the phase difference is made not only by the phase-frequency characteristics of the
optical filter, but the phase-frequency characteristic of QWLD.
Thus, it is shown that the complicate system from three constitutive equations
(Eq. 4.17), considered in Chap. 4, spanned by the positive feedback, is reduced in the
quasi-stationary mode to the system of two DEs (Eq. 5.6). It is similar to the equation
of double-circuit oscillator with inertial feeding, which was solved many times in the
specific tasks of the nonlinear oscillation theory. Specific properties of this Eq. (5.6),
which introduce new property, from the point of view of the oscillation theory of RF
and optical oscillators, exactly the inertial multiplicative laser nonlinearity. The
nonlinear functions are concretized for different pumping schemes, for instance,
for three- and four-level systems for laser pumping.
5.1.4 The Analog Model of DEs for ОЕО DM
On the base of CE system (Eq. 5.12), the analog model of OEO DM was built, which
is presented in Fig. 5.1. The model of OEO DM is constructed using the approach to
OEO consideration as the QWLD, which is spanned of the positive feedback as it is
shown in Fig. 5.1. “Feedback Chain” in this figure consists of the optical fiber OF,
the photodetector PD, the RF filter F, and the RF amplifier.
5.1.5 Abbreviated Equations of ОЕО DM
To analyze processes in the optoelectronic oscillator, we must take into consideration that the spectrum of processes is located in the relatively narrow vicinity of the
resonance frequency of the RF filter in the linear part of the system. Therefore, the
AC current in the input of the RF amplifier (A) (or in the filter (F) input) (Fig. 3.2a)
(the active element of the RF circuit) we consider as quasi-harmonic and present in
the form: u F ¼ U 10F Re [exp(2πf 0 t)], where U 10F , Φ F are the slowly changing
amplitude and phase, relatively, Δf ¼ f À f 0 is the amendment on the oscillating
frequency. Then the voltage U 10F , is equal to U 10F ¼ U 10F Re [exp(2πΔf Á t À Φ F )].
The voltage U 10F (compared to the instantaneous value u F ) has the slowly
changing complex amplitude. In OEO DM, the ratio of the frequency amendment
to the frequency Δf/f 0 ( 1. We use the Evtianov method [1] and apply the bias
theorem of the operational calculus and the similar considerations, which were used
for the deduction of abbreviated equation of OEO MZ in Chap. 3.
Let us write the second equation of the system (Eq. 5.6) in the form:
210
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .
optical filter, but the phase-frequency characteristic of QWLD.
Thus, it is shown that the complicate system from three constitutive equations
(Eq. 4.17), considered in Chap. 4, spanned by the positive feedback, is reduced in the
quasi-stationary mode to the system of two DEs (Eq. 5.6). It is similar to the equation
of double-circuit oscillator with inertial feeding, which was solved many times in the
specific tasks of the nonlinear oscillation theory. Specific properties of this Eq. (5.6),
which introduce new property, from the point of view of the oscillation theory of RF
and optical oscillators, exactly the inertial multiplicative laser nonlinearity. The
nonlinear functions are concretized for different pumping schemes, for instance,
for three- and four-level systems for laser pumping.
5.1.4 The Analog Model of DEs for ОЕО DM
On the base of CE system (Eq. 5.12), the analog model of OEO DM was built, which
is presented in Fig. 5.1. The model of OEO DM is constructed using the approach to
OEO consideration as the QWLD, which is spanned of the positive feedback as it is
shown in Fig. 5.1. “Feedback Chain” in this figure consists of the optical fiber OF,
the photodetector PD, the RF filter F, and the RF amplifier.
5.1.5 Abbreviated Equations of ОЕО DM
To analyze processes in the optoelectronic oscillator, we must take into consideration that the spectrum of processes is located in the relatively narrow vicinity of the
resonance frequency of the RF filter in the linear part of the system. Therefore, the
AC current in the input of the RF amplifier (A) (or in the filter (F) input) (Fig. 3.2a)
(the active element of the RF circuit) we consider as quasi-harmonic and present in
the form: u F ¼ U 10F Re [exp(2πf 0 t)], where U 10F , Φ F are the slowly changing
amplitude and phase, relatively, Δf ¼ f À f 0 is the amendment on the oscillating
frequency. Then the voltage U 10F , is equal to U 10F ¼ U 10F Re [exp(2πΔf Á t À Φ F )].
The voltage U 10F (compared to the instantaneous value u F ) has the slowly
changing complex amplitude. In OEO DM, the ratio of the frequency amendment
to the frequency Δf/f 0 ( 1. We use the Evtianov method [1] and apply the bias
theorem of the operational calculus and the similar considerations, which were used
for the deduction of abbreviated equation of OEO MZ in Chap. 3.
Let us write the second equation of the system (Eq. 5.6) in the form:
210
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .