constant of the active medium polarization T 2 , and the time constant of the optical
filter T 0F . Oscillations of E n in Eq. (5.62) arise only at specific threshold value of
pumping α N00 . In the right parts of Eq. (5.62), the S L ¼ α 00 E n À β 00 E n |E n |
2
expression is located, which defines the laser nonlinearity in the quasistationary mode.
We note that the equation system of OEO DM (Eq. 5.62) takes into consideration
the heterodyning properties with the help of the multiplier cos[Δϕ OF ], at that, Δϕ OF
is determined by the difference of the optical phases of the optical oscillations, which
act on the PD after passage the optical filter. The contribution into the phase
difference is made by not only the phase-frequency characteristic of the optical
filter, but the phase-frequency characteristic of QWLD.
The specific peculiarity of OEO DM is the fact that the natural optical frequency of
the optical filter depends on DC and AC components of pumping ν 0F ¼ ν 0F (α N0 ) or
ν 0F ¼ ν 0F (|E n |
2 ). The DC pumping component J 0L in OEO DM does not depend on the
state (ON-OFF) of the positive feedback loop. On the contrary, the AC pumping
component of QWLD i 1L ¼ J 1L is defined by the closed loop of OEO DM, and the
amplitude J 10L of the current i 1L and its radio frequency f 0 are the amplitude and
frequency of the OEO DM output generation, which are determined by the module
and the argument of the transfer function K DL ¼ K FBN ¼ K BZ ¼ i 1L /|E n |
2
¼ J 1L /|E n |
2 of
the positive feedback loop: |E n |
2 and Δϕ OF . Therefore, in the closed loop in OEO DM,
at cos[Δϕ OF ] ¼ 1 for QWLD α N0 ¼ α N00 Á J 0L + α N01 Á i L pumping, the expression is
true: α N0 ¼ α N00 J 0L + α N01 K DL |E n |
2
, where the DC pumping component
α N00 ¼ α N0 Á J 0L defining as the product of the DC pumping current J 0L of QWLD
by the constant coefficient α N00 , and α N01 Á i L ¼ α N01 Á K DL Á |E n |
2 .
We must take into consideration that the laser optical frequency in OEO DM
depends on the AC pumping current i L . The AC pumping current modulates of the
laser optical emission in its frequency.
At small amplitude modulation index, this tells on the accompanied frequency
modulation of the QWLD strength optical oscillation, if the DE solution with the
parametric dependence of the optical frequency leads to FM of oscillations and to
formation of two harmonics on the left and on the right of the carrier frequency.
At direct modulation of QWLD by the pumping current, oscillations of the optical
field are frequency modulated. We note that in Eq. (5.60), the natural optical
frequency ν 0P of the laser resonator (or, which is the same, the frequency of the
longitudinal generating mode of the laser resonator) depends on the inversed population N or ν 0P (N ). Therefore, the optical frequency ν of the laser generation is
dependable on the population N. At small deviations of the N population from the
average value N 0L , the frequency deviations from the mean value ν 0L are equal
ν À ν 0L ¼ C ν Á (N À N 00L ), where C ν is the constant. Since the N population depends
on pumping, as follows from Eq. (5.60), then at variation of the pumping current,
N will change. In turn, the refraction index n L of the laser active medium is
determined by the inversed population of the active medium. At variation of the
pumping current J with the frequency f ¼ f 0 and the phase φ 10L as:
J ¼ J 0L + J 10L Á cos [(2πf 0 t À φ 10L )], and the inversed population is modulated by
the harmonic oscillations of the pumping current: N ¼ N 0L + N 10L cos [(2πf 0 t À φ 10LN )],
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5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .
filter T 0F . Oscillations of E n in Eq. (5.62) arise only at specific threshold value of
pumping α N00 . In the right parts of Eq. (5.62), the S L ¼ α 00 E n À β 00 E n |E n |
2
expression is located, which defines the laser nonlinearity in the quasistationary mode.
We note that the equation system of OEO DM (Eq. 5.62) takes into consideration
the heterodyning properties with the help of the multiplier cos[Δϕ OF ], at that, Δϕ OF
is determined by the difference of the optical phases of the optical oscillations, which
act on the PD after passage the optical filter. The contribution into the phase
difference is made by not only the phase-frequency characteristic of the optical
filter, but the phase-frequency characteristic of QWLD.
The specific peculiarity of OEO DM is the fact that the natural optical frequency of
the optical filter depends on DC and AC components of pumping ν 0F ¼ ν 0F (α N0 ) or
ν 0F ¼ ν 0F (|E n |
2 ). The DC pumping component J 0L in OEO DM does not depend on the
state (ON-OFF) of the positive feedback loop. On the contrary, the AC pumping
component of QWLD i 1L ¼ J 1L is defined by the closed loop of OEO DM, and the
amplitude J 10L of the current i 1L and its radio frequency f 0 are the amplitude and
frequency of the OEO DM output generation, which are determined by the module
and the argument of the transfer function K DL ¼ K FBN ¼ K BZ ¼ i 1L /|E n |
2
¼ J 1L /|E n |
2 of
the positive feedback loop: |E n |
2 and Δϕ OF . Therefore, in the closed loop in OEO DM,
at cos[Δϕ OF ] ¼ 1 for QWLD α N0 ¼ α N00 Á J 0L + α N01 Á i L pumping, the expression is
true: α N0 ¼ α N00 J 0L + α N01 K DL |E n |
2
, where the DC pumping component
α N00 ¼ α N0 Á J 0L defining as the product of the DC pumping current J 0L of QWLD
by the constant coefficient α N00 , and α N01 Á i L ¼ α N01 Á K DL Á |E n |
2 .
We must take into consideration that the laser optical frequency in OEO DM
depends on the AC pumping current i L . The AC pumping current modulates of the
laser optical emission in its frequency.
At small amplitude modulation index, this tells on the accompanied frequency
modulation of the QWLD strength optical oscillation, if the DE solution with the
parametric dependence of the optical frequency leads to FM of oscillations and to
formation of two harmonics on the left and on the right of the carrier frequency.
At direct modulation of QWLD by the pumping current, oscillations of the optical
field are frequency modulated. We note that in Eq. (5.60), the natural optical
frequency ν 0P of the laser resonator (or, which is the same, the frequency of the
longitudinal generating mode of the laser resonator) depends on the inversed population N or ν 0P (N ). Therefore, the optical frequency ν of the laser generation is
dependable on the population N. At small deviations of the N population from the
average value N 0L , the frequency deviations from the mean value ν 0L are equal
ν À ν 0L ¼ C ν Á (N À N 00L ), where C ν is the constant. Since the N population depends
on pumping, as follows from Eq. (5.60), then at variation of the pumping current,
N will change. In turn, the refraction index n L of the laser active medium is
determined by the inversed population of the active medium. At variation of the
pumping current J with the frequency f ¼ f 0 and the phase φ 10L as:
J ¼ J 0L + J 10L Á cos [(2πf 0 t À φ 10L )], and the inversed population is modulated by
the harmonic oscillations of the pumping current: N ¼ N 0L + N 10L cos [(2πf 0 t À φ 10LN )],
246
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .