## Abstract

We have calculated photoionization rates in alkali lasers. The photoionization of alkali atoms in the gain medium of alkali lasers can significantly degrade the laser performance by reducing the neutral alkali density and with it the gain. For a ten atmosphere Rb laser and a Cs exciplex laser, the photoionization induced alkali atom loss rates are greater than 10^{5} sec^{−1}. These high loss rates will quickly deplete the neutral alkali density, reducing gain, and may require fast, possibly, supersonic flow rates to sufficiently replenish the neutral medium for CW operation.

©2011 Optical Society of America

## 1. Introduction

There has been renewed interest in alkali vapor lasers in the last decade [1] because of their potential to generate high power in a high quality beam. Lasers have been demonstrated for Potassium (n = 4), Rubidium (n = 5) and Cesium (n = 6) vapors. These lasers are pumped from the ground nS_{1/2} state to the nP_{3/2} state using D2 resonant light (see Fig. 1
with energy level diagram for Rb atom). In a standard alkali laser, a buffer gas is used to transfer population from the nP_{3/2} state to the nP_{1/2} state and to broaden the alkali vapor absorption line. If the pumping and mixing are rapid enough, a population inversion between the nP_{1/2} state and the nS_{1/2} state occurs and lasing is possible at the D1 transition.

There are several approaches in designing an efficient high power alkali lasers. One approach is to use multi-atmospheres (10 atmospheres) of helium buffer gas to mix the Rb nP states [2] and broaden the Rb absorption line to match it to a pump diode laser emission line. Another approach is to use approximately one atmosphere or less of buffer gas [3–5] for nP states mixing. In this case a narrowbanding of the pump laser linewidth to about 10 GHz is required to match it to the alkali vapor absorption line. A different approach is to pump a Cs exciplex and either lase 6P_{3/2} to 6S_{1/2} directly or use a buffer gas and lase 6P_{1/2} to 6S_{1/2} [6–8]. This approach allows the use of commercial broadband pump diode lasers. There is one problem, existing in all types of alkali lasers, which is that the nD_{5/2,} nD_{3/2} and (n + 2)S_{1/2} states can also be populated by either photon excitation from the nP states or energy pooling collisions of two nP states. The intense pump and lasing radiation can then photoionize atoms in these excited S and D states, leading to loss of neutral alkali atoms in the laser gain medium. In this paper we have calculated this photoionization rate for several types of alkali lasers and have found that this rate can be quite severe for the high pressure Rb laser and the exciplex laser. To replenish the loss of neutral alkali atoms because of photoionization, a flow of the gain medium is required with a fast, possibly supersonic, speed.

## 2. Model and calculations

The energy diagram for a standard three-level alkali laser shown in Fig. 1 and includes the relevant excited S and D states and the ionization limit. In each of the considered alkali systems, the nD_{5/2}, nD_{3/2} and (n + 2)S_{1/2} states can be populated by photon excitation from the nP_{3/2} and nP_{1/2} states by means of either D1 or D2 light and, then, ionized by the same D1 or D2 light. Table 1
lists the wavelengths (λ), lifetimes (τ), absorption cross- sections (σ) and detunings (∆) between upper level transitions for K, Rb and Cs and either D1 or D2 light. For example, the detuning ∆_{D1} in the block of the table corresponding to the nP_{3/2}→nD_{5/2} transition refers to the detuning in MHz between the frequency of nP_{3/2} → nD_{5/2} transition and the D1 light frequency (nP_{1/2} – nS_{1/2}). Similar detunings are defined for other states and for D1 or D2 light or the pump wavelength of the exciplex laser.

The stationary isolated atom cross section σ_{o} can be calculated using Eq. (1)

_{Total}is the sum of Einstein coefficients (includes both relevant states, i.e. P

_{3/2}and D

_{5/2}, decay rates) and g

_{1}and g

_{2}are the degeneracies for the lower and upper states respectively [9]. This can be calculated for each of the relevant alkali transitions in Cs, Rb or K [10–13]. In these laser systems, a buffer gas will pressure broaden the transition and the excitation rate from state j to i, E

_{ij}, due to D2 light can be expressed by Eq. (2)

_{D2}is the intensity of D2 pumping light, f

_{n}is the natural linewidth and f

_{b}is the FWHM pressure broadened linewidth. The pressure broadening for the nD

_{5/2,}nD

_{3/2}and (n + 2)S

_{1/2}states have not yet been determined for Cs, Rb, and K in the presence of helium or ethane. In a number of references [14–17], however, this broadening rate has been measured in transitions and atmospheres similar to those of interest here .These values tend to lie between 40 and 80 MHz/torr, the majority of which can be included in the range of 55 ± 15 MHz/Torr. For the purpose of this paper, we have chosen to use the mean value of 55 MHz/Torr for the different alkalis.

Once the upper excited state j is populated, the photoionization rate, P_{j}, from that state can be written as Eq. (3)

_{D1}and I

_{D2}are the intensities of the D1 lasing light and D2 pumping light respectively, σ

_{i}is the photoionization cross section [18]. The photoionization cross section is weakly dependent on the wavelength and will be similar for the (n + 2)S

_{1/2}, nD

_{3/2}, and nD

_{5/2}states in all alkalis considered. For the purposes of this paper we use the value for Rb of σ

_{i}= 2 x 10

^{−17}cm

^{2}for all the alkalis [18]. The effective ionization or loss rate per alkali atom in D1 light, R

_{L}(D1), is shown in Eq. (4).

_{P1/2}and η

_{P3/2}are the fractions of atoms in the nP

_{1/2}and nP

_{3/2}states, respectively, and τ is the lifetime of the relevant nD

_{j}or (n + 2)S

_{1/2}state. Then Eq. (5) is the total loss rate from both the D1 and D2 light.

Two different classes of standard alkali lasers have been demonstrated: K, Rb and Cs lasers with a low (≤1 atmosphere) buffer gas pressure and a Rb laser with a high (close to 10 atmospheres) helium gas pressure. Helium is not a very efficient buffer gas for nP state mixing in Rb, which is why a pressure of about 10 atmospheres is required. Helium is, however, efficient at mixing the nP states in K such that less than 1 atmosphere is sufficient [19]. A low pressure Rb or Cs laser will require another buffer gas such as ethane or methane for rapid mixing. The D1 and D2 transitions are broadened by about 20 MHz/Torr for common buffer gasses. η will be determined by the pumping rate, the nP mixing rate, the laser intensity, the relevant cross sections and the relative statistical weight factors g_{i}, which equal to 1, 2, 1, respectively for the S_{1/2}, P_{3/2} and P_{1/2} states. In the limit of strong pumping, rapid P state mixing and without lasing, η_{p3/2} equals to 0.47, 0.36 and 0.17, respectively for K, Rb and Cs at a temperature of 373K. During laser operation, η_{p3/2} will be increased as lasing will tend to equilibrate the S_{1/2} and P_{1/2} levels so that η_{p3/2} will approach 0.5 and η_{p1/2} will approach 0.25. For the exciplex laser, where there is no P state mixing, η is greater than 2/3. For calculations in this paper, we use η_{p3/2} = 0.5 and η_{p1/2} = 0.25 for a standard operating alkali laser with fast P state mixing.

## Results

The results of calculations for various types of alkali lasers are shown in Table 2
, where the calculated total effective alkali atom loss rates and the loss rate contributions from relevant transitions driven by either D1 or D2 light are presented. Included in the table are the buffer gas pressures for each alkali laser and the associated pump laser (or D2) light intensity. Here, the Rb 10 atm Diode Pumped Alkali Laser (DPAL) has an increased pump intensity compared with its 1 atm counterpart due to the fact that the cross sections and saturation intensities decrease with increasing of the buffer gas pressure. Thus, a higher pump intensity is necessary for laser operation as compared to the 1 atm alkali laser. The large pump intensity for the Cs exciplex laser (XPAL) arises from the large saturation intensity associated with pumping off resonance and typical operation uses ~5ns, 5mJ 837nm pulses [7]. For a standard alkali laser, the loss rate scales as I^{2}P/∆^{2}, where P is the buffer gas pressure and I is the average intensity. The operating intensity is linear with pressure, so that the loss rate scales as P^{3}/∆^{2}. Since ∆ is much smaller and σ_{o} larger for Rb, the loss rate is much higher than the one for K under similar operating conditions and pressures. The Rb laser operating at 10 atmospheres of helium has a loss rate increased by three orders of magnitude as compared with a 1 atmosphere Rb laser.

A different approach is that of the exciplex laser. In this case, pumping occurs away from the D2 transition. The Cs-Ar exciplex laser operates at a typical temperature of 464 K, which corresponds to a density of 1.3 x 10^{15} cm^{−3}.The transmission at the 837 nm pump wavelength is about 80%, which yields an effective absorption cross section per ground state Cs atom of σ = 1.2 x 10^{−17} cm^{2}. The effective saturation intensity, I_{sat} = 1/(σ τ), where τ is the 6P_{3/2} lifetime, is calculated to be 610 kW/cm^{2}. In the exciplex laser, the D and upper S states can also be populated by energy pooling collisions between P states, shown by Eq. (6),

_{5/2}and other upper states, as well as processes involving the 6P

_{1/2}state. The reaction rate, k, for energy pooling has been measured for Cs [20], Rb [21] and K [22]. The high pump and lasing intensities (>1MW/cm

^{2}) in the exciplex laser will rapidly ionize the 6D states in less than 100 nsec. The energy pooling excitation rate to the (n + 2)S

_{1/2}, nD

_{3/2}or nD

_{5/2}states is given by Eq. (7).where η

_{i}is the fraction of atoms in the relevant state i, N is the alkali density and k

_{ij}is the reaction rate for populating any higher state j from state i, which will then be rapidly ionized. The loss rate per alkali atom via energy pooling,

*R*, will be given by Eqs. (8a) and (8b).

For exciplex lasers, the loss rate per Cs atom due to multi-step ionization is about 10^{6} sec^{−1}. In the case of the exciplex Cs laser, the sum of the energy pooling rates, k, for populating the (n + 2)S_{1/2}, nD_{3/2} or nD_{5/2} states, is k_{total}≈2 x 10^{−9} cm^{3}/s. This corresponds to an ionization rate of 10^{4}, an increase of ~0.5% to the total loss rate. Additionally, this loss rate scales linearly with density so it will be correspondingly larger at higher temperatures. Energy pooling will also occur in standard alkali lasers. However in these cases, the rates are lower than in the exciplex laser because the intensities and densities are much lower. Ionization rates associated with energy pooling are slow and the contribution to the total loss rate by energy pooling collisions are insignificant when compared with the three-step photoionization considered above.

Though the aforementioned photoionization processes are of great significance to high power CW alkali laser operation, there are several mechanisms (recombination, diffusion, convection and flow) which can aid in restoring the neutral alkali density in the gain medium of alkali lasers. Radiative two-body and three-body recombination mechanisms [23,24] have relatively slow rates of ~10^{−12} cm^{3}s^{−1} for alkalis under conditions similar to ours. Molecular recombination, on the other hand, may be a large contributing factor to the replenishment of neutral alkali density in alkali lasers. After an alkali ion A^{+} is formed, it will quickly covert to a dimer ion in the reaction described by Eq. (9),

The recombination coefficient α_{M} for this reaction has been measured [25–27] under different experimental conditions for Cs at elevated temperatures with values in the range of 10^{−8} to 10^{−6} cm^{3} s^{−1}. Considering this range of α_{M}, a 10 atm Rb-He DPAL a with a neutral Rb density of 10^{14} cm^{−3}, where the ionization rate, R_{Total}, is about 2 x 10^{5} s^{−1}, has a steady state alkali neutral atom density in the range of ~54-93% of the original alkali density. One unknown of this mechanism is the effect the states of the newly dissociated atoms have on the behavior of the system. Upon dissociation, at least one if not both atoms can be in an excited state. With ionization occurring ~1µsec and relatively long lifetimes for some states (i.e. the 5D states in Rb with τ~235ns), ionization can occur again before the atom can decay to a state where it will be useful to the lasing system. This effect could potentially reduce the effective α_{M}, leading to a lower fraction of neutral alkali atoms in the steady state. In addition, ambipolar diffusion and electron thermalization can also play a role in reducing recombination rates [28]. These results show that photoionization may be a serious neutral atom loss mechanism in this case and that the value of α_{M} can play a critical role in the operation of alkali lasers. Also to be considered are other deleterious effects such as those seen by Tam [29] and Happer [30], where laser induced plasma demonstrated an avalanche/ hysteresis characteristic that may lead to increased or sustained plasma as well as cell degradation. Plasma formation leads to a loss of neutral atom density and also can contribute to heating and to the broadening and shifting of atomic lines. Further measurements and calculations are required to better understand dissociation, recombination and other effects induced by the laser plasma.

Convection, a circulatory motion established in a fluid with a non-uniform temperature distribution, is driven by a difference in densities of the medium. Zhanga et. al [31] has recently modeled the temperature profiles in an alkali laser cell and state that temperatures in a Cs laser operating at a pump intensity of 2000W/cm^{2} can rise to 500-800 °K, while typical ambient Cs cell temperatures are ~373 °K. As temperature is increased by ~50%, the alkali density is reduced by ~50% establishing a slow convective flow of the gain medium, replenishing the neutral alkali density. Details of convection will be geometry and operating parameter dependent and will require further analysis.

Another mechanism is diffusion of neutral alkali atoms into the gain medium mode volume from the ambient neutral surrounding vapors. We suspect that low power alkali lasers can operate CW due to diffusion of neutral alkali atoms to the gain medium, because of the ionization rates being on the same order of magnitude of the diffusion rate. A simple estimate for the diffusion time, τ_{D}, is given by Eq. (11).

_{2}. A trend of decreasing values of D with increasing atomic mass of the buffer gas is apparent in the table and with N

_{2}having roughly the same mass as an ethane molecule, we crudely estimate the value of D for Cs in ethane will be near that of N

_{2}, which is estimated to be 0.07 cm

^{2}/sec at one atmosphere. This gives a diffusion time of τ

_{D}~35ms for a typical L of 0.05cm. For the laser described in [33], there is an observed exponential decay of 26 msec in the laser output performance as a function of time. The calculated ionization rate of this laser system which is about R

_{L}= 50 s

^{−1}, or a decay time of τ = 20ms. The observed decay time is very similar to both the ionization decay time and the diffusion time. Degradation of laser performance can occur due to both photoionization and heating. Beyond these first approximations, the solutions for both of these replenishing mechanisms, convection and diffusion, in an alkali laser can become quite involved and merit further investigation to fully understand their role in the operation of an alkali laser.

Similar to the convective process, the effective loss of neutral alkali atoms can also be mitigated by force flowing the gain medium, so that new alkali atoms enter the pumping region. Typical pumping regions for an alkali laser has a transverse dimension of about d = 0.1 – 1 cm. For a transverse flow speed of v = R_{L} d, and neglecting recombination effects, the alkali density will decrease by 1/e from the upstream to downstream side, assuming flat top beams. A laser will probably require a more uniform alkali density than this, so that the flow speed will have to be several times v. Table 3
shows the calculated flow speed v for the different cases of alkali lasers with different transverse.

These results show that very high, nearly supersonic flow speeds, will be required for a 10 atmosphere Rb-He laser and for an exciplex laser. These fast flow speeds will be problematic in these lasers, especially in maintaining good optical quality and in power requirements to maintain the flow. A one atmosphere alkali laser will require flow speeds of less than 1 m/sec or less. The effect of photoionization, required alkali density uniformity and flow speed will require detailed modeling.

## Conclusion

Photoionization rates have been calculated for standard alkali lasers and for an exciplex alkali laser. For standard alkali lasers, photon excitation to the nD or (n + 2)S states with subsequent photoionization is the dominant loss mechanism of neutral alkali atoms in the gain medium. For an exciplex alkali laser, energy pooling slightly contributes to the populating of nD or (n + 2)S states, however photo excitation and photoionization remains the dominant neutral atom loss mechanism. In certain cases, such as the 10 atmosphere Rb laser, where the pressures and intensities are higher and 5P→5D transitions are nearly resonant with the D2 pump light, the loss rates can approach 10^{5} sec^{−1}, which may require supersonic flow of the gain medium to replenish the neutral atoms for high power CW operation of the laser.

## Acknowledgements

We acknowledge support of the Air Force Office of Scientific Research, the Joint Technology Office for High Energy Lasers, and the National Science Foundation.

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