coherence, ΔT M ¼ |T 2M À T 1M | is the delay difference in the phase detector
channels (or in OC1 and OC2 of the MZ modulator), ν L ¼ ν is the laser
frequency, A 2 /A 1 is the ratio of quadrature amplitude of optical harmonics: the
ffiffiffiffiffi
A 1
p
E 0L and
ffiffiffiffiffi
A 2
p
E 0L in the first harmonic and second harmonic, relatively
(Fig. 2.4b). At that, on the PD area, there is the process of statistical averaging
in time of the interference result of two optical harmonics ν 0L and ν 0L + f 0 , which
are applied to the PD area: E 10L t
ð Þ ¼
ffiffiffiffiffi
A 1
p
E 0L cos 2πν 0L t þ φ 0L þ φ 10Lm t
ð Þ
½
and
E 20L t
ð Þ ¼
ffiffiffiffiffi
A 2
p
E 0L cos 2π ν 0L þ f 0
ð
Þ t þ φ 0L þ φ 20Lm t
ð Þ
½
, with the amplitudes of
optical harmonics
ffiffiffiffiffi
A 1
p
E 0L and
ffiffiffiffiffi
A 2
p
E 0L .
The suppression of the one from three optical harmonics is the necessary condition of the noise reduction.
1. At large difference ΔT M ¼ |T 2M À T 1M |, i.e., at much larger (by 1000 times and
more) time constant T L0 of the cavity (and the G oscillation period), the frequency
discriminator plays a role of a converter of frequency fluctuations in the phase
fluctuations. At that, the correlation coefficient tends to zero.
2. At small delay difference ΔT M ¼ |T 2M À T 1M |, i.e., comparable to the cavity time
constant T c (and the G generator oscillation period), the frequency discriminator
plays a role of a correlator or the phase fluctuation suppressor, its correlation
coefficient tends to 1. The suppression degree is defined by the laser line width,
the accuracy of channel alignment on the optical power (or irregularity coefficient
of optical channel excitation), as well as the channel irregularity degree, which
influences on relative phase delays in the transverse section of the channel. In the
optical range, irregularity of the refraction index and micro-irregularities are
determinative for the quality of phase fluctuation suppression. The phase irregularity account in the transverse section can be preformed by the irregularity
coefficient, which depends on the transverse irregularity.
For instance, in OEO with a laser with the 3-kHz line width, for relative delay in
the MZ modulator of ΔT M ¼ |T 2M À T 1M | ¼ 10
À13 to 10
À12 s, T c ¼ 10
À9 to 10
À6 s,
and, as we show in Chaps. 6 and 7, the auto-correlation function value is K 21 % exp
(À2Δν L Á ΔT M ) ¼ exp(À2ΔT M /T c ) ¼ 1 with accuracy up to third character after a
comma for frequency offset 1 kHz to 1 MHz from optical carrier.
At power alignment in optical channels with 0.1-accuracy, we obtain from the
mentioned formula that the phase fluctuation dispersion in the photodetector photocurrent is less than the initial value by more than 10 times. There is a possibility of
phase modulation arrangement of laser optical emission in the optical channel
of OEO.
⁄
ä
Fig. 2.4 (continued) photodetector, NA ¼ the nonlinear amplifier, F ¼ the RF filter, C ¼ the RF
coupler, CH1, CH2 ¼ the optical channels of the MZ modulator; (b) diagrams (1–4) of optical
frequency selection; (c) L ¼ the laser, T 1M and T 2M , ¼ delay lines have delay times in channels,
“+” ¼ (adding), “” ¼ (multiplication), “Ô ¼ (conjugate operation), “
R ” ¼ (integration), F ¼ the
low-pass filter, A ¼ the amplifier
2.5 OEO with Self-Heterodyne Mixing as the Oscillator Containing the Phase. . .
33
channels (or in OC1 and OC2 of the MZ modulator), ν L ¼ ν is the laser
frequency, A 2 /A 1 is the ratio of quadrature amplitude of optical harmonics: the
ffiffiffiffiffi
A 1
p
E 0L and
ffiffiffiffiffi
A 2
p
E 0L in the first harmonic and second harmonic, relatively
(Fig. 2.4b). At that, on the PD area, there is the process of statistical averaging
in time of the interference result of two optical harmonics ν 0L and ν 0L + f 0 , which
are applied to the PD area: E 10L t
ð Þ ¼
ffiffiffiffiffi
A 1
p
E 0L cos 2πν 0L t þ φ 0L þ φ 10Lm t
ð Þ
½
and
E 20L t
ð Þ ¼
ffiffiffiffiffi
A 2
p
E 0L cos 2π ν 0L þ f 0
ð
Þ t þ φ 0L þ φ 20Lm t
ð Þ
½
, with the amplitudes of
optical harmonics
ffiffiffiffiffi
A 1
p
E 0L and
ffiffiffiffiffi
A 2
p
E 0L .
The suppression of the one from three optical harmonics is the necessary condition of the noise reduction.
1. At large difference ΔT M ¼ |T 2M À T 1M |, i.e., at much larger (by 1000 times and
more) time constant T L0 of the cavity (and the G oscillation period), the frequency
discriminator plays a role of a converter of frequency fluctuations in the phase
fluctuations. At that, the correlation coefficient tends to zero.
2. At small delay difference ΔT M ¼ |T 2M À T 1M |, i.e., comparable to the cavity time
constant T c (and the G generator oscillation period), the frequency discriminator
plays a role of a correlator or the phase fluctuation suppressor, its correlation
coefficient tends to 1. The suppression degree is defined by the laser line width,
the accuracy of channel alignment on the optical power (or irregularity coefficient
of optical channel excitation), as well as the channel irregularity degree, which
influences on relative phase delays in the transverse section of the channel. In the
optical range, irregularity of the refraction index and micro-irregularities are
determinative for the quality of phase fluctuation suppression. The phase irregularity account in the transverse section can be preformed by the irregularity
coefficient, which depends on the transverse irregularity.
For instance, in OEO with a laser with the 3-kHz line width, for relative delay in
the MZ modulator of ΔT M ¼ |T 2M À T 1M | ¼ 10
À13 to 10
À12 s, T c ¼ 10
À9 to 10
À6 s,
and, as we show in Chaps. 6 and 7, the auto-correlation function value is K 21 % exp
(À2Δν L Á ΔT M ) ¼ exp(À2ΔT M /T c ) ¼ 1 with accuracy up to third character after a
comma for frequency offset 1 kHz to 1 MHz from optical carrier.
At power alignment in optical channels with 0.1-accuracy, we obtain from the
mentioned formula that the phase fluctuation dispersion in the photodetector photocurrent is less than the initial value by more than 10 times. There is a possibility of
phase modulation arrangement of laser optical emission in the optical channel
of OEO.
⁄
ä
Fig. 2.4 (continued) photodetector, NA ¼ the nonlinear amplifier, F ¼ the RF filter, C ¼ the RF
coupler, CH1, CH2 ¼ the optical channels of the MZ modulator; (b) diagrams (1–4) of optical
frequency selection; (c) L ¼ the laser, T 1M and T 2M , ¼ delay lines have delay times in channels,
“+” ¼ (adding), “” ¼ (multiplication), “Ô ¼ (conjugate operation), “
R ” ¼ (integration), F ¼ the
low-pass filter, A ¼ the amplifier
2.5 OEO with Self-Heterodyne Mixing as the Oscillator Containing the Phase. . .
33