U 10MZ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
4S 01 =3S 03
p
Á 1 À
1
S 01 P 0L Á R FD K FODL
j
j
1=2
ð3:62Þ
If the transfer functions of MZ optical channels k 01 ¼ A and k 02 ¼ B ¼ 1 À A are
equal about 0.5, the irregularity coefficient of MZ channels’ excitation γ % 1, the
formula for radio-frequency deviations Δf ¼ Δf gen at OEO generation from its
average value at variation of the optical frequency of the laser Δν can be written
in the form:
Δ f gen ffi
m þ f EF Á T EF
T 2MZ þT 1MZ
2
þ
T 2MZ ÀT 1MZ
2
Á 1 À γ
ð
ÞÁ
Δν
f F
þ T eMZ þ T FOS þ T EF
ð3:63Þ
From Eq. (3.63), the one important OEO features follows: the dependence of
radio-frequency deviations Δf versus deviations of the laser optical generation
frequency Δν will be less at lesser delay differences in optical channels T 2 À T 1 ,
at lesser ratio of optical frequency deviations from the average frequency Δν/f EF , at
lesser difference in transfer functions of optical channels k 01 ¼ A and k 02 ¼ B ¼ 1 À A,
i.e., γ % 1, (1 À γ) % 0, and at larger the T EF time constant and at larger the delay time
in FOS T FOS .
Figure 3.19 shows the graphical solution for the OEO steady-state mode. Point
are shown determination of the OEO generation frequency as point of interception of
two functions: the PFC of the resonance circuit and the PFC of the optical fiber under
condition of amplitude balance fulfillment. The generation frequency in the OEO
steady-state mode defines its amplitude.
3.4.3.5 Resonance Characteristics of OEO with Nonlinearity
in the Form of Cosine
OEO investigation and analysis with the nonlinearity in the form of cosine will be
performed using the symbolic equations (Eq. 3.49) and replacing the operator p/
2πf 0e by s ¼ j( f/f 0e ), where j ¼
ffiffiffiffiffiffi ffi
À1
p
. Such a replacement means the transition to the
normalized generation frequency of OEO, which is represented on the complex
parameter plane along the abscissa axis.
Let us perform the replacement of the normalized current i A , x ¼ U 1MZ /U 0MZπ ,
which will be represented along the ordinate axis. Then the solution of the algebraic
equation in the form
s
2
þ 1=Q eF
ð
ÞÁs þ 1
Â
à Á x ¼ K 0FODL =Q EF
ð
ÞÁs Á cos m 00 π=2
ð
Þþβ 00 x
½
; ð3:64Þ
depends upon two variables s and x and constant parameters K 0FODL , Q EF , m 00 π/2,
β 00 . These functions of OEO normalized oscillation amplitude x versus its normalized frequency s are called resonance characteristics (resonance curves) of OEO.
Solutions are constructed, for example, in WA operation system at specification of
124 3 Modulation Methods of Laser Emission in Optoelectronic oscillator (OEO) and OEO. . .
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
4S 01 =3S 03
p
Á 1 À
1
S 01 P 0L Á R FD K FODL
j
j
1=2
ð3:62Þ
If the transfer functions of MZ optical channels k 01 ¼ A and k 02 ¼ B ¼ 1 À A are
equal about 0.5, the irregularity coefficient of MZ channels’ excitation γ % 1, the
formula for radio-frequency deviations Δf ¼ Δf gen at OEO generation from its
average value at variation of the optical frequency of the laser Δν can be written
in the form:
Δ f gen ffi
m þ f EF Á T EF
T 2MZ þT 1MZ
2
þ
T 2MZ ÀT 1MZ
2
Á 1 À γ
ð
ÞÁ
Δν
f F
þ T eMZ þ T FOS þ T EF
ð3:63Þ
From Eq. (3.63), the one important OEO features follows: the dependence of
radio-frequency deviations Δf versus deviations of the laser optical generation
frequency Δν will be less at lesser delay differences in optical channels T 2 À T 1 ,
at lesser ratio of optical frequency deviations from the average frequency Δν/f EF , at
lesser difference in transfer functions of optical channels k 01 ¼ A and k 02 ¼ B ¼ 1 À A,
i.e., γ % 1, (1 À γ) % 0, and at larger the T EF time constant and at larger the delay time
in FOS T FOS .
Figure 3.19 shows the graphical solution for the OEO steady-state mode. Point
are shown determination of the OEO generation frequency as point of interception of
two functions: the PFC of the resonance circuit and the PFC of the optical fiber under
condition of amplitude balance fulfillment. The generation frequency in the OEO
steady-state mode defines its amplitude.
3.4.3.5 Resonance Characteristics of OEO with Nonlinearity
in the Form of Cosine
OEO investigation and analysis with the nonlinearity in the form of cosine will be
performed using the symbolic equations (Eq. 3.49) and replacing the operator p/
2πf 0e by s ¼ j( f/f 0e ), where j ¼
ffiffiffiffiffiffi ffi
À1
p
. Such a replacement means the transition to the
normalized generation frequency of OEO, which is represented on the complex
parameter plane along the abscissa axis.
Let us perform the replacement of the normalized current i A , x ¼ U 1MZ /U 0MZπ ,
which will be represented along the ordinate axis. Then the solution of the algebraic
equation in the form
s
2
þ 1=Q eF
ð
ÞÁs þ 1
Â
à Á x ¼ K 0FODL =Q EF
ð
ÞÁs Á cos m 00 π=2
ð
Þþβ 00 x
½
; ð3:64Þ
depends upon two variables s and x and constant parameters K 0FODL , Q EF , m 00 π/2,
β 00 . These functions of OEO normalized oscillation amplitude x versus its normalized frequency s are called resonance characteristics (resonance curves) of OEO.
Solutions are constructed, for example, in WA operation system at specification of
124 3 Modulation Methods of Laser Emission in Optoelectronic oscillator (OEO) and OEO. . .