g
Fig. 2.5. Optical spectral structure model of a semiconductor laser after intensity modulation
608
Optoelectronic Devices and Properties
The typical frequency responses using a long optical fiber and two chirped fiber gratings are
plotted in Fig. 2.4(b), and the corresponding variations of the beat intensity with path
difference are shown in Fig. 2.4(c). When the path difference of the “two arms” reaches 20
cm, the average beat note decreases by 3 dB. During the observation time, the changes in
modulation frequency and the total fiber length can be neglected since the path difference is
less than 25 cm after 150 km propagation. The interference is extremely stable, and the
signal to noise ratio can be significantly improved because the wavelength and the
polarization of the carrier and the sidebands changes in the same direction, and the two
waves propagate in the same medium. Thus, this scheme is simple and suitable for the
spectral analysis of a laser with relatively short coherence length.
Based on the above experimental results, we propose an optical spectral structure model as
shown in Fig. 2.5, where only the carrier and the first sidebands are shown for simplicity.
This model can be summarized as follows: 1) the spectrum of a laser consists of a large
number of wave trains. Wave train is not monochromatic and its spectral linewidth is
extremely narrow (less than 1 mHz); 2) wave trains are neither identical nor of simple form,
and they have variable lengths; 3) wave trains emitting simultaneously will have random
frequency spacings, and a wave train can seed another wave train with the same frequency.
3. Frequency coherence
This section presents a new concept of frequency coherence to describe the field correlations
between two lightwaves with different frequencies. This concept is different from other
well-known coherence concepts, such as temporal coherence, spatial coherence, polarization
coherence, quantum coherence, and spectral coherence. Spatial coherence and temporal
coherence describe the correlations between beams at different space points and different
moments, respectively. Our frequency coherence describes the field correlation between two
lightwaves in the frequency-time domain. This model gives a straightforward illustration of
optical spectral structure, which is helpful for understanding the spectral structure of
semiconductor laser.
3.1 Basic concept of optical coherence
In the broadest sense, optical coherence theory is concerned with the statistical description
of the fluctuations of optical fields. Interference is a typical phenomenon that reveals
correlation between light beams. Temporal and spatial coherence have been extensively
studied in the past come of Michelson and Young’s interference experiments, respectively.
In a typical interferometer, a beam of lightwave is split into two beams, and the two light
beams are recombined together with different delay times. The two beams are perfectly
coherent when the lengths of the two paths are identical. For a certain delay difference, the
degree of coherence depends on the linewidth and wavelength stability of the light beam.
Spatial coherence describes the correlation between signals at different points in space.
Temporal coherence describes the correlation between signals observed at different
moments. There are other concepts on coherence in accordance with different physical
parameters, such as polarization coherence, quantum coherence, and spectral coherence.
Spectral correlation, which is not so widely used as temporal and spatial coherence,
describes the correlation that exists between the spectral components at a given frequency in
the light oscillations at two points in a stationary optical field (Mandel & Wolf, 1976).
Optical Spectral Structure and Frequency Coherence
609
3.2 Description in three-dimensional space
The coherence properties of two beams should be described in three-dimensional spaces.
Take coherence in space-time domain for example, the three dimensions are distance, time
delay, and amplitude. The complex degree of coherence is traditionally used to characterize
correlations in stationary fields. It is defined as normalized cross-correlation function of the
optical fields at two points. In the following, we briefly introduce the theory of the complex
degree of coherence in space-time domain (Mandel & Wolf, 1965).
Suppose that V(r1, t) and V(r2, t) are the analytic signal representations of the light
oscillations at two points with position vectors r1 and r2, the complex degree of coherence
can be expressed as
1/2
γ ( r , r ,τ ) = (
Γ r , r ,τ ) /[ I( r ) I( r )] (2)
1
2
1
2
1
2
where Γ(r1,r2,τ)= <V*(r1,t)V(r2,t+τ)> is the mutual coherence of the light, and I(r) = Γ(r,r,0) is the average intensity of the light. For all possible values, 0≤ γ(r1,r2,τ)≤1.
If the light is quasimonochromatic, the visibility of fringes at position P(r) on the
interference screen is
I ( r) − I ( r)
υ( r)
max
min
=
= γ ( r , r ,τ ) (3)
1
2
12
I
r + I
r
max ( )
min ( )
which means that |γ| is a measure of the sharpness of the interference fringes. The complex
degree of spectral coherence in the frequency-time domain (Mandel & Wolf, 1976) is given
by
1/2
μ( r , r ,ν ) = W( r , r ,ν ) /[ W( r , r ,ν ) W( r , r ,ν )] (4) 1
2
1
2
1
1
2
2
Where W(r1,r2,ν) is the cross-spectral density function (also known as the cross power
spectrum) of the two optical fields. Therefore, the coherence properties in the space-time
domain depend on position and on the delay time, and the coherence properties in the
space-frequency domain depend on position and on the frequency of the light.
3.3 Concept of frequency coherence
It is desirable to give general description of the field correlations between two lightwaves
with different frequencies, and to investigate the related phenomena and their applications.
Optical heterodyne technique using two lightwaves with different wavelengths has been
widely used to generate microwave and millimeter waves. The spectral characteristics and
correlations of the two lightwaves are critical in obtaining a stable and narrow-linewidth
microwave signal. Referring to the concept of spatial and temporal coherence in the space-
time domain and the spectral coherence in the space-frequency domain, we introduce a new
concept of frequency coherence in the frequency-time domain, which describes the field
correlation between two lightwaves with different frequencies at a given moment.
Given that two arbitrary beams a and b overlapped in wavelength, the optical fields at
optical frequencies ω 1 and ω 2 can be expressed as
− jϕ
− jϕ
− jω t
a
b
1
E (ω , t) = [ E f (ω , t) e
+ E f (ω , t) e
] e
(5)
1
1
A a
1
B b
1
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Optoelectronic Devices and Properties
− jϕ
− jϕ
− jω t
a
b
2
E (ω , t) = [ E f ω t e
+ E f ω t e
e
A a (
, )
B b (
, )
]
2
2
2
2
(6)
EA and EB are the maximal optical fields of beams a and b, respectively. fa(ω,t) and fb(ω,t) are the normalized power spectrum profiles and meet
+∞
2
∫ f ω t dω =
x = a b (7)
x (
, )
1,
,
0
The photocurrent could be written as
1
1
2
2
2
2
i( t) ∝ E f (ω , t) + E f (ω , t) + cosφ E E f (ω , t) f (ω , t)cos ϕ
Δ
(8)
A a
1
B b
1
A B a
1
b
1
2
2
1 2 2
1 2 2
+ E f ω t + E f ω t +
φ E E f ω t f ω t
ϕ
Δ
A a (
, )
B b (
, ) cos
A B a (
, ) b( , )cos
2
2
2
2
2
2
2
2
+ F(ω ) E f (ω , t) f (ω , t)cos(ω t) + F(ω ) E f (ω , t) f (ω , t)cos(ω t) m
A a
1
a
2
m
m
B b
1
b
2
m
+ cosφ F(ω ) E E f (ω , t) f (ω , t)cos(ω t + ϕ
Δ )
m
A B a
1
b
2
m
+ cosφ F(ω ) E E f (ω , t) f (ω , t)cos(ω t − ϕ
Δ )
m
A B b
1
a
2
m
where φ is the angle between the polarization directions of the two beams, F(ω b) is the
frequency response coefficient of the photodetector, and Δϕ =ϕ a-ϕ b is the phase difference
between the two beams . The first six terms in (8) represent the DC beat notes. The 7th and
8th terms indicate homodyne signals. Intensity noise exists at all frequencies simultaneously
for wide spectral optical source, and a simple method for the calibration of wide bandwidth
photoreceiver has been established (Eichen & Silletti, 1987). The 9th and 10th terms indicate
heterodyne signals. Collecting all the optical current components at frequency ωm from the
beat notes of the two beams, we have
∞
2
i
∝ ω
ω
ω
ω
ω
ω ( t)
F(
) E cos( t)∫ f ( , t) f ( , t) d
1
2
1
m
m
A
m
a
a
0
∞
2
+ F(ω E
ω t ∫ f ω t f ω t dω
m ) B cos( m )
b (
, ) b( , )
1
2
1
0
(9)
∞
+ cosφ F(ω ) E E cos(ω t + ϕ
Δ )∫ f (ω , t) f (ω , t) dω
m
A B
m
a
1
b
2
1
0
∞
+ cosφ F(ω E E
ω t − ϕ
Δ ∫ f ω t f ω t dω
m ) A B cos( m
) b( , ) a( , )
1
2
1
0
When two beams do not overlap, i.e., fa(ω 2,t)=0, and fb(ω 1,t)=0, (9) reduces to
∞
i
∝
φ ω
ω + ϕ
Δ
ω
ω
ω
ω ( t)
cos F(
) E E cos( t
)∫ f ( , t) f ( , t) d (10)
1
2
1
m
m
A B
m
a
b
0
When the light beams are optical carrier and the sidebands produced by optical intensity
modulation, ω m becomes the modulation frequency. The optical spectral distribution in the
frequency-time domain is shown in Fig. 2.5, and only the carrier and the first sidebands are
shown for simplicity. Based on the atom emission law and the time response of ultrafast
Optical Spectral Structure and Frequency Coherence
611
photonic crystal laser (Yariv, 1997), we can assume that the rising and decay times are not
identical. From our understanding of optical spectral distribution, the spectrum of a laser
consists of a large number of wave trains. It has been shown that the spectral linewidth of
wave trains is narrower than 1 mHz, and the linewidth of DFB laser used is ~ 16 MHz. Also,
the simultaneous emitted wave trains have random frequency spacings (Zhu et al., 2007).
Therefore, the probability of occurrence of two wave trains from the same lightwave source
with a frequency spacing ωm=ω2-ω1 is rather low. Thus, we can assume that
+∞
∫ f (ω, t) f (ω + ω , t d
) ω << 1,
ω ≠ 0 (11)
x
x
m
m
0
From the definition of the complex degree of coherence in the space-time and space-
frequency domains, the degree of frequency coherence of two beams in the frequency-time
domain can be expressed as
+∞
γ (ω ,ω , ) = ∫ (ω , ) (ω , ) ω, (12)
1
2 t
f
1 t f
2 t d
a
b
0
According to the definition of the power spectrum profile, 0≤|γ (ω 1, ω 2,t)| ≤1 for all possible values of γ.
The wave trains in the carrier and the corresponding wave trains in the sidebands appear
simultaneously, and their frequency intervals exactly equal the modulation frequency. Since
the carrier and the sidebands have identical intensity profile, polarization and phase, i. e.,
fb(ω 2,t)= fb(ω 1+ω m,t)= fa(ω 1,t), and considering the optical spectrum properties indicated by (7) and (11), we have γ (ω 1, ω 2,t)=1. Thus, (9) can be approximated as
i
∝ ω
ω + Δϕ
ω ( t)
F(
) E E cos( t
) (13)
m
m
A B
m
It means that the linewidth of the beat note at the modulation frequency is independent of
the optical spectral profile of the beams. In this case, the beat note at the modulation
frequency is much stronger than the beat notes at other frequencies which become random
noise. The carrier and its sidebands produced by intensity modulation are perfectly coherent
if there is no time delay. As the path difference increases the beat note at the modulation
frequency will decrease and the noise level will increase (Zhu et al., 2007). The carrier and
the sidebands may then become partially coherent.
It can be concluded that for the coherent beams the maximum intensity of the mixed beams
may exceed the sum of the intensities of the beams (Born & Wolf, 1999), and the beat note
between the two beams would have a linewidth much narrower than the sum of the beam
linewidths.
4. Experimental analysis of frequency coherence properties
In the above sections, we have presented the new hyperfine spectral structure of
semiconductor lasers and the theory of frequency coherence. In this section, frequency
coherence properties of different optical lightwaves are experimentally investigated using
optical heterodyne technique. The results indicate that the optical carrier and its intensity
612
Optoelectronic Devices and Properties
modulated sidebands are perfectly coherent and the longitudinal modes of an FP laser are
partially coherent (Zhu et al., 2009).
4.1 Modulated optical spectrum: perfectly coherent
For the beams which are coherent, the maximum intensity of the mixed beams may exceed
the sum of the intensities of the beams, and the linewidth of the beat note between the two
beams would be much narrower than the sum of the beam linewidths. These discrepancies
in intensity and linewidth are the key features in describing the coherence properties of
lightwaves. The following experiments were designed to analyze the optical and electrical
spectra of the beams and the corresponding beat notes.
Fig. 2.6 is the measurement setup. In the first experiment, five lightwave sources, which
have different spectral widths, are used, i.e., an amplified spontaneous emission (ASE)
lightwave source (an erbium-doped fiber amplifier without optical input signal), the same
ASE source together with a 28GHz optical filter and a 5GHz optical filter, an DFB laser with
a 16 MHz linewidth, and a tunable laser with a linewidth narrower than 100 kHz. These
lightwave sources are all modulated by a 15-GHz signal from a vector network analyzer
(VNA) through a LiNbO3 modulator with a bandwidth of 40-GHz. An optical fiber coupler
is used to split the output into two waves, and the optical spectrum is measured by an
optical spectrum analyzer, and a electrical spectrum analyzer (ESA) and a high-speed
photodetector is used to measure the power spectrum. The spectrum is recorded using the
“Max Hold” function of the ESA for several seconds. The peak power of the beat note is kept
the same for different lightwave sources.
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