3D CFD modeling of flowing-gas Rb DPALs: effects of buffer gas composition and of ionization of high lying Rb states

Karol Waichman; Boris D. Barmashenko; Salman Rosenwaks

doi:10.1364/JOSAB.441871

Journal of the Optical Society of America B
Vol. 38,
Issue 11,
pp. 3523-3531
(2021)
•https://doi.org/10.1364/JOSAB.441871

3D CFD modeling of flowing-gas Rb DPALs: effects of buffer gas composition and of ionization of high lying Rb states

Karol Waichman, Boris D. Barmashenko, and Salman Rosenwaks

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Karol Waichman, Boris D. Barmashenko, and Salman Rosenwaks, "3D CFD modeling of flowing-gas Rb DPALs: effects of buffer gas composition and of ionization of high lying Rb states," J. Opt. Soc. Am. B 38, 3523-3531 (2021)

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About this Article
History
- Original Manuscript: August 30, 2021
- Revised Manuscript: October 3, 2021
- Manuscript Accepted: October 5, 2021
- Published: November 1, 2021

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Abstract

A comprehensive three-dimensional computational fluid dynamics (3D CFD) modeling of flowing-gas Rb diode pumped alkali laser (DPAL) is carried out. The cases of ${\rm{He}}/{{\rm{CH}}_4}$ and pure He buffer gases are investigated, and the output power and optical efficiency are calculated for various pump powers, mole fractions of methane, buffer gas pressures, and flow velocities. The model considers the processes of excitation of high levels of Rb, ionization, ion-electron recombination, and heating of electrons, which affect the diffusion coefficient of Rb ions. Two types of Rb DPAL were studied: a low-power laboratory-scale device with pump power of several tens of watts and a high-power multi-kilowatt laser. Efficient operation of the Rb laser using pure He as buffer gas can be achieved only in a large-scale laser with a pump beam cross-sectional area of several ${\rm{cm}}^2$. The calculated results for such a device were compared with those reported by Gavrielides et al. [J. Opt. Soc. Am. B 35, 2202 (2018) [CrossRef] ], where a simplified three-level model based on the one-dimensional gas dynamics approach was applied.

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No.	Process	Cross Section	Rate Constant	Ref.
	Pumping
1	$R b (5^{2} S_{1 / 2}) + h ν_{p} \to R b (5^{2} P_{3 / 2})$	Eq. (13) from Ref. [10]		[10]
	Relaxation
2	$R b (5^{2} P_{3 / 2}) + M \to R b (5^{2} P_{1 / 2}) + M$			[11]
	$M = {C H}_{4}$	$42.3$ Å²
	$M = H e$		$[2.34 (T / 300)^{3.5} + 9.59 (T / 300)] \times 10^{- 19} m^{3} / s$	[12]
	Lasing
3	$R b (5^{2} P_{1 / 2}) + n h ν_{l} \to R b (5^{2} S_{1 / 2}) + (n + 1) h ν_{l}$	Eq. (20) from Ref. [10]		[10]
	Spontaneous emission
4	$R b (5^{2} P_{3 / 2}) \to R b (5^{2} S_{1 / 2}) + h ν_{p}$		$τ_{4} = 26.2 \times 10^{- 9} s$	[13]
5	$R b (5^{2} P_{1 / 2}) \to R b (5^{2} S_{1 / 2}) + h ν_{l}$		$τ_{5} = 27.7 \times 10^{- 9} s$	[13]
	Quenching
6	$R b (5^{2} P_{3 / 2}) + {C H}_{4} \to R b (5^{2} S_{1 / 2}) + {C H}_{4}$	$7.52 e - 3$ Å²		[11]
7	$R b (5^{2} P_{1 / 2}) + {C H}_{4} \to R b (5^{2} S_{1 / 2}) + {C H}_{4}$	$7.52 e - 3$ Å²		[11]
	Photoexcitation
8	$R b (5^{2} P_{3 / 2, 1 / 2}) + h ν_{p, l} \to R b (5^{2} D_{3 / 2, 5 / 2} 6^{2} P_{3 / 2, 1 / 2}$ and $7^{2} S_{1 / 2})$	Eq. (13) from Ref. [14] with broadening rate of 60 MHz/torr		[15,16]
	Energy pooling			[8,15]
9	$2 R b (5^{2} P_{3 / 2}) \to R b (5^{2} D_{3 / 2, 5 / 2}) + R b (5^{2} S_{1 / 2})$	$310$ Å²
10	$2 R b (5^{2} P_{3 / 2}) \to R b (7^{2} S_{1 / 2}) + R b (5^{2} S_{1 / 2})$	$32.1$ Å²
11	$2 R b (5^{2} P_{3 / 2}) \to R b (6^{2} P_{3 / 2, 1 / 2}) + R b (5^{2} S_{1 / 2})$	$0.45$ Å²
12	$2 R b (5^{2} P_{1 / 2}) \to R b (5^{2} D_{3 / 2, 5 / 2}) + R b (5^{2} S_{1 / 2})$	$78$ Å²
13	$2 R b (5^{2} P_{1 / 2}) \to R b (6^{2} P_{3 / 2, 1 / 2}) + R b (5^{2} S_{1 / 2})$	$0.45$ Å²
14	$R b (5^{2} P_{1 / 2}) + R b (5^{2} P_{3 / 2}) \to R b (5^{2} D_{3 / 2, 5 / 2}) + R b (5^{2} S_{1 / 2})$	$290$ Å²
	Spontaneous emission		$s^{- 1}$	[17]
15	$R b (5^{2} D_{5 / 2, 3 / 2}) \to R b (5^{2} P_{3 / 2}) + h ν$		$1.8 \times 1 0^{6}$
16	$R b (5^{2} D_{5 / 2},_{3 / 2}) \to R b (5^{2} P_{1 / 2}) + h ν$		$8.48 \times 1 0^{5}$
17	$R b (5^{2} D_{5 / 2},_{3 / 2}) \to R b (6^{2} P_{3 / 2, 1 / 2}) + h ν$		$1.5 \times 1 0^{6}$
18	$R b (7^{2} S_{1 / 2}) \to R b (5^{2} P_{3 / 2}) + h ν$		$3.87 \times 1 0^{6}$
19	$R b (7^{2} S_{1 / 2}) \to R b (5^{2} P_{1 / 2}) + h ν$		$2.02 \times 1 0^{6}$
20	$R b (7^{2} S_{1 / 2}) \to R b (6^{2} P_{3 / 2, 1 / 2}) + h ν$		$4.26 \times 1 0^{6}$
21	$R b (6^{2} P_{3 / 2, 1 / 2}) \to R b (5^{2} S_{1 / 2}) + h ν$		$1.59 \times 1 0^{6}$
22	$R b (6^{2} P_{3 / 2, 1 / 2}) \to R b (5^{2} P_{3 / 2}) + h ν$		$4.26 \times 1 0^{6}$
23	$R b (6^{2} P_{3 / 2, 1 / 2}) \to R b (5^{2} P_{1 / 2}) + h ν$		$2.2 \times 1 0^{6}$
	Photoionization
24	$R b (5^{2} D_{3 / 2, 5 / 2}, 6^{2} P_{3 / 2, 1 / 2}, 7^{2} S_{1 / 2}) + h ν_{p, l} \to K^{+} + e$	$0.16 A^{2}$ for $5^{2} D_{3 / 2, 5 / 2}$ $0.15 A^{2}$ for $6^{2} P_{3 / 2, 1 / 2}$		[8,15]
	Penning ionization
25	$R b (5^{2} D_{3 / 2, 5 / 2}, 6^{2} P_{3 / 2, 1 / 2}, 7^{2} S_{1 / 2}) + R b (5^{2} P_{3 / 2, 1 / 2}) \to R b^{+} + R b (5^{2} S_{1 / 2}) + e$ , total 10 reactions		$4.5 \times 1 0^{- 14} m^{3} / s$ for $5^{2} D_{3 / 2, 5 / 2}$ and $6^{2} P_{3 / 2, 1 / 2}$ $2 \times 1 0^{- 14} m^{3} / s$ for $7^{2} S_{1 / 2}$	[8,15]
	Recombination			[18]^b
26	$R b^{+} + 2 R b \to {R b}_{2}^{+} + R b$		$2.4 \times 1 0^{- 33} (293 . 15 / T) m^{6} / s$
27	$R b^{+} + R b + H e \to {R b}_{2}^{+} + H e$		$2.4 \times 1 0^{- 35} (293 . 15 / T) m^{6} / s$
28	${R b}_{2}^{+} + e \to R b (5^{2} D_{3 / 2, 5 / 2}, 6^{2} P_{3 / 2, 1 / 2}, 7^{2} S_{1 / 2}) + R b (5^{2} S_{1 / 2})$		$1 \times 1 0^{- 14} m^{3} / s$
29	$R b^{+} + e + e \to R b (5^{2} D_{3 / 2, 5 / 2}, 6^{2} P_{3 / 2, 1 / 2}, 7^{2} S_{1 / 2}) + e$		$2.4 \times 1 0^{- 22} \times T^{- 4.2} m^{6} / s$
30	$R b^{+} + e + M (H e, {C H}_{4}) \to R b (5^{2} D_{3 / 2, 5 / 2}, 6^{2} P_{3 / 2, 1 / 2}, 7^{2} S_{1 / 2}) + e$		$4 \times 1 0^{- 41} \times (T / 625)^{- 2.5} m^{6} / s$
	Quenching of the higher levels
31	$R b (5^{2} D_{5 / 2},_{3 / 2}) + {C H}_{4} \to R b (6^{2} P_{3 / 2, 1 / 2}) + {C H}_{4}$	$84 A^{2}$		[19]
32	$R b (7^{2} S_{1 / 2}) + {C H}_{4} \to R b (6^{2} P_{3 / 2, 1 / 2}) + {C H}_{4}$	$84 A^{2}$		[19]
33	$R b (6^{2} P_{3 / 2, 1 / 2}) + {C H}_{4} \to R b (5^{2} P_{3 / 2, 1 / 2}) + {C H}_{4}$	$84 A^{2}$		[19]
		$B r a n c h i n g r a t i o s 2 / 3, 1 / 3$
34	$R b (5^{2} D_{5 / 2},_{3 / 2}) + H e \to R b (6^{2} P_{3 / 2, 1 / 2}) + H e$		$2.27 \times 1 0^{- 18}$	[20]^b
35	$R b (7^{2} S_{1 / 2}) + H e \to R b (6^{2} P_{3 / 2, 1 / 2}) + H e$		$3.58 \times 1 0^{- 19}$	[20]
36	$R b (5^{2} D_{5 / 2},_{3 / 2}) + H e \to R b (5^{2} P_{3 / 2, 1 / 2}) + H e$		$2.29 \times 1 0^{- 18}$	[20]
37	$R b (6^{2} P_{3 / 2, 1 / 2}) + H e \to R b (5^{2} P_{3 / 2, 1 / 2}) + H e$		$1.64 \times 1 0^{- 19}$	[20]
38	$R b (7^{2} S_{1 / 2}) + H e \to R b (5^{2} P_{3 / 2, 1 / 2}) + H e$		$8.52 \times 1 0^{- 19}$	[20]
	Electron heating and energy losses
39	$e + R b (5^{2} P_{3 / 2, 1 / 2}) ⇄ e_{h o t} + R b (5^{2} S_{1 / 2})$			[18]
40	$e_{h o t} + {C H}_{4} \leftrightarrow e + {C H}_{4} (ν_{2}, ν_{4})$			[18]
41	$e_{h o t} + H e \to e + H e$			[18]

Pump beam	Profile	$w_{0 x}$	$w_{0 y}$	$z_{Rx}$	$z_{Ry}$
Pump beam	Gaussian	0.4 mm	0.75 mm	10 mm	10 mm
Laser beam	Profile	$w_{0 x}$	$w_{0 y}$	$z_{Rx}$	$z_{Ry}$
Laser beam	Gaussian	0.5 mm	0.5 mm	∞	∞
Optics	Output coupler reflectivity	Pump window transmission	Laser window transmission	Back mirror reflectivity	Gain length
Optics	0.6	0.95	0.98	0.99	1.5 cm
Gas at the inlet	Temperature	Pressure	Velocity	Rb vapor	${C H}_{4} / H e$
Gas at the inlet	393 [K]	1 atm.@293 K	1 [m/s]	403 [K]	1/3 molar

Pump beam	Profile	$w_{0 x}$	$w_{0 y}$	$z_{Rx}$	$z_{Ry}$
Pump beam	Top-hat	2.5 mm	2.5 mm	∞	∞
Laser beam	Profile	$w_{0 x}$	$w_{0 y}$	$z_{Rx}$	$z_{Ry}$
Laser beam	Gaussian	2.5 mm	2.5 mm	∞	∞
Optics	Output coupler reflectivity	Pump window transmission	Laser window transmission	Back mirror reflectivity	Gain length
Optics	0.2	0.95	0.98	0.99	2 cm
Gas at the inlet	Temperature	Pressure	Velocity	Rb vapor	${C H}_{4} / H e$
Gas at the inlet	423 [K]	3760 torr @293 K	50[m/s]	433 [K]	1/3 molar

