP are rare. Therefore, in order to design a high-power and high-efficiency external-cavity frequency doubling 461 nm laser, it is necessary to establish a simulation model to quantitatively analyze the thermal gradient in the crystal, and to study the thermal induced quasi-phase mismatch and the thermal lens more accurately, both theoretically and experimentally.

In this manuscript, under the condition of plane wave and small signal approximations, the interdependence between the crystal absorption of the SHG laser and the thermally induced quasi-phase mismatch in the crystal is incorporated into the process of solving the coupled wave equations. Combined with the multi-layer crystal approximation model, a theoretical method for accurately calculating the crystal absorption of the SHG laser was established. Based on the experimental parameters, the temperature gradient in the crystal, the influence of the boundary temperature on the conversion efficiency, and the focal length of the thermal lens were simulated. Then, combined with the theory of external-cavity frequency doubling, the influence of linear loss in the cavity on the conversion efficiency was analyzed. Based on these calculation results, we optimized the parameters of the frequency doubling cavity. In the experiment, a 922 nm tapered amplifier laser pumped a PPKTP frequency doubling crystal in a four-mirror bow-tie cavity and generated a 461 nm laser with a maximum output power of 382 mW, corresponding to a fundamental laser conversion efficiency of 66.2%. The theoretical calculation results were in good agreement with the experimental results. High-power, high-efficiency and high-beam-quality 461 nm frequency doubling laser technology can greatly reduce the power consumption and complexity of the blue magneto-optical trap atomic-capture optical system in the strontium atomic optical clock.

2. Theoretical Model and Simulations

2.1. Coupled Wave Equations

With small signal approximation, the coupled wave equation for the frequency doubling process based on first-order quasi-phase matching can be expressed as [47] (pp. 130–131).

$$A_2=i{B}_2{A}_1^2{\displaystyle {\int }_0^L\exp \left(i\Delta k\cdot z\right)}dz.$$

(1)

where A_n (n = 1, 2) is the laser electric field amplitude, and the subscripts 1 and 2 represent fundamental and frequency doubling lasers, respectively. $B_n=\frac{\omega }{n_{n}c}{d}_{eff}$, where ω is the angular frequency of the fundamental laser and d_eff is the crystal effective nonlinearity coefficient. n₁, n₂ are the refractive indices of fundamental and frequency doubling lasers, respectively. λ_n is the vacuum wavelength. $\Delta k=k_2-2k_1-2\pi /\Lambda$, where Λ is the polarization period. The wave vector is $k_n=\frac{2\pi n_n}{{\lambda }_n}$. A₁ is regarded as a constant, and A₂ (0) = 0. L is the length of the quasi-phase matching crystal.

In plane wave approximation, the power solution for the frequency doubling laser can be obtained by $I=\frac{1}{2}nc{\epsilon }_0{\left|A\right|}^2$ and P = I·S, shown as

$$P^{2\omega }=\frac{8{\pi }^2{L}^2{d}_{eff}^2}{n_1^2{n}_2{\lambda }_1^{2}c{\epsilon }_{0}S}{\left(P^{\omega }\right)}^2{\left[\sin \mathrm{c}\left(\frac{\Delta kL}{2}\right)\right]}^2,$$

(2)

where P^2ω is the frequency doubling laser output power and P^ω is the fundamental frequency laser input power. S is the cross-section area of the fundamental laser in the crystal. c is the speed of light in a vacuum, and ε₀ is the vacuum permittivity constant.

2.2. Optical Absorption and Temperature Distribution in Crystals

Two factors must be considered additionally in the frequency doubling process of a 922 nm laser to a 461 nm laser. Firstly, the SHG blue light absorption in nonlinear crystals is strong. For instance, PPKTP crystals exhibit a higher absorption coefficient for 461 nm lasers (approximately 0.1 cm⁻¹) compared to 922 nm lasers (about 0.003 cm⁻¹). Consequently, while longer crystals can achieve higher conversion efficiency, they also entail greater absorption. Secondly, because the optical absorption induced a thermal gradient in the beam radial direction and propagation direction in the crystal, a nonuniform refractive index distribution was generated within the crystal. Since the absorption of the fundamental frequency laser and the frequency doubling laser by the crystal belong to different cases, the principles of their heat production are discussed separately.

When the fundamental laser passes through the crystal, the input laser power is absorbed by the crystal and an attenuated power is emitted, which can be expressed as

$$P_{out}^{\omega }=P^{\omega }\exp \left(-\beta L\right)$$

(3)

where $P_{out}^{\omega }$ is the power transmitted from the crystal and P^ω is the power incident on the crystal. β is the absorption coefficient of the crystal for the fundamental frequency laser. The absorbed laser energy can be considered as thermal energy within the crystal. Once the crystal absorbs the laser energy, the thermal energy disperses uniformly in the direction of the z-axis due to heat dissipation facilitated by the crystal oven. However, owing to the Gaussian distribution of the laser, a thermal gradient emerges in the r-direction, and the thermal density is distributed as

$$Q\left(r\right)=\frac{2P^{\omega }\cdot \left[1-\exp (-\beta L)\right]}{\pi r_0^2}\exp (\frac{-2r^2}{r_0^2})$$

(4)

where Q is the thermal density and r₀ is the Gaussian radius of the laser. To simplify the calculation, the optical spot size within the Rayleigh length can be considered as uniform. In experiments, thin crystals and loose laser focus parameters are often used to reduce the thermal gradient within the crystal.

In contrast, the frequency doubling laser is gradually generated in the crystal along the direction of the fundamental laser vector and propagates inside the crystal to achieve both nonlinear gain and crystal absorption loss. Assuming that the crystal is divided into j segments along the laser vector, each layer has a sufficiently small length, denoted as dL. This ensures that the power of the frequency doubling laser can be treated as if it is uniformly distributed within each crystal layer. It should be noted that the fundamental laser is still considered to be unchanged according to the small signal approximation. The blue light power, including the absorption process, can be expressed as

$$\left\{\begin{array}{c}{P}_{out}^{2\omega }=P_L^{2\omega }-{\displaystyle \sum _{n=1}^j{P}_n^{2\omega }\left[1-\exp \left(-\alpha dL\right)\right]}\\ P_n^{2\omega }=\frac{P_0^{2\omega }}{L^2}{\left[{\displaystyle {\int }_0^{L_n}\exp \left(i\Delta kz\right)dz}\right]}^2\\ L_n=ndL, n=1,2,3\dots j\end{array}\right..$$

(5)

Here, the $P_0^{2\omega }$ represents the SHG output power without absorption or quasi-phase mismatching. $P_L^{2\omega }$ represents the SHG output power contained in the quasi-phase mismatching. Due to the significant variation in the power of blue light along the axial direction of the crystal, there is a noticeable thermal gradient in the axial direction caused by the thermal effects induced by the blue light. The heat density distribution in each piece of the crystal can be expressed as

$$Q_n\left(r,L_n,T\right)=\frac{2P_n^{2\omega }\cdot \left[1-\exp (-\alpha dL)\right]}{\pi r_0^2}\exp (\frac{-2r^2}{r_0^2}),n=1,2,3\dots j$$

(6)

Combining the total heat generated by the fundamental frequency laser and the frequency doubling laser, the overall thermal density distribution in each layer of the crystal can be expressed as

$$Q_n\left(r,L_n,T\right)=\frac{2P_n^{2\omega }\cdot \left[1-\exp (-\alpha dL)\right]+2P_n^{\omega }\left[1-\exp \left(-\beta dL\right)\right]}{\pi r_0^2}\exp (\frac{-2r^2}{r_0^2}),n=1,2,3\dots j$$

(7)

Based on small-signal approximation, the fundamental laser power in each piece of crystal is equal, represented by $P_n^{\omega }=P^{\omega }/j.$

The temperature distribution inside the crystal satisfies

$$\frac{{\partial }^{2}T\left(r\right)}{\partial r^2}=\frac{Q\left(r\right)}{K_c}.$$

(8)

K_c is the crystal thermal conductivity.

The solution for the temperature distribution can be obtained as [48]:

$$\Delta T_n(r,L_n)=\frac{\alpha dL{P}_n^{2\omega }+\beta dL{P}_n^{\omega }}{4\pi K_c}\left[\ln \left(\frac{r_b^2}{r^2}\right)+E_1\left(\frac{2r_b^2}{r_0^2}\right)-E_1\left(\frac{2r^2}{r_0^2}\right)\right],$$

(9)

where the approximation $1-\exp \left(-\alpha dL\right)\approx \alpha dL$, $1-\exp \left(-\beta dL\right)\approx \beta dL$, based on the Taylor series, is made. $\Delta T(r,z)=T(r,z)-T(r_b,z)$ is the temperature difference between the point of distance r from the crystal center and the boundary of the crystal. r_b is the distance from the crystal boundary to the optical axis, and the temperature at the boundary is the oven temperature. In Equation (9), $E_1\left(t\right)={\displaystyle {\int }_t^{\infty }\frac{e^{-x}}{x}}dx$ is the first-order integral function. Since the laser spot is much smaller than the crystal cross-section, the value of $E_1\left(\frac{2r_b^2}{r_0^2}\right)$ is very small and can be neglected.

2.3. Thermally Induced Phase Mismatching and Thermal Lens

Based on the temperature distribution solved by the equations above, the refractive index distribution inside the crystal can be obtained according to the Sellmeier equation, while the variation of the polarization period of the crystal can be derived by Λ(T) = Λ₀exp(1 + σΔT), where σ is the crystal thermal expansion coefficient. Based on the quasi-phase matching theory,

$$P^{2\omega }\propto {\left[\sin \mathrm{c}\left(\frac{\Delta k\left(T\right)L\left(T\right)}{2}\right)\right]}^2={\left[\sin \mathrm{c}\left(\pi L\left(T\right)\left(\frac{n_2\left(T\right)}{{\lambda }_2}-\frac{2n_1\left(T\right)}{{\lambda }_1}-\frac{1}{\Lambda \left(T\right)}\right)\right)\right]}^2.$$

(10)

The quasi-phase mismatch in the crystal produces gradient distribution in the radial and z-axial directions of the laser cross-section. The temperature gradient in the crystal and the frequency doubling laser power are mutually dependent and finally reach a steady state.

Another result of the thermal refractive index change is thermal lensing, which degrades the Gaussian beam space mode of the laser and even causes higher-order transverse modes. The crystal thermal lens in the external-cavity frequency doubling seriously affects the mode matching between the incident fundamental laser and the intrinsic mode of the cavity. In each crystal layer, a thin thermal lens (f_n) is generated from the absorption of the 922 nm and 461 nm lasers. The equivalent focal length f_L of the thermal lens in the crystal is

$$\left\{\begin{array}{c}\frac{1}{f_L}={\displaystyle \sum _{n=1}^j\frac{1}{f_n}}\\ f_n=\frac{K_c\pi r_0^2}{P_{abs,n}}{\left(\frac{dn}{dT}\right)}^{-1}\\ P_{abs,n}=\alpha dL{P}_n^{2\omega }+\beta dL{P}_n^{\omega }\end{array}\right..$$

(11)

2.4. Simulations of Thermal Effects

Based on the above analysis, the experimental parameters were substituted to quantitatively analyze the influence of the conversion efficiency due to the laser-induced thermal effects in the SHG process. The PPKTP crystal dimensions in the simulation calculations were 1 mm (thickness) × 1 mm (width) × 10 mm (length), and the polarization period was 5.55 μm. The thermal conductivity of the PPKTP was 3.3 W/m/K, and the thermal expansion coefficient was 0.6 × 10⁻⁶/°C. The radius of the beam waist at the center of the crystal was 25 μm for both the fundamental and frequency doubling lasers. The input power of the 922 nm laser was set at 4.5 W (close to the real experimental parameter), and the ideal quasi-phase-matching temperature of the crystal with a negligible thermal effect was set to 28.6 °C, which was also the boundary temperature of the crystal, i.e., the oven temperature. The crystal refractive index versus temperature was calculated via the Sellmeier equation given by Kato in 1992 [49].

Figure 1 depicts the simulation results, correlating the normalized conversion efficiency with the crystal boundary temperature. The temperature bandwidth when the conversion efficiency drops to 0.5 is about 2 °C, which is close to the experimental result.

The temperature distribution inside the crystal was calculated at a boundary temperature of 28.6 °C using the theoretical model, as illustrated in Figure 2. In Figure 2a, the temperature distribution along the z-direction of the entire cross section of the crystal is presented. The thermal gradient, induced by the laser power, intensified from the incident face to the exit face, with the temperature difference reaching 0.55 °C. According to the results in Figure 1, the conversion efficiency of the crystal region with the highest temperature decreased by about 18% compared to the ideal case. Figure 2b shows the temperature distribution in the crystal within the Gaussian diameter of the laser beam. The temperature difference from the center to the edge of the Gaussian spot was about 0.1 °C, and the conversion efficiency was reduced by about 0.5%. Therefore, in the experimental setup with a blue light output in the hundred-milliwatt range, the impact of radial quasi-phase mismatch on the conversion efficiency can be disregarded.

By adjusting the boundary temperature, the crystal can achieve the optimum frequency doubling output power. According to the calculation of the theory in this manuscript, Figure 3 illustrates the correlation between the crystal oven’s set temperature and the output power of the frequency doubling light. By reducing the temperature of the crystal oven to 28.0 °C, the maximum output power can reach 460.5 mW. Assuming that the influence of thermal effects on quasi-phase mismatch in the crystal is not considered, and that only the absorption of the crystal is considered, the maximum power of frequency doubling is 466.5 mW at the theoretically optimal matching temperature. This shows that the output power can be nearly restored to the ideal value by reducing the crystal boundary temperature when the absorption of laser is within a specific range.

Upon adjusting the boundary temperature to 28.0 °C, the experimental parameters mentioned above were applied to Equation (11), resulting in the calculation of the focal length of the thermal lens within the crystal as 13.8 mm. If the absorption of frequency doubled light were calculated according to the fundamental laser absorption process [35], the focal length of the thermal lens obtained under the identical parameters would be 8.8 mm.

2.5. External-Cavity-Enhanced SHG

The external-cavity frequency doubling laser output can be regarded as an SHG laser produced by the cavity-enhanced fundamental laser single-pass pumping the crystal, so

$${\eta }_{ce}=\frac{P^{2\omega }}{P_{in}^{\omega }}=\frac{{\eta }_{sp}{P}^{\omega }}{P_{in}^{\omega }},$$

(12)

and

$$P^{\omega }=\frac{m\left(1-R_1\right)P_{in}^{\omega }}{{\left[1-\sqrt{R_1(1-{\delta }_L)(1-{\eta }_{sp})}\right]}^2}.$$

(13)

In Equation (12), P^ω is the cavity-enhanced fundamental laser power before the incident end face of the crystal. It has the same meaning to Equation (2). η_ce is the cavity-enhanced frequency doubling efficiency, and η_sp is the single-pass conversion efficiency. $P_{in}^{\omega }$ is the incident fundamental laser power before the cavity input coupler mirror. In Equation (13), m is the mode matching efficiency, and R₁ is the reflectivity of the input coupling mirror. δ_L is the linear loss of the cavity for the fundamental laser, which contains the transmission loss of the high-reflection cavity mirrors and the reflection loss of the crystal end faces.

Based on the above equations, a simulation of the relationships between the cavity-enhanced SHG efficiency and the linear loss was performed, encompassing linear losses ranging from 0.5% to 4.5%. Other parameters were set as R₁ = 0.9, $P_{in}^{\omega }$ = 0–0.6 W, and m = 0.94. Figure 4 displays the outcomes of the simulation. The calculated results indicate that, considering the specified parameters, the maximum single-pass conversion efficiency within the cavity reached 9.6%. When the linear loss in the cavity was 0.5%, a high frequency doubling conversion efficiency of about 90% could also be achieved. Therefore, for the efficient enhancement of external-cavity frequency doubling, it is imperative to minimize linear loss in the experimental setup. If a further reduction in linear loss is unattainable, enhancing the single-pass conversion efficiency becomes crucial.

The efficiency of mode matching between the fundamental frequency light and the external cavity stands as a limiting factor for enhanced conversion efficiency in the external cavity. Deriving the intrinsic cavity mode involves incorporating the theoretically calculated focal length of the crystal thermal lens into the ABCD matrix. Given that the blue-light-induced equivalent thermal lens is not situated at the center of the crystal, it becomes imperative to experimentally optimize the crystal position to uphold high-quality mode matching [21,22,23]. Additionally, in scenarios where the linear loss in the cavity is minimal, moderately expanding the focusing parameter of the fundamental laser within the crystal can increase the focal length of the thermal lens. This adjustment proves beneficial for reducing the mode matching error and improving the long-term stability of the laser output.

3. Experimental Setup

A schematic diagram of the experimental setup is depicted in Figure 5. The fundamental laser, a 922 nm, high-power tapered amplifier laser initiated by an external cavity diode laser (TA pro, Toptica photonic), had a maximum output power of 1.1 W. The output beam of the tapered amplifier exhibited a high-order transverse mode. To enhance the accuracy of mode matching between the fundamental laser and the frequency doubling cavity, the 922 nm laser was directed into a single-mode, polarization-preserving fiber (PM850, OZ) during the experiment. The fiber coupling efficiency was 53%, i.e., the maximal power of the fundamental frequency laser output at the fiber end was 577 mW. Before the laser was coupled into the fiber, the polarization direction of the laser and the slow axing of the fiber aligned with the one-half-wave plate to ensure the bias-preserving characteristics of the optical fiber transmission. The beam waist of the laser output from the fiber coupler was focused by two plano-convex lenses, L1 and L2, to achieve efficient mode matching.

The frequency doubling cavity comprised a four-mirror, bow-tie configuration. It consisted of two plane mirrors (M1 and M2) and two concave mirrors (M3 and M4) with radii of curvature of 40 mm. M1 was used as the input coupling mirror of the cavity, and was coated with a 922 nm, partially reflective film. M2 was a micro-mirror with a diameter of 5 mm and a thickness of 2 mm, bonded to the top of a piezoelectric transducer (PZT) with a length of 10 mm and a diameter of 3 mm. Frequency locking between the frequency doubling cavity and the fundamental frequency laser was achieved by adjusting the length of the PZT. M2, M3, and M4 had high reflectivity levels of R > 99.5% for 922 nm, with M4 additionally coated with high-transmittivity coatings for 461 nm. A 1 mm × 2 mm × 10 mm PPKTP crystal with a polarization period of 5.55 μm was positioned between two concave mirrors. The crystal end faces were coated with high-transmittivity 922 nm and 461 nm with reflectivity values of less than 0.5%. Installed in a pure copper oven, the crystal’s temperature was precisely regulated using a thermoelectric cooler (TEC) with an accuracy of 0.01 °C. The oven was mounted on a five-dimensional adjustment frame to optimize the position and angle of the crystal. According to the Boyd–Kleinnman theory [50], the focusing parameter of the fundamental frequency laser in the crystal was designed to be 1.28 to increase the ratio between the nonlinear conversion efficiency and the intra-cavity linear loss. The radius of the fundamental beam waist in the crystal was about 25 μm. The total cavity length of the bow-tie cavity was 266 mm, and the reflection angle of the beam on the mirror was 5°. To mitigate the thermal effects of the crystal on mode matching, the crystal position was optimized based on the output power of the 461 nm laser.

The photodetector PD2, positioned behind the M3 lens in the experiment, took the 922 nm transmitted cavity mode signal and fed it into the PDD110 module within the Toptica laser controller. The error signal was generated based on lock-in frequency stabilization technology. Subsequently, the error signal was looped back to the PZT of the frequency doubling cavity via the PID110 module and the high-voltage amplifier. When the frequency of the 922 nm seed laser was slowly scanned, the frequency of the 461 nm laser could be tuned synchronously.

4. Experimental Results and Discussions

Firstly, the optimum oven temperature was measured. The M1, with a reflection rate of 85% at 922 nm, was applied, and the normalized 461 nm output laser power was measured against the crystal temperature for both high and low power levels of the 922 nm laser, as illustrated in Figure 6. At an input power of 100 mW, the crystal’s optimum temperature was found to be 28.6 °C, with a temperature bandwidth of approximately 2.5 °C, as indicated by the blue square in Figure 6. As the input power increased to 320 mW, the optimum temperature shifted to 27.8 °C. Beyond this temperature, the curve line shape of the experimental results was distorted. This was due to the obvious thermal effects in the crystal, which changed both the mode-matching and impedance-matching processes.

In order to improve the conversion efficiency, it was necessary to increase the fundamental laser power entering into the cavity. Firstly, the optimum mode matching needed to be implemented. Applying the previously outlined theoretical approach, the mode-matching parameters for the fundamental frequency light and the doubling cavity were computed. The focal length of the thermal lens was determined under initial conditions of 580 mW incident power for the fundamental frequency light and 90% reflectivity of the incident-coupled mirror. Subsequently, the intrinsic mode of the frequency doubling cavity, including the thermal lens and the Gaussian beam of the fundamental frequency light, was initially calculated for mode matching. Through fine adjustments to the crystal position, a better mode-matching efficiency was ultimately achieved. As depicted in Figure 7a, the mode-matching efficiency of the setup was 94%. The position of the crystal was adjusted by 1–2 mm opposite to the laser output direction. Secondly, the gain of the fundamental laser and the loss of the cavity were optimized for impedance matching. The assessment of impedance matching typically includes deriving the coupling efficiency from the reflected signal of the cavity mode. In this experiment, under the maximum incident fundamental laser power, an optimum coupling efficiency of 89.2% was obtained. Figure 7b displays the corresponding experimental results. Once the fundamental frequency laser had been locked to the SHG cavity, the coupling efficiency was higher than that observed in the scanning mode. When the thermally stable state was established, improved impedance matching was established within the SHG cavity.

The relationship between the output power of the SHG laser and the incident power of the fundamental laser was measured. The experimental results are shown in the data points in Figure 8, with red dots representing efficiency and blue squares representing SHG laser power. Using an input coupler with 90% reflectivity at 922 nm, a laser output power of 382 mW at 461 nm was achieved with an input power of 577 mW for the 922 nm laser, resulting in a corresponding frequency doubling conversion efficiency of 66.2%. Theoretical simulation calculations were conducted based on the external cavity frequency doubling theory and experimental parameters, as illustrated by the solid line in Figure 8. The measured linear loss in the experiment was 3.5%, primarily attributed to coating defects in the cavity mirror and the reflection of the crystal surfaces.

The beam quality of the 461 nm laser with an output power of 382 mW was measured using a beam quality analyzer (BP209-VIS/M, Thorlabs, Newton, NJ, USA), and the results are shown in Figure 9. The fitting results showed that the M² factor in the X-direction was 1.40, and the M² factor in the Y-direction was 1.44. This result also indicates that there are obvious thermal effects inside the crystal.

5. Conclusions

By introducing the interdependence between the absorption of blue light and the thermally induced quasi-phase mismatch into the solution process of coupled wave equations, as well as combining the multi-layer crystal approximation, the absorption value of the frequency doubling laser could be obtained more accurately. This approach allowed for a more accurate calculation of the temperature distribution within the crystal, subsequently enabling the determination of the focal length of the thermal lens. It is necessary to fully consider the focal length of the thermal lens and the ratio of the nonlinear loss to the linear loss in the cavity when designing a high-efficiency frequency doubling external cavity. After substituting the experimental parameters into the theoretical model, the parameters of the frequency doubling external cavity were calculated for high-efficiency mode matching. Following experimental optimization, we achieved a 94% mode matching efficiency and an 89.2% coupling efficiency. A 461 nm frequency doubling laser output of 382 mW was achieved, and the fundamental-frequency laser conversion efficiency was 66.2%. Due to the limitations imposed by linear loss in the cavity, further enhancements to the conversion efficiency were challenging. The quality of the output laser beam was influenced by thermal effects, resulting in an M² factor of approximately 1.4. This high-efficiency, external-cavity enhanced frequency doubling technology based on quasi-phase matching crystals will effectively reduce the power consumption and structure complexity of strontium atom blue magneto-optical trap optical systems.