E 10L t
ð Þ ¼
ffiffiffiffiffi
A 1
p
E 0L cos 2πν 0L t þ φ 0L þ φ 10Lm t
ð Þ
½
,
ð6:96Þ
E 20L t
ð Þ ¼
ffiffiffiffiffi
A 2
p
E 0L cos 2π ν 0L þ f 0
ð
Þ t þ φ 0L þ φ 20Lm t
ð Þ
½
,
ð6:97Þ
where coefficients A 1 , A 2 are defined by the AFC of the optical filter and PD.
Taking into consideration Eqs. (5.90), (6.96), and (6.97), PSD in the PD area
(without account of electrical oscillation influence in the MZ electrical input) can be
calculated by the approximate formula:
S η f
ð Þ ¼ 4
Z 1
0
R ηL τ
ð Þ cos 2πf τ
ð
Þdτ
¼ exp À
2ΔT M
T c
δ f
ð Þ þ S L f
ð Þ 1 À
A 1
A 2
exp À
2ΔT M
T c
!
,
ð6:98Þ
where δ( f ) is the delta-function, coefficients A 1 , A 2 are defined by the AFC of the
optical filter and PD, S L ( f ) is laser PSD, which is determined at utilization of
Eq. (3.8) by the expression: S L f
ð Þ ¼
1= Δν L
ð
Þ
1þ f =Δν L
½
2 ¼
T c
1þ 2πT c f
ð
Þ
2 .
If we take at calculation the small offsets from the optical carrier ΔT M Á 2πf ( 1,
then we can write Eq. (6.98) in the form:
S η f
ð Þ ¼ exp À
4ΔT M
T c
δ f
ð Þ þ
T c
1 þ 2πT c f
ð
Þ
2
Á 1 À
A 1
A 2
exp À
2ΔT M
T c
!
:
ð6:99Þ
From Eq. (6.99), it follows that with the growth of the ratio
ΔT M
T c
, the first term,
which defines the oscillation amplitude of the regular component (5.102), decreases
and the second term increases and aspires to S L ( f ).
Further, the convolution of S η ( f ) and S V ( f ) is:
S ηV f
ð Þ ¼ S η f
ð Þ Ã S V f
ð Þ
¼
U
2
e
2
E
4
0L
2
exp À
2τ
T c
Á δ f
ð Þ þ 1 À
A 1
A 2
exp À
2τ
T c
!
Á
U
2
e
2
E
4
0L
2
S L f
ð Þ Ã S G f
ð Þ
,
ð6:100Þ
where “Ô is convolution, and
S L f
ð Þ Ã S G f
ð Þ ¼
T c
1 þ T c f
ð Þ
2
Ã
T gen
1 þ T gen f
À
Á 2 :
ð6:101Þ
From this, the conclusion follows that S ηV ( f ) for the optimal matching, the
equality T c % T 1 is necessary and sufficient. We remind that the coherence time is
356
6 Operation Analysis of Optoelectronic oscillator (OEO) with External. . .
ð Þ ¼
ffiffiffiffiffi
A 1
p
E 0L cos 2πν 0L t þ φ 0L þ φ 10Lm t
ð Þ
½
,
ð6:96Þ
E 20L t
ð Þ ¼
ffiffiffiffiffi
A 2
p
E 0L cos 2π ν 0L þ f 0
ð
Þ t þ φ 0L þ φ 20Lm t
ð Þ
½
,
ð6:97Þ
where coefficients A 1 , A 2 are defined by the AFC of the optical filter and PD.
Taking into consideration Eqs. (5.90), (6.96), and (6.97), PSD in the PD area
(without account of electrical oscillation influence in the MZ electrical input) can be
calculated by the approximate formula:
S η f
ð Þ ¼ 4
Z 1
0
R ηL τ
ð Þ cos 2πf τ
ð
Þdτ
¼ exp À
2ΔT M
T c
δ f
ð Þ þ S L f
ð Þ 1 À
A 1
A 2
exp À
2ΔT M
T c
!
,
ð6:98Þ
where δ( f ) is the delta-function, coefficients A 1 , A 2 are defined by the AFC of the
optical filter and PD, S L ( f ) is laser PSD, which is determined at utilization of
Eq. (3.8) by the expression: S L f
ð Þ ¼
1= Δν L
ð
Þ
1þ f =Δν L
½
2 ¼
T c
1þ 2πT c f
ð
Þ
2 .
If we take at calculation the small offsets from the optical carrier ΔT M Á 2πf ( 1,
then we can write Eq. (6.98) in the form:
S η f
ð Þ ¼ exp À
4ΔT M
T c
δ f
ð Þ þ
T c
1 þ 2πT c f
ð
Þ
2
Á 1 À
A 1
A 2
exp À
2ΔT M
T c
!
:
ð6:99Þ
From Eq. (6.99), it follows that with the growth of the ratio
ΔT M
T c
, the first term,
which defines the oscillation amplitude of the regular component (5.102), decreases
and the second term increases and aspires to S L ( f ).
Further, the convolution of S η ( f ) and S V ( f ) is:
S ηV f
ð Þ ¼ S η f
ð Þ Ã S V f
ð Þ
¼
U
2
e
2
E
4
0L
2
exp À
2τ
T c
Á δ f
ð Þ þ 1 À
A 1
A 2
exp À
2τ
T c
!
Á
U
2
e
2
E
4
0L
2
S L f
ð Þ Ã S G f
ð Þ
,
ð6:100Þ
where “Ô is convolution, and
S L f
ð Þ Ã S G f
ð Þ ¼
T c
1 þ T c f
ð Þ
2
Ã
T gen
1 þ T gen f
À
Á 2 :
ð6:101Þ
From this, the conclusion follows that S ηV ( f ) for the optimal matching, the
equality T c % T 1 is necessary and sufficient. We remind that the coherence time is
356
6 Operation Analysis of Optoelectronic oscillator (OEO) with External. . .