d
2 E
dz 2 þ μ 0 ε 0
d
2 E
dt 2 þ μ 0 σ
dE
dt
¼ μ 0
d
2 P
dt 2 À μ 0
d
2 P NL
dt 2 ,
ð4:12Þ
where σ is the ohmic conductance.
At presence of the required polarization, the quasi-stationary stimulated oscillations of the electrical field can be expanded on eigenfunctions of the fundamental
modes. We are interested only by a solution for the main mode for the singlefrequency oscillation mode of generation. But for solution of the more general task
for the fundamental mode, we have the eigenfunction U n (z) ¼ sin K n z
(non-normalized for r ¼ z), and at adding over all modes n, we obtain the expansion
formula:
E z, t
ð Þ ¼
X
n
E n t
ð Þ Á sin K n z,
ð4:13Þ
where K n is the propagation constant for the mode. Amplitudes of all fundamental
oscillations are finite, and solution will be in the form of a series over eigenfunctions.
We assume that the Q-factor of the main type of oscillations is much more than
1. And one term of such a series prevails and therefore, a sum is replaced by the
single term E(z, t) ¼ E n (t) Á E n (z).
The polarization of the operating transition steps forward as the exciting force
(or current, as in generators). Using the integration formula along to resonator length
L, we obtain P n :
P n t
ð Þ ¼ 2=L
ð
Þ
Z L
0
P z, t
ð Þsin K n z
ð
Þ Þ dz:
ð4:14Þ
Introduced conductance σ is chosen so that to obtain the necessary Q-factor for
the n-mode (main mode) Q n , then the ohmic conductance is:
σ ¼ ε 0 2πν 0n =Q n ¼ ε 0 1=T OF
ð
Þ:
ð4:15Þ
If to represent the polarization P(z, t) in the form of a series: P(z, t) ¼ ∑ n P n (t) Á P n (z)
for the field strength E n (with the slowly changing amplitude E 0n (t) and phase φ L (t)).
We obtain that E n satisfies to DE with the exciting term for stimulated oscillations of
the harmonic oscillator with damping:
d
2 E n
dt 2 þ 1=T OF
ð
Þ
dE n
dt
þ 2πν 0n
ð
Þ
2 E n ¼ ε 0
ð Þ
À1 Á
d
2 P n
dt 2 ,
ð4:16Þ
in which P n is the spatial component of the active medium polarization. We would
like to attract attention to the fact that in the right part of DE, there is the second
derivative
d
2 P n
dt 2 because the electromagnetic field of the excited dipole is defined by
142
4 Semiclassical Theory and Laser Differential Equations for Optoelectronic. . .
2 E
dz 2 þ μ 0 ε 0
d
2 E
dt 2 þ μ 0 σ
dE
dt
¼ μ 0
d
2 P
dt 2 À μ 0
d
2 P NL
dt 2 ,
ð4:12Þ
where σ is the ohmic conductance.
At presence of the required polarization, the quasi-stationary stimulated oscillations of the electrical field can be expanded on eigenfunctions of the fundamental
modes. We are interested only by a solution for the main mode for the singlefrequency oscillation mode of generation. But for solution of the more general task
for the fundamental mode, we have the eigenfunction U n (z) ¼ sin K n z
(non-normalized for r ¼ z), and at adding over all modes n, we obtain the expansion
formula:
E z, t
ð Þ ¼
X
n
E n t
ð Þ Á sin K n z,
ð4:13Þ
where K n is the propagation constant for the mode. Amplitudes of all fundamental
oscillations are finite, and solution will be in the form of a series over eigenfunctions.
We assume that the Q-factor of the main type of oscillations is much more than
1. And one term of such a series prevails and therefore, a sum is replaced by the
single term E(z, t) ¼ E n (t) Á E n (z).
The polarization of the operating transition steps forward as the exciting force
(or current, as in generators). Using the integration formula along to resonator length
L, we obtain P n :
P n t
ð Þ ¼ 2=L
ð
Þ
Z L
0
P z, t
ð Þsin K n z
ð
Þ Þ dz:
ð4:14Þ
Introduced conductance σ is chosen so that to obtain the necessary Q-factor for
the n-mode (main mode) Q n , then the ohmic conductance is:
σ ¼ ε 0 2πν 0n =Q n ¼ ε 0 1=T OF
ð
Þ:
ð4:15Þ
If to represent the polarization P(z, t) in the form of a series: P(z, t) ¼ ∑ n P n (t) Á P n (z)
for the field strength E n (with the slowly changing amplitude E 0n (t) and phase φ L (t)).
We obtain that E n satisfies to DE with the exciting term for stimulated oscillations of
the harmonic oscillator with damping:
d
2 E n
dt 2 þ 1=T OF
ð
Þ
dE n
dt
þ 2πν 0n
ð
Þ
2 E n ¼ ε 0
ð Þ
À1 Á
d
2 P n
dt 2 ,
ð4:16Þ
in which P n is the spatial component of the active medium polarization. We would
like to attract attention to the fact that in the right part of DE, there is the second
derivative
d
2 P n
dt 2 because the electromagnetic field of the excited dipole is defined by
142
4 Semiclassical Theory and Laser Differential Equations for Optoelectronic. . .