6.5.14 Natural Line Width and PSD of the Phase Noise
of OEO MZ
If laser phase noises dominate above the phase noises of PD and NA, we obtain that
for σ U % 1, at the laser power growth P 0L ¼ E
2
0L , PSD of the phase noise in OEO
S PD2 (F)aspires to the value (6.76):
S PD2 F
ð Þ ¼ G 12 G 22 K
2
ΓPN Á K
2
PD
S 2βPN
P 0L F
2 T
2
0L
&
'
K
2
PD :
ð6:80Þ
Taking into consideration the mentioned assumptions, we obtain that for the
coefficient (6.80) at small offsets from the carrier, the following assumptions are
true: sin[2πν 0 (T 1M À T 2M )] ¼ 1, Y 00 /P 0L % 1, P 0L |K LZ | % 1. Then, the PSD function
of the OEO phase noise (5.79) can be presented as:
S ψ
P 0G
¼
K
2
2ΓPN C A ħvN sp
P 0G
, where C A is
the constant coefficient, K
2
2ΓPN is the coefficient depending upon the delay time in the
optical fiber and upon the laser optical power.
At small offsets (FTFOS < 1) (or within the limits of the laser line width), we
obtain PSD of the OEO phase noise S ΨAG F
ð Þ ¼
G 12 G 22 K
4
PD
1þFT FOS
½
2 Á
C A ħvN sp
P 0L F
2 T
2
0L
, which is equal:
S ΨAG F
ð Þ % G 12 G 22 K
4
PD
C A ħvN sp
P 0L F
2 T
2
0L
. We consider that PSD of the phase noise in OEO
MZ has a form of the Lorentz function, then we obtain the formula:
Δ f gen ¼
S ΨAG F ¼ Δ f gen
À
Á Á D Y
4 T EF þ T FOS
ð
Þ
2
ð6:81Þ
or S ΨAG F ¼ Δ f gen
À
Á ¼
4 T EF þT FOS
ð
Þ
2 Δ f gen
D Y
. We consider that PSD of the phase noise S L
of QWLD has the form of the Lorentz function, then we obtain the formula: Δν ¼
S L F¼Δν
ð
Þ
4T
2
0L
or we can write: S L F ¼ Δν
ð
Þ¼4T
2
0L Δν. From Eqs. (6.80) and (6.81), it
follows that at small offsets F from the nominal generation frequency, the level of
PSD of the phase noise becomes lower at decrease of the laser emission bandwidth
Δν L , as well as at reduction of the laser generation frequency, the difference in delay
times ΔT M in the MZ optical channels and the level of the laser spontaneous
emission S β and the amplification factor of NA D PN . PSD of the phase noise
decreases at growth of the delay time in RF FODL or in FOS.
Taking into account that the OEO MZ line width Δf gen ¼ Δf 0.5 is related to PSD
of the phase noise for the natural line width Δf gen ¼ Δf 0.5 , we have (taking into
account Eqs. (6.80) and (6.81)) the fundamental dependence on the natural laser line
width Δν L in the form:
342
6 Operation Analysis of Optoelectronic oscillator (OEO) with External. . .
of OEO MZ
If laser phase noises dominate above the phase noises of PD and NA, we obtain that
for σ U % 1, at the laser power growth P 0L ¼ E
2
0L , PSD of the phase noise in OEO
S PD2 (F)aspires to the value (6.76):
S PD2 F
ð Þ ¼ G 12 G 22 K
2
ΓPN Á K
2
PD
S 2βPN
P 0L F
2 T
2
0L
&
'
K
2
PD :
ð6:80Þ
Taking into consideration the mentioned assumptions, we obtain that for the
coefficient (6.80) at small offsets from the carrier, the following assumptions are
true: sin[2πν 0 (T 1M À T 2M )] ¼ 1, Y 00 /P 0L % 1, P 0L |K LZ | % 1. Then, the PSD function
of the OEO phase noise (5.79) can be presented as:
S ψ
P 0G
¼
K
2
2ΓPN C A ħvN sp
P 0G
, where C A is
the constant coefficient, K
2
2ΓPN is the coefficient depending upon the delay time in the
optical fiber and upon the laser optical power.
At small offsets (FTFOS < 1) (or within the limits of the laser line width), we
obtain PSD of the OEO phase noise S ΨAG F
ð Þ ¼
G 12 G 22 K
4
PD
1þFT FOS
½
2 Á
C A ħvN sp
P 0L F
2 T
2
0L
, which is equal:
S ΨAG F
ð Þ % G 12 G 22 K
4
PD
C A ħvN sp
P 0L F
2 T
2
0L
. We consider that PSD of the phase noise in OEO
MZ has a form of the Lorentz function, then we obtain the formula:
Δ f gen ¼
S ΨAG F ¼ Δ f gen
À
Á Á D Y
4 T EF þ T FOS
ð
Þ
2
ð6:81Þ
or S ΨAG F ¼ Δ f gen
À
Á ¼
4 T EF þT FOS
ð
Þ
2 Δ f gen
D Y
. We consider that PSD of the phase noise S L
of QWLD has the form of the Lorentz function, then we obtain the formula: Δν ¼
S L F¼Δν
ð
Þ
4T
2
0L
or we can write: S L F ¼ Δν
ð
Þ¼4T
2
0L Δν. From Eqs. (6.80) and (6.81), it
follows that at small offsets F from the nominal generation frequency, the level of
PSD of the phase noise becomes lower at decrease of the laser emission bandwidth
Δν L , as well as at reduction of the laser generation frequency, the difference in delay
times ΔT M in the MZ optical channels and the level of the laser spontaneous
emission S β and the amplification factor of NA D PN . PSD of the phase noise
decreases at growth of the delay time in RF FODL or in FOS.
Taking into account that the OEO MZ line width Δf gen ¼ Δf 0.5 is related to PSD
of the phase noise for the natural line width Δf gen ¼ Δf 0.5 , we have (taking into
account Eqs. (6.80) and (6.81)) the fundamental dependence on the natural laser line
width Δν L in the form:
342
6 Operation Analysis of Optoelectronic oscillator (OEO) with External. . .