## Abstract

A physical model combining rate, power propagation, and transient heat conduction equations for diode-pumped alkali vapor lasers (DPAL) is applied to a pulsed Rb-CH_{4} DPAL, which agrees well with the time evolution of laser power and temperature measured by K absorption spectroscopy. The output feature and temperature rise of a multi-pulse DPAL are also calculated in the time domain, showing that if we energize the pump light when the temperature rise decays to 1/2, rather than 1/e of its maximum, we can increase the duty cycle and obtain more output energy. The repetition rate of >100Hz is high enough to achieve QCW (quasi-continuous-wave) laser pulses.

© 2017 Optical Society of America

## 1. Introduction

Diode-pumped alkali lasers (DPALs) are a new type of high-power laser system with many advantages. These include high quantum efficiencies (for K, Rb, and Cs are 99.6%, 98.1%, and 95.3%, respectively), the capability to convert low-quality unphased pump light into a coherent laser beam with near diffraction-limited beam quality [1], a narrow linewidth of ~10 GHz, and the nonuse of hazardous expendable chemicals. As a result, DPALs have undergone extensive research and development since the first successful realization of an optically pumped 795 nm Rb vapor laser in 2003 [2], and of a diode-pumped 894 nm Cs vapor laser in 2005 [3]. Several groups have published demonstrations of lasing of various atomic species [2–4], pumping in longitudinal and transverse directions [5,6], and cooling by a flowing procedure [7,8]. Especially, the potential for scaling DPALs to high powers was preliminarily demonstrated in 2012 with an output power exceeding 1 kW [7]. In addition, the steady-state temperature was measured in 2015 [9] and subsequently in 2017 [10], showing a practical way to achieve diagnostics of DPALs. Meanwhile, many theoretical models of DPALs have been established [11–21], of which the focus has recently concentrated on beam size fitting [20,21] and temperature evaluation [16,17,21].

Recently, two teams proposed and demonstrated the real-time temperature measurement methods. Zhdanov et al. employed a camera with a ~1 kHz frame rate to nonperturbatively probe the distortion in the interferometric fringes caused by the variation in the refractive index, and experimentally measured the time evolution of temperatures in the gain medium of a Cs DPAL [22]. Wang et al. used a photo detector with a 14 ns rise time to directly detect the absorption signal of the K atoms added as non-disturbing tracing atoms, and demonstrated a tracing-atom-based absorption spectroscopy method for temperature measurement inside a Rb DPAL cell [23]. The limiting effects revealed by the experiments, such as output power degradation in time, require comprehensive research of the time-dependent characteristics of an alkali vapor laser. Additionally, knowledge of the temperature distribution inside the cell is essential for computation of thermal phase shift, thermal lensing, and thermally induced birefringence. Thus, study of the time evolutions of temperature and output features is very important for high-power DPAL development. In this paper, we propose a computational method to model the time-dependent three-dimensional temperature distribution in a DPAL cell. The axial distribution of beam radius and the radial distribution of intensity are both taken into account. The density distribution of alkali atoms and buffer gases, the collisional broadening rate of D1 and D2 lines, and the relaxation and quenching rates are all temperature-dependent. The rate equations of population densities, the power propagation equations, and the three-dimensional thermal conduction equation are combined to obtain the time evolution of power and temperature of both single-pulse and multi-pulses diode-pumped alkali vapor lasers.

## 2. Description of the model

The temporal and spatial divisions of a vapor cell with radius $R$ and length $L$ are shown in Fig. 1, with each divided volume element having dimensions of $\Delta x\times \Delta y\times \Delta z$. The minimum time difference of the same volume element is $\Delta t$.

#### 2.1. Rate equations and power calculation

The rate equations of population density of energy levels of alkali atoms in a volume element at time $t$ are given by

The ratio of the forward and the reverse spin-orbit relaxation rates must obey the principle of detailed balance ${\gamma}_{23}\left({}_{\text{}}{}^{2}P{}_{1/2\to 3/2}\right)=2{\gamma}_{32}\text{exp}(-\Delta E/kT)$, where ${\gamma}_{32}$, as well as the quenching rates, ${\gamma}_{31}({}_{\text{}}{}^{2}P{}_{3/2}\to {}_{\text{}}{}^{2}S{}_{1/2})$ and ${\gamma}_{21}({}_{\text{}}{}^{2}P{}_{1/2}\to {}_{\text{}}{}^{2}S{}_{1/2})$, are calculated by

where ${\sigma}_{ij}$ is the temperature-dependent collisional cross section for the transition $i\to j$ given by [24], $k$ is Boltzmann’s constant, ${m}_{r}$ is the reduced mass between the alkali atom and the ${\text{CH}}_{4}$ molecule, and ${P}_{C{H}_{4}}$ is the ${\text{CH}}_{4}$ pressure in Torr.To fit the real pump and laser intensity and beam radius inside the cell, ${f}_{p,l}$ and ${w}_{p,l}$ are expressed in the $xy$ plane:

The propagation of pump and laser power in the longitudinal dimensions, modified from [20], can be given by

Appointing an initial value of output laser power as ${P}_{l}\in [0,Pp)$, then the boundary conditions for the two way laser powers are given by ${P}_{l}^{+}\left(0\right)={P}_{l}{t}_{w}{R}_{oc}/\left(1-{R}_{oc}\right)$ and ${P}_{l}^{-}\left(0\right)={P}_{l}/{t}_{w}\left(1-{R}_{oc}\right)$. The solution for the output laser flux is found by iterating on ${P}_{l}$ until ${P}_{l}^{+}\left(L\right){t}_{w}^{2}{t}_{s}^{2}{R}_{l}={P}_{l}^{-}\left(L\right)$ (<1%), where ${R}_{oc}$ is the reflectivity of the output coupler, ${t}_{w}$ is the window transmission, and ${t}_{s}$ is the single-pass scattered transmission that is assumed to be located at the end of the laser reflector with reflectivity ${R}_{l}$.

#### 2.2. 3D thermal conduction and temperature calculation

Combining the thermal conduction equation given by Eq. (56).5) from [25] and the heat source density $\text{\Omega}$, the Tuation of heat transfer is

*C*, $\rho $, and

*K*are the specific heat in J·kg

^{−1}·K

^{−1}, the mass density in kg·m

^{−3}, and the thermal conductivity in W·m

^{−1}·K

^{−1}, respectively. The thermal conductivity of methane can be calculated by ${K}_{CH4}=0.05+0.039/200\times \left(T-400\right)$, derived from the measured data [26]. The appearance of internal currents in the gain medium caused by the absence of mechanical equilibrium due to non-uniform temperature in a gravitational field is known as free convection. For the pulsed laser, the convection speed $\upsilon $ is negligibly small on a time scale of ~ms level during temperature rise. With respect to the fact that gases show isotropic properties, we expand Eq. (10) in rectangular coordinates as

This transient heat conduction formulation includes a general temperature distribution varying both with time and position. The first term in the right-hand-side of Eq. (11) is far smaller than the second term (<0.01%) and hence is negligible. According to Fig. 1(c), we continue our treatment by making a discretization on Eq. (11) using $T(x,y,z,t)\to {T}_{i,j,k}^{t}$ and

Therefore, the momentary three-dimensional heat equation will appear as

*t*($t=0,1,2,\dots $) is an index of the real time.

The volume density of generated heat power $\Omega $ is given by

The product of *C* and $\rho $ can be calculated by

^{−1}·K

^{−1}), derived from the measured data [26]. ${n}_{C{H}_{4}}$ is the number density of ${\text{CH}}_{4}$, and ${N}_{A}$ is the Avogadro constant.

At each *t*, using the current temperature distribution ${T}_{i,j,k}^{t}$ (${T}_{i,j,k}^{0}={T}_{o}$) to solve the rate Eqs. (1)–(3) and the power propagation Eqs. (8)–(9), we can obtain the laser power ${P}_{l}$ and the 3D distribution of heat power density ${\text{\Omega}}_{i,j,k}^{t}$. Substituting ${\text{\Omega}}_{i,j,k}^{t}$ into the heat Eq. (14) obtains the temperature of the next moment ${T}_{i,j,k}^{t+1}$. These iterative processes are executed repeatedly via using multiple nested “for” loops in MATLAB to obtain both the spatial distribution and the temporal evolution of power and temperature.

## 3. Results and discussion

Due to the reason that the response time of a photodiode (for detecting the absorption signal of the K atoms) is usually much faster than CMOS or CCD sensors (for acquiring the interferometric fringes), and the error value is lower, the tracing-atom-based absorption spectroscopy method for temperature measurement is much faster and more accurate than the interference measuring method. Therefore, we applied our model to the recently-reported pulsed Rb-${\text{CH}}_{4}$ DPAL of which the real-time temperature was obtained by the absorption spectroscopy method [23]. The main experimental parameters are listed in Table 1.

It should be noted that because the power inside the beam waist was 86% of the total power, i.e., ${P}_{waist}/{P}_{total}~86\text{\%}$ [23], that ${c}_{2}=\sqrt{2}$. The alkali cell contained Rb for lasing and ${\text{CH}}_{4}$ for pressure broadening and relaxation, and K as tracing atoms. However, the number density of Rb and K ($~{10}^{19}{\text{m}}^{-3}$) is much smaller than that of ${\text{CH}}_{4}$ ($~{10}^{25}{\text{m}}^{-3}$), and hence the average thermal conductivity and the average heat capacity of the mixture are completely dominated by ${\text{CH}}_{4}$. The probe laser, precisely tuned to the central wavelength of the K D1 line so as to be absorbed by K atoms, was a single frequency DBR diode laser with a narrow linewidth of ~1 MHz, rather less than the alkali D1 line of ~10 GHz. The photo detector had a sufficiently large active area of 13 ${\text{mm}}^{2}$ to receive the total probe power, and had a sufficiently fast rise time of 14 ns to realize the real-time measurement.

The average pump (or laser) power discussed in [23] is obtained by dividing the pump pulse energy ${E}_{p}$ by the pulse duration ${\tau}_{p}$: ${\overline{P}}_{p}={E}_{p}/{\tau}_{p}$. The incident spectrally resolved pump power at $z=0$ and time *t* can be given by

*c*is the speed of light, $\text{\Delta}{\lambda}_{p}$ and $\text{\Delta}{v}_{p}$ are the linewidths (FWHM) of the pump light, and ${\lambda}_{p}$ is its central wavelength. The pump pulse signal ${S}_{p}(t)$ is given by the experiment. Under the experimental condition of ${\overline{P}}_{p}=92\text{W}$ and ${\tau}_{p}=2\text{ms}$ (FWHM), we can transform ${S}_{p}(t)$ into the pump power and calculate the temperature and power evolution as shown in Fig. 2. It can be seen that the simulation results are in good agreement with the experiment. Both laser power (signal) and temperature increase with time until they reach the maximum values, and then subsequently decrease. The reason why the laser power is not maximal at the beginning of the temperature growth, where the thermal effect appears to be the weakest, is that the pump power requires a rising process. Before the pump power reaches its maximum, because the heat deposited from spin-orbit relaxation and quenching is too large, the laser power starts to decrease linearly (in fact it should decrease exponentially if the pump signal is a square wave [27], but the still-increasing pump power has compensated part of the downward trend). The average laser power calculated by

At time = 3 ms after pump light is on, the three-dimensional distribution of temperature in the cell is shown in Fig. 3.

The experimental and simulated results of temperature rise and average laser power under different pulse durations are shown in Fig. 4(a) with ${T}_{o}=408\text{K}$ and $\overline{{P}_{p}}=85.3\text{W}$. We can see that when the pump pulse width varies from 1 ms to 5 ms, the temperature rise increases from ~130 K to ~300 K owing to the longer heat deposition time, while the average laser power decreases from 10.4 W to 9.1 W because of a more serious drop of the instant laser power induced by more accumulated heat, which can be seen in Fig. 4(b). The experimental pump pulse signal is fitted by a trapezoid function. At a pulse duration of 5 ms, the instant laser power decreases exponentially, drops to below half of its maximum before the pump light is turned off, which results in a lower average laser power than those of shorter pulse duration.

It is well known that high-power lasers operating in CW mode will suffer from power degradation owing to thermal effects in the gain medium, hence a QCW (quasi-continuous-wave) mode occurs, which has various important applications in material processing, communications, medicine, and science. In order to shorten the time required by heat dissipation, we investigate the time evolution of laser power and of temperature rise in multiple-pulse DPALs. In Fig. 5, the pump pulse is assumed to have a nearly rectangular shape [27] with an average power of 90 W. When the laser power reduces from its maximal to 1/2 of this value, the pump light is turned off; and when the temperature rise decays to 1/e (a) or 1/2 (b) of its maximum (the corresponding thermal relaxation times are indicated as ${t}_{e}$ and ${t}_{2}$), it is turned on. The average laser powers achieved in Figs. 5(a) and 5(b) are 11.2 W and 10.7 W for a single pulse (in fact they should be higher in the first pulse). The maximal temperature rises are both 224 K, and the thermal relaxation times ${t}_{e}=~5.3\text{ms}$ and ${t}_{2}=~3.8\text{ms}$. QCW operation requires a short break time between two pump pulses and more excited pulses to achieve higher output energy. At a time range of 22 ms, for ${t}_{e}$ there are only 4 pulses, the duty factor is 38% and the total output energy is 84 mJ; for ${t}_{2}$ there are 5 pulses, the duty factor is 44% and the total output power is 91 mJ. This means that we can shorten the thermal relaxation time to increase the duty cycle and obtain higher output energy. The repetition rates for Figs. 5(a) and 5(b) exceeded 100 Hz, which is high enough to achieve QCW laser pulses.

However, unlike the rectangular pump pulse used in [27], the pulse measured by [23] had a rise and fall time, which means that when the pump light was turned on, it took some time to increase, and when it was turned off, it also took some time to decrease. Therefore, we assume the pump signal to be trapezoidal with a rise and fall time of 0.5 ms, as shown in Fig. 6. The average laser powers of 9.3 W and 8.9 W achieved respectively in Figs. 6(a) and 6(b) are smaller than those of Fig. 5 owing to the higher temperature rise of 260 K. The descent of pump power resists the temperature fall, which is in agreement with experiment that even during the fast drop of pump signal, the mixed gases still experience elevated temperature. The thermal relaxation times ${t}_{e}$ and ${t}_{2}$ are 4.2 ms and 3.2 ms, respectively, which are smaller than that of Fig. 5. This is because higher temperature will result in faster heat dissipation; hence, the time the temperature takes to fall back to a certain ratio of its maximum is shorter. However, at the given time range, owing to the rising and falling processes of the pump pulse, the number of pulses for ${t}_{e}$ and for ${t}_{2}$ are 3 and 4, respectively. The total output energy of 81 mJ for Fig. 6(a) is less than that for Fig. 5(a) owing to the fewer pulses. Nevertheless, the energy for Fig. 6(b) is 100 mJ, which is even higher than Fig. 5(b). This is because, in spite of less pulses, the DPAL is still lasing during the rising and falling processes of pump power, and the pulse width is 40% larger than that in Fig. 5. However, the repetition rates of >100 Hz in Fig. 6 are still high enough for realization of QCW operation.

## 4. Conclusion

A spatiotemporal model for diode-pumped alkali vapor lasers is established by combining the rate equations, the power propagation equations, and the transient thermal conduction equation. The model shows good agreement between the calculated and measured time evolution of laser power and temperature when applied to a pulsed Rb-CH_{4} DPAL of which the temperature was obtained by using photodiode to detect the absorption signal of the K atoms added as non-disturbing tracing atoms. When the pump pulse width increases from 1 ms to 5 ms, the temperature rise increases by ~170 K, while the average laser power decreases by about 1.3 W owing to the larger heat deposition. The time-dependent output feature and temperature rise of the multi-pulses DPALs are also calculated, showing that at a time range of 22 ms, if we turn on the pump light when the temperature rise decays to half of its maximum, rather than 1/e of its maximum or lower, we are able to achieve shorter thermal relaxation time, higher duty cycle, and higher output energy. The repetition rate we investigate is higher than 100 Hz, which is enough to achieve QCW laser pulses.

## Funding

Zhejiang Provincial Natural Science Foundation (LY14A040005).

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