of fluctuations of the amplitude and the phase, which excite the laser fluctuations
m L (t), ψ m (t), can be obtained at solution of abbreviated equations (Eqs. 5.60 and
5.61) with account the fluctuation Langevinian impacts S βAN , S βPN , where (ν À ν 0 ) is
the offset from the central optical generation frequency of QWLD, B L ¼
α 0L À 2πν 0P =Q 0L
ð
Þ
½
À β 0 Á 1 þ 3E
2
0
Â
Ã
, ν 0P is the natural frequency of the optical
resonator, Q 0L is the resonator Q-factor, Δν LP0 is the half-width of the resonance
curve (or AFC) of the laser optical resonator, which is inverse proportional to the
time constant T OF ¼ T L0 of the laser resonator: Δν LP0 ¼ 1/T OF , D A , D F are the
constant coefficients, α 0L the amplification (pumping), β 0 is the constant coefficient.
We note that PSD of laser phase fluctuations S ψ L (ν), as for all self-oscillating system
with dissipation, is inverse proportional to the normalized power of optical oscillations: P L ¼ E
2
0 and T
2
L0 .
7.6.3 The Power of Spontaneous Emission and PSD of Laser
(or QWLD) Spontaneous Emission
Taking into account that the power of the LD spontaneous emission is P sp ¼ β sp Á P L ,
then we have: S βPN ¼ β sp P L /Δν P0 . The value of β sp for modern QWLD is β sp ¼ 10
À3
to 10
À4 . Beginning from the physical sense, β sp is determined roughly (with account
of the amendment in several orders) from the classic differential equations of
Einstein for the laser, as the ratio β sp % A 21 /B 21 , where B 21 , A 21 are the Einstein
coefficients for stimulated and spontaneous emission, relatively. We can consider
with well enough approximation for QWLD that β sp ¼ T 2 /T 1 , where T 2 is the time
constant of polarization (or time of the longitudinal relaxation), T 1 is the life time of
carriers (or time of the transverse relaxation). Then, S ψ L (ν) can be written as
S ψ L (ν) ¼ Δν LP0 β sp D F /(ν À ν 0 )
2 or S ψ L (ν) ¼ β sp D F /[T 0P (ν À ν 0 )
2
].
7.6.4 Estimation of Phase Noises of QWLD
According to obtained expression (if we assume that β sp ¼ T 2 /T 1 ), we obtain:
S ψ L (ν) ¼ T 2 D F /[T 1 T P0 (ν À ν 0 )
2 ] for laser PSD, we can make the rough estimation
of PSD of phase noises.
1. QWLD with the traditional Fabry–Perot optical resonator
For example, at values D F ¼ 0.1, T 2 ¼ 10
À12 s, for QWLD parameters: the
lifetime of carriers (electrons in semiconductor QWLD) T 1 ¼ τ n1 ¼ 10
À9 s, the
lifetime of photons in the optical resonator T 0P ¼ T OF ¼ τ ph ¼ 10
À12 s, at the
offset in the optical frequency from the carrier (ν À ν 0 ) ¼ 1 MHz we obtain: PSD
of the phase noise of QWLD is S ψ L ¼ 1/[10T 1 Á (ν À ν 0 )
2 ] ¼ 10
À7 dBm/Hz.
2. In QWLD with the narrowband optical resonator on the base of the Bragg cell,
at offset in the optical frequency from the carrier (ν À ν 0 ) ¼ 1 MHz for the
440
7 Optoelectronic oscillator (OEO) as the Time and Spatial Correlator of Random. . .
m L (t), ψ m (t), can be obtained at solution of abbreviated equations (Eqs. 5.60 and
5.61) with account the fluctuation Langevinian impacts S βAN , S βPN , where (ν À ν 0 ) is
the offset from the central optical generation frequency of QWLD, B L ¼
α 0L À 2πν 0P =Q 0L
ð
Þ
½
À β 0 Á 1 þ 3E
2
0
Â
Ã
, ν 0P is the natural frequency of the optical
resonator, Q 0L is the resonator Q-factor, Δν LP0 is the half-width of the resonance
curve (or AFC) of the laser optical resonator, which is inverse proportional to the
time constant T OF ¼ T L0 of the laser resonator: Δν LP0 ¼ 1/T OF , D A , D F are the
constant coefficients, α 0L the amplification (pumping), β 0 is the constant coefficient.
We note that PSD of laser phase fluctuations S ψ L (ν), as for all self-oscillating system
with dissipation, is inverse proportional to the normalized power of optical oscillations: P L ¼ E
2
0 and T
2
L0 .
7.6.3 The Power of Spontaneous Emission and PSD of Laser
(or QWLD) Spontaneous Emission
Taking into account that the power of the LD spontaneous emission is P sp ¼ β sp Á P L ,
then we have: S βPN ¼ β sp P L /Δν P0 . The value of β sp for modern QWLD is β sp ¼ 10
À3
to 10
À4 . Beginning from the physical sense, β sp is determined roughly (with account
of the amendment in several orders) from the classic differential equations of
Einstein for the laser, as the ratio β sp % A 21 /B 21 , where B 21 , A 21 are the Einstein
coefficients for stimulated and spontaneous emission, relatively. We can consider
with well enough approximation for QWLD that β sp ¼ T 2 /T 1 , where T 2 is the time
constant of polarization (or time of the longitudinal relaxation), T 1 is the life time of
carriers (or time of the transverse relaxation). Then, S ψ L (ν) can be written as
S ψ L (ν) ¼ Δν LP0 β sp D F /(ν À ν 0 )
2 or S ψ L (ν) ¼ β sp D F /[T 0P (ν À ν 0 )
2
].
7.6.4 Estimation of Phase Noises of QWLD
According to obtained expression (if we assume that β sp ¼ T 2 /T 1 ), we obtain:
S ψ L (ν) ¼ T 2 D F /[T 1 T P0 (ν À ν 0 )
2 ] for laser PSD, we can make the rough estimation
of PSD of phase noises.
1. QWLD with the traditional Fabry–Perot optical resonator
For example, at values D F ¼ 0.1, T 2 ¼ 10
À12 s, for QWLD parameters: the
lifetime of carriers (electrons in semiconductor QWLD) T 1 ¼ τ n1 ¼ 10
À9 s, the
lifetime of photons in the optical resonator T 0P ¼ T OF ¼ τ ph ¼ 10
À12 s, at the
offset in the optical frequency from the carrier (ν À ν 0 ) ¼ 1 MHz we obtain: PSD
of the phase noise of QWLD is S ψ L ¼ 1/[10T 1 Á (ν À ν 0 )
2 ] ¼ 10
À7 dBm/Hz.
2. In QWLD with the narrowband optical resonator on the base of the Bragg cell,
at offset in the optical frequency from the carrier (ν À ν 0 ) ¼ 1 MHz for the
440
7 Optoelectronic oscillator (OEO) as the Time and Spatial Correlator of Random. . .