Now we perform the linearization of the function S NE (E n ) in the right part of
Eq. (5.48) in the vicinity of above-mentioned steady-state values. We obtain for PSD
of the amplitude S ANL (F) and phase S PNL (F) noises the following expressions:
S ANL F
ð Þ ¼ S mL ¼
Q F Re À S 0E
ð
Þ
2 S SL Re þ Q FIm
ð
Þ
4 S SLIm
Q F Re À σ EL
ð
ÞQ F Re À S 0E
ð
Þþ Q FIm
ð
Þ
2
h
i 2 ;
ð5:51Þ
S PNL F
ð Þ ¼ S φL ¼
Q F Re À σ EL
ð
Þ
2 S SLIm þ Q FIm
ð
Þ
2 S SL Re
Q F Re À σ EL
ð
ÞQ F Re À S 0E
ð
Þþ Q FIm
ð
Þ
2
h
i 2 ;
ð5:52Þ
where S 0E % 1 and σ EL is 0.1–2.
In the general case, for the laser with delay in the optical feedback channel, the
abbreviated representations are true:
Q F Re
ð
Þ
2 ¼ 1 þ T 0F F
ð
Þ
2 cos
2 FT L
ð
Þ= P 0L K 0L
ð
Þ ,
ð5:53Þ
Q FIm
ð
Þ
2 ¼ 1 þ T 0F F
ð
Þ
2 sin
2 FT L
ð
Þ= P 0L K 0L
ð
Þ :
ð5:54Þ
We examine the case when values sin(FT L ) ¼ 0 and cos(FT L ) ¼ 1, then
Q FRe ¼ FT OF + 1; Q FIm ¼ 0 . In this case, amplitude fluctuations are caused by the
in-phase noise component only, while the phase fluctuations are caused by the
quadrature component only:
S ANL F
ð Þ ¼ S mL ¼
S SN Re
Q F Re À σ EL
ð
Þ
2
;
S PNL F
ð Þ ¼ S φL ¼
S SNIm
Q F Re À S 0E
ð
Þ
2
:
8
> > <
> > :
ð5:55Þ
Substituting Q FRe ¼ FT OF + 1; Q FIm ¼ 0; S 0E ¼ 1 into Eq. (5.55), we obtain
expressions for АN and PN noise of the laser:
S ANL F
ð Þ ¼ S mL ¼
S SL Re
FT OF þ 1 À σ EL
ð
Þ
2
;
S PNL F
ð Þ ¼ S φL ¼
S SNIm
FT OF þ 1 À S 0E
ð
Þ
2
¼
S SLIm
FT OF
ð
Þ
2
:
8
> > <
> > :
ð5:56Þ
Transformation of spectra of relative noise amplitude strengths, as it follows from
Eq. (5.56), occurs as in the single-chain RF RC-circuit with the time constant T OF /
(1 À σ EL ) and the transfer function on zero frequency 1/(1 À σ EL ). The value of
1 À σ EL is usually called the “limit cycle durability” in the nonlinear oscillation
theory, and typical values in oscillators are usually 0.1–2. The case 1 À σ EL ¼ 0
corresponds to the oscillation stability boundary. For considered case, when the
5.4 Fluctuation Differential Equations of OEO with the Langevinian Noise Sources
241
Eq. (5.48) in the vicinity of above-mentioned steady-state values. We obtain for PSD
of the amplitude S ANL (F) and phase S PNL (F) noises the following expressions:
S ANL F
ð Þ ¼ S mL ¼
Q F Re À S 0E
ð
Þ
2 S SL Re þ Q FIm
ð
Þ
4 S SLIm
Q F Re À σ EL
ð
ÞQ F Re À S 0E
ð
Þþ Q FIm
ð
Þ
2
h
i 2 ;
ð5:51Þ
S PNL F
ð Þ ¼ S φL ¼
Q F Re À σ EL
ð
Þ
2 S SLIm þ Q FIm
ð
Þ
2 S SL Re
Q F Re À σ EL
ð
ÞQ F Re À S 0E
ð
Þþ Q FIm
ð
Þ
2
h
i 2 ;
ð5:52Þ
where S 0E % 1 and σ EL is 0.1–2.
In the general case, for the laser with delay in the optical feedback channel, the
abbreviated representations are true:
Q F Re
ð
Þ
2 ¼ 1 þ T 0F F
ð
Þ
2 cos
2 FT L
ð
Þ= P 0L K 0L
ð
Þ ,
ð5:53Þ
Q FIm
ð
Þ
2 ¼ 1 þ T 0F F
ð
Þ
2 sin
2 FT L
ð
Þ= P 0L K 0L
ð
Þ :
ð5:54Þ
We examine the case when values sin(FT L ) ¼ 0 and cos(FT L ) ¼ 1, then
Q FRe ¼ FT OF + 1; Q FIm ¼ 0 . In this case, amplitude fluctuations are caused by the
in-phase noise component only, while the phase fluctuations are caused by the
quadrature component only:
S ANL F
ð Þ ¼ S mL ¼
S SN Re
Q F Re À σ EL
ð
Þ
2
;
S PNL F
ð Þ ¼ S φL ¼
S SNIm
Q F Re À S 0E
ð
Þ
2
:
8
> > <
> > :
ð5:55Þ
Substituting Q FRe ¼ FT OF + 1; Q FIm ¼ 0; S 0E ¼ 1 into Eq. (5.55), we obtain
expressions for АN and PN noise of the laser:
S ANL F
ð Þ ¼ S mL ¼
S SL Re
FT OF þ 1 À σ EL
ð
Þ
2
;
S PNL F
ð Þ ¼ S φL ¼
S SNIm
FT OF þ 1 À S 0E
ð
Þ
2
¼
S SLIm
FT OF
ð
Þ
2
:
8
> > <
> > :
ð5:56Þ
Transformation of spectra of relative noise amplitude strengths, as it follows from
Eq. (5.56), occurs as in the single-chain RF RC-circuit with the time constant T OF /
(1 À σ EL ) and the transfer function on zero frequency 1/(1 À σ EL ). The value of
1 À σ EL is usually called the “limit cycle durability” in the nonlinear oscillation
theory, and typical values in oscillators are usually 0.1–2. The case 1 À σ EL ¼ 0
corresponds to the oscillation stability boundary. For considered case, when the
5.4 Fluctuation Differential Equations of OEO with the Langevinian Noise Sources
241