Limitations of a Model
Lifetime in excited state of an atom or a particle is ten thousand periods of the EMF
wave. In the general case, during lifetime, the atom (or the particle) can pass (in the
space) several wavelengths. We neglect by the atom (particle) motion under action
of the electrostatic field and we neglect also by the atom motion due to the thermal
motion, and we use an analysis of the solid-state laser.
We suppose that the laser system of energy zones consists of only two states:
1 (main) and 2 (excited). For simplification, we neglect by the vector character of
EMF and consider that EMF propagates only in the single polarization plane. The
resonator is assumed as the high-Q system and it provides the single-frequency
generation mode.
In the model, we consider a variety of atoms as a medium, in which its electrical
state is described by the macroscopic polarization. A polarization is a sum of particle
(charge carriers or atoms) dipole moments normalized to the volume V 0 : P ¼ ∑ i d ei /
V 0 ,: where d ei is the quantum-mechanical average value of the dipole moment in the
space point in the time moment t, V 0 is the small volume of average near the point
under consideration, d e ¼ e Á l 0 ¼ d ei is the electrical dipole moment, e is the electric
charge of the electron, and l 0 is the dipole arm. The total electrical dipole moment of
an atom represents a sum of all atom electrons.
The problem of boundary conditions in the considered model is avoided by
consideration of the absorbing medium with a conductance (the ohmic or wave
conductance) σ, which is chosen so as to provide the observable damping of natural
oscillation owing to reflections and the diffraction.
Let us extract the high-Q mode of the laser resonator. Let it be the plane EMF
wave and propagates along the z-axis or along the i z vector and is polarized along the
x-axis or along the i x vector, and the EMF strength with amplitude E 0 and the
wavelength λ is:
E ¼ E 0 exp jz 2π=λ
ð
Þ
½
Ái x :
ð4:1Þ
In electrodynamics, the absolute value of Poynting vector is expressed through
the normalized wave strength E
2
0L . For simplicity, we use dimensionless normalized
values of the strength square E
2
L , which has a sense of the field intensity (or field
power) on the standard area. The connection of normalized E
2
L and factual E
2 , as well
as E
2
0L and E
2
0 is specified as
E
2
L ¼
c
n 0
εε 0
2
E
2 , E
2
0L ¼
c
n 0
εε 0
2
E
2
0 :
ð4:2Þ
Here ε is the medium permittivity, ε 0 the vacuum permittivity, n 0 is the refraction
index of the medium material. The normalized strength used in this book can be
defined as E L ¼
ffiffiffiffiffi ffi
E
2
L
q
.
4.1 Semiclassical Laser Equations and OEO Differential Equations
139
Lifetime in excited state of an atom or a particle is ten thousand periods of the EMF
wave. In the general case, during lifetime, the atom (or the particle) can pass (in the
space) several wavelengths. We neglect by the atom (particle) motion under action
of the electrostatic field and we neglect also by the atom motion due to the thermal
motion, and we use an analysis of the solid-state laser.
We suppose that the laser system of energy zones consists of only two states:
1 (main) and 2 (excited). For simplification, we neglect by the vector character of
EMF and consider that EMF propagates only in the single polarization plane. The
resonator is assumed as the high-Q system and it provides the single-frequency
generation mode.
In the model, we consider a variety of atoms as a medium, in which its electrical
state is described by the macroscopic polarization. A polarization is a sum of particle
(charge carriers or atoms) dipole moments normalized to the volume V 0 : P ¼ ∑ i d ei /
V 0 ,: where d ei is the quantum-mechanical average value of the dipole moment in the
space point in the time moment t, V 0 is the small volume of average near the point
under consideration, d e ¼ e Á l 0 ¼ d ei is the electrical dipole moment, e is the electric
charge of the electron, and l 0 is the dipole arm. The total electrical dipole moment of
an atom represents a sum of all atom electrons.
The problem of boundary conditions in the considered model is avoided by
consideration of the absorbing medium with a conductance (the ohmic or wave
conductance) σ, which is chosen so as to provide the observable damping of natural
oscillation owing to reflections and the diffraction.
Let us extract the high-Q mode of the laser resonator. Let it be the plane EMF
wave and propagates along the z-axis or along the i z vector and is polarized along the
x-axis or along the i x vector, and the EMF strength with amplitude E 0 and the
wavelength λ is:
E ¼ E 0 exp jz 2π=λ
ð
Þ
½
Ái x :
ð4:1Þ
In electrodynamics, the absolute value of Poynting vector is expressed through
the normalized wave strength E
2
0L . For simplicity, we use dimensionless normalized
values of the strength square E
2
L , which has a sense of the field intensity (or field
power) on the standard area. The connection of normalized E
2
L and factual E
2 , as well
as E
2
0L and E
2
0 is specified as
E
2
L ¼
c
n 0
εε 0
2
E
2 , E
2
0L ¼
c
n 0
εε 0
2
E
2
0 :
ð4:2Þ
Here ε is the medium permittivity, ε 0 the vacuum permittivity, n 0 is the refraction
index of the medium material. The normalized strength used in this book can be
defined as E L ¼
ffiffiffiffiffi ffi
E
2
L
q
.
4.1 Semiclassical Laser Equations and OEO Differential Equations
139