Optimum Cr Content in Cr, Nd: YAG Transparent Ceramic Laser Rods for Compact Solar-Pumped Lasers

Abstract

Cr content χ of 0.4 at% for a Cr doped Nd (1 at%): YAG laser rod (LR) gave a higher laser output (I_output) than that of 0.0, 0.7, and 1.0 at% in a specially designed compact solar-pumped laser (SPL) outdoors. I_outputs were measured as a function of an 808 nm pumping laser’s power indoors, changing the transmittance of the output coupler. From the obtained slope efficiencies, round-trip resonator losses Ls for the four χs were estimated, and the best-fit function L(χ) was derived. From the experimentally estimated Cr-to-Nd effective energy transfer efficiency η_Cr→Nd at the four χs, the best-fit function η_Cr→Nd(χ) was derived. Using L(χ), η_Cr→Nd(χ), and a wavelength λ- and χ-dependent absorption coefficient α(λ, χ), inferred from the literature, the power conversion efficiency η_power(χ) under 1 Sun was estimated. The estimated η_power(0.4) and η_power(0.7) were reproduced in experimentally deduced factors at the mode-matching efficiency η_mode = 0.19. The estimated maximum η_power(χ) appeared around χ = 0.2 at%, being 20% higher than that at χ = 0.4 at%. In addition to this, a composite LR (Cr, Nd: YAG core/Gd: YAG cladding) was found to achieve η_mode = 0.68 and η_power = 0.064, ranking among the highest-class SPL η_powers.

1. Introduction

Solar-pumped lasers (SPLs) are means to convert sunlight into lasers and were first reported in the 1960s [1,2,3], soon after the discovery of the laser [4]. Those initial works on SPLs mainly employed laser mediums of Nd-doped yttrium aluminum garnet (Nd: YAG) single crystals. Since then, more than 600 papers have been published on SPLs. Iodine or iodides were once major laser mediums [5,6,7]. Then, transparent Nd: YAG ceramic laser rods (LRs) were successfully fabricated and used as one of the major laser mediums [8,9]. Furthermore, Cr, Nd-doped YAG ceramic LRs have been intensively studied in connection with applications of SPLs to space solar power systems [10]. Nd-doped ZrF₄–BaF₂–LaF₃–AlF₃–NaF glasses were also studied as laser mediums for solar-pumped fiber lasers [11]. More recently, transverse excitation was introduced into a solar-pumped fiber laser system to decrease the threshold excitation power for oscillation at sunlight concentration as low as 1/15 [12]. A thin disk-shaped medium for SPL has also been proposed [13]. In addition to these, R&Ds of SPLs have been actively going on through various approaches [14,15,16,17].

SPLs can be used to transmit collected solar energy across large distances. Some examples include power transmission between an orbiting or lunar power station and the ground [10], between deep-sea and sea-level, or for powering mobile objects such as drones, robots, or electric vehicles [18]. Photo-thermal and photo-chemical applications of SPLs have also been investigated to promote a reduction of MgO in the Mg energy cycle [19]. The output laser beam was focused onto MgO, typically at an optical power density of 105 W/cm² to attain a temperature of 4000 K, at which MgO is thermally decomposed to yield metallic Mg. The resultant metallic Mg was to be utilized in the “Mg–H₂ cycle”, where lightweight metallic Mg grains are transported as an energy carrier and H₂ is produced by pouring water onto the metallic Mg grains at the required location [20,21]. The resulting MgO produced by the reduction in water is then thermally decomposed again under the focused laser light from the SPL.

In applications in space solar power systems, a solar cell–laser system can be used in place of a SPL system. Solar energy can be converted into electricity by solar cells, and electricity can be used to excite a laser to transport energy from an orbiting station to the ground. In this case, the electricity-to-laser energy conversion efficiency is practically lower than 50%. In applications in space solar power systems, SPLs can compete with the above solar cell–laser system only when the sunlight-to-laser energy conversion efficiency exceeds 50% of the solar cell energy conversion efficiency. Although the theoretical detailed balance limit of the power conversion efficiency of solar-pumped Nd³⁺: YAG-like sensitized lasers was estimated to be ~31% [22], experimentally realized power conversion efficiencies are around 5% or lower, providing a fair comparison to those of solar cells [23].

Therefore, improvement of SPL energy conversion efficiency is an important and challenging topic for many researchers. To achieve a low laser oscillation threshold while maintaining a high slope efficiency, there are many issues that must be addressed, some of which are in a trade-off relationship: the laser rod-pumping configuration and laser cavity structure, as well as the selection of suitable pumping and oscillation wavelengths. Difficulties in addressing these issues in SPLs originate from two features of sunlight: (1) the energy density is low, and (2) the spectral range is widespread, from the UV to IR spectra. To overcome issue (1), a focusing optical system with high magnification is necessary to achieve a sufficiently high pumping energy density. As for issue (2), the absorption range of the rod material should cover the spectral range of sunlight; this is referred to as the “spectrum matching problem”. Therefore, many researchers have considered that an SPL using a single rare-earth element such as Nd cannot be efficiently driven because of the extremely narrow absorption bands of rare-earth elements. To solve the “spectrum matching problem”, it is promising to introduce a sensitizer of another element that extends the absorption range considerably. It has been found that Cr³⁺ is a promising sensitizer, which has broadband absorption characteristics and transfers the absorbed energy to Nd³⁺ [13,17,20,24,25,26,27,28,29,30,31].

We have been developing compact solar-pumped lasers (μSPLs) that utilize an off-axis parabolic mirror (OAP) with diameters of ⌀50.8 mm or ⌀76.2 mm as solar concentrators, achieving a concentration ratio of 0.0001, as shown in Figure 1a [18,31,32,33,34,35,36,37,38,39,40]. In these μSPLs, we use micro laser rods that are either cylindrical with a diameter of ⌀1 mm and length of l mm, or regular tetragonal prismatic (RTP) with dimensions 1 × 1 × l mm (where l = 10 or 20), made of Cr³⁺ and Nd³⁺-co-doped YAG transparent ceramic. Compared to thick, large LRs, we can produce many more micro-LRs from single bulk transparent ceramic, making the process more cost-effective. The micro LRs are easy to cool because of their large surface area per volume. The array of μSPLs can produce the same total laser intensity as a large single SPL but with a smaller overall volume of the laser medium. We discussed the advantages of μSPLs in detail in the subsection “10.2.6. 2. 6. Merits of Miniaturization of SPLs” [32] (pp. 427–429). These LRs were manufactured by World-Lab Co., Ltd., Atsuta, Nagoya, Japan, under A. Ikesue’s supervision [8]. According to the report by Wisniewsky and Koepke (2002) [41], the existence of Cr⁴⁺ ions can be clearly distinguished from Cr³⁺ ions in absorption spectra. We observed no feature of existence of Cr⁴⁺ but only of Cr³⁺ in the absorption spectra shown in our previous report [39]. This evidence eliminates the possibility of the occurrence of self-Q-switching caused by Cr⁴⁺. It is believed that the UV component of sunlight can generate defects in LRs. These defects, called “color centers”, absorb light from a specific wavelength and degrade the performance of the LRs temporarily or permanently, depending on the composition and process conditions of the LRs. The detailed mechanisms of this degradation, called solarization [42,43], are still under investigation, and explorations of optimal composition and process conditions are key to long-term device reliability and the practical application of SPLs. To reduce UV degradation (solarization) of the LR, a UV-cut filter is attached to the entrance aperture of the OAP, although it might provide a somewhat less-than-perfect reduction in solarization. In this μSPL, an end-pumping configuration is used, where a bundle of direct solar radiation, captured by an OAP, is focused on the front end (HR-end) surface of an LR and introduced into it. The HR-end surface is coated with an optical multilayer filter (HR: high-reflectance) that transmits 95% of light between 440 nm and 850 nm and reflects 99.95% at the laser emission wavelength: λ_L = 1064 nm. An output coupler (OC) with a curvature radius of 100 mm and a diameter of 25 mm is placed 50 mm from the HR-end, forming a confocal resonator together with the HR on the HR-end. The other end of the LR, called the BBAR (broadband antireflection)-end, is coated with an optical multilayer filter that transmits 95% of light across the solar spectrum and nearly 100% at λ_L. Even under concentrated sunlight, this thin LR can be effectively cooled by natural air convection. The μSPL can operate continuously, tracking the sun for over 6.5 h from 11:00 to 17:30 on a clear summer day [37]. This duration is the longest record of continuous solar-pumped laser emission outdoors. The temperature difference between the side surface and the center region in this thin LR under concentrated sunlight was estimated to be no more than 4 degrees K assuming the coefficient of thermal diffusivity: 4.2 × 10⁻⁶ [m²/s], specific heat 602 [J/(kg∙K)], and density: 4.55 [g/cm²]. Therefore, it is considered that the effect of thermal lensing is not significant in this system. It demonstrates the potential for μSPLs to operate all day for terrestrial solar energy utilization [18]. The OAP used in a single μSPL can harvest only a limited amount of sunlight. To harvest a significant amount of solar energy, coordinated solar tracking from an array of μSPLs has been proposed [35].

Using direct natural sunlight as a pumping light for the μSPL system, outdoor oscillation experiments were conducted to examine how Cr content χ in the LR affects laser oscillation performance. RTP LRs of 1 × 1 × 10 mm (or 20 mm) with χ values of 0.0, 0.4, 0.7, or 1.0 at% and a fixed Nd content of 1.0 at% were tested. The output laser power (I_output) of the μSPL increased significantly from χ = 0.0 to 0.4 at% and then gradually decreased beyond 0.4 at% [38]. This experiment demonstrated that Cr doping effectively enhances μSPL power conversion efficiency, η_power, which is the ratio of I_output to the global solar radiation harvested by the OAP. The value of η_power represents a key performance of the SPL, irrespective of the absolute size of the SPL system. Although we know there is an optimal χ between 0.0 at% and 0.7 at%, it remains unclear whether χ = 0.4 at% is the absolute optimum.

For this μSPL in terrestrial operation, a simple analytical expression to calculate η_power from sunlight to I_output was derived based on the fundamental physics of laser oscillation [44] (2023). In this paper, we search the optimum χ using this analytical expression, taking into consideration the related experimentally obtained parameters, such as the dependence of χ on Cr³⁺ to Nd³⁺ effective energy transfer quantum efficiency, η_Cr→Nd (χ), and the dependence of χ on round-trip resonator loss, L(χ), together with a wavelength λ- and χ-dependent absorption coefficient, α(λ, χ), deduced from the data in the previously reported literature. When the light emitted from the excited Nd³⁺ ions in the LR makes a round-trip between the resonator mirrors (HR and OC in Figure 1), its intensity increases by stimulated emissions but decreases by absorption and scattering. The former is the round-trip gain, and the latter is the round-trip loss. When the gain exceeds the loss, the laser oscillation takes place. Therefore, L(χ) should be reduced as far as possible.

2. Calculation Method

2.1. Analytical Expression of η_power for μSPL

Figure 1b schematically displays the essential portion of μSPL. The analytical expression of power conversion efficiency, η_power, for this μSPL employing a Cr_χat% doped Nd (1 at%): YAG (yttrium aluminum garnet) transparent ceramic LR (laser rod) is [44]

$$\begin{array}{l}{\eta }_{power}={\displaystyle \frac{hc}{{\lambda }_L}}\cdot {\displaystyle \frac{1}{{\displaystyle \int I_{pG}(\lambda )}\cdot \frac{hc}{\lambda }d\lambda }}\cdot {\displaystyle \frac{t_{OC}}{L+t_{OC}}}\times \\ \left[\begin{array}{l}{\eta }_{Nd}\cdot {\eta }_{\mathrm{mod}e}\cdot {\eta }_{OAP}\cdot {\displaystyle {\int }_0^{{\lambda }_a}\left\{{\alpha }_{Nd,1at\%}\left(\lambda \right)+\frac{{\eta }_{Cr}}{{\eta }_{Nd}}\cdot {\eta }_{Cr\to Nd}\cdot {\alpha }_{Cr,\chi at\%}\left(\lambda \right)\right\}\cdot \frac{1-\exp \left(-\alpha \left(\lambda \right)\cdot t\right)}{\alpha \left(\lambda \right)}‧}{I}_p\left(\lambda \right)d\lambda \\ -{\displaystyle \frac{t}{C}}\cdot {\displaystyle \frac{L+t_{OC}}{2l\cdot \sigma \cdot {\tau }_{sp}}}\end{array}\right]\end{array}$$

(1)

In this relation (1), the propagations of both solar intensity and laser signal intensity along the cavity axis in LR are taken into consideration.

