p
2
þ
1
T 0F
p þ 2πν 0n
ð
Þ
2
!
p
2
þ 1=T 2
ð
Þp þ 2πν 12
ð
Þ
2
h
i
E n ¼ K α p
2 S L N Á E n
ð
Þ,
pN ¼ α N0 À
N
T 1
À G 00 N Á Re E n E
Ã
n
Â
Ã
:
8
> > <
> > :
ð5:1Þ
The left part of the first DE in Eq. (5.1) represents the laser oscillating system. The
right part of the first DE contains the nonlinear function S L (N Á E n ), which (at using
of the N n approximation in the vicinity of ν frequencies, which are close to the
average optical frequency ν 0n of the laser), takes a form:
S L N Á E n
ð
Þ¼
E n N 0
1 þ j ν À ν 0F
ð
Þ
½
T 1 þ T 1 G 00 Re E n Á E
Ã
n
Â
Ã
È
É ,
ð5:2Þ
where T 1 is the carrier lifetime, ν 0F the natural optical frequency of the laser
resonator, G 00 is the coefficient of amplification (or saturation).
The second DE in the system (Eq. 5.1) represents the equation for the population
N. The pumping α N0 , which has the DC component α N00 Á J 0L and the AC component α N01 ¼ α N001 Á i L , where α N001 is the constant coefficient, is included in the right
part of the second DE. The pumping α N0 in (Eq. 5.2) can be expressed as:
α N0 ¼ α N00 Á J 0L + α N01 Á i L , where α N00 and α N01 are constants, J 0L ¼ I 0L is DC
component of the pumping current, i L is the AC component of the pumping current.
In the general case, the AC component of the pumping current i L is defined by the
current sum of the first i 1L , second i 2L , third i 3L , . . ., nth i nL harmonics:
i L ¼ i 1L + i 2L + i 3L + Á Á Á + i nL . But since we examine the case of the narrowband
RF filter in OEO DM, we consider that the harmonic amplitudes (besides first) are
neglibly small: i 2L + i 3L + Á Á Á + i nL ¼ 0. Therefore we consider that i L ¼ i 1L ¼ J 1L .
Now we form the system of differential equations (Eq. 5.1) for OEO DM
(Fig. 5.1) for the square of strength amplitude E n of electrical component of the
electromagnetic field with the amplitude of the first harmonic E 10L and the AC
electrical component i 1L with the amplitude of the first harmonic of the laser
pumping current : J 1L : E 1n ¼ E 10L Re [exp(2πν 0 t À Φ 10L )] , i 1L ¼ J 10L Re [exp
(2πf 0 t À Φ 10L )] .
For formation of OEO DM differential equations (Fig. 3.2a), the system (Eq. 5.1)
must be added by the equation for the circuit of the positive feedback (PFB), which
encloses the laser (QWLD). The series-closed in the loop, the optical filter (OF), the
photodetector (PD), the RF amplifier (A), the RF filter (F), and the RF coupler
(C) are included into the PFB circuit (Fig. 3.2a). In order to obtain the expression,
which couples of the AC component of the pumping current i 1L ¼ J 1L of the laser
with the AC component of the strength E 1n (or intensity E
2
1n ), we transfer to the more
detailed description of PFB circuit parameters.
206
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .
2
þ
1
T 0F
p þ 2πν 0n
ð
Þ
2
!
p
2
þ 1=T 2
ð
Þp þ 2πν 12
ð
Þ
2
h
i
E n ¼ K α p
2 S L N Á E n
ð
Þ,
pN ¼ α N0 À
N
T 1
À G 00 N Á Re E n E
Ã
n
Â
Ã
:
8
> > <
> > :
ð5:1Þ
The left part of the first DE in Eq. (5.1) represents the laser oscillating system. The
right part of the first DE contains the nonlinear function S L (N Á E n ), which (at using
of the N n approximation in the vicinity of ν frequencies, which are close to the
average optical frequency ν 0n of the laser), takes a form:
S L N Á E n
ð
Þ¼
E n N 0
1 þ j ν À ν 0F
ð
Þ
½
T 1 þ T 1 G 00 Re E n Á E
Ã
n
Â
Ã
È
É ,
ð5:2Þ
where T 1 is the carrier lifetime, ν 0F the natural optical frequency of the laser
resonator, G 00 is the coefficient of amplification (or saturation).
The second DE in the system (Eq. 5.1) represents the equation for the population
N. The pumping α N0 , which has the DC component α N00 Á J 0L and the AC component α N01 ¼ α N001 Á i L , where α N001 is the constant coefficient, is included in the right
part of the second DE. The pumping α N0 in (Eq. 5.2) can be expressed as:
α N0 ¼ α N00 Á J 0L + α N01 Á i L , where α N00 and α N01 are constants, J 0L ¼ I 0L is DC
component of the pumping current, i L is the AC component of the pumping current.
In the general case, the AC component of the pumping current i L is defined by the
current sum of the first i 1L , second i 2L , third i 3L , . . ., nth i nL harmonics:
i L ¼ i 1L + i 2L + i 3L + Á Á Á + i nL . But since we examine the case of the narrowband
RF filter in OEO DM, we consider that the harmonic amplitudes (besides first) are
neglibly small: i 2L + i 3L + Á Á Á + i nL ¼ 0. Therefore we consider that i L ¼ i 1L ¼ J 1L .
Now we form the system of differential equations (Eq. 5.1) for OEO DM
(Fig. 5.1) for the square of strength amplitude E n of electrical component of the
electromagnetic field with the amplitude of the first harmonic E 10L and the AC
electrical component i 1L with the amplitude of the first harmonic of the laser
pumping current : J 1L : E 1n ¼ E 10L Re [exp(2πν 0 t À Φ 10L )] , i 1L ¼ J 10L Re [exp
(2πf 0 t À Φ 10L )] .
For formation of OEO DM differential equations (Fig. 3.2a), the system (Eq. 5.1)
must be added by the equation for the circuit of the positive feedback (PFB), which
encloses the laser (QWLD). The series-closed in the loop, the optical filter (OF), the
photodetector (PD), the RF amplifier (A), the RF filter (F), and the RF coupler
(C) are included into the PFB circuit (Fig. 3.2a). In order to obtain the expression,
which couples of the AC component of the pumping current i 1L ¼ J 1L of the laser
with the AC component of the strength E 1n (or intensity E
2
1n ), we transfer to the more
detailed description of PFB circuit parameters.
206
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .