In the optical range, the task solution becomes complicate by the fact that the
propagation condition of the plane wave is not satisfied in the general case. It can be
explained by commensurability of the emission wavelength 1.55 μm with the
geometric overall dimensions of the disk, the ball-shaped or the “ring” toroid with
the radius about 5–20 μm.
Let us consider the case of the ball-shaped resonator and perform the rough
estimation of the power density. We must bear in mind that only the part of emission
is inside the material of the optical resonator (Fig. 2.7). We estimate the density of
the optical power in the optical resonator with the volume V D ¼ 10
À15 m
3 at
introduced of the initial optical power of P opt ¼ 10
À6
– 10
À4 into the resonator:
P opt =V D ¼
10
À6 À10
À4
V D
¼
10
À6 À10
À4
10
À15
¼ 10
9
À 10
19 W=m
3 .
The circulation of the optical emission in the optical resonator with the Q-factor,
for example, Q 0F ¼ 10
10 , gives the power density value in the optical resonator in the
steady-state mode: Q 0F P opt /V D ¼ 10
19
– 10
29 W/m
3 . The one from the simple
models of the optical disk resonator at its excitation by the external optical field is
the model of the oscillator, which is under the effect of external force or the strength
with the amplitude E 100 of the electromagnetic field.
Let us pass to creation of QWLD resonance characteristics: dependences of the
oscillation amplitude versus the frequency E 10n (ν/ν 0F ). Sometimes, they are called
the Cartesian functions because they can be found out from solutions of algebraic
Cartesian equations.
On the one hand, these functions E 10n (ν/ν 0F ) are obtained from algebraic equations of the laser steady-state mode. On the other hand, we offer the second approach
to construction of E 10n (ν/ν 0F ). This approach consists in these functions construction
during the solution of the laser symbolic DEs. The second approach for obtaining of
E 10n (ν/ν 0F ) is more evident and informative, because the plots of solutions of the
laser symbolic DEs are constructed only at laser self-excitation condition satisfaction. At that, on the complex plane of parameters [y ¼ Re(.) ¼ E 10n , х ¼ Im(.) ¼ j(v n /
v 0L )], the axis of х ¼ Im(.) corresponds to the normalized laser optical frequency j(ν n /
ν 0L ) ¼ p 00 ¼ s, where the p 00 operator is the normalized operator of the laser
symbolic equation. The ordinate axis y ¼ Re(.) ¼ E 10n corresponds to the axis of
strength amplitude values E 10n . At first, we consider the construction of the laser
resonance characteristics by the first traditional approach.
4.6.2.1 Results of Steady-State Equations’ Solution
Functions presented in Fig. 4.23a, b are built at solution of the steady-state mode
equations for nonlinear characteristic of the laser active medium S E n
ð Þ ¼
α 01 E n À β 03 E
3
n . The DE solution in the steady-state gives the approximate expression
(under the condition that the generation frequency is close to the natural frequency of
the optical resonator) for the first harmonic amplitude E 10n in the steady-state mode:
4.6 Solutions of Nonlinear DEs of Fourth Order and Second Order for the Laser
181
propagation condition of the plane wave is not satisfied in the general case. It can be
explained by commensurability of the emission wavelength 1.55 μm with the
geometric overall dimensions of the disk, the ball-shaped or the “ring” toroid with
the radius about 5–20 μm.
Let us consider the case of the ball-shaped resonator and perform the rough
estimation of the power density. We must bear in mind that only the part of emission
is inside the material of the optical resonator (Fig. 2.7). We estimate the density of
the optical power in the optical resonator with the volume V D ¼ 10
À15 m
3 at
introduced of the initial optical power of P opt ¼ 10
À6
– 10
À4 into the resonator:
P opt =V D ¼
10
À6 À10
À4
V D
¼
10
À6 À10
À4
10
À15
¼ 10
9
À 10
19 W=m
3 .
The circulation of the optical emission in the optical resonator with the Q-factor,
for example, Q 0F ¼ 10
10 , gives the power density value in the optical resonator in the
steady-state mode: Q 0F P opt /V D ¼ 10
19
– 10
29 W/m
3 . The one from the simple
models of the optical disk resonator at its excitation by the external optical field is
the model of the oscillator, which is under the effect of external force or the strength
with the amplitude E 100 of the electromagnetic field.
Let us pass to creation of QWLD resonance characteristics: dependences of the
oscillation amplitude versus the frequency E 10n (ν/ν 0F ). Sometimes, they are called
the Cartesian functions because they can be found out from solutions of algebraic
Cartesian equations.
On the one hand, these functions E 10n (ν/ν 0F ) are obtained from algebraic equations of the laser steady-state mode. On the other hand, we offer the second approach
to construction of E 10n (ν/ν 0F ). This approach consists in these functions construction
during the solution of the laser symbolic DEs. The second approach for obtaining of
E 10n (ν/ν 0F ) is more evident and informative, because the plots of solutions of the
laser symbolic DEs are constructed only at laser self-excitation condition satisfaction. At that, on the complex plane of parameters [y ¼ Re(.) ¼ E 10n , х ¼ Im(.) ¼ j(v n /
v 0L )], the axis of х ¼ Im(.) corresponds to the normalized laser optical frequency j(ν n /
ν 0L ) ¼ p 00 ¼ s, where the p 00 operator is the normalized operator of the laser
symbolic equation. The ordinate axis y ¼ Re(.) ¼ E 10n corresponds to the axis of
strength amplitude values E 10n . At first, we consider the construction of the laser
resonance characteristics by the first traditional approach.
4.6.2.1 Results of Steady-State Equations’ Solution
Functions presented in Fig. 4.23a, b are built at solution of the steady-state mode
equations for nonlinear characteristic of the laser active medium S E n
ð Þ ¼
α 01 E n À β 03 E
3
n . The DE solution in the steady-state gives the approximate expression
(under the condition that the generation frequency is close to the natural frequency of
the optical resonator) for the first harmonic amplitude E 10n in the steady-state mode:
4.6 Solutions of Nonlinear DEs of Fourth Order and Second Order for the Laser
181