Flowing-gas diode pumped alkali lasers: theoretical analysis of transonic vs supersonic and subsonic devices

Eyal Yacoby; Karol Waichman; Oren Sadot; Boris D. Barmashenko; Salman Rosenwaks

doi:10.1364/OE.24.005469

1. Introduction

Diode pumped alkali lasers (DPALs) are populated via the D₂(n²S_1/2 → n²P_3/2) transition of the alkali atoms, followed by rapid relaxation (by buffer gas, helium and\or small hydrocarbon molecules) of the upper to the lower fine-structure level, n²P_3/2 to n²P_1/2 (designated as levels 3 and 2, respectively; the ground state n ²S_1/2 is designated as 1), resulting in D₁(n²P_1/2→ n²S_1/2) lasing [1,2]. In spite of the high efficiency of DPALs, high pumping power results in heat release, mainly due to the relaxation, and leads to temperature rise and decrease of the power and efficiency of the laser. Obviously, flowing the gas in the laser cell enhances the heat transfer out of the cell, impedes temperature rise and increases the power and efficiency of the laser.

Demonstration of subsonic DPALs [3,4] and modeling of both supersonic and subsonic devices [5–15] taking into account fluid dynamics and kinetic processes in the lasing medium, show the positive influence of the gas flow on the laser performance, where the highest lasing power and optical-to-optical efficiency were predicted for supersonic DPALs. In the present paper we examine transonic DPALs as a simpler alternative to supersonic devices, where complex hardware, including supersonic nozzle, diffuser and high power mechanical pump (needed for recovery of the gas total pressure which strongly drops in the diffuser), is required for continuous closed cycle operation [6] (the term “transonic” is chosen merely to imply high subsonic velocity and does not impose a new flow regime). The laser power P_lase of transonic lasers is compared with that of supersonic and subsonic lasers for optimal pressures, temperatures and buffer gas compositions in the laser section found in [5,6]. Pressure variations have the strongest influence on the laser power. On the one hand the rate of relaxation of n²P_3/2 increases with increasing pressure, resulting in increase of P_lase. On the other hand the rate of absorption on the D₂ pump transition decreases with increasing pressure for narrowband pumping (with bandwidth smaller than the absorption linewidth) resulting in decrease of P_lase. The trade-off between these two processes determines the optimal pressure (several atmospheres). For subsonic and supersonic lasers we used the same values of the Mach number M as in [5,6], (0.2 and ~2.5, respectively), whereas for transonic laser M = 0.9 was chosen, to be in the high subsonic flow regime while still sufficiently far from sonic, M = 1, conditions where transition to local supersonic may occur.

As discussed in [1,2], flowing-gas DPALs are of interest due to their scalability potential to very high power. In the present paper we compare different devices, with rather large cross sections of the flow, 5 x 5 cm², and of the pump beam, 2 x 5 cm², in order to learn if megawatt (MW) laser power can be achieved in them. The main conclusions of the comparison are valid also for lower pumping power (where the dimensions of the device are correspondingly smaller).

2. Description of the model

Schematic of a transonic device with transverse pumping dealt with in this paper is depicted in Fig. 1(a). Gas mixture of X/M (X = Cs, K and M = CH₄ or He, respectively) flows in a converging subsonic nozzle with A/A_i = H/H_i ~1.6, where A and A_i are the cross section areas of the nozzle inlet and exit, respectively, and H and H_i are the corresponding nozzle heights. A laser section of constant height H_i, width W and length L (in the flow direction x) is located downstream of the nozzle exit. Both the pump and laser beams fill the entire laser section volume. Due to different values of spin–orbit splitting ΔE, the relaxation of the n²P_3/2 levels of K and Cs atoms is carried out by different buffer gases. For the relatively small ΔE = 57.7 cm⁻¹ of K, efficient spin–orbit mixing can be provided by pure helium as a buffer gas at practical pressures ~1atm. However, for Cs atoms, where ΔE = 554.1 cm⁻¹ is rather large (compared to K), hydrocarbons (methane and ethane) buffer gases are used for efficient relaxation. A drawback of these buffer gases is that they may react with Cs atoms, contaminate the gain medium and decrease the beam quality due their relatively high refraction index [1].

Fig. 1 Schematics of the transonic (a), supersonic (b) and subsonic (c) devices with transverse pumping by rectangular pump beam of 5 x 2 cm² aperture. The flow is perpendicular to the laser and pump beams propagation axes. Note that different buffer gases are used for Cs and K.

