4.4.4
Analytical Functions for the Amplitude E 10L
and the Laser Frequency Amendment (ν À ν 0n ) . . . . . . . 167
4.5
Oscillations’ Self-Excitation and Existence in QWLD . . . . . . . . . 169
4.5.1
Conditions of Oscillations’ Self-Excitation
of QWLD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
4.5.2
Stability of the Steady-State Mode of Large
Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
4.5.3
The Operator Control Function of the Laser
at Small Oscillations in the Quasi-Stationary Mode . . . . 172
4.6
Solutions of Nonlinear DEs of Fourth Order and Second
Order for the Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
4.6.1
Results on Nonlinear DE Solution of Fourth
Order for the Laser Model . . . . . . . . . . . . . . . . . . . . . . 177
4.6.2
Resonance Characteristics for Laser DE
of Fourth Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
4.7
Solution of Nonlinear DE of Second Order for the Laser . . . . . . . 193
4.7.1
Equation of Second Order for the Laser
with the Nonlinear Element in the Form
S E 10n
ð
Þ ¼ E 10n = 1 þ T 1n G 00 E
2
10n
À
Á
. . . . . . . . . . . . . . . . . 193
4.7.2
Equation of Second Order for the Laser
with the Nonlinearity of the Cubic Polynomial
S E n
ð Þ ¼ α 00 E n À β 00 E
3
n . . . . . . . . . . . . . . . . . . . . . . . . 194
4.8
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
5 Optoelectronic oscillator (OEO) Differential Equations
as the Laser System with Modulation and Positive Feedback . . . . . . 203
5.1
Symbolic and Abbreviated Equations of OEO . . . . . . . . . . . . . . . 204
5.1.1
Symbolic Laser Equations in the OEO Structure . . . . . . . 204
5.1.2
Closed PFB Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
5.1.3
Symbolic Equations for OEO DM . . . . . . . . . . . . . . . . . 208
5.1.4
The Analog Model of DEs for ОЕО DM . . . . . . . . . . . 210
5.1.5
Abbreviated Equations of ОЕО DM . . . . . . . . . . . . . . . 210
5.2
Stability Conditions: Self-Excitation and Oscillation
Existence Conditions in ОЕО DM . . . . . . . . . . . . . . . . . . . . . . . 214
5.2.1
The Circuit of the Positive Feedback Spanned
the Laser: Differential Equations of OEO . . . . . . . . . . . . 215
5.2.2
Plots of Oscillation Stability Regions . . . . . . . . . . . . . . . 222
5.3
Dynamics of Transients in ОЕО DM and the Oscillation
Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
5.3.1
OEO Oscillation Amplitude in the Small-Signal
Single-Frequency Mode . . . . . . . . . . . . . . . . . . . . . . . . 224
5.3.2
Simplified Estimation of the Setting Time of E
2
0L
intensity and the Pumping Current J L on the Base
of OEO Abbreviated Equations . . . . . . . . . . . . . . . . . . . 225
xvi
Contents
Analytical Functions for the Amplitude E 10L
and the Laser Frequency Amendment (ν À ν 0n ) . . . . . . . 167
4.5
Oscillations’ Self-Excitation and Existence in QWLD . . . . . . . . . 169
4.5.1
Conditions of Oscillations’ Self-Excitation
of QWLD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
4.5.2
Stability of the Steady-State Mode of Large
Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
4.5.3
The Operator Control Function of the Laser
at Small Oscillations in the Quasi-Stationary Mode . . . . 172
4.6
Solutions of Nonlinear DEs of Fourth Order and Second
Order for the Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
4.6.1
Results on Nonlinear DE Solution of Fourth
Order for the Laser Model . . . . . . . . . . . . . . . . . . . . . . 177
4.6.2
Resonance Characteristics for Laser DE
of Fourth Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
4.7
Solution of Nonlinear DE of Second Order for the Laser . . . . . . . 193
4.7.1
Equation of Second Order for the Laser
with the Nonlinear Element in the Form
S E 10n
ð
Þ ¼ E 10n = 1 þ T 1n G 00 E
2
10n
À
Á
. . . . . . . . . . . . . . . . . 193
4.7.2
Equation of Second Order for the Laser
with the Nonlinearity of the Cubic Polynomial
S E n
ð Þ ¼ α 00 E n À β 00 E
3
n . . . . . . . . . . . . . . . . . . . . . . . . 194
4.8
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
5 Optoelectronic oscillator (OEO) Differential Equations
as the Laser System with Modulation and Positive Feedback . . . . . . 203
5.1
Symbolic and Abbreviated Equations of OEO . . . . . . . . . . . . . . . 204
5.1.1
Symbolic Laser Equations in the OEO Structure . . . . . . . 204
5.1.2
Closed PFB Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
5.1.3
Symbolic Equations for OEO DM . . . . . . . . . . . . . . . . . 208
5.1.4
The Analog Model of DEs for ОЕО DM . . . . . . . . . . . 210
5.1.5
Abbreviated Equations of ОЕО DM . . . . . . . . . . . . . . . 210
5.2
Stability Conditions: Self-Excitation and Oscillation
Existence Conditions in ОЕО DM . . . . . . . . . . . . . . . . . . . . . . . 214
5.2.1
The Circuit of the Positive Feedback Spanned
the Laser: Differential Equations of OEO . . . . . . . . . . . . 215
5.2.2
Plots of Oscillation Stability Regions . . . . . . . . . . . . . . . 222
5.3
Dynamics of Transients in ОЕО DM and the Oscillation
Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
5.3.1
OEO Oscillation Amplitude in the Small-Signal
Single-Frequency Mode . . . . . . . . . . . . . . . . . . . . . . . . 224
5.3.2
Simplified Estimation of the Setting Time of E
2
0L
intensity and the Pumping Current J L on the Base
of OEO Abbreviated Equations . . . . . . . . . . . . . . . . . . . 225
xvi
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