laser parameters, at which the stable oscillations exist for the field strength, which are
close to harmonic ones. Phase portraits built on the base of the DE solution have a
series of interesting features, for instance, the loop-shaped phase portrait and the
cycloid symmetric closed trajectories with singularities.
In particular, it is stated that at absence of inertial properties (in the laser model,
the carrier lifetime on the upper level is zero) for the linear-hyperbolic nonlinearity,
there exists the restricted region of values of the normalized pumping (or values of
the gain). For example, for the optical resonator and the spectral line of laser
emission Q-factors equaled to 100, for the coefficient G ¼ 10, the region of
normalized pumping, for which the quasi-harmonic oscillations are excited and
existed, is about from 1 to 2. If the carriers’ lifetime is nonzero, the region of
pumping values (at generation of quasi-harmonic oscillations) essentially widens
from 1 to 15–20, which is caused by expansion of the region, in which conditions of
single-frequency existence are satisfied at fulfillment of amplitude and phase balance
conditions.
4.6.2 Resonance Characteristics for Laser DE of Fourth
Order
Before the presentation start in this section, which is devoted to the analysis of the
QWLD resonance characteristics, we would like to tell several words about this
problem relevance. It is known that in the LIGO interferometer (International project
on investigation of gravitation waves) the ultrahigh-Q (the Q-factor Q 0F ¼ 10
10
–
10
20 ) disk optical resonators are used for registration of gravitation waves (LIGO)
[11, 12]. The problem of ultrahigh-Q optical resonator application as the discriminator in optoelectronic measuring equipment for ultrasmall physical quantities is the
extremely relevant. The influence of the field characteristic upon the behavior is
weakly theoretically studied. The Russian physicist Gorodetskiy M.L. (1966–2019),
the participant of the LIGO project [13, 14], has theoretically investigated the
phenomenon of parametric instability of the optical disk resonator (with the geometrical radius about 20–120 μm) at the ultrahigh Q-factor Q 0F ¼ 10
10
– 10
20 and at
relatively high introducing power into the optical resonator P in ¼ 10
À6
– 10
À4 W.
The mathematical model of the optical resonator (Fig. 2.7) is complicate. So, at
ultrahigh power densities of the light in the ultralow volume about V D ¼ 10
À15 m
3 ,
several processes occur:
– Nonlinear interactions of the field and the material,
– Quadratic or cubic variation of the resonator Q-factor versus the field strength
amplitude,
– The temperature heating of the resonator material due to the losses of the optical
power.
180
4 Semiclassical Theory and Laser Differential Equations for Optoelectronic. . .
close to harmonic ones. Phase portraits built on the base of the DE solution have a
series of interesting features, for instance, the loop-shaped phase portrait and the
cycloid symmetric closed trajectories with singularities.
In particular, it is stated that at absence of inertial properties (in the laser model,
the carrier lifetime on the upper level is zero) for the linear-hyperbolic nonlinearity,
there exists the restricted region of values of the normalized pumping (or values of
the gain). For example, for the optical resonator and the spectral line of laser
emission Q-factors equaled to 100, for the coefficient G ¼ 10, the region of
normalized pumping, for which the quasi-harmonic oscillations are excited and
existed, is about from 1 to 2. If the carriers’ lifetime is nonzero, the region of
pumping values (at generation of quasi-harmonic oscillations) essentially widens
from 1 to 15–20, which is caused by expansion of the region, in which conditions of
single-frequency existence are satisfied at fulfillment of amplitude and phase balance
conditions.
4.6.2 Resonance Characteristics for Laser DE of Fourth
Order
Before the presentation start in this section, which is devoted to the analysis of the
QWLD resonance characteristics, we would like to tell several words about this
problem relevance. It is known that in the LIGO interferometer (International project
on investigation of gravitation waves) the ultrahigh-Q (the Q-factor Q 0F ¼ 10
10
–
10
20 ) disk optical resonators are used for registration of gravitation waves (LIGO)
[11, 12]. The problem of ultrahigh-Q optical resonator application as the discriminator in optoelectronic measuring equipment for ultrasmall physical quantities is the
extremely relevant. The influence of the field characteristic upon the behavior is
weakly theoretically studied. The Russian physicist Gorodetskiy M.L. (1966–2019),
the participant of the LIGO project [13, 14], has theoretically investigated the
phenomenon of parametric instability of the optical disk resonator (with the geometrical radius about 20–120 μm) at the ultrahigh Q-factor Q 0F ¼ 10
10
– 10
20 and at
relatively high introducing power into the optical resonator P in ¼ 10
À6
– 10
À4 W.
The mathematical model of the optical resonator (Fig. 2.7) is complicate. So, at
ultrahigh power densities of the light in the ultralow volume about V D ¼ 10
À15 m
3 ,
several processes occur:
– Nonlinear interactions of the field and the material,
– Quadratic or cubic variation of the resonator Q-factor versus the field strength
amplitude,
– The temperature heating of the resonator material due to the losses of the optical
power.
180
4 Semiclassical Theory and Laser Differential Equations for Optoelectronic. . .