At heterodyne transformation, the optical oscillation spectrum with noises S L (S L
designates the power spectral density, which is determined by amplitude and phase
noises) is shifted to the spectrum region of the electric signal S RFL on the intermediate frequency (IF). If to add the slow fluctuation supplements φ Lm (t) to oscillation
phases of the signal and the local oscillator, then in the phase of the electric signal on
intermediate frequency, the phase fluctuations φ Lm (t) arise on the IF frequency f IF .
The signal oscillation with noises is described in the frequency domain by the
spectrum, which density is concentrated in the vicinity of the ν SL frequency.
2 The
local oscillator signal with noises is described by the spectrum, which density is
concentrated in the vicinity of the local oscillator frequency ν 0LO.L .
The value of the squared normalized EMF intensity E
2
PD t
ð Þ on the PD area, at
combined action of signal oscillation with intensity E SL and the local oscillator
signal with intensity E LO.L , is defined as:
E
2
PD t
ð Þ ¼ K C Á E SL t
ð Þ þ E LO:L t
ð Þ
½
ÁE
Ã
SL t þ τ
ð
ÞþE
Ã
LO:L t þ τ
ð
Þ
Â
Ã
:
ð3:3Þ
Here the symbol “Ô designates of the conjugate operation, and the symbol hi
designates the mathematical expectation (or the mean value) operation.
We note several features of heterodyning of the laser emission in OEO. In this
case, the laser plays both the roles: as the source of the signal oscillations and as the
local oscillator, therefore, we may speak about “self-heterodyning” of the laser
emission.
3 The intermediate frequency f IF ¼ ν SL À ν 0LO.L is in OEO the average
oscillation frequency and, as a rule, it may be, say, 10 GHz. We should add that in
the low-noise OEO, lasers are used, which operate in the single-frequency mode and
have the narrow spectral line of emission with the width Δν L ¼ 10–1000 kHz,
therefore, for the optical “frequencies of analysis” ν the following relation is true:
f IF ) ν À ν 0LO.L , where ν ¼ ν L is the optical analysis frequency.
One of features of heterodyning in OEO is the fact that the laser coherence time
and the spectral line width of laser emission are defined the spectrum of RF
oscillations. This can be proved if to consider the photodetection process in OEO
MZ, to calculate the autocorrelation function R(t, t + τ) of the laser random field E L (t)
and the spectrum of RF oscillation in OEO.
Let two emissions (Fig. 3.3b), which propagate in different optical channels of the
MZ modulator, have different delays. These emissions pass on the PD area. The laser
intensity E L ¼ E L (t) and the intensity delayed on some time Δt ¼ τ or E Lτ ¼ E L (t À Δt)
are the random correlative dependent magnitudes. As the result of photodetection of
two optical oscillations E L ¼ E L (t) with different frequencies, for example, ν 0L and
ν 0L + f, the photocurrent oscillation arises in the PD load with the average frequency
f. The spectrum of the electrical oscillation in the PD load represents the convolution
of two spectra S L1 and S L2 . Let us determine the autocorrelation function at
2 Here and later, indices S show the relation to “signals” in OEO.
3 In radio electronics, the device in such self-heterodyning mode has a special name: the selfoscillation mixer (SOM) or in Russian publication “autodynes.”
82 3 Modulation Methods of Laser Emission in Optoelectronic oscillator (OEO) and OEO. . .
designates the power spectral density, which is determined by amplitude and phase
noises) is shifted to the spectrum region of the electric signal S RFL on the intermediate frequency (IF). If to add the slow fluctuation supplements φ Lm (t) to oscillation
phases of the signal and the local oscillator, then in the phase of the electric signal on
intermediate frequency, the phase fluctuations φ Lm (t) arise on the IF frequency f IF .
The signal oscillation with noises is described in the frequency domain by the
spectrum, which density is concentrated in the vicinity of the ν SL frequency.
2 The
local oscillator signal with noises is described by the spectrum, which density is
concentrated in the vicinity of the local oscillator frequency ν 0LO.L .
The value of the squared normalized EMF intensity E
2
PD t
ð Þ on the PD area, at
combined action of signal oscillation with intensity E SL and the local oscillator
signal with intensity E LO.L , is defined as:
E
2
PD t
ð Þ ¼ K C Á E SL t
ð Þ þ E LO:L t
ð Þ
½
ÁE
Ã
SL t þ τ
ð
ÞþE
Ã
LO:L t þ τ
ð
Þ
Â
Ã
:
ð3:3Þ
Here the symbol “Ô designates of the conjugate operation, and the symbol hi
designates the mathematical expectation (or the mean value) operation.
We note several features of heterodyning of the laser emission in OEO. In this
case, the laser plays both the roles: as the source of the signal oscillations and as the
local oscillator, therefore, we may speak about “self-heterodyning” of the laser
emission.
3 The intermediate frequency f IF ¼ ν SL À ν 0LO.L is in OEO the average
oscillation frequency and, as a rule, it may be, say, 10 GHz. We should add that in
the low-noise OEO, lasers are used, which operate in the single-frequency mode and
have the narrow spectral line of emission with the width Δν L ¼ 10–1000 kHz,
therefore, for the optical “frequencies of analysis” ν the following relation is true:
f IF ) ν À ν 0LO.L , where ν ¼ ν L is the optical analysis frequency.
One of features of heterodyning in OEO is the fact that the laser coherence time
and the spectral line width of laser emission are defined the spectrum of RF
oscillations. This can be proved if to consider the photodetection process in OEO
MZ, to calculate the autocorrelation function R(t, t + τ) of the laser random field E L (t)
and the spectrum of RF oscillation in OEO.
Let two emissions (Fig. 3.3b), which propagate in different optical channels of the
MZ modulator, have different delays. These emissions pass on the PD area. The laser
intensity E L ¼ E L (t) and the intensity delayed on some time Δt ¼ τ or E Lτ ¼ E L (t À Δt)
are the random correlative dependent magnitudes. As the result of photodetection of
two optical oscillations E L ¼ E L (t) with different frequencies, for example, ν 0L and
ν 0L + f, the photocurrent oscillation arises in the PD load with the average frequency
f. The spectrum of the electrical oscillation in the PD load represents the convolution
of two spectra S L1 and S L2 . Let us determine the autocorrelation function at
2 Here and later, indices S show the relation to “signals” in OEO.
3 In radio electronics, the device in such self-heterodyning mode has a special name: the selfoscillation mixer (SOM) or in Russian publication “autodynes.”
82 3 Modulation Methods of Laser Emission in Optoelectronic oscillator (OEO) and OEO. . .