dynamics of oscillation processes. A choice of optimal type of the nonlinear function
I 1A (U PD ) for realization of the “soft” self-excitation mode, for instance, as the
polynomial of third and fifth order, becomes more complicate. This is caused by
the fact that this choice must be connected with the linear-hyperbolic (in algebraic
form) multiplicative (in the physical basis as the product of the field strength and the
population) nonlinearity of the laser.
Another feature is the fact that the equations system unifies motions for the laser
EMF in the optical range and electrical oscillations existing in the RF circuit of the
positive feedback in the radio-frequency range.
Taking into account equations for oscillations phase Φ 1L of the laser electromagnetic field and phase Φ 10E of the electrical oscillations of the pumping current, the
two modes exist: the in-phase mode and the mode of beats. With the growth of radio
frequency, theoretically, there is a possibility of the optical oscillations synchronization by the electrical oscillations. In this case, the full phase incursion of optical
and electrical oscillations in the closed loop in OEO DM must be multiple of 2π only
for “electrical oscillations” (with account that the laser emission is modulated with
the radio frequency) in the closed loop of OEO DM.
Steady-state laser modes in OEO DM will be discussed in Chaps. 6 and 7.
We should note that the system (Eq. 5.14) is obtained for the quasi-stationary
(QS) operation mode of the laser, i.e., for values of DC pumping, which is higher
than the threshold level. Owing to nonlinear functions S LCOS (E 10L ) and S LSIN (E 10L ),
this system (Eq. 5.14) takes into consideration the inertia properties of the laser
pumping, which is determined by the lifetime T 1n on the upper operation level. But
the system (Eq. 5.14) have no separate differential equation for population. In this,
there are significant restrictions of utilization of Eq. (5.14) for the quasi-stationary
mode for investigation, for example, the transients.
Before the deduction of abbreviated equations with fluctuations, which is one of
the main tasks of Chap. 5, we consider features of the abbreviated equation of OEO
DM taking into account the equation for population N(t) ¼ N 0L (t).
5.1.5.1 Symbolic and Abbreviated Equations of OEO DM with Account
of the Population Equation
We performed the system abbreviation process of initial constitutive equation
(Eq. 4.17) according the Evtianov method [1], as for Eq. (5.14), which was discussed
in Chap. 3. At that, we took into consideration the equation for population N
(t) ¼ N 0L (t) in the complete form and differential equations (Eqs. 5.4 and 5.5) for
the circuit of the positive feedback (Fig. 5.1a). We took the approximation of
functions S LCOS (E 10L ) and S LSIn (E 10L ) included in Eq. (5.14) with the help of the
third-order polynomials as it was performed in Chap. 3. For slowly changing timefunctions E 0n , Φ 10L , Φ 10J , and the population N(t) ¼ N 0L (t), we present the system of
abbreviated equations deduced from the considered differential equations for the
laser and the positive feedback circuit of OEO DM (Fig. 2.2):
5.1 Symbolic and Abbreviated Equations of OEO
213
I 1A (U PD ) for realization of the “soft” self-excitation mode, for instance, as the
polynomial of third and fifth order, becomes more complicate. This is caused by
the fact that this choice must be connected with the linear-hyperbolic (in algebraic
form) multiplicative (in the physical basis as the product of the field strength and the
population) nonlinearity of the laser.
Another feature is the fact that the equations system unifies motions for the laser
EMF in the optical range and electrical oscillations existing in the RF circuit of the
positive feedback in the radio-frequency range.
Taking into account equations for oscillations phase Φ 1L of the laser electromagnetic field and phase Φ 10E of the electrical oscillations of the pumping current, the
two modes exist: the in-phase mode and the mode of beats. With the growth of radio
frequency, theoretically, there is a possibility of the optical oscillations synchronization by the electrical oscillations. In this case, the full phase incursion of optical
and electrical oscillations in the closed loop in OEO DM must be multiple of 2π only
for “electrical oscillations” (with account that the laser emission is modulated with
the radio frequency) in the closed loop of OEO DM.
Steady-state laser modes in OEO DM will be discussed in Chaps. 6 and 7.
We should note that the system (Eq. 5.14) is obtained for the quasi-stationary
(QS) operation mode of the laser, i.e., for values of DC pumping, which is higher
than the threshold level. Owing to nonlinear functions S LCOS (E 10L ) and S LSIN (E 10L ),
this system (Eq. 5.14) takes into consideration the inertia properties of the laser
pumping, which is determined by the lifetime T 1n on the upper operation level. But
the system (Eq. 5.14) have no separate differential equation for population. In this,
there are significant restrictions of utilization of Eq. (5.14) for the quasi-stationary
mode for investigation, for example, the transients.
Before the deduction of abbreviated equations with fluctuations, which is one of
the main tasks of Chap. 5, we consider features of the abbreviated equation of OEO
DM taking into account the equation for population N(t) ¼ N 0L (t).
5.1.5.1 Symbolic and Abbreviated Equations of OEO DM with Account
of the Population Equation
We performed the system abbreviation process of initial constitutive equation
(Eq. 4.17) according the Evtianov method [1], as for Eq. (5.14), which was discussed
in Chap. 3. At that, we took into consideration the equation for population N
(t) ¼ N 0L (t) in the complete form and differential equations (Eqs. 5.4 and 5.5) for
the circuit of the positive feedback (Fig. 5.1a). We took the approximation of
functions S LCOS (E 10L ) and S LSIn (E 10L ) included in Eq. (5.14) with the help of the
third-order polynomials as it was performed in Chap. 3. For slowly changing timefunctions E 0n , Φ 10L , Φ 10J , and the population N(t) ¼ N 0L (t), we present the system of
abbreviated equations deduced from the considered differential equations for the
laser and the positive feedback circuit of OEO DM (Fig. 2.2):
5.1 Symbolic and Abbreviated Equations of OEO
213