per one passage or per the length unit of the bean path. Considered optical resonators
provide the selective functions and decrease the spectral density level of the laser
emission phase noise.
Since we are interested in OEO in resonator operation near the natural frequency
within the limits of triple excess of the pass band, we assume that, for optical
resonators used in OEO, their transfer function is defined by the following expression, which describes well their characteristics in the specified frequency band:
K OF ¼
E Lout
E Lin
¼
j2πν
ð
Þ1=T OF
ð
Þ
j2πν
ð
Þ
2 þ 1=T OF
ð
Þ j2πν
ð
Þ þ2πν 0OF
ð
Þ
2 , where T OF is the time constant of the
optical resonators, ν 0OF is the natural frequency of the resonator.
3.3.3 The Photodetector in OEO
The photodetector in OEO is the quantum-dimension device (QWPD) [12, 13]. The
effectiveness of transformation of optical emission intensity into the electrical
current in modern PDs is close to 100%. In this section, we do not pay attention of
readers to the issue of nonlinear light transformation in PD and consider it as the
inertial linear element. The PD current i PD is proportional to emission intensity
(or the square of field strength) on the PD area multiplied by the PR transformation
slope K 0PD , and the following expression is true: i PD ¼ K 0PD Á Re E L Á E
Ã
L
Â
Ã
, where
“Á” is the complex conjugation operation of the field strength E L ¼ E 0L Á Re[exp
( j2πν Á t)], where ν is the optical frequency of laser generation.
In OEO, the single-frequency optical oscillations, which are modulated by the RF
oscillation (with the radio frequency ω ¼ 2πf), pass on the PD area. Let us represent
the emission power as a sum of constant P L0 and alternate P L1 (the first harmonic
2πf) components P L ¼ P L0 + P L1 Re[exp( j2πf Á t)], where P L0 ¼ (E 0L )
2
¼ (E 12L )
2 . At
that, the current in the PD load is a sum of constant I 0PD and alternate I 1PD
component of the photocurrent I FD ¼ I 0PD + I 1PD Re[exp( j2πft)].
The complex transfer function of PD (on first harmonic) for the linearized system
can be defined as the ratio: K 1PD ( jω) ¼ [I 1PD ( jω)/P L1 ( jω)].
The voltage on the load impedance Z PD of PD is u PD ¼ Z PD Á i PD . The transfer
function of PD K 1PD is defined as K 1PD ¼
K 0PD
1þj 2πfT PD
ð
Þ ¼ K 1PD
j
jÁ exp ÀjArg K 1PD
ð
Þ
½
,
in which we extract the module K 1PD
j
j¼
K 0PD
1þ 2πfT PD
ð
Þ
2
½
1=2 , and the argument Arg
(K 1PD ) ¼ arctg(2πfT PD ), where T PD is the time constant of the photodetector.
3.3.4 The Transfer Function of the Fiber Optical System
The complex transfer function of the fiber optical system (FOS) on the amplitude of
the field strength E L ¼ E 0L Á Re[exp( j2πν Á t)], which acts in the FOS optical input,
we define as a ratio of the amplitude on the FOS optical output E 0Lout to the
3.3 Mathematical Description of Transfer Functions of OEO Components
111
provide the selective functions and decrease the spectral density level of the laser
emission phase noise.
Since we are interested in OEO in resonator operation near the natural frequency
within the limits of triple excess of the pass band, we assume that, for optical
resonators used in OEO, their transfer function is defined by the following expression, which describes well their characteristics in the specified frequency band:
K OF ¼
E Lout
E Lin
¼
j2πν
ð
Þ1=T OF
ð
Þ
j2πν
ð
Þ
2 þ 1=T OF
ð
Þ j2πν
ð
Þ þ2πν 0OF
ð
Þ
2 , where T OF is the time constant of the
optical resonators, ν 0OF is the natural frequency of the resonator.
3.3.3 The Photodetector in OEO
The photodetector in OEO is the quantum-dimension device (QWPD) [12, 13]. The
effectiveness of transformation of optical emission intensity into the electrical
current in modern PDs is close to 100%. In this section, we do not pay attention of
readers to the issue of nonlinear light transformation in PD and consider it as the
inertial linear element. The PD current i PD is proportional to emission intensity
(or the square of field strength) on the PD area multiplied by the PR transformation
slope K 0PD , and the following expression is true: i PD ¼ K 0PD Á Re E L Á E
Ã
L
Â
Ã
, where
“Á” is the complex conjugation operation of the field strength E L ¼ E 0L Á Re[exp
( j2πν Á t)], where ν is the optical frequency of laser generation.
In OEO, the single-frequency optical oscillations, which are modulated by the RF
oscillation (with the radio frequency ω ¼ 2πf), pass on the PD area. Let us represent
the emission power as a sum of constant P L0 and alternate P L1 (the first harmonic
2πf) components P L ¼ P L0 + P L1 Re[exp( j2πf Á t)], where P L0 ¼ (E 0L )
2
¼ (E 12L )
2 . At
that, the current in the PD load is a sum of constant I 0PD and alternate I 1PD
component of the photocurrent I FD ¼ I 0PD + I 1PD Re[exp( j2πft)].
The complex transfer function of PD (on first harmonic) for the linearized system
can be defined as the ratio: K 1PD ( jω) ¼ [I 1PD ( jω)/P L1 ( jω)].
The voltage on the load impedance Z PD of PD is u PD ¼ Z PD Á i PD . The transfer
function of PD K 1PD is defined as K 1PD ¼
K 0PD
1þj 2πfT PD
ð
Þ ¼ K 1PD
j
jÁ exp ÀjArg K 1PD
ð
Þ
½
,
in which we extract the module K 1PD
j
j¼
K 0PD
1þ 2πfT PD
ð
Þ
2
½
1=2 , and the argument Arg
(K 1PD ) ¼ arctg(2πfT PD ), where T PD is the time constant of the photodetector.
3.3.4 The Transfer Function of the Fiber Optical System
The complex transfer function of the fiber optical system (FOS) on the amplitude of
the field strength E L ¼ E 0L Á Re[exp( j2πν Á t)], which acts in the FOS optical input,
we define as a ratio of the amplitude on the FOS optical output E 0Lout to the
3.3 Mathematical Description of Transfer Functions of OEO Components
111