## Abstract

We present a simple all-solid-state laser source emitting 2.4 W of single-frequency light at 671 nm for laser cooling of lithium atoms. It is based on a diode-pumped solid-state laser, which is frequency doubled in a ppZnO:LN ridge waveguide with an internal doubling efficiency of 54%. We develop a simple theory for the thermal effects we observed at elevated fundamental powers, and compare the setup to a more efficient but more complex one with an external resonant frequency doubling cavity providing 5.2 W at 671 nm.

© 2017 Optical Society of America

## 1. Introduction

In the realm of quantum technology with ultracold atoms, simple, yet high-power, single-frequency laser sources are an indispensable research tool. When it comes to laser cooling of the popular lithium atom, with the necessity of addressing the *D*-line atomic transitions near 671 nm, one is left with a limited choice of suitable laser sources. One reason is the relatively weak performance of semiconductor lasers at shorter wavelengths in the red part of the visible spectrum. Such semiconductor tapered amplifiers deliver a few hundred millwatts of near-resonant light at 671 nm. Recent developments of similar chips emitting in the infrared range have made Watt-class frequency-doubled sources around 671 nm commercially available [1]. On the other hand, Ti:sapphire lasers optimized for the red part of the spectrum [2] and dye lasers are price-intensive and more cumbersome in handling.

An alternative approach is frequency doubling of infrared solid-state lasers, which offer high output power, excellent spatial mode quality, and intrinsic stability. However, this introduces the need for efficient second harmonic generation in external cavities, which need to be frequency-locked to the fundamental laser, considerably raising the complexity of such setups. Using neodymium-doped orthovanadate (Nd:YVO_{4}) as an active medium, Watt-class output power was demonstrated in our group with a laser system consisting of two cavities: The laser itself, and the resonant external frequency doubler [3]. More recently, the Kaiserslautern group demonstrated an alternative setup using an injection-locked high-power resonator yielding 5.7 W output power [4]. Even though the frequency doubling is established in a single-pass bulk nonlinear crystal, two resonators and the corresponding frequency locking are still necessary for the master laser and the power amplifier. In an effort of simplification, we demonstrated 2.1 W of emission at 671 nm from an intracavity frequency doubling of a similar laser [5], requiring one cavity only. However, this introduced the drawback of nondeterministic frequency tuning. Therefore, it is of great practical interest to simplify the frequency doubling process, without compromising the lasers’s key characteristics of stability, tunability and output power.

In contrast to free-propagation schemes, guiding of optical fields has the intrinsic advantage of small mode field diameters along the guiding structure. This allows one to increase the effects of interactions between external fields, and fields provided by the medium by a large amount. Therefore, a large number of waveguide-based devices have been developed and commercialized, reaching from passive optical fibers to fiber lasers, from waveguide EOMs to waveguide-based nonlinear-optics modules, most popular in the form of waveguide frequency doublers and mixers. Excellent performance has been demonstrated with the latter, ranging from very high nonlinear conversion efficiencies at low or intermediate power levels to frequency and phase stability down to state-of-the-art metrology levels [6].

In this article, we present a simple one-cavity, waveguide-frequency-doubled laser system. It operates in the vicinity of the lithium *D*-line resonances. Starting from a single-frequency output power of 6.6 W at 1342 nm, we obtain 2.4 W of 671 nm output using a custom-made ridge-waveguide frequency doubling module, corresponding to an internal doubling efficiency of 54%. While offering largely sufficient output powers for running an ultracold atom experiment, the light source does not require frequency locking for free-running operation, and standard frequency locking procedures can readily be implemented on the infrared source [3]. At elevated power levels, we observe thermal effects in the doubling waveguide, which we can fully explain by a simple theory that we present. It allows us to extract one- and two-photon absorption coefficients from our experimental data. Our system compares favourably to a two-cavity setup similar to the one presented in [3], for which the available 671 nm output power has now been increased to 5.2 W.

This article is organized as follows: In section 2, we describe the fundamental laser source. In section 3 we present the waveguide setup and characterization, and in section 4 we present a simple theory model for simulating the thermal effects we observe at elevated power levels. In section 5 we compare the waveguide doubling scheme to a conventional resonant frequency doubler, and we conclude in section 6.

## 2. Fundamental laser setup and characteristics

The fundamental light source we use throughout this article is a modified version of the laser presented in [3]. The following improvements have been implemented in order to minimize and compensate for thermal effects inside the laser gain medium:

- Firstly, as in [5], we use pump light at 888 nm instead of 808 nm. This lowers the quantum defect per lasing cycle of the active ions by 15%, thus reducing the heat load deposited in the laser crystal at a given pump power [7]. In addition the lower absorption coefficient at 888 nm allows to spread the thermal load over a longer crystal and a larger pump volume, therefore reducing the heat density and minimizing stress and thermal lensing in the gain medium. Moreover, this pump wavelength has the advantage that the absorption in Nd:YVO
_{4}is nearly independent on the pump light polarization at this wavelength, and hence efficient pumping is possible even by employing the partly polarized pump light derived from fiber-coupled diodes. - Secondly, the flat pump coupling mirror was replaced by a convex meniscus mirror in order to compensate for the thermal lens of the Nd:YVO
_{4}crystal. Its radius of curvature of*r*_{c}= 500 mm was chosen in accordance with the measured focal length of*f*= 170 mm of the thermal lens under lasing conditions [8,9].

Figure 1 shows the experimental setup of the optimized oscillator operating at 1342 nm. The 888 nm output of the temperature-stabilized fiber-coupled QPC BrightLase Ultra-50 diode stack (NA = 0.22, 400 μm fiber core diameter, output spectrum FWHM: 2.2 nm), emitting up to 42.6 W at 888 nm, is focused by two lenses (*f*_{1} = 75 mm, *f*_{2} = 175 mm) to a top-hat profile of radius *w*_{pump} = 970 μm inside the laser crystal. The dimensions of the actively cooled Nd:YVO_{4} crystal (a-cut, anti-reflection coated for 1342 nm and 888 nm) were chosen to be 4 × 4 × 25 mm^{3} and the ion doping was accordingly adjusted to 0.5 at. %. The cavity consists of three curved mirrors (M_{1}, M_{3} and M_{4}) and one flat mirror (M_{2}), forming a bow-tie cavity with a free spectral range of *ν*_{FSR} = 320 MHz. The pump coupling mirror M_{1} is, as mentioned above, a convex meniscus mirror with a radius of curvature of *r _{c}* = 500 mm and a thickness of 6 mm. Its concave backside has the same radius of curvature of 500 mm so that the pump beam passing through this mirror is not altered by any lensing effect (zero lens mirror). M

_{3}and M

_{4}are concave mirrors with a radius of curvature of 100 mm and 150 mm, respectively. M

_{1}, M

_{3}and M

_{4}are highly reflective coated for 1342 nm, and M

_{1}is furthermore transmitting at 888 nm. The optimal transmittance of the output coupler M

_{2}regarding the maximum output power was found to be 7.5%. Unidirectional operation of the ring laser is enforced by a Faraday rotator consisting of a cylindrical terbium-gallium-garnet crystal (TGG) of 6 mm length placed in a strong permanent magnet [10] in combination with a true-zero-order half-wave plate. Two uncoated infrared fused silica etalons of 500 μm (E

_{1}) and 4 mm (E

_{2}) thickness, the latter being temperature-stabilized, are used in order to ensure single frequency behavior and wavelength tunability. Further fine tuning of the laser frequency, for the purpose of frequency-locking its second harmonic to the lithium

*D*-line via Doppler-free saturated absorption spectroscopy [3], is established by mounting mirrors M

_{3}and M

_{4}on piezoelectric transducers (PZTs): a slow PZT (M3) displaying large displacement of around 8 μm at maximum voltage of 150 V and a fast PZT (M4) with a displacement of around ±50 nm limited by the ±15 V driver. Using a frequency-locking setup, as described in [3], these PZTs should allow to stabilize the output wavelength, providing a laser linewidth with an upper limit of 1 MHz [5].

Figure 2 shows the output power of the laser as a function of the absorbed pump power for an increasing pump power. The lasing threshold is found at an absorbed pump power *P*_{abs} = 13.9 W and the power rises linearly above *P*_{abs} = 20 W with a slope efficiency of *η*_{sl} = d*P*_{out}/d*P*_{abs} = 28.1%. A maximum output power of *P*_{out} = 6.6 W is reached for an absorbed pump power of *P*_{abs,max} = 31.3 W, which is the maximum the pump can deliver. This results in an optical-to-optical conversion efficiency of 16.3%. As can be seen in Fig. 2, the fundamental output power is not limited by thermal roll-off. Therefore, with more available pump power, we expect to see a further significant increase in single-mode fundamental power. Concerning the long-term stability of the laser output power, we have not found any signs of output power degradation during the course of the measurements presented in this article. The beam quality parameter of the IR output beam is determined through a caustic measurement and results in a value of *M*^{2} ≤ 1.2. By changing the tilt of etalon E_{1} and the temperature of etalon E_{2} the emission wavelength of the laser can be tuned. Even though the full emission spectrum of the infrared laser has not been taken systematically, a comparison with the tuning behavior of similar IR sources, as presented in [5,11], yields a perfect qualitative agreement with these former findings. In particular, the emission spectrum is centered around 1342.2 nm and has a total width of around 1.2 nm. Moreover, it is smooth except for three narrow dips at 1341.8 nm, 1342.1 nm and 1342.7 nm, which can be accounted for by absorption of water molecules in the laser cavity. The wavelength range corresponding to the lithium *D*-line resonances (1341.9 − 1342.0 nm) is well in between the two first dips, and the emitted power is close to the absolute maximum.

## 3. Waveguide second-harmonic generation

The nonlinear device applied for second harmonic generation (SHG) in this experiment is a wavelength conversion module from NTT Electronics (WH-0671-000-A-B-C) that is described in detail in [12, 13]. It contains a ridge waveguide whose core consists of periodically poled 7.5 % ZnO-doped LiNbO_{3} (ppZnO:LN) and has a nearly quadratic cross section of 14.8 × 14.2 μm^{2} and a length of *L* = 10 mm. The core material is attached to a LiTaO_{3} (LT) cladding layer with a thickness of 32 μm by using a direct bonding technique during the manufacturing process. LT was chosen as substrate material due to its lower refractive index compared to ppZnO:LN, thus improving the optical confinement of the guided modes. Furthermore both materials feature similar lattice constants and thermal expansion coefficients and therefore allow direct bonding [14]. The chosen poling period of 13.2 μm for the core material provides quasi-phase-matching (QPM) for SHG at a fundamental wavelength of 1342 nm and at a waveguide temperature of approximately 49°C. It is temperature-stabilized using a built-in NTC thermistor and thermoelectric cooling element together with a commercial regulator circuit. The entire module is mounted on a passive heat sink and enclosed in a metal housing. The temperature of the waveguide crystal can therefore be stabilized to better than ±0.1°C. The use of a ridge waveguide has some advantages compared to other waveguide designs whose fabrication is based on proton exchanged or indiffusion techniques [15]. In particular it allows higher optical confinement while avoiding substitution of lithium ions in the crystalline structure which is at the origin of additional optical losses.

The experimental setup for frequency doubling is schematically presented in Fig. 3. The waveguide module is fiber-pigtailed on the input side and includes collimating and focussing lenses ensuring efficient coupling between the fiber and the waveguide. Mode matching between the laser output and the polarization-maintaining single-mode input fiber of the waveguide is accomplished using a set of lenses. Two spherical lenses collimate the laser output beam and reduce its diameter to 1.2 mm, whereas an aspheric collimation lens (*f*_{coll} = 6.24 mm) couples the IR light into the input fiber. A half-wave plate in front of the collimation lens is used to adjust the polarization of the IR light parallel to the axis of the PM fiber corresponding to a vertical polarization of the light injected into the waveguide. The output light after the waveguide is collimated by a lens and passes through an AR-coated window (Edmund Optics P/N 47837) before exiting the waveguide module. We separate the residual fundamental radiation from the second-harmonic with a dichroic beamsplitter set featuring a total transmission of 98.0% and 93.1% for the SH and IR light, respectively.

Due to the compact design of the waveguide module and the fact that the fiber output as well as Lens_{1} and Lens_{2} are fixed in position, it is not possible to directly measure the individual coupling efficiencies of the fundamental light to the fiber and the waveguide. Nevertheless, the overall coupling efficiency to the fiber and through the ppZnO:LN crystal waveguide can be measured by comparing the IR light power before and after the module without SHG. For this purpose, the crystal mount temperature is set to 20°C which is sufficiently far detuned from the QPM temperature for the SHG process. Alternatively, the transmission at the extraordinary polarization can be directly determined by measuring the transmission at the ordinary polarization since both can safely be considered equal for our particular waveguide design. Taking into account the optical losses for the fundamental wavelength at the optical elements on the output of the waveguide module, the coupling efficiency is determined to be 78%. This value remains constant over the full available input power range.

Subsequently, the phase-matching curve of the waveguide was investigated by tuning the crystal mount temperature and monitoring the variation in SH power. Two tuning curves for different fundamental powers are presented in Fig. 4. The fundamental power injected into the input fiber is tuned by means of a *λ*/2-waveplate and a polarizer at the output of the IR laser source. For a relatively low fundamental power of 1.07 W, the curve matches very well the theoretical sinc^{2} distribution, which was first derived in [16], and the maximum SH power of 0.2 W is reached at a crystal mount temperature of 48.9°C. The measured curve is symmetric and has an acceptance width (FWHM) of Δ*T* = 5.4°C. At elevated fundamental power levels, light absorption effects heat the waveguide and modify the phase-matching curve. This results in a shift of the conversion peak by 2.5°C down to 46.4°C for the maximum fundamental power of 4.4 W and the corresponding SH power of 2.4 W. To the best of our knowledge, this is the highest single-frequency output power reached with a waveguide device. The “internal” doubling efficiency, i.e. the doubling efficiency for fundamental light coupled into the single-mode input fiber, amounts to 54%. The shape of the main peak becomes slightly asymmetric and its height is lowered with respect to the sidelobes. The acceptance width is in this case only marginally reduced to Δ*T* = 4.8°C. The observed reduction in the temperature acceptance width for higher fundamental powers can be explained by inhomogeneous heating along the waveguide core due to several light absorption mechanisms of different order. In general, these local temperature inhomogeneities tend to reduce the global temperature range for which efficient QPM can be achieved over the full length of the waveguide. Since the shape at low conversion efficiency, i.e. far from the central peak, and especially the zero-efficiency positions do not change, we attribute the observed thermal effects mainly to absorption of either the second-harmonic light, or to second-harmonic induced fundamental absorption. We note that even at the highest power levels present in the course of this article, thermal effects did not cause power instabilities or other types of output power limitations in our setup.

Figure 5 presents the adjusted crystal mount temperature maximizing the SH output power for a series of different fundamental power values. As we have already stated, the maximum SH power in the limit of low *P _{ω}* is reached for 48.9°C. While increasing fundamental power, the crystal mount temperature has to be gradually reduced and reaches its minimum value of 46.4°C at the maximum available power of 4.4 W.

The SH output power characteristics of the waveguide were measured as a function of the coupled fundamental power, as shown in Fig. 6. At high powers, the crystal mount temperature had to be adjusted at each value of *P _{ω}* in order to maximize the conversion, as discussed above. For small fundamental powers up to 0.4 W, the experimental results are well fit by a parabola with a conversion efficiency of 18.4 %/W. For elevated fundamental power, significant depletion of the fundamental wave sets in, and the data points can be very well fit by the coupled waves theory presented in [17]. Fitting the well known lossless solution

*η*

_{wg}= tanh

^{2}[(

*η*

_{0}

*P*)

_{ω}L^{2}*] to our experimental data, with*

^{1/2}*L*= 10 mm being the length of the waveguide, yields the normalized efficiency

*η*

_{0}= 19.9 %/Wcm

^{2}. We thus observe SH output power characteristics that correspond to the expected theoretical models for both low and high fundamental powers, as well as a similar conversion efficiency in both regimes. This underscores the very uncritical behavior of the SHG process even at the high powers we apply, requiring only a small readjustment of the phase matching temperature.

Furthermore, no indications of long-term degradation in terms of output mode and power have been observed during the course of these measurements. This is in accordance with the optical damage threshold of 100 MW/cm^{2} due to gray tracking which was reported for doubling of 1064 nm light in Zn:LiNbO_{3} [18], since the intensities inside our waveguide remain well below 10 MW/cm^{2} even for the highest applied fundamental powers.

In principle, the waveguide size permits the guiding of higher-order modes at both the fundamental and SH wavelength. However, during production the coupling of the fundamental light to the fundamental mode is optimized. Figure 7 presents a caustic measurement of the SH beam quality parameter which yields a value of *M*^{2} ≤ 1.1 at the output of the ridge waveguide, showing negligible coupling to higher-order modes. Moreover, the beam profile does not show any signs of ellipticity or astigmatism. The improved beam quality of the outgoing SH light compared to the incoming fundamental light can be attributed to the mode-cleaning effect of the single-mode input fiber of the waveguide, which is mainly injecting the fundamental waveguide mode for efficient SHG, as well as the waveguide design which is appropriately optimized for providing a fundamental mode with Gaussian profile for both wavelengths.

## 4. Theoretical model

We applied a simplified version of the theoretical model presented in [19] in order to simulate the SHG process and to describe quantitatively the influence of optical absorption on the phase-matching curve. In the following, the subscript index *j* denotes either the IR light for *j* = 1 or the SH light for *j* = 2. The chosen coordinate system agrees with the crystallographic axes of the nonlinear crystal, see Fig. 8. The *x* axis corresponds to the light propagation direction along the horizontal ridge waveguide axis, whereas the *y* axis denotes the other horizontal direction which is perpendicular to both the light propagation direction and the crystal optical axis c. The *z* axis is therefore chosen to be the vertical direction parallel to the crystal optical axis c.

We define the scalar value of the electric field amplitude according to [20] as *A _{j}* = (2

*P*

_{j}/S_{eff}

*n*

_{j}∊_{0}

*c*

_{0})

^{1/2}, where

*n*is the refractive index,

_{j}*c*

_{0}is the vacuum speed of light and

*∊*

_{0}is the permittivity of free space.

*S*

_{eff}denotes the effective overlap area of the electric field distributions

*E*(

_{j}*y*,

*z*) of the fundamental TM modes of the waveguide [21]:

^{2}at 1342 nm and 10.20 × 9.92 μm

^{2}at 671 nm. The mode overlap is therefore

*η*= 98.3% and the effective overlap area between these two modes is calculated to be

*S*

_{eff}= 51.06 μm

^{2}. The scalar value of the electric field amplitude

*A*is calculated by solving a system of coupled wave-equations that was derived in [16] and applied for waveguide simulation in [19]:

_{j}*A*

_{0}= 2

*ϕ*

_{1}−

*ϕ*

_{2}+ Δ

*k*

_{QPM},

*σ*

_{1}=

*ω*

_{1}

*d*

_{eff}/

*c*

_{0}

*n*

_{1},

*σ*

_{2}=

*ω*

_{1}

*d*

_{eff}/

*c*

_{0}

*n*

_{2}, where

*d*

_{eff}= 2/

*π*·

*d*

_{33}is the effective nonlinear coefficient,

*ω*is the wave frequency with

_{j}*ω*

_{2}= 2

*ω*

_{1},

*ϕ*is the electric field phase,

_{j}*α*is the linear absorption coefficient,

_{j}*δ*

_{SHIFA}=

*α*

_{SHIFA}(

*n*

_{2}∊

_{0}

*c*

_{0}/2)

^{2}and

*α*

_{SHIFA}is the second-harmonic-induced infrared absorption (SHIFA) coefficient [23, 24],

*δ*

_{2Ph.}=

*β*

_{2Ph.}

*n*

_{2}

*∊*

_{0}

*c*

_{0}/2 and

*β*

_{2Ph.}is the two-photon absorption coefficient for the second-harmonic light [25]. We assume the linear losses in the ridge waveguide to equal the linear optical absorption, since the propagation loss, deriving for example from scattering due to surface or bulk imperfections, could not be measured independently from linear absorption losses with our experimental setup. We therefore neglect the propagation loss contributions in our model.

The phase mismatch parameter is defined as

*T*) the poling period,

*β*(

_{j}*T*) =

*N*(

_{j}*T*)

*ω*

_{j}/c_{0}the waveguide fundamental mode propagation constant and

*N*(

_{j}*T*) the effective refractive index calculated with the effective index approximation [26]. This calculation uses the temperature dependent Sellmeier equations from [27] for the extraordinary refractive index

*n*of 5% MgO-doped congruent ppLN, since they provide the best characterization of our nonlinear material. In particular, the equations reported in the referenced publication result in the closest match to our experimental data for the low-power phase-matching curve in Fig. 4, when compared to other publications that study the temperature and wavelength dependence of the refractive indices for different forms and doping of lithium niobate. All of these parameters depend on the temperature

_{j}*T*which may vary along the propagation axis

*x*due to heat generation resulting from optical absorption. Note that we do not account for any inhomogeneity of the ridge waveguide height

*z*

_{R}, which is justified by the observation of a temperature dependent phase matching curve that matches the theoretical sinc

^{2}distribution very well at low intensities, see Fig. 4.

The effect of the optical absorption on the temperature distribution along the waveguide axis *x* is estimated by assuming a one-dimensional, steady-state heat diffusion model [28], represented by a corresponding differential equation for the crystal temperature of the form

*k*= 5 W/K.m is the thermal conductivity of the LiTaO

_{3}cladding layer [29] which is assumed to dominate the thermal conductivity of the entire waveguide structure,

*z*is the vertical coordinate parallel to the crystal optical axis and

*z*

_{H}= 46.2 μm is the height of the entire waveguide structure including the core and the cladding layer.

We define the heat source as a boxcar function restricted to the ridge of the waveguide with height *z*_{R} = 14.2μm

*x*along the propagation axis, and the temperature in the center of the ridge waveguide

*T*(

*x*) of the entire ridge waveguide. This result is applied in the system of coupled Eqs. (2)–(5). The heat source density depends on the IR and SH intensities and is defined by [25]

Since the presented system of differential equations has, to our knowledge, no analytical solution, it is solved by numerical simulation. In order to reproduce our experimental data concerning the shift of the optimal waveguide mount temperature with increasing fundamental power, as well as the conversion efficiency of the waveguide, we have to adapt the material dependent optical coefficients of our model by applying least-squares fitting procedures to its numerical solution. In particular, the values of the nonlinear optical coefficient *d*_{33} and the poling period Λ need to be set to 12.7 pm/V and 13.48 μm, respectively. The derived value for the nonlinear optical coefficient *d*_{33} is therefore lower than the literature value of 19.5 pm/V for SHG of 1313 nm light in congruent undoped LiNbO_{3} [31]. Also the linear optical absorption coefficients *α*_{1} and *α*_{2} are found to have relatively low values of 8.7 × 10^{−4} cm^{−1} and 6.0 × 10^{−4} cm^{−1}, respectively, compared to the literature values of 2 × 10^{−3} cm^{−1} for 1064 nm in LN [32] and 5 × 10^{−3} cm^{−1} for 671 nm in MgO:LN [33]. These discrepancies might be explained by several reasons. Firstly, our one-dimensional heat diffusion model does not comprise the heat propagation along the waveguide axis *x*, leading to an overestimation of the local rise in temperature inside the waveguide core due to optical absorption. This results in a systematic underestimation of the linear optical absorption coefficients, meaning that our model rather provides lower bounds for the actual values of these coefficients. Secondly, this underestimation of the linear optical absorption together with the neglect of other linear propagation losses has as a consequence the overestimation of the SHG efficiency in our model, which leads to a systematic underestimation of the nonlinear optical coefficient *d*_{33} when fitting the model to the experimental data. Thirdly, it was shown by [33] that the linear optical absorption coefficient of LiNbO_{3} significantly depends on the doping of the material. However, to our best knowledge, no experimental data has yet been published concerning the absorption properties of ZnO-doped LiNbO_{3}, meaning that the actual absorption coefficients might possibly be lower for ZnO:ppLN than the values found in literature.

Additionally the two-photon absorption coefficient *β*_{2Ph.} of 4.6 × 10^{−9} cm/W has to be applied, corresponding well to the literature value of 5 × 10^{−9} cm/W for 532 nm [25, 32]. In our case the intensity dependent SHIFA coefficient *α*_{SHIFA} [23] does not need to be taken into account in order to reproduce the experimental results. We attribute this to the fact that the intensities inside our waveguide remain well below 10 MW/cm^{2} even for the highest applied fundamental powers. The results of our simulation, assuming these parameter settings, are presented in Figs. 4 and 5 together with the experimental data points. Excellent agreement between our model and the measurements is observed.

## 5. Comparison to SHG in an enhancement cavity

An alternative method for efficient frequency doubling of infrared lasers is to use periodically-poled nonlinear crystals in an external enhancement cavity. Even though this approach is technically more complex than waveguide doubling, due to the requirement of a second cavity with frequency locking, we expect higher efficiencies than with the waveguide conversion module.

The doubling cavity setup has a similar design as the ones presented in [3, 34] and is illustrated in Fig. 9. As for the laser, a four-mirror folded ring-cavity is employed. The fundamental light is coupled through the plane mirror M′_{1} with reflectivity *R*_{c} = 90% optimizing the cavity impedance matching. All other mirrors are highly reflective at 1342 nm and M′_{4} has a transmission near 95% at 671 nm. M′_{2} (M′_{3}) is attached to a fast (slow) piezoelectric transducers, allowing to act on the cavity length in the 50 nm (8μm) range for the purpose of frequency-locking the cavity to the incoming fundamental light, as described in [3]. M′_{3} and M′_{4} are concave with a radius of curvature r_{c} = 150 mm and r_{c} = 100 mm, respectively. Their distance M′_{3}–M′_{4} is 156 mm. As nonlinear medium, a periodically-poled KTP bulk crystal (ppKTP) was chosen because of its high nonlinear coefficient d_{33} = 16.9 pm/V [35] and its elevated damage threshold which is higher than for LN. The crystal is placed between M′_{3}–M′_{4} in the corresponding focus with waist *w*_{0} ≃ 79 μm. The relatively large waist inside the crystal was chosen in order to reduce intensity-related detrimental effects such as two-photon and second-harmonic-induced absorption [36] and gray-tracking [37]. Mode matching between the laser output and the cavity is accomplished using a set of spherical lenses, yielding a mode coupling efficiency of 95%. The frequency doubled light is transmitted through M′_{4} and collimated to a 1/e^{2} beam radius of 1.2 mm.

After frequency-locking the cavity to the IR laser, the SH power *P*_{2ω} was measured as a function of the intra-cavity IR power *P _{ω}*, see Fig. 10(a). As expected, the conversion shows quadratic behavior and fitting yields a single-pass conversion efficiency of

*η*

_{sp}= 0.22 %/W. The SH output power follows a variation of the intra-cavity fundamental power without observable hysteresis. Moreover, no long-term degradation was observed, indicating the absence of gray-tracking, even for the highest reachable fundamental powers in our setup. The maximal SH output power

*P*

_{2ω}= 5.2 W is reached for a IR power of

*P*

_{in}= 5.9 W, corresponding to a doubling efficiency of

*η*

_{conv}= 87%, as can be seen in Fig. 10(b). The SH light at 671 nm derived from this source has already been successfully applied for implementing a new sub-Doppler laser cooling scheme for lithium atoms requiring Watt-class laser power, as described in [38].

In the following, theoretical arguments are provided which justify this higher doubling efficiency of the enhancement cavity approach compared to the single-pass waveguide setup described in section 3 and 4. In the context of second harmonic interaction of an electromagnetic wave passing through an optically nonlinear medium, the resulting conversion is in general a second harmonic output power *P*_{2ω} which, under the assumption of weak depletion of the fundamental light, obeys a quadratic dependence on the fundamental power *P _{ω}*,

*η*

_{SHG}is the conversion efficiency. For the particular case of second harmonic interaction of a focused Gaussian beam, the following expression for

*η*

_{SHG}can be derived [39],

*d*

_{eff}being the effective nonlinear coefficient of the material for a specific polarization,

*n*and

_{ω}*n*

_{2ω}the corresponding refractive indices of the material,

*L*the length of the nonlinear material,

*ε*

_{0}the vacuum permittivity and

*c*

_{0}the vacuum speed of light. The function

*h*(

*α*,

*β*) can be expressed as

*α*=

*L*/2

*z*

_{R}depends on the Gaussian beam Rayleigh length

*z*

_{R}which can be shown to be equal for both waves. The phase-matching parameter for a periodically poled sample with poling period Λ

*λ*being the fundamental vacuum wavelength, depends on the temperature. Entering values for the usual parameters of nonlinear media into these formulas yields doubling efficiencies

*η*

_{SHG}on the order of %/W, meaning that for the available continuous laser power of several Watts, single pass doubling by focusing a beam into a crystal cannot be considered as an efficient option. Therefore other approaches need to be employed. A first solution is the significant increase of the interaction length inside a waveguiding structure, see section 3 and 4. The second technique, presented in this section, consists in the resonant enhancement of the fundamental power inside a cavity. Taking into account the nonlinear conversion process, the fundamental-mode intra-cavity power

*P*at cavity resonance of the TEM

_{ω}_{00}mode is a solution of

*P*

_{in}is the fundamental pump power incident on the coupling mirror M′

_{1},

*η*

_{00}is the coupling efficiency to the the TEM

_{00}mode,

*α*

_{1}is the transmission loss of M′

_{1}and

*α*

_{tot}is the total passive round-trip loss of the cavity excluding the coupler transmission

*R*

_{c}and nonlinear conversion. The single-pass doubling efficiency

*η*

_{SHG}is calculated according to Eq. (13) with

*d*

_{eff}= 2/

*π*· 16.9 pm/V for ppKTP according to [35] and

*h*(

*α*) as given by the cavity design. The solution is calculated analytically using a computer algebra system, resulting in an expression which is not indicated here due to its length of several pages. The second-harmonic output power is then simply calculated according to Eq. (12), and we can define the actual cavity doubling efficiency as

*α*

_{tot}=

*α*

_{cr}= 1% and

*α*

_{1}= 0, the solution of Eq. (16) yields an optimal SH power output of 5.3 W at the maximum available pump power of

*P*

_{in}= 5.9 W. This optimal conversion is achieved for a coupling mirror reflectivity of

*R*

_{c}= 83%, yielding a total conversion efficiency of

*η*

_{cav}= 90%. For the chosen

*R*

_{c}= 90% coupler we would expect satisfying results over a large power range up to the maximum available pump power, where a conversion efficiency of

*η*

_{cav}= 84% should be achieved, this theoretical prediction being close to the measured maximum conversion efficiency of

*η*

_{conv}= 87%.

This result underlines the advantage of using resonant enhancement cavities for nonlinear bulk media with low single-pass conversion efficiency, since this technique allows to increase the overall conversion efficiency tremendously. Nevertheless, its implementation is technically more demanding than a single-pass approach, as appropriate locking schemes are required which furthermore introduce additional sources of noise and instability. This technical complication justifies the interest in single-pass waveguide schemes for wavelength conversion with high efficiency, since this approach implies a major simplification of SHG setups.

## 6. Conclusions

In conclusion, we have presented a simplified laser source, emitting record-level 2.4 W of output power from a waveguide SHG unit at 671 nm, with an internal doubling efficiency of 54%. It is based on a fundamental solid state laser emitting 6.6 W at 1342 nm. In spite of the elevated power levels, thermal effects are well described by the simple theory presented here, and well under control in the experiment. We do not observe evidence of second-harmonic induced fundamental light absorption. In our current setup, the output power is only limited by the available fundamental power. Furthermore, from a resonant SHG scheme, we obtain 5.2 W of output power at 671 nm with, however, an increased complexity of the setup.

## Funding

This work has been supported by the Région Ile de France (DIM IFRAF/NanoK) and the European Union (ERC Grants FERLODIM and ThermoDynaMix).

## Acknowledgment

The authors acknowledge Dr. Yoshiki Nishida from NTT Electronics for fruitful technical and scientific discussions.

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