The oscillation amplitude in the steady-state point, as it follows for the solution of
the laser abbreviated equations at low losses of the high Q-factors (of the optical
resonator and the spectral line of laser emission), is determined by the approximate
expression:
α 0L
β 0L
¼
2π ν À ν 0
ð
ÞT 1n
½
2 þ 1
n
o
exp Àj 2π ν À ν 0
ð
ÞT 1n
½
f
g
3:3
ffiffi ffi
2
p
T 1n G 00
:
ð4:47Þ
For ν % ν 0 , we obtain: α 0L ¼ 6N 0 T 1n G 00 and β 0L ¼ 20
ffiffiffiffiffi
10
p Á N 0 Á T 1n G 00
ð
Þ
2
, then
the last formula is
α 0L
β 0L
%
1
3:3
ffiffi
2
p
T 1n G 00
. Figure 4.14 shows plots explaining the fulfilled
approximation of the laser nonlinearity S L0 ¼ N 0
E 10L
1þT 1n G 00 E
2
10n
by the cubic polynomial
(curve 2) S E n
ð Þ ¼ α 0L E n À β 0L E
3
10n .
4.4.2 Abbreviated DEs for the Laser in the Quasi-Stationary
Mode
According to Evtianov approach, we can introduce the operator p 001 and obtain the
differential equations for slowly changing amplitudes and phases. As the initial, we
take Eq. (4.28) and introduce parameters K ¼ K α and S(E n ), taking into account
Eqs. (4.38), (4.45) and (4.46). We make use the ready expressions for abbreviated
immitance of single circuits. As the reference frequency, we choose 2πν 0n . Changing
p 00 by j2πν 0n + p 001 , grouping terms on the smallness order, and keeping only the
first order of smallness terms, we obtain an expression for the abbreviated conductance in the first approximation in the left part [9]. Here we take into consideration
that the detuning of the laser generation frequencies ν 0n and the natural frequency ν 12
of the amplification loop (or the natural frequency of the spectral emission line of the
laser active material) β is small for QWLD and fiber lasers β ¼ [(ν 0n À ν 12 )/
ν 12 ]
2
% 0.001.
Performing the differentiation and assuming that ( p 001 )
2
¼ 0 and β ¼ 0, we obtain
the system of two real differential equations:
dE 10L
dt
¼
2πV 0F K 0L N 0
Q 12 þ Q 0F
∂
∂E
E 10L 1 þ T 1n G 00 E
2
0n
À
Á
1 þ T 1n G 00 E
2
0n
À
Á 2 þ 2π v À v 0F
ð
ÞT 1n
½
(
)
À
2πv 0F E 10L
Q 12 þ Q 0F
E 10L
dΦ 1L
dt
¼ À2πv 0F
K 0L N 0
Q 12 Q 0F
∂
∂E
E 10L ð2π v À v 0F
ð
ÞT 1n
1 þ T 1n G 00 E
2
0n
À
Á 2 þ 2π v À v 0F
ð
ÞT 1n
½
2
(
)
:
8
> > > > > <
> > > > > :
ð4:48Þ
The abbreviated laser equations allow determination of the first harmonic
amplitude.
4.4 Abbreviated DE for the Laser in Quasi-Stationary Mode
165
the laser abbreviated equations at low losses of the high Q-factors (of the optical
resonator and the spectral line of laser emission), is determined by the approximate
expression:
α 0L
β 0L
¼
2π ν À ν 0
ð
ÞT 1n
½
2 þ 1
n
o
exp Àj 2π ν À ν 0
ð
ÞT 1n
½
f
g
3:3
ffiffi ffi
2
p
T 1n G 00
:
ð4:47Þ
For ν % ν 0 , we obtain: α 0L ¼ 6N 0 T 1n G 00 and β 0L ¼ 20
ffiffiffiffiffi
10
p Á N 0 Á T 1n G 00
ð
Þ
2
, then
the last formula is
α 0L
β 0L
%
1
3:3
ffiffi
2
p
T 1n G 00
. Figure 4.14 shows plots explaining the fulfilled
approximation of the laser nonlinearity S L0 ¼ N 0
E 10L
1þT 1n G 00 E
2
10n
by the cubic polynomial
(curve 2) S E n
ð Þ ¼ α 0L E n À β 0L E
3
10n .
4.4.2 Abbreviated DEs for the Laser in the Quasi-Stationary
Mode
According to Evtianov approach, we can introduce the operator p 001 and obtain the
differential equations for slowly changing amplitudes and phases. As the initial, we
take Eq. (4.28) and introduce parameters K ¼ K α and S(E n ), taking into account
Eqs. (4.38), (4.45) and (4.46). We make use the ready expressions for abbreviated
immitance of single circuits. As the reference frequency, we choose 2πν 0n . Changing
p 00 by j2πν 0n + p 001 , grouping terms on the smallness order, and keeping only the
first order of smallness terms, we obtain an expression for the abbreviated conductance in the first approximation in the left part [9]. Here we take into consideration
that the detuning of the laser generation frequencies ν 0n and the natural frequency ν 12
of the amplification loop (or the natural frequency of the spectral emission line of the
laser active material) β is small for QWLD and fiber lasers β ¼ [(ν 0n À ν 12 )/
ν 12 ]
2
% 0.001.
Performing the differentiation and assuming that ( p 001 )
2
¼ 0 and β ¼ 0, we obtain
the system of two real differential equations:
dE 10L
dt
¼
2πV 0F K 0L N 0
Q 12 þ Q 0F
∂
∂E
E 10L 1 þ T 1n G 00 E
2
0n
À
Á
1 þ T 1n G 00 E
2
0n
À
Á 2 þ 2π v À v 0F
ð
ÞT 1n
½
(
)
À
2πv 0F E 10L
Q 12 þ Q 0F
E 10L
dΦ 1L
dt
¼ À2πv 0F
K 0L N 0
Q 12 Q 0F
∂
∂E
E 10L ð2π v À v 0F
ð
ÞT 1n
1 þ T 1n G 00 E
2
0n
À
Á 2 þ 2π v À v 0F
ð
ÞT 1n
½
2
(
)
:
8
> > > > > <
> > > > > :
ð4:48Þ
The abbreviated laser equations allow determination of the first harmonic
amplitude.
4.4 Abbreviated DE for the Laser in Quasi-Stationary Mode
165