tical i
-7
00
0
0
0
3
Op
0
Op -60
-6
-80
Power s -8
Power s
1558 1559
155
1560 1561 1562
0
5
10
15
20
2
Wav
Wa e
v l
e en
e g
n t
g h
t (
h n
( m
n )
m
Fre
Fr q
e u
q e
u nc
e y (
nc G
y ( H
G z
H )
z
(a)
(b)
Fig. 2.14. (a) Measured optical spectra when the FP laser without an optical isolator is
modulated at 30GHz, an optical filter is used. (b) Corresponding spectra, where inset shows
the higher resolution spectrum
Normally the optical wavelengths of the FP laser may be affected by the ambient
temperature and bias current. It has been shown that when the chip temperature varies by
1°C the FP modes of the laser will change by 0.1 nm, corresponding to a frequency change of
12.5GHz at 1.55μm (Zhu et al., 2006). Although a Peltier cooler can be used to control the
temperature, it is impossible to maintain the chip temperature within 0.0001°C. From Fig.
2.14 one can see that the FP modes shift is about 610 MHz. However, from Fig. 2.12(b) it can
be observed that the changes of the FP mode spacing are within 4 MHz. This clearly shows
Optical Spectral Structure and Frequency Coherence
617
that the longitudinal mode spacing of the FP laser is relatively fixed, although the
wavelengths of the FP modes change with temperature. Therefore, the beat note between
the shifted second sidebands is much more stable.
Comparing Fig. 2.12(a) and 2.12(b) it follows that the wavelength of FP laser becomes more
stable when an optical isolator is used. Wavelength stability plays an important role in the
generation of a stable and narrow-linewidth microwave signal.
From the measured optical and electrical spectra shown in Fig. 2.14, it can be seen that the
optical power of the reference signal is 14.2dB higher than that of the higher second
sideband of FP mode 01. The beat note between the reference signal and the lower second
sideband of FP mode 02 is only 3.5 dB higher than the one between the higher second
sideband of FP mode 01 and the lower second sideband of FP mode 02. This discrepancy
(10.7dB) and 4.5 MHz linewidth of the beat note between the two sidebands also indicate
that the FP modes of the FP laser are partially coherent.
5. Elimination of frequency coherence
In the previous sections, we have theoretically and experimentally demonstrated the
concept of frequency coherence. The degree of frequency conherence mainly depends on the
spectral characteristics and correlations of the two lightwaves. Although two lightwaves
with highly frequency coherence are desirable for generating pure microwave or millimeter
wave signals, it is also needed to eliminate the frequency coherence of two lightwaves in
some cases, for example, the linewidth measurement of the lasers. In this section, the
elimination of the frequency coherence of two lightwaves is investigated experimentally and
the phenomena observed in the experiments are explained using the three-dimension
optical spectral structure model.
5.1 Three-dimension model of optical spectrum
Inte
t ns it
i y
Inte
t ns it
i y
ty
t
λ
t
λ
λ
t
λ
0
0
0
C omplete
No
f
C omplete
P art
r i
t a
i l
fm
overla
r pping
fm
overla
r pping
m
overla
r pping
λ -Δλ
λ -Δλ
λ
0
0
0
λ
L D
λ
λ
(a)
(b)
Fig. 2.15. Configuration of optical spectral structure in frequency-time domain after intensity
modulation (a) without and (b) with the time delay
Based on optical spectral structure in the frequency-time domain, the spectrum of the light
beam consists of a large mount of wave trains around the center wavelength, which have
distinct features just as the followings (Zhu et al., 2007).
a. In the frequency domain, it is not strictly monochromatic and the spectral linewidth is
much narrower than 1 mHz, corresponding to a wavelength range of 10-23 m at 1.55 μm.
b. The wave trains are emitted simultaneously with random frequency spacings. A wave
train is able to seed another wave train with the same frequency, and the probability of
618
Optoelectronic Devices and Properties
occurrence of two or more joint wave trains with the same frequency is rather high. The
subsequent wave trains may be with different frequencies, but the probability is much
lower.
c. In the time domain, the spatial and temporal intensity profiles of wave trains are
neither identical nor of simple form. The length of wave trains has a large variable
range. The intensity profile and average duration mainly depend on the laser structure,
the bias condition and the optical cavity.
When a light beam is modulated, any wave train in the carrier and the corresponding wave
train in the sidebands may appear simultaneously with perfect frequency coherence, and
their frequency interval is exactly identical to the modulation frequency. Fig. 2.15 gives the
optical spectral structure of the carrier and only one first sideband in the time-frequency
domain for simplicity. All the beat notes between wave train pairs in the sideband and
carrier are always superposed at the modulation frequency. The beat note at the modulation
frequency is much stronger than beat notes at other frequencies, which will become random
noise originated from the vicinity of the center wavelength.
If there is a time delay between these two beams, from the optical spectral structure shown
in Fig. 2.15, one can see that the corresponding wave trains in carrier and sidebands having
coherence lengths longer than the delay time are partially overlapping in time domain. That
means these trains are partially frequency coherent, and only part of the beat notes between
these wave train pairs are superposed at the modulation frequency. For the wave trains with
lengths shorter than the delay time, there is no overlapping in the time domain. The
corresponding wave trains in the carrier and sidebands have no coherence and do not beat.
If the delay time between sidebands and the carrier is long enough and longer than all wave
trains, the two beams become completely incoherent. In this case the measured linewidth of
beat note is so called spectral linewidth of light beams.
5.2 Dependence of frequency coherence on delay time
Referring to the Wiener–Khinchin theory (Richter et al., 1986), the optical spectral structure
consists of incoherent and coherent parts. We will give the formulation description (Zhu et
al. 2010). If the delay time between sideband and the carrier is τ0, the total optical field can
be expressed as:
(
E t) = E exp[ j(ω t + ϕ( t))] + β E exp [ j(ω + ω )( t +τ ) + ϕ( t +τ )] (14) 0
0
0
{ 0 m
0
0 }
where E0 is the amplitude of optical field, β is a real factor accounting for the amplitude ratio
between two fields, ω0 is the angular frequency of laser beam, ωm is the modulation
frequency, the phase section φ(t+τ0) and φ(t) introduce the phase jitter which is assumed to
be a Gaussian distribution. After the necessary formula derivation, the power spectrum S(ω)
can be expressed as:
1
(
S ω) = f δ ω − ω +
δ (
) f
(15a)
L
,
m
1 + ⎡⎣(ω −ω )
2
/ S ⎤
m
0 ⎦
where
2
4
f = β
− τ
δ
2 E exp( S ) , (15b)
0
0 0
Optical Spectral Structure and Frequency Coherence
619
⎧⎪
⎡
S
⎤⎫⎪
2 4
0
f = β E ⎨ −
S
− τ ⎢
ω − ω τ +
ω − ω τ
(15c)
L
4
1 exp(
) cos((
m )
)
sin((
) )⎥⎬ .
0
0 0
0
m
0
ω −
⎪
ω
⎩
⎣
m
⎦⎪⎭
Here only the white noise S0 originated from atom spontaneous radiation is included. In
(15a), the first term is a δ function at modulation frequency, and is the beat note between the
coherent wave trains of light beams when the average coherence length is longer than the
delay length. It can be concluded that its intensity will decrease with the increase of the
delay time. The second term is the beat note between incoherent wave trains which is
broadened by the phase random noise. It has a quasi-Lorentzian profile with a weight factor
fL located at the modulation frequency. From (15c) it can be seen that the amplitude of quasi-
Lorentzian profile will be proportional with the delay time.
Actually, when the delay fiber is long enough, both the white noise and the 1/ f noise
component are included, and the 1/ f noise component gives a similar Gaussian profile. In
this case, the two light beams will become incoherent and the δ-peak disappears. The power
spectrum will become a Voigt lineshape, which is the convolution of the Gaussian profile
and Lorentzian profile. The disappearance of the δ-peak can be regarded as the criterion of
coherence elimination and the lineshape broadening at the moment can be used as the
optical linewidth.
VNA
Mach
a -
ch Ze
Z h
e n
h der
n
f
der ilte
t r
e
Modulato
Modulat r
3dB
3d
3dB
Isolat
ol or
80%
Fiber D
r e
D lay
a
Spectrum
EDFA
20%
90%
ana
an lyz
y er
e
DFB
DF
lase
las r
10%
Optical spect
ec r
t um
analyz
y er
Fig. 2.16. Measurement scheme for lineshape analysis using delayed optical self-heterodyne
method
Up to date, lots of methods for analyzing lineshape and linewidth have been established in
the past two decades (Chan, 2007; Dawson et al., 1992; Ludvigsen et al., 1998; Richter et al.,
1986; Signoret et al., 2001; Signoret et al., 2004; Zhu et al., 2010). The method widely used for
analyzing lineshape is delayed self-heterodyne technique. The frequency fluctuations or
optical phase of the laser source under test can be converted into intensity variations by an
asymmetric Mach-Zehnder type interferometer. Enough fiber delay, which is longer than
coherence length, can make no overlapping in time domain between the carrier and
sideband, and the frequency coherence between them can be eliminated. Fig. 2.16 illustrates
the measurement setup for carrying out the lineshape analysis of beat note between the
optical carrier and shifted sidebands based on delayed optical heterodyning in the
experiment. A VNA and a LiNbO3 modulator were used to modulate the light beams from
the DFB laser and a Mach-Zehnder filter was used to separate the sidebands from the
modulated lightwaves.
620
Optoelectronic Devices and Properties
It has been shown that no matter what optical sources are used, the beat note between first
sidebands and the carrier has an extremely narrow linewidth if there is no time delay
between them (Zhu et al., 2009). With the increase of the delay line, the coherence between
the carrier and the delayed first sidebands will be reduced. Fig. 2.17 shows the measured
power spectra of the DFB laser using different delay lines. The power ratios between the δ-
peak and the Lorentzian component with different delay lengths are summarized in Table I.
It can be seen that the δ-peak decreases and the amplitude of the quasi-Lorentzian profile
originated from the noise increases with the increased delay line. When the delay fiber is
over 10 km, δ-peak disappears and the coherence between the carrier in the reference
channel and the delayed first sidebands has been eliminated completely.
L
= 2 5 k m
D e la y
L
= 1 0 k m
D e la y
)
L
= 1 .8 k m
D e la y
iv/d
5dB (
L
= 5 0 0 m
D e la y
itudenga
M
L
= 1 0 0 m
D ela y
L
= 0 m
D e la y
9 .9 9 0
9 .9 9 5
1 0 .0 0 0
1 0 .0 0 5
1 0 .0 1 0
F re q u e n c y (G H z)
Fig. 2.17. Measured power spectra of DFB laser with fiber delay changing from 0 m to 25 km
A long delay line over coherence length of the two light beams is required to completely
eliminate frequency coherence. However, long delay fiber will introduce a high insertion
loss. Although an EDFA can be used to compensate the optical power level, the amplifier
introduces noise and leads to a poor signal-to-noise ratio. In order to improve the previous
experiment, a recirculating scheme as shown in Fig. 2.18 is proposed. In experiment, the first
sidebands can be reamplified through the circulating loop. This increases the length of the
delay line efficiently. It must be mentioned that this technique is not suitable for measuring
linewidth since the output optical signal contains different circulation orders. The method
proposed by M. Han (Chen et al., 2006; Han et al., 2005) is more suitable method for
linewidth measurement.
Optical Spectral Structure and Frequency Coherence
621
Delay length (km)
0
0.1
0.5
1.8
10
Power
ratio
(dB)
7.3 4.6 3.0 2.7 0.8
10-dBlinewidth
(MHz) 1.3 3.2 5.0 5.5 5.7
Table 1. Power ratio between δ-peak and Lorentzian component, and 10-dB linewidth
VNA
Ma
M ch-Z
ch- e
Z h
e n
h der
n
f
der ilter
t
10%
Modulator
Modulato
Modulat
3dB
3dB
Is
I olat
ol or
at
90%
Isol
Is a
ol t
a or
or
80%
3dB
EDFA
Fiber
Fi
D
ber e
D lay
Spectrum
90%
Tunable
20%
analyz
y er
las
la e
s r
Optica
Op
l spec
tica
t
l spec rum
10%
analyz
y er
Fig. 2.18. Experimental configuration of optical self-heterodyne scheme with fiber delay loop
for lineshape analysis
The peak at the modulation frequency gets wider when the recirculating scheme in Fig. 2.18
is used. For this scheme, the output optical signal comprises higher circulation-order
sidebands. Thus, the beat may occur between the carrier and sidebands with a delay time in
a wide time period, in which the wavelength may shift. That means that the carrier and the
delayed sidebands are launched out from the lightwave source at different time. In this
relatively long delay time, the optical wavelength may shift due to the instability of laser
source. Consequently, broadening of the measured power spectra reveals wavelength
stability of the lightwave source in the observation period. Therefore, the measured optical
spectral linewidth depends on the observation time due to the instability of laser.
6. Narrow-linewidth microwave generation
Optical generation of frequency-tunable, narrow-linewidth, and stable microwave signals is
desirable for many applications such as in radar, wireless communications, and satellite
communication systems. Conventionally, a microwave signal can be generated in the optical
domain using optical heterodyning, in which two optical waves of different wavelengths
beat at a photodetector (PD). An electrical beat note is then generated at a PD, and its
frequency depends on the wavelength spacing of the two optical waves (Gliese et al., 1998).
This method is capable of generating microwave and millimeter wave signals. The only
frequncy limit is the bandwidth of the PD. However, the beating of two optical waves from
two independent optical sources would lead to a microwave with unstable frequency and
high phase noise since there is no frequency coherence between them.
In the previous sections, it has been shown that the linewidth of the generated microwave
signal depends only on the frequency coherence properties of the two lightwaves, not on the
spectral linewidths of the individual light beams. To generate a pure microwave signal, two
optical waves used for heterodyning must be highly frequency coherent(Zhu et al., 2009).
This section presents two typical approaches to obtaining two frequency coherent
622
Optoelectronic Devices and Properties
lightwaves. One is to make the light correlated to the light emitted in the past time from the
same active region. Another way is using two correlated lightwave sources, such as two
lasers with mutual injection or two monolithically integrated lasers.
6.1 Microwave generation using a self-injected DBR laser
It has been mentioned that the linewidth of the beat note between two light beams depends
on the frequency coherence, not on the spectral linewidths of the two beams. Therefore, we
can determine the frequency coherence of two light beams from the linewidth of their beat
note.