1. Introduction

A number of the trivalent lanthanide ions with potential as laser ions in solid hosts have promising transitions in which the prospective lower laser level (LLL) has a lifetime longer than or comparable to that of the prospective upper laser level (ULL). This is particularly true in the mid-infrared spectral region around 3-5 microns, where examples include the Dy^{3+ 6}H_11/2→⁶H_13/2 transition (∼4.3 µm), Er^{3+ 4}I_9/2→⁴I_11/2 (∼4.5 µm), Er^{3+ 4}I_11/2→⁴I_13/2 (∼2.8 µm), and Ho^{3+ 5}I₅→⁵I₆ (∼3.9 µm), among others [1–4]. The long LLL lifetimes present challenges both in laser operation and in modeling. Operationally, the slow depopulation of the LLL renders such systems self-terminating in most circumstances, so that continuous-wave (CW) lasing is not possible. (There can be exceptions to this prohibition, such as cooperative upconversion in favorable hosts and dopant concentrations to shorten the LLL lifetime [5], excited state absorption from the LLL [6], and cases in which favorable branching ratios make the ULL→LLL spontaneous decay rate lower than the LLL decay rate even if the total rate of decay from the ULL is higher [7].) The modeling challenge exists whether or not CW lasing is possible, since standard treatments of lasing in four-level systems assume the LLL lifetime is so much shorter than that of the ULL that the LLL remains essentially empty [8]. Calculations are more difficult when the LLL population is non-negligible and changes over time during pumping and lasing. A straightforward numerical integration of the rate equations can be undertaken, but due to the rapid changes in populations that occur at the onset of lasing, it can be difficult and very slow to get accurate results.

In cases where it is acceptable to restrict attention to the case of Q-switching, and if the overall efficiency rather than the time-development of the pulse is of interest, Barnes et al. have shown that a different but related problem, that of quasi-three-level lasing, can be solved in a simple way [9,10]. In this paper, we show that a similar approach works for a four-level laser system with arbitrarily long LLL lifetime, and that the resulting equation, although requiring numerical solution, can be solved very quickly. This enables easy estimates of laser threshold and slope efficiency, sufficient to guide the design of laser experiments. Portions of this work were presented at the 2023 Optica Advanced Solid-State Lasers meeting [11].

2. Approximations, rate equations, and numerical solution

It is useful first to review briefly the approach of Barnes et al. to Q-switched quasi-three-level lasers, to which our four-level model will be compared. Their approach uses several important approximations to make the analysis of the laser more tractable. The spatial dimensions of the problem are averaged – both the lateral dimensions and that along the beam axis. This simplifies the rate equations such that the various energy level populations and the photon population are functions of only one variable: time. They assume the gain element to be filled with pump light, and we modify that only by treating end-pumping and assuming that one of the parameters averaged is the radius of the pump beam in the gain medium. As is often done in the treatment of both three-level and four-level lasers, they assume the pumped level, level 3 in Fig. 1(A), to have very short lifetime compared with that of level 2, so that N₃, the population density of level 3, can be taken as essentially zero. Therefore N₁, the level 1 population density, can be replaced by (N_tot – N₂), where N_tot is the total number density of laser ions and N₂ is the level 2 population density, and thus only one equation for excited state densities is needed.

 

Fig. 1. Schematic energy levels for quasi-three-level (A) and four-level (B) laser systems. The levels in B are numbered such that the upper laser level and lower laser level have the same numbers as in A.

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The moment that the Q-switch opens can be taken as the end of useful pumping, so that during the usable pump the rate equations for the excited state population densities and photon density involve only pumping and spontaneous decays, not stimulated emission. Because the pulse develops rapidly compared with the ULL lifetime after the Q-switch opens, only stimulated emission needs to be considered in the rate equations during the laser pulse. As a result, the rate equations during the laser pulse for the quasi-three-level system become:

In these equations, N_p is the number density of laser photons, L is the length of the gain medium, L_c is the cavity length, c is the speed of light in vacuum, n is the index of refraction of the gain medium, γ = 1 + σ_a/σ_e is the inversion reduction factor, σ_a and σ_e are the absorption and stimulated emission cross sections at the laser wavelength, R_M is the reflectivity of the output coupling mirror, and T is the one-way transmission of the cavity apart from output coupling. That is, (1 – T²) is the round-trip loss due to all passive losses other than outcoupling. These equations can be rearranged and combined to eliminate time, giving:

Equation (3) can be integrated to get the pulse energy. The end of the Q-switched pulse can be taken to occur when the photon density, having grown and peaked earlier in the pulse, comes back down to its initial value, so that the left-hand side integrates to give N_pf­ – N_p0 = 0. Thus, if N₂₀ and N_2f are the ULL populations at the beginning and end of the laser pulse, respectively, integration of Eq. (3) gives:

N₂₀ is related to the pump energy, E_p, by N₂₀ = η_Pη_Cη_Aη_Qη_S×E_p×λ_L/(πa²L × hc), where a is the (average) radius of the pump beam, h is Planck’s constant, λ_L is the laser wavelength in vacuum, and the η’s are efficiency factors: η_P for conversion of electrical energy into pump light, η_C for coupling of pump light into the gain medium, η_A for the energy efficiency of absorption of the pump light, η_Q for the quantum efficiency with which absorbed photons populate the ULL, and η_S for the storage efficiency in that ULL during the pump pulse. If the pump is a “square” pulse of duration t_p and if the ULL lifetime is τ₂, then η_S = (τ₂/t_p)×[1 – exp(-t_p/τ₂)]. (If the Q-switch is opened before the pump ends, that continued pumping is not useful. Thus, t_p can be taken to be the shorter of the true pump pulse duration and the period from the start of the pump to the opening of the Q-switch.) The output energy, E_LO, of the Q-switched laser pulse is E_LO = η_MO×[ln(R_M)/ln(R_M×T²)]×πa²L×(hc/λ_L)×(N_20­-N_2f), where η_MO is the efficiency of mode overlap between the pump beam and the laser mode. In the simple case of a top-hat beam, and if the laser mode’s radius is ω, η_MO = ω²/a² when ω<a, and η_MO = 1 when ω≥a. Inserting these into Eq. (4) and rearranging, the resulting relation between the output energy E_LO and the pump energy E_p is:

Although we have chosen somewhat different notation and have arranged the relation differently, Eq. (5) is the same result as given by Barnes et al. (Eqs. (9) and (10) in Ref. [9], and the equations of section XI in Ref. [10].)

We now consider how the rate equations differ for a four-level system with a sufficiently long LLL lifetime that the LLL population density cannot be assumed to be negligible. As shown in Fig. 1(B), we choose to number these energy levels 0 through 3, so that the ULL and LLL correspond to the same level numbers as in the quasi-three-level case. This will facilitate comparison of the two systems. We again assume the population of level 3 to be negligible due to rapid decay, and we again average over the spatial dimensions to yield equations that depend only on time. After the Q-switch is opened, it is again reasonable to assume that stimulated emission develops so rapidly that spontaneous decays from both ULL and LLL and pumping, if it continues during this stage, can be neglected during the laser pulse. Thus, after the Q-switch opens dN₁/dt = - dN₂/dt, and the rate equations for the upper and lower laser level population and laser photon densities are:

Here N₁₀ and N_20­ are the LLL and ULL population densities at the moment the Q-switch is opened, respectively. Combining these equations to eliminate time gives:

Note that the only difference between Eq. (3) and Eq. (9) is that the total dopant population density, N_tot, is replaced by the sum of the ULL and LLL population densities at the moment of Q-switch opening, N₁₀+ N₂₀. Thus, the solution proceeds in the same way, except that we must look at the rate equations before the Q-switch opens to find N₁₀+ N₂₀. For the energy level scheme in Fig. 1(B), those equations are:

Here β₂₁ is the branching ratio for the total rate of decay from level 2 to level 1, including both radiative and nonradiative decay processes, and W_p is the pumping rate. These equations can be solved simply, giving:

These can then be evaluated at the moment the Q-switch opens to give N₁₀ and N₂₀. In some laser systems there are intermediate levels between the LLL and the ground state. These result in a larger set of rate equations and a more tedious solution, but it is still possible to calculate N_10­ and N₂₀ in a similar way.

In Eqs. (12) and (13), the pump rate W_p can be expressed in terms of the same efficiency factors noted in the quasi-three-level case, giving W_p = η_Pη_Cη_Aη_Qη_S×(E_p/t_p)×λ_L/(πa²L × hc), if the pump power is constant during a pulse of length t_p. Using the resulting N₁₀ and N₂₀ and integrating Eq. (9) in the same way as before, the relation between pump pulse energy, E_p, and laser output pulse energy, E_LO, is:

Due to the exponential, Eq. (14) is a transcendental equation, but it is easily solved numerically by the method of bisection. Our experience is that ten bisections are usually sufficient to determine E_LO to within an uncertainty very small compared with the experimental uncertainty of many of the input parameters. A program to implement this bisection solution executes very quickly on an ordinary personal computer. As an example of the solution’s behavior, we present results calculated for one of the materials and laser transitions of interest to us: the Er^{3+ 4}I_11/2→⁴I_13/2 transition in Er-doped Ga₂Ge₅S_13­ glass (Er:GGS). The input parameters are given in Table 1, and the calculated output energy versus pump energy for a few combinations of output coupling and estimated passive loss is given in Fig. 2. In the table, η_P, η_C, η_Q, t_p, a, η_MO and λ_L are estimates, τ₁, τ₂ and β₂₁ are from Ref. [3], and the other parameters were determined spectroscopically in the current work using the same methods and equipment as in Ref. [3]. From Fig. 2, the approximate thresholds for cases A-D are, respectively, 38, 51, 14 and 27 mJ.

 

Fig. 2. Calculated output energy, E_LO, versus incident pump energy, E_p, for potential Q-switched lasing of Er:GGS glass, using the input parameters from Table 1. A: R_M = 0.96, T² = 0.98; B: R_M = 0.96, T² = 0.96; C: R_M = 0.998, T² = 0.98; D: R_M = 0.998, T² = 0.96.

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Table 1. Input parameters for example calculations of laser efficiency of a potential Er:GGS glass laser

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3. Analytical solution for threshold and high-energy limit of slope efficiency

In two very useful limiting cases, Eq. (14) can be solved analytically. Laser threshold occurs when the term in square brackets is reduced to zero. When E_LO is sufficiently small, the fact that e^x = 1 + x for sufficiently small x can be used to make the equation linear, yielding the solution:

As Fig. 2 shows, the output versus pump curves approach a linear shape as E_p increases well above threshold. That limiting slope efficiency is approached when E_LO is so large that the exponential in Eq. (14) becomes vanishingly small. As a result, the slope efficiency, η_slope, becomes:

Thus, if only these key quantities are needed, an even faster and more accurate calculation is possible.

Note also that Eq. (16) enables a simple test of our solution. In a classic four-level system, the LLL decays so rapidly that it is always essentially empty. This is expressed in laser equations by setting the inversion reduction factor, γ, equal to one. When this is done in Eq. (16), the resulting slope efficiency is consistent with that presented in standard monographs such as Koechner [8], once adjustments are made for difference in nomenclature.

4. Comparison with direct numerical integration of rate equations

As a test of the simple solution embodied in Eq. (14) through Eq. (16), a comparison can be made with numerical integration of the free-running rate equations for the same four-level system, Fig. 1(B). If we again average over the spatial dimensions and assume the pumped level, level 3, decays so rapidly to level 2 that population N₃ is negligible, the free-running equations are:

In this case, W_p depends on N₀ (through the absorption efficiency factor used in earlier equations) and, as before, σ_a = (γ-1)σ_e. If numerical integration uses sufficiently short time-steps, the coefficients of N_p in Eq. (19) are approximately constant over one step, so that an exponential solution may be used. The other equations are solved by a standard adaptive Runge-Kutta routine, RK45 [12]. However, we find that at the onset of lasing and relaxation oscillations the populations change so rapidly that the adaptive routine drives the time-step size to impractically small values, necessitating the imposition of a minimum step size. Tests with different minimum step sizes showed that it was not practical to reach true convergence, meaning small enough steps that further decrease in step size makes no significant change. However, when the minimum step size was reduced below 1 × 10⁻⁸ seconds the changes in oscillation peak spacing were small, as were the differences in average power, whereas execution times became problematically long. Thus, that minimum time-step size was used. For a given set of input parameters, laser threshold was determined by running the numerical integration for several pump powers to find the lowest power at which lasing commenced within a pump duration of 1 ms. To estimate the slope efficiency, two different pump powers about twice the threshold value were run, and their time-integrated output powers were compared.

As would be expected since this Runge-Kutta approach must calculate population densities for many time-steps, its execution was far slower than the efficiency method with time eliminated, presented in the preceding sections. On the personal computer used, calculation of E_LO­ for 100 E_p values for a given R_M and T² using our efficiency method took about 1-2 seconds, whereas calculation for a single E_p value for a given R_M and T² using the Runge-Kutta routine required 20-30 minutes. Thus, although our efficiency method discards all time information, it has the advantage of enabling very fast results.

This RK45 method was used to calculate laser threshold and to estimate slope efficiency for the same input parameters from Table 1, and for the same R_M, T² combinations as in Fig. 2. The results for one combination are shown in Fig. 3, and the thresholds and slope efficiencies for all four combinations are presented in Table 2 for both models.

 

Fig. 3. Calculated output power versus time for potential free-running lasing of Er:GGS glass, using RK45 integration of the rate equations, with input parameters from Table 1. R_M = 0.96, T² = 0.96. Incident pump powers 80 W and 100 W.

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Table 2. Calculated laser thresholds and high-power limits of slope efficiencies, comparing the new Q-switched model results, Eq. (15) and (16), with the direct RK45 integration of the free-running rate equations. Input parameters are from Table 1

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For all four cases shown in Table 2, the threshold values are very nearly the same for our simple method of calculating Q-switched threshold and for the brute-force time-integration of the free-running equations. This is reassuring, since the photon population does not become substantial until threshold is reached, and thus the presence or absence of a Q-switch should make little difference in the laser threshold.

One would not expect close agreement in the slope efficiencies, since the physical situation for a Q-switched laser is very different from a free-running laser once lasing begins. In the Q-switched case for pump energies above threshold, lasing is actively suppressed until the desired pump energy has been delivered, during which time spontaneous decay moves some excited ions from the ULL to the LLL without any lasing, whereas in the free-running case lasing begins extracting energy as soon as threshold is reached, even as pumping continues beyond threshold. However, the fact that the slope efficiencies from the two models only differ by about a factor of two does have some significance. In planning laser experiments on candidate laser materials, it is helpful to have even approximate guidance on what material, laser cavity and pump parameters are needed. For the very fast, simple calculation approach we present here to give not only an accurate estimate of laser threshold but also even a rough estimate of slope efficiency is useful for the design of such experiments.

5. Behavior for longer pulse durations – self-termination

The efficiency calculation method presented in this paper was developed to deal with cases in which a four-level system has a long lower laser level lifetime, but the example parameters used in the preceding sections have included a pump pulse duration that is short compared with the LLL lifetime. This was based on a pump source used in our laboratory, but to see the effect of long LLL lifetime on the calculated efficiencies, we should consider longer pump pulses. It may also be noted that the branching ratio for decay from the ULL to the LLL in our example case is so small, 0.13, that according to the logic of Quimby and Miniscalco we do not expect self-termination in this system [7]. Thus, larger branching ratios should be investigated to see the effects of self-termination.

To investigate the effects of self-termination, we have used Eq. (15) and Eq. (16) to calculate the threshold and high-energy limit of the slope efficiency with R_M = 0.96 and T² = 0.98, with the input parameters of Table 1 except for the pump pulse duration, t_p, and the ULL-to-LLL branching ratio, β₂₁, which are varied. The results for pump durations extending well beyond the ULL and LLL lifetimes, and for a few different choices of the branching ratio, are shown in Fig. 4 and Fig. 5.

 

Fig. 4. Calculated laser threshold from Eq. (15) versus pump pulse duration, t_p, for several branching ratios, β₂₁. The Q-switch opens at the end of the pump pulse. Other input parameters are from Table 1. R_M = 0.96, T² = 0.98.

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Fig. 5. Calculated slope efficiency in the high-energy limit from Eq. (16) versus pump pulse duration. t_p, for several branching ratios, β₂₁. The Q-switch opens at the end of the pump pulse. Other input parameters are from Table 1. R_M = 0.96, T² = 0.98. The inset magnifies the region near a slope value of zero.

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These calculations extend to pump pulse durations more than ten times the LLL lifetime. By that time N₁₀/N₂₀, which is Eq. (13) divided by Eq. (12) evaluated at t_p, is very close to its asymptotic value of τ₁β₂₁/τ₂. For all values of the 2→1 branching ratio the threshold energy increases and the slope efficiency decreases as the pump pulse duration is increased, due to accumulation of more population in the LLL. However, for small branching ratios such as the experimentally estimated value of 0.13 for our Er:GGS glass, the growth in threshold and drop in slope efficiency are relatively slow. For sufficiently large branching ratio, such as the 0.6 value shown in Fig. 4 and Fig. 5, a pump pulse duration is reached at which the threshold goes to infinity and the slope efficiency goes negative. This is the effect of self-termination. For the branching ratio of 0.6 shown in the figure, this occurs at a pump pulse duration of 16 ms. The factor {1 – (γ-1)N₁₀/N₂₀} in Eq. (15) and Eq. (16) indicates that this failure to lase should occur if the 2→1 branching ratio is larger than or equal to τ₂/(τ₁×(γ-1)), which for the parameters of Table 1 equals approximately 0.559. The traces in Fig. 4 and Fig. 5 for this value of β₂₁ indicate that the laser threshold for this branching ratio does indeed grow rapidly but does not go to infinity within the calculated span of pulse durations, and likewise that the slope efficiency approaches zero but remains positive.

The dependence of Eq. (15) and Eq. (16) on pump pulse duration enters through their dependence on N₁₀/N₂₀, the ratio of LLL to ULL population densities at the moment the Q-switch is opened (which is either the actual end of pumping, or at least the moment beyond which further pumping does not contribute usefully). Using Eq. (12) and Eq. (13), that ratio is shown in Fig. 6 as a function of pump pulse duration for the same set of branching ratios. The curves make it clear that for sufficiently small values of the 2→1 branching ratio the population ratio due to pumping levels off at a value too low to prevent population inversion. For sufficiently large values, N₁₀/N₂₀ reaches a value too large to permit population inversion, and does so at progressively shorter pump durations as the branching ratio increases.

 

Fig. 6. Calculated population ratio at the end of the pump pulse versus pump pulse duration, t_p, for several branching ratios, β₂₁. Other input parameters are from Table 1. R_M = 0.96, T² = 0.98.

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The branching ratio required to avoid self-termination, β₂₁ < τ₂/(τ₁×(γ-1)), may be compared with the CW value predicted by Quimby and Miniscalco [7]. If the pumped energy level feeds only the ULL, as assumed in our equations, their Eq. (4) can be expressed as requiring β₂₁ < τ₂/τ₁ to avoid self-termination. The difference between their result and ours is simply that they assume the condition for lasing is N₂ > N₁, which in turn assumes that σ_e = σ_a, in which case γ = 1 + σ_a/σ_e = 2. Our expression takes into account the fact that σ_e and σ_a are often unequal, and thus the two results are consistent.

6. Summary and conclusions

In this paper, we have developed a simple, fast method for calculating the efficiency of Q-switched four-level laser systems with no restriction on the lower laser level lifetime. The approach is similar to the formulation of Barnes et al. for calculating the efficiency of a Q-switched quasi-three-level laser [9,10]. The key features of this approach are that Q-switching enables separating the pumping and lasing processes into two independent, more easily-solved sets of equations, and that eliminating time from the equations yields very simple results for the output energy versus pump energy. The resulting equation can be solved analytically to get the laser threshold and the limiting slope efficiency far above threshold, and can be solved numerically very rapidly to generate full input-output curves. The results are useful not only for predicting Q-switched laser efficiency, but also for getting estimates even for free-running pulsed lasers. Further, the dependence of the calculated laser threshold and high-power slope efficiency on parameters such as pump pulse dependence, decay rates and branching ratios elucidates in what circumstances the lower-level lifetime is so long that CW lasing is prevented by self-termination.