1. Introduction

Optically active solids can be refrigerated by excitation with monochromatic light via anti-Stokes fluorescence [1,2]. The simple cycle of absorption and subsequent fluorescence can therefore carry away thermal energy, which results in the cooling of the solid. As lasers are used as the excitation source, this technique is referred to as laser cooling of solids. This cooling technique is the basis of all-solid-state optical refrigerators [3] and radiation-balanced lasers [4,5]. Optical refrigerators consist of a laser-cooling medium and a pump laser but no mechanically moving element. They are therefore compact and intrinsically free of vibrations. These features are attractive for applications in space [6], high-precision metrology [7], and ultra-sensitive detection such as gravitational wave detectors [8].

So far, various kinds of materials from semiconductors to dielectrics have been investigated as laser-cooling media [2]. The progress using materials doped with trivalent ytterbium ions (Yb³⁺) is particularly remarkable. The simple two-level structure of Yb³⁺, originating from the nearly filled 4f electronic shell, ensures a high quantum efficiency because the absence of a level above the ²F_5/2 manifold, except 4f¹²5d levels, excludes detrimental excited-state absorption and upconversion processes. Laser cooling based on Yb³⁺ has been observed in diverse host materials, including Y₃Al₅O₁₂ (YAG) [9], KY(WO₄)₂ (KYW) [10], LiYF₄ (YLF) [11], LiLuF₄ [12], BaY₂F₈ [13], KYF₄ [14], KY₃F₁₀ [15], CaF₂, and SrF₂ [16], and fluorozirconate [17] and silica glasses [18]. Among them, fluoride single crystals are found to be particularly suited for optical refrigerators. The small Stark splitting in fluorides maintains the higher population in the upper Stark levels of the ground-state multiplet compared with oxide crystals, so this ensures the absorption at long wavelengths relevant for cooling at low temperatures even below 100 K. Laser cooling to cryogenic temperatures, below 123 K, has been accomplished using Yb:YLF crystals. The record of the lowest temperature of 87 K ever reached by anti-Stokes fluorescence cooling was demonstrated using this crystal under high-power pumping at 1020 nm [19].

Besides the development of the efficient pumping scheme with astigmatic Herriott cells [20] and the heat load management using ‘clamshells’ [11,21], the progress in laser cooling of solids is associated with the improvement of materials’ purity. Crystals of identical nominal chemical composition from different suppliers can show disparate laser-cooling results because the cooling process is sensitive to impurities that cause unwanted processes, such that they absorb pump and fluorescence photons and subsequently generate heat during their decay [22]. For a systematic evaluation of laser-cooling samples, two characteristic quantities were thus introduced as the sample-specific quality parameters [23]. The first is the external quantum efficiency η_ext, which is defined as the ratio of the number of fluorescence photons escaping the sample to the number of absorbed photons. This parameter is therefore always smaller than the internal quantum efficiency η_q, which is defined as the ratio of the number of emitted photons to the number of absorbed photons. The second, the background absorption coefficient α_b, is the coefficient of residual absorption caused by impurities. Currently, transition metal ions are believed to be the main cause of background absorption [24]. Both parameters can be determined by laser-induced thermal modulation spectroscopy (LITMoS) [24].

LITMoS has been widely applied for evaluating laser-cooling samples. However, here we raise two problems in the analysis of the experimental data of LITMoS. The first problem is that determining the external quantum efficiency requires knowledge of the mean fluorescence wavelength. However, this parameter is not readily accessible experimentally. Fluorescence emitted in different directions undergoes different degrees of reabsorption which distorts the detected fluorescence spectrum. Therefore, the mean fluorescence wavelength can be obtained only if we know the fluorescence spectrum emitted into all angles across three dimensions for a given sample shape and a pump geometry. As a compromise, the mean fluorescence wavelength has often simply been calculated from an as-measured fluorescence spectrum in a fixed detection setting; however, this approach does not provide an accurate value, in particular for anisotropic crystals. It will be shown below that any uncertainty in the mean fluorescence wavelength causes uncertainty in the external quantum efficiency. This consideration raises concerns about the reliability of the external quantum efficiencies reported so far. The second problem is that the external quantum efficiency is not an intrinsic quantity suited to compare different samples. For instance, assuming two chemically identical laser-cooling samples of different sizes, these should have the same internal quantum efficiency but their external quantum efficiencies will differ owing to the influence of factors such as size, shape, and doping level on the fluorescence escape efficiency—the fraction of generated fluorescence photons that manage to escape the sample. These two problems hinder the systematic evaluation and comparison of laser-cooling samples.

Here, we propose a computational approach using a fluorescence ray tracing simulation based on the Monte Carlo method to determine the mean fluorescence wavelength and the fluorescence escape efficiency of laser-cooling samples with arbitrary size, shape, and doping level. These computed parameters allow us to reliably determine the external quantum efficiencies as well as the internal quantum efficiency of laser-cooling samples.

This paper is structured as follows: In Section 2, we revisit the experimental method, i.e., LITMoS, and show that we can simplify the measurement by only determining the two pump wavelengths at which the cooling efficiency crosses zero. In Section 3, we present the fluorescence ray tracing simulation to determine the mean fluorescence wavelength and fluorescence escape efficiency, and the validity of the simulation results are discussed. In Section 4, we study the influence of sample geometry and refractive indices on the laser-cooling performance. In Section 5, we study the temperature dependency of mean fluorescence wavelength and fluorescence escape efficiency for the refinement of the minimal achievable temperatures of Yb:YLF and demonstrate that the reduction of sample symmetry is crucial in achieving lower cooling temperatures. Conclusions are drawn in Section 6.

2. Revisiting the experimental method: Laser-induced thermal modulation spectroscopy (LITMoS)

LITMoS is recognized as the standard characterization process for laser-cooling media. Figure 1(a) illustrates a typical experimental setup. An all-facet-polished cuboid-shaped or Brewster-cut sample is typically used for the characterization. The sample is placed in a vacuum chamber with minimized contact with its support. In our case, two optical fibers of 125 µm in diameter support the sample. A wavelength-tunable laser excites the sample and a thermal camera measures the induced temperature change of the sample ΔT. Figure 1(b) shows the induced equilibrium temperature change normalized to the absorbed pump power ΔT·P_abs⁻¹ for a 10 at.% Yb-doped YLF sample at various pump wavelengths. This sample has dimensions of 2.5 × 3.5-mm² aperture and a length of 10.5 mm, with its 2.5-mm long edge parallel to the crystal's c-axis. As the pump source, we used a home-built Yb:Lu₂O₃ thin-disk laser with an intracavity birefringent filter as the wavelength-tuning element. Details of the thin-disk laser can be found in Supplement 1.

 

Fig. 1. (a) Typical experimental setup of the LITMoS method. (b) An example of a LITMoS result for an a-cut Yb(10 at.%):YLF sample (2.5 × 3.0-mm² aperture, 10.5-mm long) under π-polarized pumping. Net cooling was observed for the pump wavelengths within the highlighted range between λ_X1 and λ_X2. The zero-crossing wavelengths λ_X1 and λ_X2 are 1011.7 ± 0.2 nm and 1075.7 ± 0.2 nm, respectively. (c, d) Laser-induced temperature change in the Yb:YLF sample as a function of time around the shorter and longer zero-crossing wavelengths, respectively. The laser pumping starts at t = 0 min.

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Since the quantity ΔT·P_abs⁻¹ is proportional to laser-cooling efficiency η_c as far as the contribution of radiative heat exchange between the sample and its surroundings is small [25], we can determine the external quantum efficiency and background absorption coefficient by fitting the data of ΔT·P_abs⁻¹(λ) with the following model equation of laser-cooling efficiency:

It is noteworthy that there is an implicit assumption behind Eq. (1) that both η_ext and α_b are independent of wavelength. The term in the bracket is referred to as the absorption efficiency η_abs in the context of laser cooling of solids.

A laser-cooling-grade sample shows two zero-crossing wavelengths, λ_X1 and λ_X2 (λ_X1 < λ_X2), at which neither cooling nor heating is observed. In such samples, the resonant absorption coefficient at the shorter zero-crossing wavelength α_r(λ_X1) is necessarily larger than the background absorption coefficient by orders of magnitudes, i.e., ${\alpha _r}({{\lambda_{\textrm{X}1}}} )\gg {\alpha _b}$. Therefore, the absorption efficiency η_abs approaches unity and the cooling efficiency at λ_X1 can be written as

The external quantum efficiency can consequently be written as the ratio of the mean fluorescence wavelength to the shorter zero-crossing wavelength as follows:

This equation tells us that accurate knowledge of λ_f is necessary to determine η_ext. We present an approach to retrieve accurate values for λ_f by simulations in the next section. Using Eq. (4), the laser-cooling efficiency at the longer zero-crossing wavelength λ_X2 can be written as

This equation no longer involves the mean fluorescence wavelength. Solving this equation for the background absorption yields

Therefore, only the ratio of the two zero-crossing wavelengths λ_X2/λ_X1 and the resonant absorption coefficient at λ_X2 are necessary to derive the background absorption coefficient. This means that even though the mean fluorescence wavelength is unknown, the background absorption coefficient can be systematically determined. In the case of the cuboid-shaped Yb(10 at.%):YLF sample, characterized in Fig. 1(b), we determined the zero-crossing wavelengths to be λ_X1 = 1011.7 ± 0.2 nm and λ_X2 = 1075.7 ± 0.2 nm, respectively, for π-polarized pumping as seen in Figs. 1(c, d). Using Eq. (6) and the resonant absorption coefficient α_r(λ_X2) = (1.7 ± 0.2)·10⁻³cm⁻¹, the background absorption coefficient α_b was determined to be (1.1 ± 0.2)·10⁻⁴cm⁻¹. We retrieved α_r(λ_X2) from the absorption spectra calculated from the emission cross-section spectra using the reciprocity (or McCumber) relation [26] since the absorption spectra from transmission spectroscopy cannot provide absorption coefficients below 10⁻²cm⁻¹. Note that the shorter zero-crossing wavelength λ_X1 is independent of the pump polarization as far as the assumption ${\alpha _r}({{\lambda_{\textrm{X}1}}} )\gg {\alpha _b}$ is valid.

Equations (4) and (6) tell us that the parameters η_ext and α_b can be determined without fitting Eq. (1) to the full form of ΔT·P_abs⁻¹(λ). This implication also suggests that the measurement points in the LITMoS can be focused on the ranges around the two zero-crossing wavelengths. Furthermore, we can refine λ_X2 in multi-pass pumping schemes to increase the pump absorption because we can waive the measurement of the absorbed power.

3. Fluorescence ray tracing simulation by Monte Carlo method

3.1 Mean fluorescence wavelength

We have shown above that the external quantum efficiency η_ext is calculated once the actual mean fluorescence wavelength λ_f and the shorter zero-crossing wavelength λ_X1 are known (Eq. (4)). However, the accurate determination of the mean fluorescence wavelength is challenging because it depends on the doping level of laser-cooling ions, the sample dimensions and shape, as well as the pumping geometry. In addition, the experimentally measured fluorescence spectrum is sensitive to the detection setting including the detection angle and the position of the detector relative to the pump beam. The dependency of the detected fluorescence on these factors is even more pronounced for non-isotropic media such as YLF because the fluorescence spectra intrinsically depend on the emitted direction. Therefore, we propose an alternative approach, the fluorescence ray tracing simulation by the Monte Carlo method. Our simulation considers the material’s anisotropy, light polarization, total internal reflection, as well as reabsorption and reemission of fluorescence. The birefringence during the propagation of fluorescence rays in anisotropic media is not considered, but its influence on the spectrum of escaping fluorescence is limited or negligible particularly in weakly birefringent crystals such as YLF (n_o = 1.45, n_e = 1.47). Figure 2 shows the flowchart of the simulation for a single ray tracing cycle.

 

Fig. 2. Flowchart of the fluorescence ray tracing simulation for mean fluorescence wavelength.

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First, a monochromatic fluorescence ray is randomly generated at a position $\vec{r}$ in the excited volume considering the spatial distribution of the inversion. The Poynting vector $\vec{k}$ of the fluorescence ray, which is equivalent to the wavevector in isotropic media, is randomly chosen, and its polarization $\vec{E}$ is set randomly but perpendicular to $\vec{k}$. The wavelength λ of the ray is chosen by the rejection sampling based on the intrinsic, i.e., reabsorption-free, fluorescence spectrum I_f0(λ) for the given polarization $\vec{E}$. The simulation then calculates the intersection of the ray with the sample’s surface. The propagation of the ray is simulated with a step size Δd until it reaches the intersection. For each step of propagation, we judge if the ray is absorbed based on the Beer-Lambert law with the absorption coefficient of the medium $\alpha ({\lambda ,\; \vec{E}} )$ by random sampling. Therefore, the probability that the fluorescence ray is absorbed during the propagation of Δd is $1 - \exp [{ - \alpha ({\lambda ,\vec{E}} )\mathrm{\Delta }d} ]$. In reality, the excited volume has a lower absorption coefficient due to bleaching by the pump laser, which is not yet considered in our model. When the ray is absorbed in the medium, to simulate the reemission process, the parameters $\vec{k}$, $\vec{E}$, and λ are reset, and ray tracing continues from the point where the absorption process occurred. Once the ray reaches the surface of the sample, the probability for reflection of the ray is calculated based on the reflectivity given by the Fresnel equations by random sampling. If the ray transmits, it is counted as an escaping ray; otherwise, the Poynting vector $\vec{k}$ is modified by the reflection and ray tracing continues. This cycle is simulated for one million fluorescence rays (N = 10⁶) to ensure a good convergence of the simulated mean fluorescence wavelength. The standard error in the mean fluorescence wavelength is calculated to be ≈0.03 nm for N = 10⁶. Note that a part of the fluorescence rays travels a long distance until escapes owing to successive total internal reflections in highly symmetric samples such as cuboids, which causes long computation times. The simulation thus interrupts the ray tracing after one hundred reflections. As the fraction of the rays that reflect more than 100 times is typically below 1%, this does not affect the resulting escaping fluorescence spectrum.

Figures 3(a, b) show the histograms of the wavelengths of generated and escaping fluorescence simulated for the 10.5-mm long a-cut Yb(10 at.%):YLF described in Section 2 pumped with a perfectly collimated 1.0-mm diameter π-polarized Gaussian beam at a wavelength of 1020 nm. Although we calculated the inversion distribution using the Beer-Lambert law with a small signal absorption coefficient, we found that the mean fluorescence remains unaffected by the inversion distribution, provided that the pump's absorption length exceeds that of the laser-cooling crystal (refer to Fig. S4 in Supplement 1). We used the experimentally acquired spectroscopic data for Yb:YLF that were taken with suppression of fluorescence reabsorption [27] as the input data for the simulation. Note that the fluorescence spectral data used for the simulation must be free of reabsorption. The absorption coefficients were calculated from the emission cross-sections using the reciprocity relation [26] and then scaled to fit with the absorption coefficient spectra obtained from the transmission spectroscopy. This approach allows us to precisely determine absorption coefficients below 10⁻²cm⁻¹ at the long wavelengths relevant for laser cooling. The histogram of the generated fluorescence rays shown in Fig. 3(a) has an identical shape to the polarization-averaged fluorescence spectrum ($2/3{I_\sigma } + 1/3{I_\pi }$). Owing to reabsorption and reemission of fluorescence, the overall spectrum of the escaping fluorescence is red-shifted. This results in a mean fluorescence wavelength λ_f of 1003.4 nm for this sample while that of the intrinsic spectra is 995.0 nm. Using Eq. (4), the external quantum efficiency of Yb:YLF sample is determined to be 99.2%.

 

Fig. 3. Histograms of the wavelength of (a) generated and (b) escaping fluorescence rays (N = 10⁶) simulated for a cuboid-shaped a-cut Yb(10 at.%):YLF sample with dimensions of 2.5 × 3.5 × 10.5 mm³ pumped with a 1-mm diameter π-polarized collimated Gaussian beam at 1020 nm. The blue dotted lines show the polarization-averaged fluorescence spectrum (1/3I_π+2/3I_σ) of Yb:YLF. The vertical dashed lines indicate the mean fluorescence wavelength of the respective spectrum which is also denoted in the figures.

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Fluorescence ray tracing simulations for different sample sizes and pumping geometries revealed the following important findings about the mean fluorescence wavelength: First, the mean fluorescence wavelength does not strongly depend on the position of the pump beam in the sample. Assuming a 10 at.% Yb:YLF sample with dimensions of 3.0 × 3.0-mm² aperture and 10.0 mm in length, simulations with a Gaussian beam pumping at the center of the aperture and at the corner, 0.5 mm from the two lateral facets (see Fig. S2 in Supplement 1), resulted in an identical mean fluorescence wavelength of 1003.4 nm although the escaping fluorescence became strongly anisotropic (see Fig. S3 in Supplement 1). This proves the challenge of an experimental determination of the mean fluorescence wavelength which requires an integration of escaping fluorescence spectra over all directions. Note that the impact of the pump beam position on the mean fluorescence wavelength is even smaller in samples with lower doping levels when the same sample size is assumed. This finding suggests that the mean fluorescence wavelength is insensitive to the pump beam shape, i.e., the mean fluorescence wavelength does not change significantly even for multiple pump beam paths, which is also seen in Fig. S4(a) in Supplement 1 showing a limited change in the mean fluorescence wavelength concerning pump beam diameter. Second, the absorption coefficient for the pump beam also has a small influence on the mean fluorescence wavelength. Figure S4(b) in Supplement 1 shows the simulated mean fluorescence wavelength for different pump absorption coefficients for the 10 at.% Yb:YLF sample (3.0 × 3.0 × 10.0 mm³). The simulation suggests that the mean fluorescence wavelength is not influenced by pump wavelength as far as one absorption length at the pump wavelength is longer than half of the sample.

To examine the validity of our Monte Carlo simulation, we first measured the fluorescence spectrum detected in the setup illustrated in Fig. 4(a). To reduce the influence of the geometry on the detected fluorescence spectrum, we collected the emitted fluorescence using a cosine corrector (CCSA2, Thorlabs) with a diameter of 4 mm and guided it to an optical spectrum analyzer (AQ6374, Yokogawa) via a multimode fiber with a core diameter of 200 µm. For calibration, the spectral throughput of the whole detection system including the cosine corrector and the multimode fiber was measured using a tungsten stripe filament lamp (WI17 G, OSRAM), a blackbody radiation source with a known spectral emissivity [28]. We positioned the cosine corrector ≈8 mm above an a-cut Yb(10 at.%):YLF sample of 2.5 × 3.5 × 10.5 mm³ under π-polarized pumping at 1020 nm. The axis of detection was parallel to the crystal’s c-axis, and the pump beam was positioned at the center of the 2.5 × 3.5-mm² aperture. We simulated the identical fluorescence detection experiment, which also considered the cosine corrector. Figure 4(b) shows the experimentally measured spectrum normalized to the simulated one. The shapes of both spectra are in good agreement, and the mean wavelength of the experimental spectrum was calculated to be 999.0 ± 0.3 nm, while it was 998.4 ± 0.03 nm for the simulated spectrum. We confirmed that the shape of the experimental fluorescence spectrum remained unchanged when the pump wavelength was varied between 1010 and 1020 nm. The small discrepancy between the experiment and simulation may be attributed to the position of the pump beam slightly off the center of the crystal in the experiment.

 

Fig. 4. (a) Setup of the fluorescence detection for a Yb(10 at.%):YLF crystal with dimensions of 2.5 × 3.5-mm² aperture and 10.5-mm long under π-polarized pumping. (b) Experimentally measured and simulated fluorescence spectra for the configuration. The dashed line indicates the mean fluorescence wavelength of the simulated spectrum corresponding to 998.4 nm.

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3.2 Fluorescence escape efficiency

The fluorescence ray tracing model described in Fig. 2 can be further modified to directly simulate the fluorescence escape efficiency. This parameter is not experimentally accessible but is required to retrieve the internal quantum efficiency. The knowledge of the fluorescence escape efficiency also allows us to study the impact of the sample size and shape for laser cooling. By additionally introducing two input parameters, the internal quantum efficiency η_q and background absorption coefficient α_b to respectively consider the non-radiative relaxation of laser-cooling ions and the parasitic absorption by impurities, the model allows simulation of the fluorescence escape efficiency η_e. The external quantum efficiency η_ext can be written using η_e as

This equation seems not straightforward when compared with the definition of external quantum efficiency ${\eta _{\textrm{ext}}} = {\eta _e}{W_{\textrm{rad}}}/({{\eta_e}{W_{\textrm{rad}}} + {W_{\textrm{nr}}}} )$ [27], where W_rad and W_nr are radiative and non-radiative relaxation rates, respectively. The mathematical derivation can be found in Supplement 1. Figure 5 shows the flowchart for simulating the fluorescence escape efficiency η_e in addition to the mean fluorescence wavelength λ_f. When a fluorescence ray is absorbed in the medium, the model judges, if the absorption is due to the laser-cooling ions or impurities by random sampling. Furthermore, to consider the non-radiative relaxation process in laser-cooling ions, another random sampling is applied to judge if a fluorescence ray is re-emitted based on the internal quantum efficiency. In the simulation model, we count the number of escaping rays N_e and disappearing rays N_dis. For a sufficiently large number of the fluorescence rays N = N_e + N_dis, the fluorescence escape efficiency η_e is given as

Again, note that η_e depends not only on the background absorption α_b and the internal quantum efficiency η_q but also on various factors including the size, shape, doping level, and refractive index of laser-cooling samples.

 

Fig. 5. Flowchart of the fluorescence ray tracing simulation by the Monte Carlo method modified considering the internal quantum efficiency and background absorption which allows to determine both the mean fluorescence wavelength and fluorescence escape efficiency.

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The internal quantum efficiency η_q can be estimated in the following steps: 1) Using Eqs. (4) and (6), we determine the external quantum efficiency and background absorption coefficient from the two experimentally determined zero-crossing wavelengths. 2) We run the simulation using the determined background absorption coefficient with varied internal quantum efficiency η_q until finding ${\eta _{\textrm{ext}}} = {\eta _q}{\eta _e}$ consistent with the experimentally determined η_ext. In the case of the Yb:YLF crystal characterized in Fig. 1(b), we found its fluorescence escape efficiency and internal quantum efficiency to be both ≈99.6%. Note that this simulation model does not consider scattering in the medium, so the simulation results may not be valid for strongly scattering samples. The simulation codes corresponding to Fig. 5 are available in Code 1 (Ref. [29])

To study the validity of the model, we demonstrate the determination of the internal quantum efficiency of two Yb:YLF samples with different dimensions but prepared from the same part of a crystalline boule of Yb(10 at.%):YLF; therefore, their internal quantum efficiency and background absorption are expected to be equal. Figure 6(a) shows a photograph of the prepared laser-cooling samples. The dimensions are 2.5 × 2.5 × 7.5 and 6.0 × 6.0 × 15.0 mm³, and their mean fluorescence wavelength λ_f^MC was simulated to be 1001.9 nm and 1007.1 nm, respectively. Figure 6(b) shows the LITMoS results of the two samples around their shorter zero-crossing wavelengths λ_X1. We applied linear fitting on the data points, displayed as solid lines, to find λ_X1. This approach is justified by the linearity of the model equation Eq. (1) concerning the pump wavelength around λ_X1, ensuring an absorption efficiency η_abs of unity. The shorter zero-crossing wavelengths λ_X1 were then determined to be 1010.2 nm and 1018.1 nm with an accuracy of ±0.2 nm, and using Eq. (4) with λ_f^MC, their η_ext was determined to be 99.2% and 98.9% for the small and large samples, respectively. The different η_ext of the samples prepared from the same crystal indicates that the fluorescence escape efficiency in the larger sample is lower owing to a stronger impact of reabsorption and re-emission processes. Based on the longer zero-crossing wavelength λ_X2, the background absorption coefficient α_b was determined to be (0.8 ± 0.2) ·10⁻⁴cm⁻¹ for the small sample. For the larger sample, we could only determine α_b to be smaller than 1.0·10⁻⁴cm⁻¹ because of the difficulty in accurately determining λ_X2, which arises from its high heat capacity resulting in a small temperature change at the available pump power. Nevertheless, these two samples are expected to exhibit an identical α_b as they are considered chemically identical. Using the modified simulation model illustrated in Fig. 5, we simulated the fluorescence escape efficiency η_e and calculated the external quantum efficiency η_ext, using Eq. (7), for the small and large samples with varied internal quantum efficiency η_q as an input parameter. Figure 6(c) shows the resulting relation between η_ext and η_q for the two samples. This relation allows us to inversely determine η_q consistent with the experimentally derived η_ext. As expected, we obtained an identical internal quantum efficiency η_q of 99.6% for both samples. The corresponding fluorescence escape efficiency η_e was 99.6% and 99.3% for the small and large samples, respectively. The determined parameters for the samples are summarized in Table 1. By obtaining an identical internal quantum efficiency η_q using two chemically identical samples of different sizes, we confirm that our model simulating the fluorescence escape efficiency η_e is capable of determining η_q of laser-cooling samples irrespective of their size. We also found that η_e is almost independent of α_b when α_b is smaller than ≈5·10⁻⁴cm⁻¹, so the uncertainty in α_b did not affect our determination of η_q.

 

Fig. 6. (a) Photograph of the prepared six-facet-polished samples from a Yb(10 at.%):YLF crystal. The dimensions of the samples are denoted in the photograph. (b) Laser-induced temperature change normalized by absorbed power ΔT·P_abs⁻¹ versus pump wavelength for the samples. The first zero-crossing wavelength λ_X1 was found at 1010.2 nm and 1018.1 nm, respectively. (c) The relation between external quantum efficiency η_ext and internal quantum efficiency η_q simulated for the two samples. The background absorption coefficient α_b in the simulation was set to 0.8·10⁻⁴cm⁻¹.

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Table 1. Determined parameters of the two Yb:YLF samples. The uncertainty of the mean fluorescence wavelength simulated by the Monte Carlo method λ_f^MC is ±0.03 nm, and that of the zero-crossing wavelengths is ±0.2 nm.

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4. Influence of sample geometry and refractive indices

The simulation model allows us to study the influence of the sample geometry on the mean fluorescence wavelength and fluorescence escape efficiency. In high-symmetry samples, such as cubes and cuboids whose facets are all perpendicular to each other, a part of fluorescence travels a long distance in the sample owing to total internal reflection. The critical angle for the total reflection of YLF is ≈43° so fluorescence rays incident on a facet with an angle in the range of 43–47° cannot escape from high-symmetry samples. Though, in reality, such fluorescence may not be permanently trapped owing to reabsorption, scattering, or surface imperfections, the total internal reflection reduces the fluorescence escape efficiency by increasing the probability of absorption by impurities.

For high-power laser-cooling experiments, Brewster-cut samples are commonly used to minimize the reflection of the pump beam at the facets. We simulated the mean fluorescence wavelength and fluorescence escape efficiency of a Yb:YLF (3.0 × 3.0-mm² aperture, 10.0-mm long) for two geometries: cuboid-shape and Brewster-cut samples. Figures 7(a, b) show the simulated mean fluorescence wavelength λ_f and fluorescence escape efficiency η_e, respectively, for varied doping levels between 2.5 and 20 at.%. Here, we set η_q = 99.0% and α_b = 5.0·10⁻⁴cm⁻¹. Note that in this section, we intentionally selected the parameters that are less favorable than those obtained in the previous section to clearly illustrate the trends. We found that λ_f for Brewster-cut samples is slightly longer than for cuboid-shaped samples for all doping levels. This is because the number of fluorescence rays escaping the sample without any reflection is higher in cuboid-shaped samples than in Brewster-cut samples. On the other hand, η_e is always higher for Brewster-cut samples, since the number of fluorescence rays on closed paths never leaving the sample owing to total internal reflection is higher in cuboid-shaped samples owing to their higher symmetry. This finding suggests that the reduction of the sample symmetry helps to lower the amount of such trapped fluorescence. To examine this, in the case of 5 at.% Yb:YLF as an example, we simulated the fluorescence escape efficiency with varied corner angle θ [see Fig. 7(c)], so that the cross-section of the sample is a trapezoid rather than a square. For both cuboid-shaped and Brewster-cut samples, η_e increases as the corner angle θ decreases from 90°, and the improvement with θ is larger for the cuboid-shaped one. Note that this improvement is purely due to the total internal reflection mitigated by the tilted facet rather than the reduction of the volume. The difference in η_e between the two shapes becomes smaller as the corner angle deviates from 90°, thanks to the symmetry reduction. Note that, as the angle θ decreased, the mean fluorescence wavelength slightly increased, e.g., by ≈0.5 nm for the cuboid-shaped sample, which caused a slight decrease in the laser-cooling efficiency as in Eq. (1). Nevertheless, the sample shape with a corner angle smaller than 90° results in a net improvement in the laser-cooling efficiency because of the higher η_e.

 

Fig. 7. Simulated (a) mean fluorescence wavelength λ_f and (b) fluorescence escape efficiency η_e versus Yb doping level for cuboid-shaped and Brewster-cut Yb:YLF samples (3.0 × 3.0-mm² aperture, 10.0-mm long) at room temperature. Fluorescence escape efficiency η_e versus the sample’s corner angle θ in the case of a cuboid-shaped and a Brewster-cut (c) 5 at.% Yb:YLF and (d) 5 at.% Yb:YAG. The solids lines are guides to the eye.

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Oxides and semiconductors possess higher refractive indices, e.g., 1.82 for Y₃Al₅O₁₂ (YAG) at 1030 nm [30] and even 3.54 for GaAs at 890 nm [31] compared with fluorides. The impact of fluorescence trapping by total internal reflection is therefore a critical issue for such high-index materials. Figure 7(d) displays the fluorescence escape efficiency for a 3.0 × 3.0 × 10.0 mm³ YAG crystal doped with 5 at.% Yb. To prepare the input spectroscopy data required for the simulation, we measured the fluorescence spectrum and absorption cross-section spectrum using a Yb(1 at.%):YAG sample. For the simulation, we assumed a refractive index of 1.816 independent of wavelength. The quantity 1 - η_e corresponds to the loss of fluorescence resulting in heating. For the cuboid-shaped and Brewster-cut samples at θ = 90°, this loss is 0.8% and 0.6% for the Yb:YLF, and 2.5% and 1.5% for Yb:YAG [cf. Figure 7(c)]; thus, the Yb:YAG shows a higher loss owing to its higher refractive index, even for identical size and quality parameters (η_q = 99.0% and α_b = 5.0·10⁻⁴cm⁻¹). On the other hand, the symmetry reduction by the corner angle θ is effective as the fluorescence escape efficiency for the Brewster-cut sample increases to 99.3% at θ = 80° which is only 0.2 percentage points lower than that of the Yb:YLF [cf. Figure 7(c)]. Thus, the reduction of sample symmetry to mitigate total internal reflection is particularly critical for high-index materials. The simulated fluorescence spectrum of a Brewster-cut Yb(5 at.%):YAG can be found in Figure S6 in Supplement 1.

5. Temperature dependency of mean fluorescence wavelength and fluorescence escape efficiency of Yb:YLF

This Monte Carlo simulation model allows us to reveal the temperature dependency of mean fluorescence wavelength and fluorescence escape efficiency. Such knowledge is critical to discussing its impact on the minimum achievable temperatures (MATs). By utilizing the temperature-dependent spectroscopic data of Yb:YLF [27] as the input data, we simulated the mean fluorescence wavelength λ_f, its ratio to the intrinsic mean fluorescence wavelength λ_f/λ_f0, and the fluorescence escape efficiency η_e for the temperature range 50–300 K for three Yb doping levels: 5, 7.5, and 10 at.%. Figures 8(a, b) show the simulation results for Brewster-cut samples with dimensions of 3.0 × 3.0 × 10.0 mm³. As temperature decreases, λ_f increases because of the corresponding increase in λ_f0, shown as cross symbols in Fig. 8(a). On the other hand, the ratio λ_f/λ_f0 approaches unity as temperature decreases. This means that the red-shift in λ_f due to reabsorption and re-emission of fluorescence becomes smaller as the spectral overlap between the absorption and fluorescence spectra diminishes at lower temperatures. Thus, the influence of the Yb doping level on the mean fluorescence wavelength, and accordingly also on the cooling efficiency, becomes smaller at lower temperatures. We also confirmed that the influence of the corner angle θ (cf. Figure 7) varied between 80° and 90° on the mean fluorescence wavelength is negligible for all three doping levels.

 

Fig. 8. (a) Temperature-dependent mean fluorescence wavelengths λ_f of Brewster-cut Yb:YLF samples with doping levels of 5, 7.5, and 10 at.%, and (b) their ratio to the intrinsic mean fluorescence wavelength λ_f0, and (e, f) fluorescence escape efficiencies η_e. This simulation assumes a sample size with a 3.0 × 3.0-mm² aperture and 10.0-mm in length (the 3.0 × 3.0 mm² cross-section is perpendicular to the crystal’s a-axis). The cross symbols in (a) show the values of intrinsic mean fluorescence wavelength λ_f0 [27].

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Figure 9(a) shows the simulated fluorescence escape efficiencies η_e for the Brewster-cut Yb(10 at.%):YLF (3.0 × 3.0 × 10.0 mm³) with corner angles θ of 80° and 90°. Here, we used an internal quantum efficiency η_q of 99.6% and a background absorption coefficient α_b of 1.0·10⁻⁴cm⁻¹ from Table 1. The sample with a corner angle θ of 80° shows a monotonic increase in η_e as temperature decreases, and it approaches 100% at 50 K. On the other hand, when the corner angle θ is 90°, η_e increases as temperature decreases from room temperature down to ≈130 K, but it decreases at lower temperatures. This trend is caused by fluorescence trapping due to the total internal reflection escalating at lower temperatures where reabsorption diminishes. In the absence of reabsorption, trapped fluorescence is inevitably absorbed by impurities in the simulation. In reality, scattering and imperfect surfaces reduce this effect. To discuss this, we introduced scattering into the model. The fluorescence ray is randomly redirected once it is judged to scatter by random sampling based on an introduced scattering coefficient. Note that the treatment of light scattering in the model is simplified because, in reality, the scattering processes show a spatial anisotropy and a dependency on the polarization. We found that scattering centers do not influence η_e when the scattering coefficient is smaller than 10⁻³cm⁻¹.

 

Fig. 9. (a) Simulated temperature-dependent fluorescence escape efficiency η_e of a Brewster-cut Yb(10 at.%):YLF sample (3.0 × 3.0 × 10.0 mm³) for two corner angles θ. For this simulation, we assumed an internal quantum efficiency η_q of 99.6% and a background absorption coefficient α_b of 0.8·10⁻⁴cm⁻¹. (b, c) Calculated cooling efficiency as a function of pump wavelength and temperature considering the temperature dependency of the fluorescence escape efficiency η_e in the case of θ = 90° and θ = 80°, respectively.

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The simulated temperature-dependent mean fluorescence wavelength λ_f(T) and fluorescence escape efficiency η_e(T) allow us to improve the accuracy of achievable cooling temperatures by calculating the cooling efficiencies as a function of pump wavelength and temperature, as shown in Figs. 9(b, c). Note that we do not consider the temperature dependency of the background absorption coefficient [19]. For the Brewster-cut Yb(10 at.%):YLF (3.0 × 3.0-mm² aperture, 10.0-mm long) with a corner angle θ of 90°, the cooling efficiency sharply drops around 100 K owing to the decrease of η_e [cf. Figure 9(a)], and the MAT is estimated to be 93 K at an optimum pump wavelength of 1019.5 nm. At a smaller corner angle θ of 80°, the MAT is found to be 86 K. Thus, mitigating the fluorescence trapping due to the total internal reflections is crucial to achieving lower cooling temperatures, and introducing a 10°-titled lateral facet is a simple but efficient approach.

6. Conclusion

We proposed a comprehensive approach for the accurate evaluation of solid-state media for laser cooling by anti-Stokes fluorescence, employing the fluorescence ray tracing simulation by Monte Carlo method. This approach prompted a revisit of the experimental method, LITMoS, revealing that key sample-specific quality parameters—external quantum efficiency and background absorption coefficient—can be retrieved from knowledge of the two zero-crossing wavelengths alone. Our simulation enables us to accurately determine two experimentally inaccessible parameters, the mean fluorescence wavelength and fluorescence escape efficiency. The knowledge of fluorescence escape efficiency allows us to retrieve the internal quantum efficiency, which is more relevant than external quantum efficiency for assessments of laser-cooling samples as the external quantum efficiency depends on the sample size, shape, as well as doping level. The reliability of our approach was confirmed through a successful validation using experimental results using Yb(10 at.%):YLF. Furthermore, we studied the impact of sample geometry and revealed that mitigating successive total internal reflections is essential for achieving high fluorescence escape efficiency. This effect is even more pronounced in materials with high refractive indices, such as oxide crystals. Our findings demonstrate that the reduction of sample symmetry is a key factor in achieving lower cooling temperatures. By providing reliable characterization for improving the crystal growth of laser-cooling materials, the proposed approach paves the way for advancements in all-solid-state optical refrigerators.