Progress In Electromagnetics Research, PIER 73, 249–275, 2007
NONLINEAR EFFECTS IN OPTICAL FIBERS: ORIGIN,
MANAGEMENT AND APPLICATIONS
S. P. Singh
†
and N. Singh
Department of Electronics and Communication
University of Allahabad
Allahabad-211002, India
Abstract—The nonlinear effects in optical fiber occur either due to
intensity dependence of refractive index of the medium or due to
inelastic-scattering phenomenon. This paper describes various types
of nonlinear effects based on first effect such as self-phase modulation,
cross-phase modulation and four-wave mixing. Their thresholds,
managements and applications are also discussed; and comparative
study of these effects is presented.
1. INTRODUCTION
The terms linear and nonlinear (Figure 1), in optics, mean intensityindependent and intensity-dependent phenomena respectively. Nonlinear effects in optical fibers (Table 1) occur due to (1) change in the
refractive index of the medium with optical intensity and, (2) inelasticscattering phenomenon. The power dependence of the refractive index
is responsible for the Kerr-effect. Depending upon the type of input signal, the Kerr-nonlinearity manifests itself in three different effects such as Self-Phase Modulation (SPM), Cross-Phase Modulation
(CPM) and Four-Wave Mixing (FWM). At high power level, the inelastic scattering phenomenon can induce stimulated effects such as Stimulated Brillouin-Scattering (SBS) and Stimulated Raman-Scattering
(SRS). The intensity of scattered light grows exponentially if the incident power exceeds a certain threshold value. The difference between
Brillouin and Raman scattering is that the Brillouin generated phonons
(acoustic) are coherent and give rise to a macroscopic acoustic wave
in the fiber, while in Raman scattering the phonons (optical) are incoherent and no macroscopic wave is generated.
†
Also with Physics Department, KNIPSS, Sultanpur, U.P., India.
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Singh and Singh
Linear
Output
Nonlinear
Input
Figure 1. Linear and nonlinear interactions.
Table 1. Nonlinear effects in optical fibers.
Nonlinear Effects in Optical Fibers
Nonlinear Refractive
Inelastic Scattering
Index Effects
SPM
CPM
Effects
FWM
SRS
SBS
Except for SPM and CPM, all nonlinear effects provide gains to
some channel at the expense of depleting power from other channels.
SPM and CPM affect only the phase of signals and can cause spectral
broadening, which leads to increased dispersion. Due to several recent
events, the nonlinear effects in optical fibers are an area of academic
research [1–4, 15, 17–20].
Progress In Electromagnetics Research, PIER 73, 2007
251
(i) Use of single mode fiber (SMF) with small cross section of lightcarrying area has led to increased power intensity inside the fiber.
(ii) Use of in-line optical amplifiers has resulted in a substantial
increase in the absolute value of the power carried by a fiber.
(iii) The deployment of multiwavelength systems together with optical
amplifier.
(iv) The deployment of high-bit-rate (>10 Gbits/s per channel)
systems.
This paper is organized as follows:
The basics of nonlinear effects are discussed in Section 2. Selfphase modulation, cross-phase modulation and four-wave mixing are
described in Sections 3, 4 and 5 respectively. Their thresholds,
managements and applications are also given in these sections. These
effects are compared in Section 6. Finally, conclusion is presented in
Section 7.
2. BASICS
For intense electromagnetic fields, any dielectric medium behaves like
a nonlinear medium. Fundamentally, origin of nonlinearity lies in
anharmonic motion of bound electrons under the influence of an
applied field. Due to this anharmonic motion the total polarization
P induced by electric dipoles is not linear but satisfies more general
relation as
P = ε0 χ(1) E + ε0 χ(2) E 2 + ε0 χ(3) E 3 + · · ·
(1)
where ε0 is the permittivity of vacuum and χ(k) (k = 1, 2, . . .) is kth
order susceptibility.
The dominant contribution to P is provided by linear susceptibility χ(1) . The second order susceptibility χ(2) is responsible for secondharmonic generation and sum-frequency generation. A medium, which
lacks inversion symmetry at the molecular level, has non-zero second
order susceptibility. However for a symmetric molecule, like silica, χ(2)
vanishes. Therefore optical fibers do not exhibit second order nonlinear refractive effects. It is worth to mention here that, the electricquadrupole and magnetic-dipole moments can generate weak second
order nonlinear effects. Defects and color centers inside the fiber core
can also contribute to second harmonic generation under certain conditions. Obviously the third order susceptibility χ(3) is responsible for
lowest-order nonlinear effects in fibers [5].
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Singh and Singh
For isotropic medium, like optical fiber, polarization vector P will
always be in direction of electric field vector E. So one may use scalar
notations instead of vector notations. For an electric field,
E = E0 cos(ωt − kz)
(2)
the polarization P becomes
P = ε0 χ(1) E0 cos(ωt − kz) + ε0 χ(2) E02 cos2 (ωt − kz)
+ε0 χ(3) E03 cos3 (ωt − kz) + · · ·
(3)
Using some trigonometric relations, equation (3) can be written as
3
1
ε0 χ(2) E02 + ε0 χ(1) + χ(3) E02 E0 cos(ωt − kz)
2
4
1
1
+ ε0 χ(2) E02 cos 2(ωt − kz) + ε0 χ(3) E03 cos 3(ωt − kz) + · · · (4)
2
4
The effect of first term is of little practical importance as it is a
constant term and gives a dc field across the medium. The second term
oscillating at frequency ω is known as first or fundamental harmonic of
polarization. The third term oscillating with frequency 2ω is called the
second harmonic of polarization. Similarly fourth term with frequency
3ω is known as third harmonic of polarization. For optical fibers, χ(2)
vanishes, and hence equation (4) becomes
P =
1
3
P = ε0 χ(1) + χ(3) E02 E0 cos(ωt − kz) + ε0 χ(3) E03 cos 3(ωt − kz) (5)
4
4
Here higher order terms are neglected because their contribution is
negligible. Due to variations in refractive index of the fiber there is lack
of phase between frequencies ω and 3ω. Due to this phase mismatch
the second term of equation (5) can be neglected and polarization can
be written as
3
(6)
P = ε0 χ(1) E0 cos(ωt − kz) + ε0 χ(3) E03 cos(ωt − kz)
4
This equation contains both linear (first term) and nonlinear
(second term) polarizations. For a plane wave represented by equation
(2), the intensity (I) is defined as,
1
(7)
I = cε0 nl E02
2
where c is velocity of light and nl is linear refractive index of the
medium at low fields. Hence,
P = ε0 χ(1) +
3 χ(3)
IE0 cos(ωt − kz)
2 cε0 nl
(8)
Progress In Electromagnetics Research, PIER 73, 2007
253
2.1. Effective Susceptibility and Effective Refractive Index
The effective susceptibility (χef f ) of the medium is defined as,
χef f =
P
3 χ(3)
I
= χ(1) +
ε0 E
2 cε0 nl
(9)
Therefore, effective refractive index (nef f ) can be written as
1
nef f = (1 + χef f ) 2
or
nef f =
(1)
1+χ
3 χ(3)
I
+
2 cε0 n2l
1
2
(10)
The last term is usually very small even for very intense light beam.
Hence above expression for nef f can be approximated with help of
Taylor’s series expansion as
nef f = nl +
3 χ(3)
I
4 cε0 n2l
(11)
or
nef f = nl + nnl I
(12)
1
2
In equation (12) first term [nl = (1 + χ(3) ) ] is linear refractive index
(3)
and second term (nnl = 34 cεχ n2 ) is nonlinear refractive index. Higher
0 l
order terms are negligible and hence neglected.
For fused silica fibers nl ≈ 1.46 and nnl ≈ 3.2 × 10−20 m2 /W.
For the propagation of a mode carrying 100 mW of power in a single
mode fiber with an effective mode area ≈ 50 µm2 , resultant intensity
is 2 × 109 W/m2 and the change in refractive index due to nonlinear
effect is,
∆n = nnl I ≈ 6.4 × 10−11
Although, this change in refractive index is very small, but due to
very long interaction length (10–10,000 km) of an optical fiber, the
accumulated effects (nonlinear) become significant. It is worth to
mention that, this nonlinear term is responsible for the formation of
solitons.
2.2. Effective Transmission Length
The nonlinear effects depend on transmission length. The longer
the fiber link length, the more the light interaction and greater the
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Singh and Singh
nonlinear effect. As the optical beam propagates along the link length,
its power decreases because of fiber attenuation. The effective length
(Lef f ) is that length, up to which power is assumed to be constant [6].
The optical power at a distance z along link is given as,
P (z) = Pin exp(−αz)
(13)
where Pin is the input power (power at z = 0) and α is coefficient of
attenuation. For a actual link length (L), effective length is defined as,
(Figure 2)
Pin Lef f =
L
P (z)dz
(14)
z=0
Using equations (13) and (14), effective link length is obtained as,
Lef f =
(1 − exp(−αz))
α
(15)
Since communication fibers are long enough so that L ≫ 1/α. This
results in Lef f ≈ 1/α.
Launched power (Pin)
Power
Real power distribution
Leff
Link length
Figure 2. Definition of the effective length.
Progress In Electromagnetics Research, PIER 73, 2007
255
In optical systems with optical amplifiers, the signal gets
amplified at each amplifier stage without resetting the effects due to
nonlinearities from previous span. Obviously the effective length in
such systems is sum of the effective length of each span. In a link of
length L with amplifiers spaced l distance apart, the effective length is
approximately given by,
Lef f =
(1 − exp(−αz)) L
α
l
(16)
Pin Leff
10000
1000
100
10
1
50
100
150
200
250
Amplifier Spacing (Km)
Figure 3. Relative value of Pin Lef f with respect to amplifier spacing.
The ordinate is the value relative to an amplifier spacing of 1 km. And
attenuation coefficient α = 0.22 dB/km.
The Figure 3 shows how Pin Lef f grows with amplifiers spacing (l).
It is clear from this figure that effects of nonlinearities can be reduced
by reducing the amplifier spacing.
2.3. Effective Cross-sectional Area
The effect of nonlinearity grows with intensity in fiber and the intensity
is inversely proportional to area of the core. Since the power is
not uniformly distributed within the cross-section of the fiber, it is
reasonable to use effective cross-sectional area (Aef f ). The Aef f is
related to the actual area (A) and the cross-sectional distribution of
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Singh and Singh
intensity I(r, θ) in following way [6],
rdrdθI(r, θ)
r θ
Aef f =
2
rdrdθI (r, θ)
(17)
r θ
where r and θ denote the polar coordinates.
definition of effective area (Aef f ).
Figure 4 provides
Launched power (Pin)
Power
Real power distribution
√Aeff/π
Radius
Figure 4. Definition of effective core area.
3. SELF-PHASE MODULATION (SPM)
The higher intensity portions of an optical pulse encounter a higher
refractive index of the fiber compared with the lower intensity portions
while it travels through the fiber. In fact time varying signal intensity
produces a time varying refractive index in a medium that has an
intensity-dependant refractive index. The leading edge will experience
a positive refractive index gradient (dn/dt) and trailing edge a negative
refractive index gradient (−dn/dt). This temporally varying index
change results in a temporally varying phase change, as shown in
Figure 5. The optical phase changes with time in exactly the same
way as the optical signal [7]. Since, this nonlinear phase modulation
Progress In Electromagnetics Research, PIER 73, 2007
257
Optical power
Front
Back
+ dn/dt
-dn/dt
-2
-1
1
0
2
time
dφ/dt
-2
Frequency chirp
-1
0
1
2
time
Figure 5. Phenomenological description of spectral broadening of
pulse due to SPM.
is self-induced the nonlinear phenomenon responsible for it is called as
self-phase modulation.
Different parts of the pulse undergo different phase shift because
of intensity dependence of phase fluctuations. This results in frequency
chirping. The rising edge of the pulse finds frequency shift in upper side
whereas the trailing edge experiences shift in lower side. Hence primary
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effect of SPM is to broaden the spectrum of the pulse [8], keeping
the temporal shape unaltered. The SPM effects are more pronounced
in systems with high-transmitted power because the chirping effect is
proportional to transmitted signal power.
The phase (φ) introduced by a field E over a fiber length L is
given by
2π
nL
(18)
φ=
λ
where λ is wavelength of optical pulse propagating in fiber of refractive
index n, and nL is known as optical path length.
For a fiber containing high-transmitted power n and L can be
replaced by nef f and Lef f respectively i.e.,
φ=
2π
nef f Lef f
λ
or
2π
(19)
(nl + nnl I)Lef f
λ
The first term on right hand side refers to linear portion of phase
constant (φl ) and second term provides nonlinear phase constant (φnl ).
