Solar-Pumped Ce:Nd:YAG Laser Amplifier Design

Abstract

A solar-pumped Ce:Nd:YAG laser amplifier design is proposed to address the challenge of scaling output power in solar-pumped laser oscillators while maintaining high beam quality. The design employs a 1.33 m² flat Fresnel lens with a 2 m focal length as a primary concentrator, which is combined with a secondary homogenizing concentrator, featuring 40 mm × 40 mm input aperture, 200 mm length, and 11.3 mm × 26 mm output aperture, to provide efficient coupling and uniform distribution of solar radiation onto a 2.9 mm thick Ce:Nd:YAG slab with 11.3 mm × 26 mm surface area and two beveled corners. This geometry enables multiple total internal reflections of a 1064 nm TEM₀₀ mode seed laser beam inside the slab, ensuring efficient interaction with the active Ce³⁺ and Nd³⁺ ions in the gain medium. Performed numerical analysis shows that the present approach can deliver a uniform solar pump power density of 2.5 W/mm² to the slab amplifier. This value is 2.05-times higher than the numerically calculated power density incident on the Nd:YAG slab of the previous solar-pumped amplifier that achieved the highest continuous-wave laser gain of 1.64. Furthermore, the optimized slab geometry with 0.44 width-to-height ratio allows the seed laser to undergo 32 internal reflections, extending its optical path length by a factor of 1.45 compared to the earlier design. These numerical achievements, combined with the Ce:Nd:YAG medium’s capacity to deliver nearly 1.57-times more laser power than Nd:YAG, reveal the potential of proposed design to yield a gain enhancement factor of 4.16, making the first solar-pumped Ce:Nd:YAG amplifier a promising solution toward energy-efficient, sustainable solutions for terrestrial and space applications.

1. Introduction

Growing concerns about climate change have pushed science and technology toward sustainable energy pathways, with solar energy playing a key role in reducing ecological footprints [1]. Extensive research has focused on advancing solar cell technologies as a sustainable means of addressing the global energy crisis [2]. Solar-pumped lasers, which directly convert a portion of solar radiation into coherent laser light, offer a complementary pathway by substantially reducing electrical demand. This makes them exceptionally attractive for space-based applications [3,4,5,6] while also offering considerable advantages for terrestrial deployment [7,8,9,10], thereby advancing greener laser solutions.

Over six decades of research and development have shaped this renewable technology [11,12], with the last four years marked by an exceptional innovation owing to the implementation of cerium-doped neodymium:yttrium aluminum garnet (Ce³⁺:Nd³⁺:YAG) as active medium in solar-pumped lasers [12,13,14,15,16,17]. Nd:YAG had long been the predominant laser medium in solar-pumped laser systems, owing to the YAG host’s excellent thermal and mechanical stability combined with the favorable spectroscopic properties of the Nd³⁺ ion. The absorption spectrum of Nd:YAG extends from the ultraviolet to the near-infrared region, partially overlapping with the solar emission spectrum. However, its absorption bands are relatively weak and narrow, resulting in a limited spectral overlap of only 14–16% with the solar spectrum [11,12]. The Ce:Nd:YAG, on the other hand, active medium features two strong and broad absorption bands centered around 340 nm and 460 nm, attributed to Ce³⁺ ions, along with additional absorption bands above 525 nm associated with Nd³⁺ ions [11,12]. It also exhibits a broad fluorescence band centered at approximately 530 nm, spanning from 500 nm to beyond 600 nm, which effectively overlaps with key Nd³⁺ absorption lines. Under broadband pumping, efficient energy transfer from Ce³⁺ to Nd³⁺ ions occurs via both radiative and non-radiative mechanisms [18,19], which may result in significantly enhanced solar-pumped laser efficiency compared to the standard Nd:YAG medium. Even though the first laser emission from a single Ce:Nd:YAG rod exposed to concentrated sunlight in an end-side configuration fell short of expectations [20], subsequent experiments with side-pumped Ce:Nd:YAG solar laser oscillator demonstrated the capacity of Ce:Nd:YAG to extract 1.57-times more laser power than Nd:YAG under the same pumping conditions [13]. The pioneer integration of Ce:Nd:YAG as the active medium in this technology, apace with advancements in multirod solar pumping architectures, has also led to notable achievements in solar-pumped laser oscillators: record-breaking solar-to-laser conversion efficiencies of 2.06% and 4.64% in fundamental and multimode regimes, respectively [15,16]; a low solar power threshold of 16.5 W for laser emission [15]; and a high tracking error compensation tolerance of 5.1° [17], contributing to improved scalability and stability of solar laser output. Although Masuda et al. [21] had demonstrated the feasibility of solar laser emission without requiring a primary concentrator or tracking system, system efficiency improves markedly when both are implemented to ensure sufficient gain in the active medium.

The pursuit of highly efficient solar-pumped lasers with superior beam quality is essential for establishing this technology as a viable solution for both Space and Earth-based applications. Gaussian TEM₀₀ mode beam is the most used laser beam shape in numerous applications [22,23], since it has the smallest beam divergence and the highest power density, leading to the highest brightness and the ability to be focused to a diffraction-limited spot [24]. Nevertheless, conventional solid-state laser oscillators continue to face challenges in scaling output power while maintaining high beam quality, primarily due to power-dependent thermal lensing in the gain medium, which may significantly affect the resonator modes at high pump powers, degrading the output beam. Efficient extraction of the TEM₀₀ mode from solar-pumped laser oscillators has so far been achieved by pumping small-diameter laser rods inside large-mode-volume resonators [15]. The laser rod functions as an aperture, allowing the suppression of higher-order transverse modes when thinner rods are used. Furthermore, their use inside long resonators, operating near the edge of the optical stability region, reinforces TEM₀₀ mode selection. However, while this configuration significantly enhances the beam quality, it does so at the expense of overall efficiency. Additionally, operation near the stability boundary makes the TEM₀₀ mode output more sensitive to pump power fluctuations [25].

Solar-pumped Ce:Nd:YAG laser amplifiers could offer a promising solution for achieving high power and stable renewable laser emission while preserving beam quality. Although both laser oscillators and laser amplifiers operate based on the principles of stimulated emission, their functions differ significantly. A laser oscillator generates coherent light with distinct phase and mode characteristics through resonant oscillation, without requiring an external input signal. In contrast, a laser amplifier enhances the power of an external laser beam without the need for a resonant cavity to sustain oscillation [26]. As a result, an amplified laser beam can exhibit increased power and intensity while retaining most of its original properties. However, despite the potential of laser amplifiers, solar-pumped laser research to date has focused mostly on improving oscillator performance, resulting only in few reports on sunlight-powered laser amplifiers [27,28,29,30], and none involving Ce:Nd:YAG medium. To date, the maximum continuous wave (CW) laser power amplification factor attained with a solar-pumped amplifier is 1.64, reported in 2013 by Guan et al. [28]. Their system employed a Nd:YAG gain medium in a slab configuration that enabled total internal reflection of the seed laser, with solar pumping provided by concentrated sunlight focused through a Fresnel lens with effective collection area of 1.33 m². Still, besides the active medium material’s limited suitability for high laser gain, this setup also lacked optical elements to homogenize the concentrated sunlight. As a result, the active medium was pumped under a non-uniform pump profile, which may limit the system’s suitability for scaling to higher laser power levels.

An alternative solar-pumped laser amplifier approach here proposed introduces Ce:Nd:YAG as an optical amplifier, with the potential to revitalize solar-pumped laser amplifier technology. The design parameters of this concept were numerically optimized by using the Zemax^® (Zemax^® is a registered trademark of Zemax LLC Copyright © Zemax LLC 1990–2014. The company is based in Kirkland, WA, USA) ray-tracing. Although the present approach adopts the same type of primary concentrator and active medium geometry as the previous Nd:YAG amplifier design under solar pumping [28], it introduces two additional key innovations beyond the active medium material:

(i)

The incorporation of a rectangular hollow secondary concentrator that allowed not only an effective coupling of the concentrated sunlight from the Fresnel lens onto the slab medium at its output, but also served as a homogenizer, ensuring uniform pump light distribution and mitigating thermal effects.

(ii)

The integration of a gain medium with a reduced width-to-height ratio, optimized to substantially increase the number of seed laser passes through the pumped region of the active material.

The combination of the Fresnel lens, with 1.3 m diameter and 2.0 m focal length, with the rectangular-shaped secondary concentrator, numerically provided a uniform incident solar power density of 2.5 W/mm² on the slab medium measuring 11.3 mm in width, 26 mm in height, and 2.9 mm in thickness. These dimensions yielded the same surface area and volume as the slab used in the previous setup [28], differing only in the width-to-height ratio. Consequently, the resulting power density incident on the slab of the proposed configuration was more than twice the value obtained in numerical simulations of the earlier solar-pumped Nd:YAG amplifier under comparable pumping conditions [28].

Pumping intensity is a key requirement for achieving laser gain, as it governs the level of population inversion and thus the amount of energy available for amplification. But no less important is the coupling efficiency between the laser beam and the excited regions of the active medium, which dictates how effectively this stored energy is transferred to the propagating laser signal [11,26]. Therefore, following an approach similar to that in [28], the slab-geometry laser medium featured two beveled corners, designed to permit a 1 W CW TEM₀₀ mode laser beam at 1064 nm to enter the amplifier perpendicularly through one beveled edge and undergone multiple internal reflections before exiting through the opposite beveled corner. The optimized slab medium, with a width-to-height ratio of 0.44, significantly lower than the 0.91 ratio used in the previous design [28], led to a 1.45-fold extension of the seed laser optical path length.

Based on these numerical developments, a 4.16-fold increase in laser gain could be predicted with the proposed solar-pumped Ce:Nd:YAG laser amplifier. Even if this configuration was applied to pump a Nd:YAG medium, it could still potentially increase laser gain by 2.66 times in relation to the previously reported maximum value of 1.64 for a Nd:YAG-based amplifier under solar pumping [28].

2. Solar-Pumped Ce:Nd:YAG Laser Amplifier Concept

Notable progress in the efficiency of solar-pumped laser oscillators, in both multimode and TEM₀₀ mode regimes, has been achieved using parabolic mirrors as primary concentrators [11,12]. Nonetheless, these systems often require elaborate and costly infrastructure, and the placement of the laser head and its supporting mechanisms in front of the parabolic mirror obstruct the incoming sunlight, limiting their practicality. In contrast, flat Fresnel lenses are lightweight and suitable for mass production [31], reducing system complexity and overall cost. Despite the inherent chromatic aberration and diffraction losses through their prism facets [32], these losses can remain well below 1% across the entire spectral range relevant to the pumping of the Ce:Nd:YAG media [33]. In addition, because the focal spot of a Fresnel lens lies in their rear side, it avoids shadowing from the laser head and facilitates easier access during experiments. For these reasons, the proposed solar-pumped laser amplifier employed a Fresnel lens as primary concentrator, as depicted in Figure 1a. It had a maximum diameter of 1.3 m (1.33 m² collection area) and a focal length of 2.0 m, focusing the incident solar radiation into a near-Gaussian spot with an approximate 1/e² width of 40 mm.

The concentrated solar radiation at the focus was then collected by the entrance aperture of the rectangular secondary concentrator which, along with the Ce:Nd:YAG slab positioned at its exit, constitutes the laser amplifier head, as schematized in Figure 1b. This hollow reflective concentrator with 40 mm × 40 mm input aperture and 200 mm length, was designed to both funnel and uniformize the concentrated solar radiation from the Fresnel lens to the Ce:Nd:YAG slab at its narrower 11 mm × 26 mm output end. It reshaped the near-Gaussian radiation with low divergence from a large-area source into a uniform light distribution with increased angular dispersion, confined to a smaller area (Figure 1b). This implied that solar flux was larger at the concentrator output end than at its entrance aperture, leading to a net light concentration and uniformity, not attainable by classical image-forming concentrators [34]. Such uniformity is critical for preventing hot spots in the active medium that could degrade both thermal and optical performance. More importantly, it also ensured uniform interaction between the seed laser and the excited regions of the medium.

To demonstrate that solar-pumped laser amplifiers employing Fresnel lenses as primary concentrators can deliver competitive performance while maintaining economic viability compared to those using parabolic mirrors, the numerical output performance of the proposed Fresnel lens-homogenizing concentrator stage was evaluated against that of a parabolic mirror-homogenizing concentrator stage. Both configurations were modeled with identical collection areas, focal lengths, and input/output homogenizer dimensions, as summarized in

Table 1. Comparative output performance of the proposed Fresnel lens–homogenizing concentrator stage and a parabolic mirror-homogenizing concentrator stage.

Parameters	Fresnel Lens-Homogenizer Stage	Parabolic Mirror-Homogenizer Stage
Primary concentrator:
Collection area	1.33 m2
Focal length	2.0 m
Secondary concentrator:
Input aperture	40 mm × 40 mm
Length	200 mm	320 mm
Output aperture	11 mm × 26 mm
Useful pump light distribution at the homogenizer input aperture
	Useful pump power: 296 W	Useful pump power: 310 W
Useful pump light distribution at the homogenizer output aperture
	Useful pump power: 230 W	Useful pump power: 203 W

. The reflective surfaces of both the parabolic mirror and secondary concentrators were specified to be silver-coated aluminum with 94% hemispheric solar reflectance [11]. The spectral reflectance remains high at 98–99% between 500 and 750 nm, before declining slightly to 96–97% at 900 nm [35].

Numerical simulations showed that the parabolic mirror yielded about 310 W of useful pump power at the homogenizer entrance, producing an input light distribution with a higher peak intensity of 0.96 W/mm² and a narrower 1/e² width of 29 mm compared to the Fresnel lens, which yielded 296 W of useful pump power, a peak intensity of 0.65 W/mm², and a 1/e² width of 40 mm. However, achieving a uniform profile at the homogenizer output with the parabolic mirror required extending its length by an additional 120 mm, which increased transmission losses. As a result, the parabolic mirror–homogenizer stage delivered nearly 12% less useful pump power (203 W) at the output than the Fresnel lens-homogenizer stage (230 W), demonstrating that the Fresnel lens-homogenizer combination is more effective at uniformly pumping the Ce:Nd:YAG amplifier medium.

To maximize the optical path length of the seed laser within the amplifier, the proposed Ce:Nd:YAG amplifier was designed in a slab geometry with dimensions of 11.3 mm in width (W), 26 mm in height (H), and 2.9 mm in thickness (T), as shown in Figure 2. Two corners of the slab were beveled, one to allow the seed laser to enter the medium and the other to enable the exit of the amplified beam. These bevels were set at 45°, an angle chosen to ensure that the laser beam, entering perpendicular to the slab surface, propagated efficiently through the medium via total internal reflection. This configuration enabled a 1064 nm TEM₀₀ mode continuous-wave seed laser, with a divergence of less than 1.2 mrad, to follow a 365 mm optical path that covered nearly the entire active region of the gain medium.

A high-reflectivity mirror for the visible spectral range was incorporated on the rear surface of the slab to enhance the absorption of the pump radiation. Thermal dissipation can be accomplished via a water-cooled heat sink set behind the slab’s rear surface [28].

3. Numerical Modeling and Analysis of the Solar-Pumped Ce:Nd:YAG Laser Amplifier

The design specifications of the present solar-pumped laser amplifier were optimized using Zemax^® ray-tracing software in non-sequential mode, with the goal of maximizing both the pump power delivered to the amplifier and the optical path length of the seed laser within it. All optical elements were defined using the non-sequential object types available in Zemax^® through the Component Editor window [11].

3.1. Solar Radiation Collection and Focusing System

To simulate the spectral characteristics of the solar input, two circular sources were defined, as indicated in

Table 2. Wavelengths and relative weights of the light source in Zemax^® for a Ce:Nd:YAG laser amplifier system.

Parameters	Source 1											Source 2
Wavelength (nm)	527	531	569	579	588	592	732	736	743	746	753	527	531	569
	758	790	793	803	805	808	811	815	820	865	880	579	588	592
Weight	0.82	1.00	0.92	0.92	0.93	0.90	0.73	0.77	0.79	0.79	0.78	1.00	0.99	0.71
	0.78	0.70	0.70	0.69	0.68	0.70	0.68	0.58	0.56	0.62	0.61	0.63	0.54	0.49

, each with an area of 1.33 m², consistent with the Fresnel lens’ collection surface. One of the light sources (Source 1) was configured to represent the most dominant solar wavelengths that overlap with the absorption bands of Nd³⁺ ions in YAG, as well as those involved in non-radiative energy transfer from Ce³⁺ to Nd³⁺ through quantum cutting processes [18]. While the other source (Source 2) was configured to capture the solar wavelengths associated with radiative transfer from Ce³⁺ to Nd³⁺. To this end, the solar spectrum was divided into two separate datasets corresponding to each source. The spectral profile for Source 1 was based on previous studies of solar-pumped Nd:YAG lasers [11], comprising 22 dominant absorption wavelengths of Nd³⁺. The relative contribution of each wavelength was weighted by its spectral irradiance within the standardized AM1.5D (Air Mass 1.5 Direct) solar spectrum [36], i.e., the solar spectral irradiance distribution defined to represent the direct sunlight reaching the Earth’s surface. For Source 2, six key wavelengths were selected, which align with Nd³⁺ absorption peaks that correspond to emission lines from Ce³⁺ fluorescence [16]. In this case, the Ce³⁺ fluorescence spectrum served as the basis for determining the relative intensity weights for each wavelength [11].

The apparent half angle of 0.27° subtended by the Sun on the Earth surface was considered on both sources. The total power contributions of each source—P_source1 and P_source2 —were calculated using the following equations [11]:

$$P_{source1}=A\times I_{\mathrm{S}}\times ({\eta }_{\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{l}\mathrm{a}\mathrm{p},{\mathrm{N}\mathrm{d}}^{3+}}+ {\eta }_{\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{l}\mathrm{a}\mathrm{p},{\mathrm{C}\mathrm{e}}^{3+}}\times {\eta }_{\mathrm{N}\mathrm{R}:{\mathrm{C}\mathrm{e}}^{3+}\to {\mathrm{N}\mathrm{d}}^{3+}})$$

(1)

$$P_{source2}=\mathrm{A}\times I_{\mathrm{S}} \times {\eta }_{\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{l}\mathrm{a}\mathrm{p},{\mathrm{C}\mathrm{e}}^{3+}}\times {\eta }_{\mathrm{R}:{\mathrm{C}\mathrm{e}}^{3+}\to {\mathrm{N}\mathrm{d}}^{3+}}$$

(2)

where

▪

A is the collection area of the Fresnel lens, set to 1.33 m²;

▪

I_S represents the terrestrial solar irradiance;

▪

${\eta }_{\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{l}\mathrm{a}\mathrm{p},{\mathrm{N}\mathrm{d}}^{3+}}$ expresses the spectral overlap efficiency between the solar spectrum and Nd³⁺ absorption bands, estimated to be around 16% for typical Nd³⁺ concentrations of 1.0–1.1 atomic percent (at.%) in YAG [11,12];

▪

${\eta }_{\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{l}\mathrm{a}\mathrm{p},{\mathrm{C}\mathrm{e}}^{3+}}$ is representative of the spectral overlap efficiency between the solar spectrum and the absorption bands of Ce³⁺ ions, which was determined to be 15.3%, based on a typical Ce³⁺ concentration of 0.1 at.% in Ce:Nd:YAG solar-pumped lasers [11,12];

▪

${\eta }_{\mathrm{N}\mathrm{R}:{\mathrm{C}\mathrm{e}}^{3+}\to {\mathrm{N}\mathrm{d}}^{3+}}$ refers to the non-radiative energy transfer efficiency from Ce³⁺ to Nd³⁺, considered to be approximately 70% [18];

▪

${\eta }_{\mathrm{R}:{\mathrm{C}\mathrm{e}}^{3+}\to {\mathrm{N}\mathrm{d}}^{3+}}$ corresponds to the radiative energy transfer efficiency from Ce³⁺ to Nd³⁺, estimated at 30% [18].

This demonstrates that the portion of the solar spectrum matching the absorption spectrum of Ce(0.1 at.%):Nd(1.0–1.1 at.%):YAG is 31.3%, nearly twice that of Nd:YAG. Assuming the maximum solar irradiance I_S of 950 W/m², typical for clear sky in Lisbon [11], the calculated power values were hence P_source1 = 310.4 W and P_source2 = 53.3 W. These calculations also considered a 92% reflectivity for heliostat mirrors [15], which redirects the incident solar radiation to the primary concentrator.

