Instability in a CO<sub>2</sub> sequence-band laser with a saturable absorber and vibration-to-vibration energy transfer processes

Toshiki Sugawara; Feng-Lei Hong; Maki Tachikawa; Tadao Shimizu

doi:10.1364/JOSAB.10.000265

Instability in a CO₂ sequence-band laser with a saturable absorber and vibration-to-vibration energy transfer processes

Toshiki Sugawara, Feng-Lei Hong, Maki Tachikawa, and Tadao Shimizu

Journal of the Optical Society of America B
Vol. 10,
Issue 2,
pp. 265-270
(1993)
•https://doi.org/10.1364/JOSAB.10.000265

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Toshiki Sugawara, Feng-Lei Hong, Maki Tachikawa, and Tadao Shimizu, "Instability in a CO₂ sequence-band laser with a saturable absorber and vibration-to-vibration energy transfer processes," J. Opt. Soc. Am. B 10, 265-270 (1993)

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Abstract

Several types of instability are observed in the passive Q-switching (PQS) pulsation of a CO₂ sequence-band laser with a saturable absorber. The observed pulse shapes are different from those of a regular-band CO₂ laser in detailed structure. We propose a novel model for the PQS instability of a CO₂ sequence-band laser by considering two vibration-to-vibration energy-transfer processes. We find that the sequence-band PQS pulse width critically depends on the rate of vibrational relaxation in the ν₃ mode. The vibration-to-vibration relaxation rate is estimated from the analysis of the pulse shapes.

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Parameter	Description
P	Rate of the pumping to the (0, 0⁰, 1) level
N₀	Population density in the ground level (0, 0⁰, 0)
R_s_,1	Rate of the vibrational relaxation from the (0, 0⁰, 2) level to the ground level
R_s_,2	Rate of the vibrational relaxation from the (1, 0⁰, 1) or (0, 2⁰, 1) level to the ground level
R_r_,1	Rate of the vibrational relaxation from the (0, 0⁰, 1) level to the ground level
R_r_,2	Rate of the vibrational relaxation from the (1, 0⁰, 0) or (0, 2⁰, 0) level to the ground level
κ₁	Rate of the transfer from the left-hand to the right-hand side in relation (1)
${\bar{κ}}_{1}$	Rate of the transfer from the right-hand to the left-hand side in relation (1)
κ₂	Rate of the transfer from the left-hand to the right-hand side in relation (2)
${\bar{κ}}_{2}$	Rate of the transfer from the right-hand to the left-hand side in relation (2)
σ_s	Cross section of the induced emission in the sequence band
σ_r	Cross section of the induced emission in the regular band
c	Light speed
f_s	Fraction of CO₂ molecules in the rotational level of the (0, 0⁰, 2) level
f_r	Fraction of CO₂ molecules in the rotational level of the (0, 0⁰, 1) level
σ_as	Cross section of the absorption for the sequence-band radiation in the saturable absorber
σ_ar	Cross section of the absorption for the regular-band radiation in the saturable absorber
L_g	Length of the laser gain tube
L	Length of the laser cavity
L_a	Length of the absorption cell
γ_s	Cavity loss rate for the sequence-band radiation
γ_r	Cavity loss rate for the regular-band radiation
r_s	Rotational relaxation rate of the absorptive levels resonant with the sequence-band laser intensity
r_r	Rotational relaxation rate of the absorptive levels resonant with the regular-band laser intensity
N_as*	Value of N_as in thermal equilibrium
N_ar*	Value of N_ar in thermal equilibrium

Normalization of the Variables	Newly Introduced Parameters
N_s,1/N₀ → N_s_,1	K₁ ≡ N₀κ₁ = N_{0 ${\bar{κ}}_{1}$}
N_s,2/N₀ → N_s_,2	K₂ ≡ = N₀κ₂N_{0 ${\bar{κ}}_{2}$}
N_r,1/N₀ → N_r_,1	F_s ≡ (L_g/L)σ_scf_sN₀
N_r,1/N₀ → N_r_,1	F_r ≡ (L_g/L)σ_rcf_rN₀
σ_rcf_rI_r → I_s	B_s ≡ σ_asc(L_a/L)N_as*
σ_rcf_rI_r → I_r	B_r ≡ σ_arc(L_a/L)N_ar*
N_as/N_as* → N_as	b_s ≡ σ_asc/(σ_scf_s)
N_ar/N_ar* → N_ar	b_r ≡ σ_arc/(σ_rcf_r)

Parameter	Sequence-Band PQS	Regular-Band PQS
P (Hz)	506.0	73.0
R_s_,1 (Hz)	0.000	0.000
R_s_,2 (MHz)	0.800	0.800
R_r_,1 (KHz)	2.400	2.400
R_r_,2 (MHz)	0.800	0.800
K₁ ≡ N₀κ₁ = N_{0 ${\bar{κ}}_{1}$} (MHz)	13.400	13.400
K₂ ≡ N₀κ₂ = N_{0 ${\bar{κ}}_{2}$} (MHz)	11.256	11.256
F_s ≡ (L_g/L)σ_scf_sN₀ (MHz)	8740	8740
F_r ≡ (L_g/L)σ_rcf_rN₀ (MHz)	3820	3820
γ_s (MHz)	5.000	—
γ_r (MHz)	—	5.000
r_s (MHz)	29.7 × pressure (Torr)	—
r_r (MHz)	—	10.261
B_s ≡ σ_asc(L_a/L)N_as* (MHz)	2.97 × pressure (Torr)	—
B_r ≡ σ_arc(L_a/L)N_ar* (MHz)	—	6.1565
b_s ≡ σ_asc/(σ_scf_s)	400.0	—
b_r ≡ σ_arc/(σ_rcf_r)	—	2000

Parameter	Description
P	Rate of the pumping to the (0, 0⁰, 1) level
N₀	Population density in the ground level (0, 0⁰, 0)
R_s_,1	Rate of the vibrational relaxation from the (0, 0⁰, 2) level to the ground level
R_s_,2	Rate of the vibrational relaxation from the (1, 0⁰, 1) or (0, 2⁰, 1) level to the ground level
R_r_,1	Rate of the vibrational relaxation from the (0, 0⁰, 1) level to the ground level
R_r_,2	Rate of the vibrational relaxation from the (1, 0⁰, 0) or (0, 2⁰, 0) level to the ground level
κ₁	Rate of the transfer from the left-hand to the right-hand side in relation (1)
${\bar{κ}}_{1}$	Rate of the transfer from the right-hand to the left-hand side in relation (1)
κ₂	Rate of the transfer from the left-hand to the right-hand side in relation (2)
${\bar{κ}}_{2}$	Rate of the transfer from the right-hand to the left-hand side in relation (2)
σ_s	Cross section of the induced emission in the sequence band
σ_r	Cross section of the induced emission in the regular band
c	Light speed
f_s	Fraction of CO₂ molecules in the rotational level of the (0, 0⁰, 2) level
f_r	Fraction of CO₂ molecules in the rotational level of the (0, 0⁰, 1) level
σ_as	Cross section of the absorption for the sequence-band radiation in the saturable absorber
σ_ar	Cross section of the absorption for the regular-band radiation in the saturable absorber
L_g	Length of the laser gain tube
L	Length of the laser cavity
L_a	Length of the absorption cell
γ_s	Cavity loss rate for the sequence-band radiation
γ_r	Cavity loss rate for the regular-band radiation
r_s	Rotational relaxation rate of the absorptive levels resonant with the sequence-band laser intensity
r_r	Rotational relaxation rate of the absorptive levels resonant with the regular-band laser intensity
N_as*	Value of N_as in thermal equilibrium
N_ar*	Value of N_ar in thermal equilibrium

Normalization of the Variables	Newly Introduced Parameters
N_s,1/N₀ → N_s_,1	K₁ ≡ N₀κ₁ = N_{0 ${\bar{κ}}_{1}$}
N_s,2/N₀ → N_s_,2	K₂ ≡ = N₀κ₂N_{0 ${\bar{κ}}_{2}$}
N_r,1/N₀ → N_r_,1	F_s ≡ (L_g/L)σ_scf_sN₀
N_r,1/N₀ → N_r_,1	F_r ≡ (L_g/L)σ_rcf_rN₀
σ_rcf_rI_r → I_s	B_s ≡ σ_asc(L_a/L)N_as*
σ_rcf_rI_r → I_r	B_r ≡ σ_arc(L_a/L)N_ar*
N_as/N_as* → N_as	b_s ≡ σ_asc/(σ_scf_s)
N_ar/N_ar* → N_ar	b_r ≡ σ_arc/(σ_rcf_r)

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