No.	Process	Cross Section	Rate Constant	Ref.
	Pumping
1	$R b (5^{2} S_{1 / 2}) + h ν_{p} \to R b (5^{2} P_{3 / 2})$	Eq. (13) from Ref. [10]		[10]
	Relaxation
2	$R b (5^{2} P_{3 / 2}) + M \to R b (5^{2} P_{1 / 2}) + M$			[11]
	$M = {C H}_{4}$	$42.3$ Å²
	$M = H e$		$[2.34 (T / 300)^{3.5} + 9.59 (T / 300)] \times 10^{- 19} m^{3} / s$	[12]
	Lasing
3	$R b (5^{2} P_{1 / 2}) + n h ν_{l} \to R b (5^{2} S_{1 / 2}) + (n + 1) h ν_{l}$	Eq. (20) from Ref. [10]		[10]
	Spontaneous emission
4	$R b (5^{2} P_{3 / 2}) \to R b (5^{2} S_{1 / 2}) + h ν_{p}$		$τ_{4} = 26.2 \times 10^{- 9} s$	[13]
5	$R b (5^{2} P_{1 / 2}) \to R b (5^{2} S_{1 / 2}) + h ν_{l}$		$τ_{5} = 27.7 \times 10^{- 9} s$	[13]
	Quenching
6	$R b (5^{2} P_{3 / 2}) + {C H}_{4} \to R b (5^{2} S_{1 / 2}) + {C H}_{4}$	$7.52 e - 3$ Å²		[11]
7	$R b (5^{2} P_{1 / 2}) + {C H}_{4} \to R b (5^{2} S_{1 / 2}) + {C H}_{4}$	$7.52 e - 3$ Å²		[11]
	Photoexcitation
8	$R b (5^{2} P_{3 / 2, 1 / 2}) + h ν_{p, l} \to R b (5^{2} D_{3 / 2, 5 / 2} 6^{2} P_{3 / 2, 1 / 2}$ and $7^{2} S_{1 / 2})$	Eq. (13) from Ref. [14] with broadening rate of 60 MHz/torr		[15,16]
	Energy pooling			[8,15]
9	$2 R b (5^{2} P_{3 / 2}) \to R b (5^{2} D_{3 / 2, 5 / 2}) + R b (5^{2} S_{1 / 2})$	$310$ Å²
10	$2 R b (5^{2} P_{3 / 2}) \to R b (7^{2} S_{1 / 2}) + R b (5^{2} S_{1 / 2})$	$32.1$ Å²
11	$2 R b (5^{2} P_{3 / 2}) \to R b (6^{2} P_{3 / 2, 1 / 2}) + R b (5^{2} S_{1 / 2})$	$0.45$ Å²
12	$2 R b (5^{2} P_{1 / 2}) \to R b (5^{2} D_{3 / 2, 5 / 2}) + R b (5^{2} S_{1 / 2})$	$78$ Å²
13	$2 R b (5^{2} P_{1 / 2}) \to R b (6^{2} P_{3 / 2, 1 / 2}) + R b (5^{2} S_{1 / 2})$	$0.45$ Å²
14	$R b (5^{2} P_{1 / 2}) + R b (5^{2} P_{3 / 2}) \to R b (5^{2} D_{3 / 2, 5 / 2}) + R b (5^{2} S_{1 / 2})$	$290$ Å²
	Spontaneous emission		$s^{- 1}$	[17]
15	$R b (5^{2} D_{5 / 2, 3 / 2}) \to R b (5^{2} P_{3 / 2}) + h ν$		$1.8 \times 1 0^{6}$
16	$R b (5^{2} D_{5 / 2},_{3 / 2}) \to R b (5^{2} P_{1 / 2}) + h ν$		$8.48 \times 1 0^{5}$
17	$R b (5^{2} D_{5 / 2},_{3 / 2}) \to R b (6^{2} P_{3 / 2, 1 / 2}) + h ν$		$1.5 \times 1 0^{6}$
18	$R b (7^{2} S_{1 / 2}) \to R b (5^{2} P_{3 / 2}) + h ν$		$3.87 \times 1 0^{6}$
19	$R b (7^{2} S_{1 / 2}) \to R b (5^{2} P_{1 / 2}) + h ν$		$2.02 \times 1 0^{6}$
20	$R b (7^{2} S_{1 / 2}) \to R b (6^{2} P_{3 / 2, 1 / 2}) + h ν$		$4.26 \times 1 0^{6}$
21	$R b (6^{2} P_{3 / 2, 1 / 2}) \to R b (5^{2} S_{1 / 2}) + h ν$		$1.59 \times 1 0^{6}$
22	$R b (6^{2} P_{3 / 2, 1 / 2}) \to R b (5^{2} P_{3 / 2}) + h ν$		$4.26 \times 1 0^{6}$
23	$R b (6^{2} P_{3 / 2, 1 / 2}) \to R b (5^{2} P_{1 / 2}) + h ν$		$2.2 \times 1 0^{6}$
	Photoionization
24	$R b (5^{2} D_{3 / 2, 5 / 2}, 6^{2} P_{3 / 2, 1 / 2}, 7^{2} S_{1 / 2}) + h ν_{p, l} \to K^{+} + e$	$0.16 A^{2}$ for $5^{2} D_{3 / 2, 5 / 2}$ $0.15 A^{2}$ for $6^{2} P_{3 / 2, 1 / 2}$		[8,15]
	Penning ionization
25	$R b (5^{2} D_{3 / 2, 5 / 2}, 6^{2} P_{3 / 2, 1 / 2}, 7^{2} S_{1 / 2}) + R b (5^{2} P_{3 / 2, 1 / 2}) \to R b^{+} + R b (5^{2} S_{1 / 2}) + e$ , total 10 reactions		$4.5 \times 1 0^{- 14} m^{3} / s$ for $5^{2} D_{3 / 2, 5 / 2}$ and $6^{2} P_{3 / 2, 1 / 2}$ $2 \times 1 0^{- 14} m^{3} / s$ for $7^{2} S_{1 / 2}$	[8,15]
	Recombination			[18]^b
26	$R b^{+} + 2 R b \to {R b}_{2}^{+} + R b$		$2.4 \times 1 0^{- 33} (293 . 15 / T) m^{6} / s$
27	$R b^{+} + R b + H e \to {R b}_{2}^{+} + H e$		$2.4 \times 1 0^{- 35} (293 . 15 / T) m^{6} / s$
28	${R b}_{2}^{+} + e \to R b (5^{2} D_{3 / 2, 5 / 2}, 6^{2} P_{3 / 2, 1 / 2}, 7^{2} S_{1 / 2}) + R b (5^{2} S_{1 / 2})$		$1 \times 1 0^{- 14} m^{3} / s$
29	$R b^{+} + e + e \to R b (5^{2} D_{3 / 2, 5 / 2}, 6^{2} P_{3 / 2, 1 / 2}, 7^{2} S_{1 / 2}) + e$		$2.4 \times 1 0^{- 22} \times T^{- 4.2} m^{6} / s$
30	$R b^{+} + e + M (H e, {C H}_{4}) \to R b (5^{2} D_{3 / 2, 5 / 2}, 6^{2} P_{3 / 2, 1 / 2}, 7^{2} S_{1 / 2}) + e$		$4 \times 1 0^{- 41} \times (T / 625)^{- 2.5} m^{6} / s$
	Quenching of the higher levels
31	$R b (5^{2} D_{5 / 2},_{3 / 2}) + {C H}_{4} \to R b (6^{2} P_{3 / 2, 1 / 2}) + {C H}_{4}$	$84 A^{2}$		[19]
32	$R b (7^{2} S_{1 / 2}) + {C H}_{4} \to R b (6^{2} P_{3 / 2, 1 / 2}) + {C H}_{4}$	$84 A^{2}$		[19]
33	$R b (6^{2} P_{3 / 2, 1 / 2}) + {C H}_{4} \to R b (5^{2} P_{3 / 2, 1 / 2}) + {C H}_{4}$	$84 A^{2}$		[19]
		$B r a n c h i n g r a t i o s 2 / 3, 1 / 3$
34	$R b (5^{2} D_{5 / 2},_{3 / 2}) + H e \to R b (6^{2} P_{3 / 2, 1 / 2}) + H e$		$2.27 \times 1 0^{- 18}$	[20]^b
35	$R b (7^{2} S_{1 / 2}) + H e \to R b (6^{2} P_{3 / 2, 1 / 2}) + H e$		$3.58 \times 1 0^{- 19}$	[20]
36	$R b (5^{2} D_{5 / 2},_{3 / 2}) + H e \to R b (5^{2} P_{3 / 2, 1 / 2}) + H e$		$2.29 \times 1 0^{- 18}$	[20]
37	$R b (6^{2} P_{3 / 2, 1 / 2}) + H e \to R b (5^{2} P_{3 / 2, 1 / 2}) + H e$		$1.64 \times 1 0^{- 19}$	[20]
38	$R b (7^{2} S_{1 / 2}) + H e \to R b (5^{2} P_{3 / 2, 1 / 2}) + H e$		$8.52 \times 1 0^{- 19}$	[20]
	Electron heating and energy losses
39	$e + R b (5^{2} P_{3 / 2, 1 / 2}) ⇄ e_{h o t} + R b (5^{2} S_{1 / 2})$			[18]
40	$e_{h o t} + {C H}_{4} \leftrightarrow e + {C H}_{4} (ν_{2}, ν_{4})$			[18]
41	$e_{h o t} + H e \to e + H e$			[18]

Pump beam	Profile	$w_{0 x}$	$w_{0 y}$	$z_{Rx}$	$z_{Ry}$
Pump beam	Gaussian	0.4 mm	0.75 mm	10 mm	10 mm
Laser beam	Profile	$w_{0 x}$	$w_{0 y}$	$z_{Rx}$	$z_{Ry}$
Laser beam	Gaussian	0.5 mm	0.5 mm	∞	∞
Optics	Output coupler reflectivity	Pump window transmission	Laser window transmission	Back mirror reflectivity	Gain length
Optics	0.6	0.95	0.98	0.99	1.5 cm
Gas at the inlet	Temperature	Pressure	Velocity	Rb vapor	${C H}_{4} / H e$
Gas at the inlet	393 [K]	1 atm.@293 K	1 [m/s]	403 [K]	1/3 molar

Abstract

Data Availability

Cited By

Figures (16)

Tables (3)

Journal of the Optical Society of America B