where N 10L is the amplitude of the modulated oscillations of N, φ 10LN is the
oscillation phase; n L also change. We take into account that at known n L and the
geometrical length L 0L of the optical resonator (for example, for the Fabry–Perot
resonator) ν 0P is determined as: ν 0P ¼
c
2n L L 0L
, where c is the light speed in vacuum.
The average optical frequency ν 0 ¼ ν 0L of the laser in the quasi-stationary mode is
determined from Eq. (5.60) at dφ/dt ¼ 0 as: ν 0 ¼ ν 0P N
ð Þ þ σ 0L þ ρ 0L E
2
0L
Â
à =2π.
Assuming that ν 0 À ν 0P ) σ 0L þ ρ 0L E
2
L
Â
Ã
, the laser generated optical frequency
ν(t) will perform oscillations from its DC value ν 0 ¼ ν 0L at small AC component of
the pumping current and also will be the periodic function:
ν(t) ¼ ν 0L + ν 0P Á cos [(2πf 0 t À φ 10LL )].
The spectrum of the field strength oscillations of the laser emission E L (t) represents oscillations, which are modulated in amplitude and in frequency. The field
strength E L (t) for central ν 0 and two side harmonics ν 0 À f 0 and f 0 + ν 0 can be
expressed as: ν 0 ¼ ν 0P N
ð Þ þ σ 0L þ ρ 0L E
2
0L
Â
Ã
=2π .
The modulation of the QWLD optical frequency can be caused by not only the
fast variation of the pumping current, but also by temperature variations of the
refraction index of the active medium and, as the result of it, by heating/cooling of
the QWLD active layer. The frequency deviation W (on level 0,7) at frequency
modulation of the laser depends on the pumping current amplitude J 10L and the slope
dν
dJ 10L
as: W ¼ J 10L Á
dν
dJ 10L
. For example, for semiconductor “volumetric” injection
lasers on the hetero-structures,
dν
dJ 10L
¼ 10
À3 THz=mA . Figure 5.7 shows the plot of
the calculated optical spectra of the current modulating of the laser diode in OEO
without fluctuations.
For modern cascade QWLD, the slope is
dν
dJ 10L
¼ 10
À6 THz=mA. The decrease
approximately by 1000 times of the QWLD frequency deviation is caused firstly, by
the factor of spatial restriction of optical emission (the transverse dimensions of the
transition zone, in which the inversed population variations occur, are about 10 nm,
the transverse dimensions of the optical channel, via which the emission propagates,
are about the laser wavelength about 1000 nm); secondly, by utilization of more
effective four-level pumping scheme (this is true for cascade QWLD with the
number of quantum wells more than 3–4); thirdly, by decrease by more than the
order of the threshold current and, accordingly, the amplitude of the AC pumping
current.
If to examine the limit case of zero slopes
dν
dJ 10L
, then at analyzing of laser
quantum-mechanical equations presented in Chap. 2, we can note that the variation
of the constant field variation leads to widening and even to splitting of the quantum
levels.
For the modulation amplitude of the laser current of 10 mА, the optical frequency
bandwidth is W ¼ 10 GHz. If this amplitude is 1 mА, the optical frequency
bandwidth is W ¼ 1 GHz. Thus, at growth of the laser modulation amplitude
(or the modulation index), the frequency modulation band will increase.
Thus, we show that the complicate system (Eq. 5.64) from three equations is
reduced in the quasi-stationary mode to relatively simple differential equations
5.4 Fluctuation Differential Equations of OEO with the Langevinian Noise Sources
247
oscillation phase; n L also change. We take into account that at known n L and the
geometrical length L 0L of the optical resonator (for example, for the Fabry–Perot
resonator) ν 0P is determined as: ν 0P ¼
c
2n L L 0L
, where c is the light speed in vacuum.
The average optical frequency ν 0 ¼ ν 0L of the laser in the quasi-stationary mode is
determined from Eq. (5.60) at dφ/dt ¼ 0 as: ν 0 ¼ ν 0P N
ð Þ þ σ 0L þ ρ 0L E
2
0L
Â
à =2π.
Assuming that ν 0 À ν 0P ) σ 0L þ ρ 0L E
2
L
Â
Ã
, the laser generated optical frequency
ν(t) will perform oscillations from its DC value ν 0 ¼ ν 0L at small AC component of
the pumping current and also will be the periodic function:
ν(t) ¼ ν 0L + ν 0P Á cos [(2πf 0 t À φ 10LL )].
The spectrum of the field strength oscillations of the laser emission E L (t) represents oscillations, which are modulated in amplitude and in frequency. The field
strength E L (t) for central ν 0 and two side harmonics ν 0 À f 0 and f 0 + ν 0 can be
expressed as: ν 0 ¼ ν 0P N
ð Þ þ σ 0L þ ρ 0L E
2
0L
Â
Ã
=2π .
The modulation of the QWLD optical frequency can be caused by not only the
fast variation of the pumping current, but also by temperature variations of the
refraction index of the active medium and, as the result of it, by heating/cooling of
the QWLD active layer. The frequency deviation W (on level 0,7) at frequency
modulation of the laser depends on the pumping current amplitude J 10L and the slope
dν
dJ 10L
as: W ¼ J 10L Á
dν
dJ 10L
. For example, for semiconductor “volumetric” injection
lasers on the hetero-structures,
dν
dJ 10L
¼ 10
À3 THz=mA . Figure 5.7 shows the plot of
the calculated optical spectra of the current modulating of the laser diode in OEO
without fluctuations.
For modern cascade QWLD, the slope is
dν
dJ 10L
¼ 10
À6 THz=mA. The decrease
approximately by 1000 times of the QWLD frequency deviation is caused firstly, by
the factor of spatial restriction of optical emission (the transverse dimensions of the
transition zone, in which the inversed population variations occur, are about 10 nm,
the transverse dimensions of the optical channel, via which the emission propagates,
are about the laser wavelength about 1000 nm); secondly, by utilization of more
effective four-level pumping scheme (this is true for cascade QWLD with the
number of quantum wells more than 3–4); thirdly, by decrease by more than the
order of the threshold current and, accordingly, the amplitude of the AC pumping
current.
If to examine the limit case of zero slopes
dν
dJ 10L
, then at analyzing of laser
quantum-mechanical equations presented in Chap. 2, we can note that the variation
of the constant field variation leads to widening and even to splitting of the quantum
levels.
For the modulation amplitude of the laser current of 10 mА, the optical frequency
bandwidth is W ¼ 10 GHz. If this amplitude is 1 mА, the optical frequency
bandwidth is W ¼ 1 GHz. Thus, at growth of the laser modulation amplitude
(or the modulation index), the frequency modulation band will increase.
Thus, we show that the complicate system (Eq. 5.64) from three equations is
reduced in the quasi-stationary mode to relatively simple differential equations
5.4 Fluctuation Differential Equations of OEO with the Langevinian Noise Sources
247