4.4.3 Steady-State Analysis
At
dE 10L
dt ¼ 0 and
dΦ 1L
dt ¼ 0, we have the system of steady-state equation from (4.48). If
ν % ν 0F , then from the first equation, the formula for the amplitude square E L (where
E L ¼ E 10L ¼ E 10n ) follows: E
2
0L ¼
1
T 1n G 00
K 0L N 0 À 1
ð
Þ .
The calculation results with utilization of steady-state equation from (4.48) are
presented below. Figure 4.15 gives the functions of the amplitude square and the
oscillation frequency amendment.
Plots presented in Fig. 4.15 are obtained from the solution of abbreviated
equations (4.48) for the laser at utilization of the nonlinearity in the form S E n
ð Þ ¼
E 10n
1þT 1n G 00 E
Ã
10n E 10n
. An analysis of these plots of E 10L permits to conclude that at increase
of the coefficient K 0L ¼ K, which plays the role of energy pumping in the laser, we
come to the “saturation” of the strength amplitude E 10L . This is the general property
for oscillators.
The plot of the function E
2
0L K
ð Þ is, in essence, the well-known for laser experts
watt–ampere characteristic of the laser. The shape change of the curve E
2
0L K
ð Þ for
different values of the G 0 coefficient gives us understanding that the linearity of the
watt-ampere characteristic is determined by the G 0 coefficient.
The G 0 coefficient including into the nonlinear laser function defines the oscillation amplitude in the steady-state mode. An analysis of the plot of frequency
deviation from the steady-state value (Fig. 4.15b) allows making conclusion that
the increase of the G 0 coefficient leads to the growth of the generation frequency
deviations from the steady-state values. Presented plots of the laser nonlinearity
S L0 K 0L N 0 ¼
E 10L
1þT 1n G 00 E
2
10n
K 0L N 0 (Fig. 4.15c, d), as the function of E 10L ¼ E n , allows
clearly interpretation for the laser of self-excitation conditions and the steady-state
conditions. The arrow shows the tangent to the plot (Fig. 4.15c) in the point of the
generation start, which coincides with the origin point. The laser generation is
possible (or the self-excitation conditions are satisfied) when the inequality
(S L0 K 0L N 0 ) > 1 is fulfilled. The second arrow in the plot in Fig. 4.15c corresponds
to the tangent of the nonlinearity function in the steady-state point. For instance, at
K 0L ¼ K ¼ 5.4, the normalized oscillation amplitude is equal about E 10L ¼ E n ¼ 6.
Remarkably that the nonlinearity function has a maximum: S L0 K 0L N 0
ð
Þ max ¼
K 0L N 0
2
ffiffiffiffiffiffiffiffiffiffi
T 1n G 00
p
in the point E 10L ¼ E n ¼
1
ffiffiffiffiffiffiffiffiffiffi
T 1n G 00
p
. Figure 4.15 shows the plot of the derivative
of nonlinear function S L0 K 0L N 0 . It is notable that the derivative minimum corresponds to the point E 10L ¼ E n ¼
ffiffi
3
p
ffiffiffiffiffiffiffiffiffiffi
T 1n G 00
p
. Thus, addressing to abbreviated equations
permitted to obtain enough exact information about the laser at its representation by
the model in the dipole approximation and at utilization of the Evtianov method,
which is developed for the radio-frequency oscillators, for the analysis of the
quantum generators.
166
4 Semiclassical Theory and Laser Differential Equations for Optoelectronic. . .
At
dE 10L
dt ¼ 0 and
dΦ 1L
dt ¼ 0, we have the system of steady-state equation from (4.48). If
ν % ν 0F , then from the first equation, the formula for the amplitude square E L (where
E L ¼ E 10L ¼ E 10n ) follows: E
2
0L ¼
1
T 1n G 00
K 0L N 0 À 1
ð
Þ .
The calculation results with utilization of steady-state equation from (4.48) are
presented below. Figure 4.15 gives the functions of the amplitude square and the
oscillation frequency amendment.
Plots presented in Fig. 4.15 are obtained from the solution of abbreviated
equations (4.48) for the laser at utilization of the nonlinearity in the form S E n
ð Þ ¼
E 10n
1þT 1n G 00 E
Ã
10n E 10n
. An analysis of these plots of E 10L permits to conclude that at increase
of the coefficient K 0L ¼ K, which plays the role of energy pumping in the laser, we
come to the “saturation” of the strength amplitude E 10L . This is the general property
for oscillators.
The plot of the function E
2
0L K
ð Þ is, in essence, the well-known for laser experts
watt–ampere characteristic of the laser. The shape change of the curve E
2
0L K
ð Þ for
different values of the G 0 coefficient gives us understanding that the linearity of the
watt-ampere characteristic is determined by the G 0 coefficient.
The G 0 coefficient including into the nonlinear laser function defines the oscillation amplitude in the steady-state mode. An analysis of the plot of frequency
deviation from the steady-state value (Fig. 4.15b) allows making conclusion that
the increase of the G 0 coefficient leads to the growth of the generation frequency
deviations from the steady-state values. Presented plots of the laser nonlinearity
S L0 K 0L N 0 ¼
E 10L
1þT 1n G 00 E
2
10n
K 0L N 0 (Fig. 4.15c, d), as the function of E 10L ¼ E n , allows
clearly interpretation for the laser of self-excitation conditions and the steady-state
conditions. The arrow shows the tangent to the plot (Fig. 4.15c) in the point of the
generation start, which coincides with the origin point. The laser generation is
possible (or the self-excitation conditions are satisfied) when the inequality
(S L0 K 0L N 0 ) > 1 is fulfilled. The second arrow in the plot in Fig. 4.15c corresponds
to the tangent of the nonlinearity function in the steady-state point. For instance, at
K 0L ¼ K ¼ 5.4, the normalized oscillation amplitude is equal about E 10L ¼ E n ¼ 6.
Remarkably that the nonlinearity function has a maximum: S L0 K 0L N 0
ð
Þ max ¼
K 0L N 0
2
ffiffiffiffiffiffiffiffiffiffi
T 1n G 00
p
in the point E 10L ¼ E n ¼
1
ffiffiffiffiffiffiffiffiffiffi
T 1n G 00
p
. Figure 4.15 shows the plot of the derivative
of nonlinear function S L0 K 0L N 0 . It is notable that the derivative minimum corresponds to the point E 10L ¼ E n ¼
ffiffi
3
p
ffiffiffiffiffiffiffiffiffiffi
T 1n G 00
p
. Thus, addressing to abbreviated equations
permitted to obtain enough exact information about the laser at its representation by
the model in the dipole approximation and at utilization of the Evtianov method,
which is developed for the radio-frequency oscillators, for the analysis of the
quantum generators.
166
4 Semiclassical Theory and Laser Differential Equations for Optoelectronic. . .