oscillations on the frequency 2f 0 of the second harmonic dominate in the level over
oscillations of the frequency f 0 of the first harmonic (Fig. 5.14с).
At that, oscillations are amplified by the RF amplifier, then pass to the input of RF
filter, which has the natural frequency, which coincides approximately with 2f 0 .
From the output of RF filter, oscillations pass to the RF frequency divider “:2.” Then
oscillations with frequency f 0 on the fundamental harmonic pass to the switch Sw.
When switch Sw is closed, these oscillations with the frequency f 0 pass to the
electrical input of the oscillator “Os.” At fulfillment of the locking conditions of the
oscillator “Os,” oscillations passed to the electrical input, and the “Os” synchronization takes place. Abbreviated equations for this structure will differ with regard to
earlier-considered by the additional differential equation for the oscillator “Os.” The
solution of abbreviated equations for this structure of OEO MZ by traditional
approach allows obtaining the locking band of the OEO MZ system with “Os,”
the amplitude and the frequency. The feature of this structure is the fact that in the
electrical input of MZ oscillation with f 0 are applies, which multiply by 2. Thus, the
opened OEO MZ system plays the role of the RF frequency multiplier.
Advantages of this structure of OEO MZ are described in Fig. 5.14, which can be
called the structure with suppression of the central optical frequency, described
earlier in Chap. 2, when we discussed the modulation methods of the laser emission.
Here we should emphasize that the transition to the structure of OEO MZ with “Os”
is substantiated only in the case when it is required to obtain the minimal level of
PSD of the phase noise of RF oscillations.
Further, we consider in detail the mathematical analysis for the OEO MZ
structure presented in Fig. 5.10.
5.5.5 The Autocorrelation Function of MZ at the Opened
Feedback Loop
We consider the correlator diagram presented in Fig. 5.11a. Let the K switch be open
and the electrical oscillation with the average frequency f 0 , passes to the electric
input of the MZ modulator. The spectrum of electrical harmonic oscillation represents the delta-function δ(f)). At that, as in the previous case, two optical oscillations
E L ¼ E L (t) and E Lτ ¼ E L (t À ΔT M ), which are delayed with regard to each other by
the time ΔT M ¼ ΔT MZ , are added on the PD area. The autocorrelation function is
R MZ τ
ð Þ % exp À
ΔT M
T c
cos 2π f 0 τ
ð
Þfor the analysis time τ > ΔT MZ .
Accordingly, the current spectrum S RFMZ in the photodetector load at τ > ΔT MZ
is determined by the formula: S RFMZ ¼ exp À
ΔT M
T c
S 0RFL
Δν L
Δν 2
L
þ f À f 0
ð
Þ
½
2 , where F is
the frequency offset from the average radio frequency f 0 of the voltage oscillations,
which acts in the MZ electrical input.
Thus, at opened feedback loop, the spectrum S RFMZ has the Lorentzian shape, is
determined by the laser phase noise, and the maximal value of S RFMZ is defined by
266
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .
oscillations of the frequency f 0 of the first harmonic (Fig. 5.14с).
At that, oscillations are amplified by the RF amplifier, then pass to the input of RF
filter, which has the natural frequency, which coincides approximately with 2f 0 .
From the output of RF filter, oscillations pass to the RF frequency divider “:2.” Then
oscillations with frequency f 0 on the fundamental harmonic pass to the switch Sw.
When switch Sw is closed, these oscillations with the frequency f 0 pass to the
electrical input of the oscillator “Os.” At fulfillment of the locking conditions of the
oscillator “Os,” oscillations passed to the electrical input, and the “Os” synchronization takes place. Abbreviated equations for this structure will differ with regard to
earlier-considered by the additional differential equation for the oscillator “Os.” The
solution of abbreviated equations for this structure of OEO MZ by traditional
approach allows obtaining the locking band of the OEO MZ system with “Os,”
the amplitude and the frequency. The feature of this structure is the fact that in the
electrical input of MZ oscillation with f 0 are applies, which multiply by 2. Thus, the
opened OEO MZ system plays the role of the RF frequency multiplier.
Advantages of this structure of OEO MZ are described in Fig. 5.14, which can be
called the structure with suppression of the central optical frequency, described
earlier in Chap. 2, when we discussed the modulation methods of the laser emission.
Here we should emphasize that the transition to the structure of OEO MZ with “Os”
is substantiated only in the case when it is required to obtain the minimal level of
PSD of the phase noise of RF oscillations.
Further, we consider in detail the mathematical analysis for the OEO MZ
structure presented in Fig. 5.10.
5.5.5 The Autocorrelation Function of MZ at the Opened
Feedback Loop
We consider the correlator diagram presented in Fig. 5.11a. Let the K switch be open
and the electrical oscillation with the average frequency f 0 , passes to the electric
input of the MZ modulator. The spectrum of electrical harmonic oscillation represents the delta-function δ(f)). At that, as in the previous case, two optical oscillations
E L ¼ E L (t) and E Lτ ¼ E L (t À ΔT M ), which are delayed with regard to each other by
the time ΔT M ¼ ΔT MZ , are added on the PD area. The autocorrelation function is
R MZ τ
ð Þ % exp À
ΔT M
T c
cos 2π f 0 τ
ð
Þfor the analysis time τ > ΔT MZ .
Accordingly, the current spectrum S RFMZ in the photodetector load at τ > ΔT MZ
is determined by the formula: S RFMZ ¼ exp À
ΔT M
T c
S 0RFL
Δν L
Δν 2
L
þ f À f 0
ð
Þ
½
2 , where F is
the frequency offset from the average radio frequency f 0 of the voltage oscillations,
which acts in the MZ electrical input.
Thus, at opened feedback loop, the spectrum S RFMZ has the Lorentzian shape, is
determined by the laser phase noise, and the maximal value of S RFMZ is defined by
266
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .