theory: to transfer to the system of two nonlinear DE in ordinary derivatives, first of
each has the fourth order for field strength, while the second is DE of the first order
for population. Main parameters of these Des are the laser and the optical resonators
Q-factors, the carrier lifetime on the upper level and gain. This DE system can be
concerned to two-resonance system with inertial pumping.
Owing to studying of the population saturation effect on upper energy level under
pumping action, we extract the main nonlinear element in the form of linearhyperbolic function with inertia property caused by carrier time delay on upper
level. The analysis of this two-resonance system allowed determination of the smallsignal operator transfer function and the characteristic equation describing the
oscillation stability. DEs obtained permitted to form the laser analog structure with
two feedback loops for the field strength and a population.
After this, we obtained the abbreviated DEs, which show the oscillating transients. The transient of the field strength has a time delay with regard to population
transient, which is caused by the carrier lifetime.
Fig. 4.34 Solutions of the differential equation for QWLD (4.83) at growth of the pumping level
K ¼ α 0 for the scaled EMF strength y ¼ E n for nonlinearity of the cubic polynomial S E n
ð Þ ¼
α 00 E n À β 00 E
3
n , E n ¼ E 10n , for α 00 ¼ 0.5, β 00 ¼ 0.125 and scaled values 2πν 0n ¼ 1, (1/T OF ) ¼ 0.1
and K ¼ 0.2 (а), 0.4 (b), 2.4 (с)
198
4 Semiclassical Theory and Laser Differential Equations for Optoelectronic. . .
each has the fourth order for field strength, while the second is DE of the first order
for population. Main parameters of these Des are the laser and the optical resonators
Q-factors, the carrier lifetime on the upper level and gain. This DE system can be
concerned to two-resonance system with inertial pumping.
Owing to studying of the population saturation effect on upper energy level under
pumping action, we extract the main nonlinear element in the form of linearhyperbolic function with inertia property caused by carrier time delay on upper
level. The analysis of this two-resonance system allowed determination of the smallsignal operator transfer function and the characteristic equation describing the
oscillation stability. DEs obtained permitted to form the laser analog structure with
two feedback loops for the field strength and a population.
After this, we obtained the abbreviated DEs, which show the oscillating transients. The transient of the field strength has a time delay with regard to population
transient, which is caused by the carrier lifetime.
Fig. 4.34 Solutions of the differential equation for QWLD (4.83) at growth of the pumping level
K ¼ α 0 for the scaled EMF strength y ¼ E n for nonlinearity of the cubic polynomial S E n
ð Þ ¼
α 00 E n À β 00 E
3
n , E n ¼ E 10n , for α 00 ¼ 0.5, β 00 ¼ 0.125 and scaled values 2πν 0n ¼ 1, (1/T OF ) ¼ 0.1
and K ¼ 0.2 (а), 0.4 (b), 2.4 (с)
198
4 Semiclassical Theory and Laser Differential Equations for Optoelectronic. . .