# Periodically extended architecture for power scaling of diode-pumped solid-state lasers: proof of concept with Pr:LiYF

Remi Soulard, Jean-Louis Doualan, Amandine Baillard, Pavel Loiko, Patrice Camy

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Remi Soulard, Jean-Louis Doualan, Amandine Baillard, Pavel Loiko, Patrice Camy. Periodically extended architecture for power scaling of diode-pumped solid-state lasers: proof of concept with Pr:LiYF. Optics Express, 2025, 33, pp.574742. 10.1364/oe.574742. hal-05416862

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Submitted on 15 Dec 2025

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# Periodically extended architecture for power scaling of diode-pumped solid-state lasers: proof of concept with Pr:LiYF₄

REMI SOULARD,¹ JEAN-LOUIS DOUALAN,¹ AMANDINE BAILLARD,¹ PAVEL LOIKO,¹ ID AND PATRICE CAMY¹, *

¹Centre de Recherche sur les Ions, les Matériaux et la Photonique (CIMAP), UMR 6252 CEA-CNRS-ENSICAEN, Université de Caen Normandie, 6 Boulevard Maréchal Juin, 14050 Caen, France
*patrice.camy@ensicaen.fr

**Abstract:** We report on a power-scaling concept for blue diode-pumped visible lasers based on a periodically extended multi-crystal cavity architecture. This design is validated using a continuous-wave praseodymium laser employing three 0.5 at.% Pr:LiYF₄ crystals, delivering 7.14 W of output power at 639.7 nm (in the red) with a slope efficiency of 30.8% (relative to the incident pump power). The laser exhibits linear polarization (σ) and near-diffraction-limited beam quality (M²x,y < 1.15) at an incident pump power of 25.2 W provided by six 444 nm free space GaN laser diodes. We demonstrate the linear scaling of total cavity losses and an almost constant optical conversion efficiency with the number of laser crystals. The design criteria for periodically extended laser cavities are formulated based on resonator stability conditions and mode-matching efficiency, considering multiple thermal lenses. The proposed architecture can be applied to high-power laser operation on different transitions, as well as to laser amplifiers.

Published by Optica Publishing Group under the terms of the [Creative Commons Attribution 4.0 License](https://creativecommons.org/licenses/by/4.0/). Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

## 1. Introduction

The ongoing drive to scale the energy and power output of solid-state laser sources continues to broaden their range of applications. Optical pumping enables the conversion of one wavelength (color) into another, while also transforming a low-brightness beam, such as that from semiconductor laser diodes, into a high-brightness, low-divergence output. Inevitably, a certain fraction of the pump power is lost (fundamentally, due to the quantum defect, the difference between the pump and laser photon energy) and transferred into heat dissipated in the active medium [1]. This spatially non-uniform heating gives rise to a wide range of thermo-optic effects including thermal lensing, stress-induced fracturing, and depolarization losses [1–3], which collectively alter the operating conditions of the laser resonator and limit the extractable power or energy. Over the past decades, numerous gain medium architectures have been developed to mitigate these thermal effects while maintaining high laser efficiency. These designs optimize parameters such as crystal aspect ratios, pump geometries, and beam-guiding mechanisms. A common objective among them is to maximize the heat surface exchange area for a given active volume containing the necessary concentration of laser-active ions to enhance heat dissipation to the coolant and thus reduce the thermal load.

Designing a power-scalable laser involves meeting several key criteria: i) minimal thermo-optic effects through efficient heat dissipation, ii) high optical gain, iii) excellent beam quality with fundamental transverse mode operation, iv) a high laser-induced damage threshold, and v) suppression of nonlinear effects, see Fig. 1.

The first group of gain medium architectures includes those in which the laser mode is primarily shaped by the optical resonator, beginning with conventional bulk solid-state lasers
<table><caption>**Fig. 1.** Pro and contra arguments of various geometries employed for power scaling of solid-state laser sources.</caption><thead><tr><th>Parameter / geometry</th><th>Bulk media</th><th>Fibers</th><th>Crystal-fiber</th><th>Thin disks</th><th>Slabs</th><th>Multi-crystal</th></tr></thead><tbody><tr><td>Thermal effects</td><td>Strong</td><td>Very weak</td><td>Weak</td><td>Weak</td><td>Weak</td><td>Weak</td></tr><tr><td>Gain</td><td>Low</td><td>Very high</td><td>High</td><td>Very low</td><td>Moderate</td><td>Moderate</td></tr><tr><td>Beam quality</td><td>Good</td><td>Excellent</td><td>Moderate</td><td>Good</td><td>Poor</td><td>Good</td></tr><tr><td>Fracture / damage risk</td><td>Low</td><td>High</td><td>Moderate</td><td>Moderate</td><td>Low</td><td>Low</td></tr><tr><td>Non-linear effects</td><td>Very low</td><td>Very high</td><td>Low</td><td>Very low</td><td>Low</td><td>Very low</td></tr><tr><td>Fabrication complexity</td><td>Very easy</td><td>Easy</td><td>Complex</td><td>Complex</td><td>Moderate</td><td>Very easy</td></tr></tbody></table>

which operation is strongly affected by thermal lensing [4–6]. By ultimately changing the aspect ratio (thickness to lateral dimensions), this approach evolves into the thin-disk, or “active mirror”, geometry [7]. This configuration promotes unidirectional heat flow, enabling excellent thermal management. Thin-disk lasers employing multi-pass pumping have emerged as a practical solution for kilowatt-class continuous-wave and ultrafast oscillators. However, the underlying enabling technologies, such as the fabrication of ultra-thin disks and the dielectric crystal-to-metal soldering, are technically demanding and costly.

The second group encompasses architectures in which mode formation is governed by total internal reflection dominating any thermal variation of the optical path length, enabling waveguiding and extended light propagation paths that distribute thermal loads more evenly. A prominent example is fiber lasers, which use carefully engineered core/cladding geometries and are now a well-established platform for high-power laser systems in a spectral band ranging roughly from 1 to 2 µm [8]. Nevertheless, these systems face two significant limitations: the high risk of damage to glass fibers under intense power levels (especially for fluoride fibers [9]) and the pronounced nonlinear effects that accumulate along the fiber length at high intracavity intensities. There also exist solutions which combine both approaches, such as slab [10] and crystal fiber lasers [11], comprising both waveguiding for the pump (laser) modes, and optimized surface / volume ratios for better heat removal.

Over the past years, there has been a raising interest in solid-state coherent light sources directly emitting in the visible under pumping in the blue, e.g., by GaN diode lasers [12]. This is because of a variety of applications of such lasers, covering scientific (Raman spectroscopy, optical trapping, interferometry, quantum optics), medical (flow cytometry, fluorescence microscopy), industrial (laser projection and display, material processing) and consumer (laser pointers, light shows) sectors. Such sources also simplify access to UV radiation [13] by reducing the number of frequency conversion steps (as compared to IR lasers).

Multiple rare-earth ions emit in the visible and direct laser emission has been demonstrated under blue GaN diode pumping addressing various colors such as cyan (Pr³⁺) [14], green (Pr³⁺, Tb³⁺, Ho³⁺) [15–17], yellow (Dy³⁺, Tb³⁺) [17,18], orange (Pr³⁺, Sm³⁺) [19], red (Pr³⁺) [20], and deep red (Pr³⁺, Ho³⁺) [21,22]. In terms of the laser host materials, fluoride crystals and glasses (in the form of fibers) have proven to be optimum as they combine low phonon energies leading to weak non-radiative relaxation rates, weak crystal fields reducing the risk of interconfigurational $4f^n \rightarrow 4f^{n-1}5d^1$ excited-state absorption [23], good thermal properties (in case of crystals), weak photodarkening (for fibers) and high rare-earth solubility. So far, the field of visible diode-pumped lasers has been dominated by fluoride crystals (LiYF₄) [24] and glass fibers (ZBLAN) [25] doped with trivalent praseodymium (Pr³⁺) ions, mainly because of the diversity of accessible colors due to multiple laser transitions in the visible (with the highest luminescence branching ratio corresponding to the red), and high absorption cross-sections in the blue due to spin-allowed transitions. The crystalline and fluoride fiber Pr lasers are currently the two competing technologies for high-power, multi-Watt level visible lasers.
Table 1 summarizes the best results in terms of the output power reported so far for red continuous-wave Praseodymium lasers pumped in the blue. Metz *et al.* explored the laser efficiency of Pr:LiYF₄ solid-state laser pumping with a high-brightness, frequency doubled 479.1 nm optically pumped semiconductor laser (2ω-OPSL) generating 2.8 W at 639 nm with a slope efficiency η of 61% (relative to the incident pump power) and a laser threshold of only ~20 mW [26]. Under diode pumping, lower laser efficiencies were achieved due to the compromised mode matching efficiency. Ijima *et al.* explored the double-crystal geometry for power scaling of diode-pumped Pr:LiYF₄ bulk lasers (note that the cavity was not periodically extended) and achieved 2.9 W of red output with η = 22.1% and P<sub>th</sub> ~ 0.9 W [27]. Further power scaling was reached by relying on long (~20 mm), lightly doped (0.1–0.3 at.% ) rods of Pr:LiYF₄ [28,29]: Xue *et al.* generated 10.8 W at 640 nm with η = 27.7% and P<sub>th</sub> = 3.12 W [29]. However, the fabrication of such rods of high quality and homogeneity is difficult, and special attention should be paid to the spatial overlap of the laser and pump modes.

<table><caption>**Table 1.** Output Performance<sup>a</sup> of Red Praseodymium Lasers Employing Various Geometries Reported So Far</caption><thead><tr><th>Geometry</th><th>C<sub>Pr</sub>,<br>mol%</th><th>P<sub>out</sub>,<br>W</th><th>λ<sub>L</sub>,<br>nm</th><th>P<sub>th</sub>, W</th><th>η, %</th><th>η<sub>opt</sub>, %</th><th>M<sup>2</sup></th><th>Ref.</th></tr></thead><tbody><tr><td>Bulk<sup>b</sup></td><td>0.5</td><td>2.8</td><td>640</td><td>0.02<sup>Inc</sup></td><td>61<sup>Inc</sup></td><td>58<sup>Inc</sup></td><td>&lt;1.2</td><td>[26]</td></tr><tr><td rowspan="2">Bulk, rod</td><td>0.3</td><td>6.7</td><td>640</td><td>~0.5<sup>Inc</sup></td><td>42.3<sup>Inc</sup></td><td>39.3<sup>Inc</sup></td><td>&gt;&gt;1</td><td>[28]</td></tr><tr><td>0.1</td><td>10.8</td><td>640</td><td>3.12<sup>Inc</sup></td><td>27.7<sup>Inc</sup></td><td>23.0<sup>Inc</sup></td><td>1.26</td><td>[29]</td></tr><tr><td>Bulk, two-crystal</td><td>-</td><td>2.9</td><td>640</td><td>~0.9<sup>Inc</sup></td><td>22.1<sup>Inc</sup></td><td>20.7<sup>Inc</sup></td><td>-</td><td>[27]</td></tr><tr><td>Bulk, multi-crystal</td><td>0.5</td><td>7.14</td><td>639</td><td>3.14<sup>Inc</sup></td><td>30.8<sup>Inc</sup></td><td>28.3<sup>Inc</sup></td><td>1.1</td><td>This work</td></tr><tr><td>Thin-disk</td><td>3.6</td><td>0.88</td><td>644</td><td>~1.7<sup>Inc</sup></td><td>7.2<sup>Inc</sup></td><td>5.8<sup>Inc</sup></td><td>-</td><td>[33]</td></tr><tr><td>Crystal-fiber</td><td>0.1</td><td>5.45</td><td>639</td><td>1.84<sup>Inc</sup></td><td>21.4<sup>Inc</sup></td><td>19.1<sup>Inc</sup></td><td>1.16</td><td>[31]</td></tr><tr><td>Fiber</td><td>0.8</td><td>9.1</td><td>635</td><td>1.32<sup>Launch</sup></td><td>27<sup>Launch</sup></td><td>25.3<sup>Launch</sup></td><td>1.1</td><td>[30]</td></tr><tr><td>Waveguide<sup>b</sup></td><td>0.3</td><td>0.50</td><td>639</td><td>0.04<sup>Inc</sup></td><td>53.2<sup>Inc</sup></td><td>51.6<sup>Inc</sup></td><td>1.66</td><td>[32]</td></tr></tbody></table>

Among the guided-wave configurations, remarkably, a Pr:ZBLAN double-clad fiber laser pumped by a fiber-coupled GaN diode module delivered 9.1 W at 635 nm with η = 27% (versus the launched pump power) and P<sub>th</sub> = 1.32 W [30]. Further power scaling was however limited by the damage of the fiber tip. Xiong *et al.* demonstrated a lightly-doped (0.1 at.%) crystal-fiber Pr:LiYF₄ diode-pumped laser delivering 5.45 W of red output with a moderate η = 21.4% [31]. The technology of fabrication of such laser elements is challenging, especially for soft fluoride crystals. Highly efficient laser operation was achieved in a Pr:LiYF₄ channel (ridge) waveguide laser under 2ω-OPSL pumping, benefiting from excellent mode overlap conditions, yielding 0.50 W with η = 53.2% and P<sub>th</sub> ~ 40 mW [32]. Direct diode pumping of such waveguides represents a challenge for efficient coupling of low-brightness output into ridges.

Figure 2 illustrates the concept of a periodically extended multi-crystal solid-state laser (PESSL) and laser amplifier. The linear laser cavity comprises *n* identical single-crystal units, the back mirror and the output coupler. Each unit includes the gain medium (various pumping geometries are possible, e.g., longitudinal or side pumping), and beam focusing optics, represented by a lens or a dielectric mirror. The pumping induces a thermal lens in each laser crystal. The mode shaping and the resonator stability conditions are determined by both the beam focusing optics and the thermal lenses. However, the cavity is designed such that adding *n* + 1 unit (and, accordingly, another thermal lens) does not disturb the cavity stability and mode matching. Moreover, the periodic design ensures the same transverse and longitudinal mode distribution

<footnote><sup>a</sup>C<sub>Pr</sub> – Pr doping level, P<sub>out</sub> – continuous-wave output power, λ<sub>L</sub> – laser wavelength, P<sub>th</sub> – laser threshold; η – slope efficiency, η<sub>opt</sub> – optical efficiency (laser output power divided by the incident (launched, in case of fiber) pump power), M<sup>2</sup> – beam quality parameter. <sup>b</sup>High-brightness pumping.</footnote>
within each unit. The heat load is evenly distributed over *n* crystals. Altogether, this enables linear scaling of the output power with the number of crystals. The same concept is further extended to a laser amplifier. For the same spatial characteristics of the seed laser mode as the mode of the equivalent laser cavity, the same beam propagation is expected.

<figure><img src="image_3.png" /><figcaption>**Fig. 2.** Concept of a periodically extended multi-crystal solid-state laser / amplifier.</figcaption></figure>

The idea of power scaling of solid-state lasers by implementing multiple gain media into the same stable laser cavity was proposed a long time ago and developed for flashlamp pumped laser rods [34,35]. Note that here we exclude aperiodic multi-crystal cavity designs employing two or several gain crystals in a non-periodic way which allow for certain power scaling as compared to single-crystal cavities but cannot be further extended to accommodate any number of gain elements [36,37]. The theory of cavity stability employing multiple laser elements with thermal lenses was developed based on the ray transfer matrix formalism [38]. Spühler *et al.* presented a periodically extended mode-locked laser employing three diode side pumped Nd:YAG laser heads [39]. Recently, Yang *et al.* reported on a periodically extended Nd:YAG master oscillator power amplifier (MOPA) operating in the picosecond regime [40]. The amplifier consisted of three identical stages each comprising two side-pumped laser rods and pairs of plane folding mirrors and lenses for beam reshaping between the stages. Based on the well-developed Nd:YAG laser technology, this source delivered an output pulse energy of 71.1 mJ at a repetition rate of 1 kHz and a pulse duration of ~64 ps accompanied with a high beam quality $M^2$ ~ 2–3.

In the present work, we demonstrate an innovative approach to power scaling of diode-pumped solid-state lasers directly emitting in the visible based on a periodically extended laser cavity architecture, using Pr:LiYF₄ as a reference material for red laser emission. This approach utilizes building blocks with low technological requirements (namely, bulk laser crystals and free-space blue GaN diode lasers) and enables power scaling simply by adding single-crystal cavity units. As a result, dedicated thermal management for each individual laser crystal is not required. The proposed design benefits from the advantages of simple linear single-crystal cavities, namely high optical conversion efficiency and high spatial beam quality of the laser emission.

