J 0L ¼ I 0L is the DC pumping current. Such abbreviated equation has the following
form:
dE 10L
dt
¼ 2πν 0F Á
K 0L Á N 0
Q 12 þ Q 0F
d
2
dE
2
10L
S LCOS E 10L
ð
Þ
½
ŠÀ2πν 0F Á
E 10L
Q 12 þ Q 0F
, ð5:12Þ
E 10L
dΦ 1L
dt
¼ À2πν 0F Á
K 0L N 0
Q 12 Q 0F
d
2
dE
2
10L
S LSIn E 10L
ð
Þ
½
Š ,
ð5:13Þ
where nonlinear functions S LCOS (E 10L ) and S LSIN (E 10L ) have the form as in
Chap. 4: S LCOS E 10L
ð
Þ¼
E 10L 1þT 1n G 00 E
2
0n
½
Š
1þT 1n G 00 E
2
0n
ð
Þ
2 þ 2π νÀν 0F
ð
Þ T 1n
ð
Þ
2
Â
Ã
and S LSIN E 10L
ð
Þ¼
E 10L 2π νÀν 0F
ð
Þ T 1n
½
Š
1þT 1n G 00 E
2
0n
ð
Þ
2 þ 2π νÀν 0F
ð
Þ T 1n
ð
Þ
2
Â
à .
Now we take into consideration that in our structural diagram the feedback
coefficient k fb is the transfer function of RF FODL, i.e., K RFFODL ¼ U PD /U L . But
the transfer function of the feedback network K FBN ¼ K DL for OEO DM can be
defined as, K DL ¼ K FBN ¼
i L
E n
j j
2 ¼
i L
E L
j j
2 , where E L is the normalized strength in the
QWLD output, which is equal in magnitude to the value in the FOS input, i L ¼ i m is
the AC current modulation component in the QWLD in the OEO DM.
For simplicity, we assume that the module of the RF FODL transfer function is
constant and equal to K 0FODL , and then K 0FODL ¼ E
2
0L K FODL .
Taking into consideration Eqs. (5.11)–(5.13), we write the complete system of
abbreviated equations for OEO DM:
f
dE 10L
dt
¼ 2πν 0F
K 0L N 0 ðJ 10L Þ
Q 12 þ Q 0F
S LCOS ðE 10L Þ À 2πν 0F Á
E 10L
Q 12 þ Q 0F
,
E 10L
dΦ 1L
dt
¼ À2πν 0F
K 0L N 0 ðJ 10L Þ
Q 12 Q 0F
S LSIN ðE 10L Þ,
dJ 1L
dt
¼ 2π f 0e
E
2
10L K FOLD
2R L Q EF
I 1A ðE 10L Þcos½2π f 0e T FOS Š À 2π f 0e
J 1L
2Q EF
,
dΦ 10E
dt
¼ 2π f 0e
E
2
10L K FOLD
2Q EF
I 1A ðE 10L Þsinð2π f 0e T FOS Þ:
ð5:14Þ
We obtain the system of abbreviated equations (Eq. 5.14), owing to which we can
investigate transient and steady-state processes in OEO DM. The analysis of these
equations is much simpler than the symbolic equation. The AC component of the
laser pumping current (or the current first harmonic in the laser input) J 1L influences
on the population N 0 (J 1L ), which determines solutions for the amplitude E 10L and
phase Φ 1L of the laser field strength in the first and second equations. The laser
power (or the square of the normalized amplitude) E
2
10L is included in the third and
fourth equations of the system (Eq. 5.14) and, relatively, determines its solutions for
J 1L and the phase Φ 10E . The peculiarity of this self-oscillating system is the presence
of the laser nonlinearities S LCOS (E 10L ), S LSIN (E 10L ) and the RF amplifier nonlinearity
I 1A (U PD ). In OEO DM, the presence of mentioned nonlinearities complicates the
212
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .