The total delay of oscillations in RF FODL T FODL is T FODL ¼ T MZ + T FOS + T PD ,
where T MZ , T PD , T FOS are, relatively, oscillation delays in MZ, PD, and in optical
fiber.
We introduce the symbolic coefficient K a of the current influence in the input I a of
the active element upon the voltage in the input U a of this active element.
If to substitute expressions (for the structure in Fig. 6.12) and to take into account
the steady-state equation, then we obtain the complex fluctuation abbreviated equation for OEO MZ:
Y a p
ð ÞU a ¼ I b U a
ð Þ À K a I a þ I nb þ ξ YYn :
ð6:37Þ
6.5.4 Fluctuation Deviations of Amplitudes and Phase
Here we designate the relative fluctuation deviations (6.37) of the amplitude U a and
the phase φ U from the own steady-state values as: U a ¼ U a0 Á (1 + m U ) and
φ U ¼ φ 0U + ψ U , where m U are fluctuations of the amplitude U a , ψ U are phase
fluctuations from its mean values, φ 0 is the initial constant phase shift. The complex
amplitude U a can be presented in the form:
U a ¼ U a0 1 þ m U
ð
Þexp j2πΔft þ jφ 0U þ jψ U
ð
Þ :
ð6:38Þ
To simplify equations deduction, taking into account the smallness of deviations
ψ U , the following relation is true for small observation time: exp( jψ U ) ¼ 1 + jψ U .
Then, the complex amplitude is: U a ¼ U a0 (1 + m U + jψ U ). The slowly changing
amplitude of the first harmonic U a ¼ U a0 (1 + m U ) exp ( j2πΔft + jφ 0U + jψ U ) can be
written with account of fluctuations in the form: U a ¼ U a0 (1 + m U + jψ U )
exp ( j2πΔft + jφ 0U ). Having linearized functions in the vicinity of steady-state
values U a and φ U , we obtain expressions:
I a ¼ I a0 U a
ð Þexp j2πΔft þ φ u
ð
Þ
¼ I a0 U a0 1 þ m U þ jψ U
ð
Þ exp j2πΔft þ φ u
ð
Þ
½
Á exp j2πΔft þ φ u
ð
Þ
:
ð6:39Þ
For record simplification, we introduce the relative local slope of the oscillating
characteristic or coefficients σ aU , σ bU ¼ σ U , which can be defined as σ aU ¼
dI a =dU a
I a =U a
,
σ bU ¼ σ U ¼
dI b =dU a
I b =U a
. The coefficient σ U has a sense of a ratio of the relative local
slope of the AE oscillating characteristic dI b /dU a reduced to the ratio I b1 /U a1 , which
is constructed taking into account the noninertial property of NA. In economics, to
explain of the profit growth at variation of the price, we can introduce the concept of
the function elasticity. The coefficient σ U has the similar sense. It shows how fast the
function I a (U a ) decreases or increases in the steady-state mode point. Taking into
consideration the last notation for I a , we can write: I a ¼ I a0 (1 + σ aU m U + jψ U ).
316
6 Operation Analysis of Optoelectronic oscillator (OEO) with External. . .
where T MZ , T PD , T FOS are, relatively, oscillation delays in MZ, PD, and in optical
fiber.
We introduce the symbolic coefficient K a of the current influence in the input I a of
the active element upon the voltage in the input U a of this active element.
If to substitute expressions (for the structure in Fig. 6.12) and to take into account
the steady-state equation, then we obtain the complex fluctuation abbreviated equation for OEO MZ:
Y a p
ð ÞU a ¼ I b U a
ð Þ À K a I a þ I nb þ ξ YYn :
ð6:37Þ
6.5.4 Fluctuation Deviations of Amplitudes and Phase
Here we designate the relative fluctuation deviations (6.37) of the amplitude U a and
the phase φ U from the own steady-state values as: U a ¼ U a0 Á (1 + m U ) and
φ U ¼ φ 0U + ψ U , where m U are fluctuations of the amplitude U a , ψ U are phase
fluctuations from its mean values, φ 0 is the initial constant phase shift. The complex
amplitude U a can be presented in the form:
U a ¼ U a0 1 þ m U
ð
Þexp j2πΔft þ jφ 0U þ jψ U
ð
Þ :
ð6:38Þ
To simplify equations deduction, taking into account the smallness of deviations
ψ U , the following relation is true for small observation time: exp( jψ U ) ¼ 1 + jψ U .
Then, the complex amplitude is: U a ¼ U a0 (1 + m U + jψ U ). The slowly changing
amplitude of the first harmonic U a ¼ U a0 (1 + m U ) exp ( j2πΔft + jφ 0U + jψ U ) can be
written with account of fluctuations in the form: U a ¼ U a0 (1 + m U + jψ U )
exp ( j2πΔft + jφ 0U ). Having linearized functions in the vicinity of steady-state
values U a and φ U , we obtain expressions:
I a ¼ I a0 U a
ð Þexp j2πΔft þ φ u
ð
Þ
¼ I a0 U a0 1 þ m U þ jψ U
ð
Þ exp j2πΔft þ φ u
ð
Þ
½
Á exp j2πΔft þ φ u
ð
Þ
:
ð6:39Þ
For record simplification, we introduce the relative local slope of the oscillating
characteristic or coefficients σ aU , σ bU ¼ σ U , which can be defined as σ aU ¼
dI a =dU a
I a =U a
,
σ bU ¼ σ U ¼
dI b =dU a
I b =U a
. The coefficient σ U has a sense of a ratio of the relative local
slope of the AE oscillating characteristic dI b /dU a reduced to the ratio I b1 /U a1 , which
is constructed taking into account the noninertial property of NA. In economics, to
explain of the profit growth at variation of the price, we can introduce the concept of
the function elasticity. The coefficient σ U has the similar sense. It shows how fast the
function I a (U a ) decreases or increases in the steady-state mode point. Taking into
consideration the last notation for I a , we can write: I a ¼ I a0 (1 + σ aU m U + jψ U ).
316
6 Operation Analysis of Optoelectronic oscillator (OEO) with External. . .