Download Full Size | PDF

For comparing the transonic DPAL performance with that of subsonic and supersonic, the same alkali number density, n_X_,i = 8.4 × 10¹³ cm⁻³, at the laser section inlet is assumed. Based on this density and Mach number at the laser section inlet, M_i, the inlet temperature is calculated as follows. The values of T_i, p_i and n_X_,i at the inlet to the laser section are related to their respective stagnation values (with the subscript “stag”) by the isentropic flow equations [16]:

\frac{T_{i}}{T_{s t a g}} = {(\frac{p_{i}}{p_{s t a g}})}^{\frac{γ - 1}{γ}} = {(\frac{n_{X, i n}}{n_{X, s t a g}})}^{γ - 1} = {(1 + \frac{γ - 1}{2} M_{i}^{2})}^{- 1},

where γ is the ratio of heat capacities at the operating temperatures of ~400 −550 K, γ_He = 1.67, γ_CH4 = 1.26-1.20 and T_i, p_i are the temperature and pressure at the laser section inlet, respectively. We assume that the temperature of the alkali source is equal to the stagnation temperature, T_stag, of the flow and thus find T_i which satisfies the condition n_sat(T_stag) > n_X,stag where n_sat(T_stag) is the saturated vapor number density at the vapor source. This condition is necessary to prevent alkali vapor nucleation. The saturated alkali number density is:

n_{s a t} = \frac{p_{s a t}}{k_{B} T_{s t a g}} \approx \frac{10^{a - \frac{b}{T_{s t a g}}}}{k_{B} T_{s t a g}} .

where p_sat is the saturated vapor pressure given in the usual exponential form with constants a and b according to the alkali in question and

k_{B}

is Boltzmann constant [1]. As follows from Eq. (1) the alkali number density at the stagnation conditions is:

n_{X, s t a g} = n_{X, i} {(1 + \frac{γ - 1}{2} M_{i}^{2})}^{\frac{1}{γ - 1}}

T_stag is found from the solution of the transcendental equation n_sat = n_X,stag, Eqs. (2) and (3), and the corresponding T_i is calculated from Eq. (1). M at the nozzle inlet can also be calculated from the nozzle area ratio A/A_i and M at the nozzle exit. At the nozzle inlet

A / A * = (A / A_{i}) {(A_{i} / A *)}_{M = 0.9}

, where

A *

is the critical cross section of the nozzle corresponding to M = 1. Using the isentropic flow relations [16] one finds for both DPALs studied, K and Cs, that the nozzle inlet M is ~0.4. For given M_i, T_i, and n_X_,i optimal values of p_i corresponding to the maximum output power were used in the calculations (this point is dealt with in the Introduction).

To avoid condensation of X, the nozzle walls are heated to a higher temperature than that of the saturated X vapor. The laser resonator consists of a reflector and an output coupler with reflectivity r₁ (close to 100%) and r₂, located outside the laser cell. The schematic of the supersonic and subsonic devices also studied in this paper is presented in Figs. 1(b) and 1(c), respectively.

The model considers transverse flow in the x direction perpendicular to the laser optical axis z and transverse pumping configuration where the pump beams with spectral bandwidth ∆ν_p propagate perpendicularly to the optical axis z in the y direction (Fig. 1).

The flow and beam parameters in the laser section are determined from a three-dimensional computational fluid dynamics (3D CFD) model based on an ANSYS commercial code that calculates the gas dynamics and chemistry in the active medium. The flow-field is computed by a set of 3D conservation Eqs. (-) mass and momentum conservation for the pressure and velocity fields, and energy conservation for the gas temperature field. The molecular transport and chemical kinetics are defined by the species conservation equations, where the species represent the different energy states of the alkali atoms. The transport equations and the kinetics processes in K and Cs are presented in more detail in [6,15].

The computational mesh for the resonator section consists of n_x x n_y x n_z hexahedral cells (n_x = 30, n_y = 80, n_z = 80) and is shown in Fig. 2. The flow boundary conditions were the mass flow rate at the “flow in” and the pressure outlet at the “flow out” boundaries. The boundary conditions allow for the required flow velocity, gas temperature, mixture composition and gas pressure in the medium. For the walls we assumed constant temperature.

Fig. 2 The mesh for the rectangular transverse-pumping beam, consists of n_x x n_y x n_z hexahedral cells (n_x = 30, n_y = 80, n_z = 80).

Download Full Size | PDF

The model of Fabry-Perot resonator, often employed in flowing-gas lasers with wide mirror aperture [17] was applied for calculating the laser intensity in the resonator. The laser beam is divided into n_x x n_y very narrow beams propagating independently in the optical axis direction z. The intensity of each part of the rectangular beam depends on both the x and y coordinates. The reason is that the local gain coefficients for the pump and laser beams are not uniform in their cross sections, resulting in non-uniform distributions of the pump and laser intensities. Similarly to the laser beam, the pump beams are also divided into n_x x n_z very narrow beams propagating independently in the y direction. The pump and laser intensities are described by the Beer-Lambert law [6,15]. In the case of broadband pumping the spectral distribution of the pump intensity $I_{p}^{\pm} (x, y, z, v)$ over the frequency v should be taken into account; the superscripts + and – indicate the pump beams propagating in the opposite directions + z and –z. The rates of changes of $I_{p}^{\pm} (x, y, z, ν)$ with z are given by

\frac{\partial I_{p}^{\pm} (x, y, z, ν)}{\partial y} = \pm g_{31} (x, y, z, ν) I_{p}^{\pm} (x, y, z, ν),

where

g_{31} (x, y, z, ν)

is the local gain coefficient (usually negative) of the pump D₂ transition at a location (x, y, z) given by:

g_{31} (x, y, z, ν) = σ_{31} (x, y, z, ν) [n_{3} (x, y, z) - 2 n_{1} (x, y, z)],

where

σ_{31} (x, y, z, ν) = σ_{31} (x, y, z, ν_{p}) f_{31} (ν)

is the stimulated emission cross section for pumping the D₂ transition at frequency v _,

σ_{31} (x, y, z, ν) = σ_{31, a t} \frac{Δ ν_{31, n}}{Δ ν_{31} (x, y, z)},

n_i is the number density of the alkali specie at energy level i = 1, 2, 3 obtained from the kinetic equations as described in [6,15],

f_{31} (ν) = {(Δ ν_{31} / 2)}^{2} / [{(ν_{} - ν_{p})}^{2} + {(Δ ν_{31} / 2)}^{2}]

and ν_p is the pump frequency.

In Eq. (7) σ₃₁( $x, y, z$ ,ν_p) and σ_31,_at are the stimulated emission and atomic cross section, respectively, in the line center and Δν_31,_n and Δν₃₁(x, y, z) are the natural and the pressure broadened widths of the D₂ transition, respectively, where the spatial variations of σ₃₁( $x, y, z$ ,ν_p) are caused mainly by the variations of the gas temperature. In Eq. (8), f₃₁(ν) is the pressure broadened Lorentzian spectral distribution.

The boundary condition for $I_{p}^{+}$ at z = 0 and $I_{p}^{-}$ at z = W are

I_{p}^{\pm} (ν) = \frac{P_{p}}{W L} t f_{p} (ν),

where P_p is the pump power, t is the windows transmission and

f_{p} (ν) = {(\frac{4 \ln 2}{π Δ ν_{p}^{2}})}^{1 / 2} \exp [- 4 \ln 2 {(\frac{ν - ν_{p}}{Δ ν_{p}^{}})}^{2}]

is the pump normalized Gaussian spectral distribution of full width at half maximum (FWHM) Δν_p [15].

The laser intensities in both directions $I_{l}^{\pm} (x, y, z)$ , where z = 0 is assumed to be located at the inner surface of the window closer to the output coupling mirror with reflectivity r₂, are assumed to be monochromatic and their spatial variations are described by equations similar to (4):

\frac{\partial I_{l}^{\pm} (x, y, z)}{\partial z} = \pm g_{21} (x, y, z) I_{l}^{\pm} (x, y, z),

where

g_{21} (x, y, z)

is the local gain coefficient of the laser D₁ transition given by:

g_{21} (x, y, z) = σ_{21} (x, y, z, ν_{l}) [n_{2} (x, y, z) - n_{1} (x, y, z)],

where

σ_{21} (x, y, z, ν_{l}) = σ_{21, a t} \frac{Δ ν_{21, n}}{Δ ν_{21} (x, y, z)}

is the stimulated emission cross section for the lasing D₁ transition at frequency ν_l_{, $σ_{21, a t}$} the atomic cross section in the line center and

Δ ν_{21, n}

and

Δ ν_{21} (x, y, z)

are the natural and the pressure broadened widths of the D₁ transition, respectively.

The boundary conditions for the two way laser intensities $I_{l}^{+}$ and $I_{l}^{-}$ at z = 0, were calculated by ray tracing of the given output laser flux, $I_{l a s e} (x, y)$ : $I_{l}^{-} (x, y, 0) = I_{l a s e} (x, y) / [t (1 - r_{2})]$ and $I_{l}^{+} (x, y, 0) = I_{l}^{-} (x, y, 0) t^{2} r_{2} .$ The solution for the output laser flux was found by iterating I_lase until $I_{l}^{-} (x, y, z = W) / I_{l}^{+} (x, y, z = W) = t^{2} r_{1} .$ The output power $P_{l a s e}$ is given by

P_{l a s e} = \iint_{0 < x < L, 0 < y < H} I_{l a s e} (x, y) d x d y .

3. Results and discussion

Figure 3 shows the dependence of P_lase on the pump power, P_p, for subsonic (M ~0.2) transonic (M ~0.9) and supersonic (M ~2.7) Cs DPALs with CH₄ as a buffer gas, and with optimal parameters that are summarized in Table 1. We found out that variations of M by ~ ± 10% for each of the flow regimes at a given temperature in the laser section, result in a very weak changes of P_lase ~ ± 1%. Thus, calculations made for the chosen values of M show the main trends for the laser power dependence on the flow regime which are by and large independent of the exact values of M and depend mainly on p_i and T_i.

Fig. 3 Dependence of P_lase on P_p for subsonic (M ~0.2) transonic (M ~0.9) and supersonic (M ~2.7) Cs DPALs with CH₄ as a buffer gas, and with optimal parameters summarized in Table 1.

Download Full Size | PDF

Table 1. Optimal parameters of different Cs DPALs with CH₄ as a buffer gas, transversely pumped with 5 x 2 cm² rectangular pump beam, with reflector r₁ = 0.95, output coupler r₂ = 0.35 and windows transmission t = 0.98. For transonic flow p_stag = 10 atm, T_stag = 425K and for supersonic p_stag = 70 atm, T_stag = 493 K.

View Table | View all tables in this article

Calculations were first performed for narrowband pumping of Cs with pump bandwidth ∆ν_p = 10 GHz (examples 1-3 in Table 1). The maximum achievable power of ~1.8 MW for the supersonic device obtained at P_p ~3.5 MW is higher than that of the transonic and subsonic devices by only ~3% and ~10%, respectively, and the value of the maximum optical-to-optical efficiency of ~80% is about the same for the three cases (cases 1-3 in Table 1). Thus, in the case of Cs DPALs the supersonic operation mode has only small advantage over transonic. At the same time for the subsonic laser the maximum power is by 7% lower than for the transonic, showing a larger advantage of the transonic over the subsonic operation mode. Extremely large ~60% losses of alkali atoms due to ionization at P_p = 3.5 MW result in substantial decrease of the power and the efficiency of the laser. The reason for these losses is a very high intra-resonator radiation intensity of I_p ~2 × 10⁹ W/m² near the pump beam entrances. The active medium temperature rise in both supersonic and transonic cases is negligible (~10 K) due to their high flow velocities of ~1160 m/s and ~450 m/s, respectively, and the main reason for the small difference between the curves is the lower inlet temperature of the former.

The maximum values of the pump power exceeding hundreds of kW and having narrowband spectrum are not easily achievable for the present state-of-the-art. That is why we also performed computations for more realistic case of broadband pumping (example 4 in Table 1). The results are shown in Fig. 3 where the dependence P_lase(P_p) is presented for the transonic Cs DPAL with ∆ν_p = 200 GHz (case 4 in Table 1). For the transonic laser with ∆ν_p = 10 GHz, P_las_e is larger than that calculated for ∆ν_p = 200 GHz over the whole range of P_p, its maximum value ~1.75 MW achievable at P_p ~3.5 MW being larger than that calculated for ∆ν_p = 200 GHz by only ~10%. The reason for this relatively small difference is the high gas mixture pressure (6 atm) that results in increase of the pumping beam absorption due to the increase of the D₂ transition spectral broadening.

Figure 4 shows the dependence P_lase(P_p) for subsonic (M ~0.2) transonic (M ~0.9) and supersonic (M ~2.4) K DPALs with He as a buffer gas and with optimal parameters that were found in [5,6], and are summarized in Table 2. In this case the maximum achievable power of ~2.8 MW in the supersonic K DPAL is by more than 45% larger than ~1.9 MW obtained for the transonic laser with the same resonator and K vapor density at the inlet. These powers are higher than the power achievable in the subsonic laser, by ~80% and ~20%, respectively, showing considerable advantages of the supersonic device over the transonic and of the transonic device over the subsonic. The fact that the power increase for the K supersonic laser is much larger than for the Cs device is explained by a much smaller fine-structure splitting of the ²P state (~58 cm⁻¹ for K and ~554 cm⁻¹ for Cs). Indeed, for rapid relaxation between the fine structure levels 3 and 2 the ratio between the populations of these levels is given by Boltzmann distribution:

\frac{n_{3}}{n_{2}} = 2 \exp (- \frac{Δ E}{k_{B} T}) .

High T results in increased

n_{3}

and decreased

n_{2}

and thus in decreased pump absorption and lasing rate, i.e., low optical-to-optical efficiency. The influence of the temperature on the population ratio is stronger for smaller

Δ E

, resulting in a much stronger effect of the temperature decrease in the supersonic K DPAL.

Fig. 4 Dependence of P_lase on P_p for subsonic (M ~0.2) transonic (M ~0.9) and supersonic (M ~2.4) К DPALs with He as a buffer gas, and with optimal parameters summarized in Table 2.

Download Full Size | PDF

Table 2. Optimal parameters of different K DPALs with He as a buffer gas, transversely pumped with 5 x 2 cm2 rectangular pump beam, with reflector r₁ = 0.95, output coupler r₂ = 0.35 and windows transmission t = 0.98; the pump bandwidth $Δ v_{p}$ = 10 GHz. For transonic flow p_stag = 11 atm, T_stag = 485K and for supersonic p_stag = 90 atm, T_stag = 586 K.

View Table | View all tables in this article

4. Summary

Comparative 3D CFD modeling of supersonic, transonic and subsonic DPALs pumped by large cross section (5 x 2 cm²) beams, taking into account fluid dynamics and kinetic processes in the lasing medium, is reported. For Cs DPALs the supersonic operation mode has no substantial advantage over the transonic one, the main processes limiting the power being saturation of the D₂ transition and large ~60% losses of alkali atoms due to ionization, whereas the influence of gas heating is negligible. At the same time for K DPALs the maximum value of the power in the supersonic laser is higher than that calculated for the transonic laser by more than 45%, showing a considerable advantage of the supersonic device over the transonic; the power of the transonic laser is by 20% higher than that of the subsonic laser showing an advantage of transonic device over subsonic. For broadband (~200 GHz) pumping of transonic Cs DPALs, the maximum achievable P_lase is smaller than that for the narrowband (10 GHz) transonic case by only ~10%. The reason for this relatively small difference is the high optimal pressure of the gas mixture that increases the pumping beam absorption by increasing the D₂ transition spectral broadening.

Acknowledgments

Effort sponsored by High Energy Laser Joint Technology Office (HEL-JTO) and the European Office of Aerospace Research and Development (EOARD) under grant FA9550-15-1-0489 and by Israel Science Foundation (ISF) under grant 893/15.

References and links

1. W. F. Krupke, “Diode pumped alkali lasers (DPALs) — A review (rev1),” J. Progr. Quantum Electron. 36(1), 4–28 (2012). [CrossRef]

2. B. V. Zhdanov and R. J. Knize, “Review of alkali laser research and development,” Opt. Eng. 52(2), 021010 (2012). [CrossRef]

3. A. V. Bogachev, S. G. Garanin, A. M. Dudov, V. A. Eroshenko, S. M. Kulikov, G. T. Mikaelian, V. A. Panarin, V. O. Pautov, A. V. Rus, and S. A. Sukharev, “Diode-pumped caesium vapour laser with closed-cycle laser-active medium circulation,” J. Progr. Quantum Electron. 42(2), 95–98 (2012). [CrossRef]

4. B. V. Zhdanov, M. D. Rotondaro, M. K. Shaffer, and R. J. Knize, “Potassium diode pumped alkali laser demonstration using a closed cycle flowing system,” Opt. Commun. 354, 256–258 (2015). [CrossRef]

5. B. D. Barmashenko and S. Rosenwaks, “Feasibility of supersonic diode pumped alkali lasers: model calculations,” Appl. Phys. Lett. 102(14), 141108 (2013). [CrossRef]

6. E. Yacoby, K. Waichman, O. Sadot, B. D. Barmashenko, and S. Rosenwaks, “Modeling of supersonic diode pumped alkali lasers,” J. Opt. Soc. Am. B 32(9), 1824–1833 (2015). [CrossRef]

7. Z. Yang, H. Wang, Q. Lu, Y. Li, W. Hua, X. Xu, and J. Chen, “Modeling, numerical approach, and power scaling of alkali vapor lasers in side-pumped configuration with flowing medium,” J. Opt. Soc. Am. B 28(6), 1353–1364 (2011). [CrossRef]

8. B. D. Barmashenko and S. Rosenwaks, “Modeling of flowing gas diode pumped alkali lasers: dependence of the operation on the gas velocity and on the nature of the buffer gas,” Opt. Lett. 37(17), 3615–3617 (2012). [CrossRef] [PubMed]

9. B. D. Barmashenko and S. Rosenwaks, “Detailed analysis of kinetic and fluid dynamic processes in diode-pumped alkali lasers,” J. Opt. Soc. Am. B 30(5), 1118–1126 (2013). [CrossRef]

10. S. Rosenwaks, B. D. Barmashenko, and K. Waichman, “What can we gain from supersonic operation of diode pumped alkali lasers: model calculations,” Proc. SPIE 8898, 88980X (2013). [CrossRef]

11. B. D. Barmashenko, S. Rosenwaks, and K. Waichman, “Model calculations of kinetic and fluid dynamic processes in diode pumped alkali lasers,” Proc. SPIE 8898, 88980W (2013). [CrossRef]

12. B. D. Barmashenko, S. Rosenwaks, and K. Waichman, “Kinetic and fluid dynamic processes in diode pumped alkali lasers: semi-analytical and 2D and 3D CFD modeling,” Proc. SPIE 8962, 86920C (2014).

13. S. Rosenwaks, B. D. Barmashenko, and K. Waichman, “Semi-analytical and 3D CFD DPAL modeling: Feasibility of supersonic operation,” Proc. SPIE 8962, 896209 (2014). [CrossRef]

14. K. Waichman, B. D. Barmashenko, and S. Rosenwaks, “CFD DPAL modeling for various schemes of flow configurations,” Proc. SPIE 9251, 92510U (2014).

15. K. Waichman, B. D. Barmashenko, and S. Rosenwaks, “Computational fluid dynamics modeling of subsonic flowing-gas diode pumped alkali lasers: comparison with semi-analytical model calculations and with experimental results,” J. Opt. Soc. Am. B 31(11), 2628–2637 (2014). [CrossRef]

16. A. H. Shapiro, The Dynamics and Thermodynamics of Compressible Fluid Flow, Vol. 1 (Ronald, 1953), pp. 73–152.

17. B. D. Barmashenko and S. Rosenwaks, “Analysis of the optical extraction efficiency in gas-flow lasers with different types of resonator,” Appl. Opt. 35(36), 7091–7101 (1996). [CrossRef] [PubMed]

No.	Flow	T_i(K)	p_i(atm)	M	u_i(m/s)	∆ν_p(GHz)
1	Subsonic	403	6	0.2	100	10
2	Transonic	385	6	0.9	450	10
3	Supersonic	253	3	2.7	1160	10
4	Transonic	385	6	0.9	450	200

No.	Flow	T_i(K)	p_i(atm)	n_X,i(cm^-3)	M	u_i(m/s)
1	Subsonic	460	5.7	$8.4 \cdot 10^{13}$	0.2	250
2	Transonic	380	6	$8.4 \cdot 10^{13}$	0.9	1150
3	Supersonic	200	6	$8.4 \cdot 10^{13}$	2.4	2000

No.	Flow	T_i(K)	p_i(atm)	M	u_i(m/s)	∆ν_p(GHz)
1	Subsonic	403	6	0.2	100	10
2	Transonic	385	6	0.9	450	10
3	Supersonic	253	3	2.7	1160	10
4	Transonic	385	6	0.9	450	200

No.	Flow	T_i(K)	p_i(atm)	n_X,i(cm^-3)	M	u_i(m/s)
1	Subsonic	460	5.7	$8.4 \cdot 10^{13}$	0.2	250
2	Transonic	380	6	$8.4 \cdot 10^{13}$	0.9	1150
3	Supersonic	200	6	$8.4 \cdot 10^{13}$	2.4	2000

Flowing-gas diode pumped alkali lasers: theoretical analysis of transonic vs supersonic and subsonic devices

Abstract

1. Introduction

2. Description of the model

3. Results and discussion

4. Summary

Acknowledgments

References and links

Cited By

Figures (4)

Tables (2)

Equations (15)

Optics Express