K FODL ¼ P 0L Á M Z Á M eZ Á K FOS Á K PD , K DL
j
j ¼ K BZ
j
j
¼ M Z
j j M eZ
j
jK FOS
j
jK PD
j
j,
ð6:10Þ
argK DL ¼ argK BZ ¼ argM Z þ argM eZ þ argK FOS þ argK PD ,
ð6:11Þ
where P 0L is the DC component of the laser power, S PD is the slope of the watt–
ampere characteristic of PD, K FOS is the transfer function of the optical fiber with
account of the matching elements. T MZ is the coefficient of the voltage conversion
into the light delay in the MZ optical channel or T MZ ¼ T M ¼ d arg M Z /dU 0MZ .
Executed analysis of MZ in OEO and formulas obtained allow the transfer to the
amplitude and frequency determination in the steady-state mode of the OEO MZ
generation.
6.4.3 Abbreviated Equations of OEO MZ
With account of used designations, here we use the approach for deduction of
abbreviated equations of the OEO oscillator, which is described in Chaps. 3 and 5.
Here we shall consider all RF circuits (besides the RF filter) included into the RF part
of OEO, as wideband circuits. At that, abbreviated differential equations of the OEO
MZ for the slowly changed amplitude U 1L ¼ U a and phase ψ 1L ¼ ψ U of the first
harmonic oscillations (in the output of OEO MZ) have the form:
T 1e
dU 1L
dt
¼ U 1L Á Re K FOLD R con S NA0
ð
Þ À U 1L
U 1L T 1e
dψ 1L
dt
¼ U 1L 2π f res À f
ð
Þ T 1e þ U 1L Im K FOLD R con S NA0
ð
Þ
8
> <
> :
,
ð6:12Þ
where R con is the control OEO resistance, S NA0 is the average slope of NA, T F ¼ T 1e
is the time constant of the RF filter, and f res is its resonance frequency.
For the following nonlinear characteristic of the active element i(u) ¼ S 01 u À S 03 u
3
and the average slope of this characteristic of AE in the NA S 1 (U ) ¼ S 01 À (3/4)
S 03 U
2
, we search the solution of the differential equation. Then, from the first
equation of the system (6.12), we have got the expression for the amplitude U of
OEO MZ oscillations:
U ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
4S 01 =3S 03
p
Á 1 À
1
S 01 P 0L R PD M Z
j j M eZ
j
j K FOS
j
jK PD
j
j K F
j j
1=2
:
ð6:13Þ
From abbreviated equation (6.12), we can obtain the equations of the phase and
amplitude balance for the steady-state mode in the form:
306
6 Operation Analysis of Optoelectronic oscillator (OEO) with External. . .
j
j ¼ K BZ
j
j
¼ M Z
j j M eZ
j
jK FOS
j
jK PD
j
j,
ð6:10Þ
argK DL ¼ argK BZ ¼ argM Z þ argM eZ þ argK FOS þ argK PD ,
ð6:11Þ
where P 0L is the DC component of the laser power, S PD is the slope of the watt–
ampere characteristic of PD, K FOS is the transfer function of the optical fiber with
account of the matching elements. T MZ is the coefficient of the voltage conversion
into the light delay in the MZ optical channel or T MZ ¼ T M ¼ d arg M Z /dU 0MZ .
Executed analysis of MZ in OEO and formulas obtained allow the transfer to the
amplitude and frequency determination in the steady-state mode of the OEO MZ
generation.
6.4.3 Abbreviated Equations of OEO MZ
With account of used designations, here we use the approach for deduction of
abbreviated equations of the OEO oscillator, which is described in Chaps. 3 and 5.
Here we shall consider all RF circuits (besides the RF filter) included into the RF part
of OEO, as wideband circuits. At that, abbreviated differential equations of the OEO
MZ for the slowly changed amplitude U 1L ¼ U a and phase ψ 1L ¼ ψ U of the first
harmonic oscillations (in the output of OEO MZ) have the form:
T 1e
dU 1L
dt
¼ U 1L Á Re K FOLD R con S NA0
ð
Þ À U 1L
U 1L T 1e
dψ 1L
dt
¼ U 1L 2π f res À f
ð
Þ T 1e þ U 1L Im K FOLD R con S NA0
ð
Þ
8
> <
> :
,
ð6:12Þ
where R con is the control OEO resistance, S NA0 is the average slope of NA, T F ¼ T 1e
is the time constant of the RF filter, and f res is its resonance frequency.
For the following nonlinear characteristic of the active element i(u) ¼ S 01 u À S 03 u
3
and the average slope of this characteristic of AE in the NA S 1 (U ) ¼ S 01 À (3/4)
S 03 U
2
, we search the solution of the differential equation. Then, from the first
equation of the system (6.12), we have got the expression for the amplitude U of
OEO MZ oscillations:
U ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
4S 01 =3S 03
p
Á 1 À
1
S 01 P 0L R PD M Z
j j M eZ
j
j K FOS
j
jK PD
j
j K F
j j
1=2
:
ð6:13Þ
From abbreviated equation (6.12), we can obtain the equations of the phase and
amplitude balance for the steady-state mode in the form:
306
6 Operation Analysis of Optoelectronic oscillator (OEO) with External. . .