## Abstract

A simple optical model of K DPAL, where Gaussian spatial shapes of the pump and laser intensities in any cross section of the beams are assumed, is reported. The model, applied to the recently reported highly efficient static, pulsed K DPAL [Zhdanov *et al*, Optics Express 22, 17266 (2014)], shows good agreement between the calculated and measured dependence of the laser power on the incident pump power. In particular, the model reproduces the observed threshold pump power, 22 W (corresponding to pump intensity of 4 kW/cm^{2}), which is much higher than that predicted by the standard semi-analytical models of the DPAL. The reason for the large values of the threshold power is that the volume occupied by the excited K atoms contributing to the spontaneous emission is much larger than the volumes of the pump and laser beams in the laser cell, resulting in very large energy losses due to the spontaneous emission. To reduce the adverse effect of the high threshold power, high pump power is needed, and therefore gas flow with high gas velocity to avoid heating the gas has to be applied. Thus, for obtaining high power, highly efficient K DPAL, subsonic or supersonic flowing-gas device is needed.

© 2015 Optical Society of America

## 1. Introduction

Among the different Diode pumped alkali lasers (DPALs), extensively studied during the last decade [1,2], the K DPAL is of special interest. It has very high quantum efficiency (99.6%) and can operate with low pressure He buffer gas [1]. Cs and Rb DPALs on the other hand operate with hydrocarbon buffer (usually with some He added), and Rb also with high pressure (several atmospheres) He. Recently a highly efficient static, pulsed K DPAL with slope efficiency of 52% was demonstrated in USAFA [3]. In spite of the high efficiency, its threshold power was rather high, 22 W. As shown below this value cannot be fitted by simple standard models [1,4–6] assuming uniform intensities of the pump and laser beams in the transverse direction. In this paper we report on a model of K DPAL where Gaussian spatial shapes of the pump and laser intensities in any cross section of the beams are assumed and show that these assumptions can explain the measured dependence of the lasing power on the pump power. In addition, the influence of the spatial distribution of the spontaneous emission across the beam on the threshold power is studied in detail.

## 2. Description of the model

K DPAL operates at frequency *ν _{l}* of the D

_{1}(4

^{2}

*P*

_{1/2}→ 4

^{2}

*S*

_{1/2}) transition of K. It is pumped via absorption of radiation of diode laser at frequency

*ν*of the D

_{p}_{2}(4

^{2}

*S*

_{1/2}→ 4

^{2}

*P*

_{3/2}) transition, followed by rapid relaxation (by He) of the upper to the lower fine-structure level, 4

^{2}

*P*

_{3/2}to 4

^{2}

*P*

_{1/2}(designated as levels 3 and 2, respectively; the ground state 4

^{2}

*S*

_{1/2}is designated as 1). The model considers typical configuration of a DPAL with end-pump geometry studied in [3] and shown in Fig. 1. A pump beam with total power ${P}_{p,0}$ enters a cylindrical laser cell of length

*l*through windows with transmission

*t*. The lasing medium consists of a mixture of alkali vapor K and He for broadening the D

_{2}transition and mixing between the fine-structure levels 3 and 2. The walls of the cell are heated to the temperature

*T*~180-190 C. The laser resonator of length

*L*consists of a concave reflector of radius

*R*and a plane output coupler with reflectivity

*r*

_{1}(close to 100%) and

*r*

_{2}, respectively, located outside the laser cell. Both the pump and laser beams propagate along the optical axis

*Z*of the resonator.

Unlike our previous models [6,7] the pump and laser intensities, ${I}_{p}$ and ${I}_{l}^{\pm}$, respectively, are not assumed to be uniform in the beam cross section *xy* and their spatial distributions have the form:

*xy,*slightly dependent of

*z*, so that

The spectral distribution of the pump power${P}_{p}(z,\nu )$over the frequency *ν* was taken into account, since in the experiments [3] broad band pumping was applied. The laser powers ${P}_{l}^{\pm}(z)$ are assumed to be monochromatic with frequency *ν _{l}*; the superscripts + and – indicate laser beams propagating in the opposite directions +

*z*and –

*z*. Only the pump beam propagating in the +

*z*direction was included into the model since checking the power of the pump beam propagating in the -

*z*direction we found it should be negligibly small. The reason is that due to almost complete absorption of the pump beam during the passage of the cell in the +

*z*direction followed by the reflection from the concave mirror only small fraction of the initial pump energy returns to the cell. The rates of changes of ${P}_{p}(z,\nu )$ and ${P}_{l}^{\pm}(z)$ with

