oscillation are disturbed. The results of numerical calculations on the base of
solution of the resonance characteristic equations in the steady-state laser mode for
the linear-hyperbolic, quadratic-hyperbolic, and linear-cubic nonlinearities.
Calculated functions of the QWLD oscillation amplitudes versus frequency
(resonance characteristics) allow determination of the threshold values of the
A coefficient, the region of stable steady-state single-frequency laser generation.
We offer the research method, which allows determination for the laser model how
does the amplitude-frequency function change at the power level growth in the
optical resonator or on the active medium. These functions obtained from solution
of symbolic equations take into account the gain on the feedback loop, the lifetime,
the saturation coefficient, Q-factors of the resonator, and the spectral emission line of
the active medium. The modeling of optical emission power variation in the optical
resonator is executed by means of introduction of the A parameter, at variation of
which the Q-factor of the resonator varies in nonlinear manner. We considered the
quadratic and cubic parametric variation of the resonator Q-factor. Laws on Q-factor
variation are determined for quadratic Q-factor variation as Q 0F ¼
1= 0:001 Á 1 À A Á E
2
10n
À
Á
Â
Ã
, and for cubic equations Q 0F ¼ 1/[0.001 Á (1 À A Á E 10n
3 )].
As the result of symbolic equation solutions in quasi-stationary mode, we show that
at critical value of A ¼ 0.0081, branches of stable operation and branches of unstable
operation “merge into one node” corresponding to disordered oscillation motions in
the wide frequency region. The stability loss of the single-frequency generation is
also manifested at growth of the A parameter at cubic parametric impact. At that,
critical value of the A parameter is about A ¼ 0.0034.
Determination of critical values is the important contribution in the theory of
ultrahigh-Q disk resonators and Bragg resonators.
Fulfilled investigation of resonance characteristics widens our knowledge about
oscillation processes and enables to make the well-founded conclusion. It consists in
the fact that growth of Q-factors cannot increase without limit in ultrasmall volumes
of optical structures. Decrease of the spectral line width and reduction of the laser
phase noises can be performed with utilization of optical frequency control systems
and PLL systems with application of ultrahigh-Q resonators as discriminators in the
additional laser optical channel. At that, the power density in resonators can be
significantly decreases by several ten times.
4.7 Solution of Nonlinear DE of Second Order for the Laser
4.7.1 Equation of Second Order for the Laser
with the Nonlinear Element in the Form
S E 10n
ð
Þ= E 10n = 1 þ T 1n G 00 E
2
10n
À
Á
At the high-Q optical resonator in the laser (when the Q-factor exceeds the Q-factor
of the spectral emission line by several hundred times and more), the conditions are
created for transfer from the nonlinear differential equation of fourth order for
4.7 Solution of Nonlinear DE of Second Order for the Laser
193
solution of the resonance characteristic equations in the steady-state laser mode for
the linear-hyperbolic, quadratic-hyperbolic, and linear-cubic nonlinearities.
Calculated functions of the QWLD oscillation amplitudes versus frequency
(resonance characteristics) allow determination of the threshold values of the
A coefficient, the region of stable steady-state single-frequency laser generation.
We offer the research method, which allows determination for the laser model how
does the amplitude-frequency function change at the power level growth in the
optical resonator or on the active medium. These functions obtained from solution
of symbolic equations take into account the gain on the feedback loop, the lifetime,
the saturation coefficient, Q-factors of the resonator, and the spectral emission line of
the active medium. The modeling of optical emission power variation in the optical
resonator is executed by means of introduction of the A parameter, at variation of
which the Q-factor of the resonator varies in nonlinear manner. We considered the
quadratic and cubic parametric variation of the resonator Q-factor. Laws on Q-factor
variation are determined for quadratic Q-factor variation as Q 0F ¼
1= 0:001 Á 1 À A Á E
2
10n
À
Á
Â
Ã
, and for cubic equations Q 0F ¼ 1/[0.001 Á (1 À A Á E 10n
3 )].
As the result of symbolic equation solutions in quasi-stationary mode, we show that
at critical value of A ¼ 0.0081, branches of stable operation and branches of unstable
operation “merge into one node” corresponding to disordered oscillation motions in
the wide frequency region. The stability loss of the single-frequency generation is
also manifested at growth of the A parameter at cubic parametric impact. At that,
critical value of the A parameter is about A ¼ 0.0034.
Determination of critical values is the important contribution in the theory of
ultrahigh-Q disk resonators and Bragg resonators.
Fulfilled investigation of resonance characteristics widens our knowledge about
oscillation processes and enables to make the well-founded conclusion. It consists in
the fact that growth of Q-factors cannot increase without limit in ultrasmall volumes
of optical structures. Decrease of the spectral line width and reduction of the laser
phase noises can be performed with utilization of optical frequency control systems
and PLL systems with application of ultrahigh-Q resonators as discriminators in the
additional laser optical channel. At that, the power density in resonators can be
significantly decreases by several ten times.
4.7 Solution of Nonlinear DE of Second Order for the Laser
4.7.1 Equation of Second Order for the Laser
with the Nonlinear Element in the Form
S E 10n
ð
Þ= E 10n = 1 þ T 1n G 00 E
2
10n
À
Á
At the high-Q optical resonator in the laser (when the Q-factor exceeds the Q-factor
of the spectral emission line by several hundred times and more), the conditions are
created for transfer from the nonlinear differential equation of fourth order for
4.7 Solution of Nonlinear DE of Second Order for the Laser
193