## 2. Periodically extended laser cavity: design and stability

### 2.1. Laser architecture

To prove the viability of the concept of a periodically extended laser architecture for power scaling of blue diode-pumped rare-earth solid-state lasers directly emitting in the visible, we have selected the praseodymium (Pr³⁺) doped lithium fluoride crystal (LiYF₄) as gain medium, which is currently recognized as state-of-the-art material for this kind of coherent light sources. The laser was operated in the red (the high-gain ³P<sub>J</sub> → ³F<sub>2</sub> transition). The layout of the cavity is depicted in Fig. 3. Each single-crystal *n*-th cavity unit comprises a V-shaped design with a plane
dichroic mirror DM$_{n}$ (coated for high transmission (HT, $T = 90\%$) at 444 nm and high reflection (HR) at 640 nm), and two HR curved folding mirrors CM$_{n1}$ and CM$_{n2}$ (radius of curvature, $RoC = -200$ mm) forming the waist of the laser mode on the dichroic mirror. The DM to CM distance is set to $190 \pm 1$ mm. The adjacent units are sharing these curved mirrors. The cavity is ended by a plane HR rear mirror on one side, and on another side – by a plane output coupler with a transmission at the laser wavelength $T_{OC}$ in the range of $4\%$ to $25\%$. The folding angle $\theta$ is set at $8^\circ$ to reduce the astigmatism. The total cavity length (for 3 units) is 178 cm.

<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Layout of the periodically extended diode-pumped red Pr:LiYF$_4$ laser: (a) Experimental setup with $n = 3$ single-crystal units; (b) a single unit; (c) orientation of the laser crystal. LD$_{n1(2)}$ – blue GaN diode lasers, CL$_{n1(2)}$ – cylindrical lenses, FL$_{n1(2)}$ – focusing lenses, DM$_{n}$ – plane dichroic mirrors, CM$_{n1(2)}$ – curved folding mirrors, HR – plane rear mirror, OC – plane output coupler, *dashed rectangle* indicates the periodically extended single-crystal unit.</figcaption></figure>

The laser elements are cut from 0.5 at.% Pr:LiYF$_4$ crystal for light propagation along the **a**-axis (**a**-cut), with an aperture of $5(c) \times 5$ mm$^2$ and a thickness $t$ of 5 mm. They are polished on both sides to laser grade quality and antireflection (AR, $R < 0.2\%$) coated at both the pump and laser wavelengths. The crystals are mounted in Cu-holders using an Indium foil for better heat removal. The holders are cooled by circulating water (12 °C). The crystals were placed next to the DMs leaving a small airgap of less than 1 mm.

To optically pump the laser elements, we used a set of 2n free space blue GaN diode lasers (*Nichia*) each delivering a maximum output power of $5.0 \pm 0.3$ W at a central wavelength of $444.0 \pm 0.3$ nm (at the maximum driving current) with a laser linewidth (full width at half maximum) of ~1-2 nm. The pump wavelength addressed the $^3$H$_4 \rightarrow ^3$P$_2$ transition of Pr$^{3+}$ ions. The unwanted temperature drift of the diode wavelength was avoided by stabilizing the diode temperature using individual Peltier elements and cooling the Cu diode mounts by circulating water (18 °C). The diode output was collimated using a short focal length *LIMO* cylindrical lens (focal lengths: $f = 1 \times 6$ mm). The measured beam quality parameters $M^2$ of the pump radiation were 2 by 10 along the “fast” and “slow” axes, respectively. The diodes provided a linearly polarized emission which corresponded to $\pi (E \parallel c)$ in the optically uniaxial Pr:LiYF$_4$ crystal leading to higher pump absorption efficiency. Each crystal was symmetrically pumped from one side (through the DM) by 2 diode lasers following the V-shaped geometry of the individual periodic cavity unit. The pump radiation was focused by an AR coated spherical lenses ($f = 100$ mm, Thorlabs). This focal length provides an optimal compromise between spot size and divergence within the crystal. A shorter focal length would reduce ellipticity but increase divergence, thereby decreasing overlap, whereas a longer focal length would enlarge the spot and increase the mismatch with the laser mode. The pumping was in a single pass, and the measured small signal pump absorption was $87 \pm 2\%$. The residual pump was evacuated through the CMs.
The pump beam had a rectangular stripe shape, and the measured pump spot size (diameter) in the focus at the $1/e^2$ level was 25 ± 5 µm and 70 ± 10 µm in the vertical and horizontal directions, respectively.

The spectra of laser emission were measured using an optical spectrum analyzer (Yokogawa AQ6373E) with a resolution of 0.05 nm. The spatial profile of the laser mode in the far-field was captured using a CCD-camera (Edmund Optics). The $M^2$ parameters of laser emission were determined by an ISO-standard method using a collimating lens ($f = 300$ mm). The crystal temperature was monitored by a thermal imaging camera (FLIR E5).

## 2.2. Laser stability analysis

Thermo-optic effects are generally considered as the main limiting factor for power scaling and efficient operation of diode-pumped solid-state lasers. Thermal lens of the laser element is an ideal lens which action on the laser mode and cavity stability is equivalent to the integral action of the thermo-optically distorted laser element. Thermal lens is power-dependent, and it can be described using the model of a thin or a thick lens depending on the element geometry. Below we consider two effects caused by thermal lens: i) it changes the resonator stability conditions (potentially bringing the cavity to the instability zone and causing a power roll-over or even laser ceasing), and ii) it affects the size of the laser mode in the active element, even when the cavity remains within the stability zone (which changes the mode-matching efficiency and can potentially induce higher-order-mode generation). Thermal lens is astigmatic (even for optically isotropic materials), inducing the ellipticity of the laser beam. The optical power of the thermal lens $D$ (inverse of its focal length $f$) could be positive or negative, depending on the material properties.

A linear laser resonator with a single laser element (and a single thermal lens) could be represented on the equivalent $g$-diagram by a straight line. The actual point on this line depends on the optical power of the thermal lens $D$. The power-dependent thermal lens drives the resonator along this straight line resulting in the existence of two stability zone (in general case). The multi-crystal cavity is however represented by a curve which can reverse its direction resulting in multiple, narrowed stability zones which complicate the laser design. The criteria for the design of a periodically extended cavity with $n$ single-crystal units could be formulated as follows (in terms of an addition of $n + 1$-th unit): i) it should not bring the resonator into the instability zone considering the power-dependent thermal lens; ii) it should keep the laser mode size in all the gain media within the desired range; iii) it should tolerate small changes in the length of the cavity arms and small variation of the thermal lens parameters between the laser crystals.

It is thus relevant to consider the effect of multiple crystals on cavity stability and laser mode size. This was done within the ray transfer matrix (so-called ABCD) formalism. A thin thermal lens was considered. Due to the 8° folding angle within the laser crystal, it was assumed that each LD induced its own independent thermal lens. The astigmatism of the thermal lens was neglected. The TEM$_{00}$ mode was considered (only the astigmatism induced by the cavity design was considered). The results of this analysis are presented in Fig. 4 for periodically extended cavities depicted in Fig. 3 with a different number of single-crystal units $n$ ranging from 1 to 7. First, the cavity stability parameter $g = 1 - (A+D)^2/4$, where A and D are the diagonal elements of the round-trip ray transfer matrix of the cavity, was calculated versus optical power of a positive (focusing) thermal lens (the cavity is stable for $0 < g \le 1$). Second, the laser mode radius $w_L$ at the pump face of the laser crystal was calculated for the sagittal and tangential planes.

On increasing the number of crystals, in terms of cavity stability, the maximum affordable optical power of the thermal lens decreases, the stability zones split into sub-zones and become narrower, which constrains the laser resonator design, in particular in terms of tolerances for the length of the periodic cavity arms. The dashed area in Fig. 4(a) marks the operation range of the designed cavity at full diode power, indicating that the resonator stays around the center of the
<figure><img src="image_5.png" /><figcaption>Fig. 4. Stability analysis for a periodically extended diode-pumped Pr:LiYF₄ laser with a different number of single-crystal units *n* ranging from 1 to 7: (a) calculated cavity stability parameter *g*; (b) calculated mean laser mode size at the pump face of the laser crystals *w*<sub>L</sub>. Dashed area marks the operation range of the designed laser (see Appendix A for the thermal lens calculation).</figcaption></figure>

stability zone independent of the number of single-crystal units *n*. In terms of the laser mode size, stronger thermal lens leads to a slight beam expansion (the effect becomes more significant for *n* > 3). Solid-state lasers operating on the four-level scheme (without reabsorption at the laser wavelength which can contribute to additional spatial mode filtering) are prone to generation of higher-order modes as soon as the size of the fundamental mode (TEM₀₀) is smaller than the pump spot size. In this regard, the above-mentioned action of the thermal lens is favorable for preserving the high beam quality at high pump levels.

### 3. Proof of concept: power scaling of red laser

The power transfer curves for the periodically extended ($n=3$) diode-pumped Pr:LiYF₄ laser are shown in Fig. 5(a). Each point corresponds to adding an additional GaN laser diode ($2n$ in total) operated at the maximum available power. In the true continuous-wave regime, the laser generated a maximum output power of 7.14 W in the red corresponding to a slope efficiency $\eta$ of 30.8% / 35.4% (versus incident/absorbed pump power) and an approximated laser threshold $P_{th}$ of 3.14 W (for the highest tested output coupling $T_{OC}$ of 25%). The input-output dependence was linear up to at least the maximum applied pump power of 25.2 W. The optical blue-to-red conversion efficiency amounted to 28.3%.

To analyze the limitations for the maximum slope efficiency achievable in the designed laser, the loss analysis was performed. For a four-level laser, the upper limit for $\eta$ with respect to the absorbed pump power is set by the expression [41]:

$$ \eta < \eta_{St,L} \times \eta_q \times \eta_{mode} \times \eta_{OC}, \quad (1) $$

where $\eta_{St,L}$ is the Stokes efficiency under lasing conditions, $\eta_q \approx 1$ is the quantum efficiency (being close to unity for Pr³⁺-doped fluoride laser materials), $\eta_{mode}$ is the mode overlap efficiency, and $\eta_{OC} = \gamma_{OC}/(\gamma + \gamma_{OC})$ is the output-coupling efficiency, where $\gamma_{OC} = -\ln(1 - T_{OC})$ and $\gamma = -\ln(1 - 2L)$ are the output-coupling and passive loss, respectively, $2L$ is the round-trip propagation loss. The round-trip passive loss was estimated by applying the Caird analysis, i.e., by plotting inverse of the slope efficiency versus inverse of the output-coupling loss: $1/\eta = 1/\eta_0(1 + \gamma/\gamma_{OC})$,
<figure><img src="image_6.png" /><figcaption>**Fig. 5.** Output performance of the periodically extended ($n = 3$) diode-pumped red Pr:LiYF$_4$ laser: (a) input-output dependences for various output coupling, $\eta$ – slope efficiency; (b) the Caird analysis of overall roundtrip propagation losses $L$; (c) simulation of the output power versus output coupling $T_{OC}$ for $n$-crystal cavities ($n = 1$ to $7$) at the maximum available incident pump power from $2n$ GaN diodes, points – experimental data; (d) typical spectra of laser emission: effect of the uniform increase of the incident pump power for each laser diode. Laser polarization is $\sigma$.</figcaption></figure>

where $\eta_0 \approx \eta_{St,L} \times \eta_{mode}$ is the intrinsic slope efficiency, see Fig. 5(b). This analysis yields the best-fit parameters of $2L = 4.2 \pm 0.5\%$ (this value includes scattering and residual Fresnel losses at laser crystals, as well as losses at the cavity mirrors) and $\eta_0 = 44.2 \pm 5\%$. Consequently, the overlap efficiency between the pump and laser modes within the multiple gain media is extracted, $\eta_{mode} = 63.7 \pm 5\%$. This value originates from the non-perfect overlap between the stripe-like mode profiles from the blue GaN diodes and the nearly Gaussian red laser mode.

Based on the determined cavity parameters (the overall roundtrip loss and the mode overlap efficiency), the dependence of the output power on the output coupling was calculated for the multi-crystal ($n = 3$) laser, see Appendix A. The results are shown in Fig. 5(c). The model predicts an optimum output coupling of about 20%, close to the one used in the present work. By assuming a linear variation of loss with the number of crystals (see below), and a nearly constant mode overlap efficiency, Eqs. (1)-(2) were used to predict the maximum extractable output power and the optimum output coupling for $n = 1$ to $7$ crystals, Fig. 5(c). The model predicts a gradual increase of the optimum $T_{OC}$ with the number of crystals, from 6% to 45%, respectively, due to the gradually increased losses in the cavity. For the periodically extended cavity with $n = 7$, one could expect the generation of >17 W of red power for about 70 W from $2n$ blue GaN diode lasers.

The red laser emission was linearly polarized, and the polarization state ($\sigma$) was naturally selected by the anisotropy of the gain. The typical spectra of laser emission, monitored at different output power levels, are presented in Fig. 5(d). On increasing the pump power, the laser wavelength experiences a slight red shift, from 639.5 to 639.7 nm, and the spectrum broadens (laser linewidth at full output power: 0.2 nm). To explain this behavior, the temperature dependence of the stimulated-emission (SE) properties of Pr$^{3+}$ ions in the LiYF$_4$ crystal was studied. Figure 6(a) depicts the SE cross-section, $\sigma_{SE}$, spectra for the $^3$P$_J \rightarrow ^3$F$_2$ transition for $\sigma$-polarized light. On increasing the temperature from 25 °C to 200 °C, the peak emission wavelength experienced a red shift from 639.52 nm to 639.79 nm, the peak $\sigma_{SE}$ dropped from $3.10 \times 10^{-19}$ cm$^2$ to $1.77 \times 10^{-19}$ cm$^2$, and the emission bandwidth broadened from 0.75 nm to
1.13 nm. The luminescence lifetime of the emitting state $^3$P$_J$ was nearly constant over the studied temperature range, $\tau_{lum} = 49.7$ µs, Fig. 6(b). This analysis allowed us to estimate the maximum temperature in the hot (pumped) zone of the crystal, about 180 °C.

<figure><img src="image_7.png" /><figcaption>**Fig. 6.** Temperature-dependent emission properties of Pr$^{3+}$ in LiYF$_4$: (a) stimulated-emission cross-sections, $\sigma_{SE}$, for the $^3$P$_J \rightarrow ^3$F$_2$ transition, $\sigma$-polarization; (b) luminescence lifetime of the $^3$P$_J$ states.</figcaption></figure>

The red laser beam profiles in the far-field (at 40 cm from the OC) were captured for the optimum output coupling, and the effect of increasing pump level was studied, see Fig. 7. First, the output of each blue GaN diode was adjusted simultaneously resulting in a uniform variation

<figure><img src="image_8.png" /><figcaption>**Fig. 7.** Spatial characteristics of laser emission from the periodically extended ($n=3$) diode-pumped red Pr:LiYF$_4$ laser: (a,b) laser mode profiles in the far-field: (a) effect of the uniform increase of the incident pump power for each laser diode; (b) effect of subsequent switching on of different laser diodes operated at full power; (c) evaluation of the beam quality parameters $M^2_{x,y}$ at the maximum pump level; (d) the corresponding 1D intensity profiles (symbols) and their Gaussian fits (curves). The laser polarization is vertical.</figcaption></figure>
of the thermal lens of each laser crystal. As expected, near the laser threshold, the laser mode is almost circular (beam ellipticity: $e = w_x/w_y = 0.97$) and close to the TEM$_{00}$ one, and on increasing the overall incident pump power, it experienced a significant deformation around $P_{inc}$ of 20 W, then returning to a slightly elliptic TEM$_{00}$ one ($e = 0.71$), see Fig. 7(a). This is related to the fact that the laser resonator passes through an instability region, cf. Figure 4. In the second experiment, the laser diodes operating at full power were switched on one by one, Fig. 7(b). In this case, the resonator contains thermal lenses in hot (pumped) laser crystals, and no lensing appears in cold (unpumped) ones. This affects cavity stability in a severe way, as evidenced by a strongly multimode output observed over a broader range of pump powers (physically, due to the broadening of the instability zone).

The evaluation of the beam quality factor at the maximum available pump level is presented in Fig. 7(c), yielding $M_x^2 = 1.13 \pm 0.05$ and $M_y^2 = 1.12 \pm 0.05$, indicating operation on the fundamental mode. The corresponding 1D intensity profiles alongside their Gaussian fits are depicted in Fig. 7(d). These observations confirm laser operation on the fundamental transverse mode.

## 4. Discussion: from single-crystal to multi-crystal cavity

In this section, we systematically study the effect of the number of single-crystal units $n$ in a periodically extended diode-pumped Pr:LiYF$_4$ laser on its performance characteristics, Fig. 8. The maximum output power was found to scale linearly with $n$ (for output coupling optimized for a given number of crystals), see Fig. 8(a). This allowed for the extraction of 2.3 W of red emission per crystal (or, equivalently, per two GaN diodes), projecting into >16 W of output power for $n = 7$ oscillator. The corresponding optical efficiency with respect to absorbed pump power was nearly constant, $31.7 \pm 1\%$, Fig. 8(b).

The mode overlap efficiency $\eta_{mode}$ estimated from the Caird analysis remained in the range $64\% - 73\%$, being strongly affected by variations in the beam quality for each GaN laser diode, Fig. 8(c). These values are in line with the calculation using the model of Laporta and Brussard [42] for elliptical pump beams having different mode sizes $w_{Pa} < w_{Pb}$ within the two principal orthogonal meridional planes (labelled as the “a” and “b” semi-axes). The laser threshold and the mode overlap efficiency are primary determined by two dimensionless parameters: $\alpha = (<w_L>/<w_{Pa}>)^2$, the squared ratio of the mean size of the laser mode, $<w_L>$, and the mean size of the pump mode along its smaller semiaxis, $<w_{Pa}>$, and $\beta = <w_{Pa}>/<w_{Pb}>$ ($0 < \beta \le 1$, the beam is circular for $\beta = 1$), inverse of the beam ellipticity.

The laser threshold increases almost linearly with $\alpha$ (when the laser mode size increases with respect to the pump beam size). For a given $\alpha$, the laser threshold also increases with the pump beam ellipticity. Thus, ideally, lower thresholds are attained for circular pump beams, tight pump beam focusing and laser mode sizes being smaller than the pump beam waist. Such behavior is ultimately achieved under high brightness pumping, e.g., with 2ω-OPSL blue lasers. The mode overlap efficiency is initially a rapidly increasing function of $\alpha$ (for $\alpha < 1$) and then this trend diminishes with $\eta_{mode}$ approaching unity. This effect arises from the spatial variation of both the laser and pump fields. As the laser mode size becomes larger, the laser intensity is higher over a larger cross-sectional area of the pumped material volume, leading to a more efficient extraction of energy and higher slope efficiency.

To conclude, the mode overlap efficiency can be improved by expanding the laser mode cross-section to encircle the elliptical pump beam, but at the expense of a higher threshold. The best tradeoff in terms of output power is dependent on the pump level. For the designed laser, we followed this strategy, by targeting an optimized maximum output power rather than the lowest laser threshold. Further optimization of $\eta_{mode}$ could be attained using a pair of cylindrical lenses employed for better pump beam reshaping for each diode (at the expense of increased pump module complexity).
<figure><img src="image_9.png" /><figcaption>**Fig. 8.** Effect of the number of crystals *n* on the performance characteristics of periodically extended diode-pumped red Pr:LiYF<sub>4</sub> laser: (a) maximum output power; (b) optical efficiency vs. absorbed pump power; (c) mode overlap efficiency η<sub>mode</sub>; (d) roundtrip passive losses 2L; (e) measured average temperature of a given laser crystal versus the number of employed laser diodes.</figcaption></figure>

The round-trip passive loss 2L scaled nearly linearly with the number of crystals, measuring 1.6 ± 0.5% per crystal, affected by the small variations in quality between individual laser crystals, see Fig. 8(d). By monitoring the leak of red laser emission related to the residual reflectivity of the AR coating on the laser crystal, it was estimated to be R < 0.2%. Considering two passes of laser radiation (during a roundtrip), we conclude that the determined passive loss 2L is mainly coming from the dielectric coatings. Thus, this parameter can be further optimized – down to the limit set by scattering losses in the laser crystal (2L<sub>scatter</sub> ≈ 0.1% per crystal) – through careful design of the antireflection coatings. A multi-crystal cavity consisting of *n* crystals of thickness *t* is expected to incur greater cumulative passive losses than a cavity employing a single gain medium of equivalent total thickness (*n* × *t*), as in the case of a rod-shaped laser element. On the contrary, increasing the thickness of a single rod will inevitably complicate the mode matching conditions for the pump and laser radiation due to the strong divergence of the pump, the problem which is greatly avoided in the multi-crystal approach.

The average temperature of a given laser crystal within the periodically extended cavity was monitored as a function of the number of added GaN laser diodes (a total of 6 diodes for n = 3), Fig. 8(e). A nearly linear temperature rise was observed when pumping the considered crystal with one and two laser diodes, while the temperature of this crystal was almost independent of the pumping of other ones. This clearly proves the positive effect of the multi-crystal cavity on the thermal management of each particular laser element, making their thermal management independent while being able to multiply the laser power extracted from the oscillator.
## 5. Conclusion

We report on an innovative laser architecture for power scaling of diode-pumped solid-state lasers relying on the periodically extended resonator design consisting of n identical single-crystal units. Each unit utilizes technologically simple elements, such as bulk laser crystals without dedicated thermal management, and open space diode lasers. This ensures an even distribution of overall heat load over multiple laser crystals, reducing thermo-optic aberrations and the risk of thermal fracture. The proposed design benefits from advantages of linear single-crystal cavities, such as high optical conversion efficiency and good beam quality, which are maintained for a high number of single-crystal units n. In the case of a visible solid-state laser, this multistage PESSL concept allows an efficient conversion of a low-brightness laser diode into a high-brightness visible laser.

This concept is validated using a Pr:LiYF₄ gain medium for laser emission in the red. By subsequently implementing one, two or three single-crystal units each pumped by two open space GaN diode lasers, and adapting the output coupling, we demonstrate a linear variation of the roundtrip passive losses, an almost constant optical conversion efficiency and independent thermal management of each unit. At full pump power, the laser delivers >7 W of output power in the red in the true continuous-wave regime featuring a high optical efficiency ~28% and a near diffraction limited beam quality. Our analysis indicates the possibility of further power scaling above 10 W thus exceeding the best results achieved with the competitive technology of Pr:ZBLAN fluoride fiber lasers by adding more single-crystal units into the resonator. The proposed design could be easily extended to other laser transitions (green, orange or deep red) of Pr³⁺ ions (which represents a difficulty for fiber lasers in which a strong gain competition with the red transition is always present) or other rare-earth ions and host crystals.

We propose criteria for the design of periodically extended laser cavities, by considering the resonator stability conditions and the mode matching efficiency impacted by the multiple thermal lenses of the gain crystals. The main challenges of the multi-crystal laser architecture are as follows: i) the rapid increase in overall round-trip passive losses arising from the growing number of crystal interfaces with dielectric coatings, which constitute the dominant loss mechanism. Addressing this requires a compromise between power extraction from each crystal – through optimized doping level and/or geometry – and the total number of crystals employed; ii) the narrowing of cavity stability ranges with the addition of further single-crystal units, strongly influenced by variations in the thermal lens properties of each crystal, thereby limiting the practically achievable number of units; and iii) the increased overall system complexity, including the need for precise control of cavity arm lengths to maintain stability and mode overlap efficiency, as well as independent alignment of multiple optical elements comprising both the cavity optics and the pump beam reshaping stages. Taken together, these considerations suggest that periodically extended cavities incorporating up to ten single-crystal units may still remain feasible.

## Appendix A.

### Thermal lensing in a diode-pumped crystal

The sensitivity factor of the thermal lens, $M = dD/dP_{abs}$, where $P_{abs}$ is the absorbed pump power, is given by [1,43]:

$$M = \frac{\eta_h}{2S_P \kappa \chi}, \chi \approx dn/dT + (n-1)(1 + v^*)\alpha, \quad (A-1)$$

where $\eta_h$ is the fractional heat loading which can be approximated as the quantum defect, $1 - \lambda_P/\lambda_L$, $\lambda_P(L)$ are the pump and laser wavelengths, $S_P$ is the pump mode area, $\kappa$ is the thermal conductivity, and $\chi$ is the “generalized” thermo-optic coefficient. By omitting the term related to the photoelastic effect, the latter parameter constitute two terms: the temperature coefficient of
the refractive index $dn/dT$, and the end-bulging term, $(n-1)(1+v^*)\alpha$, where $n$ is the refractive index, $v^*$ is the effective Poisson ratio [44], and $\alpha$ is the coefficient of the linear thermal expansion along the direction of light propagation.

The thermo-optic coefficient $dn/dT$ of the LiYF$_4$ crystal was measured in this work by the laser beam deviation method under linear thermal gradient [45], yielding $-6.8 \times 10^{-6} \text{ K}^{-1}$ at 633 nm for $\sigma$-polarized light. Negative $dn/dT$ values are common for all fluoride crystals. The thermal expansion coefficient $\alpha$ was measured by the dilatometry method, giving $13.3 \times 10^{-6} \text{ K}^{-1}$ along the $\mathbf{a}$-axis. The “generalized” thermo-optic coefficient is very small and positive, $1.0 \times 10^{-6} \text{ K}^{-1}$, due to the counteraction of the two above mentioned terms, indicating a focusing thermal lens, in line with the previous studies [46,47]. For the designed pump architecture, we estimate $M$ to be $1.5 \pm 0.5 \text{ m}^{-1}/\text{W}$. This value is used in the resonator stability analysis.

### Output power for a four-level laser

The dependence of the output laser power $P_{out}$ on the output coupling $T_{OC}$ was calculated as:

$$P_{out} = \eta \times (P_{abs} - P_{th}), \qquad (A-2)$$

where $\eta$ is the laser slope efficiency [41]:

$$\eta = \eta_{St,L} \times \eta_{q} \times \eta_{mode} \times \eta_{OC}, \qquad (A-3)$$

where $\eta_{St,L} = \lambda_P/\lambda_L$ is the Stokes efficiency under lasing conditions, $\lambda_{P(L)}$ are the pump / laser wavelengths, respectively, $\eta_q \approx 1$ is the quantum efficiency (being close to unity for Pr$^{3+}$-doped fluoride laser materials), $\eta_{mode}$ is the mode overlap efficiency, and $\eta_{OC} = \gamma_{OC}/(\gamma + \gamma_{OC})$ is the output-coupling efficiency, where $\gamma_{OC} = -\ln(1 - T_{OC})$ and $\gamma = -\ln(1 - 2L)$ are the output-coupling and passive loss, respectively, $2L$ is the round-trip propagation loss, and $P_{th}$ is the laser threshold given by:

$$P_{th} = \eta_{mode} \frac{h\nu_P \gamma_{OC} + \gamma \pi w_L^2}{\tau_{lum} \sigma_{SE}^2}, \qquad (A-4)$$

where $h$ is the Planck constant, $\nu_P = c/\lambda_P$ is the pump frequency, $c$ is the speed of light, $\tau_{lum}$ is the luminescence lifetime of the upper laser level, $\sigma_{SE}$ is the stimulated-emission cross-section at $\nu_P$, and $w_L$ is the laser mode size in the gain medium.

### Expressions for optimization of the mode overlap efficiency

The mode overlap efficiency for the case of an elliptical pump beam is given by [42]:

$$\eta_{mode} = \frac{\alpha\beta[(\alpha\beta^2 + 2)(\alpha + 2)]^{1/2}}{[(\alpha + 1)(\alpha\beta^2 + 1)]}, \qquad (A-5)$$

where $\alpha = (<w_L>/<w_{Pa}>)^2$, $<w_L>$ is the mean size of the laser mode, and $<w_{Pa}>$ is the mean size of the pump mode along its smaller semiaxis, and $\beta = <w_{Pa}>/<w_{Pb}>$ (inverse of the beam ellipticity). The average mode size inside the gain medium is obtained using the following expression:

$$w_{Pa(b)} = \left[ \frac{1}{l^*} \int_0^{l^*} w_{Pa(b)}^2(z) dz \right]^{1/2}, \qquad (A-6)$$

where $z$ is the axial coordinate, and $l^*$ is an effective length related to the absorption length $l$/$\alpha_{abs}$ of the pump radiation ($\alpha_{abs}$ – absorption coefficient at the pump wavelength) and to the divergence angle $\theta_{Pa(b)}$ of the pump beam inside the crystal.

**Funding.** Agence Nationale de la Recherche (ANR-22-CE08-0025-01, ANR-23-LCV2-0001); Région Normandie.
**Disclosures.** The authors declare no conflicts of interest.

**Data availability.** Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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