## Abstract

Diode pumped alkali vapor amplifier (DPAA) is a potential candidate in high power laser field. In this paper, we set up a model for the diode double-side-pumped alkali vapor amplifier. For the three-dimensional volumetric gain medium, both the longitudinal and transverse amplified spontaneous emission (ASE) effects are considered and coupled into the rate equations. An iterative numerical approach is proposed to solve the model. Some important influencing factors are simulated and discussed. The results show that in the case of saturated amplification, the ASE effect can be well suppressed rather than a limitation in power scaling of a DPAA.

©2011 Optical Society of America

## 1. Introduction

As first demonstrated in 2006 [1], diode pumped alkali vapor lasers (DPALs) have gained much attention and fast development in recent years. As a new kind of optically pumped gas laser, DPAL offers a new way to high-power, high-efficiency, high-brightness and compact laser systems due to its many significant characteristics. The quantum efficiency is high (95.3% for Cs, 98.1% for Rb and 99.6% for K), which is important for increasing the optical to optical efficiency and minimizing thermal problems. And the thermal problems can be further reduced since the gaseous gain medium can be flowed to remove the heat. Good thermal management as well as the stability at high pump intensity [2] provides no significant single aperture power scaled limitation for DPALs. For the fast recycling time of alkali atoms in the lasing process [3], it is promising to extract high power from small gain volumes. These advantages make DPAL a potential candidate in the future high power laser field. Till now, a great number of DPAL experiments have demonstrated high efficiency and good beam quality [4–11].

As compared with a single oscillator, another power scaled method is to use the MOPA (master oscillator power amplifier) configuration, by adding a single or chain of diode-pumped alkali vapor amplifiers (DPAAs). As being successfully applied in solid-state lasers, the MOPA configuration can simplify the pump arrangement, disperse the thermal effect and keep good beam quality. Till now, some DPAA experiments have been made [12–14]. Hostutler *et al.* obtained 7.9dB amplification in a 2cm long Rb amplifier with seed laser of 50mW [12]. Zhdanov *et al.* obtained an amplification factor of 145 for low power Cs laser radiation [13]. The maximal DPAA output was obtained by Zhdanov *et al*., with 25W output power, when the seed laser was 5W under 280W diode pump power [14]. Till now, all these DPAA experiments were made in longitudinally-pumped configuration, and a corresponding model was set up by B. Pan *et al.* which has agreed well with experimental results [15].

Another more scalable pump scheme is the side-pumped configuration, which has been applied in DPALs [7, 10]. It can simplify the pump arrangement, decrease the demand of pump brightness, enhance the pump uniformity, lighten the power burden on optical components and be more convenient in structure design when the gaseous medium is flowed. In design of a side-pumped DPAA, a key issue has to be considered, especially for such three-dimensional volumetric medium with high gain, is the amplified spontaneous emission (ASE). The ASE will decrease the gain, and usually becomes more serious when the volume of the medium is larger and the gain is higher. So in power scaling of DPALs and DPAAs, the ASE effect must be seriously considered to see if it will be a limitation. Till now, many researchers have made studies on ASE for different kinds of high gain lasers, except for such recently developed alkali lasers. Most of the papers have discussed ASE for one-dimensional, often pencil-like geometries, such as laser rods [16, 17] or fibers [18]. For these configurations, only longitudinal ASE is considered, and for every point inside the gain medium one can define a solid angle, into which photons that are emitted have the most significant contribution to ASE. For three-dimensional amplifier with arbitrary shape, the solid angle is hard to define, and a Monte Carlo type numerical method has been applied to simulate the ASE from all photons spontaneously emitted inside the gain medium [19,20]. In this paper, the dealing of ASE is a key issue in our model. To adapt to the division scheme of the amplifier, which is needed in side-pumped configuration, we develop a new method to calculate the ASE. In our method, the longitudinal and transverse ASE effects are considered individually, and both of them are coupled into the rate equations. The results can give the range of ASE efficiency, which will be helpful for the laser design.

In section 2, the kinetic model considering both the longitudinal and transverse ASE effects for a symmetrically double-side pumped DPAA is introduced. In section 3, the numerical approaches are described. In section 4, some important influencing factors are simulated and discussed.

## 2. Modeling of an alkali vapor amplifier

The schematic diagram of a DPAA is shown in Fig. 1
. Because we are only interested in high power laser operation, the alkali vapor cell is pumped symmetrically by multiple diode stacks from two sides. The cell contains a homogeneous mix of alkali vapor (rubidium or cesium) and buffer gases (helium and hydrocarbons, for example methane or ethane) at operation temperature on the order of 100${}^{\circ}C$. To match the beam shape of diode lasers, the cell is designed into a rectangular box ($H\times W\times L$). The pump light enters into the cell through *x-z* plane. In fast-axis (along *x*-axis), the pump light can be collimated by beam shaping optics. In slow-axis (along *z*-axis), the small spreading can be neglected for pumping by multiple diode stacks. The pump intensity along *x-z* plane can be assumed homogeneous. The seed laser enters into the cell from *x-y* plane and propagates along *z*-axis to be amplified. In the entrance plane (*x*O*y* plane), the seed laser intensity is also assumed to be homogeneous.

To adapt to the pump structure, we make a two-dimension division of the alkali gain medium along *x* and *y* axes. Each divided volume element has dimensions of $\Delta x\times \Delta y\times L$ that meet the condition of $\Delta x,\Delta y\ll L$. The equations that govern the distribution of population densities in a volume element are

*x*-axis, we assume the population densities also to be homogeneous along

*x*-axis, despite the transverse ASE effect may cause some variations. Under this assumption, the population densities are only

*y*dependent denoted as ${n}_{i}(y)(i=1,2,3)$. ${\Gamma}_{p}$ and ${\Gamma}_{l}$ are the stimulated pump absorption and laser emission rates respectively. ${\Gamma}_{ASE}$ is the averaged ASE induced population decay rate of ${}^{2}{P}_{1/2}$ level (‘ASE rate’ for short). ${\gamma}_{32}$ is the fine structure mixing rate that relaxes the population from ${}^{2}{P}_{3/2}$ to ${}^{2}{P}_{1/2}$, which is usually enhanced by adding some small hydrocarbons, for example methane or ethane. ${A}_{21}({A}_{31})$ are the spontaneous emission rates, and ${Q}_{21}({Q}_{31})$ are the quenching rates that introduced by hydrocarbons. $\Delta E$ is the energy gap between ${}^{2}{P}_{1/2}$ and ${}^{2}{P}_{3/2}$ levels, $k$the Boltzmann constant and $T$the absolute temperature.

The pump light is assumed to be in Gaussian spectral profile. The incident spectrally resolved pump powers at two sides are given by

The expression of pump absorption rate ${\Gamma}_{p}(y)$ is similar as in reference [21]

The laser emission rate ${\Gamma}_{l}(y)$ is given by

Now we start to deduce the expression of ${\Gamma}_{ASE}(y)$. In step 1, because each volume element satisfies the condition of $\Delta x,\Delta y\ll L$, we can only consider the longitudinal (one-dimension along *z*-axis) ASE effect in each element. In step 2, because the spontaneous emission from volume elements will affect each other, we consider such transverse ASE effect to get the final expression of ${\Gamma}_{ASE}(y)$.

#### Step 1. Longitudinal ASE effect in an individual volume element

The schematic diagram of longitudinal ASE effect in a volume element is shown in Fig. 2
. We further divide the volume element along *z*-axis into many sub-segment with length of $\Delta z$. Each sub-segment has a solid angle relative to the end face denoted as $\Omega $. For example, the solid angle of the *n*-th sub-segment is

_{${P}_{ASE}(y)$}is the total ASE power that emitted from an end of the volume element. The contribution to ${P}_{ASE}(y)$ from the

*n*-th sub-segment is

The total ASE power ${P}_{ASE}(y)$ is then the sum of contributions from all the sub-segments

In the deduction process, we have used the averaged solid angle $\overline{\Omega}=(\Delta x\Delta y)/{(L/2)}^{2}$ instead of the different solid angles for different sub-segments. ${V}_{l}={\eta}_{\mathrm{mod}e}\Delta x\Delta yL$ is the lasing volume in a volume element, and ${\eta}_{\mathrm{mod}e}$ is the mode overlap factor.

The ASE rate in a volume element is then described as

The subscript $1\to 1$ means that the averaged ASE rate in the volume element is induced by the gain medium inside this volume element itself. The factor 2 is added because the ASE power emits from the two sides. We also express this ASE rate as length dependent (*L*), which will be used later.

#### Step 2. Transverse ASE effect among different volume elements

First, we consider the ASE rate in volume element 1 that induced by the ASE emitted from volume element 2 (see Fig. 3
). *O* and *E’* are two arbitrary points in volume elements 1 and 2. *E* is another point that satisfies the condition of *OE = OE’.* Now in order to simplify the derivation and calculation, we make another assumption: although we have considered the population densities ${n}_{i}(y)(i=1,2,3)$ as *y* dependent, in the calculation of transverse ASE effect, this transverse population variation is neglected. In fact, for a well designed DPAA, the transverse population variation should also be limited to a relatively low value, which will benefit the uniformity of laser intensity in the cross section as well as the heat management. Under this assumption, the influence of ASE on *O* that emitted from *E’* is equivalent to that emitted from *E*. Furthermore, the influence of ASE on *O* that emitted from the whole volume element 2 (*A’D’*) is equivalent to that emitted from lines *AB* and *CD*. It should be noticed that the points *A* and *D* have already located out of the range of volume element 1.

It is seen that for an arbitrary point *O* in volume element 1, we can find a range of lines in this element itself (*AB + CD = L’-2h*), whose ASE affect on *O* is equivalent to that induced by the whole volume element 2. We also noticed that as the position of *O* changes along volume element 1, the length of *AD* (*L’*) changes and *BC* (*2h*) keeps constant. When *O* locates at the end, *L’* reaches its maximal value of ${L}_{\mathrm{max}}^{\text{'}}={({L}^{2}+{h}^{2})}^{1/2}+h$; and when *O* locates in the middle, *L’* reaches its minimal value of ${L}_{\mathrm{min}}^{\text{'}}=2{[{(L/2)}^{2}+{h}^{2}]}^{1/2}$. For these two extreme values of *L’*, we can obtain the range of the ASE rate

The subscript $2\to 1$ means that the ASE rate in volume element 1 is induced by volume 2. And the total ASE rate in volume element 1 is ${\Gamma}_{ASE1\to 1}(y)+{\Gamma}_{ASE2\to 1}(y)$ (parameter $L$ in ${\Gamma}_{ASE1\to 1}(y,L)$ and subscripts *max*, *min* are neglected).

Now we can calculate the total ASE rate in a volume element inside the volumetric alkali gain medium (see Fig. 4
). For an arbitrary volume element (*p,q*), each element will give contribution to its ASE rate, and the total ASE rate in (*p,q*) is

*p,q*) that induced by (

*m,n*), the range of which can be calculated by Eq. (12)-(14). From the geometric point of view, when (

*p,q*) locates at the corner (

*a*in Fig. 4), ${\Gamma}_{ASE(p,q)}(y)$ reaches its maximal value, and when (

*p,q*) locates at the center (

*b*in Fig. 4), ${\Gamma}_{ASE(p,q)}(y)$ reaches its minimal value. So if we calculate ${\Gamma}_{ASE(m,n)\to (p,q)}$ by Eq. (14) and assume (

*p,q*) at the corner, we can obtain the maximal value of the total ASE in a volume element; if we calculate ${\Gamma}_{ASE(m,n)\to (p,q)}$ by Eq. (13) and assume (

*p,q*) at the center, we obtain its minimal value. The range of ASE rate is described as

## 3. Numerical approaches

In this section, we first describe the numerical approach for an individual volume element (in longitudinal dimension) and then introduce the numerical approach that connects different volume elements in this symmetrically double-side pumped configuration (in transverse dimension).

#### 3.1 Numerical approach in longitudinal dimension

For we only consider the case of cw operation, all the derivatives in Eqs. (1)-(3) are set zero. By Eqs. (2) and (4) we can obtain the relation between ${n}_{2}$ and ${n}_{3}$ (parameter *y* is neglected here)

By use of searching algorithm, for a fixed value of ${n}_{3}$, we can obtain ${n}_{2}$ and ${n}_{1}$ in terms of ${n}_{3}$, which are denoted as ${n}_{2}({n}_{3})$ and ${n}_{1}({n}_{3})$. Take ${n}_{1}({n}_{3})$ and ${n}_{2}({n}_{3})$ into Eq. (3), and still use the searching algorithm, we can solve ${n}_{3}$, and then obtain the population distribution ${n}_{i}(y)(i=1,2,3)$.

When the population densities are obtained, we can calculate the amplified laser power and other pump power inverted channels by

where ${P}_{laser}(y)$ is the output amplified laser power, ${P}_{fluorescence}(y)$, ${P}_{quenching}(y)$,${P}_{heat}(y)$ and ${P}_{ASE}(y)$ are fluorescence, quenching, waste heat and ASE powers in a volume element respectively.#### 3.2 Numerical approach in transverse dimension

In transverse dimension, we have developed an iterative algorithm, which is described below:

- (a) First, we calculate the transverse population distribution for the single-side pumped configuration, that is, with no consideration of ${P}_{p}^{-}(y,\lambda )$. The propagation equation of ${P}_{p}^{+}(y,\lambda )$ is given by
Take the already known ${P}_{p}^{+}(0,\lambda )$ together with numerical approach 3.1 and Eq. (25), we can solve an initial transverse population distribution ${n}_{i}^{(0)}(y)(y\in [0,W],i=1,2,3)$.

- (b) For symmetrically double-side pumped configuration, the population densities in steady state should also be symmetrically distributed. Here, the symmetrically distributed population density is assumed to be
- (c) Let the pump light ${P}_{p}^{+}(y,\lambda )$ pass the gain medium through the population distribution ${n}_{i}^{(0)*}(y)$ by Eq. (25), we can obtain the unabsorbed pump power ${P}_{p}^{+(0)}(W,\lambda )$ at the other side of the gain medium.
- (d) In steady state, the unabsorbed pump power for both forward and backward propagating pump lights should be equal, that is ${P}_{p}^{+}(W,\lambda )={P}_{p}^{-}(0,\lambda )$. Here we assume the amplifier to be in steady state, and the total pump power at one side should be
Now the laser can be seen as single-side pumped with ${P}_{p}^{tot(1)}(0,\lambda )$ as a new pump light. As similar as in step (a), we can use approach 3.1 to calculate a new population distribution ${n}_{i}^{(1)}(y)(y\in [0,W],i=1,2,3)$. In the calculation, the propagation equation for ${P}_{p}^{+}(0,\lambda )$ is Eq. (25), and for ${P}_{p}^{-(0)}(0,\lambda )$ is

- (e) By step (b), we can obtain a symmetrically distributed population ${n}_{i}^{(1)}{}^{*}(y)$. Compare ${n}_{i}^{(1)}{}^{*}(y)$ with ${n}_{i}^{(0)}{}^{*}(y)$, if ${n}_{i}^{(1)}{}^{*}(y)={n}_{i}^{(0)}{}^{*}(y)(y\in [0,W],i=1,2,3)$, then ${n}_{i}^{(1)}{}^{*}(y)$ is the final solution, and other laser information has been obtained in the calculation process. If ${n}_{i}^{(1)}{}^{*}(y)\ne {n}_{i}^{(0)}{}^{*}(y)(y\in [0,W],i=1,2,3)$, then substitute ${n}_{i}^{(0)}{}^{*}(y)$ by ${n}_{i}^{(1)}{}^{*}(y)$, and repeat steps (c)-(e) to continue the iterative process for a final solution.

## 4. Simulation results and discussion

In this section, we have simulated the influence of some important factors, and the results could give us a further comprehension of the characteristics of DPAAs.

#### 4.1 Influence of the temperature

The operation temperature decides the number density of the alkali atoms, which will dramatically affect the DPAA’s characteristics. In the calculation, the parameters are chosen as follows: the dimensions of the alkali gain medium are set as $L\times W\times H=20cm\times 5cm\times 5cm$. The buffer gases include 400torr methane and 1120torr helium (measured at 20${}^{\circ}C$), which result in an atomic absorption linewidth (FWHM) of 0.09nm at a typical temperature of 120${}^{\circ}C$. The pump intensity is set as a moderate value of ${I}_{p}^{}=5kW/c{m}^{2}$ with linewidth (FWHM) of 0.2nm, which could be realized by volume Bragg gratings (VBGs). The line shapes of the atomic absorption and pump light are assumed to be in Lorentzian and Gaussian profiles with overlapped center wavelengths. The seed laser intensity is set as a relatively weak value of ${I}_{s}=100W/c{m}^{2}$.

The simulation results are shown in Fig. 5 . We can see that for a DPAA, an optimal operation temperature exists for a maximal amplification factor [Fig. 5(a)]. As the temperature increases, the alkali atom concentration increases which enhances the pump absorption fraction, but the laser extraction efficiency ${\eta}_{opt-abs}$ decreases [Fig. 5(b)]. For the absorbed pump power, it has five main inverted channels: laser, fluorescence, ASE, heat and non-radiative transition loss (quenching). From Fig. 5(c), we can see that the fluorescence and ASE efficiency (the maximal value calculated here) will increase as temperature increases, which induce the decrease of the laser extraction efficiency. The heat efficiency is always below 1.8%, which is in correspondence with the Rb quantum defect (1.9%).

#### 4.2 Influence of seed laser and pump intensities

As a three-level laser system, the DPAA needs to be pumped at relatively high intensities, usually in a range of $1~10kW/c{m}^{2}$. Figure 6 shows the influence of pump intensity at different seed laser intensities. For the temperature characteristic of the DPAA, the simulation results are given at temperatures which are optimized for maximal laser extraction efficiencies (${\eta}_{opt-opt}$). Other parameters are the same as in 4.1. We notice that as the intensity changes, the total seed or pump powers will change correspondingly, and we only care about the efficiencies rather than the power values. It is seen that for a constant seed laser intensity, as the pump intensity increases, both the amplification factor and the laser extraction efficiency increase [Fig. 6(a) and (b)]. And for a higher pump intensity, a higher optimal temperature is needed because more alkali concentration is required for sufficient pump power absorption [Fig. 6(c)]. A comparison shows that when the seed laser is weak, the amplification factor is high but the laser extraction efficiency is low, that is, a large fraction of pump power is wasted. As a contrast, when the seed laser is strong, the amplification factor is relatively low but the laser extraction efficiency dramatically increases, and becomes more sensitive to the change of pump intensity.

To get a further understanding of the intensity influence, we also calculated the pump absorption efficiency and some main pump inverted channels (fluorescence and ASE) in Fig. 7 . We can see that, when the seed laser is weak, the pump absorption fraction is low [Fig. 7(a)]. This is because a low seed laser intensity is not sufficient to pull down the population from ${}^{2}{P}_{1/2}$ to ${}^{2}{S}_{1/2}$ energy levels through stimulated emission process. Thus the recycling of an alkali atom (${}^{2}{S}_{1/2}{\to}^{2}{P}_{3/2}{\to}^{2}{P}_{1/2}{\to}^{2}{S}_{1/2}$) is bottlenecked in the stimulated emission process, which will in turn affect the pump power absorption. As the same with solid or fiber laser amplifiers, when the seed laser is weak, the ASE effect becomes strong [Fig. 7(b)]. For ${I}_{s}=1W/c{m}^{2}$, the ASE efficiency reaches 80% at ${I}_{p}^{}=10kW/c{m}^{2}$, while for ${I}_{s}=1000W/c{m}^{2}$, the ASE efficiency is always kept below 2%. This indicates that in the case of saturated amplification, the ASE can be well suppressed and not become a limitation in power scaling of a DPAA. We also notice that the ASE efficiency increases as the pump intensity increases. Although a higher pump intensity will induce a stronger ASE effect, it can suppress the fluorescence efficiency dramatically [Fig. 7(c)]. For the case of saturated amplification, the degree of fluorescence suppression by a high pump intensity is stronger than the lightly increased ASE effect. In addition, a high pump intensity will enhance the pump absorption, so we should focus the diode pump light into high intensity for a high efficient DPAA.

We also studied the range of ASE efficiency that calculated by Eq. (16)-(18) at different seed laser intensities. The results show that when the ASE is well suppressed, the range is also small. For example, the ASE range is about 18% for ${I}_{s}=1W/c{m}^{2}$ and only 0.7% for ${I}_{s}=1000W/c{m}^{2}$. Because in optimal design, the ASE should be suppressed to a relatively low value, so the range estimation of the ASE could be rather accurate. For simplification, all the ASE efficiencies are showed as the maximal values in this paper.

#### 4.3 Influence of the gain medium’s dimensions

The width influence was shown in Fig. (8) . The pump intensity is set as ${I}_{p}^{}=5kW/c{m}^{2}$, the seed laser intensity ${I}_{s}=100W/c{m}^{2}$, and other parameters the same. The results show that as the width increases, both the amplification factor and the laser extraction efficiency decreases [Fig. 8(a) and (b)]. The main reason is the obvious increase of ASE, while the efficiencies of other pump inverted channels nearly keep constant [Fig. 8(c)]. The optimal temperature experiences a decrease as the width increases, because lower alkali atom concentration is required when the pump absorption distance becomes longer. In real design, when the ASE effect can be strongly suppressed with saturated amplification, the width could be relatively longer with lower optimal temperature, which will benefit the chemical stability and lifetime for the laser systems.

The length influence is shown in Fig. (9) . The width is set as 5cm with other parameters the same. It should be noticed that for a constant pump intensity, the total pump power increases as length increases. The results show that a longer length can effectively enhance the pump absorption [Fig. 9(b)] and suppress the ASE effect [Fig. 9(c)], thus results in a higher laser extraction efficiency. The reason is that, as compared with a short gain medium, the amplified seed laser will be much stronger in the longer part, which will enhance the pump absorption [Fig. 7(a)] and ASE suppression [Fig. 7(b)] in this longer part as well as for the whole amplifier. Other pump inverted channels (fluorescence, heat and quenching) and the optimal temperature are not sensitive to the change of the length. From these results, we can come to a conclusion that a longer gain medium will benefit both the amplification and laser extraction efficiency with enhanced ability to suppress the ASE effect.

The height of the gain medium is not specially discussed here. Because we assume the pump and laser intensity uniform along the height dimension (*x*-axis), and it only affects the calculation of the ASE effect here. In real design of a DPAA, the height adjustment is equivalent with the adjustment of the pump intensity when the pump power is fixed, except for a different transverse ASE effect. And another design consideration of the height is to adapt to the beam shape of the seed laser.

## 5. Conclusion

In this article, a model is set up for a DPAA in symmetrically double-side pumped configuration. For the three-dimensional volumetric alkali medium with high gain, the ASE may become a key issue in power scaling. And we have developed a new method to calculate the range of ASE efficiency, which has considered both the longitudinal and transverse ASE effects. To solve the model, an iterative numerical approach is proposed. To study the characteristics of the DPAA, some important influencing factors are simulated and discussed, including the operation temperature, pump and seed laser intensities and the gain medium’s dimensions. From the simulation results, we have obtained some useful conclusions: as the same with DPALs, an optimal temperature exists for DPAAs for maximal laser extraction efficiency; at constant pump and seed laser intensities, the shape of a narrower but longer gain medium will benefit the suppression of ASE; a high pump intensity is required to effectively suppress the fluorescence; when the seed laser is weak, ASE will dominate the stored pump power, and a strong seed intensity is needed to effectively suppress the ASE; in power scaling of DPAAs, the ASE effect will not become a limitation but can be effectively suppressed by use of the saturated amplification scheme. Till now, no experiment of a side-pumped DPAA has been reported, and further experimental studies are necessary to verify the validity of our model.

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