Pavel Loiko

Ludovic Gauthier-Manuel

Gurvan Brasse

Alain Braud

Abdelmjid Benayad

Patrice Camy
email: patrice.camy@ensicaen.fr

Efficient wavelength-tunable deep-diced ridge waveguide lasers in bulk Yb 3+ :CaF 2 crystal

Keywords:

Channel (ridge) waveguides are produced in bulk 4.5 at.% Yb 3+ :CaF 2 crystals by "deep" diamond saw dicing. They possess a rectangular profile with high aspect ratio (a width of 10-50 μm and a depth of 200 μm) and good verticality. The r.m.s. surface roughness of the diced side walls is only 0.5 nm. The waveguide propagation losses are as low as 0.13 ± 0.05 dB/cm at ~1 μm (for the 40 μm wide waveguide and TM-polarization) and they slightly increase for narrower structures. A continuous-wave Yb 3+ :CaF 2 waveguide laser generated a maximum output power of 723 mW at 1029-1051 nm with a slope efficiency η of 76.0% (vs. the absorbed pump power), a laser threshold of 165 mW, and slightly multimode laser output (M 2 x = 1.05, M 2 y = 1.3). The waveguides exhibited lower propagation losses and better laser performance for TM polarized pump (as compared to TE polarized one). The laser emission was partially polarized. The waveguide laser allowed for continuous wavelength tuning between 1009 and 1072 nm (tuning range: 63 nm). The developed waveguides are promising for compact and integrated mode-locked sources at ~1 μm.

Introduction

In integrated optics, dicing is a process by which a die (a functional block of material) is separated from the wafer. One of the most common ways of dicing is mechanical sawing using a machine called a dicing saw. Such a machine uses a high rotation speed extremely thin diamond blade and produces cuts with high surface quality and well-controlled geometry. This technology was mainly applied to semiconductor wafers (e.g., Si or GaAs). In the recent years, there is a raising interest for application of this technology to micro-structuring of optical materials. One attractive application here is the fabrication of optical waveguides (WGs) [START_REF] Courjal | High aspect ratio lithium niobate ridge waveguides fabricated by optical grade dicing[END_REF][START_REF] Volk | Low loss ridge waveguides in lithium niobate thin films by optical grade diamond blade dicing[END_REF]. The advantages of diamond saw dicing for waveguide fabrication are the following: (i) well-controlled geometrical shape with μm-scale precision [START_REF] Courjal | High aspect ratio lithium niobate ridge waveguides fabricated by optical grade dicing[END_REF], (ii) small surface roughness of cuts [3] and, thus, low scattering losses and high optical quality of the surfaces [START_REF] Courjal | Low-loss LiNbO 3 tapered-ridge waveguides made by optical-grade dicing[END_REF], (iii) the availability of commercially available dicing saws and (iv) the variety of suitable materials such as thin films [START_REF] Kifle | Diamond saw dicing of thulium channel waveguide lasers in monoclinic crystalline films[END_REF], polymers [START_REF] Chen | Low loss nanostructured polymers for chip-scale waveguide amplifiers[END_REF], glasses [START_REF] Zhang | Planar and ridge waveguides in Yb 3+ -doped silicate glasses fabricated by proton implantation and precise diamond blade dicing[END_REF] and single-crystals [START_REF] Courjal | High aspect ratio lithium niobate ridge waveguides fabricated by optical grade dicing[END_REF][START_REF] Volk | Low loss ridge waveguides in lithium niobate thin films by optical grade diamond blade dicing[END_REF].

By diamond saw dicing of laser-active materials, one can produce active WGs for laser emission [START_REF] Jia | Ridge waveguide lasers in Nd:YAG crystals produced by combining swift heavy ion irradiation and precise diamond blade dicing[END_REF][START_REF] Waeselmann | Nd:sapphire channel waveguide laser[END_REF][START_REF] Loiko | Watt-level Tm:LiYF 4 channel waveguide laser produced by diamond saw dicing[END_REF][START_REF] Brüske | Efficient ridge waveguide amplifiers and lasers in Er-doped lithium niobate by optical grade dicing and three-side Er and Ti in-diffusion[END_REF][START_REF] Li | Monolithic waveguide laser mode-locked by embedded Ag nanoparticles operating at 1 μm[END_REF]. Such WGs having a well-defined transverse geometrical profile are classified as surface channel (ridge) ones. Surface WGs are of interest for bio-and environmental sensing applications as the intracavity mode can interact with the surrounding material (e.g., gas, liquid or deposited film) via evanescent-field coupling [START_REF] Khonina | Evanescent field ratio enhancement of a modified ridge waveguide structure for methane gas sensing application[END_REF][START_REF] Charrier | Evanescent wave optical micro-sensor based on chalcogenide glass[END_REF].

So far, laser-active surface WGs were produced by diamond-saw dicing in different materials. The general feature of these studied is that before the dicing, a waveguiding layer of material was formed (e.g., by liquid phase epitaxy (LPE) [START_REF] Loiko | Watt-level Tm:LiYF 4 channel waveguide laser produced by diamond saw dicing[END_REF], pulsed laser deposition (PLD) [START_REF] Waeselmann | Nd:sapphire channel waveguide laser[END_REF], swift heavy ion irradiation [START_REF] Jia | Ridge waveguide lasers in Nd:YAG crystals produced by combining swift heavy ion irradiation and precise diamond blade dicing[END_REF], ion in-diffusion [START_REF] Brüske | Efficient ridge waveguide amplifiers and lasers in Er-doped lithium niobate by optical grade dicing and three-side Er and Ti in-diffusion[END_REF] etc.), so that the dicing was used to change the laser geometry from planar to channel waveguide one. Waeselmann et al. patterned Nd 3+ :α-Al 2 O 3 (sapphire) thin films deposited by PLD; the channel waveguide laser with a cross-section of 41(x) × 2.6(y) μm 2 generated 322 mW at ~1.09 μm with a slope efficiency of ~12% corresponding to relatively high propagation losses of ~6 dB/cm [START_REF] Waeselmann | Nd:sapphire channel waveguide laser[END_REF]. Recently, Loiko et al. fabricated square 30(x) × 30(y) μm 2 waveguides in Tm 3+ :LiYF 4 LPE films yielding much better power transfer characteristics (an output power of 1.30 W at 1.88 μm with a slope efficiency of 80%) and lower propagation losses (0.28 dB/cm) [START_REF] Loiko | Watt-level Tm:LiYF 4 channel waveguide laser produced by diamond saw dicing[END_REF].

Another approach is to apply dicing to microstructuring of bulk crystals with a goal of fabricating surface WGs with high aspect ratios (depth/width, or y/x). This approach was referred as deep diamond saw dicing resulting in nearly inverted planar waveguide geometry. It was first applied to surface micro-structuring of bulk nonlinear crystals: Courjal et al. fabricated ridges in LiNbO 3 with high aspect ratios up to y/x = 500 (1(x) × 500(y) μm 2 ), good side-wall verticality and propagation losses down to 0.5 dB/cm (for a 6(x) × 35(y) μm 2 waveguide) [START_REF] Courjal | High aspect ratio lithium niobate ridge waveguides fabricated by optical grade dicing[END_REF]. Very recently, we applied this approach to bulk laser crystals using Tm 3+ :LiYF 4 as a reference material [START_REF] Loiko | Channel waveguide lasers in bulk Tm:LiYF 4 produced by deep diamond-saw dicing[END_REF]. Despite the relatively low propagation losses (0.1-0.4 dB/cm, depending on the aspect ratio), the laser performance of such WGs was affected by thermal effects (the laser operated in quasi-continuous-wave regime). Moreover, the localization of the laser mode was not fully clarified.

Among the laser host materials suitable for power-scalable applications, cubic calcium fluoride (CaF 2 ) is attracting a lot of attention [START_REF] Druon | On Yb:CaF 2 and Yb:SrF 2 : review of spectroscopic and thermal properties and their impact on femtosecond and high power laser performance[END_REF]. From the point of view of thermo-mechanical properties, it provides high thermal conductivity (~9.7 W m -1 K -1 ) [START_REF] Slack | Thermal conductivity of CaF 2 , MnF 2 , CoF 2 , and ZnF 2 crystals[END_REF], isotropic thermal expansion and weak and negative thermal lensing [START_REF] Druon | On Yb:CaF 2 and Yb:SrF 2 : review of spectroscopic and thermal properties and their impact on femtosecond and high power laser performance[END_REF]. CaF 2 can be easily doped with rare-earth ions such as trivalent ytterbium (Yb 3+ ) and it features strong ion clustering already at low doping levels (>0.1 at.%) leading to significant inhomogeneous broadening of spectral bands (a glassy-like behavior) [START_REF] Petit | Spectroscopy of Yb 3+ :CaF 2 : from isolated centers to clusters[END_REF]. The latter is attractive for broadband wavelength tuning [START_REF] Lucca | High-power tunable diode-pumped Yb 3+ :CaF 2 laser[END_REF][START_REF] Petit | CW and tunable laser operation of Yb 3+ doped CaF 2[END_REF] and especially generation [START_REF] Machinet | High-brightness fiber laser-pumped 68 fs-2.3 W Kerr-lens mode-locked Yb:CaF 2 oscillator[END_REF] and amplification [START_REF] Ricaud | Short-pulse and high-repetition-rate diode-pumped Yb:CaF 2 regenerative amplifier[END_REF] of ultrashort pulses, including high-power laser facilities [START_REF] Siebold | Terawatt diode-pumped Yb:CaF 2 laser[END_REF]. Thus, Yb 3+ :CaF 2 crystals appear as excellent candidates for power-scalable waveguide lasers.

In the present work, we aimed to demonstrate first deep diamond saw diced active WGs in bulk Yb 3+ :CaF 2 exhibiting highly-efficient and wavelength-tunable laser emission.

To date, there are very few reports on ytterbium calcium fluoride WGs. Petit et al. reported on fabrication of planar and ridge WGs in bulk Yb 3+ :CaF 2 by H + ion implantation without laser action [START_REF] Petit | Active waveguides produced in Yb 3+ :CaF 2 by H + implantation for laser applications[END_REF]. Ren et al. fabricated buried channel WGs in bulk Yb 3+ ,Na + :CaF 2 by ultrafast laser inscription (ULI) and achieved 174 mW at ~1.01 μm with an optical-to-optical efficiency of 45.3% [START_REF] Ren | Switchable single-dual-wavelength Yb,Na:CaF 2 waveguide lasers operating in continuous-wave and pulsed regimes[END_REF]. Molecular beam epitaxy (MBE) [START_REF] Daran | Er-Yb codoped CaF 2 thin films grown by molecular beam epitaxy[END_REF] and LPE [START_REF] Peña | Yb:CaF 2 grown by liquid phase epitaxy[END_REF] growth of Yb 3+ -doped CaF 2 films was reported and laser operation in planar waveguide geometry was achieved: Loiko et al. obtained 114 mW at 1.04 nm with a slope efficiency of 12.9% [START_REF] Loiko | Ytterbium calcium fluoride waveguide laser[END_REF]. However, these studies have not fully exploited the power scaling capabilities of calcium fluoride and intrinsically high slope efficiencies of Yb 3+ laser scheme based on resonant pumping.

Fabrication and characterization of waveguides

Crystal growth

As a laser gain material, we used bulk ytterbium-doped calcium fluoride (Yb 3+ :CaF 2 ) crystal. The 4.5 at.% Yb 3+ :CaF 2 crystal (in the crystal, Yb 3+ ion density: N Yb = 11.03 × 10 20 cm -3 ) was grown using a conventional Bridgman technique with a homemade furnace. A mixture of high purity (4 N) CaF 2 and YbF 3 powders was introduced into a graphite crucible. A good vacuum (<10 -5 mbar) was realized before refilling the growth chamber with a mixture of Ar and CF 4 gases to avoid the formation of oxyfluoride phase. The furnace temperature was set to about 50 K higher than the melting point of CaF 2 . After soaking, the charge was thoroughly melted, and the crystal growth was carried out with a rate of 2-4 mm/h. When the growth was completed, the crystal was cooled down to room temperature (RT, 293 K) within 48 h.

Deep diamond saw dicing

For diamond saw dicing, a rectangular sample was cut from the Yb 3+ :CaF 2 crystal. It was oriented for light propagation along the [111] crystallographic direction employing a Laue diffractometer. The sample thickness t was 6.0 mm. The input and output surfaces, as well as the top surface of the sample were polished to laser quality with good parallelism. They remained uncoated.

The top surface of the sample was subjected to precision diamond saw dicing. For processing calcium fluoride, we developed a specific process to ensure good surface quality and minimize the stresses by reducing revolution speeds of the blades and diamond sizes. These parameters were optimized in a set of test experiments to avoid edge chipping. We have also developed a calibration process to control the ridge geometry and thickness. The selected dicing parameters ensured gentle polishing of the ridge side walls leading to their small roughness. The dicing depth was 200 μm and the ridges with the widths of 10-50 μm (with a step of 10 μm) were produced, resulting in the aspect ratios y/x in the range 4-20. The ridges were separated by a distance of 500 μm.

No additional polishing was applied to the input and output facets of the sample.

Confocal microscopy

The geometry and surface morphology of the fabricated WGs were studied using a confocal laser microscope (Sensofar S-neox) equipped with a set of microscope objectives (10 × , 20 × and 100 × ) and a blue light-emitting diode (LED, λ = 405 nm). It operated in the bright-field regime and reflection mode.

First, we observed one of the end-facets with different magnifications, Fig. 1(a-d). In Fig. 1(a), one can see several ridges of different width (10, 20 and 30 μm), as well as part of the top sample surface which remained undiced. There are no cracks propagating in the bulk crystal volume (the dark lines are due to small surface imperfections originating from the polishing process; they are highlighted because of the reflection mode of observation which is very sensitive to the surface quality). In Fig. 1(b), one ridge with an intermediate width (30 μm) is shown. One can see that there is no chipping of the end-facet which may appear when the revolution speed of the blade is too high. In Fig. 1(c), we observed the upper part of the same ridge having almost perfect rectangular profile with vertical side walls. However, in the bottom part of the waveguide, Fig. 1(d), a certain rounding is observed. It is slightly asymmetric at two opposite side walls. This shape is typical for diamond saw dicing, as the blade quality slowly decreases with time resulting in rounded bottom. No μm-range dicing debris are observed (some sub-μm-sized particles are visualized as bright points in Fig. 1(c andd)). Furthermore, we observed the top surface of the sample near one of the end-facets, see Fig. 1(d). The WGs have a straight geometry with no cracks nor chipping along all their length. The ridges propagated through the entire length of the sample. A close view on one of the WGs is shown in Fig. 1(e). When observing the sample from the top, the rectangular ridge geometry is well confirmed.

To conclude about the roughness of different surfaces of fabricated ridges, we employed the confocal mode. For this, the ridge with a width of 40 μm was used. First, we looked at the top surface of the ridge, Fig. 2(a). Its quality is determined by polishing prior to the dicing process. The root mean square (r.m.s.) roughness of this surface in the longitudinal direction (along the waveguide) is 1.2 nm, Fig. 3(a).

Then, we observed one of the side walls (its surface quality is determined by soft polishing during the dicing process), Fig. 2(b). The corresponding r.m.s. roughness along the waveguide is even further decreased with respect to the top surface, down to 0.5 nm, Fig. 3(b). This indicates the role of dicing in maintaining the good quality of the side walls which is relevant for reducing the waveguide propagation losses. Another possibility to observe the same surface is along the vertical direction. Within upper 3/4 (150 μm) of the ridge height, very good verticality is preserved, as the maximum deviation does not exceed 15 nm, Fig. 3(c). In the lower part (1/4) of the ridge, the above mentioned rounding appears.

We have also observed the diced (bottom) area between two adjacent WGs, Fig. 2(c). It has a characteristic lined morphology with the direction of lines being parallel to the dicing direction. While the r.m.s. surface roughness studied across these lines, Fig. 3(d), is increased to 40 nm, it is still relatively low. Note that the guided light does not interact with this surface.

Finally, we have observed one of the end-facets of the waveguide, Fig. 2(d). Its quality is again determined by pre-polishing before the dicing process. The corresponding roughness plot is shown in Fig. 3(e) corresponding to a r.m.s. devitation of 4.5 nm. In addition, a certain rounding (variation of roughness up to 20 nm) was observed near the top surface of the ridge and it originates from the polishing of the bulk crystal. As expected, Fig. 2(d) also reveals the above mentioned asymmetry of the waveguide base which is characteristic of the dicing process.

Waveguiding properties

For studying the waveguiding properties and optical losses of the fabricated WGs, we used a CW Ti:Sapphire laser (3900 S, Spectra Physics) emitting in the fundamental mode (TEM 00 , M 2 ≈ 1) with a linear polarization. Its output was coupled into the WGs using a spherical uncoated CaF 2 lens (focal length, f = 40 mm) resulting in a pump spot 2w P of 27 ± 5 μm. The transmitted light was collimated using another spherical CaF 2 lens (f = 15 mm) and the profile of the guided mode at the output facet was captured using a CCD camera (BladeCam-XHR, DataRay Inc.). The scale calibration was performed by illuminating the WGs with known dimensions using a near-IR light source (LED) placed before the focusing lens.

The near-field profiles of the guided pump modes at the wavelength of λ p = 979.5 nm are shown in Fig. 4. This study was performed for the vertical laser polarization. The coupled light is well confined within the ridges with almost negligible leakage into the bulk crystal volume. The pump mode is clearly multimode for all the WGs: for the 50 μm wide ridge, it is TM 23 and for the 10 μm one, it is TM 01 . There is a small asymmetry of the mode in the bottom part of the waveguide which matches the rounding revealed using the confocal microscopy. For the performed experiment, the Rayleigh length for the laser beam in the crystal z R = (πw P 2 n)/(λ p M 2 ) is 0.84 mm, so that in the conditions of free propagation, the beam diameter at the output facet of the sample would be 2w ~200 μm (along the vertical direction), i.e., filling the whole volume of the ridge.

U N C O R R E C T E D P R O O F
To study the waveguide propagation losses, the Ti:Sapphire laser was tuned out of the main absorption peak of Yb 3+ ions, to λ p = 912.0 nm. The intensity of the scattered pump along the waveguide was observed from its top surface using a CCD-camera and a microscope objective, as shown in Fig. 5. The experiments were performed for two pump polarizations, vertical and horizontal. The incident power was set to be very low (<5 mW) to avoid absorption saturation, i.e., the experiment was performed in the small-signal regime. The on-axis intensity of the pump radiation was fitted using the formula I(z) = I 0 × exp(-δ tot z), where I 0 is the intensity near the input facet of the waveguide, δ tot = α abs + δ loss is the total loss coefficient, α abs = σ P abs N Yb = 1.07 cm -1 is the small-signal absorption coefficient at the pump wavelength, where σ P abs = 0.97 × 10 -21 cm 2 , δ loss is the coefficient of passive losses and z is the axial coordinate. The best-fits are shown in Fig. 5 and the obtained δ loss values expressed in dB/cm are summarized in Table 1.

The propagation losses tend to increase for narrower WGs for both laser polarizations, e.g., from 0.17 dB/cm for 50 μm wide waveguide up to 0.52 dB/cm for the 10 μm wide one (for the vertical polarization). We assign this to two reasons: (i) stronger mode overlap with the side walls for smaller WGs and (ii) higher surface roughness and worse parallelism for the side walls of the smaller WGs. The latter effect can originate from stronger vibrations of the blade which is in contact with smaller volume of the material. Second, the propagation losses are almost two times higher for horizontally polarized light, e.g., δ loss = 0.34 dB/cm for the 50 μm wide waveguide and 0.91 dB/cm for the 10 μm one. Note that cubic calcium fluoride is an optically isotropic material. The observed polarization anisotropy of scattering losses for guided light may originate from the dicing process itself as it creates slightly asymmetric ridge profile.

The pump coupling efficiency η coupl = P coupl /P inc was determined in a separate set of experiments at 892.0 nm (out of the Yb 3+ absorption) by monitoring the power at the output facet of the waveguide. As expected from the geometrical overlap of the pump beam and the waveguide cross-section, η coupl decreased for smaller widths of the WGs, from 77.9% for the 50 μm wide waveguide down to 43.0% for the 10 μm wide one. Note that these values include Fresnel loss at the uncoated input facet of the ridge (T Fr = 96.1%, calculated using the refractive index n = 1.4927 [START_REF] Malitson | A redetermination of some optical properties of calcium fluoride[END_REF]).

Laser operation

Laser set-up

The scheme of the laser set-up is shown in Fig. 6(a). The Yb 3+ :CaF 2 crystalline sample with the diced WGs was mounted on a passively-cooled Cu-holder using a silver paint to improve the heat dissipation. The sample was placed in a linear plano-plano cavity formed by a flat pump mirror (PM) coated for high transmission (HT, T > 97%) at 0.88-0.99 μm and for high reflection (HR, R > 99.9%) at 1.02-1.20 μm, and a set of flat output couplers (OCs) having a transmission in the range of 2%-50% (±0.5%) at 1.0-1.1 μm. Both the PM and the OCs were placed close to the sample end-facets with minimum air gaps; no index-matching liquid was used to avoid the optical damage. The geometrical cavity length was ~6.0 mm. As a pump source, we employed the CW Ti:Sapphire laser delivering up to ~2.0 W at 979.5 nm corresponding to the fundamental mode (M 2 ≈ 1) and linear laser polarization. The incident pump power was varied by a rotatable half-wave plate and a Glan-Taylor polarizer setting the pump polarization to be vertical or horizontal. The pump was focused into the WGs using a spherical uncoated CaF 2 lens (f = 40 mm, T = 93.9% at ~0.98 μm) through the PM. The measured pump spot size in the focus 2w P was 27 ± 5 μm. The pumping was in a single-pass. The pump beam was focused near the top of the ridge waveguides. The pump absorption at the threshold pump power was determined from pump-transmission measurements under non-lasing conditions. The laser output was collimated using another spherical CaF 2 lens (f = 15 mm). The residual pump was filtered out using a cut-off filter (FEL1000, Thorlabs). The profiles of the pump and laser modes in the near field were captured with the CCD camera. The scale calibration was performed as described above. The laser spectra were mea sured using an optical spectrum analyzer (OSA, Ando AQ6315-E) with a spectral resolution of 0.1 nm.

A photograph of the waveguide laser is shown in Fig. 6(b). The waveguides exhibited weak green upconversion luminescence owing to impurity Er 3+ ions, Fig. 6(c).

CW laser performance

At first, we studied the laser performance of the intermediate (40 μm wide) waveguide for the optimum pump wavelength λ p = 979.5 nm (corresponding to zero-phonon-line (ZPL) of Yb 3+ ions in CaF 2 ). The laser generated a maximum output power of 723 mW at 1029-1051 nm (broad spectrum) with a slope efficiency η of 76.0% (vs. the absorbed pump power P abs ) and a laser threshold of P th = 165 mW (for T OC = 20%), Fig. 7(a). The maximum optical-to-optical efficiency vs. the pump power incident on the crystal η opt was 38.9%. With increasing the output coupling, the laser threshold gradually increased, from P th = 52 mW (T OC = 1%) to 394 mW (T OC = 50%). No thermal roll-over was observed up to the maximum available pump power. No fracture of the waveguide was observed as well. No lasing without the OC (i.e., relying on the Fresnel losses at the output facet) was observed probably due to the low refractive index of CaF 2 (n = 1.429 at ~1.04 μm [START_REF] Malitson | A redetermination of some optical properties of calcium fluoride[END_REF]) leading to an equivalent T OC as high as 96.9%. The pump polarization in these studies was set to be vertical.

No lasing was achieved when pumping into the bulk area of the sample. This is due to the negative thermal lens in Yb 3+ :CaF 2 [START_REF] Druon | On Yb:CaF 2 and Yb:SrF 2 : review of spectroscopic and thermal properties and their impact on femtosecond and high power laser performance[END_REF] which drives a plano-plano cavity to an instability region. This observation confirms the waveguide geometry of the laser. On the other hand, negative thermal lens may promote the mode leakage into the substrate contributing to the total passive losses. However, this effect is expected to be diminished by reabsorption in unpumped regions of the waveguide.

The typical spectra of laser emission from the waveguide laser are shown in Fig. 7(b). The spectra are broad due to the smooth and broad gain spectra of Yb 3+ ions in CaF 2 and contains multiple lines originating from the Fabry-Perot (etalon) effects at the multiple crystal/air and mirror/air interfaces. With increasing the output coupling, the central emission wavelength experienced a blue-shift, from ~1048 nm (T OC = 1%) to ~1029 nm (T OC = 50%) due to the quasi-three-level nature of the 2 F 5/2 → 2 F 7/2 transition.

The Caird analysis [START_REF] Caird | Quantum electronic properties of the Na 3 Ga 2 Li 3 F 12 :Cr 3+ laser[END_REF] was applied, Fig. 7(c). The inverse of the slope efficiency, 1/η, was plotted vs. the inverse of the output-coupling loss, 1/γ OC , and the experimental data were fitted using the formula: [START_REF] Courjal | High aspect ratio lithium niobate ridge waveguides fabricated by optical grade dicing[END_REF] where γ OC and γ (the loss coefficient) are given by:

(2a) (2b)
where η 0 is the intrinsic slope efficiency and L is the single-pass passive loss. The use of Eq. ( 1) yields η 0 = 75.0 ± 4%. The waveguide propagation losses are then calculated from L:

(3) leading to the value of 0.13 ± 0.05 dB/cm at ~1.04 μm. This value agrees with that determined from the pump-transmission measurements (0.21 ± 0.06 dB/cm at ~0.91 μm) considering longer laser wavelength.

The determined δ loss is smaller than the value corresponding to Fresnel losses at the crystal/air interfaces. Indeed, for a single pass and two reflections at the interfaces, L Fr = 6.1% and the equivalent δ loss,Fr = 0.44 dB/cm. This is due to the fact that the input and output faces of the crystal were polished with a good parallelism (value) so that the crystal itself acted as an etalon, so that the generated longitudinal modes experienced much lower losses.

The measured single-pass pump absorption, η abs,NL , as a function of the incident pump power is plotted in Fig. 7(d). It gradually increased from the small-signal value, η abs0 : [START_REF] Courjal | Low-loss LiNbO 3 tapered-ridge waveguides made by optical-grade dicing[END_REF] where σ p abs is the absorption cross-section at the pump wavelength (0.54 × 10 -20 cm 2 at 979.5 nm) indicating the effect of ground-state bleaching. In our case, according to Eq. ( 4), η abs0 = 97.2%. With increasing the output coupling, due to increased inversion in the waveguide, the pump absorption at the threshold pump power η abs,th dropped from 92.7% (T OC = 1%) to 68.2% (T OC = 50%).

The laser excitation curve (the plot of the output power vs. the pump wavelength) was measured for the 40 μm wide waveguide ensuring a constant value of the incident pump power, as shown in Fig. 8(a). Laser operation was achieved for λ p in the range of 908.9-996.3 nm. The laser excitation curve well matched with the absorption cross-section spectrum for the 2 F 7/2 → 2 F 5/2 Yb 3+ transition in CaF 2 . The emission linewidth for the Ti:Sapphire laser was <0.1 nm. We also tried pumping of the waveguide into another absorption peak of Yb 3+ ions at 922.0 which is much broader, Fig. 8(b). In this case, the laser generated a maximum output power of only 521 mW at 1019-1038 nm with a maximum η opt = 22.5%. Moreover, a thermal roll-over was observed for P inc > 1.7 W. Such deterioration of the laser performance with respect to pumping at 979.5 nm is mainly due to higher fractional heat loading: [START_REF] Kifle | Diamond saw dicing of thulium channel waveguide lasers in monoclinic crystalline films[END_REF] where η St,L is the Stokes efficiency under laser conditions and λ L is the laser wavelength. Indeed, η h = 10.7% and 5.2% for pumping at the short-wavelength part of the absorption spectrum and at the ZPL, respectively. We have directly compared the laser performance of all WGs (widths: 10-50 μm), see Fig. 9(a). In general, for narrower WGs, the pump coupling efficiency decreased while the ground-state bleaching increased due to the higher pump intensity. Both effects limited the power scaling capabilities. Indeed, for the same output coupling (T OC = 20%), η abs,th dropped from 84.3% (50 μm waveguide) to 76.7% (10 μm one). The performance of the 40 μm and 50 μm waveguides was similar. For the intermediate size waveguide (30 μm), the laser operated with even higher slope efficiency η = 79.5% and reduced laser threshold (P th = 98 mW) while the power scaling was limited to 635 mW at 1039-1051 nm as explained above. For even smaller structures (10-20 μm), the laser performance gradually deteriorated and the threshold was nearly unchanged. Moreover, for the 10 μm waveguide, a thermal roll-over was observed. There are two main reasons for this behavior: (i) higher propagation losses for smaller ridges, cf. Table 1, and (ii) worse thermal management of thin waveguides which provide almost no heat flow in the vertical direction.

The spectra of laser emission for WGs with different width are compared in Fig. 9(b). They are similar except of the smallest structures (10-20 μm) for which the central wavelength experienced a slight blue-shift to ~1034 nm due to increased passive losses.

The near-field mode profiles for all the studied WGs are shown in Fig. 10(a). The laser modes are localized near the top part of the waveguide. With decreasing the waveguide width, the mode ellipticity: along the vertical direction gradually increases, i.e., from e = 0.41 for 50 μm wide waveguide to e = 0.88 for 10 μm wide one. For larger waveguides (40-50 μm), the mode was well localized within the ridge (e.g., the mode size 2w Lx = 38 μm for the 50 μm wide waveguide) and for smaller ones (10-20 μm), the mode overlap with the side walls of the ridge increased (as indicated by the mode size 2w Lx = 12 μm for the 10 μm wide waveguide). These changes in the mode profile may explain higher propagation losses in the latter case.

The typical 1D intensity profiles of the laser mode from the 40 μm wide waveguide measured in the near-field are shown in Fig. 11(a). In the horizontal direction, the profile is well fitted with a Gaussian function (goodness of the fit: R 2 > 0.99) and the mode size 2w Lx = 26 μm.

In the vertical direction, the beam is slightly multimode with 2w Ly = 62 μm (e = 0.58). The beam quality factors were measured in the far field by the ISO-standard method [START_REF]Lasers and Laser-Related Equipment -Test Methods for Laser Beam Widths, Divergence Angles and Beam Propagation Ratios -Part 1: Stigmatic and Simple Astigmatic Beams[END_REF] using an uncoated spherical CaF 2 lens (f = 75 mm) placed after the output coupler, see Fig. to the pump polarization E p which was fixed to be vertical) varied. This is revealed in Fig. 12 where the power transmitted through a rotating Glan-Taylor polarizer is plotted as a function of its rotation angle θ. The angle θ = 0 corresponds to horizontal orientation of the polarizer. The curves are well fitted using the formula: [START_REF] Zhang | Planar and ridge waveguides in Yb 3+ -doped silicate glasses fabricated by proton implantation and precise diamond blade dicing[END_REF] suggesting partially polarized light. The polarization degree is then:

For example, for the 20 μm wide waveguide, P = 96% and ϕ = 5°( high polarization degree, almost vertical linear polarization well matching the pump polarization state) and for the 50 μm one, P = 92% and ϕ = 53°(lower polarization degree, stronger rotation of the preferential orientation of the electric field for laser emission with respect to the pump polarization). One possible explanation for this tendency can be the observed anisotropy of waveguide propagation losses for TM and TE polarizations, Table 1. As this anisotropy becomes more significant for narrower waveguides, predominantly vertically polarized laser emission is generated.

To further reveal the anisotropic polarization response of the fabricated ridges, we studied the laser performance of the same waveguide (40 μm wide) under two different pump polarizations (vertical and horizontal). In the latter case, the Yb 3+ :CaF 2 waveguide laser generated an output power of 457 mW at 1033-1044 nm with η of only 52.9% and P th of 238 mW, as shown in Fig. 13(a andb). The observed lower slope efficiency and increased laser threshold (as compared to the case of vertical pump polarization) are in line with the higher propagation losses for TE pumping, see Table 1. The corresponding pump and laser modes are shown in Fig. 13(c-f). For the horizontal pump polarization, the leakage of the pump mode into the bulk area of the sample is more pronounced. The laser mode also exhibits stronger ellipticity in the vertical direction.

Tunable operation

To demonstrate wavelength tuning in the developed waveguide laser, the flat OC was replaced by a concave one (radius of curvature, RoC, of -100 mm) with a similar coating (T OC = 2 ± 0.5% at 1.0-1.1 μm) placed at ~94 mm apart from the output facet of the sample (a nearly hemispherical cavity design). As a tuning element, we used an uncoated quartz birefringent plate BIR1020, Newlight Photonics) with a diameter of 1″ and a thickness of 2.0 mm. It was a-cut with the optical axis (c-axis) lying in the plane of the plate. The plate was placed at the Brewster angle near the output facet of the sample. The pump polarization was vertical and the pump wavelength λ p was 979.5 nm.

Prior to the tuning experiment, we studied the laser operation in the free-running regime (when removing the filter from the cavity). The laser generated a maximum output power of 194 mW at 1036-1057 nm with a slope efficiency of 23.1% and a laser threshold of 150 mW. The laser performance slightly deteriorated as compared to the plano-plano cavity with the same OC, cf. Fig. 7(a), namely 329 mW at 1038-1059 nm with η = 27.9% and P th = 55 mW, due to higher intracavity losses related to the back-coupling efficiency at the output facet of the ridge.

After inserting the birefringent plate, a continuous wavelength tuning between 1009.2 and 1072.4 nm was demonstrated, as shown in Fig. 14(a). This corresponded to a tuning range of ~63 nm. The maximum of the tuning curve was observed at 1052 nm for an output power of 118 mW. This value is lower than that for the free-running operation because of the insertion loss of the BRF. One may also argue that the

Discussion

The results on Yb 3+ -doped calcium fluoride waveguide lasers achieved so far are summarized in Table 2 focusing in particular on the output power, slope efficiency and optical-to-optical efficiency. Two methods were applied previously to fabrication of laser-active WGs, namely, ULI also using bulk doped crystals (as in the present work) and LPE growth of thin single-crystalline layers on bulk undoped substrates. In the present work, we report on the highest output power and slope efficiency (achieved with high output coupling, T OC = 20%) and the smallest laser threshold (for small T OC = 2%) as compared to the previous works. We also report on the relatively low propagation losses compared to those for the LPE method. In terms of the overall optical-to-optical efficiency vs. the input pump power, our work still leaves a space for further improvement. We believe that η opt may exceed 50% via optimization of the pump coupling (which can be potentially limited only by the Fresnel losses considering large cross-sectional area of the waveguides and high refractive index contrast at the crystal/air interfaces) and using longer waveguides (>1 cm) and/or higher Yb 3+ doping levels ensuring nearly complete pump absorption.

The tuning range achieved for the Yb 3+ :CaF 2 waveguide laser (~63 nm) is comparable with the best results obtained with the corresponding bulk crystals. Petit et al. reported on a Ti:Sapphire pumped bulk 1.7 at.% Yb 3+ :CaF 2 laser with a Lyot filter which was continuously tunable between ~1000 and 1060 nm [START_REF] Petit | CW and tunable laser operation of Yb 3+ doped CaF 2[END_REF]. Lucca et al. demonstrated tuning of a diode-pumped 5 at.% Yb 3+ :CaF 2 laser between 1017.8 and 1072.1 nm (tuning range: 54 nm) with a power maximum at ~1052 nm [START_REF] Lucca | High-power tunable diode-pumped Yb 3+ :CaF 2 laser[END_REF] which agrees with the present work.

Recently, we applied the same concept of deep diamond saw dicing for Tm 3+ :LiYF 4 crystal [START_REF] Loiko | Channel waveguide lasers in bulk Tm:LiYF 4 produced by deep diamond-saw dicing[END_REF]. The following mechanisms of mode guiding in the fabricated ridges were suggested: (i) high refractive index contrast at three crystal-air interfaces, (ii) reabsorption losses in the unpumped (non-bleached) regions of the waveguide, and (iii) possible variation of the refractive index due to the stresses induced by dicing via the photo-elastic effect. Note that tetragonal LiYF 4 is optically anisotropic and exhibits natural birefringence. Indeed, the polarization state in the Tm 3+ :LiYF 4 waveguide laser was determined by the gain anisotropy (vertical, π). The inherent optical anisotropy of LiYF 4 may promote the waveguiding properties of diamond saw diced ridges. Here, we prove that deep diamond saw dicing can be also applied to fabrication of ridge WGs in optically isotropic (fluoride) materials. We also observe additional indications of anisotropic stress fields in the microstructure sample (i.e., the polarization-sensitive response of the waveguides) which can be promoted by slightly asymmetric shape of the bottom part of the ridges.

One may argue that the observed lasing is related to the bulk crystal geometry (i.e., not the waveguide one). When focusing the pump into the bulk (non-diced) area of the sample, no laser action was observed. This is because Yb 3+ :CaF 2 exhibits a negative (defocusing) thermal lens due to the negative dn/dT coefficient, -11.3 × 10 -6 K -1 [START_REF] Druon | On Yb:CaF 2 and Yb:SrF 2 : review of spectroscopic and thermal properties and their impact on femtosecond and high power laser performance[END_REF]. Negative thermal lens makes the plano-plano cavity unstable. As for the waveguide laser experiments, negative thermal lens seems to be weaker than the index-guiding mechanism. However, it may contribute to the mode leakage into the bulk area of the sample.

As a final comment, in the previous work with Tm 3+ :LiYF 4 [START_REF] Loiko | Channel waveguide lasers in bulk Tm:LiYF 4 produced by deep diamond-saw dicing[END_REF], we detected significant thermal effects preventing power scaling. In the present work, the waveguide laser operated in true CW regime and no thermal roll-over was observed up to at least P inc ~1.2 W (when pumping into the zero-phonon-line absorption peak of Yb 3+ ions and avoiding the use of very narrow waveguides with the widths below 20 μm). There are two main reasons for that. First, Tm 3+ laser scheme provides higher heat loading (η h ~28% for Tm 3+ :LiYF 4 [START_REF] Schellhorn | High-power diode-pumped Tm: YLF laser[END_REF]) as compared to the in-band pumped Yb 3+ lasers (η h = 5.2%, see above). Second, the thermal conductivity of CaF 2 is higher [START_REF] Slack | Thermal conductivity of CaF 2 , MnF 2 , CoF 2 , and ZnF 2 crystals[END_REF].

In the present study, the power scaling of the Yb 3+ :CaF 2 waveguide laser was limited by the available pump (the Ti:Sapphire laser). We believe that further power scaling reaching multi-watt output in true continuous-wave regime is feasible when using other high-brightness pump sources, e.g., fiber lasers, or even fiber-coupled InGaAs laser diodes. In the latter case, broader ridges (50 μm) would be preferable. The main limitation for power scaling in CW regime would be the thermal roll-over due to the limited heat removal from the ridge (which is possible only along the vertical direction, towards the bulk/undiced area) but not the damage of the waveguide input faces. The excessive heating of the ridge may induce additional losses owing to the mode leakage induced by the negative thermal lens. We suppose that it can be overcame to a certain extent by active cooling of the sample.

Conclusions

To conclude, deep diamond saw dicing (featuring high height-to-width aspect ratios of the microstructures) is a powerful approach to fabricate surface (ridge) waveguides in bulk ytterbium-doped calcium fluoride crystals for laser applications. Such waveguides ensure relatively low propagation losses (down to 0.13 ± 0.05 dB/cm at ~1 μm), small surface roughness of the side walls (about 0.5 nm), good verticality of the ridges (within 15 nm for the upper 3/4 part of the ridge) and the lack of cracking/chipping, as well as dicing debries. No post-processing polishing is required for such microstructures.

The combination of good thermal properties of CaF 2 with the broadband emission of Yb 3+ ions which are subjected to strong ion clustering determine the power scaling capabilities of deep diced Yb 3+ :CaF 2 waveguide lasers, as well as the possibility of broadband wavelength tuning. In the present work, using deep-diced Yb 3+ :CaF 2 ridge WGs, we achieved the output powers up to 723 mW at ~1.04 μm with a slope efficiency of 76.0% (for high output-coupling) and the laser thresholds as low as 55 mW (for small output coupling). These parameters represent the record-high values for ytterbium calcium fluoride waveguide lasers. Moreover, we demonstrated continuous wavelength tuning of the waveguide laser between 1009.2 and 1072.4 nm which is comparable with the best results for the corresponding bulk crystals.

The nature of waveguiding in deep-diced ridges has to be further clarified. In the present work with the cubic (optically isotropic) CaF 2

Fig. 1 .

 1 Fig. 1. Confocal laser microscopy study of deep-diced ridge waveguides in bulk 4.5 at.% Yb 3+ :CaF 2 : (a-d) end-facet view, microscope objective: (a) 10 × , (b) 20 × , (c,d) 100 × , (e,f) top view, microscope objective: (e) 10 × , (f) 100 × . Bright field, reflection mode, λ = 405 nm.

Fig. 2 .

 2 Fig. 2. Morphology of different surfaces of the Yb 3+ :CaF 2 crystal sample containing deep-diced ridge WGs studied with a confocal microscope in the interferometric mode: (a) top surface of the guide (pre-polished); (b) one of the waveguide side walls (diced); (c) diced area between two adjacent guides; (d) input facet (pre-polished). WG width: 40 μm.

UFig. 3 .

 3 Fig. 3. Roughness profiles for different surfaces of the Yb 3+ :CaF 2 crystal sample containing deep-diced surface WGs: (a) top surface of the guide, observed along the dicing direction (pre-polished); (b,c) one of the waveguide side walls (diced): (b) observed along the dicing direction, (c) observed in the transverse direction; (d) diced area between two adjacent guides, observed in the transverse direction (diced); (e) input facet of the guide, observed in the vertical direction (pre-polished). WG width: 40 μm, r.m.s. -root mean square value.

UFig. 4 .

 4 Fig. 4. Near-field profiles of the guided pump modes for the deep-diced Yb 3+ :CaF 2 ridge waveguides: WG width is (a) 50 μm, (b) 40 μm, (c) 30 μm, (d) 20 μm and (e) 10 μm. Pump polarization: vertical, λ p = 979.5 nm. White lines marking the air/crystal interfaces are drawn as guides for eyes.

UFig. 5 .

 5 Fig. 5. Evaluation of the waveguide propagation losses δ loss (at λ = 912 nm) in deep-diced ridge WGs in Yb 3+ :CaF 2 : the pump polarization is (a) vertical and (b) horizontal. Symbols experimental data on light intensities along the guides, linestheir exponential fits considering the total loss (δ loss + δ abs ).

Fig. 6 .

 6 Fig. 6. Deep-diced Yb 3+ :CaF 2 ridge waveguide laser: (a) scheme of the laser, λ/2half-wave plate, P -Glan-Taylor polarizer, PMpump mirror, OCoutput coupler, FL and CLfocusing and collimating lenses, respectively, F -cut-off filter; (b) photograph of the laser; (c) top view photograph of the pumped guide.

UFig. 7 .

 7 Fig. 7. Laser performance of the deep-diced Yb 3+ :CaF 2 waveguide (width: 40 μm): (a) input-output dependences, η -slope efficiency; (b) typical spectra of laser emission measured at maximum P abs ; (c) Caird analysis of propagation losses: a plot of inverse of the slope efficiency, 1/η, vs. inverse of the output-coupling loss, 1/γ OC ; (d) measured pump absorption under non-lasing conditions (single-pass), η abs,NL . The small-signal absorption, η abs,0 , is given for comparison. Pump polarization: vertical, λ p = 979.5 nm.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Effect of the pump wavelength on the performance of the deep-diced Yb 3+ :CaF 2 waveguide laser (width: 40 μm): (a) laser excitation curve (output power vs. the pump wavelength), P inc = 0.58 W, the absorption cross-section, σ abs , is shown for comparison, vertical arrows indicate two selected pump wavelengths; (b) input-output dependences for λ p = 979.5 nm and 922.0 nm. Pump polarization: vertical, T OC = 20%.

UFig. 10 .

 10 Fig. 10. Near-field profiles of the laser modes for the deep-diced Yb 3+ :CaF 2 ridge waveguides: WG width is (a) 50 μm, (b) 40 μm, (c) 30 μm, (d) 20 μm and (e) 10 μm. Pump polarization: vertical, λ p = 979.5 nm, P abs ~0.5 W, T OC = 20%. White lines marking the air/crystal interfaces are drawn as guides for eyes.

Fig. 11 .

 11 Fig. 11. Beam quality analysis for the deep-diced Yb 3+ :CaF 2 waveguide laser (WG width: 40 μm): (a) 1D intensity profiles of the laser mode in the near-field: symbolsexperimental data, curves -Gaussian fits; (b) evaluation of the beam quality factors M 2 x,y in the far-field, T OC = 20%, P abs ~0.5 W. Pump polarization: vertical.horizontal polarization of laser emission determined by the Brewster-oriented BRF leads to slightly higher propagation losses. In the same figure, we show the gain spectra for Yb 3+ ions in CaF 2 . Yb 3+ ions represent a quasi-three-level laser scheme with reabsorption, so that the gain cross-se[START_REF] Waeselmann | Nd:sapphire channel waveguide laser[END_REF] are calculated to conclude about the laser wavelength. Here, β = N 2 ( 2 F 5/2 )/N Yb is the inversion ratio and N 2 is the population of the upper laser level. Owing to the strong ion clustering in Yb 3+ :CaF 2 , it

UFig. 12 .

 12 Fig. 12. Analysis of the polarization state of emission of the deep-diced Yb 3+ :CaF 2 waveguide lasers (WG width: 20-50 μm): transmitted output power after the Glan-Taylor polarizer vs. its angular position. T OC = 20%, P inc ~0.75 W. Pump polarization: vertical (90°).

UFig. 13 .

 13 Fig. 13. Effect of the pump polarization (E p ) on the laser performance of the deep-diced Yb 3+ :CaF 2 waveguide lasers (WG width: 40 μm): (a) input-output dependences, η -slope efficiency; (b) typical spectra of laser emission measured at maximum P abs ; (c,d) near-field profiles of the pump modes; (e,f) near-field profiles of the laser modes, P abs ~0.5 W λ p = 979.5 nm, T OC = 20%. In (c-f), white lines marking the air/crystal interfaces are drawn as guides for eyes.crystal, we observed a notable polarization-response of the fabricated waveguides, namely, a great difference of the propagation losses for TM and TE polarized light, as well as linearly polarized laser emission. These observations may suggest an anisotropic stress field in the ridges modifying the refractive index.Further work should focus on the optimization of the transverse profile of the ridges for simultaneously maintaining good mode confinement, keeping low propagation losses and reaching single-transverse-mode operation.The fabricated waveguides are promising for compact mode-locked oscillators at ~1 μm operating at high repetition rates (GHz-range).

UFig. 14 .

 14 Fig. 14. Tunable deep-diced Yb 3+ :CaF 2 ridge waveguide laser (WG width: 40 μm): (a) tunability curve, T OC = 2%, P abs = 1.25 W, gain cross-section, σ g , spectra for several inversion ratios β = N 2 ( 2 F 5/2 )/N Yb are shown for comparison; (b) typical spectra of laser emission from the tunable laser; the spectrum for the free-running laser is shown for comparison.

Table 1

 1 Measured

	WG width	η coupl , %	δ loss , dB/cm
			E || vertical	E || horizontal
	50 μm 40 μm 30 μm 20 μm 10 μm	77.9 ± 0.5 74.2 ± 0.5 67.9 ± 0.5 57.6 ± 0.7 43.0 ± 1.0	0.17 ± 0.05 0.21 ± 0.06 0.28 ± 0.07 0.35 ± 0.1 0.52 ± 0.1	0.34 ± 0.07 0.39 ± 0.07 0.44 ± 0.1 0.46 ± 0.1 0.91 ± 0.2

pump coupling efficiency η coupl (at λ = 892 nm) and waveguide propagation losses δ loss (at λ = 912 nm) for deep-diced ridge waveguides in Yb 3+ :CaF 2 .

Table 2 .

 2 Output characteristics* of Yb 3+ :CaF 2 waveguide lasers reported so far.

	Method **	Yb, at.%	P out , W	η, %	P th , W	λ L , nm	η opt , %	η coupl , %	δ loss , dB/cm	Ref
	ULI LPE Dicing	2.0 1.4 4.5	0.17 0.11 0.72 0.33	~56 12.9 76.0 27.9	0.074 0.24 0.165 0.055	1014, 1028 1037 1029-1051 1038-1059	44.3 ~6 38.9 17.6	-73 74.2	0.5 0.14 ± 0.05 0.13 ± 0.05	[25] [28] ***
	P

outoutput power, η -slope efficiency vs. absorbed pump power, P thlaser threshold, λ Llaser wavelength, η optoptical-to-optical efficiency, η couplpump coupling efficiency. **Fabrication method: ULI -Ultrafast Laser Inscription, LPE -Liquid Phase Epitaxy. ***This work.

P. Loiko et al. Optical Materials xxx (xxxx) xxx-xxx

Acknowledgements

This work received funding from the Normandy Region, France (project CRISTO, CPER-LABEX EMC3).

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Pavel Loiko

Ludovic Gauthier-Manuel

Gurvan Brasse

Esrom Kifle

Lauren Guillemot

Alain Braud

Abdelmjid Benayad

Vivien Menard

Patrice Camy
email: patrice.camy@ensicaen.fr

Channel waveguide lasers in bulk Tm:LiYF 4 produced by deep diamond-saw dicing

We report on a novel approach to fabricate channel (ridge) waveguides (WGs) in bulk crystals using precision diamond saw dicing. The channels feature a high depth-to-width aspect ratio (deep dicing). The proof-of-the-concept is shown for a Tm 3+ :LiYF 4 fluoride crystal. Channels with a depth of 200 μm and widths of 10-50 μm are diced and characterized by confocal laser microscopy revealing a r.m.s. roughness of the walls well below 100 nm. The channels obtained possess waveguiding properties at ~815 nm with almost no leakage of the guided mode having a vertical stripe intensity profile into the bulk crystal volume and relatively low propagation losses (0.20-0.43 dB/cm). Laser operation is achieved in quasi-CW regime by pumping at 780 nm. The maximum peak output power reaches 0.68 W at ~1.91 μm with a slope efficiency of 53.3% (in σ-polarization). The proposed concept is applicable to a variety of laser crystals with different rare-earth dopants.

Introduction

Fluoride crystals doped with rare-earth ions (RE 3+ ) are attractive for development of efficient power-scalable waveguide (WG) lasers emitting in the Short Wavelength Infrared (SWIR) spectral range, an, in particular, at ~2 μm and beyond) [START_REF] Loiko | In-band pumping of Tm:LiYF 4 channel waveguide: a power scaling strategy for ∼2 μm waveguide lasers[END_REF][START_REF] Bolaños | 2.8 W end-pumped Yb 3+ :LiYF 4 waveguide laser[END_REF][START_REF] Ren | Switchable single-dualwavelength Yb,Na:CaF 2 waveguide lasers operating in continuous-wave and pulsed regimes[END_REF][START_REF] Loiko | Tm,Ho:LiYF 4 planar waveguide laser at 2.05 μm[END_REF][START_REF] Loiko | Ytterbium calcium fluoride waveguide laser[END_REF]. As host matrices, they provide good thermal properties (e.g., the thermal conductivity of undoped LiYF 4 , <κ> = 6.0 Wm -1 K -1 ) [START_REF] Aggarwal | Measurement of thermo-optic properties of Y 3 Al 5 O 12 , Lu 3 Al 5 O 12 , YAlO 3 , LiYF 4 , LiLuF 4 , BaY 2 F 8 , KGd(WO 4 ) 2 , and KY(WO 4 ) 2 laser crystals in the 80-300 K temperature range[END_REF], broadband transparency (0.15-7.5 μm for LiYF 4 ), low refractive index (n o = 1.443 and n e = 1.465 for LiYF 4 at ~1.9 μm) [START_REF] Barnes | Temperature variation of the refractive indices of yttrium lithium fluoride[END_REF] and low phonon energies (hν ph = 446 cm -1 for LiYF 4 ) [START_REF] Miller | Lattice vibrations of LiYF 4[END_REF]. The latter determines weak non-radiative relaxation and long upper laser level lifetimes of the RE 3+ ions [START_REF] Walsh | Branching ratios, cross sections, and radiative lifetimes of rare earth ions in solids: Application to Tm 3+ and Ho 3+ ions in LiYF 4[END_REF]. An example of efficient fluoride laser material is RE 3+ -doped lithium yttrium fluoride crystal, LiYF 4 [START_REF] So | A power-scaling strategy for longitudinally diode-pumped Tm:YLF lasers[END_REF][START_REF] Loiko | Comparative spectroscopic and thermo-optic study of Tm:LiLnF 4 (Ln = Y, Gd, and Lu) crystals for highly-efficient microchip lasers at ~2 μm[END_REF][START_REF] Loiko | Efficient Tm:LiYF 4 lasers at ~2.3 μm: Effect of energy-transfer upconversion[END_REF]. This material belongs to the tetragonal crystal class (sp. gr. I4 1 /a) and it exhibits natural birefringence [START_REF] Barnes | Temperature variation of the refractive indices of yttrium lithium fluoride[END_REF]. It also offers a single rare-earth site (the Y 3+ one) and can be doped with the RE 3+ ions in high concentrations [START_REF] Soulard | Laser operation of highly-doped Tm:LiYF 4 epitaxies: towards thin-disk lasers[END_REF]. The technology of growth of bulk RE 3+ -doped LiYF 4 crystals by the Czochralski (Cz) method is well-developed [START_REF] Cockayne | The Czochralski growth and laser characteristics of Li(Y,Er,Tm,Ho)F 4 and Li(Lu,Er,Tm,Ho)F 4 scheelite single crystals[END_REF].

Nowadays, a common way to produce crystalline RE 3+ -doped LiYF 4 WGs is the Liquid Phase Epitaxy (LPE) [START_REF] Starecki | Rare earth doped LiYF 4 single crystalline films grown by liquid phase epitaxy for the fabrication of planar waveguide lasers[END_REF]. In this way, high optical quality thin crystalline films of RE 3+ :LiYF 4 are achieved on undoped oriented bulk LiYF 4 substrates resulting in a planar WG geometry [START_REF] Bolanos | Tm:LiYF 4 planar waveguide laser at 1.9 μm[END_REF]. Consequently, a microstructuring step is required to fabricate channel WGs with a welldefined transverse profile of the refractive index leading to the single-transverse-mode operation. One known method of microstructuring of LPE-grown crystalline films is the ion beam milling leading to relatively low propagation losses (δ loss ~0.11 dB/cm) [START_REF] Geskus | Microstructured KY(WO 4 ) 2 :Gd 3+ ,Lu 3+ ,Yb 3+ channel waveguide laser[END_REF]. However, it is rather complicated from the technological point of view. Recently, we demonstrated the suitability of precision diamond saw dicing for microstructuring of LPE-grown Tm 3+ -doped LiYF 4 crystalline thin-films [START_REF] Loiko | Watt-level Tm:LiYF 4 channel waveguide laser produced by diamond saw dicing[END_REF]. A Tm 3+ :LiYF 4 channel WG laser generated 1.30 W at 1880 nm with a slope efficiency of 80% with respect to the absorbed pump power and a low laser threshold of 80 mW.

In the present work, we aimed to extend this technology to direct fabrication of optical WGs in bulk crystals. A special geometry of WGs with a high depth-to-width aspect ratio (referred as deep dicing) was selected. In this way, the mode confinement is expected to be provided by the refractive index contrast at the crystal / air interfaces and an additional refractive index variation due to the photo-elastic effects originating from stresses induced by mechanical dicing.

So far, precision diamond saw dicing was used for fabrication of optical WGs in various optical materials, Table 1. Chen et al. applied this method for fabrication of ridges in polymer films deposited on SiO 2 /Si wafers and doped with fluoride nanoparticles (Er 3+ ,Yb 3+ :NaYF 4 ); optical gain at ~1.5 μm was reported [START_REF] Chen | Low loss nanostructured polymers for chipscale waveguide amplifiers[END_REF]. Waeselmann et al. patterned Nd 3+ :α-Al 2 O 3 (sapphire) thin films produced by pulsed laser deposition (PLD). Laser operation was achieved: the WG laser generated 322 mW at 1092 nm with a slope efficiency of ∼12% probably due to the high losses of ~6 dB/cm [START_REF] Waeselmann | Nd:sapphire channel waveguide laser[END_REF]. Earlier, Jia et al. fabricated ridge WGs in bulk Nd 3+ :Y 3 Al 5 O 12 singlecrystals subjected to swift heavy ion irradiation and extracted 84 mW of output power at 1064 nm with a slope efficiency of 43% corresponding to lower δ loss = 1.7 dB/cm [START_REF] Jia | Ridge waveguide lasers in Nd:YAG crystals produced by combining swift heavy ion irradiation and precise diamond blade dicing[END_REF]. Kifle et al. structured LPE-grown anisotropic Tm:KY 1-x-y Gd x Lu y (WO 4 ) 2 thin films and generated 262 mW at 1833 nm with a high slope efficiency of 82.6% and moderate δ loss = 1.1 dB/cm [START_REF] Kifle | Diamond saw dicing of thulium channel waveguide lasers in monoclinic crystalline films[END_REF]. Finally, WGs in irradiated LiNbO 3 and KTiOPO 4 crystals were also fabricated by diamond saw dicing [START_REF] Sun | 466 mW green light generation using annealed proton-exchanged periodically poled MgO: LiNbO 3 ridge waveguides[END_REF][START_REF] Courjal | High aspect ratio lithium niobate ridge waveguides fabricated by optical grade dicing[END_REF], featuring WG propagation losses of ~1 dB/cm. These structures were used for secondharmonic generation leading to green emission. Note that all the previous studies [START_REF] Chen | Low loss nanostructured polymers for chipscale waveguide amplifiers[END_REF][START_REF] Waeselmann | Nd:sapphire channel waveguide laser[END_REF][START_REF] Jia | Ridge waveguide lasers in Nd:YAG crystals produced by combining swift heavy ion irradiation and precise diamond blade dicing[END_REF][START_REF] Kifle | Diamond saw dicing of thulium channel waveguide lasers in monoclinic crystalline films[END_REF][START_REF] Chen | Second harmonic generation of diamond-blade diced KTiOPO 4 ridge waveguides[END_REF][START_REF] Sun | 466 mW green light generation using annealed proton-exchanged periodically poled MgO: LiNbO 3 ridge waveguides[END_REF] focused on near-surface dicing (with the depth not exceeding few tens of μm) corresponding to small aspect ratios (depth / width, or y/x) of about unity or much less. High aspect ratio (e.g., top width: 1 μm, depth: 500 μm) WGs were reported only in lithium niobate [START_REF] Courjal | High aspect ratio lithium niobate ridge waveguides fabricated by optical grade dicing[END_REF], however, reasonably low propagation losses of 0.5 dB/cm were observed for less exaggerated structures, cf. Table 1.

Fabrication of waveguides

Diamond saw dicing

As a reference material for proof-of-the-concept, we selected the tetragonal Tm 3+ :LiYF 4 crystal. It was grown by the conventional Czochralski (Cz) method and doped with 4.0 at.% Tm (the actual ion concentration: N Tm = 5.50×10 20 cm -3 ). A rectangular sample was cut for light propagation along the crystallographic a-axis (a-cut) with a thickness (t) of 7.0 mm and an aperture of 3.5(c)×10 mm 2 . The input and output faces and the top surface of the sample were polished to laser quality and remained uncoated.

Then, the top surface was subjected to precision diamond saw dicing resulting in fabrication of surface channels (ridges) with a depth of 200 μm along the c-axis and a varied width of 10-50 μm (with a step of 10 μm), Fig. 1. Such widths are similar to those achieved in LPE films or single crystals by other micro-structuring methods, such as ion beam milling or fs direct laser writing. The step of 10 μm was selected from the point of view of convenience in discriminating the guides from each other, there were no technical limitation for this. The channels propagated through the whole length of the sample. For this purpose, very low grit size blades were selected and calibration processes occurred to ensure the dimensions of the ridges. The channels were separated by about 500 μm, which is sizable but comes from the use of a large blade: it enforces a better stability and minimize vibrations during deep dicing. The end-facets of the crystal sample were not repolished after the dicing, but the dicing process may additionally improve the end-facet quality by cutting the material orthogonally to the ridge thus reducing roughness and ensuring a good verticality of the input/output surfaces. The quality and geometry of channels were inspected with a confocal laser microscope (Sensofar S-neox) equipped with a blue light-emitting diode (LED, λ = 405 nm), Fig. 2. First, we studied one of the end-facets of the sample, showing vertical "pillows" separated by diced regions, Fig. 2(a). No cracks propagating into the guides nor into the bulk crystal volume are observed (the dark lines are due to the polishing of the end-facet of the sample before dicing and they are enhanced because of the observation geometry in reflected light which is sensitive to the surface quality). A close view on one of the fabricated guides (with an intermediate width of 30 μm), Fig. 2(b), reveals vertical walls within about 3/4 of its height. The flatness of the input facet of the guide is similar to that of the bulk crystal, so that no repolishing is required.

In the bottom part of the same guide (at about 1/4 of its height), Fig. 2(c), a rounding is observed and it is asymmetric from left and right diced areas. In this part, small bright spots with a size <0.1 μm are visible and they are interpreted as writing debris. Such rounding in the bottom part of the channels is characteristic for diamond saw dicing and it was also observed in guides diced at small depths [START_REF] Loiko | Watt-level Tm:LiYF 4 channel waveguide laser produced by diamond saw dicing[END_REF][START_REF] Kifle | Diamond saw dicing of thulium channel waveguide lasers in monoclinic crystalline films[END_REF].

Furthermore, the top surface of the sample was observed, Fig. 2(d). The fabricated guides are straight and there are no cracks preventing the light propagation through the whole length of the sample. A lose view on the top surface of two guides (with a width of 30 μm and 10 μm) is shown in Fig. 2(e,f). The ridges have smooth side walls. The r.m.s. surface roughness of the channel walls is well below 100 nm (this value is an upper estimation, limited by the precision of the used method). According to the studies of ridges in lithium niobate, even lower roughness down to ~5 nm is expected.

Waveguiding properties

The passive waveguiding properties of the fabricated channels were studied at the wavelength of ~815 nm. It is close to the pump wavelength for Tm 3+ :LiYF 4 (780 nm, the 3 H 6 → 3 H 4 Tm 3+ transition). The power of the pump beam was maintained to be low (~20 mW) to operate in the small-signal regime avoiding bleaching of the Tm 3+ absorption. As a laser source, we used a Ti:sapphire laser (3900S, Spectra Physics). Its output (beam quality parameter: M 2 ≈ 1) was focused into the WGs using an uncoated CaF 2 lens (focal length: f = 40 mm) resulting in a measured spot size 2w P of 30±5 μm (at the 1/e 2 level). The mode profile at the output facet of the waveguide was reimaged to a CCD camera (BladeCam-XHR, DataRay Inc.) using a short focal length CaF 2 lens (f = 15 mm). The scale calibration for the camera was performed by illuminating the WGs with a known size using a near-IR light source (LED) placed before the focusing lens. The obtained calibration images well matched those obtained with the confocal microscopy yielding an estimated scaling error of less than 3 µm. The polarization of the laser beam in the crystal corresponded to π (E || c, vertical). This polarization was selected because it corresponds to higher absorption and stimulated-emission cross-sections for Tm 3+ ions in LiYF 4 . In the first set of experiments, the pump was focused in the central part (with respect to the height) of each guide. The guided modes had a vertical stripe intensity profile well confined within the cross-section of the channel (representing a kind of a rotated planar WG geometry immersed in the air), Fig. 3. The pump modes were spatially multimode for all the guides. For example, for the 10 μm and 20 μm wide WGs, the pump modes were assigned as TM 1,0 and TM 2,0 , respectively. The appearance of high-order modes was justified by the M 2 measurement along the vertical direction. For wider WGs (30 -50 μm), the beam was spatially multimode also along the horizontal direction and an asymmetry in the mode was detected: the maximum of light intensity was localized near one of the side walls, Fig. 3(b,c). This is attributed to the slight asymmetry of the waveguide cross-section induced by dicing, see Fig. We studied the same WG (with a width of 10 μm) focusing the pump beam in different positions (along the vertical axis) of the input facet, Fig. 4. When focusing the pump near the top of the WG, a vertically extended TM 0,0 mode was observed, Fig. 4(a). By shifting the position of the pump spot to the center of the guide, the mode extended in the vertical direction corresponding to TM 1,0 , Fig. 4(b). Still, almost no leakage of the pump into the bulk volume was observed. Only when focusing the pump close to the bottom part of the guide, see Fig. 4(c), the modal profile was distorted strongly leaking into the bulk, and a small nearly-circular mode guides under the "pillow" was visible. This distortion is due to the asymmetric rounding of the guide described above.

Let us discuss the guiding mechanism of the fabricated channels. Tetragonal LiYF 4 is an optically uniaxial crystal; its optical axis is parallel to the c-axis [START_REF] Barnes | Temperature variation of the refractive indices of yttrium lithium fluoride[END_REF]. There are two principal light polarizations, denoted as π (E || c) and σ (E ⊥ c) and corresponding to the principal refractive indices n e and n o . Both polarizations are available for the selected crystal cut (a-cut). From the side walls and the top surface, the guiding is provided by the very high refractive index contrast between the crystal (for undoped LiYF 4 , n e = 1.4724 at 815 nm) and the air.

The pump geometry (M 2 ≈ 1, 2w P = 30 μm) corresponded to a Rayleigh length z R of 1.28 mm in the crystal. Thus, at the output facet of the WG, the pump beam would have a diameter of (2w P ) out ~ 165 μm which agrees with our observations. However, this does not explain the modal profile of the pump along the vertical direction. For the mode confinement of the pump radiation along the vertical direction, we propose two mechanisms: (i) a refractive index variation (increase) in the guide due to the photoelastic effect (a change of the refractive index owing to permanent stresses, in our case, induced by dicing) and (ii) the absorption at the pump wavelength which is stronger for the "wings" of the pump mode which are not able to saturate the Tm 3+ ions [START_REF] Van Dalfsen | Thulium channel waveguide laser with 1.6 W of output power and ∼80% slope efficiency[END_REF]. For Tm 3+ :LiYF 4 , the absorption cross-section at 815 nm σ P abs is 0.41×10 -21 cm 2 in π-polarization corresponding to a small-signal absorption loss in the studied WGs of 0.7 dB (in a single-pass).

In a similar manner, one may explain the guiding of the laser mode. Note that the 3 F 4 → 3 H 6 laser transition of Tm 3+ ions represents a quasi-three-level laser scheme with reabsorption. For example, for lasing in π-polarization, the stimulated-emission (SE) cross-section at the peak emission wavelength of ~1880 nm σ SE is 4.0×10 -21 cm 2 and the reabsorption cross-section σ L abs is 0.41×10 -21 cm 2 . Thus, the gain-guiding (the reabsorption in the non-pumped areas of the WG) can be in part responsible for the mode confinement along the vertical direction [START_REF] Mackenzie | 15 W diode-side-pumped Tm:YAG waveguide laser at 2 μm[END_REF][START_REF] Shepherd | Highpower planar dielectric waveguide lasers[END_REF]. The second possible reason is the photo-elastic effect. In Fig. 5, we analyzed the 10 μm wide WG using a mode calculator (a variational mode solver based on quasi-analytical vectorial slab mode expansion) [START_REF] Ivanova | A variational mode solver for optical waveguides based on quasianalytical vectorial slab mode expansion[END_REF]. The minimum refractive index variation (increase) in the upper part of the WG with respect to its bottom part and the bulk volume Δn which can support the fundamental mode (TM 0,0 ) at 1880 nm is ~2.5×10 -3 , Fig. 5(b). The corresponding pump mode at 780 nm (TM 1,0 ) is shown in Fig. 5(a) in agreement with the experiment. The WG propagation losses δ loss at 815 nm were estimated by observing the top surface of the sample with a CCD camera and detecting the intensity of the scattered pump light along the WG (in the longitudinal direction). An example of this evaluation for the 20 μm and 50 μm wide WGs is shown in Fig. 6. The on-axis intensity of the pump radiation changed according to the equation I(z) = I 0 ×exp(-δ tot z), where δ tot = α abs + δ loss is the total loss coefficient, α abs = σ P abs N Tm = 0.23 cm -1 is the small-signal absorption coefficient at the pump wavelength and z is the axial coordinate. The origin of strong dispersion of the data points in the end of the guides is non-uniform reflection of scattered light from the silver paint used for mounting the sample. The δ loss monotonously increases for more narrow WGs, from 0.20±0.03 dB/cm for the 50 μm wide WG to 0.43±0.04 dB/cm for the 10 μm wide one, Table 2. This is explained by stronger interaction of the pump mode with the side walls of the WGs containing small defects causing light scattering. Under pumping at 780 nm, the channels provided an intense polarized luminescence at ~1.9 μm due to the 3 F 4 → 3 H 6 Tm 3+ transition. The spectral properties were similar to those of the bulk crystal.

Laser operation

Laser set-up

The scheme of the WG laser is shown in Fig. 7. As a pump source, we used a CW Ti:Sapphire laser delivering about 3.2 W of linearly polarized output at 780 nm in a fundamental mode (M 2 ≈ 1). This wavelength corresponded to the local peak in the absorption spectrum of Tm 3+ :LiYF 4 crystal (the 3 H 6 → 3 H 4 Tm 3+ transition). The pump polarization was set to be vertical (π-polarization in the crystal). The incident pump power was varied by a rotatory λ/2 plate and a Glan-Taylor polarizer. The pump beam was focused by an uncoated CaF 2 lens (f = 40 mm, T = 93.8%) providing a pump spot diameter at the input face of the WG 2w P of 30±5 μm. For quasi-CW pumping, the pump beam was modulated using a mechanical chopper (frequency: 10 Hz, duty cycles ranging from 1:2 to 1:12). The Tm 3+ :LiYF 4 crystal with the surface guides was mounted on a Cu-holder using a silver paste to improve the thermal contact from the bottom part of the sample. The holder was passively cooled. The crystal was placed in a simple linear plano-plano cavity. It was formed by a flat pump mirror (PM) coated for high transmission (HT, T = 99.4%) at ~0.78 μm and for high reflection (HR, R > 99.9%) at 1.60-2.02 μm, and a set of flat output couplers (OCs) with a transmission at the laser wavelength T OC ranging from 2% to 50%. Both cavity mirrors were gently pressed towards the WG end-facets. No index-matching liquid was used to avoid damaging the optical surfaces. The geometrical cavity length was 7.0 mm. The pump beam was focused into the WGs through the PM.

To filter out the residual (non-absorbed or non-coupled) pump after the OC, a long-pass filter (FEL900, Thorlabs) was used. The laser emission spectra were measured using an optical spectrum analyzer (AQ6375B, Yokogawa). To determine the pump coupling efficiency, we coupled into the WGs the laser beam from the Ti:Sapphire laser tuned to 830 nm (out of the Tm 3+ absorption) and monitored the power at the output facet. The pump coupling η coupl = P launch /P inc (P launch is the launched pump power) was calculated excluding the propagation losses δ loss estimated in Section 2.2. The η coupl value includes Fresnel losses at the uncoated input facet of the crystal (T Fr = 96.4%, as calculated using a refractive index of LiYF 4 : n e = 1.4722 at 830 nm). It is in the range of 80.5-46.1% for the 50-10 μm wide WGs and it is relatively close to the value obtained considering only the Fresnel losses and the geometrical overlap of the WG end-facet and the pump beam. As expected, η coupl decreased for smaller widths of the guides. For the 4.0 at.% Tm 3+ :LiYF 4 crystal, the small-signal pump absorption η abs,0 at 780 nm is close to unity. Indeed, η abs,0 = 1exp(-σ P abs N Tm t) = 96.2%, where σ P abs = 0.85×10 -20 cm 2 is the absorption cross-section at the pump wavelength λ P for π-polarized light. The pump absorption under non-lasing conditions, η abs,NL = P abs /P launch (P abs is the absorbed pump power) was determined from the pump-transmission (end-fire) measurements, see Fig. 8 for the example of the 20 μm and 50 μm wide WGs. The η abs,NL slowly decreased with the incident pump power due to the ground-state bleaching. At very small pump powers, η abs,NL ≈ η abs,0 . For estimating the absorbed pump power under lasing conditions P abs , we have taken the η abs,NL value at the laser threshold (for each of the studied OCs). For example, for the 50 μm wide WG, the pump absorption slightly decreased from 94.5% for T OC = 2% down to 93.9% for the highest studied T OC = 50%. The pump saturation for the WG with a smaller width (20 μm) was stronger, as expressed by smaller values of η abs,NL for the same pump level, which is due to the higher light intensity in the guide. The values of pump absorption for the same T OC = 50% and different WG width are compared in Table 2.

Laser performance

The laser operation was achieved for all the WGs when focusing the pump beam in the upper part of each of the guides. No laser emission was detected when the pump was focused in the bulk part of the sample. The experiments were performed in quasi-CW regime (duty cycle: 1:2) to diminish the thermal effects in thin passively-cooled ridges. The input-output dependences for the largest studied WG (width: 50 μm) are shown in Fig. 9(a). For T OC = 50%, the laser generated a maximum output peak power of 381 mW at 1901-1929 nm (a broad emission spectrum) with a slope efficiency η of 42.2% with respect to the absorbed pump power P abs . The laser threshold was at P abs = 0.25 W and the optical-to-optical efficiency η opt was 18.6% (as calculated vs. the pump power incident on the guide). For smaller output coupling, the laser threshold gradually decreased reaching P th = 0.16 W for T OC = 2%. For P abs exceeding ~1 W, a thermal roll-over was observed in the input-output dependences. Thus, P abs was limited to about 1.5 W to avoid thermal fracture of the guides.

The typical spectra of laser emission for the 50 μm wide WG and various transmissions of the output coupler are shown in Fig. 9(b). For all the OCs, the laser emission was linearly polarized (σ); the polarization was naturally selected by the gain anisotropy. The emission occurred at around ~1.91 μm for all the studied OCs in agreement with the gain spectra of Tm 3+ :LiYF 4 for σ-polarization and small inversion ratios (see below).

For the sake of comparison, in Fig. 9(c,d), we show the input-output dependences and the typical spectra of laser emission for the smaller guide (width: 20 μm). The best performance for this WG corresponded to a maximum output peak power of 153 mW at 1884-1888 nm (in π polarization) with lower η = 25.3% and higher laser threshold P th = 0.29 W (as compared to the 50 μm wide WG, for T OC = 50%). The optical-to-optical efficiency was thus only 2.9%. The spectral behavior of the 20 μm wide WG laser was different from the 50 μm wide one, Fig. 9(d). For small output coupling, the laser operated in σ-polarization corresponding to the emission at ~1.91 μm (e.g., 1906-1911 nm for T OC = 5%). For intermediate T OC = 7 -30%, two polarizations coexisted within the whole studied range of the absorbed pump power. As a result, the spectra contained emission lines within several regions (e.g., 1878-1882 & 1912-1918 nm for T OC = 10%, with the short wavelength emission in π-polarization and the long-wavelength one in σ-polarization). Finally, for high T OC = 50%, only π-polarized emission occurred.

Such polarization-switching is common for optically anisotropic (birefringent) laser crystals which exhibit similar gain cross-sections for two orthogonal principal light polarizations [START_REF] Druon | Magic mode switching in Yb:CaGdAlO 4 laser under high pump power[END_REF][START_REF] Loiko | Thermal-lens-driven effects in N g -cut Yb-and Tm-doped monoclinic KLu(WO 4 ) 2 crystals[END_REF]. The switching between two polarizations can be promoted by change of the total intracavity losses (e.g., via the output coupling) or by polarization-dependent thermal lensing affecting the mode overlap efficiency or cavity stability. As a result, there may exist such levels of the cavity losses or pump powers for which two polarizations coexist leading to dualwavelength (or multi-color) emission.

As pointed out above, the 3 F 4 → 3 H 6 laser transition is a quasi-three-level scheme. The gain cross-sections, σ gain = βσ SE -(1 -β)σ abs , where β = N 2 ( 3 F 4 )/N Tm is the inversion ratio and N 2 is the population of the upper laser level ( 3 F 4 ), are thus calculated to conclude about the selected polarization and, consequently, the laser wavelength. Such a calculation for Tm 3+ :LiYF 4 and π and σ polarizations is performed in Fig. 10. For small inversion ratios β < 0.20, the gain in σpolarization is higher. A local peak in the gain spectra is found at ~1.91 μm and it is relatively broad (the gain bandwidth, determined as full width at half maximum, FWHM, Δλ g is ~60 nm for β = 0.20). For intermediate 0.20 < β < 0.30, the peak gain cross-sections at ~1.91 (σ) and ~1.88 μm (π) are close. Finally, for high inversion ratios of β > 0.35, π-polarization corresponds to higher gain. The local peak at ~1.88 μm is narrower (as expressed by Δλ g ~35 nm).

These observations well explain the observed spectral behavior for narrow (20 μm) WG, considering that higher output coupling will lead to higher β. We believe that polarizationswitching in such WGs can be controlled via a precise control of the propagation losses which can be reached, e.g., by managing the cross-section profile of the guides or optimizing the dicing conditions. The waveguide propagation losses for the 20 μm and 50 μm wide WGs were estimated by means of the Caird analysis [START_REF] Caird | Quantum electronic properties of the Na 3 Ga 2 Li 3 F 12 :Cr 3+ laser[END_REF]: inverse of the slope efficiency, 1/η, was plotted as a function of inverse of the output-coupling losses, 1/γ OC , where γ OC = -ln(1 -T OC ). The experimental points were fitted using the equation (1/η) = (1/η 0 )(1 + 2γ/γ OC ) [START_REF] Morris | 1.9 µm waveguide laser fabricated by ultrafast laser inscription in Tm:Lu 2 O 3 ceramic[END_REF], where η 0 is the intrinsic slope efficiency and γ = -ln(1 -L), L is the single-pass passive loss. The best-fit values yields the propagation losses δ loss = 4.34×L/t = 0.32±0.2 dB/cm (20 μm WG) and 0.13±0.2 dB/cm (50 μm WG). The determined values of δ loss agree well with the estimations (cf. Table 2) from the pumppropagation measurements considering the difference in the light wavelength and modal profile (less transverse modes are expected for the laser emission because of much longer λ). We also confirm that the losses tend to increase for smaller guides.

In Fig. 12, we compare the laser performance for all the guides (10-50 μm) and the same output coupling (T OC = 50%). The output power and the slope efficiency gradually decreased and the laser threshold increased for the smaller guides. For example, the 10 μm wide WG laser generated a peak power of 15 mW at 1878-1886 nm with η = 4.4% and a high laser threshold of 0.31 W. There exist several factors affecting the laser performance of smaller guides. First, the WG propagation losses increase as described above leading to both decreased η and higher P th . Second, narrow guides exhibit severe thermal problems, so that the thermal roll-over in the output dependences occurs earlier. Finally, smaller guides provide higher intracavity laser intensities which, together with higher inversion needed to compensate for the higher passive losses, enhances the upconversion. Energy-transfer upconversion in Tm lasers is known to have a serious effect on the laser threshold [START_REF] So | A power-scaling strategy for longitudinally diode-pumped Tm:YLF lasers[END_REF]. Indirectly, it further enhances the heat dissipation.

The spectral behavior of the WGs lasers, Fig. 12(b), agrees with these considerations and the gain spectra. For small guides (10-20 μm) with higher passive losses, the laser operated in π-polarization and for broader guides (30-50 μm) -in σ-polarization. The difference in the gain bandwidths also explains broader laser emission in the latter case. To verify our consideration about the thermal effects in the guides, we studied the laser performance of the 50 μm WG with the same output coupling (T OC = 50%) but under different pump regimes, ranging from true CW to quasi-CW with various pump duty cycles (between 1:2 to 1:12), Fig. 13. The CW laser generated only 34 mW and the lasing was ceased for P abs > 0.5 W. For the quasi-CW operation regimes with decreasing the duty cycle down to 1:12 (thus reducing the heat loading), the input-output dependence gradually approached the linear one, so that the power scaling was limited only by the available pump. For the 1:12 duty cycle, the maximum peak output power reached 684 mW with = 53.3%. Note that the laser threshold was almost independent on the pump modulation.

The determined value of the slope efficiency exceeds the Stokes limit under lasing conditions, η St,L = λ P /λ L = 40.8% (λ L ~1910 nm is the laser wavelength) indicating the action of cross-relaxation for adjacent Tm 3+ ions increasing the pump quantum efficiency [START_REF] Van Dalfsen | Thulium channel waveguide laser with 1.6 W of output power and ∼80% slope efficiency[END_REF].

Conclusions

To conclude, we report on a novel approach to fabricate ridge WGs with a large surface area (high depth-to-width aspect ratio, well above 1) in bulk fluoride crystals based on precision diamond saw dicing. This method has the following advantages: (i) it is relatively simple from a technological point of view, (ii) it provides an appropriate precision, an easy control of the geometrical profile and a fast writing speed, (iii) it ensures strong guiding both in the passive and laser-active regimes of light propagation without leakage of the modes in the bulk volume, (iv) it provides low roughness of the guide walls and, consequently, relatively low passive losses, potentially down to 0.1 dB/cm, and (v) it can be applied for a wide variety of materials with different RE 3+ dopants. A single-transverse-mode laser operation is expected with a proper design of the geometrical profile of the channels; it will be also facilitated by the gain guiding in the case of quasi-three-level lasers. Further studies are however needed to clarify the nature and the absolute value of the expected refractive index variation in the guides with respect to the bulk volume.

In the particular case of deep-diced Tm 3+ :LiYF 4 WGs, the main limitation for power scaling is the thermal issues. It can be overcame by better thermal management, e.g., by applying active cooling. Other strategies may involve using higher Tm 3+ doping levels for more efficient crossrelaxation reaching a pump quantum efficiency of 2 (for the conventional pumping at ~780 nm) or employing in-band pumping directly to the 3 F 4 upper laser level. Bulk Tm 3+ :LiYF 4 crystals of laser quality with high doping levels (7 -15 at.% Tm) are easily accessible. An optimization of the writing depth may also greatly improve the thermal behavior of the guides.

Fig. 1 .

 1 Fig. 1. Scheme of the deep diced ridge WGs in bulk Tm 3+ :LiYF 4 crystal (the end-facet view).

Fig. 2 .

 2 Fig. 2. Confocal laser microscope images of the Tm 3+ :LiYF 4 crystal subjected to precision diamond-saw dicing: (a-c) the end-facet view showing (a) (from left to right) 40 μm, 30 μm, 20 μm and 10 μm wide WGs, (b) a close view in the 30 μm WG and (c) the bottom part of the 30 μm WG; (d-f) the top surface view on (d) (from left to right) 40 μm, 30μm, 20 μm and 10 μm WGs, (e,f) the input facet of (e) 30 μm and (f) 10 μm WG. Reflection mode, λ = 405 nm.

Fig. 3 .

 3 Fig. 3. 2D intensity profiles of the guided pump mode at the output WG facet: the WG width is (a) 10 μm, (b) 30 μm and (c) 50 μm. While lines showing the crystal surface are drawn as guides for eyes. λ P = 815 nm. Pumping in the central part of the input facet. The laser polarization (π) is vertical.

  2(b),(c) for example. No leakage of the guided modes into the bulk crystal volume was detected.

Fig. 4 .

 4 Fig. 4. 2D intensity profiles of the guided pump mode at the output WG facet: focusing of the pump beam in (a) the top part, (b) the medium part and (c) bottom part of the 10 μm wide WG. While lines showing the crystal surface are drawn as guides for eyes. Red dashed circles indicate the position of the pump beam at the input facet. λ P = 815 nm. The laser polarization (π) is vertical.

Fig. 5 .

 5 Fig. 5. Simulated 2D intensity mode profiles in the 10 μm wide WG at the light wavelength of (a) 780 nm (pump, TM 1,0 mode) and (b) 1880 nm (laser, TM 0,0 mode). The light polarization is vertical. The refractive index increase in the guide Δn is ~2.5×10 -3 , n e = 1.4728 (780 nm) and n e = 1.4652 (1880 nm).

Fig. 6 .

 6 Fig. 6. (a) Longitudinal distribution of the intensity of pump radiation at 815 nm in the 20 μm and 50 μm wide WGs: symbolsexperimental data, curvestheir exponential fits using a total loss coefficient δ tot = α abs + δ loss ; (b) The corresponding 2D intensity map from a CCD-camera for the 50 μm wide WG.

Fig. 7 .

 7 Fig. 7. (a) Scheme of the waveguide laser set-up: λ/2half-wave plate, P -Glan-Taylor polarizer, FL and CLfocusing and collimating lenses, respectively, PMpump mirror, OCoutput coupler, Fcut-off filter; (b) top view photograph of the 50 μm wide WG pumped at 780 nm, P inc = 1.0 W, the bright trace corresponds to Tm 3+ upconversion luminescence.

Fig. 8 .

 8 Fig. 8. Measured pump absorption under non-lasing conditions, η abs,NL , in the 20 μm and 50 μm wide ridge WGs in 4 at.% Tm 3+ :LiYF 4 as a function of the incident pump power. Blue circlesmall-signal pump absorption.

Fig. 9 .

 9 Fig. 9. Deep-diced Tm 3+ :LiYF 4 ridge waveguide laser: (a,c) input-output dependences, ηslope efficiency; (b,d) typical laser emission spectra measured at (b) P abs ~ 1.0 W and (d) P abs ~ 0.6 W. Quasi-CW regime of operation with a duty cycle of 1:2. The channel width is (a,b) 50 μm and (c,d) 20 μm. The laser polarization is (a,b) σ and (c,d) σ or π.

Fig. 10 .

 10 Fig. 10. Gain cross-sections, σ gain = βσ SE -(1 -β)σ abs , for the 3 F 4 ↔ 3 H 6 transition of Tm 3+ in LiYF 4 crystal for π and σ light polarizations. β = N 2 ( 3 F 4 )/N Tm is the inversion ratio, N 2 is the population of the upper laser level ( 3 F 4 ).

Fig. 11 .

 11 Fig. 11. Caird analysis: a plot of inverse of the slope efficiency, 1/η, vs. inverse of the outputcoupling losses, 1/γ OC , for 20 μm and 50 μm wide WGs: symbolsexperimental data, linestheir fits for the determination of the propagation losses δ.

Fig. 12 .

 12 Fig. 12. Effect on the guide width on the laser performance of the deep-diced Tm 3+ :LiYF 4 ridge waveguide laser: (a) input-output dependences, ηslope efficiency; (b) typical laser emission spectra measured at intermediate P abs . Quasi-CW regime of operation with a duty cycle of 1:2. T OC = 50%. The laser polarization is σ or π.

Fig. 13 .

 13 Fig. 13. Evidence of thermal effects in deep-diced Tm 3+ :LiYF 4 ridge waveguide laser: inputoutput dependences measured for true CW and quasi-CW operation regimes. WG with a width of 50 μm, T OC = 50%. ηslope efficiency for the duty cycle of 1:12.

Table 1 . Optical Waveguides Produced by Diamond Saw Dicing

 1

	Material	Form	Size, μm	Aspect ratio (y/x)	δloss, dB/cm	Ref.
	Polymer + Er,Yb:NaYF4	Spin-coated film	10(x)×3(y)	0.30	2.9	[19]
	Nd:α-Al2O3	PLD film	41(x)×2.6(y)	0.06	~6	[20]
	Nd:Y3Al5O12	Irradiated crystal	22(x)×9(y)	0.41	1.7	[21]
	Tm:KY1-x-yGdxLuy(WO4)2 LPE film	61(x)×15.3(y)	0.25	1.1	[22]
	Tm:LiYF4	LPE film	30(x)×30(y)	1.0	0.28	[18]
	KTiOPO4	Irradiated crystal	25(x)×30(y)	1.2	1.04	[23]
	MgO:LiNbO3	Irradiated crystal	8.5(x)×30(y)	3.5	2.14	[24]
	LiNbO3	Ion-indiffused crystal 6(x)×>20(y)	>3.3	0.5	[25]
	Tm:LiYF4	Bulk crystal	10-50(x)×200(y)	4-20	0.1-0.4 *
	*This work.

Table 2 . Pump Coupling Efficiency, Propagation Losses and Pump Absorption in the Studied WGs

 2

	Guide width, μm	ηcoupl, % (incl. δloss, meas.)	ηcoupl, % (geom., calc.)	δloss, dB/cm (meas.)	ηcoupl, % (excl. δloss, estim.)	δloss, dB/cm (Caird, estim.)	ηabs,L, %, TOC = 50% (meas.)
		Pump (0.83 μm)	Pump	Pump (0.81 μm)	Pump (0.81 μm)	Laser (1.91 μm)	Pump (0.78 μm)
	10	43.0±0.5	40.9	0.43±0.04	46.1±0.5	-	79.1±1
	20	50.9±0.5	81.8	0.41±0.03	54.4±0.5	0.32±0.2	94.2±1
	30	67.9±0.5	96.4	0.40±0.09	72.4±1	-	96.5±0.5
	40	76.7±0.5	96.4	0.36±0.05	81.3±0.7	-	89.7±0.5
	50	77.9±0.5	96.4	0.20±0.03	80.5±0.5	0.13±0.2	93.9±0.5

Acknowledgments

This work was partly supported by the French RENATECH network and its FEMTO-ST technological facility.

Funding

French Agence Nationale de la Recherche (ANR) (CEPAGE ANR-AAP-CE2; LabEx EMC3 (ANR-10-LABX-09-01); SPLENDID2 (ANR-19-CE08-0028)); Normandie region; European Community funds FEDER ("NOVAMAT").

Disclosures

The authors declare no conflicts of interest.