If we combine Eq. (5.3) and the formula (Eq. 3.48) for y l ( jω) and replace the
symbolic operator jω by the differential operator in the time domain d/dt and ( jω)
2
by d
2 /dt
2 , we obtain the differential nonlinear equation of the second order.
Now we take into consideration that in our structural diagram, the feedback
coefficient is the transfer function of RF FODL, i.e., k fb ¼ K RFFODL ¼ U PD /U L .
For simplicity we shall consider that the module of the RF FODL transfer
function is constant and equals to K 0FODL , and K 0FODL ¼ E
2
0L K FODL . We take into
account that the directional coupler C has the transfer function on voltage and on
current, which are equal to 1, and then u F % u MZ. Having considered jω as the
differentiation operator d/dt, and taking into account that u L ¼ Z L i 1L , we can rewrite
the OEO equation in the operator form as:
p
2
þ 1=T EF
ð
Þp þ 2π f 0e
ð
Þ
2
h
i
i L ¼ 1= Z L T EF
ð
Þ
½
K 0FOLD pi A u PD
ð Þexp ÀpT FOS
ð
Þ ;
ð5:4Þ
and in the time domain as:
d
2 i L
dt 2 þ
1
T EF
Á
di L
dt
þ 2π f F0
ð
Þ
2 i L ¼
1
Z L T EF
K 0FODD
di A u PD À T FOS
ð
Þ
dt
:
ð5:5Þ
At closed positive feedback loop in OEO, the instantaneous AC current value
(in the electric laser input) from Eq. (5.4) i L ¼
E n
j j
2 1=T 0EF
ð
Þ K OF ÁK PD ÁpÁ exp ÀpT FOS
ð
Þ Á S NY u PD
ð Þ
p 2 þ 1=T 0EF
ð
Þ Á pþ 2π f 0e
ð
Þ
2
½
:
Let us remind that we can determine (as we already mentioned in Chap. 3 of this
book) the feedback circuits “from the laser optical fiber to its electrical input” and the
transfer function of this feedback. 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.
Taking into account Eq. (5.4) in the PFB loop, we transfer to determination of
symbolic equations for OEO DM.
5.1.3 Symbolic Equations for OEO DM
Let us examine the single-frequency mode of the laser generation and the dynamic
single-frequency mode of the radio-frequency generation. Taking into attention
Eq. (5.4), the system of differential equations for the strength amplitude E n of
electrical component of the electromagnetic field (outside the optical resonator) of
the QWLD output emission and for the instantaneous value of the pumping current
i L is:
208
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .
symbolic operator jω by the differential operator in the time domain d/dt and ( jω)
2
by d
2 /dt
2 , we obtain the differential nonlinear equation of the second order.
Now we take into consideration that in our structural diagram, the feedback
coefficient is the transfer function of RF FODL, i.e., k fb ¼ K RFFODL ¼ U PD /U L .
For simplicity we shall consider that the module of the RF FODL transfer
function is constant and equals to K 0FODL , and K 0FODL ¼ E
2
0L K FODL . We take into
account that the directional coupler C has the transfer function on voltage and on
current, which are equal to 1, and then u F % u MZ. Having considered jω as the
differentiation operator d/dt, and taking into account that u L ¼ Z L i 1L , we can rewrite
the OEO equation in the operator form as:
p
2
þ 1=T EF
ð
Þp þ 2π f 0e
ð
Þ
2
h
i
i L ¼ 1= Z L T EF
ð
Þ
½
K 0FOLD pi A u PD
ð Þexp ÀpT FOS
ð
Þ ;
ð5:4Þ
and in the time domain as:
d
2 i L
dt 2 þ
1
T EF
Á
di L
dt
þ 2π f F0
ð
Þ
2 i L ¼
1
Z L T EF
K 0FODD
di A u PD À T FOS
ð
Þ
dt
:
ð5:5Þ
At closed positive feedback loop in OEO, the instantaneous AC current value
(in the electric laser input) from Eq. (5.4) i L ¼
E n
j j
2 1=T 0EF
ð
Þ K OF ÁK PD ÁpÁ exp ÀpT FOS
ð
Þ Á S NY u PD
ð Þ
p 2 þ 1=T 0EF
ð
Þ Á pþ 2π f 0e
ð
Þ
2
½
:
Let us remind that we can determine (as we already mentioned in Chap. 3 of this
book) the feedback circuits “from the laser optical fiber to its electrical input” and the
transfer function of this feedback. 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.
Taking into account Eq. (5.4) in the PFB loop, we transfer to determination of
symbolic equations for OEO DM.
5.1.3 Symbolic Equations for OEO DM
Let us examine the single-frequency mode of the laser generation and the dynamic
single-frequency mode of the radio-frequency generation. Taking into attention
Eq. (5.4), the system of differential equations for the strength amplitude E n of
electrical component of the electromagnetic field (outside the optical resonator) of
the QWLD output emission and for the instantaneous value of the pumping current
i L is:
208
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .