Having used the last formula, we obtain the expression for connection of squared
amplitude of the field strength E PD on the PD area (the PD optical input) with the
squared field amplitude on the laser output E 0L :
E PD ¼ 2E 0L Á k 0 Á I 0 A 0
ð Þ cos Δϕ 0e
½
h
iexp ÀΔν L Á t
½
:
ð3:25Þ
Here we used the formula for mathematical expectation calculation:
cos 2πtf þ Δϕ 0e þ φ em t
ð Þ
½
h
i¼ cos 2πtf
½
Á cos Δϕ 0e þ φ em t
ð Þ
½
h
i
À sin 2πtf
½
Á cos Δϕ 0e þ φ em t
ð Þ
½
h
i
¼ cos Δϕ 0e
½
h
iexp ÀΔν L Á t
½
, ð3:26Þ
where hi designate the operation of average value finding (averaging in time).
The output electrical current of the PD is defined as i PD t
ð Þ ¼ E PD Á E
Ã
PD . If
nonlinearity of PD is not taken into account, the output current of PD is defined
by the second multiplier with the fluctuating DC component and the first harmonic of
current:
i PD t
ð Þ ¼ I PD0 Á cos Δϕ 0e þ φ em t
ð Þ
½
þ I PD10 Á cos 2πft þ Δϕ 0e þ φ em t
ð Þ
½
: ð3:27Þ
The output current of PD (taking into account the PD nonlinearity) includes the
DC component I PD0 defining by the level of DC component of the laser emission on
the PR area and by components of the first I PD10 , second I PD20 (in the general case,
the other possible harmonics), which level is determined by nonlinear characteristics
of PD is:
i PD t
ð Þ ¼ I PD0 Á cos Δϕ 0e þ φ em t
ð Þ
½
þI PD10 Á cos 2πft þ Δϕ 0e þ φ em t
ð Þ
½
þI PD20 Á cos 2 Á 2πft þ 2 Á Δϕ 0e þ 2 Á φ em t
ð Þ
½
:
ð3:28Þ
At selection by the rejection optical filter (Fig. 3.12b), the DC component is
suppressed, and the radio frequency of the photocurrent is multiple to 2: 2 Á 2πf, and
constant components I PD0 and I PD10 are close to zero:
i PD t
ð Þ ¼ I PD20 Á cos 2 Á 2πft þ 2Δϕ 022e þ 2φ em t
ð Þ
½
,
ð3:29Þ
From the last mathematical formulas, we can make the following conclusions.
We should note the main point: the suppression of the optical carrier frequency
ν 0 significantly decreases the phase fluctuations. A degree of the fluctuation suppression is defined by the level of the carrier suppression. The PD photocurrent in
OEO DM is a result of self-heterodyning; the level of harmonic components is
defined by the difference phase incursion of the first and second optical harmonics
Δϕ 0e , and laser phase fluctuations φ em (t), in the end are defined by the spectrum of
3.2 Methods of Modulation and Heterodyning of Laser Emissions at DM and MZ. . .
101
amplitude of the field strength E PD on the PD area (the PD optical input) with the
squared field amplitude on the laser output E 0L :
E PD ¼ 2E 0L Á k 0 Á I 0 A 0
ð Þ cos Δϕ 0e
½
h
iexp ÀΔν L Á t
½
:
ð3:25Þ
Here we used the formula for mathematical expectation calculation:
cos 2πtf þ Δϕ 0e þ φ em t
ð Þ
½
h
i¼ cos 2πtf
½
Á cos Δϕ 0e þ φ em t
ð Þ
½
h
i
À sin 2πtf
½
Á cos Δϕ 0e þ φ em t
ð Þ
½
h
i
¼ cos Δϕ 0e
½
h
iexp ÀΔν L Á t
½
, ð3:26Þ
where hi designate the operation of average value finding (averaging in time).
The output electrical current of the PD is defined as i PD t
ð Þ ¼ E PD Á E
Ã
PD . If
nonlinearity of PD is not taken into account, the output current of PD is defined
by the second multiplier with the fluctuating DC component and the first harmonic of
current:
i PD t
ð Þ ¼ I PD0 Á cos Δϕ 0e þ φ em t
ð Þ
½
þ I PD10 Á cos 2πft þ Δϕ 0e þ φ em t
ð Þ
½
: ð3:27Þ
The output current of PD (taking into account the PD nonlinearity) includes the
DC component I PD0 defining by the level of DC component of the laser emission on
the PR area and by components of the first I PD10 , second I PD20 (in the general case,
the other possible harmonics), which level is determined by nonlinear characteristics
of PD is:
i PD t
ð Þ ¼ I PD0 Á cos Δϕ 0e þ φ em t
ð Þ
½
þI PD10 Á cos 2πft þ Δϕ 0e þ φ em t
ð Þ
½
þI PD20 Á cos 2 Á 2πft þ 2 Á Δϕ 0e þ 2 Á φ em t
ð Þ
½
:
ð3:28Þ
At selection by the rejection optical filter (Fig. 3.12b), the DC component is
suppressed, and the radio frequency of the photocurrent is multiple to 2: 2 Á 2πf, and
constant components I PD0 and I PD10 are close to zero:
i PD t
ð Þ ¼ I PD20 Á cos 2 Á 2πft þ 2Δϕ 022e þ 2φ em t
ð Þ
½
,
ð3:29Þ
From the last mathematical formulas, we can make the following conclusions.
We should note the main point: the suppression of the optical carrier frequency
ν 0 significantly decreases the phase fluctuations. A degree of the fluctuation suppression is defined by the level of the carrier suppression. The PD photocurrent in
OEO DM is a result of self-heterodyning; the level of harmonic components is
defined by the difference phase incursion of the first and second optical harmonics
Δϕ 0e , and laser phase fluctuations φ em (t), in the end are defined by the spectrum of
3.2 Methods of Modulation and Heterodyning of Laser Emissions at DM and MZ. . .
101