We substitute n L in the first equation of Eq. (5.22). Then we obtain the system of
two symbolic equations of:
p þ 1=T 0F À G 0 N 00
ð
Þ p þ 1=T 1
ð
ÞþE
2
n0
À
Á
e
2
L ¼ G 0 E
2
0L α N0 i mmL À G 0 N 00 e
2
L
À
Á
,
Q 00E i mmL ¼ p E 10L
j
j
2 K 0DL S NY0 e
2
L
À Á :
(
ð5:23Þ
Now we take into consideration that:
p þ 1=T 0F À G 0 N 00
ð
Þ p þ 1=T 1
ð
ÞþE
2
n0
À
Á
¼ p
2
þ 1=T 0F
ð
Þþ 1=T 1
ð
ÞþE
2
n0 À G 0 N 00
Â
Ã
p þ 1=T 0F À G 0 N 00
ð
Þ 1=T 1
ð
ÞþE
2
n0
Â
à :
ð5:24Þ
We introduce designations: ω
2
0L ¼ 1=T 0F
ð
ÞÀG 0 N 00
½
1=T 1
ð
ÞþE
2
n0
Â
Ã
, and
α P ¼ 1=T 0F
ð
Þþ 1=T 1
ð
ÞþE
2
0L À G 0 N 00 , ω
2
0e ¼ 2π f eF0 Æ ν 0F
ð
Þ
½
2
.
Then Eq. (5.23) can be written as:
p
2
þ α P p þ ω
2
0L
Â
Ã
e
2
L ¼ G 0 E
2
0L α N0 i mmL À G
2
0 E
2
0L N 00 e
2
L ,
i mmL ¼ Q 00E
½
À1 pE
2
0L K 0DL S NY0 e
2
L
À Á :
(
ð5:25Þ
Substituting i mL into the first equation of Eq. (5.25), we obtain for the variable e
2
L ,
which has a sense of the AC component of the laser optical emission intensity, the
OEO symbolic equation of fourth order in the small-signal mode:
p
2
þ α P p þ ω
2
0L
À
Á
p
2
þ
1
T eF
p þ ω
2
0e
e
2
L
¼ p
2
α N0 G 0 E
4
0L K 0DL S NY0 e
2
L
À Á À Q 00E G
2
0 E
2
0L N 00 e
2
L :
ð5:26Þ
Finally, we obtain the symbolic equation of OEO DM, which coincides in
mathematical notation with the equation for the tube or transistor autonomous
oscillator with two resonators (with different and not-multiple natural frequencies)
located in the feedback circuit .
The left part of the obtained equation (Eq. 5.26) is the oscillating characteristic. It
consists of two multipliers, each of which can be called the operator conductance.
Here ω
2
0e and ω
2
0L are the natural resonance frequencies, relatively, in the RF filter
and in QWLD. The ω
2
0L frequency is determined by laser parameters and depends on
the DC pumping level. The second frequency ω
2
0e is the natural frequency of the RF
resonator. It is desirable to note that the damping decrement α P , which is
α P ¼ [(1/T 0F ) + (1/T 1 ) À G 0 N 00 ], depends on the level of the inversed population
N 00 and the lifetime T 1 and T 0F . In the right part of Eq. (5.26), the first term reflects
the action of the nonlinear RF amplifier (A), the circuit of positive FB consisting of
218
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .
two symbolic equations of:
p þ 1=T 0F À G 0 N 00
ð
Þ p þ 1=T 1
ð
ÞþE
2
n0
À
Á
e
2
L ¼ G 0 E
2
0L α N0 i mmL À G 0 N 00 e
2
L
À
Á
,
Q 00E i mmL ¼ p E 10L
j
j
2 K 0DL S NY0 e
2
L
À Á :
(
ð5:23Þ
Now we take into consideration that:
p þ 1=T 0F À G 0 N 00
ð
Þ p þ 1=T 1
ð
ÞþE
2
n0
À
Á
¼ p
2
þ 1=T 0F
ð
Þþ 1=T 1
ð
ÞþE
2
n0 À G 0 N 00
Â
Ã
p þ 1=T 0F À G 0 N 00
ð
Þ 1=T 1
ð
ÞþE
2
n0
Â
à :
ð5:24Þ
We introduce designations: ω
2
0L ¼ 1=T 0F
ð
ÞÀG 0 N 00
½
1=T 1
ð
ÞþE
2
n0
Â
Ã
, and
α P ¼ 1=T 0F
ð
Þþ 1=T 1
ð
ÞþE
2
0L À G 0 N 00 , ω
2
0e ¼ 2π f eF0 Æ ν 0F
ð
Þ
½
2
.
Then Eq. (5.23) can be written as:
p
2
þ α P p þ ω
2
0L
Â
Ã
e
2
L ¼ G 0 E
2
0L α N0 i mmL À G
2
0 E
2
0L N 00 e
2
L ,
i mmL ¼ Q 00E
½
À1 pE
2
0L K 0DL S NY0 e
2
L
À Á :
(
ð5:25Þ
Substituting i mL into the first equation of Eq. (5.25), we obtain for the variable e
2
L ,
which has a sense of the AC component of the laser optical emission intensity, the
OEO symbolic equation of fourth order in the small-signal mode:
p
2
þ α P p þ ω
2
0L
À
Á
p
2
þ
1
T eF
p þ ω
2
0e
e
2
L
¼ p
2
α N0 G 0 E
4
0L K 0DL S NY0 e
2
L
À Á À Q 00E G
2
0 E
2
0L N 00 e
2
L :
ð5:26Þ
Finally, we obtain the symbolic equation of OEO DM, which coincides in
mathematical notation with the equation for the tube or transistor autonomous
oscillator with two resonators (with different and not-multiple natural frequencies)
located in the feedback circuit .
The left part of the obtained equation (Eq. 5.26) is the oscillating characteristic. It
consists of two multipliers, each of which can be called the operator conductance.
Here ω
2
0e and ω
2
0L are the natural resonance frequencies, relatively, in the RF filter
and in QWLD. The ω
2
0L frequency is determined by laser parameters and depends on
the DC pumping level. The second frequency ω
2
0e is the natural frequency of the RF
resonator. It is desirable to note that the damping decrement α P , which is
α P ¼ [(1/T 0F ) + (1/T 1 ) À G 0 N 00 ], depends on the level of the inversed population
N 00 and the lifetime T 1 and T 0F . In the right part of Eq. (5.26), the first term reflects
the action of the nonlinear RF amplifier (A), the circuit of positive FB consisting of
218
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .