4.5.3.1 The Locus Method for Obtaining of QWLD Steady-State Modes
This method uses the superposition in the complex plane of the oscillating system
locus and the dynamic locus of the active element (the laser). The last locus is built
on the base of the function:
S L ¼ S L j ν=ν 0F
ð
Þ, E 10L , K 0L N 0
½
¼ S L Re þ jS LIm ,
ð4:68Þ
where E 10L is determined from the steady-state equation. The oscillating system
locus of QWLD can be constructed on the formula:
Y L j ν=ν 0F
ð
Þ
½
¼ ν=ν 0F
ð
Þ
4 À jа 10 ν=ν 0F
ð
Þ
3 À а 20 ν=ν 0F
ð
Þ
2 þ jа 30 ν=ν 0F
ð
Þþ1 ð4:69Þ
We represent this formula as
Y L j ν=ν 0F
ð
Þ
½
¼Y L Re þ jY LIm ν=ν 0F
ð
Þ:
ð4:70Þ
The intersection of two these loci (at satisfying of the geometrical stability
criterion) determines the points of the laser steady-state mode. Figure 4.19 illustrates
an application of the locus method to investigation of the steady-state stability of the
laser model in the dipole approximation. Positive directions of loci are shown by
arrows and correspond to the current frequency ν growth. In the general case, the
locus of the active element is considered as dynamic. Modes are stable for those part
of the loci, where the derivative
dS E 10n
ð
Þ
dE 10n
< 0. This is the condition of the amplitude
stability of steady-state mode.
As we see from Fig. 4.19, the unstable modes fall in those part of the locus, where
dY LIm
dν < 0. Such points are designated in Fig. 4.19 by numbers 1 and 2. This is the
condition of phase or frequency instability. The point 3 in this figure happens to be
unstable. The phase stability condition takes the form:
dY LIm =dν
dY L Re =dν >
dS LIm =dE 10n
dS L Re =dE 10n
.
According to this criterion, the part of the locus with the negative slope of Y LIm
can become the stable and on the contrary, the phase instability can appear on those
parts where
dY LIm
dν < 0, even if this function has no parts with the negative slope.
The module (1) and the argument (2) of the oscillating system and the laser
inertial nonlinearity and its loci for two different values of the gain (the pumping
levels) are presented in Fig. 4.20 for demonstration of the locus method.
The investigation of oscillation excitation conditions from small amplitudes and
the oscillation existence of the large steady-state amplitudes leads to the following
brief conclusions. These investigations with utilization of the Gurvitz criterion
enable to write a system of inequalities for main parameters of the oscillating
systems. Regions of gain values in the laser feedback loop, of Q-factors of the
optical resonators and of the spectral line of laser emission are extracted. Special
attention in the analysis of the laser system is paid to examination of the inertial
nonlinear QWLD and studying of laser operation in the quasi-stationary mode with
the large exceed of the pumping current over its threshold value. Utilization of the
4.5 Oscillations’ Self-Excitation and Existence in QWLD
175
This method uses the superposition in the complex plane of the oscillating system
locus and the dynamic locus of the active element (the laser). The last locus is built
on the base of the function:
S L ¼ S L j ν=ν 0F
ð
Þ, E 10L , K 0L N 0
½
¼ S L Re þ jS LIm ,
ð4:68Þ
where E 10L is determined from the steady-state equation. The oscillating system
locus of QWLD can be constructed on the formula:
Y L j ν=ν 0F
ð
Þ
½
¼ ν=ν 0F
ð
Þ
4 À jа 10 ν=ν 0F
ð
Þ
3 À а 20 ν=ν 0F
ð
Þ
2 þ jа 30 ν=ν 0F
ð
Þþ1 ð4:69Þ
We represent this formula as
Y L j ν=ν 0F
ð
Þ
½
¼Y L Re þ jY LIm ν=ν 0F
ð
Þ:
ð4:70Þ
The intersection of two these loci (at satisfying of the geometrical stability
criterion) determines the points of the laser steady-state mode. Figure 4.19 illustrates
an application of the locus method to investigation of the steady-state stability of the
laser model in the dipole approximation. Positive directions of loci are shown by
arrows and correspond to the current frequency ν growth. In the general case, the
locus of the active element is considered as dynamic. Modes are stable for those part
of the loci, where the derivative
dS E 10n
ð
Þ
dE 10n
< 0. This is the condition of the amplitude
stability of steady-state mode.
As we see from Fig. 4.19, the unstable modes fall in those part of the locus, where
dY LIm
dν < 0. Such points are designated in Fig. 4.19 by numbers 1 and 2. This is the
condition of phase or frequency instability. The point 3 in this figure happens to be
unstable. The phase stability condition takes the form:
dY LIm =dν
dY L Re =dν >
dS LIm =dE 10n
dS L Re =dE 10n
.
According to this criterion, the part of the locus with the negative slope of Y LIm
can become the stable and on the contrary, the phase instability can appear on those
parts where
dY LIm
dν < 0, even if this function has no parts with the negative slope.
The module (1) and the argument (2) of the oscillating system and the laser
inertial nonlinearity and its loci for two different values of the gain (the pumping
levels) are presented in Fig. 4.20 for demonstration of the locus method.
The investigation of oscillation excitation conditions from small amplitudes and
the oscillation existence of the large steady-state amplitudes leads to the following
brief conclusions. These investigations with utilization of the Gurvitz criterion
enable to write a system of inequalities for main parameters of the oscillating
systems. Regions of gain values in the laser feedback loop, of Q-factors of the
optical resonators and of the spectral line of laser emission are extracted. Special
attention in the analysis of the laser system is paid to examination of the inertial
nonlinear QWLD and studying of laser operation in the quasi-stationary mode with
the large exceed of the pumping current over its threshold value. Utilization of the
4.5 Oscillations’ Self-Excitation and Existence in QWLD
175