The first equation of the mentioned system is the nonlinear equation of the fourth
order for the instantaneous value of the normalized field strength. The second
equation is the equation of the first order for the population.
The main parameters of such a system of DE are the time constant T 2 (or the Qfactor of the spectral line of laser emission Q 02 ¼ νT 02 ) and the time constant T 0F
(or the Q-factor of the optical resonator Q OF ¼ νT OF ), the lifetime of carriers on the
upper operating level T 1n , and the gain (of the saturation coefficient) G 00 . We note
that this SOS system has double-circuit (or double-resonance) network with the
inertial pumping (energy feed). The operation feature is the presence of automatic
control of the laser generation level by means of variation of laser pumping (or feed).
At decrease or increase of the laser generation amplitude, pumping provides its
recovering up to steady-state value. This control mode is similar to regulation of the
laser power at sensing with the help of the photodetector. The power control is
performed by variation of the pumping current.
The term in the right side of the first equation describes of the exciting force,
which is proportional to the product N Á E n . At that, oscillations of the field strength
E n are the “fast” function. The “slow” function N reflects the population difference
on the upper level and depends on pumping.
From the second equation, we can express the function N(p) transferring to the
formal symbolic notation: N p
ð Þ ¼ α N0 = p þ
1
T 1
þ G 00 Á Re E n E
Ã
n
Â
Ã
n
o
. With the purpose to determine the nonlinearity in this system of DEs, we write the product of the
field strength by population in the time domain: N(t) Á E n (t) ¼ N 00 Á E 0n (t)
cos (2πν n t + φ Ln (t) + 2πν n T OY ).
The laser in this equation can be interpreted as the generator with the active
element, which is characterized by the gain K α and the nonlinearity S L (E n ). In the
general case, at large densities of the field strength, the function S L (E n ) has inertial
properties: S LT (E n ) ¼ S L (E n ) Á exp (Àj2πν 0n T OY ). By the mathematical notation,
these equations are similar to differential equations of the autonomous RF oscillator
with two stages, each of which is loaded on the oscillating circuit.
The natural frequencies of oscillating circuits are equal (up to thousandth parts).
The DE system is also similar to the equation of the autonomous oscillator with
inertial feed [9]. The laser pumping plays a role of the laser “power supply.” To take
into consideration the delay in the FB loop, the right term can be supplemented by
the multiplier exp(Àj2πν 0n T R ). In differential equations, we can take into account the
nonlinearity and inertial properties of the active medium arising at strong fields.
Let us write the term in the right side of the first DE introducing the gain:
K ¼
2 Á 2d
2
e
3 Á ε 0 2πh
N 00 :
ð4:29Þ
Then we extract the nonlinear function S L (N Á E n ) ¼ (N/N 00 ) Á E n . Now we can
write the right term in DE as:
4.3 Laser Kinetic Equations and the Pumping System
155
order for the instantaneous value of the normalized field strength. The second
equation is the equation of the first order for the population.
The main parameters of such a system of DE are the time constant T 2 (or the Qfactor of the spectral line of laser emission Q 02 ¼ νT 02 ) and the time constant T 0F
(or the Q-factor of the optical resonator Q OF ¼ νT OF ), the lifetime of carriers on the
upper operating level T 1n , and the gain (of the saturation coefficient) G 00 . We note
that this SOS system has double-circuit (or double-resonance) network with the
inertial pumping (energy feed). The operation feature is the presence of automatic
control of the laser generation level by means of variation of laser pumping (or feed).
At decrease or increase of the laser generation amplitude, pumping provides its
recovering up to steady-state value. This control mode is similar to regulation of the
laser power at sensing with the help of the photodetector. The power control is
performed by variation of the pumping current.
The term in the right side of the first equation describes of the exciting force,
which is proportional to the product N Á E n . At that, oscillations of the field strength
E n are the “fast” function. The “slow” function N reflects the population difference
on the upper level and depends on pumping.
From the second equation, we can express the function N(p) transferring to the
formal symbolic notation: N p
ð Þ ¼ α N0 = p þ
1
T 1
þ G 00 Á Re E n E
Ã
n
Â
Ã
n
o
. With the purpose to determine the nonlinearity in this system of DEs, we write the product of the
field strength by population in the time domain: N(t) Á E n (t) ¼ N 00 Á E 0n (t)
cos (2πν n t + φ Ln (t) + 2πν n T OY ).
The laser in this equation can be interpreted as the generator with the active
element, which is characterized by the gain K α and the nonlinearity S L (E n ). In the
general case, at large densities of the field strength, the function S L (E n ) has inertial
properties: S LT (E n ) ¼ S L (E n ) Á exp (Àj2πν 0n T OY ). By the mathematical notation,
these equations are similar to differential equations of the autonomous RF oscillator
with two stages, each of which is loaded on the oscillating circuit.
The natural frequencies of oscillating circuits are equal (up to thousandth parts).
The DE system is also similar to the equation of the autonomous oscillator with
inertial feed [9]. The laser pumping plays a role of the laser “power supply.” To take
into consideration the delay in the FB loop, the right term can be supplemented by
the multiplier exp(Àj2πν 0n T R ). In differential equations, we can take into account the
nonlinearity and inertial properties of the active medium arising at strong fields.
Let us write the term in the right side of the first DE introducing the gain:
K ¼
2 Á 2d
2
e
3 Á ε 0 2πh
N 00 :
ð4:29Þ
Then we extract the nonlinear function S L (N Á E n ) ¼ (N/N 00 ) Á E n . Now we can
write the right term in DE as:
4.3 Laser Kinetic Equations and the Pumping System
155