6.5.6 Symbolic Expressions for Fluctuations
From Eq. (6.32), we obtain the equation for fluctuations of amplitudes and phases.
For this, we can express U a through I b0 using the connection: U a ¼ I b0 Á
1
Y a0 jΔf
ð
Þ ,
where at p ¼ 0 we used the equality Y a ( p + jΔf ) ¼ Y a0 ( jΔf ). Let us present of the
normalized symbolic conductance Y Na ( p) ¼ Y a ( p + jΔf )/Y a0 ( jΔf ). Then
Y p þ jΔf
ð
Þ
Y jΔf
ð
Þ
Á m U þ jψ U
ð
Þ¼σ U Á m U þ jψ U þ μ a Re þ jμ aIm :
ð6:43Þ
Now we can present of the normalized conductance in the form: Y Na ¼
Y pþjΔf
ð
Þ
Y jΔf
ð
Þ ¼ Y Na Re þ jY NaIm . We substitute this expression Y Na in Eq. (6.43) and
equating the real and imaginary parts from the right and left sides, we have:
Y Na Á m U þ jψ U
ð
Þ¼σ U Á m U þ jψ U þ μ a Re þ jμ aIm :
ð6:44Þ
We would like to note that for fluctuating components, the following relation is
true: pm U ¼ m U À ψ U . Then, the symbolic equations for the real and imaginary parts
of the fluctuating quantities will take the form:
Y Na Re p
ð Þ Á m U À Y NaIm p
ð Þ Á ψ U ¼ σ U Á m U þ μ a Re ,
ð6:45Þ
Y NaIm p
ð Þ Á m U þ Y Na Re p
ð Þ Á ψ U ¼ ψ U þ μ aIm :
ð6:46Þ
Converting to the matrix view, according to the Kramer rule, we find determinants, and the following expressions can be found for m U and ψ U :
m U ¼
μ a Re Á Y a Re À 1
ð
Þþμ aIm Á Y aIm
Y a Re p
ð Þ À σ U
½
ÁY a Re p
ð Þ À 1
½
þY aIm p
ð Þ
½
2
,
ð6:47Þ
ψ U ¼
Y a Re À σ U
ð
ÞÁμ aIm À Y aIm Á μ a Re
Y a Re p
ð Þ À σ U
½
ÁY a Re p
ð Þ À 1
½
þY aIm p
ð Þ
½
2
:
ð6:48Þ
In the general case for OEO, PSD of fluctuations of the amplitude S mOEO and the
phase S ΨOEO have the complicate analytical form and each of them are defined by the
in-phase S μRe (ω) and the quadrature S μIm (ω) components.
The influence of the OEO power and the natural bandwidth of the laser spectral
line upon the noise properties shows approximately the following expressions for
S mAG and S ΨAG :
318
6 Operation Analysis of Optoelectronic oscillator (OEO) with External. . .
From Eq. (6.32), we obtain the equation for fluctuations of amplitudes and phases.
For this, we can express U a through I b0 using the connection: U a ¼ I b0 Á
1
Y a0 jΔf
ð
Þ ,
where at p ¼ 0 we used the equality Y a ( p + jΔf ) ¼ Y a0 ( jΔf ). Let us present of the
normalized symbolic conductance Y Na ( p) ¼ Y a ( p + jΔf )/Y a0 ( jΔf ). Then
Y p þ jΔf
ð
Þ
Y jΔf
ð
Þ
Á m U þ jψ U
ð
Þ¼σ U Á m U þ jψ U þ μ a Re þ jμ aIm :
ð6:43Þ
Now we can present of the normalized conductance in the form: Y Na ¼
Y pþjΔf
ð
Þ
Y jΔf
ð
Þ ¼ Y Na Re þ jY NaIm . We substitute this expression Y Na in Eq. (6.43) and
equating the real and imaginary parts from the right and left sides, we have:
Y Na Á m U þ jψ U
ð
Þ¼σ U Á m U þ jψ U þ μ a Re þ jμ aIm :
ð6:44Þ
We would like to note that for fluctuating components, the following relation is
true: pm U ¼ m U À ψ U . Then, the symbolic equations for the real and imaginary parts
of the fluctuating quantities will take the form:
Y Na Re p
ð Þ Á m U À Y NaIm p
ð Þ Á ψ U ¼ σ U Á m U þ μ a Re ,
ð6:45Þ
Y NaIm p
ð Þ Á m U þ Y Na Re p
ð Þ Á ψ U ¼ ψ U þ μ aIm :
ð6:46Þ
Converting to the matrix view, according to the Kramer rule, we find determinants, and the following expressions can be found for m U and ψ U :
m U ¼
μ a Re Á Y a Re À 1
ð
Þþμ aIm Á Y aIm
Y a Re p
ð Þ À σ U
½
ÁY a Re p
ð Þ À 1
½
þY aIm p
ð Þ
½
2
,
ð6:47Þ
ψ U ¼
Y a Re À σ U
ð
ÞÁμ aIm À Y aIm Á μ a Re
Y a Re p
ð Þ À σ U
½
ÁY a Re p
ð Þ À 1
½
þY aIm p
ð Þ
½
2
:
ð6:48Þ
In the general case for OEO, PSD of fluctuations of the amplitude S mOEO and the
phase S ΨOEO have the complicate analytical form and each of them are defined by the
in-phase S μRe (ω) and the quadrature S μIm (ω) components.
The influence of the OEO power and the natural bandwidth of the laser spectral
line upon the noise properties shows approximately the following expressions for
S mAG and S ΨAG :
318
6 Operation Analysis of Optoelectronic oscillator (OEO) with External. . .