5.5.3.1 The Correlation Function of the Spectral Density of the Phase
Noise and the QWLD Spectrum
At arbitrary analysis time, in order to ACF calculate at known laser phase fluctuations, we can use the following expression [3, 5], which can be written as:
R L00 τ
ð Þ ¼ K φ12 τ
ð Þ
¼ exp ÀΔν L τ
ð
Þ cos ν À ν 0L
ð
Þτ
½
þΔν L = ν À ν 0L
ð
Þ
½
sin ν À ν 0L
ð
Þ τ
½
f
g :
ð5:85Þ
The spectrum of the laser generation with account of fluctuations is determined as
the Fourier transform of the expression (Eq. 5.83) and is written as:
S 1 f
ð Þ ¼ exp À2πΔν L τ D
ð
Þ δ f
ð Þ þ
Δν L
Δν 2
L þ f À f 0
ð
Þ
½
2
 1 À exp À2πΔν L τ D
ð
ÞÁ cos 2π f 0 τ D
ð
Þþ
Δν L
f 0
sin 2π f 0 τ D
ð
Þ
!
&
'
,
ð5:86Þ
in which δ( f ) is the Dirac delta-function, τ D ¼ τ is the observation time or the time of
analysis of the oscillation process of the laser generation, f the radio frequency of the
analysis.
At small time of observation τ D (less than 10
À6 s), the QWLD spectrum shape
(taking into account the smallness of quantities Δν L τ D and fτ D ) is written as:
S 1 f
ð Þ ¼ exp À2πΔν L τ D
ð
Þ δ f
ð Þ þ
Δν L
Δν 2
L þ f À f 0
ð
Þ
½
2
 1 À exp À2πΔν L τ D
ð
Þ
½
Á
ð 5:87Þ
Figure 5.12 illustrates the calculated plots of the autocorrelation functions K φ12
(Eq. 5.85) as the function of the frequency offset F ¼ 2π( f À f 0 ) from the carrier
frequency f 0 at different observation times τ D ¼ τ for the laser with the half-width
spectral line Δν L ¼ 1 kHz at different observation times τ ¼ τ D ¼ ΔT M .
From plots in Fig. 5.12, we can make the conclusion that for the laser with the
emission spectral line width Δν L ¼ 1 kHz, at observation times less than 10
À5 s at
offset F/(2π) ¼ f À f 0 ¼ 10 kHz, the laser ACF R L00 (τ, F) ¼ K φ12 (τ, F) takes the
values less than 0.909, which relates with the finite coherent time of the laser.
262
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .
Noise and the QWLD Spectrum
At arbitrary analysis time, in order to ACF calculate at known laser phase fluctuations, we can use the following expression [3, 5], which can be written as:
R L00 τ
ð Þ ¼ K φ12 τ
ð Þ
¼ exp ÀΔν L τ
ð
Þ cos ν À ν 0L
ð
Þτ
½
þΔν L = ν À ν 0L
ð
Þ
½
sin ν À ν 0L
ð
Þ τ
½
f
g :
ð5:85Þ
The spectrum of the laser generation with account of fluctuations is determined as
the Fourier transform of the expression (Eq. 5.83) and is written as:
S 1 f
ð Þ ¼ exp À2πΔν L τ D
ð
Þ δ f
ð Þ þ
Δν L
Δν 2
L þ f À f 0
ð
Þ
½
2
 1 À exp À2πΔν L τ D
ð
ÞÁ cos 2π f 0 τ D
ð
Þþ
Δν L
f 0
sin 2π f 0 τ D
ð
Þ
!
&
'
,
ð5:86Þ
in which δ( f ) is the Dirac delta-function, τ D ¼ τ is the observation time or the time of
analysis of the oscillation process of the laser generation, f the radio frequency of the
analysis.
At small time of observation τ D (less than 10
À6 s), the QWLD spectrum shape
(taking into account the smallness of quantities Δν L τ D and fτ D ) is written as:
S 1 f
ð Þ ¼ exp À2πΔν L τ D
ð
Þ δ f
ð Þ þ
Δν L
Δν 2
L þ f À f 0
ð
Þ
½
2
 1 À exp À2πΔν L τ D
ð
Þ
½
Á
ð 5:87Þ
Figure 5.12 illustrates the calculated plots of the autocorrelation functions K φ12
(Eq. 5.85) as the function of the frequency offset F ¼ 2π( f À f 0 ) from the carrier
frequency f 0 at different observation times τ D ¼ τ for the laser with the half-width
spectral line Δν L ¼ 1 kHz at different observation times τ ¼ τ D ¼ ΔT M .
From plots in Fig. 5.12, we can make the conclusion that for the laser with the
emission spectral line width Δν L ¼ 1 kHz, at observation times less than 10
À5 s at
offset F/(2π) ¼ f À f 0 ¼ 10 kHz, the laser ACF R L00 (τ, F) ¼ K φ12 (τ, F) takes the
values less than 0.909, which relates with the finite coherent time of the laser.
262
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .