DC component of the optical emission intensity of the first harmonic with the
average optical frequency of ν 0 , and R MZ1 , R MZ2 , relatively, are ACF from the
impact of the first (with radio frequency f 0 ) and the second (with 2f 0 ) harmonics:
R MZ u, t, t þ τ
ð
Þ¼
E
4
0L
4
γ
2
f K γ À cos φ 0MZ
ð
ÞÁ 1 À
x
2
4
! 2
 cos 2πν 0L τ þ β 1 À β 2
ð
Þ
h
i
À K γ sin φ 0MZ
ð
ÞÁ
x
2
À
x
3
16
! 2
 cos 2πν 0L τ þ 2π f 0 τ þ β 1 À β 2
ð
Þ
h
i
À K γ cos φ 0MZ
ð
ÞÁ
x
2
8
! 2
 cos 2πν 0L τ þ 2 Á 2π f 0 τ þ β 1 À β 2
ð
Þ
h
i g ,
ð5:90Þ
where K γ is the excitation coefficient of optical channels, φ 0MZ is the constant phase
shift in MZ defining by the DC bias voltage U 0MZπ , x ¼ U 1MZ /U 0MZπ , and x 2 [0; 1],
U 1MZ is the amplitude of AC voltage harmonic passed to the MZ electrical input,
β 1 À β 2 is the difference of the phase noise fluctuation values.
The important point in ACF determination of u g (t) ¼ U 10 cos [2πft + ϕ 0e + φ em (t)]
is the determination of the average value hcos(2πν 0L τ + 2πf 0 τ + β 1 À β 2 )i, in which
the difference β 1 À β 2 is the fluctuating term and values of β 1 and β 2 are relatively:
β 1 ¼ φ Lm (t À T 1M ) À φ Lm (t À T 2M ), β 2 ¼ φ Lm (t À T 1M À τ) À φ Lm (t À T 2M À τ).
In order to find the average difference of random quantities we use the formula:
cos 2πν 0L τ þ 2π f 0 τ þ β 1 À β 2
ð
Þ
¼ cos 2πν 0L τ þ 2π f 0 τ
ð
Þ Á cos β 1 À β 2
ð
Þ
h
i À sin 2πν 0L τ þ 2π f 0 τ
ð
Þ Á cos β 1 À β 2
ð
Þ
h
i
¼ cos β 1 À β 2
ð
Þ
h
i cos 2πν 0L τ þ 2π f 0 τ
ð
Þ ¼exp ÀΔν L ΔT M
j
j
ð
Þ cos 2πν 0L τ þ 2π f 0 τ
ð
Þ :
ð5:91Þ
Taking Eq. (5.91) and last transformation into account, the expression for the
product cos(2πν 0L τ + β 1 ) Á cos (2πν 0L τ + β 2 ) can be written in the form:
cos 2πν 0L τ þ β 1
ð
Þ Ácos 2πν 0L τ þ β 2
ð
Þ ¼
1
2
cos β 1 À β 2
ð
Þþcos 2 Á 2πν 0L τ þ β 1 þ β 2
ð
Þ
½
¼
1
2
cos β 1 À β 2
ð
Þþcos β 1 þ β 2
ð
Þcos 2 Á 2πν 0L τ
ð
ÞÀsin β 1 þ β 2
ð
Þsin 2 Á 2πν 0L τ
ð
Þ
½
¼
1
2
cos β 1 À β 2
ð
Þþcos β 1 þ β 2
ð
Þcos 2 Á 2πν 0L τ
ð
Þ
½
:
ð5:92Þ
With account of Eq. (5.92), R MZ (u, t, t + τ) takes the different shape for the
different observation time |τ| > |ΔT M | and |τ| |ΔT M |:
268
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .
average optical frequency of ν 0 , and R MZ1 , R MZ2 , relatively, are ACF from the
impact of the first (with radio frequency f 0 ) and the second (with 2f 0 ) harmonics:
R MZ u, t, t þ τ
ð
Þ¼
E
4
0L
4
γ
2
f K γ À cos φ 0MZ
ð
ÞÁ 1 À
x
2
4
! 2
 cos 2πν 0L τ þ β 1 À β 2
ð
Þ
h
i
À K γ sin φ 0MZ
ð
ÞÁ
x
2
À
x
3
16
! 2
 cos 2πν 0L τ þ 2π f 0 τ þ β 1 À β 2
ð
Þ
h
i
À K γ cos φ 0MZ
ð
ÞÁ
x
2
8
! 2
 cos 2πν 0L τ þ 2 Á 2π f 0 τ þ β 1 À β 2
ð
Þ
h
i g ,
ð5:90Þ
where K γ is the excitation coefficient of optical channels, φ 0MZ is the constant phase
shift in MZ defining by the DC bias voltage U 0MZπ , x ¼ U 1MZ /U 0MZπ , and x 2 [0; 1],
U 1MZ is the amplitude of AC voltage harmonic passed to the MZ electrical input,
β 1 À β 2 is the difference of the phase noise fluctuation values.
The important point in ACF determination of u g (t) ¼ U 10 cos [2πft + ϕ 0e + φ em (t)]
is the determination of the average value hcos(2πν 0L τ + 2πf 0 τ + β 1 À β 2 )i, in which
the difference β 1 À β 2 is the fluctuating term and values of β 1 and β 2 are relatively:
β 1 ¼ φ Lm (t À T 1M ) À φ Lm (t À T 2M ), β 2 ¼ φ Lm (t À T 1M À τ) À φ Lm (t À T 2M À τ).
In order to find the average difference of random quantities we use the formula:
cos 2πν 0L τ þ 2π f 0 τ þ β 1 À β 2
ð
Þ
¼ cos 2πν 0L τ þ 2π f 0 τ
ð
Þ Á cos β 1 À β 2
ð
Þ
h
i À sin 2πν 0L τ þ 2π f 0 τ
ð
Þ Á cos β 1 À β 2
ð
Þ
h
i
¼ cos β 1 À β 2
ð
Þ
h
i cos 2πν 0L τ þ 2π f 0 τ
ð
Þ ¼exp ÀΔν L ΔT M
j
j
ð
Þ cos 2πν 0L τ þ 2π f 0 τ
ð
Þ :
ð5:91Þ
Taking Eq. (5.91) and last transformation into account, the expression for the
product cos(2πν 0L τ + β 1 ) Á cos (2πν 0L τ + β 2 ) can be written in the form:
cos 2πν 0L τ þ β 1
ð
Þ Ácos 2πν 0L τ þ β 2
ð
Þ ¼
1
2
cos β 1 À β 2
ð
Þþcos 2 Á 2πν 0L τ þ β 1 þ β 2
ð
Þ
½
¼
1
2
cos β 1 À β 2
ð
Þþcos β 1 þ β 2
ð
Þcos 2 Á 2πν 0L τ
ð
ÞÀsin β 1 þ β 2
ð
Þsin 2 Á 2πν 0L τ
ð
Þ
½
¼
1
2
cos β 1 À β 2
ð
Þþcos β 1 þ β 2
ð
Þcos 2 Á 2πν 0L τ
ð
Þ
½
:
ð5:92Þ
With account of Eq. (5.92), R MZ (u, t, t + τ) takes the different shape for the
different observation time |τ| > |ΔT M | and |τ| |ΔT M |:
268
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .