If to separate from the polarization a part connected with pumping P P and a part
connected with nonlinearity P NL , then the wave equation can be rewritten in the
form:
∇ Â ∇ Â E þ
1
с 2
∂
2 E
∂t 2 ¼
1
ε 0 с 2
d
2 P P
dt 2 À
1
ε 0 с 2
d
2 P NL
dt 2 ,
ð4:9Þ
where P P describes a polarization caused by the pumping field taking into account a
population: P NL ¼ ε 0 χ
3 (ν)N|E|
2 Á E, where χ is the medium susceptibility (here we
consider the cubic nonlinearity). As a rule, we can neglect by nonlinearity P NL in
connection with its smallness. It manifests at nonlinear optical phenomena, for
instance, at harmonic doubling, at two- and three-photon nonlinear interaction, the
Brillouin dispersion, etc. Such phenomena have the threshold character and are
demonstrated at high power densities (more than 10
10 mW/m
3 ).
At pumping action, the polarization P P is proportional the population difference
between levels N(t) and depends on the dipole moment p
2
e ¼ d
2
e : P P ¼
d
2
e
2Á2πh Á N Á E.
We assume that non-excited field for the main (or zero) mode is described by the
equation in the linear case:
∇ Â ∇ Â e m À
ν
2
Á n
2
ν
ð Þ
с 2
e m ¼ 0,
ð4:10Þ
Note that the e m vector defining the polarization plane of oscillations is normalized to the maximum so as max(e m ) ¼ 1, and integration of |e m |
2 over the volume
V is: V 0 ¼
R
|e m |
2 dV.
Let us represent the main mode field in the form:
E ¼
1
2
 E 0n t
ð Þ Á e m r
ð Þ Á exp Àj2πνt þ jφ L t
ð Þ
ð
Þ þ E 0n t
ð Þ Á e m r
ð Þ Á exp j2πνt þ jφ L t
ð Þ
ð
Þ
½
,
ð4:11Þ
where ν is the mode optical frequency close to the natural frequency ν 0L of the laser
resonator, r ¼ z is the radius-vector, E 0n (t) and φ L (t) is slowly changing amplitude
and phase.
4.1.4 Differential Equations of QWLD
We substitute Eq. (4.11) into Eq. (4.9), multiply it by e m and integrate over the whole
volume. As the result, we obtain DEs connecting strengths and polarizations:
4.1 Semiclassical Laser Equations and OEO Differential Equations
141
connected with nonlinearity P NL , then the wave equation can be rewritten in the
form:
∇ Â ∇ Â E þ
1
с 2
∂
2 E
∂t 2 ¼
1
ε 0 с 2
d
2 P P
dt 2 À
1
ε 0 с 2
d
2 P NL
dt 2 ,
ð4:9Þ
where P P describes a polarization caused by the pumping field taking into account a
population: P NL ¼ ε 0 χ
3 (ν)N|E|
2 Á E, where χ is the medium susceptibility (here we
consider the cubic nonlinearity). As a rule, we can neglect by nonlinearity P NL in
connection with its smallness. It manifests at nonlinear optical phenomena, for
instance, at harmonic doubling, at two- and three-photon nonlinear interaction, the
Brillouin dispersion, etc. Such phenomena have the threshold character and are
demonstrated at high power densities (more than 10
10 mW/m
3 ).
At pumping action, the polarization P P is proportional the population difference
between levels N(t) and depends on the dipole moment p
2
e ¼ d
2
e : P P ¼
d
2
e
2Á2πh Á N Á E.
We assume that non-excited field for the main (or zero) mode is described by the
equation in the linear case:
∇ Â ∇ Â e m À
ν
2
Á n
2
ν
ð Þ
с 2
e m ¼ 0,
ð4:10Þ
Note that the e m vector defining the polarization plane of oscillations is normalized to the maximum so as max(e m ) ¼ 1, and integration of |e m |
2 over the volume
V is: V 0 ¼
R
|e m |
2 dV.
Let us represent the main mode field in the form:
E ¼
1
2
 E 0n t
ð Þ Á e m r
ð Þ Á exp Àj2πνt þ jφ L t
ð Þ
ð
Þ þ E 0n t
ð Þ Á e m r
ð Þ Á exp j2πνt þ jφ L t
ð Þ
ð
Þ
½
,
ð4:11Þ
where ν is the mode optical frequency close to the natural frequency ν 0L of the laser
resonator, r ¼ z is the radius-vector, E 0n (t) and φ L (t) is slowly changing amplitude
and phase.
4.1.4 Differential Equations of QWLD
We substitute Eq. (4.11) into Eq. (4.9), multiply it by e m and integrate over the whole
volume. As the result, we obtain DEs connecting strengths and polarizations:
4.1 Semiclassical Laser Equations and OEO Differential Equations
141