1. Introduction

Mid-infrared fiber lasers have been followed with interest owing to the growing demands in the fields of biomedicine, industry, research, and defense accompanied with the versatility, compactness, and flexibility of fiber lasers [1,2]. Although a variety of achievements in rare-earths doped mid-infrared fiber lasers operating both in CW and pulsed region have been witnessed [3–12], the potential mid-infrared output of solid-core fiber lasers is constrained by absorption in silica and the poor power handling capabilities of soft glass host materials. Optically pumped gas lasers are an alternative technology to emit abundant and wide mid-infrared region by vibrational-rotational transitions. The notable drawback of short interaction length in gas lasers can be improved by the ideal interaction environment of the HCF. Thus gas-filled HCF lasers where the gas gain medium is contained in the hollow core region emerge as the times require. Much of the success is owed to the development of silica-based HCFs with low attenuation at mid-infrared wavelengths [13–17]. In addition, the gas-filled HCF lasers exhibit attractive properties including good heat mitigation, high damage thresholds, and weaker nonlinearity, enabling simple and portable mid-infrared emission sources.

The initial experimental demonstration of gas-filled HCF lasers based on population inversion shows 3.12 µm and 3.16 µm nanosecond laser pulses in the single-pass amplified spontaneous emission regime [18], opening a new era in gas and fiber laser research. Since then, a variety of suitable gas gain media, including I₂, CO, CO₂, HCN, N₂O, and HBr, are utilized to fill into the HCF to extend a wide wavelength range, from near-infrared to mid-infrared region [19–24]. The first demonstration of CW gas laser in HCF is realized with molecular iodine in 1280∼1340 nm region. Added with a feedback HCF, the acetylene-filled HCF lasers with cavity structure are also achieved either in CW or synchronously pumped configuration with reduced pump threshold [21]. The mid-infrared laser beyond 4 µm which is difficult to achieve in rare-earths doped fiber lasers due to limited rare-earth ions species and fiber failure, can be available in N₂O, CO₂, and HBr gas molecules filled in HCF [22–24]. And both CW and pulsed laser operation beyond 4 µm is included, extending the mid-infrared wavelength range of fiber lasers outstandingly. Although a series of experimental demonstrations are of rapid development, the thorough mechanism of lasing and models of gas-filled HCF laser performance need to be investigated further. Currently, only a numerical model is built to understand the primary physical processes that affect the output and efficiency of pulsed gas amplifiers in acetylene-filled HCF [25]. This time-dependent model is developed for low-power nanosecond scale pulsed operation, showing a challenging and nontrivial interplay between the gas and light propagation physical processes. Recently, we have also developed a theoretical model that simplifies complex rotational relaxation owing to a low rotational relaxation rate and such a simplified model shows agreement with the HBr-filled HCF laser operating in CW region [24] but not applicable to pulsed region. In addition, the phenomena of rotational relaxation related to energy transfer which are helpful in exploring the molecular dynamics are not reported in gas-filled HCF lasers until now.

In this work, we have extended the previous four-energy level model of HBr-filled HCF laser from CW pumping to pulsed pumping. The characteristics of simulated population density and power distribution along HCF with CW pumping are investigated. The simulated evolutions of the pump pulse and laser pulse at different positions along the HCF are studied. In addition, through the experiment, the rotational relaxation phenomena that the populations of the upper level redistribute to other rotational levels are recorded, showing the type of HBr isotope pumped, changing the pump power and tuning pump wavelength can influence the rotational relaxation under the CW pumping condition.

2. HBr molecule and numerical model description

A typical HBr-filled HCF laser configuration is sketched in Fig. 1(a). The CW or pulsed pump light is coupled into the HCF through optical elements like mirrors and lenses. The HCF is enclosed at each end using a gas cell that also provides optical access through a window. The gas cells and HCF are evacuated and then filled with low pressure HBr. Since the gain is large enough to produce laser-like output, lasing starts from spontaneous emission in a single-pass configuration without an external cavity providing feedback [22]. At the output end of the HCF, the mid-infrared laser is separated from the residual pump through dichroic mirrors.

 

Fig. 1. (a) A typical experimental setup of the HBr-filled HCF gas laser without feedback and the pump light can be CW or pulsed. (b) Transition processes of rotational-vibrational energy level diagram of the HBr molecules, shown in the left and simplified transition model just considering one of the absorption lines pumping, shown in the right. The ΔJ=+1 and ΔJ = -1 transitions are plotted in light blue and light green.

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As a diatomic molecule, the HBr molecule is known to have only one vibrational normal mode denoted by v. The vibrational normal mode v can only take a positive integer value from 0 and the group of horizontal lines for each vibrational state indicates the manifold of rotational states, expressed by J, as shown in the left of Fig. 1(b). The molecular rotation is independent from the vibration and the rotational level is like a ladder with increasing energy level spacing. The HBr molecules can be excited from the ground vibrational state v = 0 to the v = 2 vibrational state, namely the first overtone absorption. Then emission occurs on P-branch and R-branch transitions from the v = 2 to v = 1 vibrational state. According to selection rules, the R-branch (P-branch) transitions refer to as ΔJ=+1 (ΔJ = -1). The R-branch and P-branch transitions are usually labeled as R(J) and P(J) where J is the rotational quantum number of the lower state, as illustrated in the left of Fig. 1(b). Although the HBr molecules have two isotopes, namely H⁷⁹Br and H⁸¹Br with almost equal natural abundance, the characteristics of the two isotopes are similar, just with 50 GHz energy level mismatch [26]. The process of generating laser from any absorption line with vibration state v = 0→v = 2 is considered and a four-energy level system is established, as plotted on the right of Fig. 1(b). Among them, E₀ level is the pumping lower level on the vibration state v = 0 and E₂ level is the pumping upper level on the vibration state v = 2, which is also the lasing upper level. E_1P and E_1R levels are the lasing lower levels in the vibration state v = 1, corresponding to the P-branch transition and R-branch transition respectively. N₀, N₂, N_1P, and N_1R are the population density at the corresponding energy level respectively.

The total population density of all energy levels remains unchanged, set as N_total:

Spontaneous emission, stimulated emission, and stimulated absorption are considered in energy level transition processes. In addition, owing to the small rate, the rotational relaxation is not included particularly but considered together with coefficients of non-radiative transitions τ_{1R_0}, τ_{1P_0}, and τ_{2_0}, which represent the energy level lifetime from E_1R, E_1P, and E₂ to E₀ by non-radiative transitions. A_{2_0} is the Einstein A coefficient from E₂ to E₀, representing the probability of spontaneous transition. Similarly, A_{2_1P} and A_{2_1R} are the spontaneous transition probability from E₂ to E_1P, and E_1R respectively, and the reciprocal of Einstein A coefficient also implies the energy level lifetime determined by spontaneous emission. The specific Einstein A coefficient can be achieved from HITRAN database [27] and only a little difference exists between the H⁷⁹Br and H⁸¹Br. σ_{2_0} and σ_{0_2} are the emission cross section and absorption cross section between the energy level E₂ and E₀. σ_{2_1P} and σ_{1P_2} are the emission cross section and absorption cross section between the energy levels E₂ and E_1P. σ_{2_1R} and σ_{1R_2} are the emission cross section and absorption cross section between the energy levels E₂ and E_1R. According to Fig. 1(b), the population density at each energy level with respect to time can be obtained:

When the pump light is coupled into the HCF, it propagates along the HCF and interacts with HBr. The pump light intensity I_p, the P-branch laser light intensity I_sP and the R-branch laser light intensity I_sR will change along the HCF, affected by the gain, loss and spontaneous emission. Among them, the gain g has a relationship with the emission cross-section and the inversion population density ΔN (taking the transition between E₂ and E₀ as an example): g_{2_0}=σ_{2_0}ΔN =σ_{2_0}(N₂-$\frac{{{{f}_\textrm{2}}}}{{{{f}_\textrm{0}}}}$N₀). Where f₂ and f₀ are the degeneracy degrees corresponding to the E₂ and E₀. Then the variation of pump light intensity and laser light intensity along HCF can be written as:

The direction along the HCF is z direction and the HCF is divided up into infinitesimal elements with length dz. α_p, α_sP, and α_sR are the transmission losses of the pump light, P-branch laser and R-branch laser in the HCF. Ω is a factor describing only a fraction of spontaneous emission that will transmit along the HCF direction, acting as the initial seed laser. f_1P and f_1R are the degeneracy degrees corresponding to the E_1P and E_1R. Under room temperature 25°C, HBr pressure 1 mabr and pumped by R(2) absorption line, the numerical values of associated parameters are listed in Table 1.

Table 1. The values of mentioned parameters in the numerical model

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3. Simulation results

3.1 CW pumping

Under the condition of CW pumping, the population density in each energy level is in steady state. In the Eqs. (2)–(4), the dN₀/dt, dN_1P/dt, and dN_1R/dt can be considered equal to zero. The initial condition is the incident CW pump power given at the initial position of the HCF. The I_sP and the I_sR are zero and all the population density is concentrated at the energy level E₀. Using Eqs. (1)–(4) and the steady state condition, the density population of each energy level at the initial position can be solved. By Eqs. (5)–(7) and Euler's method, the I_p, I_sP, and I_sR at the next micro element position are solved. Then the population density distribution of each energy level can also be solved. The cyclic calculation continues until the I_p, I_sP, and I_sR of each micro element position of HCF and the population density distribution of each energy level are solved.

Figure 2 plots the distributions of population density and the power along the HCF under different gas pressures, when pumped by R(2) absorption line of H⁷⁹Br isotope. Under low pressure of 1 mbar in Fig. 2(a), the excited state populations do not return to the ground state E₀ completely. Thus N₀ is always in an increasing state, and the pump power is not fully absorbed. There is still residual pump power at the end of the HCF, and the laser power is also always in an increasing state, indicating that the low pressure HBr cannot absorb the pump power effectively under the current HCF length. With the transmission of the pump along the HCF, the populations in the lasing upper level E₂ gradually transit to the lasing lower levels E_1P and E_1R through stimulated emission. Therefore, N₂ gradually decreases and N_1P and N_1R gradually increase. Owing to the larger Einstein A coefficient and larger emission cross section [27], the growth rate of N_1P is faster than that of N_1R and the threshold of P-branch laser is lower. In addition, it can be seen from Fig. 2 that the decrease of N₂ and the increase of N_1P occur at the same time. N_1R starts to be accumulated after N_1P has been accumulated for a period of time, and the corresponding P-branch laser is generated earlier than R-branch laser. The populations that transition to the lower lasing level E_1P and E_1R will return to the ground state E₀ through non-radiative transitions with an unchanged rate. With the gradual decrease of N₂, the populations from the E₂ to E_1P and E_1R decrease, while the rate of populations from the E_1P and E_1R to E₀ through non-radiative is unchanged. Therefore, the accumulated N_1P and N_1R will gradually decrease after reaching the peak. Since the P-branch transition has a larger emission cross section, the population N_1P reaches the peak earlier than N_1R with higher peak. From the pump power and laser power distribution, the pump power has been absorbed along the HCF and continuously attenuated, while the laser power gradually increases and the P-branch laser power is larger than that of the R-branch laser due to a larger emission cross-section. The increased HBr pressure leads to increased pump absorption. At 5 mbar and 10 mbar in Fig. 2(b) and Fig. 2(c), it can be seen that the pump power is completely absorbed at the position of 2.5 m and 1.5 m of the HCF respectively. And the corresponding populations N₀, N_1P, and N_1R have returned to the ground state E₀. N₀ is close to the total population density N_total. After this length of HCF, the transmission of P-branch and R-branch lasers is only affected by the transmission loss of HCF, and the laser power gradually decreases. Therefore, it can be seen that there is an optimum gas pressure for the HCF with a certain length. Under the optimum gas pressure, the pump power is completely absorbed just at the end of the HCF.

 

Fig. 2. The simulated population density and power distribution along HCF under (a) 1 mbar, (b) 5 mbar and (c) 10 mbar pumped by R(2) absorption line. The incident pump power is set to 10 W and the length of HCF is set to 5 m.

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The output residual pump power and laser power as a function of the incident pump power under different pressures pumped by R(2) absorption line is plotted in Fig. 3. For the case of 1 mbar of gas pressure in Fig. 3(a), there is about 2.6 W residual pump power remained with 10 W incident pump power. With the increase of the HBr pressure in Fig. 3(b) and Fig. 3(c), the pump power is completely absorbed at 5 mbar pressure and above, showing the obvious influence of HBr pressure on the pump power absorption and the output laser power. In addition, the laser threshold also increases with the increase of HBr pressure owing to that intensified molecular collisions lead to the reduction of the non-radiative transition life time of E₂. The threshold of R-branch laser is much higher than that of P-branch laser. It can be predicted that when the HBr pressure further rises, the pump power cannot reach the threshold of R-branch laser, and only P-branch laser is included in the output laser. Therefore, the pure output spectra with single spectral line can be achieved by controlling the pump power and pressure. Although the increase of HBr pressure intensifies the collision and increases the loss, it also enhances absorption. More populations participate in the generation of lasers, increasing the gain. Therefore, the output laser power is the result of the joint determination of these two factors. When the HBr pressure increases from a lower level, the gain takes the dominant factor. The total laser output power increases from ∼1.4 W in Fig. 3(a) to ∼3 W in Fig. 3(b). After reaching the optimum HBr pressure, the loss takes the dominant factor with increasing pressure. The laser power decreases to ∼2.8 W in Fig. 3(c) and it can be predicted that with the further increase of HBr pressure, the laser output power will decrease. Additionally, in order to compare the theoretical and experimental results, the green and purple star dots are plotted in Fig. 3 as the measured residual pump power and total output laser power respectively. It can be seen that the measured total output laser power is much lower than the corresponding simulated results. Accordingly, the measured residual pump power is higher than the simulated results. In the numerical model, the loss caused by the complex molecular collisions is simply considered together with coefficients of non-radiative transitions. We think the loss is underestimated, causing the obvious discrepancy between simulated data and experimental results. In addition, the HBr pressure in the experiment is considered to be the value displayed by the pressure gauge of experimental setup (not shown in Fig. 1(a)). However, the actual HBr pressure in HCF is estimated to be lower than the value displayed by the pressure gauge.

 

Fig. 3. The simulated output laser power and residual pump power as a function of the incident pump power under (a) 1 mbar, (b) 5 mbar, and (c) 10 mbar. The length of HCF is set to 5 m. The plotted star dots are corresponding measured experimental data for comparison.

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3.2 Pulsed pumping

The basic model of pulsed pumping is consistent with that of CW pumping, both of which are four-energy level systems as shown in Fig. 1(b), but there are also obvious differences between them. For pulsed pumping, the laser generation process ends before reaching equilibrium due to the short pulse duration. The populations and photons generated in each energy level are in sharp change, and the system is in an unsteady state. Therefore, the steady state assumption of dN₀/dt = 0, dN_1P/dt = 0, and dN_1R/dt =0 for CW pumping is no longer applicable, and the differential Eqs. (1)–(7) can be solved simultaneously using the finite-difference time-domain method [28]. The incident pump pulse is set in Gaussian shape, decomposed into a large number of time elements dt. The initial condition is similar to that of CW pumping. The intensity of the incident pump pulse time element is set as the initial pump intensity. Similarly, The I_sP and I_sR are initially zero and all the populations are initially concentrated on the energy level E₀. The central difference method is used to replace the differential expression on the left side of Eqs. (2)–(7). Then the expressions of the I_p, I_sP, I_sR, N₀, N_1P, N_1R, and N₂ are inversely solved along each position of HCF. Among them, the I_p, I_sP, and I_sR at the end of HCF are saved as standby. At this time, the distribution of each position of the I_p, I_sP, I_sR, N₀, N_1P, N_1R, and N₂ under the condition of the time element is solved. With the evolution of time, the results of the next time element are gradually solved until all time elements within the set pump pulse range are solved. The saved I_p, I_sP, and I_sR at the end of HCF under each time element are accumulated to form the output pump pulse and generated laser pulse profile. The concrete pulse profile evolutions of the residual pump and generated laser at different positions are shown in Fig. 4. Similar to CW pumping, the HBr pressure is set at 1 mbar and the pump wavelength is set as R(2) absorption line of H⁷⁹Br isotope. In addition, the average power of the pump light is set at 10 W with a repetition rate of 1 MHz and a pulsed duration of 20 ns. At the position of 0 m, no laser is generated and the pulse profile plotted with black curves is namely the set pump pulse. The peak position of the set pump pulse is at t = 30 ns, and the corresponding peak power is about 470 W. It should be pointed out that the black curves shown in Fig. 4 are the pump pulse profile propagating to different positions only considering the transmission loss of HCF, which is convenient for comparison with other pulses.

 

Fig. 4. The simulated evolution of pump pulse and laser pulse at different positions along the HCF. All pulses are plotted with the same horizontal and vertical coordinates with the same scale. It should be noted that the black curve is the transmitted pump pulse in HCF without HBr filled, namely only considering the transmission loss of HCF for comparison. And other curves are corresponding pulse evolutions in HBr-filled HCF.

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It can be found that due to the transmission loss of HCF, the peak power of the transmitted pump pulse plotted with black curves gradually decreases. The central peak position of the pulse t = 30 ns gradually increases at an interval of 10 ns corresponding to exactly the increase of the distance at an interval of 3 m. Under the transmitted pump pulse, with the increase of time, the laser peak power gradually increases. The P-branch laser pulses appear first due to a larger emission cross section. Then the following R-branch laser pulses occur accordingly. The P-branch and R-branch laser pulses mainly consume the rising edge of the transmitted pump pulse where the generated laser is also mainly located. It can be seen that the generated laser pulse profile is not in Gaussian shape with an abrupt rising edge. The unconsumed falling edge of the transmitted pump pulse mainly contributes to the residual pump pulse so that the green curves representing the residual pump pulse are closer to the falling edge of the transmitted pump pulse. With the increase of the distance from the incident end of the HCF, the pump pulse is always absorbed and consumed, and the residual pump pulse is always in a downward trend until it almost disappears, as shown in Fig. 4. The generated laser pulse gradually rises, and remains almost unchanged at 6∼9 m. The further drops of the laser pulse can be predicted with the increase of the distance. This is the joint effect of the gain from consuming the pump pulse and the transmission loss in HCF. At a closer distance, the laser generating rate under the high peak power pump is higher than the transmission loss, and the laser pulse gradually rises. The laser generating rate decreases with the consuming pump pulse. Thus, a downward trend can be predicted with the further increasing distance.

The comparison of the influence of HCF length on (average) output and residual pump power with the (pulsed) CW pumping is shown in Fig. 5. The pump wavelength is set as R(2) absorption line of H⁷⁹Br isotope. The pump absorption is enhanced with the increasing HCF length. Although the average residual pump power in Fig. 5(a) has almost the same trend with the residual power in Fig. 5(b) at 1 mbar pressure, the higher simulated output laser power in CW pumping shows that the laser transition is more efficient than that in pulsed pumping with specific pulse duration of 20 ns and repetition rate of 1 MHz. The star dots in Fig. 5(b) are the experimental data compared with the simulated results. Similar to Fig. 3, the measured output laser is lower while the measured residual pump power is higher. From Fig. 5(c), the influence of pressure on laser generation is obvious. With the increase of HBr pressure, the optimum HCF length moves towards a shorter length. With CW pumping, the maximum output laser power is from 1.69 W (9 m HCF length and 1 mbar pressure) to 2.73 W (5 m HCF length and 2 mbar pressure) and 3.19 W (3 m HCF length and 3 mbar pressure). With pulsed pumping, the maximum average output laser power is from 1.23 W (9 m HCF length and 1 mbar pressure) to 1.82 W (7 m HCF length and 2 mbar pressure) and 2.18 W (5 m HCF length and 3 mbar pressure). It can be seen that the laser power scaling is more obvious in CW pumping and the optimum HCF length of pulsed pumping is longer than that of CW pumping at 2∼3 mbar. However, the loss resulting from the rotational relaxation is not included specifically in the model which would decrease the laser power in CW pumping but has little effect on the pulsed pumping owing to lower pulse duration than relaxation time.

 

Fig. 5. (a) The simulated average output power and average residual pump power with respect to the HCF length under pulsed pumping with 10 W incident average pump power and 1 mbar pressure; (b) simulated output power and residual pump power as a function of the HCF length under CW pumping with 10 W incident pump power and 1 mbar pressure. The plotted star dots are corresponding measured experimental data for comparison; (c) comparison of output power in CW pumping and average output power in pulsed pumping versus the HCF length at different pressure.

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4. Observed rotational relaxation phenomena in the experiment

Due to the thermal motion of molecules, the HBr molecules collide with other molecules or walls all the time. The energy transfer induced by inelastic collisions can change the population density of energy levels. Among them, the rotational relaxation is a process in which the population density in a rotational state transits to other rotational states in the same vibrational state through non-radiative transitions. The absorption and emission line intensity is determined by the Boltzmann distribution and the degeneracy of the rotational energy level. At room temperature, the strongest absorption lines are found near J = 3 for both P and R branch. Through the experiment, the output spectra caused by the rotational relaxation are measured under the conditions of CW pumping and higher HBr pressure (at 8 mbar and above). For the absorption lines (including R(0), R(7), and R(11) absorption lines) away from the strongest absorption lines pumping, the rotational relaxation occurs by changing the pump power. However, for the absorption lines (including R(2), R(3), and R(5) absorption lines) near the strongest absorption lines pumping, the rotational relaxation cannot be observed by changing the pump power while through other methods of tuning the pump wavelength from the center of the absorption line, the rotational relaxation occurs. In other words, using the absorption lines away from the strongest absorption lines and tuning the pump wavelength deviating from the center of the absorption line makes the rotational relaxation in HBr-filled HCF occur easily. It should be noted that such interesting rotational relaxation phenomena have not yet been reported in gas-filled HCF lasers. The potential explanations should consider more specific relaxation rates which are simplified in our numerical models.

Specifically, the measured output spectra varying with the increasing pump power when pumped by R(0) absorption line of two HBr isotopes are shown in Fig. 6. From Fig. 1(b), the targeted laser lines are R(0) and P(2) pumped by R(0) absorption line. But owing to the high threshold of R-branch laser line at 8.3 mbar, as illustrated in Fig. 3, the R(0) laser line is not measured. The targeted laser line P(2) generation rate is related to the pump power. Under low pump power, the rotational relaxation is dominant and the excited HBr molecules in J = 1 rotational state relax to J = 3-4 rotational state. Interestingly, the two isotopes H⁷⁹Br and H⁸¹Br should have similar characteristics just with a little difference of mass. But the rotational relaxation phenomena are different. For isotope H⁷⁹Br, a single relaxation transition line P(5) is observed in Fig. 6(a) while the relaxation transition line P(4) occurs for isotope H⁸¹Br in Fig. 6(b). With the increasing pump power, the targeted laser line P(2) dominates the output spectra gradually. It is owing to the J = 1 rotational state that is directly pumped from the ground state, determined by the pump power while the relaxation transition lines P(4) and P(5) are originated from the rotational relaxation which depends on the HBr pressure (namely population density). Due to the unchanged rotational relaxation rate, the gain of the P(4) and P(5) transition reaches saturation more easily, inhibiting the growth of the P(4) and P(5) laser power. However, for the P(2) laser line, the stimulated emission is strengthened with increasing pump power. Therefore, the P(2) transition dominates with increasing power in the competition.