Fresnel lenses are generally flat, with one smooth surface and the other etched with multiple concentric grooves. These grooves function as individual prisms, each replicating a narrow, curved section of an original conventional lens of the same diameter, but without the full mass of its solid structure. By segmenting the lens in this way, significantly less optical material is required, resulting in lower system costs compared to traditional lenses. The specifications of the modeled flat Fresnel lens in Zemax^® are listed in

Table 3. Specifications of the Fresnel lens primary concentrator.

Fresnel Lens Parameters	Value
Material	PMMA
Diameter	1.3 m
Focal length	2.0 m
Rim angle	18°
Thickness	5 mm
Pitch angle	12°
Depth	0.3 mm
Conic	−0.7
Focal spot width (1/e2)	40 mm
Transmission efficiency of useful pump radiation	81.3%

. It was designed using polymethyl methacrylate (PMMA), a robust and lightweight polymer with a density of around 1.18 g/cm³. This material exhibits excellent transmission within the solar spectrum, enhancing the lens’s durability under solar exposure and its thermal stability [31].

The Fresnel lens size, yielding a collection area of 1.33 m², was selected to enable a fair comparison with the highest previously reported CW amplification results from a solar-pumped laser amplifier that utilized the same collection area for its primary concentrator [28]. Regarding the choice of focal length, it was optimized to maximize the transmission efficiency of the pump radiation, while also accounting for typical commercial parameters. As shown in Figure 3, relatively short focal lengths, and consequently wider rim angles, led to a reduction in collected power due to increased reflection losses [37,38], particularly at focal distances lower than 1.5 m. Beyond 1.8 m, transmission efficiency nearly stabilizes, with increases of less than 1% observed up to a focal length of 2 m.

Based on these considerations, a focal length of 2 m was selected for a Fresnel lens with a 1.3 m collection diameter, yielding a transmission of 81.3% within the 1/e² focal spot of 40 mm. This corresponded to useful pump power of 296 W at the focus. Given that the useful power density for pumping Ce:Nd:YAG accounts for 31.3% of the total solar spectrum intensity (as discussed above), the Fresnel lens is expected to concentrate about 946 W of solar power in the focal region.

The pitch angle defines the tilt of the inactive faces of the prisms relative to the optical axis (z-axis). A pitch angle of a few degrees is typically added to Fresnel molds to make extraction of the molded part easier [39]. In this case, a pitch angle of 12° was specified. A groove depth of 0.3 mm was also defined [11], and Zemax^® automatically calculated the corresponding radial positions of the grooves using an iterative search algorithm. A conic constant of −0.7 was also set [11].

3.2. Ce:Nd:YAG Amplifier Head

Variations in the slab’s dimensions and geometry strongly influence the seed laser’s propagation through the gain medium, thereby affecting the resulting optical gain. For this reason, two amplifier head configurations with distinct slab width-to-height ratios, and thus different secondary concentrator output apertures, were numerically modeled and compared, as shown in Figure 4. Configuration 1 was first modeled and optimized to efficiently transmit and homogenize the pump radiation collected by the Fresnel lens toward a Ce:Nd:YAG slab with dimensions of 16.3 mm width, 18 mm length, and 2.9 mm thickness, yielding a width-to-length ratio of 0.91. These dimensions were initially chosen to replicate those used in the previous solar-pumped Nd:YAG amplifier that reported the highest continuous-wave (CW) amplification factor of 1.64 [28]. Configuration 2 was subsequently modeled as an enhanced version of Configuration 1, aiming to extend the seed laser’s optical path inside the gain medium. To achieve this, the slab’s dimensions were adjusted to 11.3 mm width, 26 mm length, maintaining the same cross-sectional area but reducing the width-to-length ratio to 0.44.

As shown in Figure 4a, the secondary concentrator’s input aperture was identical for both configurations (40 mm × 40 mm), thereby collecting the same concentrated solar power.

A silver-coated aluminum surface with 94% solar reflectance was assumed for the inner walls of both concentrators, characterized by a spectral reflectance of 98–99% between 500 and 750 nm, then decreasing slightly to 96–97% at 900 nm [11,35]. The concentrator’s length is a key design parameter to ensure uniform light delivery to the gain medium [40]. A length of 200 mm was found to offer an effective compromise between uniformity and transmission efficiency at the output of both secondary concentrators. Their output apertures were dimensioned to match the corresponding slab width-to-height ratios. Consequently, different output sizes of 16.3 mm W × 18 mm H and 11.3 mm W × 26 mm H were, respectively, adopted for Configurations 1 and 2, ensuring uniform incident pump distribution across the slabs, as demonstrated in Figure 4b.

To collect this data, a detector rectangle comprising approximately 115,000 pixels was positioned on the top surface of the slabs, maintaining the same width-to-height ratio in both physical size and pixel resolution. The number of analysis rays is also a crucial parameter for the accuracy of the numerical results and image resolution of the detector. In this case, 2 × 10⁸ analysis rays were assigned to each solar source. Red indicates the regions of maximum incident pump flux on the slabs, which was uniform across their entire surface as expected, whereas blue corresponds to areas with little or no incident pump flux, observed at the beveled corners. Useful pump power incident on the slabs totaled 247 W and 230 W for configurations 1 and 2, respectively, corresponding to transmission efficiencies of approximately 83.5% and 77.7% for their respective secondary concentrators. Based on these numerical values, the total incident solar powers on the slabs were estimated to be 790 W and 734 W, respectively, resulting in solar power fluxes of 2.69 W/mm² and 2.50 W/mm². Thanks to the uniform distribution of incident pump power on the slabs, homogeneous absorption patterns were also achieved, as depicted in Figure 4c, a condition essential for enabling effective and consistent interaction between the seed laser and the active regions of the slabs.

