S mOEO ω
ð Þ ¼
S aμ Re Á Y a Re À 1
ð
Þ
2 þ S aμIm Á Y aIm
½
2 þ 2 Y a Re À 1
ð
Þ Y aIm Re S aμReIm
Â
Ã
Y a Re À σ U
f
Á Y a Re À 1
½
þ Y aIm
½
2 g 2
,
ð6:49Þ
S ΨOEO ω
ð Þ ¼
Y a Re À σ U
ð
Þ
2 Á S aμIm þ Y
2
aμIm Á S aμ Re À 2 Y a Re À σ U
ð
ÞÁY aIm Re S aμReIm
Â
Ã
Y a Re À σ U
½
ÁY a Re À 1
½
þ Y aIm
½
2
n
o 2
,
ð6:50Þ
where S aμRe is PSD of mutual fluctuations of the amplitude S aμ Re ¼
4
R 1
0 μ a Re t
ð Þμ a Re t À τ
ð
Þcos 2πf τ
ð
Þdτ , S aμIm is PSD of mutual fluctuations of the
phase S aμIm ¼ 4
R 1
0 μ aIm t
ð Þμ aIm t À τ
ð
Þcos 2πf τ
ð
Þdτ, PSD of mutual fluctuations the
amplitude and the phase S aμReIm ¼ 4
R 1
0 μ a Re t
ð Þμ aIm t À τ
ð
Þcos 2πf τ
ð
Þdτ . Expressions (6.49) and (6.50) have a close form to the similar expressions in for the
transistor oscillators [3] and OEO MZ [1, 4]. But, the sense of the symbolic
conductance and the in-phase S μRe (ω) and the quadrature S μIm (ω) components is
different.
6.5.7 An Analysis of Formulas for PSD of Amplitude
and Phase Noises
The quadrature S LIm (ω) and the in-phase S LRe (ω). PSD included in Eqs. (6.49) and
(6.50) are determined by the earlier obtained expressions (6.26), (6.27) and (6.20),
(6.21). As we see from Eqs. (6.26) and (6.27), the quadrature S LIm (ω) and the
in-phase S LRe (ω) PSD are not equal in the general case. Here we consider the specific
case to obtain the evident, relatively simple formulas for PSD of the amplitude and
phase noises. We assume that S LIm (ω) % S LRe (ω) % S aμReIm (ω). Let us examine the
case when G 12 ¼ 1 À (A 1 /A 2 ) Á exp (ÀT FOS /T c ) % 1 and sin[2πν 0 (T 2M À T 1M )] ¼ 1.
Such an equality 2πν 0 (T 2M À T 1M ) ¼ πm is possible in the quadrature operation
mode of MZ and at that, sin[2πν 0 (T 2M À T 1M )] ¼ 1, i.e., 2πν 0 (T 2M À T 1M ) ¼ mπ/2,
and then S μAM % S μAM ‐ PM , at condition fulfillment that the laser coherence time T c
is much more than the delay time T FODL ¼ T FOS in RF FODL, i.e.,
G 12 ¼ 1 À exp (ÀT FOS /T c ) % 1. For this case, when S LIm (ω) % S LRe (ω), expressions
U a ¼ U a0 (1 + m U ) exp ( j2πΔft + jφ 0U + jψ U ), exp( jψ U ) ¼ 1 + jψ U are written in the
following form:
S mOEO F
ð Þ ¼ S aμ Re Á
Y a Re À 1 þ Y aIm
Y a Re À σ U
½
ÁY a Re À 1
½
þ Y aIm
½
2
"
# 2
,
ð6:51Þ
6.5 Differential Fluctuation Equations of OEO MZ
319
ð Þ ¼
S aμ Re Á Y a Re À 1
ð
Þ
2 þ S aμIm Á Y aIm
½
2 þ 2 Y a Re À 1
ð
Þ Y aIm Re S aμReIm
Â
Ã
Y a Re À σ U
f
Á Y a Re À 1
½
þ Y aIm
½
2 g 2
,
ð6:49Þ
S ΨOEO ω
ð Þ ¼
Y a Re À σ U
ð
Þ
2 Á S aμIm þ Y
2
aμIm Á S aμ Re À 2 Y a Re À σ U
ð
ÞÁY aIm Re S aμReIm
Â
Ã
Y a Re À σ U
½
ÁY a Re À 1
½
þ Y aIm
½
2
n
o 2
,
ð6:50Þ
where S aμRe is PSD of mutual fluctuations of the amplitude S aμ Re ¼
4
R 1
0 μ a Re t
ð Þμ a Re t À τ
ð
Þcos 2πf τ
ð
Þdτ , S aμIm is PSD of mutual fluctuations of the
phase S aμIm ¼ 4
R 1
0 μ aIm t
ð Þμ aIm t À τ
ð
Þcos 2πf τ
ð
Þdτ, PSD of mutual fluctuations the
amplitude and the phase S aμReIm ¼ 4
R 1
0 μ a Re t
ð Þμ aIm t À τ
ð
Þcos 2πf τ
ð
Þdτ . Expressions (6.49) and (6.50) have a close form to the similar expressions in for the
transistor oscillators [3] and OEO MZ [1, 4]. But, the sense of the symbolic
conductance and the in-phase S μRe (ω) and the quadrature S μIm (ω) components is
different.
6.5.7 An Analysis of Formulas for PSD of Amplitude
and Phase Noises
The quadrature S LIm (ω) and the in-phase S LRe (ω). PSD included in Eqs. (6.49) and
(6.50) are determined by the earlier obtained expressions (6.26), (6.27) and (6.20),
(6.21). As we see from Eqs. (6.26) and (6.27), the quadrature S LIm (ω) and the
in-phase S LRe (ω) PSD are not equal in the general case. Here we consider the specific
case to obtain the evident, relatively simple formulas for PSD of the amplitude and
phase noises. We assume that S LIm (ω) % S LRe (ω) % S aμReIm (ω). Let us examine the
case when G 12 ¼ 1 À (A 1 /A 2 ) Á exp (ÀT FOS /T c ) % 1 and sin[2πν 0 (T 2M À T 1M )] ¼ 1.
Such an equality 2πν 0 (T 2M À T 1M ) ¼ πm is possible in the quadrature operation
mode of MZ and at that, sin[2πν 0 (T 2M À T 1M )] ¼ 1, i.e., 2πν 0 (T 2M À T 1M ) ¼ mπ/2,
and then S μAM % S μAM ‐ PM , at condition fulfillment that the laser coherence time T c
is much more than the delay time T FODL ¼ T FOS in RF FODL, i.e.,
G 12 ¼ 1 À exp (ÀT FOS /T c ) % 1. For this case, when S LIm (ω) % S LRe (ω), expressions
U a ¼ U a0 (1 + m U ) exp ( j2πΔft + jφ 0U + jψ U ), exp( jψ U ) ¼ 1 + jψ U are written in the
following form:
S mOEO F
ð Þ ¼ S aμ Re Á
Y a Re À 1 þ Y aIm
Y a Re À σ U
½
ÁY a Re À 1
½
þ Y aIm
½
2
"
# 2
,
ð6:51Þ
6.5 Differential Fluctuation Equations of OEO MZ
319