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Abstract
Investigation of the kinetic principle of a diode-pumped alkali vapor laser (DPAL) is key to achieve multifunctional DPALs. In this work, we propose a spatiotemporal model, which combines the time-dependent rate equations of population densities and propagation equations of energies to study the dynamic processes from turn-on to steady-state in DPALs. Time evolution of population densities and pump and laser intensity are resolved on a picosecond time scale to study the build-up process of laser oscillations. For nanosecond-pulse pumping, we obtain a laser pulse of 1.6 ns and a delay time of 2.6 ns at an incident pulse width of 2 ns. This pulse can be stretched by increasing the pump pulse width and delayed by applying a more extended cavity. For CW operation, spiking and relaxation oscillations resulting from a dynamic balance of gain and losses are demonstrated to be much faster than other types of lasers.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Diode-pumped alkali vapor lasers (DPALs) have many advantages, including high power, good beam quality, and no hazardous expendable chemicals, which can be potentially used in laser cooling, material processing, directional energy transmission, etc. Worldwide groups have published DPAL demonstrations with various atomic species [1–3], different pumping configurations [4–7], flowing procedure [8,9], and temperature diagnostics [10–12]. Among these demonstrations, time-related researches including CW K DPAL, pulsed K DPAL [13,14], and static and flowing Cs DPAL [15], play an essential role in revealing millisecond power drops caused by thermal accumulation and window contamination. Meanwhile, many researchers have made tremendous inroads toward understanding the mechanism of DPALs by proposing promising models that take into account the kinetic and fluid dynamic processes in the medium [16–23]. However, accurate simulation of the time-dependent cavity features of a DPAL remains a significant challenge. Our previous work has demonstrated the time evolution of power and temperature in single-pulse and multi-pulse DPALs on a millisecond time scale and reproduced the reported experimental results [24]. Nevertheless, for nanosecond-pulse pumping configurations, which have broad applications in laser engraving, drilling, photoablation, etc., the rate equations cannot be treated as steady-state. The laser signal in the resonant cavity builds up exponentially with time before the electron transitions reach dynamic equilibrium. Thus, sub-nanosecond-resolved rates equations should be taken into consideration to solve the dynamic processes. Besides, although a time-dependent finite-volume model named “BLAZE-V” was used to perform the multi-dimensional computation of an exciplex pumped alkali laser (XPAL) system [25], it required specific codes to run time-consuming simulations. Time evolution of various parameters of nanosecond-pulse DPALs on a picosecond time scale, which can deeply reveal laser oscillation establishment, remains insufficient in research.
Furthermore, CW lasers inevitably undergo spiking and relaxation oscillation processes before the intensity reaches a steady-state oscillation level [26]. These processes generally occur on a microsecond time scale but maybe faster in gas lasers, which creates a demand for a high-temporal-resolution simulation. Therefore, we propose a picosecond-resolved DPAL model, which couples the time-dependent rate equations of populations and propagation equations of the pump and laser energy by a spatiotemporal iterative algorithm, to investigate the cavity features and output performance of nanosecond-pulse and CW DPALs. Staring from spontaneous emission with a few photons lying within the effective frequency bandwidth, the signal in the resonant cavity grows exponentially until the laser intensity circulating inside the cavity reaches a pulsed or steady-state output, generating billions of photons. For CW operation, the coupled nonlinear rate-equations exhibit fluctuation due to the dynamic balance between gain and losses, and the laser intensity will experience spiking and relaxation oscillations in the evolution toward a steady-state level.
2. Description of the model
The schematic diagram of the studied DPAL is shown in Fig. 1. The pump beam reflected by a polarizing beam splitter (PBS) enters the cylindrical cell, which consists of a mixture of alkali vapor (Rb) and buffer gases ($\textrm{C}{\textrm{H}_4}$). The laser operates at a temperature of T. The laser photons generated by spontaneous emission travel between the output coupler (OC) and high reflector (HR) to be multiplied.
Fig. 1. Schematic diagram of a DPAL. The pump and laser beams are indicated by red and pink colors, respectively. ${L_C}$: cell length, ${L_O}$: the distance between OC and the left side of the cell, ${L_H}$: the distance between HR and the right side of the cell.
The optical axis (the $z$-axis) is divided into small segments with a length of $\Delta z = {L_C}/\textrm{n}$, where n is the divided number of the cell. The simulated time interval $\Delta t = \Delta z/c$, where c is light speed. At every divided volume element, the time-resolved differential rate-equations (TDRs) of population density of energy levels of alkali atoms are given by
The pump, laser and fluorescent powers between two divided small segments ($z$ and $z + \mathrm{\Delta}z$) follow the Beer-Lambert law:
The incident pump power can be expressed as
The generated fluorescence/laser will be reflected by both HR and OC and hence $P_l^ + $ and $P_l^ - $ will convert to each other continually:
A flowchart of the total spatiotemporal iteration algorithm is given in Fig. 2.
3. Results and discussion
Nanosecond-pulse configuration is one of the promising developments for DPALs. However, laser oscillation is still building up on this time scale according to most cavity lengths (about 2 ns optical path). This brings demand for time-dependent rate equations and photon equations to simulate the characteristic variation of the laser. Thus, we first applied our model to a pulsed Rb-$\textrm{C}{\textrm{H}_4}$ DPAL of which the parameters are given as the operating temperature $T$ = 408 K, the partial pressure of methane at room temperature ${P_{C{H_4}}}$ = 500 Torr, the window transmission $\eta $ = 0.96, the reflectivity of output coupler ${R_{OC}}$ = 0.5, the reflectivity of high reflector ${R_{HR}}$ = 0.96, the length of cell ${L_C}$ = 5 cm, the distance between the left cell window and the output coupler ${L_O}$ = 30 cm, and the distance between the right cell window and the high reflector ${L_H}$ = 20 cm. The spectral width of pump light $\mathrm{\Delta}{\lambda _p}$ = 0.2 nm [29], and the temporal pulse width of pump light ${\tau _p}$ = 2 ns. The 33-ps-resolved time evolution of pump and laser intensities, as well as the population densities, are shown in Fig. 3.
Fig. 3. Time evolution of pump and laser intensities and average population densities with different pump pulse energies. (a) and (b) ${E_p}$ = 10 µJ, (c) and (d) ${E_p}$ = 1 mJ.
The laser pulse exhibits a Gaussian-like profile at a low incident energy of 10 µJ. For a pump pulse width of ${\tau _p}$ = 2 ns, the time shift between the peak of the pump and laser pulse, ${t_s}$ = 2.6 ns. This shift is related to the cavity length that delays the fluorescence to form a laser resonance. The laser pulse width, ${\tau _l}$, is calculated to be 1.6 ns. The time evolution of the various population densities of energy levels (averaging over $z$) is presented in Fig. 3(b). One can see that after the incidence of the pump pulse, ${n_1}$ decreases due to the pump absorption while ${n_2}$ and ${n_3}$ increase accordingly. They reach an extremum value just before the generation of the laser output (gray line). The maximum drop of ${n_1}$ can reach approximately 36% of its peak value, and the donated population contributes 52% and 48% to ${n_2}$ and ${n_3}$, respectively, to build up the laser oscillation. After this drop (rise), the concentrations will return to their initial level tardily. For ${E_p}$ = 10 µJ and ${\tau _p}$ = 2 ns, the recovery time is approximately 60 ns. When increasing the applied pump pulse energy to a higher level, such as 1 mJ, the nonlinear behavior of the atomic reaction becomes complicated. Multiple irregular-shaped laser pulses inside an envelope (dashed line) are obtained (Fig. 3(c)). The wrapped profile exhibits a long duration of 12 ns, whereas the split pulses maintain approximately 1.6 ns in width. These separated oscillations may be related to or enhance Rabi splitting when a robust signal is applied to atomic transitions [30,31]. The lower ground state ${n_1}$ and upper laser energy level ${n_2}$ reach a high inversion, ${n_2} - {n_1}$, of $0.4 \times {10^{19}}\cdot {\textrm{m}^{ - 3}}$ on average (Fig. 3(d)) and $1 \times {10^{19}}\cdot {\textrm{m}^{ - 3}}$ in maximum. When the respective statistical weights of the energy levels are taken into consideration, the maximum population inversion reaches 22% of the total density. This inversion does not form an output laser pulse instantaneously because of an oscillation delay of approximately 2.2 ns exhibiting between the population inversion and oscillation buildup.
We next investigated the reason for the delay of the laser output and the stretch of the laser pulse by calculating the dependencies of pulse shift ${t_s}$ and laser pulse width ${\tau _l}$ on pump pulse width ${\tau _p}$ and cavity length $L = {L_C} + {L_O} + {L_H}$ as presented in Fig. 4(a). The relationship between ${t_s}$ and ${\tau _p}$ is non-monotonic, exhibiting an extreme value of ${t_s}$ = 4.8 ns at ${\tau _p}$ = 20 ns, followed by a small decline, and then a linear rise. In the meanwhile, ${\tau _l}$ increases monotonically with ${\tau _p}$ from 1.6 ns at ${\tau _p}$ = 2 ns to 75 ns at ${\tau _p}$ = 100 ns, demonstrating that a longer pump pulse can stretch the laser pulse. If a CW pump light (i.e., infinite ${\tau _p}$) is applied, the output ${P_l}$ will experience relaxation oscillations to reach a steady-state level, which is to be discussed next. For the maximum simulated ${\tau _p}$, ${t_s}$ is calculated to be approximately 7.2 ns. Such long pump pulse width and short pulse shift result in an envelope of the pump profile over the laser profile (Fig. 4(b)), demonstrating that the laser finishes output ahead of the complete injection of the pump pulse, which is consistent with the result given in XPAL [25]. However, a more extended cavity can be applied to separate these two pulses. We investigated the influence of the cavity length L on ${t_s}$ and ${\tau _l}$ as given in Fig. 4(c). One can see that ${t_s}$ increases linearly from 2 ns at $L$ = 40 cm to 4 ns at $L$ = 100 cm, demonstrating that a more extended cavity can delay the laser output. This is comprehensible because it takes a longer time for the laser light to travel circularly in a longer cavity. The intersection of the two lines (${t_s} = {\tau _p} + {\tau _l} = $ 3.6 ns) indicates a cavity length of 87 cm to separate the pump and laser pulses efficiently (Fig. 4(d)), whereas for the higher ${\tau _p}$ = 100 ns, a super long L exceeding 30 m should be applied. In this case, separation using PBS, instead of using an unpractical long cavity, is recommended.
Fig. 4. (a) Dependence of pulse shift and laser pulse width on pump pulse width with ${E_p}$ = 10 µJ. (b) Time evolution of pump and laser intensity with ${\tau _p}$ = 100 ns. (c) Dependence of pulse shift and total pulse width (${\tau _p}$+${\tau _l}$) on cavity length with ${\tau _p}$ = 2 ns. (d) Time evolution of pump and laser intensity with ${\tau _p}$ = 2 ns and $L$ = 87 cm.
When a DPAL operates at CW mode with a pump rise time of 2 ns, the laser signal in the resonant cavity follows a 2.6-ns turn-on of laser gain. It increases exponentially and oscillatingly with time as shown in Fig. 5. The initial faint fluorescent level at 1.6 ns is 8 photons per cubic meter generated by pump absorption and spontaneous emission in the medium; however, after 350 ns the accumulated number of laser photons can reach a steady-state value of approximately 4.5×1018 m-3 due to stimulated emission. The steady-state value corresponds to the resonant level when the laser gain is saturated enough to equal the sum of internal loss and output coupling loss [27]. Nevertheless, before the stable output, the laser intensity will experience spiking (discrete, sharp, and large-amplitude pulses, which usually occur during the initial startup of lasers) and relaxation oscillations (small amplitude, quasi-sinusoidal, and exponential decay) due to the dynamic balance of gain and losses. Spiking and relaxation oscillations are characteristic phenomena of most solid-state lasers, semiconductor lasers, and some other laser systems [26,27], but are hardly observed in gas lasers. This is because the upper-state lifetime is so short as compared to the decay time of the resonant cavity that the phenomena occur within 1 microsecond, much faster than the processes in solid-state lasers. For instance, the spiking we predict exhibits 27 ns before the laser intensity reaches the maximum value, and subsequently followed by the attenuated relaxation oscillations which last approximately 300 ns until the instantaneous laser intensity approaches a roughly steady-state oscillation level. The oscillation frequency of 0.25 GHz is similar to the inverse period of the spikes, and the width of the quasi-sinusoidal pulses of 2 ns (inset of Fig. 5) is much shorter than that of solid-state lasers (microseconds). The final optical-to-optical efficiency is calculated to be approximately 25%, more significant than the measured value of the recently-reported Rb-$\textrm{C}{\textrm{H}_4}$ DPAL [29].
Fig. 5. Time evolution of laser intensity when applying a CW pump light with ${E_p}$ = 100 µJ (energy variation resulting in the spiking and relaxation oscillations can be seen in Visualization 1).
4. Conclusion
A spatiotemporal model for a nanosecond and CW DPALs is established by combining the transient rate equations of population densities and the propagation equations of pump and laser energy. Time evolutions of population densities and pump and laser intensity are computed on a picosecond time scale to study the build-up process of laser oscillation. The results reveal the influence of pump energy, pump pulse width and cavity length on pump-to-laser pulse shift and laser pulse width under nanosecond-pulse pumping, and demonstrate the fast spiking and relaxation oscillations in CW DPALs and the difference of these processes between this type of laser and other lasers. Additionally, this basic model can be integrated with other complicated processes including mode competition effect (longitudinal and transverse modes), thermal effect (millisecond-scale, investigated by our previous model [24]) and nonlinear effects (e.g. Kerr effect and harmonic generations), which holds promise for diverse applications in high-energy pulse lasers. To conclude, the study of the picosecond time evolution of DPALs deepens the understanding of the formation mechanism of relaxation oscillation process and laser oscillation buildup. It can beF used for the design of promising nanosecond-pulse DPALs with expected pulse shift, width, and energy.
Funding
National Natural Science Foundation of China (61525503, 61620106016, 61835009, 81727804); Postdoctoral Research Foundation of China (2019M653000); Department of Education of Guangdong Province (2016KCXTD007).
Disclosures
The authors declare no conflicts of interest.
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