*z*are described by the Beer-Lambert law, where

*z*= 0 is assumed to be located at the inner surface of the window closer to the output coupling mirror with the reflectivity

*r*

_{2}:

_{2}transition and the gain coefficient for the D

_{1}transition, respectively, averaged over the beam cross section

*xy*, ${n}_{1}$, ${n}_{2}$and ${n}_{3}$are the number densities of K atoms in levels 1, 2 and 3, respectively, whereas

_{1}and D

_{2}transitions, respectively, ${\sigma}_{{\text{D}}_{1}({\text{D}}_{2}),at}={\lambda}_{l(p)}^{2}/2\pi $ are the atomic cross sections, ${\lambda}_{l(p)}$- the wavelengths and $\Delta {\nu}_{{D}_{1}({D}_{2}),n}$ and $\Delta {\nu}_{{D}_{1}({D}_{2})}$ - the natural and the pressure broadened widths, respectively, of these transitions. In Eq. (9), it is assumed that the line shape of the D

_{2}transition is described by pressure broadened Lorentzian spectral distribution [4–7].

The values of${n}_{1},$${n}_{2}$and ${n}_{3}$are found from the rate equations for the 3-level system presented in [8]. As shown in [9–11] the rates of the excitation of K atoms to higher electronic levels followed by their ionization are negligibly small for pump intensities <10 – 20 kW/cm^{2}. Hence, these levels are not included in the model. Since the pulse length, ~30 μs, applied in [3] was much larger than the spontaneous emission times of the levels 2 and 3, ~30 ns [12], the steady state solutions of the rate equations were used in the computations. Unlike [8] we did not use a monochromatic approximation, assuming that the effective absorption cross section is calculated by the convolution of the pumping laser and absorption spectrums but took into account the spectral distribution of ${P}_{p}(z,\nu )$ in different locations in the laser cell. In this case${n}_{1},$ ${n}_{2}$and ${n}_{3}$ are given by the following expressions:

The boundary condition for ${P}_{p}$at *z* = 0 is

The boundary condition for the two way laser powers at *z* = 0, ${P}_{l}^{+}$ and ${P}_{l}^{-},$ were calculated by ray tracing of the given output laser power${P}_{lase}$: ${P}_{l}^{-}(0)={P}_{lase}/\left[t(1-{r}_{2})\right]$ and ${P}_{l}^{+}(0)={P}_{l}^{-}(0){t}^{2}{r}_{2}.$ The solution for the output laser flux was found by iterating on ${P}_{lase}$ until ${P}_{l}^{-}(z=l)/{P}_{l}^{+}(z=l)={t}^{2}{r}_{1}$ [13]. The system of the differential Eqs. (4) and (5) with aforementioned boundary conditions was solved numerically using “ode 45” Matlab computer program.

## 3. Results and discussion

#### 3.1. Calculation of the output laser power

Parameters of K DPAL for which the calculations were performed are listed in Table 1. The broadening rates ${\Gamma}_{{\text{D}}_{1}}$and ${\Gamma}_{{\text{D}}_{\text{2}}}$of the D_{1} and D_{2} transitions are used for calculating the pressure broadened widths of these transitions. Normalized spectral distribution ${g}_{p}(\nu )$of the pump laser at the inlet to the laser cell is shown in Fig. 2. The smooth curve is obtained by fitting the measured spectrum with the superposition of the two Lorentzian functions. The spectrum has a bimodal shape and the distance between the two maximums depends on the diode laser power varying from 8.5 GHz at low powers to 12.5 GHz at high powers due to heating of the diode. We assumed that the distance between the peaks is equal to the average of the above values, 10.5 GHz. Figure 2 also shows normalized spectral distribution of the absorption on the pump D_{2} transition. The central frequency of the absorption line was locked between the peaks of the pump laser spectrum and we assumed that it was located at equal distances from the peaks.

It was assumed that spatial distributions,${f}_{p,l}(x,y,z)$, of the pump and laser intensity over the beam cross section *xy* have Gaussian shape:

*x*and

*y*directions, respectively, whereas ${w}_{l}(z)$is the laser beam radius (HWHM). ${w}_{x}(z)$and${w}_{y}(z)$are taken from the experimental measurements and ${w}_{l}(z)$calculated for the fundamental transverse mode (which is axially symmetrical) of the stable resonator with parameters presented in Table 1 and shown in Fig. 3. The assumption of a Gaussian spatial distribution of the pump and laser beams is consistent with the experimentally observed bell-shaped intensity spatial profile in the laser spot, and with the fact that when the pump power exceeds the threshold by less than a factor of two, only the fundamental Gaussian transverse mode lases. Other distributions ${f}_{p,l}(x,y,z)$ are tested below.

To test the validity of our assumptions on the absorption and on the pump laser spectra, we calculated the dependence of the small pump signal transmission through the cell on the temperature in the absence of lasing and compared the results with the experimental data [3]. Just as in [3], a very low pump intensity of 1 W/cm^{2} was assumed to avoid saturation of the D_{2} transition. The transmission $Tr$ of the pump beam is defined as the ratio

Figure 5 shows the experimental (obtained in [3]) and calculated (using the present model) dependence of the output laser power ${P}_{lase}$on the incident pump power ${P}_{p,0}$for two values of *T*, 185 C and 193 C. For simplicity the gas temperature *T* was assumed to be spatially uniform and equal to that of the laser cell walls. This assumption is justified since according to our recent computational fluid dynamics calculations [16] for the 30 μs pump pulses the temperature changes are < 10K, i.e., very small in comparison to the average gas temperature of ~450 K. Unlike the aforementioned computations of the small pump signal transmission shown in Fig. 4, the temperature of the source of K vapor was assumed to be equal to *T* due to its heating by the pump and laser beams and heat transfer from the bulk of the gas. The values of ${P}_{lase}$ calculated for *T* = 185 C (the temperature of the cell windows before the application of the pumping beam [3]), are close to the measured values but a little differ from them at low${P}_{p,0}$showing lower threshold power${P}_{th}$ = 16 W and slope efficiency. By changing the temperature we found that the best fitting to the measured ${P}_{lase}$ is achieved at *T* = 193 C (see Fig. 5) which is by only 8 C higher that the initial *T* = 185 C. This temperature increase can be explained by the heating of the gas experimentally observed in [9] and caused by the heat release due to the aforementioned relaxation of level 3 to 2. In this case the calculated slope efficiency and ${P}_{th}$are 53% and 22 W, respectively, being in excellent agreement with the measured values.

Since the spatial distributions,${f}_{p,l}(x,y,z)$, of the pump intensity were not accurately measured in [3] we did computations for two other cases where ${f}_{p,l}(x,y,z)$were different from those given by Eqs. (14) and (15). In the first case hat-top pump beam was tested with

In the second case cylindrical hat-top pump and laser beams with radiuses of ${r}_{p}$ and ${r}_{l}$, respectively, and uniform transverse distributions of the intensity were tested:

*z*inside the cell. Such pump and laser beams were studied in the standard semi-analytical models of DPALs [4–6]. As shown in Fig. 5, the calculated${P}_{lase}$in this case are larger than the experimental values, whereas ${P}_{th}$ = 5 W was much smaller than the measured 22 W value. Thus, agreement between the calculated and measured ${P}_{lase}$can be reached only assuming Gaussian spatial distributions of the intensity (Eqs. (14) and (15)) in both the pump and laser beams.

#### 3.2. Influence of the spatial distribution of the spontaneous emission on the threshold power

The high threshold power ${P}_{th}$ = 22 W shown in Fig. 5 cannot be explained by standard semi-analytical models of the DPAL [4–6]. Actually, according to these models ${P}_{th}$ is given by

*T*= 193 C the saturated K vapor density${n}_{0}={10}^{14}{\text{cm}}^{-3},$ assuming ${n}_{3}=0.5{n}_{0}$and ${n}_{2}=0.25{n}_{0}$ [4,9], and for ${\eta}_{\text{abs}}=1$ and $V=5{\text{x10}}^{-3}{\text{cm}}^{3}$(found from Fig. 3 showing the size of the pump beam) we found from Eq. (20) that ${P}_{\text{th}}$is about 4 W. This value is close to ${P}_{th}=5\text{W}$ found for cylindrical hat-top pump and laser beams with uniform transverse distributions of the intensity, both values being much lower that the measured ${P}_{th}$ = 22 W. The only possible reason for the difference between the estimated and measured values of${P}_{th}$could be that the assumption that $V$is close to the volume occupied by the pump beam is not correct. To check this possibility the spatial distribution of the spontaneous emission intensity

*z*was calculated. Figure 6 shows ${I}_{sp}(y)$and the spatial distributions of the pump and laser beams ${f}_{p}(y)/{f}_{p}(y=0)$and${f}_{l}(y)/{f}_{l}(y=0),$ respectively, at $x=0.$ It is seen that the width of the spontaneous emission region is more than three times larger than the widths of the pump and laser beams. That means that the volume $V$occupied by K atoms contributing to the spontaneous emission is much larger than the volume of the pump beam which contradicts our initial assumption. The large width of the spontaneous emission region is due to strong excitation of K atoms in the wings of the Gaussian pump and laser beams. In spite of the relatively low light intensity of the beams in the wings, the excitation rates of levels 2 and 3 is still much higher than the spontaneous emission rate. As a result, the steady state populations of these states, established mainly due to the balance between the processes of the absorption/stimulated emission on the D

_{1}and D

_{2}transitions and the very fast relaxation of level 3 to 2, are almost the same as in the optical axis. We note that the large values of

*n*

_{2}in the wings do not result in lasing on the high order transverse modes which have big size in the transverse direction. Actually, our calculations show that the saturated population inversion on this transition,

*n*

_{2}-

*n*

_{1}, is positive only near the optical axis at the region with $\left|y\right|<0.6$mm where the fundamental transverse mode oscillates and is negative in the wings and hence hindering the lasing on the high order transverse modes. The last statement is in line with the aforementioned assumption that only the fundamental Gaussian transverse mode participates in the lasing. As follows from Eq. (20) large value of $V$ results in the high threshold power ${P}_{th}$observed in the K DPAL.

High ${P}_{th}$is the main obstacle for obtaining high${P}_{lase}$and optical-to-optical efficiency of the K DPAL. It is very difficult to decrease ${P}_{th}$which depends on ${n}_{2}$and${n}_{3}$, proportional to the K atoms density ${n}_{0}$ (see Eqs. (10)–(12)) and on the volume and shape of the pump and laser modes (see Eq. (20)). Decrease of ${n}_{0}$by decreasing *T* results in the decrease of absorption on the D_{2} transition and of the slope efficiency, as shown in Fig. 5 where it is seen that the slope efficiency at 180 C is lower than that at 193 C. It is also very difficult to decrease the volume and shape of the pump and laser modes since the pump beam diameter is already < 1 mm. So taking into account the fact that spontaneous emission does not affect the slope efficiency of the DPALs [4–6] we conclude that for increasing ${P}_{lase}$and the optical-to-optical efficiency it is necessary to increase ${P}_{p,0}$which should be several times larger than${P}_{th}$, i.e., > 100 W. To avoid the heating of the medium at such high ${P}_{p,0}$ gas flow with high gas velocity should be applied. Hence, the only possible way for obtaining high power, highly efficient K DPAL, subsonic [17] or supersonic [10] flowing-gas device has to be employed. For example, as calculated in [10], at ${P}_{p,0}=10$kW and gas velocity of 20 m/s the optical-to-optical efficiency of the subsonic K DPAL is as high as 63%. Supersonic mode of operation has an additional advantage since the flowing gas also cools down due to the supersonic expansion [10]. We have recently carried out further calculations (unpublished) where we compared subsonic, transonic and supersonic modes of operation and the advantage of supersonic operation was corroborated.

## 4. Summary

A simple optical model of K DPAL, where Gaussian spatial shapes of the pump and laser intensities in any cross section of the beams are assumed, is reported. The model, applied to the static, pulsed K DPAL with high slope efficiency of 52% demonstrated in [3], shows good agreement between the calculated and measured dependence of ${P}_{lase}$on the incident pump power${P}_{p,0}$. In particular, the model calculates correct value of the threshold pump power ${P}_{th}$of 22 W (corresponding to pump intensity of 4 kW/cm^{2}) which is much higher than that predicted by the standard semi-analytical models of the DPAL [4–6]. Other assumed spatial distributions of the pump and lasing intensities, in particular those with the hat-top pump and Gaussian laser beams and with cylindrical hat-top pump and laser beams, are unable to calculate the observed values of${P}_{lase}$. It is shown that the reason for the large values of ${P}_{th}$is that the volume occupied by the excited K atoms contributing to the spontaneous emission is much larger than the volumes of the pump and laser beams, resulting in very large energy losses due to the spontaneous emission. The large volume of the spontaneous emission region is due to strong excitation of K atoms in the wings of the Gaussian pump and laser beams. High ${P}_{th}$is the main obstacle for obtaining high${P}_{lase}$and optical-to-optical efficiency of the K DPAL. Since it is very difficult to decrease${P}_{th}$, it turns out that for increasing ${P}_{lase}$and the optical-to-optical efficiency it is necessary to increase ${P}_{p,0}$and to apply gas flow with high gas velocity to avoid heating the gas. Hence, for obtaining high power, highly efficient K DPAL, employment of subsonic [16] or supersonic [10] flowing-gas device is needed.

## Acknowledgment

Effort sponsored by the High Energy Laser Joint Technology Office (HEL JTO) and the European Office of Aerospace Research and Development (EOARD) under grant FA8655-13-1-3072.

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