S ΨOEO F
ð Þ ¼ S aμIm Á
Y a Re À σ U À Y aIm
Y a Re À σ U
½
ŠÁY a Re À 1
½
Šþ Y aIm
½
Š
2
"
# 2
:
ð6:52Þ
When we use the single oscillating circuit with the natural frequency ω 0 , the filter
time constant T F as the RF filter and we use in RF FODL (with the total effective
delay T FODL ) of the solitary single-mode optical fiber, the following expressions are
true for the real Y a Re ¼
y M 1þFT EF
½
Š cos FT FOS
½
Š
P 0L M Z
j jK PD
j
jK F
j j
and the imaginary Y aIm ¼
y M 1þFT EF
½
Š sin FT FOS
½
Š
P 0L M Z
j jK PD
j
jK F
j j parts, taking into account F ¼ ω À ω 0 ¼ 2πf À 2πf 0 . We introduce
the designation Y 00 ¼
y M 1þFT EF
½
Š
M Z
j jK PD
j
jK F
j j . After substitution Y aRe and Y aIm (determined in
Eq. (6.36)), using Y 00 (6.36), the expressions (6.51) and (6.52), we can write the
expression for S mOEO (ω), S ΨOEO (F) is:
S mOEO ω
ð Þ ¼ S μ Re Á K
2
ΓAN , S ΨOEO ω
ð Þ ¼ S μIm Á K
2
ΓPN ,
ð6:53Þ
where coefficientsK
2
ΓAN andK
2
ΓPN are:
K
2
ΓAN ¼
Y
P 0L
ffiffi ffi
2
p
sin FT FOS þ π=4
ð
Þ
Â
à À 1
n
o 2
Y 00
P 0L
h i 2 À
Y 00
P 0L
Á 1 þ σ U
ð
Þcos FT FOS
½
Šþσ U
&
' 2 ,
ð6:54Þ
K
2
ΓPN ¼
Y
P 0L
ffiffi ffi
2
p
sin π=4 À FT FOS
ð
Þ À σ U
n
o 2
Y 00
P 0L
h i 2 À
Y 00
P 0L
Á 1 þ σ U
ð
Þcos FT FOS
½
Šþσ U
&
' 2 :
ð6:55Þ
We used for Eqs. (6.54) and (6.55) the trigonometric formula for the sum:
cos FT FOS
½
Šþ sin FT FOS
½
Š¼
ffiffi ffi
2
p
sin π=4 þ FT FOS
ð
Þ , and the trigonometric formula
for difference: cos FT FOS
½
ŠÀ sin FT FOS
½
Š¼
ffiffi ffi
2
p
sin π=4 À FT FOS
ð
Þ .
Figures 6.13, 6.14, 6.15, and 6.16 show the plots constructed on the expressions
(6.54) and (6.55) of the suppression coefficients of the amplitude K
2
ΓAN F
ð Þ and the
phase K
2
ΓPN F
ð Þ noise in OEO MZ for various values of σ U and
P 0L
Y 00
.
Plots of suppression coefficients of the amplitude K
2
ΓAN σ U
ð Þ and the phase
K
2
ΓPN σ U
ð Þ noise of OEO MZ are presented in Fig. 6.17 depicted by expressions
(6.54) and (6.55) for different σ U at the normalized offset F ¼ 3.
Plots depicted in Figs. 6.13, 6.14, 6.15, 6.16, and 6.17 by expressions (6.54) and
(6.55) allows the following conclusions. Suppression coefficients of the amplitude
and the phase noises are completely determined of PSD of the amplitude and phase
noises on OEO MZ.
At growth of the module of the local slope coefficient σ U up to 10, we can
suppress the amplitude noises by more than 100 times in the normalized offset
region F > 2. At that, the phase noises are suppressed insignificantly by the value not
320
6 Operation Analysis of Optoelectronic oscillator (OEO) with External. . .