T eF
dJ 1L
dt
¼ J 1L Re K BZ R NA S NA0 À J 1L þ μ a Re ,
J 1L T eF
dψ 1L
dt
¼ J 1L 2π f res À f
ð
Þ T eF þ J 1L ImK BZ R NA S NA0 þ μ aIm ,
8
> <
> :
ð7:60Þ
where μ aRe is in-phase and μ aIm is quadrature components of the noise.
Let us examine the small-signal modulation mode of the laser in OEO and at that,
E
2
0L ¼ E
2
00L þ e l , N 0L ¼ N 00L + n l , φ ¼ φ 00L + φ L , J 1L ¼ J 00L + α 00l , where E
2
00L ,
N 00L , φ 00L are steady-state solutions of the differential equations, and e l , n l , φ L , α 00l
are small deviations from the steady-state values, relatively, for AC components
E 00L , N 00L , φ 00L , J 1L (i.e., the inequalities E 00L ) e l , N 00L ) n l , φ 00L ) φ L ,
J 1L ) α 00l are fulfilled). Now we consider the features of OEO DM operation
under conditions of existence of the positive selective feedback and the fulfillment
of the limit cycle stability condition.
At first, we make normalization of differential equations (Eqs. 7.59 and 7.60). We
introduce the normalized time τ ¼ t/T 1L and suppose that G ¼ G 0 T 1L . Using the
Laplace transform, we pass to the parameter d/dt ¼ p; j2π( f À f 0 ) ¼ j(ω À ω 0 ), where
ω 0 is the generated frequency of OEO DM close to the frequency of the RF filter.
Now we investigate Eqs. (7.59) and (7.60) under conditions of the quasi-static mode
in the steady-state point: E
2
00L , N 00L , φ 00L . We neglect by the second order and
designates the small deviations of the field strength amplitude e l , the amplitude
noises m ξ , the noise component of the pumping current α 00l , the population noises n ξ
for the population n l , for the optical phase φ L the noises of the laser optical phase φ ξ .
Now we obtain the system of symbolic equations for the small deviation of the
amplitude e l , the population n l , the phase φ L , the amplitude of the pumping current
α 00l of OEO DM with fluctuation Langevinian components for the amplitude of the
laser optical oscillations ξ Le , for carriers in the laser active medium ξ
c
Ln , for the phase
of the optical oscillations ξ
c
Lφ :
À α 00l À 1
ð
Þ n ξ þ pm ξ À pφ ξ ¼ ξ Le ,
p þ α 00l
ð
Þ n ξ À Re K S À G
ð
Þ m ξ À pφ ξ ¼ ξ
c
Ln ,
2π f 00P C N T 1L n ξ À T 1L ρ 0L þ ImK S
ð
Þ m ξ þ pφ ξ þ φ ξ À φ ξ Re K S À G
ð
Þ¼ξ
c
Lφ :
8
> <
> :
ð7:61Þ
In the equation system (Eq. 7.61), for fluctuation parameters in the third equation,
the term 2πf 00P C N T 1L n ξ defines the contribution in PSD of OEO amplitude and
phase noises from carrier population n ξ in the laser active medium; the term
(T 1L ρ 0L + Im K S ) Á m ξ —the contribution of laser amplitude noises m ξ ; the term
[pφ ξ + φ ξ À φ ξ Á (ReK S À G)]—the contribution of laser phase noises φ ξ . We remind
here that C N is constant, f 00P ¼ ν 00P À ν, where ν 0P is the natural frequency of the
optical resonator, T 1L ¼ T 1 is the lifetime of carriers.
Let us obtain also the system of symbolic equations of OEO DM for small
deviations of the pumping current amplitude α 00l , the phase of RF oscillations
ψ L1 , the noise components of noisy in-phase μ aRe , and quadrature μ aIm components:
7.6 Power Spectrum Density of Amplitude and Phase Noise in OEO DM
443
dJ 1L
dt
¼ J 1L Re K BZ R NA S NA0 À J 1L þ μ a Re ,
J 1L T eF
dψ 1L
dt
¼ J 1L 2π f res À f
ð
Þ T eF þ J 1L ImK BZ R NA S NA0 þ μ aIm ,
8
> <
> :
ð7:60Þ
where μ aRe is in-phase and μ aIm is quadrature components of the noise.
Let us examine the small-signal modulation mode of the laser in OEO and at that,
E
2
0L ¼ E
2
00L þ e l , N 0L ¼ N 00L + n l , φ ¼ φ 00L + φ L , J 1L ¼ J 00L + α 00l , where E
2
00L ,
N 00L , φ 00L are steady-state solutions of the differential equations, and e l , n l , φ L , α 00l
are small deviations from the steady-state values, relatively, for AC components
E 00L , N 00L , φ 00L , J 1L (i.e., the inequalities E 00L ) e l , N 00L ) n l , φ 00L ) φ L ,
J 1L ) α 00l are fulfilled). Now we consider the features of OEO DM operation
under conditions of existence of the positive selective feedback and the fulfillment
of the limit cycle stability condition.
At first, we make normalization of differential equations (Eqs. 7.59 and 7.60). We
introduce the normalized time τ ¼ t/T 1L and suppose that G ¼ G 0 T 1L . Using the
Laplace transform, we pass to the parameter d/dt ¼ p; j2π( f À f 0 ) ¼ j(ω À ω 0 ), where
ω 0 is the generated frequency of OEO DM close to the frequency of the RF filter.
Now we investigate Eqs. (7.59) and (7.60) under conditions of the quasi-static mode
in the steady-state point: E
2
00L , N 00L , φ 00L . We neglect by the second order and
designates the small deviations of the field strength amplitude e l , the amplitude
noises m ξ , the noise component of the pumping current α 00l , the population noises n ξ
for the population n l , for the optical phase φ L the noises of the laser optical phase φ ξ .
Now we obtain the system of symbolic equations for the small deviation of the
amplitude e l , the population n l , the phase φ L , the amplitude of the pumping current
α 00l of OEO DM with fluctuation Langevinian components for the amplitude of the
laser optical oscillations ξ Le , for carriers in the laser active medium ξ
c
Ln , for the phase
of the optical oscillations ξ
c
Lφ :
À α 00l À 1
ð
Þ n ξ þ pm ξ À pφ ξ ¼ ξ Le ,
p þ α 00l
ð
Þ n ξ À Re K S À G
ð
Þ m ξ À pφ ξ ¼ ξ
c
Ln ,
2π f 00P C N T 1L n ξ À T 1L ρ 0L þ ImK S
ð
Þ m ξ þ pφ ξ þ φ ξ À φ ξ Re K S À G
ð
Þ¼ξ
c
Lφ :
8
> <
> :
ð7:61Þ
In the equation system (Eq. 7.61), for fluctuation parameters in the third equation,
the term 2πf 00P C N T 1L n ξ defines the contribution in PSD of OEO amplitude and
phase noises from carrier population n ξ in the laser active medium; the term
(T 1L ρ 0L + Im K S ) Á m ξ —the contribution of laser amplitude noises m ξ ; the term
[pφ ξ + φ ξ À φ ξ Á (ReK S À G)]—the contribution of laser phase noises φ ξ . We remind
here that C N is constant, f 00P ¼ ν 00P À ν, where ν 0P is the natural frequency of the
optical resonator, T 1L ¼ T 1 is the lifetime of carriers.
Let us obtain also the system of symbolic equations of OEO DM for small
deviations of the pumping current amplitude α 00l , the phase of RF oscillations
ψ L1 , the noise components of noisy in-phase μ aRe , and quadrature μ aIm components:
7.6 Power Spectrum Density of Amplitude and Phase Noise in OEO DM
443