1. INTRODUCTION

Ever since the invention of the laser, designers have pursued new ways to increase its power. Over several decades of research and development, ultrashort pulsed operation has pushed peak laser powers to the petawatt level. However, increasing average laser power has proven more challenging. After extensive development efforts, even the brightest cw lasers produce roughly 10 kW with good beam quality. The principal obstacle to high average power laser operation is the management of the waste heat generated within the gain medium.

Thermal management places strong restrictions on the design of a high average power laser. Gain medium must be selected for their thermo-optic as well as radiative properties. Heat removal forces laser mode volumes to be kept very small with extreme aspect ratios. For example, the mode volume of a typical kilowatt fiber laser is only $0.003{\mathrm{cm}}^3$ and has an aspect ratio over ${10}^6$ to 1. Even with such extreme mode geometries, thermally induced instability is still the principal limitation to power scaling [1–3]. Reduction of the thermal loading density allows for larger mode volumes. Larger mode volumes allow for reduced intensities and parasitic nonlinearities. Having more active ions also reduces pump saturation effects allowing for energy storage and high peak power operation.

In previous work, we have explored the theory and practice of incorporation of optical cooling to reduce or eliminate laser heating. This paper takes a theoretical approach to the wider problem of optimizing laser performance. The goal of this paper is to explore the fundamental limitations of low quantum defect lasers. The approach is to simplify the analysis by considering only effects inside the laser medium. By tracking the local power density in the laser medium, insights can be obtained about radiative efficiency and waste heat generation. This analysis applies to the most common type of high average power lasers: the steady-state quasi-three-level systems. After developing the basic definitions and relations in a lossless idealized material, the analysis will be extended to include the most common loss processes; quenching, background absorption, and fluorescence reabsorption. Several possible optimization approaches will be considered and compared. The well-characterized laser material Yb:YAG will be used to illustrate these relations and optimization strategies.

2. POWER FLOW IN AN IDEAL LASER MATERIAL

High average power lasers are typically based on transitions between the lowest energy and the first excited manifold of a dopant ion such as trivalent ytterbium. These lasers are often referred to as quasi-three-level systems, but they consist of only two well separated energy bands made up of several nondegenerate levels, which rapidly thermal equilibrate with the host matrix. These laser systems avoid detrimental energy pooling and allow for low quantum defect laser schemes. We begin our analysis by considering an ideal laser system with only radiative losses and no radiative trapping. Here, the power flow of the fluorescent field will be explicitly included. After deriving some basic results, the model will be extended to handle more realistic cases.

We start by defining the total active ion density of the laser medium as $N_T$. Optical transitions are presumed to occur between lower and upper manifolds giving rise to overlapping, thermally broadened, absorption and emission spectra. These spectra are characterized by effective absorption and emission cross sections labeled ${\sigma }_A(\lambda ,T)$ and ${\sigma }_E(\lambda ,T)$ at a wavelength $\lambda$ and a temperature $T$. It will be useful to define a fractional cross section as

Ion densities in the lowest energy manifold, $N_1$, are excited to the upper energy manifold, $N_2$, by a pump field at wavelength, ${\lambda }_P$, and intensity, $I_P$. Ions in the upper manifold radiate with a lifetime ${\tau }_R$ or are stimulated to emit decay by a laser field at wavelength, ${\lambda }_L$, and intensity, $I_L$. With these definitions and mass conservation, we can write the local rate equation as

The first term on the right describes the pump rate, the second term describes the stimulated emission rate, and the last term describes the total spontaneous decay rate of the upper manifold.

Applying energy conservation to this laser system, we recognize that the difference between the absorbed power and the emitted power must be the local heat generation rate:

For simplicity, we have used the shorthand notations:

Note that the heat generation depends on the mean fluorescence wavelength, ${\lambda }_F$, defined as

It will be convenient to work with scaled excitation densities and intensities defined as

It is apparent from Eq. (2) that the saturation intensities should be defined as $I_{P\text{sat}}=hc{\beta }_P/{\lambda }_P{\tau }_R{\sigma }_P$ and $I_{L\text{sat}}=hc{\beta }_L/{\lambda }_L{\tau }_R{\sigma }_L$. It will also be convenient to define the quantity ${\beta }_{PL}$ as

The above quantities can be evaluated using the well-known approximations of Fuchtbauer–Ladenburg and the McCumber reciprocity relations. With these approximations, $\beta (\lambda ,T)$ and ${\lambda }_F$ can be computed from the energy level structure and effective absorption cross sections [4–6]:

Here, $Z_1(T)$ and $Z_2(T)$ are the Boltzmann thermal partition functions of the lower and upper manifolds and ${\epsilon }_2$ is the minimum energy of the upper manifold. The quantity ${\epsilon }_2$ is often referred to as the zero phonon energy.

With the above definitions, we can write the steady-state excitation density that satisfies Eq. (2) simply as

The steady-state pump absorption and laser gain coefficients are, simply,

The above relations make clear the physical significance of the fractional cross sections, $\beta (\lambda ,T)$ and ${\beta }_{PL}/{\beta }_P$. From Eq. (2), we can see that $\beta (\lambda ,T)$ is simply the excitation fraction of the active ion at which absorption at wavelength $\lambda$ changes over to gain. From Eq. (10), it is clear that ${\beta }_{PL}/{\beta }_P$ is equal to the scaled pump intensity needed to reach this threshold excitation fraction.

We will illustrate the above relations using the radiative properties of Yb:YAG. While Yb:YAG is usually lased at its emission peak at 1030 nm, here we have chosen ${\lambda }_L=1050\mathrm{nm}$, which corresponds to the lowest energy transition from the ${}^2{\mathrm{F}}_{5/2}$ level of Yb:YAG. The mean fluorescence wavelength and radiative lifetime of Yb:YAG are reported to be 1010 and 0.979 ms at room temperature [7,8]. The peak absorption cross section of the 1050 nm line is $6.0\mathrm{E}-23{\mathrm{cm}}^2$, leading to a saturation intensity of $70\mathrm{kW}/{\mathrm{cm}}^2$.

Figure 1 illustrates how the $\beta$’s varies with the choice of operating conditions in the Yb:YAG system. While the $\beta$’s are bounded between 0 and 1, they vary by several orders of magnitude for reasonable choices of operating wavelengths and temperature. Saturation intensities and $\beta (\lambda ,T)$ fall rapidly with wavelength and temperature. Reduced threshold excitation densities and lower saturation intensities are prime advantages of low temperature operation. Note that all $\beta (\lambda ,T)$ cross at 968 nm, the zero-phonon transition energy in Yb:YAG.

 

Fig. 1. Plot of $\beta ({\lambda }_P,T)$ (dotted lines) and ${\beta }_{PL}/{\beta }_P$ (solid lines) for the Yb:YAG system. Values are calculated as discussed in the text for several temperatures assuming ${\lambda }_L=1050\mathrm{nm}$; 300 K (red); 200 K (green); 100 K (blue).

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The right side of Fig. 1 illustrates how the threshold pump intensity, ${\beta }_{PL}/{\beta }_P$, depends on the wavelength. As the pump wavelength blue shifts, ${\beta }_{PL}/{\beta }_P$ approaches ${\beta }_L$. At room temperature, the minimum pump threshold in Yb:YAG is 0.023 of the scaled intensity, but at 100 K the minimum threshold has fallen to $8.9\mathrm{E}-6$. As the pump wavelength is redshifted, thresholds grow monotonically and then diverge to infinity as the pump wavelength approaches the laser wavelength.

To complete the steady-state solution, solve for the powers flowing through the optical medium. The optical power densities cycling through the laser medium in the pump and laser fields are

The steady-state power converts directly into heat through the laser’s effective quantum defect, which is found in Eq. (3) to be

Unlike the radiative power densities, heat builds up in the lasing medium, leading to temperature gradients and detrimental thermo-optic distortions. For a solid-state laser, temperature gradients must build until equilibrium is established. In a real laser system, additional power and hardware will be required to remove heat from the system. It is therefore reasonable to reduce heating while maintaining laser performance. The next section will examine laser optimization strategies using this simple model.

3. OPTIMIZING POWER FROM A STEADY-STATE LASER MATERIAL

The steady-state power densities derived in the previous section allow us to find the optimal performance conditions of the ideal laser medium. For this purpose, we define a laser material efficiency as the emitted laser power divided by the absorbed pump power:

We must distinguish ${\eta }_m$ from the more commonly used laser optical efficiency. The laser optical efficiency will be proportional to ${\eta }_m$ but will also depend on beam propagation through an optical system.

In order to compare the magnitude of the laser internal heating, we will define a laser material heating parameter:

Recognizing that the heat generation can be positive or negative, we define ${\mathrm{\Xi }}_m$ in terms of the absolute magnitude. Note that ${\mathrm{\Xi }}_m$ is defined in terms of laser power and not absorbed pump power. The heating parameter more commonly used in the literature would be the product of ${\mathrm{\Xi }}_m$ and ${\eta }_m$. The definitions used here will lead more directly to an optimization procedure.

Equations (13) and (14) describe the performance of a purely radiative cw laser. Laser power can theoretically be optimized by either maximizing ${\eta }_m$ or minimizing ${\mathrm{\Xi }}_m$. This would be accomplished through the appropriate selection of the operating intensities and wavelengths. Note that the heating fraction depends strongly on wavelength separations, while the laser efficiency is independent of the fluorescence wavelength.

Consider first the approach of improving laser performance by maximizing the laser material efficiency. Equation (13) reveals that whenever the laser material is pumped far above inversion, the material efficiency will be approach ${\lambda }_P/{\lambda }_L$. Lower intensities reduce the material efficiency and a longer pump wavelength generally increase it. However, we can see in Fig. 1 that, as ${\lambda }_P$ approaches ${\lambda }_L$, ${\beta }_{PL}/{\beta }_P$ increases rapidly, thus raising the laser’s threshold and decreasing efficiency.

This optimization approach is illustrated in Fig. 2 for the idealized Yb:YAG lasing at 1050 nm. As expected, laser material efficiency initially rises linearly with pump wavelength and then falls to zero as ${\lambda }_P$ approaches ${\lambda }_L$. Good efficiencies require that intensities exceed the saturation value. At the highest intensities considered, the room temperature material efficiency reaches 88% when the pump wavelength is 1005 nm. Cooling is seen to improve material efficiency somewhat at longer wavelengths. For the highest intensity plotted, the 100 K material efficiency is predicted to reach a maximum of 91.5% when pumped at 1028 nm.

 

Fig. 2. Plot of the material efficiency for an ideal Yb:YAG laser with ${\lambda }_L=1050\mathrm{nm}$ and several values of the scaled intensity. For the purpose of this illustration, the scaled pump and laser intensities are set equal. Solid curves are for 300 K and the dashed curves are for 100 K.

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An alternative approach to improving laser performance would be to minimize the heating parameter ${\mathrm{\Xi }}_m$. This approach will generate the least amount of waste heat per Watt of laser output power. Figure 3 shows ${\mathrm{\Xi }}_m$ calculated for the room temperature conditions that were plotted in Fig. 2. Figure 3 illustrates the large variation in the heating parameter, which is in contrast to the relatively flat laser efficiency. There is a null in the heat generation, which shifts to longer pump wavelengths as the intensities rise. As the pump wavelength is tuned through this point, heat generation switches over to optical cooling. For longer pump wavelengths, ${\mathrm{\Xi }}_m$ rises rapidly, thus exceeding unity just above laser threshold.

 

Fig. 3. Plot of laser material heating parameter for idealized Yb:YAG. Lasing is at 1050 nm and the temperature is 300 K. The vertical line marks the position of the mean fluorescence wavelength. As before, the scaled pump and laser intensities are set equal.

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The null in the heating efficiency is referred to as radiation balanced lasing. At this operating point, fluorescence cooling precisely offsets the laser quantum defect heating [9,10]. The RBL resonances shown in Fig. 3 have 5 to 10 nm spectral width and are redshifted by roughly one linewidth as the intensities are doubled. Thus, algebra will show that, at the RBL point, Eq. (11) simplifies to

Thus, as expect, when the pump wavelength approaches ${\lambda }_L$, the RBL laser efficiency approaches unity but only as the pump intensity becomes extremely large.

Comparing Figs. 2 and 3 reveals that optimums for the two approaches do not occur at the same pump wavelength. The RBL points occur at longer wavelengths for the same intensity. Taking the $i_P=i_L=2$ case, for example, an ideal 1050 nm Yb:YAG laser pumped at 970 nm is predicted to have ${\eta }_m=0.593$ and ${\mathrm{\Xi }}_m=0.102$. The same laser pumped at 1026 nm, would see ${\eta }_m=0.402$ with no heat generation. At the highest intensities plotted, the maximum possible efficiency is 0.882, while the RBL efficiency is 0.745. Since thermal loading requires active cooling to maintain temperature and beam quality, these systems’ overall efficiencies are likely to be comparable.

A third approach for improving laser performance is motivated by the special case of no lasing. When laser intensities are small, the laser material heating efficiency simplifies to become proportional to ${\lambda }_P-{\lambda }_F$. Thus, if the laser medium is pumped at its mean fluorescent wavelength, there will be no heat generated by the pump at any intensity. This approach might be indicated when the stored energy is not fully extracted from the laser medium, as often occurs in bulk laser geometries or during pulsed mode operation. For a laser operating with the condition of ${\lambda }_P={\lambda }_F$, the fractional laser heating simplifies to the laser’s quantum defect,

4. POWER FLOW IN A NONIDEAL LASER MATERIAL

A more accurate model of laser power flow must include realistic material losses. In the previous section, the laser material was assumed to be perfectly radiative, linear, and optically thin. Now we consider what happens when those assumptions are only mostly correct.

Let us begin by including the possibility of weak quenching of the laser excitation. This could occur through electron–phonon deactivation or through an ion-to-ion energy transfer mechanism [11]. To the first-order, these quenching processes introduce another linear loss term into Eq. (1). This linear loss term is normally incorporated by replacing the radiative lifetime, ${\tau }_R$, with a measured fluorescence lifetime, ${\tau }_F$. Nonexponential decay routes, such as ionic energy transfer quenching, are commonly handled by adding a quadratic loss term with a characteristic quenching lifetime, ${\tau }_Q$. For simplicity, we will assume that the quenching processes are completely nonradiative; quenched excitations are converted completely into heat.

The second loss process to be added to the model is background absorption. Background absorption is present at low levels in even the best optical materials [12]. Optical cooling studies have shown that these weak losses become important for low quantum defect pumping [13]. The background absorption coefficient, ${\alpha }_B$, will be assumed to be due to some trace impurities, so the excitation model is not directly modified by this process. Here, we choose to scale the background absorptions by the ionic absorption at the same wavelength: $\delta {\alpha }_P={\alpha }_B/{\sigma }_P{N}_T$ and $\delta {\alpha }_L={\alpha }_B/{\sigma }_L{N}_T$. As we will see, even small background absorptions can significantly reduce efficiency and increases heating.

The last process needed for a more realistic laser model is reabsorption of the fluorescence inside the laser medium. Often referred to as radiation trapping, this process will occur whenever absorption or reflection prevents some of the spontaneous emission from escaping the laser medium. We will assume that the reabsorption is due solely to the active ions. Thus, while trapping can enhance other nonradiative losses by increasing excitation densities, here it is a purely radiative phenomenon.

The addition of radiative trapping significantly complicates the modeling of the laser materials. It introduces nonlocal source terms for the fluorescent field, which requires intense numerical calculation for the solution. For this analysis, we will invoke an approximate solution developed in an earlier paper [14]. This approximate solution assumes that the laser medium traps light uniformly throughout and that the amount of trapping can be described simply by a cavity photon lifetime. This approximation allows the fluorescent field to be removed from the rate equation. The rate equation then simplifies to a quadratic in terms of the excitation density with modified values for fluorescence wavelength, ${\lambda }_F^{\prime }$, and lifetimes, ${\tau }_F^{\prime }$ and ${\tau }_Q^{\prime }$. These are the effective values that would be measured external to the radiatively trapped laser medium.

It should be noted here that it is not usually possible to measure the radiative lifetime, ${\tau }_R$, directly. In the literature, it is generally calculated from the absorption or emission spectra. When quenching is weak, a more accurate measurement of ${\tau }_R$ is possible by measuring nonradiative losses directly. To obtain the radiative lifetime of Yb:YAG quoted above from [8], we used a low power tunable laser to perform photothermal deflection on bulk samples. This technique allows an accurate measurement of the pump wavelength at which optical heating crosses over to optical cooling, ${\lambda }_C$. The radiative lifetime can then be computed by correcting the measured fluorescence lifetime for nonradiative losses using the relation, ${\tau }_R={\tau }_F{\lambda }_C/{\lambda }_F$. Best fluorescence lifetime measurements are acquired using thin powdered samples immersed in an index-matching medium to avoid fluorescence trapping.

Incorporating these loss processes gives a revised rate equation and allows us to solve for the power densities in a realistic laser material. We will distinguish its solution from that of the ideal material by adding a prime. The revised rate equation now becomes

With the assumption of weak losses, the steady-state solution can be written in terms of the ideal solution, $n_2$, given by Eq. (9):

This expression makes it is clear that the correction to the steady-state excitation densities depends principally on the change in the fluorescence lifetime. A decrease in the lifetime from linear quenching will generate a proportional decrease in the excitation density. Conversely, an increase in ${\tau }_F^{\prime }$ due to radiative trapping will increase excitation densities. In a well-designed high-power laser, quenching has been minimized, but the laser medium is optically thick. This suggests that the impact of radiative trapping would be larger than quenching, leading to longer fluorescence lifetimes and higher excitation densities than the “ideal” system considered in the last section. Note that, since $n_2\ll 1$, the quadratic quenching term will have little effect on the density unless ${\tau }_Q^{\prime }\approx {\tau }_R$.

Using the same fractional definition as before, the laser material efficiency is modified by the revised excitation fraction and by the additional absorption:

In the approximation of weak losses, Eq. (19) becomes

Expressed in this way, it is clear that the laser material efficiency increases with excitation density and decreases with background absorption. The contribution from background absorption, however, is reduced by $\beta$. The impact of these losses on efficiency increases substantially when the material is operated near laser threshold or pump saturation.

To illustrate the impact of including material losses, we again use Yb:YAG as the example. This time we estimated losses to the material parameters. In our previously reported experiments on Yb:YAG, the lasers were designed to reduce radiative trapping. Nevertheless, it was found that trapping increased the fluorescence lifetime to 1.25 ms, and the mean fluorescence wavelength shifted to 1017 nm. These values were measured using a $2\mathrm{mm}\times 2\mathrm{mm}\times 120\mathrm{mm}$ slab of 2%Yb:YAG, which was fusion bonded to a large plate of high purity sapphire [15]. We assume weak quadratic quenching by setting ${\tau }_Q^{\prime }$ to 20 ms and weak background absorption with $\delta {\alpha }_L=\delta {\alpha }_P={10}^{-3}$. These values are consistent with values observed in samples of high purity 2%Yb:YAG [12,14]. Figure 4 shows the result of evaluation of Eq. (19) for the same conditions used in Fig. 2. With the more realistic material parameters given above, the efficiencies are higher than for the ideal case. This is due to the increased fluorescence lifetime produced by the radiative trapping. The impact is larger at lower intensities where efficiencies increase by approximately 10%. At the highest intensity, room temperature optimal efficiency rises only slightly to 0.895 at 1008 nm. For operation at 100 K, the optimal efficiency is 0.927 at 1029 nm. Still, the efficiency curves are relatively flat, thus allowing wide latitude in the selection of the pump wavelength.

 

Fig. 4. Plot of the predicted laser material efficiency for the same conditions used in Fig. 2. Material parameters have been modified as described in the text.

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The form of Eq. (19) also suggests a strategy to maintain efficiency in the face of higher background absorption. To a limited extent, background absorptive losses will be offset by radiative trapping. Fluorescent trapping significantly increases storage lifetime, excitation density, and laser material efficiency.

In a similar way, we can revise the laser heating parameter, which incorporates the weak loss processes in the laser material. Heat generation will be modified differently by each of the loss processes. Since radiative trapping is assumed to be purely radiative, its contribution to laser heating can be computed directly from the steady-state solution in Eq. (12) using the measured effective material properties. The mean fluorescence wavelength, ${\lambda }_F^{\prime }$, should be measured using the operational laser medium to fully account for the geometrical and saturation effects of trapping. The measured fluorescence lifetime should be corrected to remove quenching using the relation, ${\tau }_R^{\prime }={\tau }_F^{\prime }{\lambda }_C/{\lambda }_F$. Substituting ${\lambda }_F^{\prime }$ and ${\tau }_R^{\prime }$ into Eq. (12) will yield the quantum defect heat generation corrected for the fluorescent reabsorption, $H_F^{\prime }$.

In addition to the quantum defect heating, we identify specific contributions for the heat generated by quenching, $\delta H_Q^{\prime }$, and by background absorption, $\delta H_A^{\prime }$:

The total heat generation is the sum of these contributions leading to the revised heating parameter, ${\mathrm{\Xi }}_m^{\prime }\equiv |H_F^{\prime }+\delta H_Q^{\prime }+\delta H_A^{\prime }|/L^{\prime }$ or

In Eq. (22), the first term on the right accounts for radiative trapping. The second term accounts for quenching and the third accounts for background absorption. We have incorporated the approximation, $n_2^{\prime }\simeq n_2$, which was found to be valid from Eq. (18).

Figure 5 plots ${\mathrm{\Xi }}_m^{\prime }$ for the Yb:YAG example for the same conditions as Fig. 4. Comparing this plot with that of the idealized Yb:YAG in Fig. 3 reveals the impact of adding realistic material losses. Linear and quadratic quenching redshifts the crossover wavelength by 5 nm, and the background absorption adds another 1 nm shift. When radiative trapping is added to the model, thermal loading decreases, and the redshift of the crossover wavelength increases another 9 nm. This narrows and disrupts the RBL resonance at lower intensities. Background absorption heating is stronger at higher intensities. For the highest intensities, a fractional background absorption of $\delta {\alpha }_L=\delta {\alpha }_P={10}^{-3}$ disrupts the RBL resonance. A fourfold increase in background absorption was found to overwhelm fluorescent cooling, thus disrupting the RBL resonances at all pump wavelengths.

 

Fig. 5. Plot of the realistic room temperature laser material heating parameter for the same conditions used in Fig. 4. The vertical dashed line marks the position of the effective crossover from fluorescent heating to fluorescent cooling, ${\lambda }_C^{\prime }$.

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Comparing Figs. 4 and 5 clarifies the impact of weak losses on optimal material performance. The large redshifts in the RBL resonances significantly reduce material efficiency at minimum ${\mathrm{\Xi }}_m^{\prime }$. Best efficiency with no heat generation is 0.40 when the pump is 1040 nm with intensities of $i=4$. Operation away from the exact RBL resonance allows material efficiencies to recover. If we restrict ${\mathrm{\Xi }}_m^{\prime }\le 0.01$, a best efficiency 0.62 occurs for $i=8$ with the pump at 1040 nm.

To compare the above results with experiments, we have examined the literature on high-efficiency Yb:YAG lasers. Matsubara et al. have demonstrated a record high efficiency of 81% optical-to-optical in a cw room temperature microchip laser. This 1048 nm laser was end-pumped with a high intensity Ti:sapphire laser at 940 nm. Assuming a pump saturation intensity at 940 nm of $26\mathrm{kW}/{\mathrm{cm}}^2$, Fig. 4 indicates this efficiency would be possible with an intracavity pump intensity of $400\mathrm{kW}/{\mathrm{cm}}^2$, which is somewhat higher than the reported incident intensity.

Finally, we can compare the calculations above with our own previous experimental results on radiation balanced lasing [16]. In these experiments, a $2\mathrm{mm}\times 2\mathrm{mm}\times 110\mathrm{mm}$ slab of Yb:YAG was pump at 1030 nm and lased at 1050 nm. The laser was configured as a single-end pumped resonator with an 80% reflection mirror output coupler. Pump powers as high as 4.8 kW were incident leading to scaled pump intensities of $i_P\le 13.3$. Laser powers at 1050 nm up to 1.6 kW were generated under pump-limited conditions, leading to scaled intensities of $i_L\le 3.8$. At a laser power of 600 W, the laser slope efficiency was 0.46, and the measured heating parameter dropped to below 0.001. This operating point corresponded to $i_P=7.3$ and $i_L=2.1$. The parameters used for the production of Fig. 5 above appear to overestimate the material losses, predicting ${\eta }_m^{\prime }=0.55$ and ${\mathrm{\Xi }}_m^{\prime }=0.021$. The actual laser performance appears to be closer to the ideal case, which predicts ${\eta }_m=0.49$ and ${\mathrm{\Xi }}_m=0.0012$. This indicates that the RBL resonance in the experiment was not redshifted as much as predicted in this approximate model.

5. SUMMARY

A power flow analysis has been developed for steady-state low quantum defect laser materials. The analysis differs from previous studies by explicitly including the important effect of optical cooling in the laser material. Simple analytic expressions were developed for predicting material efficiency and heating in the laser material. The model was extended to include important loss processes: quenching, background absorption, and radiative trapping. Analyzing the laser material power flow separately from the optical propagation allows for clear insight into the limits of laser performance. The equations developed here allow for straightforward evaluation of how the laser performance depends on the material parameters and operating conditions. This local model can be readily extended to laser systems using finite element analysis.

The laser material model was used to evaluate several approaches for optimizing laser performance. The impact of material losses and operating conditions were compared using Yb:YAG as an example. Theoretical material efficiency was found to peak as pump wavelength approached that of the laser. Optimal room temperature material efficiencies in Yb:YAG were found to be roughly 90% in the presence of realistic losses. Optimal thermal loading was found to require lower quantum defect pumping and can yield comparable overall efficiencies. Calculated efficiencies and thermal loading are in approximate agreement with reported values. Cryogenic operation was predicted to improve material efficiency at longer wavelengths and lower quantum defects. Heat generation was found to be much more sensitive to material losses than was efficiency. Radiation balanced operation was contrasted in ideal and realistic materials and compared with previous laser measurements. Background absorption and fluorescent reabsorption were found to significantly redshift and disrupt the RBL resonances leading to reduced efficiency.