a series of important conclusions follows: the optical MZ channels are nonuniform
and nonideal, the emission in MZ strongly disperses on the media boundaries, the
emission shape in the near zone in the MZ output is closed to an ellipse owing to the
addition of coherent emissions, which are dispersed on channel boundaries. Taking
into account the phase delays, we observe the minima zones in the form of dark
circles located along the OC1 and OC2 channels with the period closed to the
wavelength of emission.
In addition, we note that to compensate the laser phase fluctuations, we must
fulfill the condition of equality with accuracy less than 0.1–0.001 from intensities in
the outputs of the first and second channels in the transverse sections in the aperture.
Otherwise, fluctuations’ compensation is not fulfilled. We must take into consideration that detection of the optical emission occurs in the area of PD. The conditions
of intensity equality in the transverse section with mentioned accuracies require the
fulfillment of strict requirements to the material nonuniformity and to the drift of the
geometrical sizes of channels in the modern MZ modulators.
Thus, the significant difference of heterodyning in the optical range from the
similar operation of the radio-frequency range is the following. In the optical range,
dimensions of the photodetection zone are commensurable with the laser wavelength. In the radio-frequency range, the geometrical dimensions of the detection
zone of two electromagnetic oscillations at heterodyning can be simply performed as
essentially less than the wavelength of radio oscillations, which participate in the
process of its mixing. In this connection, the fluctuation compensation at heterodyning in the radio-frequency range is performed at equalization of the oscillation
powers.
The above-mentioned can be illustrated by the fact that as the result of mixing on
the PD area of the optical emissions, which passed the appropriate different optical
MZ channels and having the power spectral densities S 1L (F) and S 2L (F) in the PD
photo-current, we obtain: S PD2 F
ð Þ ¼ S 1L F, R
ð
ÞþS 2L F, R
ð
ÞÀ2
A 2
A 1
K 21 S 12L F, R
ð
Þ,
where S 1L (F), S 2L (F) are PSD of optical phase fluctuations of emission, relatively, in
optical channels OC1 and OC2, S 12L (F, R) is mutual spectral density of PSD S 1L (F),
S 2L (F); K 21 is the autocorrelation function of the random process
K 21 % exp (À2Δν L Á ΔT M ) ¼ exp (À2ΔT M /T c ); Δν L % 1/T c is the natural width
of the spectral line of the reference generator (a laser) emission, T c is the time of laser
coherence, S 1L (F), S 2L (F) are PSD of the optical phase fluctuations, relatively, in the
optical channels OC1 and OC2, ΔT M ¼ |T 2M À T 1M | is the delay difference in MZ
channels and in FOS, A 2 /A 1 is the ratio of quarter 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. 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, 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
ffiffiffiffiffi
A 1
p
E 0L and
ffiffiffiffiffi
A 2
p
E 0L .
292
6 Operation Analysis of Optoelectronic oscillator (OEO) with External. . .
and nonideal, the emission in MZ strongly disperses on the media boundaries, the
emission shape in the near zone in the MZ output is closed to an ellipse owing to the
addition of coherent emissions, which are dispersed on channel boundaries. Taking
into account the phase delays, we observe the minima zones in the form of dark
circles located along the OC1 and OC2 channels with the period closed to the
wavelength of emission.
In addition, we note that to compensate the laser phase fluctuations, we must
fulfill the condition of equality with accuracy less than 0.1–0.001 from intensities in
the outputs of the first and second channels in the transverse sections in the aperture.
Otherwise, fluctuations’ compensation is not fulfilled. We must take into consideration that detection of the optical emission occurs in the area of PD. The conditions
of intensity equality in the transverse section with mentioned accuracies require the
fulfillment of strict requirements to the material nonuniformity and to the drift of the
geometrical sizes of channels in the modern MZ modulators.
Thus, the significant difference of heterodyning in the optical range from the
similar operation of the radio-frequency range is the following. In the optical range,
dimensions of the photodetection zone are commensurable with the laser wavelength. In the radio-frequency range, the geometrical dimensions of the detection
zone of two electromagnetic oscillations at heterodyning can be simply performed as
essentially less than the wavelength of radio oscillations, which participate in the
process of its mixing. In this connection, the fluctuation compensation at heterodyning in the radio-frequency range is performed at equalization of the oscillation
powers.
The above-mentioned can be illustrated by the fact that as the result of mixing on
the PD area of the optical emissions, which passed the appropriate different optical
MZ channels and having the power spectral densities S 1L (F) and S 2L (F) in the PD
photo-current, we obtain: S PD2 F
ð Þ ¼ S 1L F, R
ð
ÞþS 2L F, R
ð
ÞÀ2
A 2
A 1
K 21 S 12L F, R
ð
Þ,
where S 1L (F), S 2L (F) are PSD of optical phase fluctuations of emission, relatively, in
optical channels OC1 and OC2, S 12L (F, R) is mutual spectral density of PSD S 1L (F),
S 2L (F); K 21 is the autocorrelation function of the random process
K 21 % exp (À2Δν L Á ΔT M ) ¼ exp (À2ΔT M /T c ); Δν L % 1/T c is the natural width
of the spectral line of the reference generator (a laser) emission, T c is the time of laser
coherence, S 1L (F), S 2L (F) are PSD of the optical phase fluctuations, relatively, in the
optical channels OC1 and OC2, ΔT M ¼ |T 2M À T 1M | is the delay difference in MZ
channels and in FOS, A 2 /A 1 is the ratio of quarter 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. 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, 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
ffiffiffiffiffi
A 1
p
E 0L and
ffiffiffiffiffi
A 2
p
E 0L .
292
6 Operation Analysis of Optoelectronic oscillator (OEO) with External. . .