Analysis and optimization of lifetime thermal loading in continuous-wave Cr<sup>4+</sup>-doped solid-state lasers

Alphan Sennaroglu

doi:10.1364/JOSAB.18.001578

Journal of the Optical Society of America B
Vol. 18,
Issue 11,
pp. 1578-1586
(2001)
•https://doi.org/10.1364/JOSAB.18.001578

Analysis and optimization of lifetime thermal loading in continuous-wave Cr⁴⁺-doped solid-state lasers

Alphan Sennaroglu

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Alphan Sennaroglu, "Analysis and optimization of lifetime thermal loading in continuous-wave Cr⁴⁺-doped solid-state lasers," J. Opt. Soc. Am. B 18, 1578-1586 (2001)

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Abstract

A systematic and rigorous method for the analysis and optimization of ${Cr}^{4 +}$ -doped solid-state lasers subject to lifetime thermal loading is described. First, a figure of merit is derived to identify the important parameters that influence the strength of this effect. Next, a theoretical model based on rate-equation analysis is presented for threshold and efficiency calculations. The method is then applied to the analysis of ${Cr}^{4 +} : forsterite$ and ${Cr}^{4 +} : YAG$ lasers. Experimental pump absorption, laser threshold, and laser efficiency data are evaluated to determine the best-fit values of the absorption, emission, and excited-state absorption cross sections for the two laser media. Best-fit cross section values are then used to determine the optimum crystal length, crystal absorption, and resonator reflectivity that maximize the laser output power. Finally, the optimization algorithm is applied to the study of a hypothetical solid-state gain medium to investigate how the optimum crystal and resonator parameters vary as a function of absorption and emission cross sections.

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Parameter	Symbol	Unit	${Cr}^{4 +} : Forsterite$	${Cr}^{4 +} : YAG$	${Ti}^{3 +} : Sapphire$
Heat conductivity	κ	W/cm-K	0.05a	0.12b	0.35c
Lifetime at $T_{r}$	$τ_{f 0}$	µs	3.87	5.35	3.60
Thermal lifetime coefficient	$τ_{fT}$	µs/°C	0.0490	0.0416	0.0175
Radiative lifetime	$τ_{r}$	µs	27d	48e	3.9
Lifetime at $T_{b}$	$τ_{f} (T_{b})$	µs	2.9	4.5	3.2
Pump wavelength	$λ_{p}$	µm	1.06	1.06	0.532
Laser wavelength	$λ_{L}$	µm	1.26	1.46	0.8
Heating fraction	$η_{h}$	dimensionless	0.91	0.93	0.44
Lifetime figure of merit	$Φ_{τ}$	dimensionless	6.7	29	290

Parameter	Symbol	Unit	${Cr}^{4 +} : Forsterite$	${Cr}^{4 +} : YAG$
Absorption cross section	$σ_{a}$	cm²	$1.36 \times 10^{- 19}$	$11.2 \times 10^{- 19}$
Emission cross section	$σ_{e}$	cm²	$1.16 \times 10^{- 19}$	$0.47 \times 10^{- 19}$
Excited-state absorption	$σ_{esa}$	cm²	$0.18 \times 10^{- 19}$	$0.093 \times 10^{- 19}$
Refractive index	$n_{0}$	dimensionless	1.635a	1.8b
Crystal figure of merit	FOM	dimensionless	26	110
Pump beam waist	$ω_{p 0}$	µm	34	51
Cavity beam waist	$ω_{L 0}$	µm	38	38
Crystal length	$L_{0}$	cm	2	2
Pump waist location in the crystal	$z_{fp}$	cm	1	1
Cavity waist location in the crystal	$z_{fL}$	cm	1	1
Effective crystal radius	$r_{0}$	µm	1780	2500
Small-signal absorption coefficient	$α_{p 0}$	cm^-1	0.57	1.1
Crystal boundary temperature	$T_{b}$	°C	12–30	10–30
Reflectivity of the output coupler	$R_{L}$	dimensionless	0.976	0.988

Name	Symbol	Unit	Value
Normalized excited-state absorption	$f_{L}$	dimensionless	0.5
Refractive index	$n_{0}$	dimensionless	1.8
Heat conductivity	κ	W/cm-K	0.1
Lifetime at $T_{r}$	$τ_{fo}$	µs	5
Thermal lifetime coefficient	$τ_{fT}$	µs/°C	0.04
Pump wavelength	$λ_{p}$	µm	1.06
Laser wavelength	$λ_{L}$	µm	1.5
Heating fraction	$η_{h}$	dimensionless	1
Crystal figure of merit	FOM	dimensionless	100
Pump beam waist	$ω_{p 0}$	µm	35
Cavity beam waist	$ω_{L 0}$	µm	35
Pump waist location in the crystal	$z_{fp}$	cm	$L_{0} / 2$
Cavity waist location in the crystal	$z_{fL}$	cm	$L_{0} / 2$
Effective crystal radius	$r_{0}$	µm	2500
Crystal boundary temperature	$T_{b}$	°C	15

Parameter	Symbol	Unit	${Cr}^{4 +} : Forsterite$	${Cr}^{4 +} : YAG$	${Ti}^{3 +} : Sapphire$
Heat conductivity	κ	W/cm-K	0.05a	0.12b	0.35c
Lifetime at $T_{r}$	$τ_{f 0}$	µs	3.87	5.35	3.60
Thermal lifetime coefficient	$τ_{fT}$	µs/°C	0.0490	0.0416	0.0175
Radiative lifetime	$τ_{r}$	µs	27d	48e	3.9
Lifetime at $T_{b}$	$τ_{f} (T_{b})$	µs	2.9	4.5	3.2
Pump wavelength	$λ_{p}$	µm	1.06	1.06	0.532
Laser wavelength	$λ_{L}$	µm	1.26	1.46	0.8
Heating fraction	$η_{h}$	dimensionless	0.91	0.93	0.44
Lifetime figure of merit	$Φ_{τ}$	dimensionless	6.7	29	290

Parameter	Symbol	Unit	${Cr}^{4 +} : Forsterite$	${Cr}^{4 +} : YAG$
Absorption cross section	$σ_{a}$	cm²	$1.36 \times 10^{- 19}$	$11.2 \times 10^{- 19}$
Emission cross section	$σ_{e}$	cm²	$1.16 \times 10^{- 19}$	$0.47 \times 10^{- 19}$
Excited-state absorption	$σ_{esa}$	cm²	$0.18 \times 10^{- 19}$	$0.093 \times 10^{- 19}$
Refractive index	$n_{0}$	dimensionless	1.635a	1.8b
Crystal figure of merit	FOM	dimensionless	26	110
Pump beam waist	$ω_{p 0}$	µm	34	51
Cavity beam waist	$ω_{L 0}$	µm	38	38
Crystal length	$L_{0}$	cm	2	2
Pump waist location in the crystal	$z_{fp}$	cm	1	1
Cavity waist location in the crystal	$z_{fL}$	cm	1	1
Effective crystal radius	$r_{0}$	µm	1780	2500
Small-signal absorption coefficient	$α_{p 0}$	cm^-1	0.57	1.1
Crystal boundary temperature	$T_{b}$	°C	12–30	10–30
Reflectivity of the output coupler	$R_{L}$	dimensionless	0.976	0.988

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Journal of the Optical Society of America B