For defining the gain medium material in Zemax^®, the dominant absorption wavelengths of Nd³⁺, mentioned in Section 3.1, and the corresponding absorption coefficients [11] were incorporated into the glass catalog of the software. Figure 4c shows the absorbed pump flux distributions in the central transverse cross-sections of the Ce(0.1 at.%):Nd (1.1 at.%):YAG laser medium for both configurations. To obtain this data and achieve the desired image resolution, a detector volume comprising 560,000 voxels was used for each slab, while 2 × 10⁸ analysis rays were also set for each solar source. As observed in Figure 4b,c, Configuration 1, which features a slab and secondary concentrator with a higher width-to-height ratio of 0.91, enabled increased incident pump power on the slab and, consequently, higher absorption of pump photons, which may enhance the population inversion compared to Configuration 2. Nevertheless, it is worth noting that the incident and absorbed pump fluxes obtained from Configuration 1 were only 1.08- and 1.12-times higher, respectively, than those from Configuration 2, which in turn significantly improved the overlap region between the seed laser beam and the excited areas of the medium, as demonstrated in Figure 4d. To study the trajectory of the incident beam through the Ce:Nd:YAG amplifiers, a radiant source was modeled in Zemax^®, replicating the characteristics of a 1064 nm diode-pumped solid-state laser operating in TEM₀₀ mode, with a continuous-wave output power of 0.8–1 W, a beam diameter of 1.5 mm at 1/e², and a divergence angle below 1.2 mrad, that are suitable for effectively entering and exiting the beveled corners of both modeled slabs without losses. While in Configuration 1 the seed laser underwent 20 reflections within the slab medium, resulting in a total optical path length of approximately 251 mm, the modified slab in Configuration 2, with a lower width-to-height ratio of 0.44 but the same surface area and volume, enabled 32 reflections, extending the total optical path to around 365 mm, which is 1.45-times longer than that permitted by the slab in Configuration 1.

3.3. Thermal-Induced Effects in the Solar-Pumped Ce:Nd:YAG Amplifier

To confirm that the slab medium in the proposed solar-pumped amplifier system, with cooling applied only to the rear surface, can endure high-intensity solar pumping without incurring damage, a thermal analysis was conducted using LASer Cavity Analysis and Design (LASCAD™) software (version 3.3.5). This software allows users to import data from Zemax^®, containing the three-dimensional distribution of absorbed power density in the active medium, for evaluation of the thermal effects and optimizing laser resonator parameters [15,16,33], either for rod or slab geometries. Therefore, the absorbed flux distribution data for each laser slab, as shown in Figure 4c, was subsequently imported into LASCAD™ 3.3.5, which performs thermal and structural analysis using finite element analysis. The calculations involve determining the heat load distribution and solving the three-dimensional heat conduction and structural deformation differential equations. To accomplish this, LASCAD™ 3.3.5 discretizes the gain medium into a user-defined three-dimensional mesh and iteratively solves the governing equations. The analysis incorporates the gain material properties, pump configuration, and cooling geometry. Accordingly, the gain medium parameters were predefined in LASCAD™ 3.3.5, including a fluorescence lifetime of 230 µs, a stimulated emission cross-section of 2.8 × 10⁻¹⁹ cm², and typical absorption and scattering losses of 0.002 cm⁻¹ [16]. Additionally, a mean absorbed and intensity-weighted solar radiation wavelength of 660 nm was adopted [41]. The slab dimensions and cooling boundaries were also specified beforehand. Particularly, only the rear surfaces of the slab media, measuring 16.3 mm × 18 mm and 11.3 mm × 26 mm for configurations 1 and 2, respectively, were set to be refrigerated, with a constant reference temperature of 300 K. A film coefficient of 0.2 W mm⁻² K⁻¹ was applied to model fluid convective heat transfer. The accuracy of the gain medium’s thermal performance results depends directly on the chosen mesh resolution and simulation time. To balance accuracy with computational efficiency, the mesh consisted of about 440,000 elements, with up to 300 iterations performed for both thermal and structural analyses [11]. Figure 5 shows the thermally induced effects in the slab medium for the two configurations, simulated using LASCAD™ 3.3.5.

As anticipated, the slab with a width-to-height ratio of 0.91, subjected to higher incident and absorbed pump flux (see Figure 4b,c), exhibited a greater peak stress intensity of 26.2 N/mm², whereas the slab with a lower ratio of 0.44 reached a reduced maximum of 21.6 N/mm². As also expected, in both cases, the maximum stress occurred at the top surface of the slab, since it was directly exposed to the incident light from the secondary concentrator output and most distant from the cooling interface. Nevertheless, the incorporation of a homogenizing secondary concentrator enabled efficient redistribution of stress intensity across the slab surface in both configurations. Notably, the slab with a lower width-to-height ratio (0.44) exhibited a more homogeneous stress profile throughout its volume, whereas the slab with the higher ratio (0.91) showed localized stress relief in its central region. Additionally, both configurations revealed reduced stress intensity near the slab corners, where enhanced heat dissipation and reduced mechanical constraints mitigate thermal stress buildup. Overall, the results indicate that the proposed slab geometries maintain moderate stress levels relative to those reported for earlier solar-pumped Ce:Nd:YAG laser oscillators [15,16,33], well below the tensile strength of Ce:Nd:YAG, which ranges from approximately 130 to 260 N/mm² [42]. Thanks to the homogenized pump intensity, the slabs also exhibited a uniform temperature distribution across their transverse section, while moderate temperature rises developed along the thickness, as depicted in Figure 5c,d. These increases reached maximum values of approximately 357 K for the slab with a width-to-height ratio of 0.91 and 352 K for the slab with a ratio of 0.44, occurring at the surfaces most directly exposed to pumping. The resulting temperature profiles are therefore consistent with the stress patterns observed in Figure 5a,b.

The combination of homogenized pumping and optimally designed slab geometries thus ensures effective thermal management, providing a robust foundation for stable and efficient laser amplification.

4. Estimated Laser Gain Factor of the Proposed Solar-Pumped Ce:Nd:YAG Laser Amplifier

Increasing the pumping intensity of the amplifying medium leads to more particles being excited, thereby strengthening the population inversion in the upper laser level, essential for stimulated emission, a key requirement for enhancing laser gain. Equally important is maximizing the seed laser’s optical path through the excited regions of the active medium.

As demonstrated by the numerical modeling of both solar-pumped Ce:Nd:YAG laser amplifier configurations in Section 3, Configuration 2 exhibited greater potential for laser gain. Although its gain medium received approximately 6.9% less incident pump flux, potentially resulting in a lower population inversion compared to Configuration 1, this limitation was offset by a 45% longer interaction length of the seed laser in the amplifying medium.

For relatively weak input laser signal I_in, the output intensity I_out grows exponentially with the product of stimulated emission cross-section σ, population inversion density n, and propagation length l of the seed laser inside the active medium, as indicated by Equation (3) for a four-level laser system [11]:

$$I_{out}(l)=I_{in}{e}^{\sigma nl}$$

(3)

Through this relationship, it is possible to estimate the potential laser gain of the present approach, particularly when compared to previously reported results under similar solar pumping conditions. Therefore, the numerical performance of the proposed Configuration 2, specifically the incident pump density on the slab and the seed laser’s optical path length, was compared to that of the previous amplifier with the highest CW laser gain under solar pumping [28]. For a fair and accurate comparison, and to assess the power density deliverable to the gain medium in the earlier solar-pumped laser amplifier [28], that system was also modeled and numerically analyzed in Zemax^® under pumping conditions similar to those of the proposed amplifier. The results of this comparative analysis are summarized in

Table 4. Comparison of the performance between the proposed solar-pumped Ce:Nd:YAG laser amplifier and the previously reported solar-pumped Nd:YAG amplifier [28], based on similar numerical pumping conditions.

Parameters	Previous Solar-Pumped Nd:YAG Laser Amplifier [28]	Proposed Solar-Pumped Ce:Nd:YAG Laser Amplifier (Configuration 2)	Improvement Over [28]
Primary concentrator:
Type	Flat Fresnel lens		—
Collection area	1.33 m2		—
Focal length	1.2 m	2.0 m	—
Numerical calculated transmission efficiency (40 mm × 40 mm detector area)	59.8%	81.3%	1.36 times
Secondary concentrator:
Type	No secondary concentrator	Reflective homogenizer	—
Input aperture		W 40 mm × H 40 mm	—
Length		200 mm	—
Output aperture		W 11.3 mm × H 26 mm	—
Transmission efficiency		77.7%	—
Gain medium:
Material	Nd:YAG	Ce:Nd:YAG	—
Geometry	Slab		—
Dimensions (Width-to-height ratio)	W 16.3 mm × H 18 mm × T 2.9 mm (0.91)	W 11.3 mm × H 26 mm × T 2.9 mm (0.44)	—
Pumping performance with optimal alignment at the focal spot (Δα = 0.0°, Δh = 0.0°)
Incident pumping profile	Non-uniform	Uniform	Much more uniform
Average incident solar density on the slab (950 W/m2 solar irradiance)	1.22 W/mm2	2.50 W/mm2	2.05 times
Pumping performance with typical solar tracking error of Δα = ±0.2° (Δh = ± 0.06°)
Incident pumping profile	Non-uniform	Less Uniform	Much more uniform
Average incident solar density on the slab (950 W/m2 solar irradiance)	1.08 W/mm2 (11.5% reduction)	2.37 W/mm2 (5.2% reduction)	2.19 times
Seed laser path:
Number of total internal reflections	20	32	1.6 times
Path length $\mathit{l}$	~251 mm	~365 mm	1.45 times
Laser gain ${\mathit{I}}_{\mathit{o}\mathit{u}\mathit{t}}/{\mathit{I}}_{\mathit{i}\mathit{n}}$ (Δα = 0.0°, Δh = 0.0°)	1.64 (Experimental)	6.82 (Estimated)	4.16-fold increase

.

The proposed solar-pumped Ce:Nd:YAG laser amplifier employed the same type of primary concentrator and the same collection area (1.33 m²) as the previous Nd:YAG amplifier design [28]. It also featured a slab gain medium with identical surface area and volume. However, the adoption of a flat Fresnel lens with a longer focal length, offering higher transmission efficiency (81.3%), combined with a secondary concentrator, enabled the proposed system to deliver a maximum solar power density of 2.5 W/mm² on the slab medium. This was more than double the value numerically achieved by the earlier amplifier system [28] under the same solar irradiance of 950 W/m². Moreover, the rectangular secondary concentrator in the proposed design also functioned as a homogenizer, distributing the incident concentrated solar power evenly across the slab and thus preventing hot spots in the gain medium. In contrast, the previous amplifier design [28], which lacked a homogenizing element, led to a non-uniform irradiance profile on the slab that could hinder the coupling efficiency between the laser to be amplified and the active regions of the gain medium. As shown in Table 4, the homogenizing secondary concentrator was also advantageous in compensating for solar tracking errors. These errors are directly related to the solar azimuth (α) and altitude (h) angles, with deviations in the azimuth angle having the most significant impact, especially near solar noon [11,17]. For instance, around local solar noon, an azimuth error of 0.2° (Δα) corresponds to only about 0.06° of error in the altitude angle (Δh) [17]. Consequently, the apparent focal spot displacement is more pronounced in the azimuthal direction (x-axis), as observed in the incident pumping profile on the gain medium of the previous amplifier scheme under a typical 0.2° azimuth tracking error. In this case, the incident solar density dropped to 1.08 W/mm², representing an 11.5% reduction compared to the 1.22 W/mm² achieved under optimal alignment at the focal spot (Δα = 0.0°, Δh = 0.0°). However, with the secondary homogenizer in the proposed amplifier approach, despite a slight degradation in the uniformity of the pumping density on the gain medium, the density was reduced by only 5.2% under the same tracking error, reaching 2.37 W/mm², 2.19-times higher than that of the previous amplifier [28].

In addition to the twofold increase in incident power density on the active medium achieved with the proposed scheme with optimal alignment at the focal spot, the reduction in the slab’s width-to-height ratio from 0.91 to 0.44 resulted in a significant rise in the number of total internal reflections of the seed laser within the slab, from 20 to 32. Consequently, its optical path increased from 251 mm to 365 mm, representing a 1.45 enhancement [28].

Taking into account the previously mentioned relationship given by Equation (3), the numerical achievements proved that the present configuration, originally proposed for pumping a Ce:Nd:YAG gain medium, could still potentially enhance laser output power by 2.66 times if applied to a Nd:YAG slab medium. This would correspond to a power amplification factor of 4.35.

Previous experimental work with a side-pumped Ce:Nd:YAG laser oscillator demonstrated that Ce:Nd:YAG medium can extract 1.57-times more laser power than Nd:YAG under identical pumping conditions [13]. Hence, replacing Nd:YAG with Ce:Nd:YAG in the proposed design may yield a comparable improvement in laser gain, leading to an estimate gain factor of 6.82, equivalent to a 4.16-fold increase over the gain factor of 1.64 previously achieved with a solar-pumped Nd:YAG amplifier [28].

5. Conclusions

A solar-pumped Ce:Nd:YAG laser amplifier approach was proposed for advancing solar-pumped laser technology. This design stood out from the previous Nd:YAG-based amplifier, which held the best CW laser amplification performance under solar pumping, not only because it introduced an active material with superior solar absorption for the first time in solar-pumped laser amplifiers, but also due to two additional features: (i) the combination of a Fresnel lens with a longer focal length with a homogenizing secondary concentrator, enabling effective and uniform coupling of solar radiation onto the slab medium, a vital condition for future scalability to higher laser output powers; and (ii) a redesigned slab geometry with a reduced width-to-height ratio, that substantially enhanced the interaction between the seed laser beam and the active medium.

In this numerical study, two configurations, primarily differing in the width-to-height ratio of the slab gain medium, with the secondary concentrator output dimensions adjusted accordingly, were here presented to clarify how this geometric factor guided the optimization of the system parameters for potentially achieving high laser gain. The configuration employing a slab medium with the same width-to-height ratio of 0.91 as the previously reported Nd:YAG amplifier [28] allowed the TEM₀₀ mode seed laser to undergo 20 internal reflections within the slab, resulting in a total optical path length of approximately 251 mm. In this case, a uniform solar power density of 2.69 W/mm² was delivered to the slab. In the configuration using a slab with reduced width-to-height ratio of 0.44, the incident solar flux was also distributed uniformly, albeit reduced to 2.50 W/mm². Still, this result was more than double that achieved in the numerical simulation of the previous amplifier design [28] under identical pumping conditions. Most notably, this configuration allowed the seed laser to complete 32 passes through the active medium, resulting in a total optical path length of 365 mm, which is 1.45-times longer.

By combining these enhancement factors with Ce:Nd:YAG’s superior laser extraction efficiency, reported to be up to 1.57-times greater than that of Nd:YAG [13], the proposed solar-pumped Ce:Nd:YAG laser amplifier is expected to significantly exceed the previous CW laser power amplification factor of 1.64 [28], potentially enhancing it by nearly 4.16 times. This advancement may open new possibilities for high-efficiency, renewable-powered laser technologies.