Chapter 4
Semiclassical Theory and Laser Differential
Equations for Optoelectronic
oscillator (OEO) Analysis
This chapter is the logical continuation of OEO investigation, in which we consider
that the laser is the main element defining the dynamic and noise properties of OEO.
In this interpretation, OEO is represented as the quantum generator spanned by the
positive feedback: the optoelectronic network containing the optical fiber, the
photodetector, the RF filter, and the RF amplifier.
Section 4.1 is devoted to model formation of the quantum generator, for which we
use the semiclassical theory based on the dipole representation of the laser in doublelevel approximation. For formation of differential equations (DE) in quasi-stationary
mode, we consider in Sect. 4.2 the DE system, which consists of three differential
equations. This DE system establishes a connection between the active medium
polarization, the electromagnetic field strength, the inversed population of particles
on upper and lower energy levels, and pumping, taking into account the Q-factors of
the optical resonator and the emission spectral line of the active medium and the
population inertial properties. Examination of this system of nonlinear DEs with
inertial properties allows formation of the laser analog model in the dipole
approximation.
The analysis of the self-oscillating system (SOS) of the laser model is performed
in Sect. 4.3, for which we introduce the operator transfer function, the nonlinear
element (NE) and execute the analysis of nonlinear properties. The special role is
paid (Sect. 4.4) at laser system analysis to investigation of inertial nonlinear QWLD
at its operation in quasi-stationary mode with the high excess of pumping current
over its threshold value. Investigation of oscillation excitation conditions and oscillation existence conditions in the OWLD steady-state is performed in Sect. 4.5.
Section 4.6 is devoted to results’ presentation of the nonlinear fourth-order DE
solution for the laser model in the dipole approximation. The plots of realizations of
oscillating processes for two types of nonlinearities are analyzed. Results of computer solution for the quasi-stationary mode of laser generation are presented for
fourth-order DE at the linear-hyperbolic inertial nonlinearity of the active medium.
Numerical calculation results on the base of the symbolic equation solution are
© The Editor(s) (if applicable) and The Author(s), under exclusive license to
Springer Nature Switzerland AG 2020
A. A. Bortsov et al., Laser Optoelectronic Oscillators, Springer Series in Optical
Sciences 232, https://doi.org/10.1007/978-3-030-45700-6_4
133
Semiclassical Theory and Laser Differential
Equations for Optoelectronic
oscillator (OEO) Analysis
This chapter is the logical continuation of OEO investigation, in which we consider
that the laser is the main element defining the dynamic and noise properties of OEO.
In this interpretation, OEO is represented as the quantum generator spanned by the
positive feedback: the optoelectronic network containing the optical fiber, the
photodetector, the RF filter, and the RF amplifier.
Section 4.1 is devoted to model formation of the quantum generator, for which we
use the semiclassical theory based on the dipole representation of the laser in doublelevel approximation. For formation of differential equations (DE) in quasi-stationary
mode, we consider in Sect. 4.2 the DE system, which consists of three differential
equations. This DE system establishes a connection between the active medium
polarization, the electromagnetic field strength, the inversed population of particles
on upper and lower energy levels, and pumping, taking into account the Q-factors of
the optical resonator and the emission spectral line of the active medium and the
population inertial properties. Examination of this system of nonlinear DEs with
inertial properties allows formation of the laser analog model in the dipole
approximation.
The analysis of the self-oscillating system (SOS) of the laser model is performed
in Sect. 4.3, for which we introduce the operator transfer function, the nonlinear
element (NE) and execute the analysis of nonlinear properties. The special role is
paid (Sect. 4.4) at laser system analysis to investigation of inertial nonlinear QWLD
at its operation in quasi-stationary mode with the high excess of pumping current
over its threshold value. Investigation of oscillation excitation conditions and oscillation existence conditions in the OWLD steady-state is performed in Sect. 4.5.
Section 4.6 is devoted to results’ presentation of the nonlinear fourth-order DE
solution for the laser model in the dipole approximation. The plots of realizations of
oscillating processes for two types of nonlinearities are analyzed. Results of computer solution for the quasi-stationary mode of laser generation are presented for
fourth-order DE at the linear-hyperbolic inertial nonlinearity of the active medium.
Numerical calculation results on the base of the symbolic equation solution are
© The Editor(s) (if applicable) and The Author(s), under exclusive license to
Springer Nature Switzerland AG 2020
A. A. Bortsov et al., Laser Optoelectronic Oscillators, Springer Series in Optical
Sciences 232, https://doi.org/10.1007/978-3-030-45700-6_4
133