the acceleration of dipole charges (or the second derivative of the dipole charge
displacement).
Further, we shall address to the known system of semiclassical equations, in
which the first equation is already defined by us.
4.2 Constitutive Equations of a Laser
4.2.1 DE for the Single-Frequency Single-Mode Regime
The generally accepted DE system for the single-frequency single-mode (the only
type of spatial oscillations) laser generation regime [8]. The system of stimulated
oscillation equations of the resonator filled by the active medium has a form for the
dimensionless normalized strength of EMF for the main mode E n :
d
2 E n
dt 2 þ
1
T 0F
dE n
dt
þ 2πv 0n
ð
Þ
2 E n ¼
1
ε 0
d
2 P n
dt 2
d
2 P n
dt 2 þ
1
T 2
dP n
dt
þ 2πv 12
ð
Þ
2 P n ¼
2d
2
e
3 Á 2πh
NE n
dN
dt
¼ α N0 À
N
T 1
À G 00 N Á Re P n Á E
Ã
n
Â
Ã
,
8
> > > > > > <
> > > > > > :
ð4:17Þ
where G 00 ¼
1
T 1
G 0
1þκE n E
Ã
n
ð
Þ
, G 0 is the gain, κ is the nonlinear coefficient. Variables
including in Eq. (4.17) are listed in Table 4.1.
Solution of the system of “complete” differential equations Eq. (4.17) even with
utilization of numerical methods and modern computers is the labor-intensive but
solvable task. This is explained by the fact that the ratio of particle lifetime on the
upper level T 1 to the oscillation period of the laser field T L ¼ 1/ν L (where v L is the
oscillation frequency) is about T 1 /T L ¼ 10
5 , and the total number of calculated
values may exceed 10
8
–10
12 . The solution of Eq. (4.17) is possible to perform
applying the analog modeling.
4.2.2 Nonlinear Symbolic Differential Equations for QWLD
Introducing the operator p ¼
d
dt , taking p ¼ j2πν, where ν is the oscillation frequency
of the laser fundamental harmonic, we perform the Laplace transform for our
equations. Then we introduce K D ¼
2d
2
e
3Á2πh and fulfilling the algebraic transformations,
we have:
4.2 Constitutive Equations of a Laser
143
displacement).
Further, we shall address to the known system of semiclassical equations, in
which the first equation is already defined by us.
4.2 Constitutive Equations of a Laser
4.2.1 DE for the Single-Frequency Single-Mode Regime
The generally accepted DE system for the single-frequency single-mode (the only
type of spatial oscillations) laser generation regime [8]. The system of stimulated
oscillation equations of the resonator filled by the active medium has a form for the
dimensionless normalized strength of EMF for the main mode E n :
d
2 E n
dt 2 þ
1
T 0F
dE n
dt
þ 2πv 0n
ð
Þ
2 E n ¼
1
ε 0
d
2 P n
dt 2
d
2 P n
dt 2 þ
1
T 2
dP n
dt
þ 2πv 12
ð
Þ
2 P n ¼
2d
2
e
3 Á 2πh
NE n
dN
dt
¼ α N0 À
N
T 1
À G 00 N Á Re P n Á E
Ã
n
Â
Ã
,
8
> > > > > > <
> > > > > > :
ð4:17Þ
where G 00 ¼
1
T 1
G 0
1þκE n E
Ã
n
ð
Þ
, G 0 is the gain, κ is the nonlinear coefficient. Variables
including in Eq. (4.17) are listed in Table 4.1.
Solution of the system of “complete” differential equations Eq. (4.17) even with
utilization of numerical methods and modern computers is the labor-intensive but
solvable task. This is explained by the fact that the ratio of particle lifetime on the
upper level T 1 to the oscillation period of the laser field T L ¼ 1/ν L (where v L is the
oscillation frequency) is about T 1 /T L ¼ 10
5 , and the total number of calculated
values may exceed 10
8
–10
12 . The solution of Eq. (4.17) is possible to perform
applying the analog modeling.
4.2.2 Nonlinear Symbolic Differential Equations for QWLD
Introducing the operator p ¼
d
dt , taking p ¼ j2πν, where ν is the oscillation frequency
of the laser fundamental harmonic, we perform the Laplace transform for our
equations. Then we introduce K D ¼
2d
2
e
3Á2πh and fulfilling the algebraic transformations,
we have:
4.2 Constitutive Equations of a Laser
143