2 Theoretical model for the Tm ion emission dynamics

In order to simulate the steady-state dynamics of the $\mathrm{Tm}{:}{\mathrm{Lu}}_2{\mathrm{O}}_3$ ceramic laser, we consider the energy levels and the transitions shown in Figure 1. The rate equations can be obtained from the ones in Ref. [Reference Albalawi, Varas, Chiasera, Gebavi, Albalawi, Blanc, Balda, Lukowiak, Ferrari and Taccheo17] as follows:

(1) $$\begin{align}\frac{\mathrm{d}{N}_4}{\mathrm{d}t}&={W}_{14}{N}_1-{W}_{41}{N}_4-\frac{N_4}{\tau_4}-{P}_{41}{N}_4{N}_1+{P}_{22}{N}_2^2, \end{align}$$

(2) $$\begin{align}\frac{\mathrm{d}{N}_3}{\mathrm{d}t}&=-\frac{N_3}{\tau_3}+\frac{\beta_{43}{N}_4}{\tau_4},\end{align}$$

(3) $$\begin{align}\frac{\mathrm{d}{N}_2}{\mathrm{d}t}&=2{P}_{41}{N}_4{N}_1-2{P}_{22}{N}_2^2-\frac{N_2}{\tau_2}+\frac{\beta_{42}{N}_4}{\tau_4}+\frac{\beta_{32}{N}_3}{\tau_3}\nonumber\\& \quad +{A}_{\mathrm{L}}{N}_1-{E}_{\mathrm{L}}{N}_2,\end{align}$$

(4) $$\begin{align}\frac{\mathrm{d}{N}_1}{\mathrm{d}t}&={W}_{41}{N}_4-{W}_{14}{N}_1+{P}_{22}{N}_2^2-{P}_{41}{N}_4{N}_1+\frac{N_2}{\tau_2}\nonumber\\& \quad +\frac{\beta_{41}{N}_4}{\tau_4}+\frac{\beta_{31}{N}_3}{\tau_3}+{E}_{\mathrm{L}}{N}_2-{A}_{\mathrm{L}}{N}_1,\end{align}$$

where ${N}_1$ , ${N}_2$ , ${N}_3$ and ${N}_4$ are the population densities of the levels ${}^3{\mathrm{H}}_6$ , ${}^3{\mathrm{F}}_4$ , ${}^3{\mathrm{H}}_5$ and ${}^3{\mathrm{H}}_4$ respectively. The sum ${N}_1+{N}_2+{N}_3+{N}_4=N$ is given by the total ion density, which can be inferred by the doping level. The spontaneous emission lifetime of the i-level is given by ${\tau}_i$ , while ${\beta}_{ij}$ represents the $i\to j$ -level branching ratio, with ${\sum}_j{\beta}_{ij}=1$ . The pump rates are defined by ${W}_{14}(t)={\sigma}_{\mathrm{a}}{I}_{\mathrm{p}}(t)/h{\nu}_{\mathrm{p}}$ and ${W}_{41}(t)={\sigma}_{\mathrm{e}}{I}_{\mathrm{p}}(t)/h{\nu}_{\mathrm{p}}$ , where ${\sigma}_{\mathrm{a}}$ and ${\sigma}_{\mathrm{e}}$ are respectively the absorption and emission pump cross-section obtained from the measurements reported in Figure 2, ${I}_{\mathrm{p}}$ is the pump intensity, h is the Planck constant and ${\nu}_{\mathrm{p}}$ is the frequency of the pump laser. The functions ${E}_{\mathrm{L}}(t)={\sigma}_{\mathrm{eL}}{I}_{\mathrm{L}}(t)/h{\nu}_{\mathrm{L}}$ and ${A}_{\mathrm{L}}(t)={\sigma}_{\mathrm{aL}}{I}_{\mathrm{L}}(t)/h{\nu}_{\mathrm{L}}$ specify the lasing rates with ${\sigma}_{\mathrm{aL}}$ and ${\sigma}_{\mathrm{eL}}$ estimated from the data reported in Figure 7 shown later , which are similar to those found in Ref. [Reference Baylam, Canbaz and Sennaroglu18]. Direct and inverse CRs are taken into account through the parameters ${P}_{41}$ and ${P}_{22}$ , respectively.

Figure 1 Scheme of the energy levels used to model the laser dynamics, drawn following the nomenclature used in Ref. [Reference Albalawi, Varas, Chiasera, Gebavi, Albalawi, Blanc, Balda, Lukowiak, Ferrari and Taccheo17].

Figure 2 Measured absorption spectrum of the ceramic sample used in this work.

The systems of Equations (1)–(4) can be coupled with an optical cavity (modeled as a Fabry–Pérot resonator) by considering the equation obtained starting from^[ Reference Baylam, Canbaz and Sennaroglu ^{18 ]}

(5) $$\begin{align}\frac{\partial {I}_{\mathrm{L}}(t)}{\partial t}=2\left({\alpha}_{\mathrm{L}}(t)l-\frac{T}{2}-L\right)\frac{I_{\mathrm{L}}(t)+{I}_{\mathrm{s}}(t)}{T_{\mathrm{R}}},\end{align}$$

where ${\alpha}_{\mathrm{L}}={\sigma}_{\mathrm{eL}}{N}_2-{\sigma}_{\mathrm{aL}}{N}_1$ is the amplification coefficient, l is the medium thickness, $T=1-{R}_1$ is the output coupler transmission and ${T}_{\mathrm{R}}=l/c$ is the cavity roundtrip half time, where ${R}_1$ , ${R}_2$ and D are the cavity mirror reflectivities and cavity length, respectively, and $L$ represents the combined residual cavity losses. Equation (5) is initialized by the spontaneous emission intensity term ${I}_{\mathrm{s}}(t)={N}_2(t)h{\nu}_{\mathrm{L}} l\varOmega /\left(4{\pi \tau}_2\right)$ , with $\varOmega$ being the smaller solid angle defined by the mirrors and ${\nu}_{\mathrm{L}}$ the frequency of the emitted laser. The output laser intensity is given by ${I}_{\mathrm{out}}(t)={TI}_{\mathrm{L}}(t)$ .

The total ${\mathrm{Tm}}^{3+}$ ion density with a doping percentage of ${\eta}_{\mathrm{d}}$ (atomic fraction) is given by $N\left({\eta}_{\mathrm{d}}\right)=2.8{\eta}_{\mathrm{d}}\times {10}^{28}\ {\mathrm{m}}^{-3}$ where, in our case, we have ${\eta}_{\mathrm{d}}=0.04$ ( $N=1.12\times {10}^{27}\ {\mathrm{m}}^{-3}$ ). According to Refs. [Reference Albalawi, Varas, Chiasera, Gebavi, Albalawi, Blanc, Balda, Lukowiak, Ferrari and Taccheo17,Reference Gebavi, Milanese, Balda, Chaussedent, Ferrari, Fernandez and Ferraris19], we use ${\tau}_i\left({\eta}_{\mathrm{d}}\right)={\tau}_{i0}/\left(1+{A}_i{\eta}_{\mathrm{d}}^2\right)$ , where ${\tau}_{20}=3.4\;\mathrm{ms}$ and ${\tau}_{40}=0.6\;\mathrm{ms}$ ^[ Reference Albalawi, Varas, Chiasera, Gebavi, Albalawi, Blanc, Balda, Lukowiak, Ferrari and Taccheo ^{17 ,} Reference Baylam, Canbaz and Sennaroglu ^{18 ,} Reference Loiko, Koopmann, Mateos, Serres, Jambunathan, Lucianetti, Mocek, Aguiló, Díaz, Griebner, Petrov and Kränkel ^{20 ]}. In our experimental conditions ${\tau}_2\left(4\%\right)\simeq 1.22\;\mathrm{ms}$ and ${\tau}_4\left(4\%\right)\simeq 63\;\mu \mathrm{s}$ , while ${\tau}_3\approx 2\;\mu \mathrm{s}$ is assumed to be independent of the dopant concentration. The branching ratio coefficients are given by ${\beta}_{31}=0.9793$ , ${\beta}_{32}=0.0207$ , ${\beta}_{41}=0.9035$ , ${\beta}_{42}=0.0762$ and ${\beta}_{43}=0.0203$ ^[ Reference Antipov, Golovkin, Gorshkov, Zakharov, Zinov’ev, Kasatkin, Kruglova, Marychev, Novikov, Sakharov and Chuprunov ^{21 ]}. The CR mechanism is dominated by the direct ${P}_{22}/{P}_{41}=0.03-0.08$ ^[ Reference Albalawi, Varas, Chiasera, Gebavi, Albalawi, Blanc, Balda, Lukowiak, Ferrari and Taccheo ^{17 ,} Reference Tao, Huang, Yu, Yang, Chen and Ye ^{22 ,} Reference Falconieri, Lanzi, Salvetti and Toncelli ^{23 ]}. Following Ref. [Reference Tao, Huang, Yu, Yang, Chen and Ye22], the coefficient is given by ${P}_{41}\left({\eta}_{\mathrm{d}}\right)=B{\eta}_{\mathrm{d}}^2/\left({\eta}_{\mathrm{d}}^2+{\eta}_0^2\right)$ , where ${\eta}_0=4.3$ %^[ Reference Rustad and Stenersen ^{24 ]} is the characteristic dopant concentration and $B=2.8\times {10}^{-22}\;{\mathrm{m}}^3\ {\mathrm{s}}^{-1}$ is obtained from Refs. [Reference Tao, Huang, Yu, Yang, Chen and Ye22,Reference Falconieri, Lanzi, Salvetti and Toncelli23]. We find ${P}_{41}\left(4\%\right)=6.28\times {10}^{-29}\;{\mathrm{m}}^3\;{\mu \mathrm{s}}^{-1}$ , with ${\left({P}_{41}\left(4\%\right)N\left(4\%\right)\right)}^{-1}\simeq 14.2\;\mu \mathrm{s}$ . The CR parameter ${\eta}_{\mathrm{CR}}$ can be numerically evaluated by considering the ratio ${\eta}_{\mathrm{CR}}\left({P}_{\mathrm{eff}}\right)={N}_2/{N}_2^{\mathrm{cr}}$ under stationary conditions, where ${N}_2^{\mathrm{cr}}$ are the results of the system of Equations (1)–(5) when the CR is forcefully turned off ( ${P}_{41}={P}_{22}\equiv 0$ ). Above a certain threshold, which is approximately given by the lasing threshold of the system with no CR, the efficiency parameter becomes almost independent of the pump power and thus ${\eta}_{\mathrm{CR}}\left({P}_{\mathrm{eff}}\right)\to {\eta}_{\mathrm{CR}}$ .

Numerical simulations are performed considering a perfect overlapping between the pump and laser waist, so that the retrieved pump power actually corresponds to the effective absorbed pump power ${P}_{\mathrm{eff}}$ . The slope efficiency is defined by ${\eta}_{\mathrm{sl}}={P}_{\mathrm{out}}/\left({P}_{\mathrm{eff}}-{P_{\mathrm{eff}}}^{\mathrm{th}}\right)$ , where ${P_{\mathrm{eff}}}^{\mathrm{th}}$ is the effective pump threshold. This value is related to ${\eta}_{\mathrm{CR}}$ through ${\eta}_{\mathrm{sl}}\simeq {\eta}_{\mathrm{CR}}\left({\lambda}_{\mathrm{P}}/{\lambda}_{\mathrm{L}}\right){R}_1/\left({R}_1+L\right)$ , where ${\lambda}_{\mathrm{P}}$ and ${\lambda}_{\mathrm{L}}$ are the pump and laser wavelength, respectively. The limit is given by ${\eta}_{\mathrm{sl}}\le 0.8$ for ${\eta}_{\mathrm{CR}}=2$ . An alternative set of equations is considered in Ref. [Reference Baylam, Canbaz and Sennaroglu18], where ${P}_{41}={P}_{22}\equiv 0$ , while the measured ${\eta}_{\mathrm{CR}}$ is directly introduced in Equation (5) by using ${\eta}_{\mathrm{CR}}{\sigma}_{\mathrm{eL}}{N}_2-{\sigma}_{\mathrm{aL}}{N}_1$ instead of ${\alpha}_{\mathrm{L}}$ . The simulation parameters are reported in Table 1.

Table 1 Simulation parameters.

