Numerical modeling of multimode laser resonators

Philipp Wittmuess; Stefan Piehler; Tom Dietrich; Marwan Abdou Ahmed; Thomas Graf; Oliver Sawodny

doi:10.1364/JOSAB.33.002278

Journal of the Optical Society of America B
Vol. 33,
Issue 11,
pp. 2278-2287
(2016)
•https://doi.org/10.1364/JOSAB.33.002278

Numerical modeling of multimode laser resonators

Philipp Wittmuess, Stefan Piehler, Tom Dietrich, Marwan Abdou Ahmed, Thomas Graf, and Oliver Sawodny

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Philipp Wittmuess, Stefan Piehler, Tom Dietrich, Marwan Abdou Ahmed, Thomas Graf, and Oliver Sawodny, "Numerical modeling of multimode laser resonators," J. Opt. Soc. Am. B 33, 2278-2287 (2016)

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Abstract

A novel approach for the simulation of the performance of multimode lasers is presented which enables the computation of the optical mode composition of a wide range of macroscopic cw resonators with a given pump distribution. Based on the steady-state solution of the rate equation for the population inversion density, an analytical relation for the intensities of the individual modes is developed. A novel numerical algorithm is presented that efficiently solves this relation for the fraction of total power of each individual mode, taking into account the gain saturation caused by the other simultaneously oscillating modes. As a by-product, important laser design criteria, such as heat load in the laser crystal, output beam parameters, and optical efficiency, can be calculated. This approach is validated by comparison of simulation results with measurements taken from an experimental thin-disk multimode laser.

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$Δ N$	population inversion density	$m^{- 3}$
$Δ N^{0}$	zero population inversion density at current temperature, taking into account the thermal population of the lower laser level ( $E_{1}$ )	$m^{- 3}$
$r_{P}$	normalized pump distribution	$m^{- 3}$
$R$	pump rate	$s^{- 1}$
$f_{1}$	fractional population density of the lower laser level ( $E_{1}$ ) with respect to the total population density of the lower manifold at current temperature	1
$f_{2}$	fractional population density of the upper laser level ( $E_{2}$ ) with respect to the total population density of the upper manifold at current temperature	1
$f$	$= f_{1} + f_{2}$	1
$τ$	lifetime in the upper manifold	s
$σ$	absorption and emission cross section	$m^{- 2}$
$n$	refractive index of the laser active medium	1
$c$	speed of light	$m \cdot s^{- 1}$
$k_{B}$	Boltzmann constant	$J \cdot K^{- 1}$
$s_{j}$	number of photons in mode $j$	1
$T$	round-trip time of one photon	s

$c$	$2.997 \cdot 10^{8} m \cdot s^{- 1}$	speed of light
$E_{0}$	0 J	energy of level 0
$E_{1}$	$1.21509 \cdot 10^{- 20} J$	energy of level 1
$E_{2}$	$2.0507 \cdot 10^{- 19} J$	energy of level 2
$E_{3}$	$2.1097 \cdot 10^{- 19} J$	energy of level 3
$τ$	$9.51 \cdot 10^{- 4} s$	lifetime in upper manifold [27,28]
$σ$	$2.59 \cdot 10^{- 24} m^{- 2}$	emission/absorption cross section [28]
$η_{Q}$	0.975	quantum efficiency of fluorescence (value was taken from [27], which addressed Yb:YAG lasers)
$n$	1.82	refractive index of the laser medium [27]
$N_{doping}$	$1.39 \cdot 10^{27} m^{- 3}$	doping density
$Δ N^{0}$	$- 1.0399 \cdot 10^{26} m^{- 3}$	zero population inversion density at room temperature (calculated from doping density and energy levels)
$f$	0.8470	sum of fractional populations in level 1, 2 with respect to the lower and upper manifolds, respectively, at room temperature
$T$	$1.268 \cdot 10^{- 8} s$	resonator round-trip time (from resonator geometry)

$Δ N$	population inversion density	$m^{- 3}$
$Δ N^{0}$	zero population inversion density at current temperature, taking into account the thermal population of the lower laser level ( $E_{1}$ )	$m^{- 3}$
$r_{P}$	normalized pump distribution	$m^{- 3}$
$R$	pump rate	$s^{- 1}$
$f_{1}$	fractional population density of the lower laser level ( $E_{1}$ ) with respect to the total population density of the lower manifold at current temperature	1
$f_{2}$	fractional population density of the upper laser level ( $E_{2}$ ) with respect to the total population density of the upper manifold at current temperature	1
$f$	$= f_{1} + f_{2}$	1
$τ$	lifetime in the upper manifold	s
$σ$	absorption and emission cross section	$m^{- 2}$
$n$	refractive index of the laser active medium	1
$c$	speed of light	$m \cdot s^{- 1}$
$k_{B}$	Boltzmann constant	$J \cdot K^{- 1}$
$s_{j}$	number of photons in mode $j$	1
$T$	round-trip time of one photon	s

$c$	$2.997 \cdot 10^{8} m \cdot s^{- 1}$	speed of light
$E_{0}$	0 J	energy of level 0
$E_{1}$	$1.21509 \cdot 10^{- 20} J$	energy of level 1
$E_{2}$	$2.0507 \cdot 10^{- 19} J$	energy of level 2
$E_{3}$	$2.1097 \cdot 10^{- 19} J$	energy of level 3
$τ$	$9.51 \cdot 10^{- 4} s$	lifetime in upper manifold [27,28]
$σ$	$2.59 \cdot 10^{- 24} m^{- 2}$	emission/absorption cross section [28]
$η_{Q}$	0.975	quantum efficiency of fluorescence (value was taken from [27], which addressed Yb:YAG lasers)
$n$	1.82	refractive index of the laser medium [27]
$N_{doping}$	$1.39 \cdot 10^{27} m^{- 3}$	doping density
$Δ N^{0}$	$- 1.0399 \cdot 10^{26} m^{- 3}$	zero population inversion density at room temperature (calculated from doping density and energy levels)
$f$	0.8470	sum of fractional populations in level 1, 2 with respect to the lower and upper manifolds, respectively, at room temperature
$T$	$1.268 \cdot 10^{- 8} s$	resonator round-trip time (from resonator geometry)

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Equations (26)

Journal of the Optical Society of America B