different length, which are connected in the input by Y- and in the output by
X-optical directional couplers. The input Y-coupler executes the function of a
divider of the laser emission with the electrical field strength E L into two channels
OC1 and OC2. The X-coupler located in the output serves for gathering and
directing of optical emission, which comes out from the channels OC1 and OC2
to the optical channel located after the Х-coupler and further directing of this
emission to the photodetector.
Let us describe the results of analysis of the radio-frequency control for OEO with
the differential RF FODL formed by single-mode light guides and connected as
shown in Fig. 7.24a. Authors consider and analyze the transfer function of such a
differential RF FODL K TR ( jω) for the control of excitation coefficients of FOS1 and
FOS2. The expression for the transfer function of such a RF FODL can be presented
as: K( jω) ¼ α(r)K 1 ( jω) + β(r)K 2 ( jω), where α(r), β(r) are coefficients of excitation
relatively of FOS1 and FOS2, K 1 ( jω) ¼ exp [Àα 01 (n/с)(T 0FOS + T 1FOS ) À jω
(T 0FOS + T 1FOS )], K 2 ( jω) ¼ exp [Àα 01 (n/с)(T 0FOS + T 2FOS ) À jω(T 0FOS + T 2FOS )],
n is the refraction index of the material of the light-guiding threat of FOS0, FOS1,
and FOS2, с is the light speed, α 01 , α 02 are specific light losses in FOS1 and FOS2,
relatively.
The normalized transfer function of differential RF FODL is |K NTR ( jω)| ¼ |
K TR ( jω)|/[S LD S PD μ exp (Àγ 0 L 0 )], where S LD is the slope of the watt–ampere
characteristic of the laser diode, S PD is the slope of the volt–ampere characteristics
of PD, μ are optical losses on the FOS matching. The phase-frequency characteristics
of RF FODL φ(ω) are obtained in assumption of the light losses absence in FOS1
and FOS2, when γ 1 L 1 ¼ γ 2 L 2 % 0 and μ ¼ 1.
At variation of the excitation coefficient of the FOS1 light guide α ¼ А from 0 to
1, AFC of the differential RF FODL deforms and has the comb character with clearly
expressed minima at ω(Т 2FOS À Т 1FOS ) ¼ π + 2πm; m ¼ 1, 2,. . .; the PFC of
differential RF FODL for values α % 0.5 is characterized by the presence of
periodically repeated sharp portions, which are degenerated into jumps by π at
α ¼ 0.5.
In the region of ω(Т 2FOS À Т 1FOS ) < π/2, PFC is close to the linear function. It
should be noted that in the limit cases (α ¼ 0 and α ¼ 1), PFC of the differential RF
FODL represents the straight lines φ ¼ ωT 1FOS and φ ¼ ωT 2FOS .
The adjusting curves of the frequency retuning in OEO with the differential RF
FODL are shown in Fig. 7.24b, c. Taking into consideration of obtained transfer
functions of the differential RF FODL, the solution of OEO abbreviated equations
gives a possibility to obtain of frequency and amplitude functions versus the external
impact and to find the regions of stable stationary generation of OEO with the
differential RF FODL.
The comb character of |K NTR ( jω)| of differential RF FODL applies features on the
generation mode at impact indication in such types of devices. Figure 7.24b, c shows
the plots of f gen (α) and U( f gen ) at varying α depicted for different values of the smallsignal gain in the open loop χ. The growth of χ leads to the qualitative variation of
the function U( f gen ), and at small χ 1 , the zone generation is observed.
7.4 Frequency Control in OEO with RF FODL with Two Optical Fibers
421
X-optical directional couplers. The input Y-coupler executes the function of a
divider of the laser emission with the electrical field strength E L into two channels
OC1 and OC2. The X-coupler located in the output serves for gathering and
directing of optical emission, which comes out from the channels OC1 and OC2
to the optical channel located after the Х-coupler and further directing of this
emission to the photodetector.
Let us describe the results of analysis of the radio-frequency control for OEO with
the differential RF FODL formed by single-mode light guides and connected as
shown in Fig. 7.24a. Authors consider and analyze the transfer function of such a
differential RF FODL K TR ( jω) for the control of excitation coefficients of FOS1 and
FOS2. The expression for the transfer function of such a RF FODL can be presented
as: K( jω) ¼ α(r)K 1 ( jω) + β(r)K 2 ( jω), where α(r), β(r) are coefficients of excitation
relatively of FOS1 and FOS2, K 1 ( jω) ¼ exp [Àα 01 (n/с)(T 0FOS + T 1FOS ) À jω
(T 0FOS + T 1FOS )], K 2 ( jω) ¼ exp [Àα 01 (n/с)(T 0FOS + T 2FOS ) À jω(T 0FOS + T 2FOS )],
n is the refraction index of the material of the light-guiding threat of FOS0, FOS1,
and FOS2, с is the light speed, α 01 , α 02 are specific light losses in FOS1 and FOS2,
relatively.
The normalized transfer function of differential RF FODL is |K NTR ( jω)| ¼ |
K TR ( jω)|/[S LD S PD μ exp (Àγ 0 L 0 )], where S LD is the slope of the watt–ampere
characteristic of the laser diode, S PD is the slope of the volt–ampere characteristics
of PD, μ are optical losses on the FOS matching. The phase-frequency characteristics
of RF FODL φ(ω) are obtained in assumption of the light losses absence in FOS1
and FOS2, when γ 1 L 1 ¼ γ 2 L 2 % 0 and μ ¼ 1.
At variation of the excitation coefficient of the FOS1 light guide α ¼ А from 0 to
1, AFC of the differential RF FODL deforms and has the comb character with clearly
expressed minima at ω(Т 2FOS À Т 1FOS ) ¼ π + 2πm; m ¼ 1, 2,. . .; the PFC of
differential RF FODL for values α % 0.5 is characterized by the presence of
periodically repeated sharp portions, which are degenerated into jumps by π at
α ¼ 0.5.
In the region of ω(Т 2FOS À Т 1FOS ) < π/2, PFC is close to the linear function. It
should be noted that in the limit cases (α ¼ 0 and α ¼ 1), PFC of the differential RF
FODL represents the straight lines φ ¼ ωT 1FOS and φ ¼ ωT 2FOS .
The adjusting curves of the frequency retuning in OEO with the differential RF
FODL are shown in Fig. 7.24b, c. Taking into consideration of obtained transfer
functions of the differential RF FODL, the solution of OEO abbreviated equations
gives a possibility to obtain of frequency and amplitude functions versus the external
impact and to find the regions of stable stationary generation of OEO with the
differential RF FODL.
The comb character of |K NTR ( jω)| of differential RF FODL applies features on the
generation mode at impact indication in such types of devices. Figure 7.24b, c shows
the plots of f gen (α) and U( f gen ) at varying α depicted for different values of the smallsignal gain in the open loop χ. The growth of χ leads to the qualitative variation of
the function U( f gen ), and at small χ 1 , the zone generation is observed.
7.4 Frequency Control in OEO with RF FODL with Two Optical Fibers
421