 

Fig. 6. The measured output spectra varying with the increasing pump power pumped by R(0) absorption line of isotope (a) H⁷⁹Br and (b) H⁸¹Br. The HBr pressure is 8.3 mbar

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Figure 7 plots the measured output spectra varying with the tuning pump wavelength deviating from the center of absorption line when pumped by R(3) absorption line of two HBr isotopes. The R(3) absorption line near the strongest absorption lines and the rotational relaxation phenomena are not observed by changing the pump power in the experiment. Using the fine-tunable CW pump light, we find that the pump wavelength is tuned to deviate the central absorption wavelength a little so that the pump power can be absorbed partly. In that case, the rotational relaxation can occur. For isotope H⁷⁹Br pumped by R(3) absorption line, the P(5) transition line is not only the targeted laser line but also the prefered relaxation transition line. Thus with the tuning pump wavelength, a single line P(5) is observed in Fig. 7(a). As it should be, with further deviation of the pump wavelength from the center of the absorption line, the pump power is not absorbed totally and no spectra can be observed. For isotope H⁸¹Br pumped by R(3) absorption line in Fig. 7(b), the targeted laser line is P(5) while the prefered relaxation transition line is P(4). When the pump wavelength is tuned at the center of the absorption line, a pure targeted laser line P(5) occurs without any relaxation transition line. With the pump wavelength tuned deviating from the center of the absorption line, the relaxation transition line P(4) occurs and dominates gradually. Similarly, with further deviation of the pump wavelength, any spectra cannot be observed as the pump power is not absorbed totally.

 

Fig. 7. The measured output spectra varying with the tuning pump wavelength towards the center of absorption line pumped by R(3) absorption line of isotope (a) H⁷⁹Br and (b) H⁸¹Br. The HBr pressure is 8.2 mbar

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All the mentioned phenomena of rotational relaxation are recorded in CW operation as the molecules have sufficient time to relax to other energy states. But for the pulsed pumping with about dozens of ns pulse duration, the emission lines caused by rotational relaxations are not observed even at 37 mbar in the experiment. It verifies that the rotational relaxation time is more than the pulse duration of dozens of ns and there is not enough time for the population of the upper level to redistribute to other rotational levels.

5. Conclusion

In conclusion, we have demonstrated a four-energy level system model of optically pumped HBr-filled HCF lasers. Due to the small value, the rotational relaxation rate is not included particularly but dealt together with coefficients of non-radiative transitions. The simulated laser performance is carried out both in CW pumping and pulsed pumping conditions. For CW pumping, the steady state is included while the finite-difference time-domain method is used to solve the rate equations in pulsed pumping. The simulated results indicate that the HBr pressure and the HCF length are two factors that significantly affect the laser output, threshold and residual pump. In addition, the phenomena of rotational relaxation in HBr-filled HCF lasers are described experimentally for the first time, showing the type of HBr isotope pumped, changing the pump power and tuning pump wavelength can influence the rotational relaxation under CW pumping condition. For pulsed pumping with about dozens of ns pulse duration, the rotational relaxation cannot be observed owing to the insufficient time for the population of the upper level to redistribute to other rotational levels. The demonstration provides a deeper understanding of the laser performance of such gas-filled HCF lasers and is helpful to further power scaling.