3 Experimental setup

The ceramic sample used in our study, with size 5 mm × 5 mm × 3.1 mm, was produced by Konoshima Chemicals Co., Japan.

As for its optical quality, the sample appeared transparent and clear. Its scattering coefficient was measured (using a p-polarized laser beam) in the visible region at a wavelength of $543\kern0.22em \mathrm{nm}$ (where the Tm absorption is negligible). The overall transmission of the sample at the measurement wavelength was about 74%, to be compared to the theoretical value of 82% that can be calculated on the basis of the Fresnel reflection. Therefore, the losses determined by scattering on the ceramics defects were about 8%. Upon assuming an exponential law for the propagation into the sample, and by taking into account the reflection off the entrance and exit faces (uncoated) and the direction of propagation into the sample due to refraction, an absorption coefficient ${\alpha}^{\left(543\mathrm{nm}\right)}\simeq 0.3\kern0.22em {\mathrm{cm}}^{-1}$ can be retrieved.

Furthermore, a spectrophotometer was used to test the sample absorption at different wavelengths against known values. An example of such a measure is reported in Figure 2, and is in good agreement with those reported in the literature^[ Reference Baylam, Canbaz and Sennaroglu ^{18 ,} Reference Loiko, Koopmann, Mateos, Serres, Jambunathan, Lucianetti, Mocek, Aguiló, Díaz, Griebner, Petrov and Kränkel ^{20 ]}. A set of similar measurements was performed at different transverse positions of the sample, which showed excellent sample transverse homogeneity.

In the measurements described here, the ceramic sample is mounted on a water cooled copper block. The thermal contact is ensured by a thin indium foil mounted between the lateral surface of the ceramic sample and the copper block itself. We use two working temperatures of 13°C and $23{}^{\circ}\mathrm{C}$ ; the temperature is monitored by two one-wire sensors (Maxim, model DS18B20) with a $0.5{}^{\circ}\mathrm{C}$ sensitivity mounted on the sample holder.

Our test laser cavity is based on a three-mirror layout, as depicted in Figure 3, in a similar optical scheme to that in Ref. [Reference Feng, Toci, Patrizi, Pirri, Hu, Chen, Wei, Pan, Li, Zhang, Su, Vannini and Li25]; in particular, it features an end mirror (EM in Figure 3), through which the (longitudinal) pumping occurs, an output coupling mirror (OM) and a $100\kern0.22em \mathrm{mm}$ curvature radius folding spherical mirror (SM). The pumping beam is obtained by a laser diode emitting at a measured wavelength of $794.6\pm 0.4\kern0.22em \mathrm{nm}$ , hence in the proximity of the absorption peak in Figure 2.

Figure 3 Scheme (not to scale) of the experimental apparatus: the achromatic doublets (ADs) are used to focus the pump beam from the optical fiber to the sample; the cavity is made up of three mirrors (see text); the dichroic entry mirror (EM) and the spherical mirror (SM) feature a high-reflectivity (HR) coating for approximately $2\kern0.22em \mu \mathrm{m}$ radiation and an anti-reflectivity (AR) coating for the pump wavelength. A 90% or 97% reflectivity output coupler mirror (OM) is used throughout the measurement. Both the pump and the laser beams are monitored in terms of power and spectrum with photodiodes, power meters and spectrometers. In the inset, the laser spot is captured at a distance of $500\kern0.22em \mathrm{mm}$ from the output coupler mirror with a Dataray WinCamD camera.

The laser diode is coupled to a multimode optical fiber of $200\;\mu \mathrm{m}$ in diameter and a numerical aperture of 0.22. The beam emerging from the optical fiber is focused by two plano-convex $f=50\kern0.22em \mathrm{mm}$ optical doublets arranged in a $4f$ scheme. The ${M}^2$ of the pump is measured using a similar procedure to that described, for instance, in Ref. [Reference Siegman26]. The intensity profiles of the beam at several planes are found to be well fitted by a Gaussian function^[ Reference Moulton ^{27 ]}; on applying a standard ${M}^2$ corrected Gaussian propagation model to fit the observed widths as a function of the propagation distance, the beam waist is estimated to be ${w}_{\mathrm{P}}\simeq 160\kern0.22em \mu \mathrm{m}$ , and ${M}^2\simeq 150$ ; this is in rather good agreement (within $10\%$ ) with what can be estimated from theoretical considerations: ${M}_{\left(\mathrm{th}\right)}^2\simeq 170$ (see, for instance, Ref. [Reference Griessmann, Martinez-Becerril and Lundeen28]).

We point out that, given the full width at half maximum of $2\kern0.22em \mathrm{nm}$ of the pump diode laser emission, we did not observe any significant change in the behavior of the $\mathrm{Tm}{:}{\mathrm{Lu}}_2{\mathrm{O}}_3$ due to the slight detuning of the pump laser with respect to the exact absorption peak of $796.2\kern0.22em \mathrm{nm}$ , except for a slight decrease in the radiation absorbed by the sample that is taken into account by the measurement procedure we used, which is described below.

We operate the pump laser with pulses lasting $10\;\mathrm{ms}$ repeated at a frequency of 10 Hz and we observe that the laser emission peaked at both 1965 and $2065\kern0.22em \mathrm{nm}$ , as predicted in Refs. [Reference Loiko, Koopmann, Mateos, Serres, Jambunathan, Lucianetti, Mocek, Aguiló, Díaz, Griebner, Petrov and Kränkel20,Reference Koopmann, Peters, Petermann and Huber29] and observed in Refs. [Reference Baylam, Canbaz and Sennaroglu18,Reference Koopmann, Peters, Petermann and Huber29]. In our case we observe only the $1965\kern0.22em \mathrm{nm}$ wavelength emission when using the 90% reflectivity output coupler while using the 97% reflectivity output coupler and both emission wavelengths or just the $2065\kern0.22em \mathrm{nm}$ one is visible, depending on the alignment of the cavity, and hence on the cavity losses; see Figure 4. All the data that are presented below in the text are taken with a well-aligned laser emitting only at $2065\kern0.22em \mathrm{nm}$ .

Figure 4 Laser spectra for the two 90% and 97% reflectivity output coupler mirrors. With the 97% reflectivity we observe a change in the emission spectra as a function of the cavity losses due to its alignment.

4 Experimental results

We calculate the cavity beam size by means of the ABCD formalism^[ Reference Kogelnik and Li ^{30 ]}, obtaining a waist of $144\pm 5\;\mu \mathrm{m}$ roughly constantly across the ceramic sample.

Using the measured pump beam spot size and the laser beam model we can calculate the average pump rate $\left\langle {R}_{\mathrm{P}}\right\rangle$ considering the volumetric overlap between the two beams in the lasing medium^[ Reference Svelto ^{31 ]}, as follows:

(6) $$\begin{align}\left\langle {R}_{\mathrm{p}}\right\rangle =\frac{\alpha {P}_{\mathrm{Inc}}}{h{\nu}_{\mathrm{p}}}\frac{\int_0^d\frac{w_{\mathrm{L}}{(z)}^2}{w_{\mathrm{L}}{(z)}^2+{w}_{\mathrm{p}}{(z)}^2}{\mathrm{e}}^{-\alpha z}\mathrm{dz}}{\frac{\pi }{2}{\int}_0^d{w}_{\mathrm{L}}{(z)}^2\mathrm{dz}},\end{align}$$

where ${P}_{\mathrm{inc}}$ is the pump laser power inside the sample. Assuming the laser beam has a constant spot size ${w}_{\mathrm{L}}$ inside the ceramic sample of length $d$ we can calculate the effective absorbed pump laser power P _eff as follows:

(7) $$\begin{align}{P}_{\mathrm{eff}}=\left\langle {R}_{\mathrm{p}}\right\rangle \kern0.1em \pi {w}_{\mathrm{L}}^2d=\chi {P}_{\mathrm{inc}},\end{align}$$

which in our case results in $\chi =0.47\pm 0.06$ .

It is worth noting at this point the main limitations of the above procedure. Firstly, the laser beam profile inside the cavity is retrieved by the ABCD simulation; although this is a well-consolidated procedure in such experiments (see, for instance, Refs. [Reference Baylam, Canbaz and Sennaroglu18,Reference Loiko, Koopmann, Mateos, Serres, Jambunathan, Lucianetti, Mocek, Aguiló, Díaz, Griebner, Petrov and Kränkel20]), it can result in some uncertainty. Secondly, Equation (7) holds in the case of a negligible depletion of the ground state of the medium, since the absorption coefficient considered could otherwise vary during laser operation. For this reason, we calculate the absorbed pump power as the difference between the power incident upon the active medium and the one transmitted through it. Since a direct measurement of the transmitted power directly downstream of the active medium (i.e., inside the cavity) would inhibit the lasing condition, making the procedure inconsistent, such a measurement is actually carried out using a silicon photodiode placed behind the SM (see Figure 3), in the direction of the pump beam; such a photodiode was preliminarily calibrated in order to retrieve an absolute figure for the transmitted pump power, under no lasing condition, using a power meter placed just after the active medium. Then, our actual measurements are carried out by simultaneously acquiring the laser power signal, using a power meter placed right at the exit of the cavity (behind the output coupler), and the absolutely calibrated photodiode signal from which the transmitted pump power can be retrieved. Finally we obtain the absorbed pump power as follows:

(8) $$\begin{align}{P}_{\mathrm{abs}}={P}_{\mathrm{inc}}-{P}_{\mathrm{tras}}\frac{V_{\mathrm{on}}}{V_{\mathrm{off}}},\end{align}$$

where ${V}_{\mathrm{on}}$ and ${V}_{\mathrm{off}}$ refer to the photodiode signal with and without lasing, respectively, and considering as well the reflections at the ceramic faces. With this procedure, we observe a difference of up to 10% in the transmitted pump power with or without lasing. From the linear best fit to the data shown in Figure 5 we can observe that we always work below the saturation intensity; hence, the absorbed pump laser power can be written as follows:

(9) $$\begin{align}{P}_{\mathrm{tras}}=\left(1-g\right){P}_{\mathrm{inc}},\end{align}$$

resulting in $g=0.6\pm 0.02$ in accordance with the fraction calculated using the small signal absorption coefficient $\alpha =322\;{\mathrm{m}}^{-1}$ .

Figure 5 Pump laser power transmitted as a function of the incident pump laser power for the two different working temperatures of 13°C and $23{}^{\circ}\mathrm{C}$ . Straight lines are the results of a best fit calculation that we use to obtain the pump absorption ratio $g$ in Equation (9).

It should be noted that the value ${P}_{\mathrm{abs}}$ we obtain with this procedure gives us the pump radiation absorbed over the whole volume described by the pump beam, which is larger than the laser beam; therefore, we calculate ${P}_{\mathrm{eff}}=\left(\chi /g\right){P}_{\mathrm{abs}}$ , which matches Equation (7) with the substitution ${{P}_{\mathrm{abs}}={gP}_{\mathrm{inc}}}$ .

The measured laser power as a function of P _eff for the two working temperatures reported in Figure 6 shows a very small difference between the two datasets.

Figure 6 Laser power as a function of the effective absorbed pump power for the two working temperatures of 13°C and $23{}^{\circ}\mathrm{C}$ ; the straight lines are the result of a best fit calculation that provides both the laser threshold power and the slope efficiency.

With our definition of P _eff we obtain the slope efficiency of the laser by means of a best fit calculation of the experimental data in Figure 6 with the equation ${P}_{\mathrm{L}}=m\left({P}_{\mathrm{eff}}-{P}_{\mathrm{thr}}\right)$ resulting in ${m}_{13}=0.75\pm 0.02$ and ${m}_{23}=0.74\pm 0.02$ , which gives CR parameters ${\eta}_{\mathrm{CR}}=\left({\lambda}_{\mathrm{L}}/{\lambda}_{\mathrm{P}}\right)\times m$ of 1.960.05 and 1.910.05, respectively, at 13°C and $23{}^{\circ}\mathrm{C}$ . We point out that these values are in agreement with values reported in the recent literature^[ Reference Baylam, Canbaz and Sennaroglu ^{18 ,} Reference Loiko, Koopmann, Mateos, Serres, Jambunathan, Lucianetti, Mocek, Aguiló, Díaz, Griebner, Petrov and Kränkel ^{20 ]}, provided that the higher doping level of our sample and the dependence of ${\eta}_{\mathrm{CR}}$ upon the Tm concentration given in Ref. [Reference Zheng, Wu, Zhang, Luo, Pan, Wang, Hao and Zhang32] are taken into account. To measure the laser threshold power we added to the apparatus a neutral density filter between the two lenses of the pump beam optics to obtain a lower pump laser power. The resulting laser power is reported in Figure 7 with the same best fit calculation resulting in a laser threshold of ${P}_{\mathrm{thr}}=0.6\pm 0.02\;\mathrm{W}$ and in a slope efficiency $m=0.73\pm 0.02$ and a CR coefficient of ${\eta}_{\mathrm{CR}}=1.89\pm 0.05$ with a working temperature of $23{}^{\circ}\mathrm{C}$ . Data in Figure 7 are superimposed on the numerical simulation performed using the model described in Section 2, where the parameter $L=1.1\%$ results from the best fit on our data. The simulation results in a laser threshold of ${P}_{\mathrm{thr}}=0.64\;\mathrm{W}$ and a CR parameter ${\eta}_{\mathrm{CR}}=1.91$ , in agreement with the experimental data.

Figure 7 Experimental and theoretical laser power as a function of P _eff. The dashed line is a linear fit of the data, while the solid line is obtained with the model in Equations (1)–(5). The cavity energy loss $L$ is tailored to 1.3% for the model to match the data.

To further validate the theoretical model we simulated the pulsed behavior of our system and compared the numerical results with the experimental one obtained by modulating the amplitude of the pump laser with rectangular pulses of tunable time duration at a fixed repetition frequency of 1 kHz. The data obtained are shown in Figure 8 superimposed to the simulation obtained using the term ${W}_{41}$ in Equations (1)–(4) for the pump waveform recorded with a power calibrated silicon photodiode. The laser power is recorded with an indium gallium arsenide (InGaAs) photodiode and the signal obtained is scaled using the experimental slope efficiency of the laser. The raw data are filtered with a numerical low-pass filter with the cutoff frequency set at the sampling rate of the oscilloscope used to record the signals. In this case the pulsed dynamics of the laser intensity over a millisecond timescale is well reproduced by the simulation. As a matter of fact, the exact temporal dynamics at the rising edge of the pulse depends critically on the actual pump laser intensity rising profile, whose behavior over approximately $10\kern0.22em \mu \mathrm{s}$ timescales is not perfectly captured by our experimental apparatus. Moreover, rather complex emission dynamics, possibly involving both the emission wavelengths reported above, was also experimentally observed in Ref. [Reference Baylam, Canbaz and Sennaroglu18]; we are not accounting for this short timescale behavior, as it does not affect our comparison with the experimental data over the millisecond timescale considered in this paper. It is worth noticing that the delay asymptotically equal to $100\;\mu \mathrm{s}$ between the pump laser rising wavefront and the laser emission in the first pulse is independent of the pulse duration, while in the subsequent pulses, the delay is shorter but depends on time between the pulses. As a further remark, again from Figure 8 it can be noted that the laser emission amplitude reaches the steady state within the few initial pulses, that is, in a timescale comparable with the fluorescent time of the laser excited state ${\tau}_{40}$ .

Figure 8 Experimental and theoretical laser power as a function of time obtained for pump pulse width of $150\;\mu \mathrm{s}$ in the top panel and $700\;\mu \mathrm{s}$ in the bottom panel.

Finally, we can use the lifetime ${\tau}_{40}$ of the excited manifold ${}^3{\mathrm{H}}_4$ , the ${P}_{41}$ coefficient and thulium concentration $N$ to calculate the CR coefficient. As reported^[ Reference Loiko, Koopmann, Mateos, Serres, Jambunathan, Lucianetti, Mocek, Aguiló, Díaz, Griebner, Petrov and Kränkel ^{20 ,} Reference van Dalfsen, Aravazhi, Grivas, García-Blanco and Pollnau ^{33 ]}, ${{\eta}_{\mathrm{CR}}=\frac{P_{41}N}{1/{\tau}_{40}+{P}_{41}N}}$ and, with our parameters, it results in ${\eta}_{\mathrm{CR}}=1.97$ , in good agreement with our experimental and simulated values as well as with the values reported in the literature.