5.1 Symbolic and Abbreviated Equations of OEO
To construct the analog models of OEO DM and OEO MZ, we consider structural
diagrams (Figs. 3.2 and 5.1a, b) and form and analyze of symbolic DEs of OEO.
At first, we take the symbolic differential equations of the laser, which we
described in Chap. 4.
In the diagram of Fig. 3.2, the structures of OEO DM and OEO MZ with
allocation of Feedback Chain and RF FODL are presented. For OEO MZ, two
different approaches of the positive feedback spanned of modulated laser emission
source “Laser-MZ” are shown in Fig. 5.1b, с. In the case of Fig. 3.2b, the one-port
Feedback Chain has the optical input and electrical output. In the case of Fig. 3.2с,
the one-port RF FODL has the electrical input and the electrical output. In the
structure of Fig. 3.2b, “Laser” is enclosed by the feedback by the “DC” block,
which fulfills the function of the laser optical frequency of the output power with
utilization of adjustment by the laser DC pumping current. On the structure of ОEO
DM in Fig. 3.2a, instantaneous values of voltages, relatively, on the load resistance
Z L of photodetector u PD (t), on electrical output of the RF amplifier u A (t), of the RF
filter u F (t), of the RF coupler u C (t), and instantaneous values of the laser pumping
current i L ¼ u L /Z L are shown. In the structure of ОEO MZ in Fig. 3.2b, the
instantaneous values of the field oscillation strength in the optical input of MZ
(or in the laser output) E n and in the optical output of E nMZ , and the instantaneous
voltage values in the electrical input u MZ are shown. In the structure of ОEO MZ in
Fig. 3.2с, the instantaneous voltage values, relatively, in the load resistance Z L of PD
u PD (t), in electrical outputs of the RF amplifier u A (t), of the RF filter u F (t), of the RF
coupler u C (t), and instantaneous voltage value on the electrical input of MZ are
shown.
In order to obtain the connection of the AC pumping current i 1L ¼ J 1L with the
AC component of the strength E 1n (or intensity E
2
1n ), we transfer to more detailed
description of parameters of the positive feedback circuit.
5.1.1 Symbolic Laser Equations in the OEO Structure
In Chap. 4, we examined and analyzed the symbolic differential equations (Eq. 4.28)
and the abbreviated DEs for the laser model in the dipole approximation.
Let the strength of linear-polarized electrical field of optical carrier, which passes
in the optical input of the optical fiber from the laser output, be expressed as
(Fig. 3.2a): E n (t) ¼ E L (t) ¼ E 0L cos (2πν 0L t + φ 0L ) , where ν 0L ¼ ν 0n is the average
frequency of laser generation, and E 0L is the amplitude of the laser oscillation. φ 0L is
the initial phase. The pumping current α N0 affects the laser electric input (Fig. 3.2a).
According to Eq. (4.28), for E n and the population N of the model in dipole
approximation, we have the following system of symbolic DEs ( p is the symbolic
operator of differentiation):
204
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .
To construct the analog models of OEO DM and OEO MZ, we consider structural
diagrams (Figs. 3.2 and 5.1a, b) and form and analyze of symbolic DEs of OEO.
At first, we take the symbolic differential equations of the laser, which we
described in Chap. 4.
In the diagram of Fig. 3.2, the structures of OEO DM and OEO MZ with
allocation of Feedback Chain and RF FODL are presented. For OEO MZ, two
different approaches of the positive feedback spanned of modulated laser emission
source “Laser-MZ” are shown in Fig. 5.1b, с. In the case of Fig. 3.2b, the one-port
Feedback Chain has the optical input and electrical output. In the case of Fig. 3.2с,
the one-port RF FODL has the electrical input and the electrical output. In the
structure of Fig. 3.2b, “Laser” is enclosed by the feedback by the “DC” block,
which fulfills the function of the laser optical frequency of the output power with
utilization of adjustment by the laser DC pumping current. On the structure of ОEO
DM in Fig. 3.2a, instantaneous values of voltages, relatively, on the load resistance
Z L of photodetector u PD (t), on electrical output of the RF amplifier u A (t), of the RF
filter u F (t), of the RF coupler u C (t), and instantaneous values of the laser pumping
current i L ¼ u L /Z L are shown. In the structure of ОEO MZ in Fig. 3.2b, the
instantaneous values of the field oscillation strength in the optical input of MZ
(or in the laser output) E n and in the optical output of E nMZ , and the instantaneous
voltage values in the electrical input u MZ are shown. In the structure of ОEO MZ in
Fig. 3.2с, the instantaneous voltage values, relatively, in the load resistance Z L of PD
u PD (t), in electrical outputs of the RF amplifier u A (t), of the RF filter u F (t), of the RF
coupler u C (t), and instantaneous voltage value on the electrical input of MZ are
shown.
In order to obtain the connection of the AC pumping current i 1L ¼ J 1L with the
AC component of the strength E 1n (or intensity E
2
1n ), we transfer to more detailed
description of parameters of the positive feedback circuit.
5.1.1 Symbolic Laser Equations in the OEO Structure
In Chap. 4, we examined and analyzed the symbolic differential equations (Eq. 4.28)
and the abbreviated DEs for the laser model in the dipole approximation.
Let the strength of linear-polarized electrical field of optical carrier, which passes
in the optical input of the optical fiber from the laser output, be expressed as
(Fig. 3.2a): E n (t) ¼ E L (t) ¼ E 0L cos (2πν 0L t + φ 0L ) , where ν 0L ¼ ν 0n is the average
frequency of laser generation, and E 0L is the amplitude of the laser oscillation. φ 0L is
the initial phase. The pumping current α N0 affects the laser electric input (Fig. 3.2a).
According to Eq. (4.28), for E n and the population N of the model in dipole
approximation, we have the following system of symbolic DEs ( p is the symbolic
operator of differentiation):
204
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .