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Abstract
Mixing cross sections of Cs between the 62P1/2 and 62P3/2 levels and quenching cross sections from these levels to the ground state 62S1/2 were measured, with methane, ethane, and propane as collision partners. We excited Cs to the 6P3/2 level with a tunable nanosecond pulse laser and measured the fluorescence at the 62P1/2 level. The measured mixing cross sections with methane, ethane, and propane were (1.39 ± 0.16), (5.67 ± 0.85), and (7.91 ± 0.93)×10−15 cm2, respectively, and the statistically averaged quenching cross sections were (1.5 ± 0.25), (10 ± 2.0), and (25 ± 5.3)×10−18 cm2, respectively. The quenching cross section of Cs-propane is reported for the first time.
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1. Introduction
Diode-pumped alkali lasers (DPALs) are optically pumped, near-infrared, continuous-wave (CW) gas lasers [1–3]. The DPAL medium consists of a vaporized alkali metal, such as potassium, rubidium, or cesium, mixed with helium buffer gas at near atmospheric pressure. The lowest electronically excited state of alkali metals is split into two sublevels, 2P3/2 and 2P1/2, due to spin-orbit interactions. Transitions from these levels to the ground state (2S1/2) are traditionally called D2 (2P3/2-2S1/2) and D1 (2P1/2-2S1/2), respectively. DPALs operate on a simple principle: ground state atoms are optically pumped by a narrow-band laser diode (LD) tuned at the D2 transition, and atoms in the 2P3/2 sublevel are then transferred to the 2P1/2 sublevel by collisional energy exchange with a buffer gas. Lasing action occurs at the D1 transition.
An exciplex pumped alkali laser (XPAL) is a DPAL variant that exploits the exciplex formation of alkali atoms and rare gases at high temperatures [4–6]. In contrast to DPALs, the wide absorption band of the alkali metal-rare gas exciplex eliminates the need for narrow-band LDs; however, challenges remain due to the low partial pressure of the exciplex, which results in poor coupling with pump light. For both DPALs and XPALs, Cs-based systems require the addition of light hydrocarbon gases such as methane [7–9], ethane [4,5,10,11], or propane [12–14] to facilitate upper-state mixing due to the large energy gap between the two 2P states.
Hydrocarbon gases not only facilitate upper-state mixing but are also a source of energy loss due to collisional quenching from the upper states. To accurately model DPAL performance and thermal effects, these values must be measured precisely. Despite the importance of these values, reports of measurements are surprisingly scarce.
Until recently, the mixing cross sections used in theoretical studies of Cs DPALs were measured in the 1970s by Walentynowicz et al. [15,16], but no reliable data for the quenching cross sections were available. Early theoretical works on Cs DPALs estimated the quenching loss of Cs atoms by hydrocarbon buffer gas from the value of Rb [17,18] or ignored it as insignificant [19].
In 2011, Pitz et al. measured the mixing and quenching cross sections of Cs with various gases [20]. For the first time, they gave the upper bound of the quenching cross sections: (1.4 ± 0.6)×10−16 cm2 for Cs-CH4 and (2.2 ± 0.6)×10−16 cm2 for Cs-C2H6. However, theoretical studies such as those by Endo et al. [13] and Yacoby et al. [21] questioned these values by comparing model calculations and experimental results, and these values were found to be at least an order of magnitude overestimated.
Recently, Gearba et al. reported very reliable measurements of the mixing and quenching cross sections between Cs and CH4 using an ultrafast laser pulse excitation of Cs and then observed the time evolution of the fluorescence by using a time-correlated single-photon counting technique [22]. The measured quenching cross section was (1.57 ± 0.03)×10−18 cm2, which was similar to the values suggested by Refs. [13] and [21].
At the time of writing, the only reliable mixing and quenching cross sections were for CH4; the quenching cross section of C2H6 was questionable, and the mixing cross section of C3H8 had not been reported since the 1970s. In addition, to the best of our knowledge, no reports of quenching cross sections between Cs and C3H8 have been published.
Considering the above, we thought it would be useful to measure and report the mixing and quenching cross sections of three gases used in DPALs and XPALs. In this paper, we report the measurement of the mixing and quenching cross sections of Cs with methane, ethane and propane for a total of six values using nanosecond pulse laser excitation of Cs at the D2 transition and observation of the time evolution of the D1 transition at room temperature.
2. Theory
2.1 Rate equation model
We used the same measurement scheme as Gearba et al. [22], who used a pulsed laser to pump Cs atoms to one of the 2P levels and then observed the time history of the fluorescence of the other 2P level. However, because the pulse width of our pump laser is relatively long (17 ns full width at half maximum, FWHM), the observed data must be treated differently.
We assume that the Cs atom takes only the 2S1/2, 2P1/2, and 2P3/2 states during the measurement. This is justified because higher-level excitation states are strongly quenched in the presence of even a small amount of buffer gas [22]. The number densities of the 2S1/2, 2P1/2, and 2P3/2 states are denoted n0, n1, and n2, respectively. Figure 1 shows the processes considered in this work. We followed the naming rules of Ref. [22]. The mixing rates are Rij, where i and j represent states 1 and 2, respectively, the spontaneous emission rates are γi0, and the quenching rates are Qi0. Furthermore, we consider the pumping rates P0j in our model.
Then, the time variations of n1 and n2 are given by the following simultaneous differential equations:
The mixing rates R12 and R21 are related by the principle of detailed balance:
where g2 = 4 and g1 = 2 are the degeneracies of the 2P3/2 and 2P1/2 levels, respectively, and ΔE is the fine-structure splitting of the 2P states. The spontaneous emission rates γ20 and γ10 are precisely known [23–25].The pumping rate P0j is expressed as
The objective of this study is to determine σMij and σQi0 between Cs and hydrocarbon gases by observing the time history of the fluorescence. Therefore, it is necessary to relate these values to Eqs. (1) and (2).
2.2 Determination of the mixing cross section
Let us assume the following. A pump light pulse is delivered to a collection of Cs atoms with buffer gas. We take the D2 line as the pump light wavelength, although the explanation is the same if the D1 transition is taken as the pump transition. The intensity of the pump light pulse is strong enough to saturate ground-state absorption, and the pump pulse width is shorter than both the spontaneous emission lifetime and the characteristic time of the quenching reaction (Qi0−1), but it cannot be approximated as zero. The width is typically of the order of 10 ns. In this case, when pump light is present, the removal of Cs(2P1/2) by spontaneous emission and quenching reactions can be ignored.
Then, the phenomena can be divided into a “mixing dominant” region and a “relaxation dominant” region, depending on the ratio of Rij to γi0. Note that Qi0 is always two orders of magnitude smaller than Rij when considering methane, ethane or propane.
In the relaxation dominant region, i.e., where Rij << γi0, we can assume that n0 + n2 >> n1 when the pump light is on. If the stimulated absorption and emission of the D2 line are equilibrated, i.e., σS20np2c >> γ20, the approximation n2 = 2n0, or n2 = 2n/3, holds. Calculating with our experimental condition (described in Section 3), σS20np2c / γ20 becomes 105. In this case, Eq. (2) is reduced to
and if the pump pulse width (with intensity well above the saturation intensity) is τ, the number density n1 at the end of the pump pulse will beAfter this, the time evolution of n1 obeys Eq. (2), with P01 = 0 and ${R_{21}}{n_2} = \frac{{2{R_{21}}n}}{3}\exp ( - {\gamma _{20}}t)$, namely,
Solving Eq. (12) with the initial condition ${n_1}(0) = \frac{{2{R_{21}}n\tau }}{3}$ yields the following expression:
Because $\textrm{exp}\{{\textrm{(}{\gamma_{10}} - {\gamma_{20}})t} \}$ changes at a much slower rate than $\exp ( - {\gamma _{10}}t)$, n1(t) is approximated by a single exponential decay beginning at t = 0. In other words, n1 peaks immediately after the pump pulse ends and has a value of $\frac{{2{R_{21}}n\tau }}{3}$.
Since R21 = nHCσM21vrel (Eq. (4)), the maximum number density of the 2P1/2 state in the relaxation dominant region is
In conclusion, in the relaxation dominant region, when the Cs-hydrocarbon mixture is strongly pumped at the D2 transition line, the peak signal intensity of the D1 transition is proportional to the hydrocarbon number density (nHC).
On the other hand, in the mixing dominant region, where Rij >> γi0, both the stimulated absorption and emission of the D2 transition and the mixing reactions are in equilibrium states, and therefore, the ratio n0:n1:n2 is constant when the pump light is on. The ratio is described by
by defining a nondimensional fraction f as [26,27] where f denotes the fraction of the 2P1/2 state number density to the total 2P state number density in the equilibrium state. Therefore, n1 approaches $\frac{{2fn}}{{f - 3}}$, and the peak signal intensity of the D1 transition does not depend on nHC.When the buffer gas pressure is gradually increased from zero while Cs atoms are pumped by successive pulses, the peak value of each D1 signal increases in proportion to the buffer gas pressure in the relaxation dominant region, saturates at a certain point, and becomes constant in the mixing dominant region. If we normalize the D1 signal peak values to their maximum values, the slope of the normalized signal intensity I versus nHC in the relaxation dominant region, dI/dnHC, becomes
Thus, we obtain information about the mixing reaction cross section σM21.
This analytical approach is not exact and thus cannot be directly used to determine σM21. In this study, we developed a point-source rate equation based on the existing DPAL simulation code [28], which takes into account the pressure broadening of Cs against hydrocarbon gases, the time history of the pump light intensity, the spectral properties of both the pump light and Cs atom absorption, the quenching reaction, and spontaneous emission. More details of the simulation code are provided in the Supplement 1. The simulations were run for different values of σM21 (and σM12), and I was calculated as a function of nHC for each. The experimental σM21 value was chosen as the value that agreed the best with the series of simulation runs.
2.3 Determination of the quenching cross section
The quenching cross section σQi0 is measured in the completely mixing dominant region, i.e., at high buffer gas pressures. In this region,
holds. Adding Eqs. (1) and (2) and then substituting in Eq. (18) yields the time domain differential equation of the total upper level density, nu = n1 + n2, in the absence of pump light asThis equation is easily solved to obtain
If we define the statistically averaged γ and Q as
n1(t) is expressed asWe cannot extract Q10 and Q20 independently in this context; however, because they operate in the mixing dominant region, this is not a major obstacle in the theoretical study of DPALs. Because of the relationship shown in Eq. (23), we can determine Qav by observing the decay constant of the D1 signal in the mixing dominant region, plotting (γav+Qav) against nHC, and finding the slope. The statistically averaged quenching cross section σQav is defined as
3. Experimental setup
Figure 2 shows a schematic diagram of the experimental setup. The apparatus consists of a pump laser, a Cs cell connected to a gas handling system, pressure transducers, high-speed photodetectors, an energy meter, and an oscilloscope. Figure S1 shows its photograph. The pump source is an optical parametric oscillator (OPO) (LOTIS TII LT-2211) pumped by a second-harmonic Nd:YAG laser (LOTIS TII LS-2134) that produces pulses of 852 nm (D2 transition) or 895 nm (D1 transition). The pulse widths are 17 ns (at 852 nm) and 28 ns (at 895 nm) FWHM, and the spectral bandwidth is 0.2 nm. The output pulse energy is reduced to 100 µJ/pulse by a half-wave plate (HWP) and a polarization beam splitter (PBS). The pulse repetition rate is set to 10 Hz. The output beam has an elliptical shape and a Gaussian intensity distribution with a cross section of 0.5 cm2.
The Cs cell is modified from a laser cavity used in previous investigations [29]. To avoid radiation trapping, the distance between the cell and the observation port window was reduced from 22 mm to 1.5 mm. The effect of this modification on the measurement results will be discussed below. Because the cell was originally developed for DPALs, it could be heated, allowing for high Cs number densities to be maintained. However, in this work, all measurements were performed at room temperature (298 K).
Pressure in the cell was measured with a capacitance manometer (MKS 622A11TBD) if it was 10 Torr or less or with a semiconductor pressure gauge (Nagano Keiki, CG61) if it was between 10 and 2000 Torr. The hydrocarbon gases used were >99.999% methane (Grade 1, Japan Fine Products), >99.9% ethane (Grade UHP, Sumitomo Seika Chemicals), and >99.99% propane (Grade S, Takachiho Chemical Industrial).
Fluorescence was measured with Photodetector 1 (Si avalanche photodiode, ThorLabs APH430A2), which has a bandwidth (−3 dB) of DC – 400 MHz. The collecting optics was a single condenser lens (f = 20 mm) used in a 1:1 imaging system with NA = 0.3. A pair of bandpass filters (BPFs) are used to prevent the pump light from being detected; the BPFs are ThorLabs FB850-10 (850 ± 5 nm) for D2 signal transmission and Andover 895FS10-25 (895 ± 5 nm) for D1 signal transmission. In both cases, it was confirmed that no unwanted pump radiation was detected at the maximum sensitivity of the detection system.
The pump light passing through the Cs cell is dumped by an energy meter (ThorLabs ES111C), and the reflected light is observed by Photodetector 2 (Si photodiode, New Focus 1621) to acquire the temporal pulse shape. Although the waveform of the pump light is used in the simulation, not every waveform is used. Instead, a standard waveform is used for the input without considering pulse-by-pulse variations. The signals from Photodetectors 1 and 2 are recorded by a high-speed oscilloscope (Teledyne Lecroy WaveSurfer 10) with a sampling rate and bandwidth of 10 Gs/s and 1 GHz, respectively.
The experiment is carried out as follows. The Cs cell is charged with several tens of µg of Cs under an Ar atmosphere before being evacuated to less than 1 mTorr by an oil rotary vacuum pump. Then, one of the three buffer gases is introduced into the cell. The measurement of σQav starts at 1800 Torr for CH4, 1200 Torr for C2H6, and 600 Torr for C3H8. For σQav measurements, we are only interested in the decay part of the fluorescent signal, and the oscilloscope gain is set to saturate the peak. After 10 seconds (100 traces) of acquisition at each pressure, the vacuum shut valve was opened to partially exhaust the gas, and fluorescent signal acquisition was then performed at the next pressure.
The measurement of σMij starts at 1000 Torr for CH4 and 100 Torr for C2H6 and C3H8. The measurement method is the same as above; however, in this case, the peak value of the signal is of interest, and 100 traces are acquired at each pressure by changing the oscilloscope range as needed to appropriately acquire the entire waveform. The pressure range spans several orders of magnitude, with the lowest pressures being 1 Torr (for CH4) and 0.1 Torr (for C2H6 and C3H8).
4. Results and discussion
4.1 Typical waveforms of the fluorescence
Figure 3 shows a graph of typical measurement results for the determination of σM21. Pump light (D2 line) and fluorescence (D1 line) time histories are shown. The gas used is C2H6, and the pressures are 0.5, 1, and 2 Torr. The actual traces are noisier, but they are smoothed with a low-pass filter (−3 dB at 40 MHz) to avoid taking noise as a peak. As described in Section 2, there is a linear rise in the D1 signal while the pump light is on and a rollover immediately after the pump light is turned off.
For C2H6, the transition from relaxation dominant to mixing dominant (R21 = γ10) occurs at approximately 3 Torr, so the fluorescence peak should be proportional to the pressure, as shown in the graph. Peak fluorescence intensity was measured for 100 traces at each pressure and then averaged and plotted against the buffer gas pressure.
Figure 4 shows a graph of typical measurement results for the determination of σQav. The pump light (D2 line) and fluorescence (D1 line) time histories are shown. The gas used is C2H6. The signals are normalized to their maximum value and plotted on a logarithmic scale (Np). Because no low-pass filtering is used in this measurement, relatively high noise is observed in the fluorescence intensity. To calculate the decay constant, the fluorescence waveform is first corrected for offset value and processed with a low-pass filter, and then the interval from −1.0 Np to −2.5 Np is used for the calculation. The effect of removing the offset from the calculation of σQav is discussed later. The decay constant is calculated individually for 100 traces at each pressure and then averaged and plotted against the buffer gas pressure.
4.2 Quenching cross section determination
Figure 5 shows the decay constant vs. pressure relationship for C2H6. The same graphs for CH4 and C3H8 are shown in Fig. S4. The error bars represent the standard deviation of the 100 traces measured at each pressure. If the measurements are fitted with a linear function, the y-intercept should be the decay constant in the absence of buffer gas, i.e., γαv, which is calculated to be 2.93×107 s−1 at 298 K (f = 0.879). However, the linear function does not intersect at y = γαv if no offset is applied to the traces before calculating the decay constants. We determined the offset value such that the linear function intersects exactly at y = γαv. Then, we calculated the slope (dy/dp) of this line, where p is the buffer gas pressure (in Torr). In Fig. 5, the optimum offset value was determined to be +0.73% of the full-scale (0 Np).
The statistically averaged quenching cross section σQav is then found by
The above measurements were repeated 30 to 60 times with each buffer gas, and the average was taken as the measured σQav. The histograms of all measurements are shown in Fig. S5. Although we performed the measurements with the D1 line as the pump transition and the D2 line as the fluorescence, the statistical error was much larger because of an order of magnitude lower fluorescence intensity. Therefore, the D1 line excitation results will not be discussed further.
The measured values of the quenching reaction cross sections are summarized in Table 1. Table 1 also displays values from previous works. The table also shows the quenching cross sections that were estimated from a comparison of DPAL numerical simulations and experimental results in our previous work [13] and Yacoby et al. [21].
Table 1. Measured quenching cross sections σQav
The values of [20] are an order of magnitude larger than those in other works, implying that they are overestimated. The present measurements are in good agreement with the precise measurements of [22]; thus, the validity of our measurement method is validated. It should also be noted that the quenching cross sections of C2H6 and C3H8 agree well with the results of [13]. This validates the accuracy of our simulation model.
The errors in the current work are the standard deviations of multiple measurements. Because the precision of the pressure gauge and oscilloscope are less than 1%, they cannot explain the standard deviation of the measurement results (approximately 20% of the mean value). The primary reason for the large fluctuations is attributed to an insufficient signal-to-noise ratio (SNR) of the fluorescence signal. In addition, day-to-day and morning-to-evening systematic errors were observed. This suggests that some uncontrolled factors, such as contamination in the gas handling or Cs cell or electrical noise sources, are to blame for the deviations in the results.
4.2 Mixing cross section determination
Next, the mixing reaction cross section was measured. As an example, Fig. 6 shows the relationship between the fluorescence peak intensity and pressure when C2H6 is used as the buffer gas, demonstrating a smooth transition from a linear increase in the relaxation dominant region to a constant value in the mixing dominant region, as explained in Section 2. The error bars shown in the plots are the standard deviation of 100 measured peaks. The different colored lines represent the results of the simulation. The same graphs for CH4 and C3H8 are shown in Fig. S6. In the simulation runs, the quenching reaction cross sections measured in the previous subsection were used.
Figure 6, which fits the simulation to the experimental results, takes σM21 to minimize the residual mean, as shown in the following equation:
The residual ε varies from measurement to measurement, but it typically reaches 10−5 and occasionally reaches 10−6. This indicates that the model accurately represents the experimental results. The reduced chi-squared, which measures the goodness of the theoretical model [30], is 1.0-1.1, which indicates very good consistency. Details of the chi-squared analysis are provided in the Supplement 1. For each buffer gas, we measured and determined σM21 50 to 80 times, and the average value was taken as the measured value. The histograms of all measurements are shown in Fig. S8.
The measured values of the mixing reaction cross sections are summarized in Table 2. The values of previous studies are also shown in the table. The results of the present study are in reasonable agreement with previous results. In particular, the mixing cross section of CH4 agrees well with the precise measurement of [22], which indicates the validity of our measurement method.
Table 2. Measured mixing cross sections σM21
The errors of the current work are the standard deviations of multiple measurements. In addition to the statistical errors, the systematic errors mentioned above were also seen in the σM21 measurements. Pulse-to-pulse variations in the fluorescence peak were observed, particularly in the low-pressure, relaxation dominant region, as shown in Fig. 6. This is mainly due to the pulse-to-pulse width variation of the pump light. The measured pulse width was 17.3±0.9 ns FWHM, which is a variation of ±5% from the average value. Equation (11) shows that the peak fluorescence intensity is proportional to the pulse width, but it is insensitive to the pump intensity. This effect was partially mitigated by recording 100 peak fluorescence intensities at each pressure. Because the variation in the average value over n measurements is inversely proportional to $\sqrt n$, the effect of the pulse width variation was reduced to ±0.5%.
4.3 Mitigation of radiation trapping
We checked for the effects of radiation trapping in our setup. To avoid radiation trapping, Gearba et al. [22] used a gas tube 2 mm in diameter for their measurements. However, our cell is relatively thick, with a diameter of 1 cm. Nevertheless, we believe our setup is free from radiation trapping.
Radiation trapping appears in sodium D lines when k0L > 1 [31], where k0 is the absorption coefficient at the line center and L is the optical path length. Because k0 of the Cs D1 line is 1 cm−1 at room temperature and is in the Doppler broadening regime, our experimental conditions are not far below the above criteria. To confirm the effect of radiation trapping experimentally, we measured and compared quenching cross sections using cells with and without modifications. Figure 7 shows the Cs cell with and without modifications. Because of the small number of measurements taken, this result cannot be considered definitive, but we did observe a 20% higher σQav in the unmodified cell than in the modified cell. The detailed result is shown in Table S1. According to radiation trapping theory [32], radiation trapping affects the decay time to the longer side. The reason for the larger σQav can be explained as follows. In the calculation shown in Fig. 5, the offset is determined such that the y-intercept coincides with γav; however, radiation trapping causes the offset to be excessive. As a result, the slope increases by 20%, and the resulting σQav is overestimated. In other words, the effect of radiation trapping on the measured σQav at 15X optical path is at most 20%, which indicates that radiation trapping had no effect on the measured cross sections in our experiments.
5. Conclusions
We measured the mixing and quenching cross sections of Cs with various hydrocarbons, including CH4, C2H6, and C3H8, which are commonly used for DPAL operation. The cross sections were obtained by using an OPO laser with a pulse width of 17 ns (FWHM) to excite the D2 line; then, the fluorescence intensity of the D1 line was measured, and these values were used in the differential equation of the time variation of the Cs number density.
The quenching cross section was determined by using the fact that the decay constant of the D1 line is a linear function of the buffer gas pressure when the buffer gas pressure is high. The measured, statistically averaged cross sections (σQav) for methane, ethane, and propane were (1.5±0.25), (10±2.0), and (25±5.3)×10−18 cm2, respectively, with the variation being the standard deviation of multiple measurements. The obtained σQav for CH4 is in good agreement with Gearba et al. [22], suggesting the validity of this study. The results were also consistent with our previous work [13], in which we estimated σQav by comparing the experimental DPAL output power and relevant simulations by varying the buffer gas pressure. The σQav of C3H8 was reported for the first time. The effect of radiation trapping was experimentally shown to be negligible.
The mixing cross section was obtained by using the fact that the fluorescence peak is proportional to the buffer gas pressure when the buffer gas pressure is low. Because this proportional relationship is approximate, we developed a point-source numerical simulation code to calculate the peak signal intensity by using the pump pulse waveform, spectral properties of the pump pulse and medium, relaxation due to spontaneous emission, and collisional quenching with the relevant hydrocarbon. Then, the mixing cross section (σM21) was calculated by finding the best fit between the experimentally obtained results and the simulations. The measured results for methane, ethane, and propane were (1.39±0.16), (5.67±0.85), and (7.91±0.93)×10−15 cm2, respectively, which are in good agreement with previous measurements. In particular, σM21 of CH4 agrees well with [22].
The present measurements were made with a relatively inexpensive apparatus that can be used not only for Cs but also for other alkali atoms and for the temperature dependences of the cross sections. We hope that the results of this study will contribute to understanding not only alkali lasers but also the physics of collisional relaxation between alkali atoms and buffer gases.
Funding
Japan Society for the Promotion of Science (JP20K05366).
Acknowledgments
M. Endo thanks Taro Yamamoto of Numazu Sanso Kogyo for helping with experiments and supplying consumables.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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