## Abstract

An analytical model is formulated to support understanding and underpin experimental development of laser action in the promising diode end-pumped Alexandrite system. Closed form solutions are found for output power, threshold, and slope efficiency that for the first time incorporate the combined effects of laser ground state absorption and excited state absorption (laser ESA), along with pump excited state absorption (pump ESA), in the case of an end-pumping geometry. Comparison is made between model predictions and experimental results from a fiber-delivered diode end-pumped Alexandrite laser system, showing the impact of wavelength tuning, crystal temperature, laser output coupling, and intracavity loss. The model is broadly applicable to other quasi-three-level lasers with combined laser and pump ESA. A condition for bistable operation is also formulated.

© 2016 Optical Society of America

## 1. INTRODUCTION

Alexandrite (${\mathrm{Cr}}^{3+}\text{:}{\mathrm{BeAl}}_{2}{\mathrm{O}}_{4}$) is a solid-state laser crystal with excellent thermomechanical properties and a long upper-state lifetime ($\sim 260\text{\hspace{0.17em}}\mathrm{\mu s}$) [1]. It can operate on a vibronic transition, giving continuously wavelength tunable operation with a demonstrated range of 701–858 nm [2].

Early work on Alexandrite used flashlamp pumping [1–4]; however, its broad absorption bands also permit pumping with red laser diodes [5]. Recent advancements in high-power red laser diodes has enabled efficient and multiwatt diode-pumped operation [6,7]. The benefits of diode pumping include an order of magnitude increase in laser efficiency and the ability to more easily control the delivery of the pump energy to the gain medium, as in end pumping, achieving a better ${\mathrm{TEM}}_{00}$ mode overlap [7]. Diode-pumped Alexandrite is a promising prospect as a tunable, high efficiency laser source for lidar [5], among other applications.

The energy structure of Alexandrite is shown in Fig. 1. Several broad bands exist, consisting of continuous vibrational energy levels that are thermally populated. They are the ${}^{4}{\mathrm{A}}_{2}$ ground state, ${}^{4}{\mathrm{T}}_{2}$ upper lasing level, and additional higher-lying bands. The pump is absorbed through a vibronic transition from the ground state, where the excited ions then decay nonradiatively to the upper metastable laser level. Laser vibronic emission initiates from the ${}^{4}{\mathrm{T}}_{2}$ level into the vibrational ground levels, giving rise to the broad emission band of Alexandrite.

A particular feature of Alexandrite is a long-lived energy state ${}^{2}\mathrm{E}$ that is positioned just below the ${}^{4}{\mathrm{T}}_{2}$ level by an energy separation of just $\mathrm{\Delta}E=800\text{\hspace{0.17em}}{\mathrm{cm}}^{-1}$ [1]. The ${}^{2}\mathrm{E}$ and ${}^{4}{\mathrm{T}}_{2}$ levels are in thermal equilibrium, such that at higher temperatures a greater fraction of excited population is in the ${}^{4}{\mathrm{T}}_{2}$ level, which increases the effective emission cross section ${\sigma}_{e}$ of the total population [8]. The ${}^{2}\mathrm{E}$ and ${}^{4}{\mathrm{T}}_{2}$ levels individually have different fluorescence lifetimes of ${\tau}_{E}=1.54\text{\hspace{0.17em}}\mathrm{ms}$ and ${\tau}_{T}=6.6\text{\hspace{0.17em}}\mathrm{\mu s}$, respectively [1]. This gives the total excited population a combined fluorescence lifetime ${\tau}_{f}$ of the form [9]

Ground state absorption (GSA) of the laser radiation occurs in Alexandrite due to thermal filling of the ground vibrational levels. Due to its fast nonradiative dynamics [10,11], the relationship between the emission cross section ${\sigma}_{e}$ and GSA cross section ${\sigma}_{a}$ is given by the extended McCumber theory [1,12] as

where $E=hc/{\lambda}_{l}$ is the laser photon energy, $h$ is the Planck constant, $c$ is the speed of light, ${\lambda}_{l}$ is the laser wavelength, and ${E}^{*}$ is the effective no-phonon energy ($E<{E}^{*}$). The McCumber relationship, Eq. (2), reveals the exponential nature of the wavelength and temperature dependence of the GSA cross section due to the varying Boltzmann filling of the ground vibrational levels.Due to higher-lying energy bands, radiation can also be absorbed from excited ions in the upper laser level, giving rise to excited state absorption (ESA). In Alexandrite, this occurs across its laser wavelength band (laser ESA) [13] and throughout its pump bands (pump ESA) [14]. After ESA, the higher lying excited ions decay nonradiatively back to the upper laser level, releasing their energy in the form of heat.

In addition to the temperature effects, the optical transition cross sections vary significantly within the wavelength tuning range. The result is a complex system of parameters that requires a suitable model to understand the details of Alexandrite laser performance. Previous models for Alexandrite have been for flashlamp pumped systems [3,4], but the efficiency of energy transfer from the pump to gain medium was not calculated, leaving pump ESA unquantified. A theory has been developed for a four-level end-pumped laser with pump ESA [15], but this is only applicable to Alexandrite in the spectral lasing region where laser ESA and GSA are negligibly small (approximately 760–780 nm).

In this work, a new theoretical model of laser action is developed, which for the first time, to the best of our knowledge, includes the effects of laser ESA and GSA, along with pump ESA, in the case of the end-pumping geometry. Having closed-form solutions for the efficiencies and thresholds, with the addition of pump ESA to a quasi-three-level end-pumped system, makes this theory unique. It is valid for any end-pumped quasi-three-level laser with pump and laser ESA, and is therefore applicable throughout the Alexandrite output wavelength spectrum. With the inclusion of temperature-dependent cross sections, this is a complete laser model for the diode end-pumped Alexandrite system.

The efficiency of a diode end-pumped tunable Alexandrite laser was investigated, with the aim of comparison to the model to explain its behavior when changing various parameters, such as laser wavelength, crystal temperature, laser output coupling, and cavity loss.

## 2. DIODE-PUMPED, WAVELENGTH-TUNABLE ALEXANDRITE

A wavelength-tunable Alexandrite laser was constructed to experimentally investigate the behavior of tunable and temperature-dependent laser operation. There can be identified three regimes of operation in Alexandrite. These are a consequence of the wavelength dependences of the laser emission, GSA, and ESA cross sections, shown in Fig. 2 at a temperature of 28°C.

In the short wavelength regime, from 700 to 760 nm, both laser ESA and GSA are present. Laser emission and ESA both increase toward shorter wavelengths, but GSA becomes most dominant due to its exponential increase and determines the short wavelength lasing limit. In a central wavelength region, from 760 to 780 nm, the laser is approximately a four-level system with little laser GSA or ESA occurring. In the long wavelength region, greater than 780 nm, laser GSA is vanishingly small, and the system can be increasingly considered purely four-level, but with increasing laser ESA. This becomes the dominant factor and creates the long wavelength lasing limit when the ESA and emission cross sections are equal. It should be noted that the cross sections are all functions of crystal temperature, so the relative significances of the different processes and wavelength regimes are also temperature-dependent.

The experimental end-pumped Alexandrite laser system was built with the cavity and pump configuration shown in Fig. 3. The Alexandrite was a nominally 0.24 at. % Cr-doped rod, with length 4 mm and diameter 2 mm. The crystal faces were antireflection (AR)-coated for both the laser and pump wavelengths. It was mounted in a thermoelectric cooler (TEC)-controlled copper mount with a temperature range of 10°C–105°C, which allowed investigation of the temperature-dependent behavior of Alexandrite.

The pump source was a 5.7 W, 637 nm, fiber-delivered diode module that was linearly polarized with a cube polarizer and aligned parallel to the b-axis of the Alexandrite with a half-wave plate (HWP), giving an available pump power of 3.2 W. The pump was focused into the crystal with an ${f}_{p}=20\text{\hspace{0.17em}}\mathrm{mm}$ aspheric lens, resulting in a beam waist $1/{e}^{2}$ diameter at the crystal of 150 μm, Rayleigh length 0.6 mm, and ${M}^{2}=45$. Despite propagation through a multimode fiber, the transverse pump radiation distribution at its focus was near Gaussian.

The plane back mirror (BM) in the cavity was dichroic, having high reflectance for the laser and high transmission for the pump, through which the crystal was end-pumped. The output coupler (OC) was also planar, with a reflectance of $R=99.4\%$ initially used. Stability of the cavity was achieved with an intracavity lens of focal length $f=100\text{\hspace{0.17em}}\mathrm{mm}$, which had a super-V AR coating centered at 760 nm. For wavelength tuning, a 0.5 mm thick quartz birefringent filter (BiFi) mounted at Brewster’s angle was used, which had a free spectral range of approximately 100 nm.

The laser output power against input pump power is shown in Fig. 4 for three different laser wavelengths, corresponding to the three regimes of operation, at a crystal temperature of 60°C. The laser was most efficient operating at 760 nm with a peak output power of 785 mW, lowest threshold of 0.45 W, and maximum slope efficiency of 30%. The effect of tuning to 790 nm was a slightly increased threshold of 0.63 W and reduced slope efficiency of 22%. The least efficient operation was at 727 nm, with the highest threshold of 0.88 W and a significantly lower slope efficiency of 7%. The laser output had excellent spatial quality in all cases and with an ${M}^{2}=1.02$ at 760 nm; the beam profile is shown in Fig. 4. The output spectrum typically had a FWHM bandwidth of 2 GHz, which was narrower than prior work due to the presence of the BiFi [6].

The effect of laser wavelength tuning on the output power is shown in Fig. 5, at crystal temperatures of 10°C and 60°C. The optimum wavelengths of 765 nm and 770 nm at 10°C and 60°C, respectively, are typical for Alexandrite [1,7,16]. Elevated temperatures extended the upper tuning range, as has been previously reported [2], but in turn increased the lower bound by 7 nm. The higher temperatures resulted in improved output powers above 750 nm.

It is the complex interplay of effects in Alexandrite lasers, with the changing efficiencies across the laser output spectrum seen in the experimental data, that makes it important to understand the various factors at play to aid in optimization.

## 3. MEASURING PUMP ESA

Of crucial importance in quantifying the effect of pump ESA is knowing the ratio of the pump ESA, ${\sigma}_{1}$, to pump GSA, ${\sigma}_{0}$, cross sections: $\gamma ={\sigma}_{1}/{\sigma}_{0}$. This parameter has previously been measured for Alexandrite, but it is useful to verify the accuracy of the measurement, because it has only been reported once before [14].

The measurement can be made through pulsed pump-probe measurements by comparing the difference in transmission between the pumped and unpumped cases. Alternatively, the saturating transmission of a sample can be used to determine the pump ESA fraction [14,17].

The simplest way of using transmission saturation would be to compare the small signal transmission to that of a highly saturated sample as in [14], but due to the limited brightness and power of the available pump source, instead the rate of crystal transmission saturation with incident intensity was used.

The pump source was the same fiber-delivered diode module of the laser. Due to wavelength changes with the diode current, its power control was achieved passively via rotation of a $\lambda /2$ wave plate between two polarizers. The pump was then focused on the crystal with an $f=26\text{\hspace{0.17em}}\mathrm{mm}$ aspheric lens, resulting in a pump beam on the crystal of diameter 170 μm, Rayleigh length 0.84 mm, and ${M}^{2}=45$. The crystal was a 0.22 at. % Cr Alexandrite slab with a depth of 2.08 mm, mounted on a water-cooled copper heat sink held at a temperature of 20°C. The crystal faces were uncoated, so the Fresnel reflections were corrected for in the analysis.

The transmitted power was measured through a $635\pm 10\text{\hspace{0.17em}}\mathrm{nm}$ filter to minimize any fluorescence detection. The crystal transmission, $T$, against incident peak intensity, ${I}_{0p}$, normalized to the pump saturation intensity, ${I}_{s}$, is shown in Fig. 6. In order to use these data to extract the pump ESA parameter, a theoretical model of the saturable transmission of the pump beam was needed.

The transmission, ${T}_{p}$, of a plane wave pump beam of intensity ${I}_{0}$ through a saturable absorber with pump ESA is given by [17]

The intensity as a function of radius, ${I}_{0}(r)$, for an incident Gaussian beam is

where ${I}_{0p}$ is the peak intensity and $w$ is the $(1/{e}^{2})$ beam radius. To find the transmission of a Gaussian beam, ${T}_{G}$, the ratio of output to input power is calculated asTo find the transmission of a Gaussian beam, it is assumed that it is nondivergent; the integral of ${T}_{G}$ is then found. This must be calculated numerically due to the transcendental nature of Eq. (3).

The theoretical transmissions were calculated using Eqs. (3) and (6), with ${\tau}_{f}=281\text{\hspace{0.17em}}\mathrm{\mu s}$ and ${\sigma}_{0}=7.11\times {10}^{-24}\text{\hspace{0.17em}}{\mathrm{m}}^{2}$ at 20°C [1,14]. The parameters were chosen to give the best fit to the experimental data, with the resulting theoretical plots shown alongside the measurements in Fig. 6. The small signal transmission was ${T}_{0}=0.357\pm 0.003$, which gave a predicted active ion density of $N=6.96\times {10}^{25}\text{\hspace{0.17em}}{\mathrm{m}}^{-3}$. The modeling shows a close fit to the data, with the result $\gamma =0.76\pm 0.02$. This value is consistent with the previously reported measurement of $\gamma =0.75\pm 0.01$ [14].

## 4. THEORETICAL LASER MODEL

The energy level model considered in this analysis is shown in Fig. 7. The pump excites population from level 0 to 2 with cross section ${\sigma}_{0}$, which then decays nonradiatively to level 1 with a lifetime of ${\tau}_{21}$. Gain at the laser wavelength is by stimulated emission from level 1 to 0 with cross section ${\sigma}_{e}$, and GSA occurs with cross section ${\sigma}_{a}$ from level 0 to 1. There is also ESA from level 1 to 3 at both the pump and laser wavelengths with cross sections ${\sigma}_{1}$ and ${\sigma}_{1a}$, respectively. This excited population then decays nonradiatively down to level 1 with a lifetime ${\tau}_{31}$. There is also fluorescent decay from level 1 to 0 with a lifetime ${\tau}_{f}$.

Different gain media can be modeled by the energy level system in Fig. 7, either by directly corresponding to the system, or through appropriately combining energy levels to match those of the model. For example, in Alexandrite, the population in the upper ${}^{4}{\mathrm{T}}_{2}$ and ${}^{2}\mathrm{E}$ states can be combined and treated as the single population of ${n}_{1}$. The thermal and spectral dynamics are implemented by using effective parameters for the fluorescence lifetime and cross sections.

Different kinds of quasi-three-level systems can be modeled by altering the definition of ${\sigma}_{a}$. For example, when laser gain is from a vibronic transition and thermalization is sufficiently fast, as in Alexandrite, the GSA cross section is given by the extended McCumber theory of Eq. (2). Alternatively, if the terminal laser level is well-defined (as in Yb:YAG), the GSA cross section can be a simple fraction of the emission cross section [18], given by the Boltzmann filling of the level.

The laser rod considered in this analysis is end-pumped from the rod end position $z=0$, resulting in a longitudinal variation in the pump radiation intensity $I(z)$, and active ion excitation density ${n}_{1}(z)$. The pump and laser radiation are taken to be collinear, monochromatic, nondivergent, and have constant transverse intensity profiles. The nonradiative transition rates are assumed to be fast, such that $({\tau}_{21},{\tau}_{31})\to 0$; therefore, ${n}_{2}={n}_{3}=0$ and ${n}_{0}+{n}_{1}=N$. Under these conditions, the equations governing the system are

The equations are linked to a specific laser cavity through the laser round-trip threshold gain condition of

where $R$ is the OC reflectivity, $L$ is the round-trip cavity loss, and ${G}_{\mathrm{th}}$ is the single pass gain at threshold.Integrating Eq. (8) along the length of the medium, $l$, gives the single-pass gain

Solving Eq. (7) in the steady state gives the solution

where ${I}_{s}=h{\nu}_{p}/({\sigma}_{0}{\tau}_{f})$ is the pump saturation intensity, ${\phi}_{a}=1/(c{\sigma}_{a}{\tau}_{f})$ is the laser GSA saturation photon density, and ${\phi}_{s}=1/(c{\sigma}_{e}{\tau}_{f})$ is the laser emission saturation photon density.To complete the solution, Eqs. (9) and (13) are combined to find $F$ under the assumption of constant laser cavity photon density, $\phi (z)=\phi $, which is accurate for all but high-gain cavities [19,20]. The derivation is given in Appendix A, which yields the relation between the input pump intensity $I(0)$ and intracavity laser photon density $\phi $, given by

The laser threshold pump intensity ${I}_{0\mathrm{th}}$ is obtained from Eq. (14) by setting $\phi =a=0$ at a threshold $F$, giving

To understand the laser efficiency, the two mechanisms of pump absorption, GSA and ESA, must be evaluated. The power absorbed through GSA constitutes useful absorption, as it generates inversion, whereas power absorbed as ESA is a loss, because it is absorption from already excited population. The total power absorbed ${P}_{a}$ is found using Eq. (9), giving

where ${P}_{0}=AI(0)$ is the pump input power. The useful pump power absorbed, ${P}_{u}$, is found by summing GSA pump absorption in the medium asTo quantify the overall laser efficiency, the fraction of pump power above threshold that is converted to laser power, the slope efficiency ${\eta}_{s}$ is commonly used. Equation (14) shows that the laser photon density $\phi $, and hence the laser output power, is a nonlinear function of input pump intensity ${I}_{0}=I(0)$; therefore, the slope efficiency is not a constant. However, it is asymptotically constant for $\phi \gg {\phi}_{s}$, where $a\to {\sigma}_{a}/{\sigma}_{e}$ and can be assumed to be constant. The slope efficiency is given by ${\eta}_{s}=\frac{1}{2}h{\nu}_{l}c[-\mathrm{ln}(R)]\frac{\mathrm{d}\phi}{d{I}_{0}}$, which yields

The slope efficiency has now assumed a standard form involving the output coupling, Stokes, and pump absorption efficiencies (including the pump quantum efficiency). Another term in experimental systems is the mode overlap efficiency, the spatial matching of the laser and pump modes. This is not considered in the current analysis, but will be discussed in a later section when comparison is made to experimental data.

The Stokes efficiency can normally be used as the intrinsic maximum slope efficiency, ${\eta}_{0}$, of a laser gain medium. This efficiency would occur with perfect spatial overlap and a high output coupling fraction, such that passive cavity losses are negligible. However, in the presented model, there are reductions to the slope efficiency that are not affected by cavity design in both the output coupling efficiency, ${\eta}_{\mathrm{oc}}$, and pump quantum efficiency, ${\eta}_{p}$, terms. These result in a modified intrinsic efficiency of

Of particular interest in high-power laser systems is the amount of heat generated in the medium. This has varying consequences, in particular, thermally induced lensing. Heating is from the conversion of input pump energy to heat. This is always partly from the quantum defect fraction of the Stokes efficiency, but can also be due to the nonradiative decay from ESA that converts photons into heat.

The heating fraction ${\eta}_{H}$ is the fraction of absorbed pump power that is converted into heat. A useful approximation is that the laser GSA is small compared to the output coupling and cavity losses, and the laser is operating far above threshold, giving

where the Stokes heating factor is easily identified as ${\lambda}_{p}/{\lambda}_{l}$, pump ESA heating is the ${\eta}_{p}$ factor, and laser ESA heating contributes through ${\gamma}_{l}$. The derivation is given in Appendix B.#### A. Analysis of Model Solutions

The previous section gives the results of the model in its full forms. These are useful for precise computations, but to understand the effect of different variables on the laser parameters, approximate forms can be more revealing. The first approximation that can be made is that the input pump power is totally absorbed, $T\to 0$. This is a good approximation in most lasers, as a nonabsorbed pump can reduce overall efficiency. The second is that ${\alpha}_{1}F\ll 1$, which is the case of low inversion. Under these conditions, the pump threshold and quantum efficiency are

A plot of pump quantum efficiency against output laser power is shown in Fig. 8 for different ratios of ${\sigma}_{a}/{\sigma}_{e}$, from four-level (${\sigma}_{a}/{\sigma}_{e}=0$) to three-level (${\sigma}_{a}/{\sigma}_{e}=1$) lasing, but with fixed laser and pump ESA. The pump quantum efficiency is at a minimum at threshold, its value approximated by Eq. (27) when $a=0$. It asymptotes to a maximum efficiency as laser power increases, and $a\to {\sigma}_{a}/{\sigma}_{e}$.

The cause of the increasing pump quantum efficiency is clarified in Fig. 9, which plots the population inversion distribution at different laser output powers. As laser output power increases, laser GSA causes the inversion of less pumped regions to increase. However, to satisfy the laser round-trip gain condition, there must be a fixed integrated inversion $F$, so the inversion at the pump input is correspondingly decreased. A lower population inversion over the main pump absorption region results in a smaller pump ESA loss and an increase in the pump quantum efficiency.

The increasing efficiency with laser power causes interesting effects when output laser power is compared to input pump power, as in Fig. 10. For low GSA ratios (${\sigma}_{a}/{\sigma}_{e}$), the power curves approximate to the linear form previously shown for end-pumped quasi-three-level lasers. However, at higher GSA strengths, the rate of change in pump quantum efficiency is large enough that multiple steady-state solutions for the output laser power occur, giving bistable operation. When $T=0$, the condition for bistability is

which is found through differentiation of Eq. (14) and is an equality for a cavity on the edge of bistability.For bistable operation to occur, the change in population distribution must be large enough to have a sufficiently large effect on the pump quantum efficiency. This can be achieved, for example, through making the change in population distribution larger by increasing the strength of GSA, ${\sigma}_{a}/{\sigma}_{e}$, or having larger areas of unpumped gain medium by increasing its length, $l$. Alternatively, the pump quantum efficiency change can be enhanced through a larger $\gamma $. Increasing these parameters gives power curves identical to those seen in Fig. 10.

Bistable power operation has been observed in CW lasers, including a saturable absorber or nonlinear dispersive medium [22], but has only been seen intrinsic to two classes of solid-state laser. It has been shown in the Tm, Ho [23], and Yb [24] classes of lasers, where the medium acts as its own saturable absorber. However, to the best of our knowledge, this is the first identification of bistable behavior in a solid-state system due to a changing efficiency of pump absorption in the medium, and should occur in any medium that has both pump ESA and laser GSA.

## 5. COMPARISON OF MODEL TO ALEXANDRITE SYSTEM

A comparison was made between the analytical model predictions and experimental measurements from an Alexandrite laser system. Equations (15) and (22) were used to find the thresholds and slope efficiencies, respectively, using the parameters in Table 1. The corresponding experimental results for threshold and slope efficiency were taken from the system described in Section 2.

The results are shown in Fig. 11, with experimental measurements in Figs. 11(a) and 11(c), and modeling in Figs. 11(b) and 11(d), at crystal temperatures of 10°C and 60°C, respectively. The effect of output coupling was investigated at three laser wavelengths corresponding to the three regimes of lasing in Alexandrite, those distinguished in Section 2. The theoretical predictions were computed with (solid lines) and without (dashed lines) pump ESA, to clarify the effect of pump ESA on the laser behavior.

The experimental thresholds were accurately computed by the model across the parameter range. For example, at 727 nm and 60°C, Figs. 11(c) and 11(d), the thresholds were correctly modeled as being higher than those of the other wavelengths at high OC reflectance, but becoming lower than at 790 nm for reduced reflectances.

The theoretical slope efficiency curves exhibited the same trends as the experimental system. For example, it predicted an intersection between the 730 and 790 nm slope efficiency curves at 10°C, Fig. 11(b). This is apparent in the experimental data, Fig. 11(a), where the efficiency is initially higher at 790 nm, but then rapidly decreases. The magnitudes of the theoretical slope efficiencies are in general higher than those of the experimental system. The modeling did not account for mismatch between the spatial profiles of the laser and pump beams, which is the most likely source of the majority of the discrepancy and would have caused a reduction in efficiency if included.

The modeling predicts that pump ESA causes an approximately quadratic increase in threshold with OC reflectance, Eq. (26). This is seen throughout the modeling by comparing the thresholds with and without pump ESA in Figs. 11(b) and 11(d) by the solid and dashed lines, respectively. The quadratic relationship is also seen in the experimental data, being particularly clear at 760 nm at 10°C, Fig. 11(a). The nonlinear thresholds in the experiment can therefore be confidently attributed to pump ESA effects.

The experimental slope efficiencies had an optimum OC reflectance for maximum slope efficiency, for example at 99% OC reflectance, 760 nm, and 10°C in Fig. 11(a). An optimum reflectance is apparent in the modeling, but only with the inclusion of pump ESA. This is due to a reduction in pump quantum efficiency for decreasing OC reflectance, Eq. (27), with an approximate linear form. The optimum slope efficiency in Alexandrite is therefore highly likely to be from pump ESA effects and has been observed in previous work [6,7].

Pump ESA gives a second-order correction to the laser threshold for OC reflectance, whereas for the slope efficiency, it is first-order in reflectance. It is therefore clearer to determine the accuracy of the pump quantum efficiency expression through comparison of the slope efficiencies, rather than the thresholds. In the experimental measurements, the slope efficiency decreased more quickly by lowering the OC reflectance than in the model, being particularly clear at 760 nm and 60°C, Figs. 11(c) and 11(d). This discrepancy can be explained by a lower pump quantum efficiency in the experimental system compared to the model.

The pump quantum efficiency is reduced when inversion is concentrated in one area due to the nonlinear nature of the loss. There are two factors that are most likely to have caused an overestimation of the pump quantum efficiency in the model when compared to experiment. The first is the approximation of constant transverse intensity profiles. The Gaussian profiles in the experimental system had a higher intensity in the center than the wings and would concentrate the inversion there. The second is the assumption of a nondivergent pump beam. In the experimental laser, the 4 mm crystal length was longer than the confocal parameter of the pump beam, so the intensity varied longitudinally from divergence in addition to the absorption, and would have concentrated the inversion toward the beam waist.

Although the pump quantum efficiency of the model appears to be not accurately matched in magnitude, it exhibits the same behaviors with changing laser parameters as the experimental system. The relations found from this analytical model, such as Eqs. (26) and (27), give insight into how the various parameters affect laser efficiency and can aid in laser cavity design, with only the sacrifice of accuracy. Additionally, due to the nonlinear nature of pump ESA loss, for a model including the transverse spatial variation, analytical solutions would not be possible, making numerical solutions necessary. Any insight into the effects of different laser parameters would need to be inferred through numerical calculation.

## 6. CONCLUSION

In this work, the efficiency characteristics of a wavelength-tunable diode end-pumped Alexandrite laser were investigated. Alexandrite exhibits laser GSA and ESA, alongside pump ESA, with the relative significances dependent on many factors, such as laser wavelength and crystal temperature. To have a complete model of the Alexandrite system, an analytical model was newly formulated that yielded expressions for output power, laser threshold, and slope efficiency. The model is applicable to Alexandrite across its wavelength tuning range and can be applied to other systems with pump and laser ESA by using the appropriate parameters.

The constituent factors of the slope efficiency were identified, giving a modified output coupling efficiency that includes the effect of laser ESA, and a pump quantum efficiency factor that gives the fraction of ESA in the absorbed pump energy. A relation for the intrinsic slope efficiency was found. The heating fraction of absorbed pump power was also formulated to include the effects of pump and laser ESA, processes that deposit absorbed energy as heat.

The ratio of pump ESA to GSA cross section was measured for the Alexandrite crystal sample, giving a ratio of $\gamma =0.76\pm 0.02$, which was consistent with previous work [14].

The results of the analytical model were compared to experimental measurements of slope efficiency and threshold of an Alexandrite laser when changing the crystal temperature, laser wavelength, and OC reflectance. The model was able to successfully replicate and explain the trends of the experimental system. Pump ESA was found to affect both the threshold and slope efficiency by increasing the threshold and lowering the slope efficiency when using lower reflectance output coupling.

It was found that the pump quantum efficiency was potentially being overestimated by the model. This would likely be due to simplifications made in the spatial profile of the pump and laser beams. Despite this, the model was able to match the behavior of the experimental system, and so is a useful tool in prediction of the behavior of an Alexandrite system, giving insight into how to optimize the system.

## APPENDIX A: DERIVATION OF LASER MODEL

To find an equation governing the population inversion distribution, Eq. (13) is used to eliminate $I$ in Eq. (9), giving

where $a=(\phi /{\phi}_{a})/(1+\phi /{\phi}_{s})$ is a laser saturation factor. This can be integrated, yieldingTo find the relationship between the incident pump and laser output intensity, the integral relation for $F$ is calculated through a change of variables using Eq. (A1), giving

where ${f}_{0}=f(0)$. Performing the integral and using Eq. (13) for simplification yields## APPENDIX B: HEATING FRACTION

It has been previously shown that with just pump ESA (${\sigma}_{1a}=0$), the heating fraction is ${\eta}_{H}=(1-{\eta}_{q}{\eta}_{p})$ [15], which continues to be valid when laser GSA is included by using Eq. (20) for ${\eta}_{p}$. To include the effect of laser ESA heating, the expression for ${\eta}_{H}$ must be altered.

The laser ESA power is obtained from ${P}_{\mathrm{l},\mathrm{ESA}}=\frac{1}{2}\phi [2{\alpha}_{1a}F]h{\nu}_{l}cA$, which is the same as that for the laser output power expression, except that the output coupling $-\mathrm{ln}(R)$ is replaced by $2{\alpha}_{1a}F$. This leads to an output coupling fraction of laser ESA, ${\eta}_{\mathrm{l},\mathrm{ESA}}$, of

## Funding

Imperial College London (ICL).

## Acknowledgment

One of the authors (WK-J) acknowledges support from an Imperial College President’s Ph.D. Scholarship.

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