Table 1. Definitions of variables and constants in relation (1).

λ	Wavelength
c	Velocity of light in a vacuum
h	Planck’s constant
IpG(λ)	Spectral power of the global solar radiation harvested by the OAP
ηpower	Power conversion efficiency: the ratio of Ioutput to the global solar radiation harvested by the OAP (the λ-integrated IpG(λ))
λL	Laser emission wavelength
λa	Absorption edge wavelength: the longest sunlight wavelength absorbed in the laser medium, which contributes to laser emission
ηNd	Quantum efficiency to form a Nd3+ ion excited to 4F3/2 level by a photon absorbed by a Nd3+ ion.
αNd,1at%(λ)	Portion of the spectral absorption constant in the laser medium contributed by 1 at% Nd3+ ions.
ηCr	Quantum efficiency to form a Cr3+ ion excited to 2F3/2 level by a photon absorbed by a Cr3+ ion.
ηCr→Nd	Effective energy transfer quantum efficiency from an optically excited Cr3+ ion at the 2E level to an Nd3+ ion to make an Nd3+ ion excited to the 4F3/2 level in LR during laser oscillation, which contributes to Ioutput.
αCr,χat% (λ)	Portion of the spectral absorption constant in the laser medium contributed by χ at% doped Cr3+ ions.
α(λ)	αNd,1at%(λ) + αCr,χ at%(λ)
t	Length of the laser medium along the propagation direction of the pumping light
ηmode	The ratio of the volume in which laser oscillation takes place to the volume in which absorption of the pumping light takes place in the laser medium
Ip(λ)	Spectral direct solar radiation harvested by the OAP
ηOAP	The ratio of the direct solar irradiation power focused on the front end of the laser medium to the direct solar power harvested by the OAP.
C	The concentration ratio of the direct solar radiation harvested by the OAP
Nt	Number of Nd3+ ions excited up to 4F3/2 level per second per unit volume
τsp	Spontaneous emission lifetime of a Nd3+ ion at 4F3/2 level, which is ideally equal to the inverse of the decay rate
σ	Emission cross-section of the laser medium at λL
l	Length of the laser medium along the laser resonator
L	Round-trip resonator loss at λL, i.e., 2αL∙l + d, where αL is the distributed loss constant in the LR and d is the diffraction loss between the BBAR-end of the LR and the OC. In the μSPL, d is negligibly small because the radius of the OC is sufficiently large.
tOC	Transmittance of OC at λL

summarizes the definitions of variables and constants in relation (1). The first term in the square bracket is related to the slope efficiency η_slope, and the second term in the square bracket is associated with the threshold power I_th.

Although relation (1) stands on a simple laser oscillation model in contrast to detailed numerical analytical approaches such as by LASCAD^TM (e.g., [32,37]), this approach gives an intuitive perspective and suggestions to experimentalists on contributions of fundamental factors such as l, L, α(λ), χ, t_OC, C, η_Cr /η_Nd, η_Cr→Nd, η_mode, (cf. Table 1) and cavity configurations to η_power. Here, η_mode stands for the ratio of the volume in which laser oscillation takes place to the volume in which absorption of the pumping light takes place in LR. It is preferred to design the laser resonator configuration so that η_mode increases and approaches unity as far as possible.

Solar concentrators employed in SPLs cannot converge diffuse solar radiation, such as that scattered from clouds or the blue sky, on LRs. Thus, SPLs can only utilize direct solar radiation. Therefore, the ratio of I_output to the power of the direct solar radiation captured by the solar concentrator stands for the η_power of an SPL, usually. On the other hand, the ratio of electric output power to the sum of direct and diffuse solar radiation on solar cells usually stands for the η_power of solar cells. This difference in the definition of η_power between SPLs and solar cells is confusing when they are compared with each other as means for solar energy utilization. Therefore, we use the direct solar radiation I_p(λ) in the numerator and the global (the direct + the diffuse) solar radiation I_pG(λ) in the denominator in relation (1). According to the reference solar spectra, ASTM G173-03 (https://www.nrel.gov/grid/solar-resource/spectra-am1.5, accessed on 17 May 2025, the total direct solar radiation in the wavelength region between 280 nm and 4000 nm is 900.2 W/m². If we also consider the contribution of diffuse solar radiation, it increases up to 1000.4 W/m². Therefore, relation (1) gives 90% smaller η_power than the usually reported η_powers under the conventional definition of SPLs.

In the case of the present μSPL, there is a loss factor: η_OAP of the concentrated solar radiation. It is partly caused by a UV cut filter located at the entrance aperture of the OAP. Another contribution to η_OAP is the reflectance of the OAP surface. In the present system, η_OAP is around 0.78.

2.2. Dependence of Spectral Absorption Coefficient on Cr Content χ: α(λ, χ)

The χ dependence of the spectral absorption coefficient α(λ, χ) for Cr (χ at%), Nd (1.0 at%)-co-doped transparent ceramic YAG (yttrium aluminum garnet) rod, such as at χ = 0.4, 0.7, and 1.0, for example, were numerically synthesized [44] from the experimentally obtained α(λ, 3.0) for Cr (3.0 at%), Nd (1.0 at%)-co-doped transparent ceramic YAG rod by Saiki et al. [45].

2.3. Laser Oscillation Mode-Matching Efficiency η_mode

In the configuration of a confocal resonator [46] indicated in Figure 1b, the beam waist of a radius ω₀: 0.130 mm is at the HR (high-reflection) end of the LR (laser rod). Defining z as the distance from the position of the beam waist (the HR-end of the LR) towards the OC (output coupler), the beam spot size ω(z) increases with z. The beam spot radius is as follows:

The ω (20 mm) on the BBAR (broadband antireflection)-end surface of the LR is 0.152 mm, and the ω (50 mm) on the OC is 0.184 mm. Here, we neglect the effect of the refractive index of the LR because the beam crosses the BBAR-end surface of the LR almost perpendicularly. If we model the laser modal domain as a truncated cone defined by the beam waist radius on the HR-end surface of the LR as ω₀ for an upper base circle, the beam spot radius at the BBAR-end surface of the LR as ω (l) for a lower base circle, and the length of the LR as l as a height, η_mode is calculated to be 0.063 in the case of a homogeneously excited 1 × 1 × l (=20 mm) LR. If we take 2ω₀ as the radius of the beam spot on the HR-end surface of the LR and 2ω (20 mm) as the radius of the beam spot on the BBAR-end surface of the LR, η_mode is 0.250 in a 1 × 1 × l (=20 mm) LR. If we take 3ω₀ as the radius of the beam spot on the HR-end surface of the LR and 3·ω (l = 20 mm) as the radius of the beam spot on the BBAR-end surface of the LR, η_mode is 0.564 in a 1 × 1 × l (=20 mm) LR. In the actual experimental procedure, the location of the OC is adjusted manually from the exact confocal configuration to give the maximum I_output. We do this adjustment because the actual distribution of absorbed input sunlight power is not monotonic and not axisymmetric in the LR due to the geometry of the OAP, as shown in [32,37]. To obtain the χ dependence of η_power by relation (1), η_mode should be surveyed to give the best fit to the several previously experimentally obtained η_powers. The concept of estimations described here was explained with an inserted figure in “Table 1” in our previous report [44].

2.4. Dependence of Effective Energy Transfer Efficiency from Cr³⁺ to Nd³⁺ in the LR on Cr Content χ: η_Cr→Nd (χ)

The values for Cr³⁺ to Nd³⁺ energy transfer quantum efficiency in Cr (χ at%₎, Nd (1 at%): YAG (yttrium aluminum garnet) transparent ceramic laser mediums have been measured by several methods and reported in the literature: [31,47,48,49,50,51,52]. These values shall be collectively expressed as ς_Cr→Nd here. Related to the effective energy transfer quantum efficiency, η_Cr→Nd in an LR during laser oscillation, values corresponding to (η_Cr/η_Nd)·η_Cr→Nd have been obtained by Kato et al. (2020) [39] to be 0.448, 0.213, and 0.166 for χ = 0.4, 0.7, and 1.0 at%, respectively, in solar-pumped laser emission experiments outdoors using the μSPL system shown in Figure 1 [39]. Here, η_Cr and η_Nd are the quantum efficiencies forming a Cr³⁺ ion excited to ²F_3/2 level by a photon absorbed by a Cr³⁺ ion, and the quantum efficiency to form a Nd³⁺ ion excited to ⁴F_3/2 level by a photon absorbed by a Nd³⁺ ion, respectively. Substituting a collective lens for the OC (output coupler) in the same μSPL system shown in Figure 1b and collecting fluorescence from the solar-pumped LR (laser rod) with the lens outdoors, values corresponding to (η_Cr/η_Nd)·η_Cr→Nd have also been experimentally obtained to be 0.675, 0.623, and 0.625 for χ = 0.4, 0.7, and 1.0 at%, respectively, in the same work by Kato et al. [39]. In this latter case, we express the value corresponding to η_Cr→Nd as ξ_Cr→Nd to distinguish it from η_Cr→Nd.

The ratio η_Cr/η_Nd is not known. For the first-order approximation, it can be unity, because both η_Cr and η_Nd represent similar properties of the isolated metal ions doped in YAG ceramics. For the second-order approximation, the ratio can be a value slightly less than unity. In contrast to transition metal ions such as Cr³⁺, the energy levels of rare-earth ions, such as Nd³⁺, are not significantly influenced by the surrounding electronic structure of YAG matrix atoms. The very narrow absorption peaks of Nd³⁺ ions and the broad absorption peaks of Cr³⁺ ions [45] support this idea. Therefore, the efficiency of relaxation of the excited Nd³⁺ ions in higher energy levels than the metastable ⁴F_3/2 levels, η_Nd, will be higher than that of the excited transition metal ions, such as Cr³⁺ ions, to their ⁴T₂ or ²E levels, η_Cr. As a value for η_Cr/η_N slightly less than unity, 0.7 and 0.9 were tested in the previous work [44]. In this work, we performed the calculations supposing η_Cr/η_Nd = 0.9.

Figure 2 shows the plots of the values for η_Cr→Nd, ξ_Cr→Nd, and ς_Cr→Nd against the Cr content χ. In Figure 2, the data points with the first author’s name in Italic type represent the values obtained by fluorescence measurements in spontaneous emission. Those with the first author’s name in Roman type represent the values obtained in stimulated emission under the laser oscillation. Figure 2 shows that the values for ξ_Cr→Nd are larger than the values for η_Cr→Nd. This phenomenon might come from (a) a phenomenon that Nd³⁺ ions in an unexcited state decrease and it becomes difficult for excited Cr³⁺ ions to find Nd³⁺ ions in an unexcited state to transfer their energy in the case of η_Cr→Nd, while there may be a sufficient amount of Nd³⁺ ions in the unexcited state in the case of ξ_Cr→Nd, or (b) a phenomenon that only Nd³⁺ ions in a laser oscillation mode in LR contribute η_Cr→Nd while there is no limitation to the laser oscillation mode in the case of ξ_Cr→Nd. As for the data points of ς_Cr→Nd with Italic authors’ names, the spontaneous fluorescent emissions from the laser media have been measured more efficiently by employing integrating spheres than collective lenses. Therefore, ς_Cr→Nd can be larger than ξ_Cr→Nd. In Figure 2, however, the values for ξ_Cr→Nd indicated by Italic Kato (2020) [39] are almost at the same level as the values for ς_Cr→Nd.

Figure 2 shows that η_Cr→Nd decreases significantly with an increase in χ. This result might come from (c) a phenomenon wherein the probability for an excited Cr³⁺ ion to find an unexcited Nd³⁺ ion to which the Cr³⁺ ion transfers its energy decreases with an increase in χ under an inverted population condition in laser emission when the Nd³⁺ content is constant at 1 at% as in the present work, or (d) a phenomenon wherein the probability of energy transfers from excited Cr³⁺ ions to unexcited Cr³⁺ ions increases with an increase in χ, and consequently η_Cr→Nd decreases with an increase in χ. However, (d) seems to be unlikely because ξ_Cr→Nd does not decrease as significantly with an increase in χ as η_Cr→Nd. Supposing that η_Cr→Nd takes the simplest form, exp(−β·χ), the least-square-fit (LSF) of the three experimentally obtained data points, 0.448/(η_Cr/η_Nd) at χ = 0.4 at%, 0.213/(η_Cr/η_Nd) at χ = 0.7 at%, and 0.166/(η_Cr/η_Nd) at χ = 1.0 at% gives β·of 1.805 (1/at%) assuming η_Cr/η_Nd = 0.9. In Figure 2, the curve η_Cr→Nd (χ) = exp(−1.805∙χ) is also drawn. In Figure 2, the values obtained in the condition of laser oscillation, such as those by Endo et al. (2010) [48] at χ = 0.1 and by Hasegawa et al. (2015) [31] at χ = 0.4 appear near the LSF curve, η_Cr→Nd (χ) = exp(−1.805∙χ). The values by Mares et al. (1991) [47] appear significantly lower than the LSF curve. Judging from the very low χ of 0.04 or 0.08, Mares et al. might have used single-crystal YAG laser media, in which it is difficult to dope Cr in high concentrations. Therefore, it may not be appropriate to compare their data with the others in Figure 2. The value by Endo et al. (2010) [48] at χ = 3.0 appears at a much higher η_Cr→Nd than the LSF curve. The laser medium of χ = 3.0 showed much higher α_L than the laser medium of χ = 0.1. In the summary of Endo et al. [48], only the ς_Cr→Nd of the laser medium at χ = 0.1 was mentioned, but not at χ = 0.3. Therefore, it may not be appropriate to compare the value at χ = 3.0, with a much higher α_L, with the others in Figure 2. Considering these facts, it should be reasonable to use η_Cr→Nd (χ) = exp(−1.805∙χ) in the present calculation of η_power by relation (1).

2.5. Dependence of Round-Trip Resonator Loss on Cr Content χ: L(χ)

As indicated in relation (1), L can be derived using either the slope efficiency, η_slope or the threshold power, I_th. We often describe the method to derive L using the threshold power, I_th, as ‘Findlay–Clay analysis’ [53] and that used to derive L using slope efficiencies η_slope as ‘Caird analysis’ [54]. Hereafter, Ls derived by Findlay–Clay analysis and Caird analysis are expressed as L_Findlay-Clays and as L_Cairds, respectively, for convenience.

The output power, I_outputs, of μSPLs employing Cr (χ at%), Nd (1 at%): YAG (yttrium aluminum garnet) transparent ceramic RTP (regular tetragonal prismatic) rods of 1 × 1 × 10 mm and 1 × 1 × 20 mm with for different χ: 0, 0.4, 0.7, and 1.0 were measured by pumping Nd³⁺ directly using an 808 nm (λ_a) laser diode in the three different configurations of OCs (output couplers) of the measured t_OCs: 0.007, 0.049, 0.105. The experimental configuration was essentially the same one as reported in a figure by Hasegawa et al. [36]. Figure 3a–h show the results. In each figure, we obtained η_slopes from portions of the linear increase in the output power I_output with the input laser power I_p(λ_a). We indicated the obtained values of the slope efficiency, η_slope, in the figures for each line. We derived the threshold laser powers, I_ths, from the intercepts of the extended lines of the linear portions and the abscissa axis, where I_output = 0. The obtained values of I_th are indicated with filled triangles, together with corresponding figures in mW.

From relation (1), we derived the following relation:

$$\begin{array}{l}{I}_{th}=G\times \left\{L_{Findley-Clay}-\ln \left(1-t_{OC}\right)\right\}\end{array}$$

(2)

Here, G is a constant value, and t_OC is a transmittance of OC at λ_L. If I_th (1) is measured using an OC of t_OC = t_OC (1), and I_th (2) is measured using an OC of t_OC = t_OC (2), L_Findley-Clay is

$$L_{Findley-Clay}={\displaystyle \frac{-I_{th}(1)\times \ln \left\{1-t_{OC}\left(2\right)\right\}+I_{th}(2)\times \ln \left\{1-t_{OC}\left(1\right)\right\}}{\left\{I_{th}(2)-I_{th}(1)\right\}}}$$

(3)

Although we can obtain L_Findlay-Clay using relation (3), we often obtain L_Findlay-Clay from the −ln(1 − t_OC) axis-intercept of the LSF (least-square-fit) line defined by more than three values of I_th plotted against −ln(1 − t_OC) to minimize the effect of measurement errors. Figure 4 shows Findlay-Clay plots made from Figure 3 based on relation (2).

From relation (1), we can derive the following relation:

$${\displaystyle \frac{\left(1-t_{OC}\right)}{{\eta }_{slope}}}={\displaystyle \frac{1}{{\eta }_0}}\times \left\{{\displaystyle \frac{\exp \left(L_{Caird}\right)-1}{t_{OC}}}+1\right\}$$

(4)

If we measure η_slope (1) using an OC of t_OC = t_OC (1) and η_slope (2) using an OC of t_OC = t_OC (2), L_Caird is

$$L_{Caird}=\ln ⟦1-{\displaystyle \frac{t_{OC}\left(2\right)\times {\eta }_{slope}\left(2\right)\times \left[t_{OC}\left(1\right)\times \left\{1-t_{OC}\left(1\right)\right\}\right]-t_{OC}\left(1\right)\times {\eta }_{slope}\left(1\right)\times \left[t_{OC}\left(2\right)\times \left\{1-t_{OC}\left(2\right)\right\}\right]}{{\eta }_{slope}\left(2\right)\times \left[t_{OC}\left(1\right)\times \left\{1-t_{OC}\left(1\right)\right\}\right]-{\eta }_{slope}\left(1\right)\times \left[t_{OC}\left(2\right)\times \left\{1-t_{OC}\left(2\right)\right\}\right]}}⟧$$

(5)

Although we can obtain L_Caird using relation (5), we often obtain L_Caird from the slope (exp (L_Caird) − 1)/η₀ of the LSF line defined by more than three values of (1 − t_OC)/η_slope plotted against 1/t_OC to minimize the effect of measurement errors. In this process, we obtain 1/η₀ from the (1 − t_OC)/η_slope axis-intercept of the LSF line. Figure 5 shows Caird plots made from Figure 3 based on relation (4).

As for Findlay-Clay plots, the three points in Figure 4a,d,h are approximately on the LSF line obtained from the three values of I_th plotted against −ln(1 − t_OC). However, the three points in Figure 4b,c,g are well deviated from the LSF line. In such cases, I_ths were obtained from lines connecting two points for t_OC = 0.007 and 0.049, omitting the largest t_OC. The relation (4) indicates that (1 − t_OC)/η_slope increases with 1/t_OC in Caird plots, as Figure 5a,c–h. However, (1 − t_OC)/η_slope slightly decreases with 1/t_OC in Figure 5b. The Cr 0.4 at% doped LR (laser rod) was used most frequently for outdoor experiments tracking the sun in our laboratory [37] because it yielded the highest I_output among the four Cr (χ at%), Nd (1 at%): YAG transparent ceramic LRs in the μSPL system [38]. Therefore, accumulated damage in this LR by solarization may have been predominant in comparison with the other LRs, even with the UV cut filter. This accumulated damage may have caused the unreasonable negative slope in Figure 5b. Only in Figure 5h, the three points appear near the LSF line. L cannot be obtained using relation (5) because (1 − t_OC)/η_slope at t_OC = 0.049 is less than that at t_OC = 0.105 in Figure 5b,d,f,g, resulting in negative slopes. Therefore, L was calculated from relation (5) using (1 − t_OC)/η_slope at t_OC = 0.049 and 0.005, omitting the largest t_OC.

Table 2. L_Findlay-Clay obtained by Findlay-Clay plots from Figure 4 and L_Caird obtained by Caird plots from Figure 5.

	LFindlay-Clay		LCaird
	3 Points	2 Points	3 Points	2 Points
(a) Cr 0 at%	0.0519		0.0014	0.0017
(b) Cr 0.4 at%	0.0678	0.1406	−0.0002
(c) Cr 0.7 at%	0.0455	0.0576	0.0043	0.0042
(d) Cr 1.0 at%	0.0872		0.0066	0.0078
(e) Cr 0 at%	0.0597	0.0676	0.0030	0.0030
(f) Cr 0.4 at%	0.0485	0.0593	0.0018	0.0022
(g) Cr 0.7 at%	0.0709	0.0502	0.0043	0.0060
(h) Cr 1.0 at%	0.0759		0.0143

summarizes L_Findlay-Clays and L_Cairds obtained from the LSF lines, from relation (3), and relation (5) using two points at t_OC = 0.007 and 0.049. Figure 6a shows the correlation of L_Findlay-Clays and L_Cairds thus obtained. The L_Cairds are one order of magnitude smaller than L_Findlay-Clays.

Table 3. L_Findlay-Clay via I_th by using relation (3) and L_Caird via η_slope by using relation (5).

	LFindlay-Clay via Ith by Using Relation (3)					LCaird via ηslope by Using Relation (5)
	tOC			average	corrected average	tOC			average	corrected average
	0.007 0.049	0.049 0.105	0.105 0.007			0.007 0.049	0.049 0.1	0.1 0.007
(a)	0.0500	0.0533	0.0511	0.0515	0.0506	0.0017	−0.0039	0.0011	−0.0003	0.0014
(b)	0.1406	0.0322	0.0765	0.0831	0.1406	−0.0002	−0.0010	−0.0003	−0.0005	−0.0002
(c)	0.0576	0.0334	0.0483	0.0464	0.0576	0.0042	0.0058	0.0044	0.0048	0.0042
(d)	0.0794	0.0964	0.0857	0.0872	0.0794	0.0078	−0.0043	0.0058	0.0068	0.0078
(e)	0.0676	0.0436	0.0579	0.0563	0.0676	0.0030	0.0017	0.0029	0.0025	0.0030
(f)	0.0593	0.0375	0.0509	0.0492	0.0593	0.0022	−0.0041	0.0016	−0.0001	0.0022
(g)	0.0502	0.1069	0.0655	0.0742	0.0502	0.0060	−0.0117	0.0033	−0.0008	0.0060
(h)	0.0740	0.0781	0.0756	0.0759	0.0740	0.0140	0.0170	0.0146	0.0152	0.0140

shows L_Findlay-Clays obtained from relation (3) and L_Cairds obtained from relation (5), using two cases out of three t_OCs: 0.007, 0.049, and 0.105. The “average” stands for the average of the three Ls obtained from three combinations of t_OC: (0.007, 0.049), (0.049, 0.105), and (0.105, 0.007). The “corrected average” stands for the average of the two Ls obtained from two combinations of t_OC, (0.007, 0.049) and (0.105, 0.007), or L obtained from the combination of t_OC, (0.007, 0.049), depending on the cases. Slopes obtained from the combination of t_OC (0.049, 0.105) are often considerably different from the slopes obtained from the other combinations of t_OC, (0.007, 0.049) and (0.105, 0.007), as shown in Figure 4 and Figure 5. Therefore, we do not use them for the derivation of the average value. Figure 7b shows the correlation of L_Findlay-Clays and L_Cairds in the columns of “corrected average” in Table 3, except for the case of χ = 0.4 and l = 10 mm in which the calculated L_Caird is negative. Figure 7b also shows that L_Caird is one order of magnitude less than L_Findlay-Clay. Figure 8a,b show bar charts representing the obtained L_Findlay-Clay and L_Caird for each LR (rod rength l = 10 or 20 mm) of different Cr content χ, respectively. Here, the open bars represent the values in the column, “3 points”, in Table 2, and the filled bars represent the values in the column, “corrected average”, in Table 3. It should be noted that the scales of the bar length are one order of magnitude different between Figure 7a,b.

L_Findlay-Clay for l = 10 mm and χ = 0.4 at% in Figure 7a obtained using relation (3) is unusually large compared to L_Findlay-Clays for the other l and χ. This inconsistency may also originate from accumulated damage caused by solarization in the LR with l = 10 mm and χ = 0.4 at%. Ignoring the data at l = 10 mm and χ = 0.4 at%, the average of the open bar and the filled bar at each χ appears to decrease from χ = 0.0 at% to 0.4 at% and then increase from χ = 0.4 at% to 1.0 at%, although this trend is not very pronounced. In contrast, L_Cairds in Figure 7b show clear increases with χ from 0.4 to 1, except for χ = 0.4 at l = 10 mm. This behavior of L_Caird is reasonable because Cr³⁺ dopants can cause scattering loss. It is also logical that L_Cairds for l = 20 mm generally show larger values than those for l = 10 mm. Excluding L_Findlay-Clay and L_Caird for (l = 10 and χ = 0.4), the average of the open bar and the filled bar at each χ is plotted against χ for each l, as shown in Figure 8. We also provided LSF as a quadratic function for each L_Findlay-Clays and L_Cairds and each l. The quadratic functions L_quad.(χ)s are downward convex. L_quad.(χ)s for l = 10 mm may be of lower accuracy because of the lack of data at χ = 0.4 near the lowest value in comparison with those defined by four data points, with that being at χ = 0.4. Therefore, η_powers shall be calculated only for l = 20 mm in the following sections.

It is reasonable that L_Caird and L_Findlay-Clay increase with χ, as shown in the experimental data at χ = 0.4, 0.7, and 1.0 in Figure 8 because an increase in χ should increase optical scattering sites in the LR (laser rod). Viewed in this light, however, it is not reasonable that the experimentally obtained L_Caird and L_Findlay-Clay at χ = 0 are larger than those at χ = 0.4. Although the detailed process of the LRs, including combinations of hot isostatic pressing and thermal annealing, has not been disclosed because of the know-how involved [8], the process of the LR of χ = 0 without Cr may be different from those of the Cr-doped LRs of χ = 0.4, 0.7, and 1.0 because of the difference in the segregation coefficients between Cr and Nd [9]. Then it may be better to consider that the χ dependence of L_Caird or L_Findlay-Clay: L(χ) exhibits a monotonic increase with χ except at χ = 0, in contrast to L_quad.(χ)s being convex downward, as shown in Figure 8. Exponential functions L_exp.(χ)s obtained by LSF to the experimental L_Caird or L_Findlay-Clay at χ = 0.4, 0.7, and 1.0 in the case of l = 20 mm are shown in Figure 9. L_exp.(χ)s were not obtained in the case of l = 10 mm because there were only two data points for χ = 0.7 and 1.0.

3. Results of Calculations of η_power

3.1. The Mode-Matching Efficiency η_mode to Give the Best Fit to the Experimental Power-Conversion Efficiency η_power

Figure 10 shows the measured laser output powers, I_outputs, as a function of the incident sunlight power into the OAP (off-axis parabolic mirror), I_pG, at different Cr contents, χ = 0, 0.4, 0.7, and 1 at%, in the case of the transmittance of OC at λ_L, t_OC = 0.01, and rod length, l = 20 mm, redrawn from Figure 3 in Ref. [39]. The maximum I_pG in all the measured points in Figure 10 is 3.9 W. The maximum I_pG should be 4.5 W in ideal experimental conditions at 1 Sun (100 mW/cm²) because the aperture radius of the OAP is 38 mm. Figure 10 shows four extended dashed lines connecting two measured points of the maximum and the second maximum I_outputs for each. The four intersections of these dashed lines and the perpendicular lines I_pG = 4.5 W give I_output as 11.0, 29.0, 59.3, and 4.8 mW for each χ: 0, 0.4, 0.7, and 1.0, respectively.

3.2. Estimated η_powe as a Function of the Cr Content χ

Figure 11 compares η_powers calculated for η_mode = 0.19 and l = 20 mm using the L_quad.(χ)s for L_Findley-Clays and L_Cairds shown in Figure 8b, with η_powers experimentally deduced at 1 Sun. The η_powers calculated using the L_quad.(χ) for L_Findley-Clay are far below the experimentally deduced η_powers. In contrast, η_powers calculated using the L_quad.(χ) for L_Cairds are in good agreement with the experimentally deduced η_powers. This result shows that L_Cairds give η_powers closer to the truth than L_Findley-Clays. Figure 12 shows comparisons of Cr content χ dependencies of η_powers, η_power(χ) for η_mode = 0.15, 0.17, 0.19, and 0.21, calculated using (a) L_quad.(χ) for L_Caird, (b) using L_exp.(χ) for L_Caird with those experimentally deduced for 1 Sun. It is shown that η_mode = 0.15 and 0.17 give lower η_powers and η_mode = 0.21 gives higher η_powers than those experimentally deduced. The η_mode = 0.19 gives η_powers in good agreement with those experimentally deduced. As for agreement with experimentally deduced η_powers, L_quad.(χ) seems to be more successful than L_exp.(χ). However, the difference is not significant. η_power reaches its maximum at 0.0154 at around χ = 0.23 in the case of L_quad.(χ), and 0.0160 at around χ = 0.18 in the case of L_exp.(χ). Experimentally deduced η_powers for χ = 0 and 1 at% are obviously below the calculated η_powers for η_mode = 0.19. These discrepancies may arise from the fact that we measured the maximum I_output for χ = 0 and 1 at% at I_pG = 3.5 W and I_pG = 3.9 W, respectively, not far from the threshold I_pG located at around 3 W. The slopes of the extended dashed lines connecting two measured points of the maximum and the second maximum I_outputs for χ = 0 and 1 could be larger if we had measured I_outputs at I_pG = 4.5 W. Then the experimentally deduced η_powers for χ = 0 and 1 at% would be much closer to the calculated η_powers for η_mode = 0.19.

In our previous experiment [39], the LR of χ = 0.4 at% gave a higher I_output than the LRs of χ = 0, 0.7, and 1.0 at%. The calculations shown in Figure 12 reproduce this tendency. In addition to this, the calculations show that the highest η_power, as high as 0.0154–0.0160, appears around χ = 0.2.

4. Discussion

4.1. Contribution of Cr Content, χ, Dependence of Energy-Transfer Efficiency, η_Cr→Nd, and Resonator Loss, L, to χ Dependence of Energy Conversion Efficiency, η_power

The contributions of η_Cr→Nd(χ) as shown in Figure 2 and L_quad.(χ) as shown in Figure 8b to η_power(χ) can be explained by either η_Cr→Nd(χ) or L_quad.(χ) being held constant, or both being constant in the calculations of η_power(χ)s. Figure 13 presents four cases of η_power(χ) calculations where L is constant, 0.003 in the two cases, and η_Cr→Nd is constant, 0.5 in the two cases. In the case where both L and η_Cr→Nd are constant, the calculated η_power increases with χ. When we use L_quad.(χ) and η_Cr→Nd is constant, η_power(χ) has a peak at around χ = 0.35. When we use η_Cr→Nd (χ) and keep L constant, η_power(χ) has a peak at χ = 0.19. With both contributions, L_quad.(χ) and η_Cr→Nd (χ), η_power has a peak at around χ = 0.23.

4.2. Comparison of L_Findlay-Clay and L_Caird

Both relation (2) and relation (4) are derived from the fundamental relation of laser oscillation (1). Therefore, L_Caird should be equal to L_{Findlay-Clay.} However, L_Findlay-Clays are almost ten times larger than L_Cairds in the present work. According to Figure 11, the values of L_Cairds are appropriate for actual laser emission from the μSPL pumped by natural sunlight. In contrast, the values of L_Findlay-Clays are too large for measurable laser emission from the μSPL under natural sunlight pumping.

The discrepancy between L_Findlay-Clay and L_Caird has been reported in the previous literature. Cornacchia et al. (2007) [55] reported laser oscillations at 640.2, 721.5, 607.2, and 522.8 nm in Czochralski-grown Pr doped LiLuF₄ single crystals under 479.5 nm laser pumping. They obtained L_Findlay-Clay and L_Caird for each laser oscillation wavelength. The values for L_Caird/L_Findlay-Clay were 1.14, 1.26, 0.26, and 1.02 for 640.2, 721.5, 607.2, and 522.8 nm, respectively. Therefore, L_Caird was nearly equal to L_Findlay-Clay in the cases of 640.2, 721.5, and 522.8 nm. However, L_Findlay-Clay was approximately five times larger than L_Caird in the case of 607.2 nm. In their report, they described that the possible reason was still under investigation. To the best of our knowledge, “the possible reason” has not been reported by them since 2007.

We found similar examples in several other previous reports. Values for L_Caird/L_Findlay-Clay: (a) 0.42, (b) 0.12, (c) 0.14, (d) 0.02, (e) 0.092, (f) 0.068, (g) 0.139, and (h) 0.391 can be obtained from the laser emission properties in the following reports: (a) Shoji et al. (2000) [56], (b) and (c) Liang and Almeida (2011) [43], (d) Nasser at al. (2019) [57], and (e)–(h) Saiki et al. (2022) [58], respectively. In six out of eight examples (a)–(h), L_Findlay is nearly ten times larger than L_Caird. On the other hand, values of (i) 5.61, (j) 3.65, (k) 1.04 for L_Caird/L_Findlay-Clay had been reported in Weksler and Shwartz (1988) [59] ((i) and (j)), and in Sanghera et al. (2011) [60] ((k)), respectively. Although the possible cause for this discrepancy has not been reported yet, we should check both L_Findlay-Clay and L_Caird as Cornacchia et al. (2007) [55] did.

4.3. Recipes to Improve the Power Conversion Efficiency, η_power

As mentioned in Section 3.2, decreasing Cr content from χ = 0.4 at% to 0.2 at% is one route to improving η_power. In addition to this, improvement of the mode-matching efficiency η_mode is key to increasing η_power because it is evident that the present η_mode is as small as 0.19, as shown in Figure 12. A truncated cone with a circular area of 0.226 mm (1.74 times the beam waist radius 0.130 mm) in radius on the HR (high reflectance)-end of the LR (laser rod) and a circular area of 0.264 mm in radius on the BBAR (broadband antireflection)-end (1.74 times larger than the beam radius when the beam radius on the HR-end is 130 nm) reproduces the mode volume of η_mode = 0.19 in the 1 × 1 × 20 mm RTP-LR (regular tetragonal prismatic-laser rod) in the present μSPL configuration. If a composite LR with a 0.53 × 0.53 × 20 mm RTP-core of Cr_χat%, Nd_1at%: YAG (yttrium aluminum garnet) transparent ceramic and a refractive-index-matched cladding of Gd_{5at %}: YAG transparent ceramic [36] as shown in the inserted figure in Figure 14 is employed, the mode-matching efficiency: η_mode increases up to 0.68. In this configuration, the calculated η_power (χ) is shown in Figure 14. The calculated η_power(χ) takes a maximum as high as 0.063 at around χ = 0.225 at% at the transmittance of OC at λ_L: t_OC = 0.02 using L_quad.(χ), and 0.064 at around χ = 0.2 at% at t_OC = 0.02 using L_exp.(χ).

5. Conclusions

For a 1 × 1 × 20 mm Cr co-doped Nd_1at%: YAG (yttrium aluminum garnet) transparent ceramic laser rod employed in a specially designed μSPL,

(a)

We derived Cr content χ dependence of the resonator loss L(χ) from experimentally obtained output-laser-power as a function of an 808 nm pumping laser power.

(b)

We obtained χ dependence of Cr³⁺ to Nd³⁺ effective energy transfer efficiency η_Cr→Nd(χ) in our previous outdoor μSPL experiment.

(c)

We deduced a spectral absorption coefficient as a function of χ, α (λ, χ), from the data in the previous literature.

Using L(χ), η_Cr→Nd(χ), and α (λ, χ), a numerical expression of η_power(χ) for the μSPL was established. The η_power(χ) reproduces the experimentally deduced η_power (0.4) at 1 Sun around 0.013 and η_power(0.7) around 0.0064. The η_power(χ) indicated that maximum η_power(χ) = 0.0154–0.0160 will be attained at around χ = 0.2. The reproduction of the experimental data also elucidated that the mode-matching efficiency η_mode was as low as 0.19. To improve this low η_mode, a composite laser rod with a 0.53 × 0.53 × 20 mm Cr and Nd-doped core and a peripheral cladding in which Gd was doped in YAG to match the refractive index with that of the core was proposed. By employing this composite laser rod, the analytical estimation indicates that η_mode increases to 0.68. Then, η_power increases to 0.063–0.064 (0.07–0.071 in the conventional SPL definition, cf. Section 2.1). This four times or higher η_power than the present one ranks in the highest class of η_power for SPLs [23] and encourages further improvement of the μSPL.