If intensity is time dependent i.e., the wave is temporally
modulated then phase (φ) will also depend on time [9]. This variation
in phase with time is responsible for change in frequency spectrum,
which is given by
dφ
(20)
ω=
dt
In a dispersive medium a change in the spectrum of temporally
varying pulse will change the nature of the variation. To observe
this, consider a Gaussian pulse, which modulates an optical carrier
frequency ω (say) and the new instantaneous frequency becomes,
φ=
ω ′ = ω0 +
dφ
dt
(21)
The sign of the phase shift due to SPM is negative because of the minus
sign in the expression for phase, (ωt − kz) i.e.,
φ=−
2π
Lef f (nl + nnl I)
λ
And therefore ω becomes,
ω ′ = ω0 −
dI
2π
Lef f nnl
λ
dt
(22)
Progress In Electromagnetics Research, PIER 73, 2007
Clearly at leading edge of the pulse
dI
dt
> 0 hence
ω ′ = ω0 − ω(t)
And at trailing edge
dI
dt
259
(23)
< 0 so,
ω ′ = ω0 + ω(t)
(24)
where,
dI
2π
Lef f nnl
(25)
λ
dt
This shows that the pulse is chirped i.e., frequency varies across the
pulse. This chirping phenomenon is generated due to SPM, which
leads to the spectral broadening of the pulse. Figures 6 and 7 show
the variation of I(t) and dI/dt for a Gaussian pulse.
ω(t) =
I(t)
0
t
Figure 6. For a pulse with intensity varying as function of time.
There is broadening of the spectrum without any change in
temporal distribution in case of self-phase modulation while in case
of dispersion, there is broadening of the pulse in time domain and
spectral contents are unaltered. In other words, the SPM by itself leads
only to chirping, regardless of the pulse shape. It is dispersion that is
responsible for pulse broadening. The SPM induced chirp modifies the
pulse broadening effects of dispersion.
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dI(t)/dt
0
t
0
Figure 7. For a pulse with dI/dt varying as function of time.
3.1. Thresholds and Management
SPM arises due to intensity dependence of refractive index.
Fluctuation in signal intensity causes change in phase of the signal.
This change in phase induces additional chirp, which leads to dispersion
penalty. This penalty will be small if input power is less than certain
threshold value. The appropriate chirping of the input pulses can also
be beneficial for reducing the SPM effects. For this, chirped RZ or
CRZ modulation can be adopted.
The power dependence of nonlinear phase constant (φnl ) is
responsible for SPM impact on communication systems [5, 20]. To
reduce this impact, it is necessary to have φnl ≪ 1. Nonlinear phase
constant (φnl ) can be written as
φnl = knl Pin Lef f
2π nnl
λ Aef f .
Pin ≪ kαnl .
where nonlinear propagation constant knl =
So, with Lef f ≈ α1 ; one may obtain,
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261
Therefore, to have φnl ≪ 1 is equivalent to Pin ≪ kαnl .
Typically α = 0.2 dB/km at λ = 1550 nm and knl = 2.35 ×
10−3 1/mW. The input power should be kept below 19.6 mW.
The move to increase the span between in-line optical amplifiers,
more power must be launched into each fiber. This increased power
increases SPM effect on lightwave systems, which results in pulse
spreading. The use of large-effective area fibers (LEAF) reduces
intensity inside the fiber and hence SPM impact on the system.
The chirp produced by SPM, which causes broadening [10],
depends on the input pulse shape and the instantaneous power level
within the pulse. For Gaussian shaped pulse, the chirp is even and
gradual, and for a pulse that involves an abrupt change in power level
(e.g., square pulse) the amount of chirp is greater. Therefore a suitable
input pulse shape may be able to reduce the chirp and hence SPM
induced broadening.
In general, all nonlinear effects are weak and depend on long
interaction length to build up. So any mechanism that reduces
interaction length decreases the effect of non-linearity. The damage
due to SPM-induced pulse broadening on system performance depends
on the power transmitted and length of the link. An estimation of this
is shown in Figure 8, which shows that pulse can be twice as wide at
the end of 200 km transmission as it was at the start.
The performance of self-phase modulation-impaired system can
2.0
Relative phase
width (t(z)/to) 1.8
1.4
1.0
0.8
10mW
1mW
20mW
0.4
0
50
150
200
Link length (km)
Figure 8. Pulse spreading caused by SPM as a function of distance.
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Singh and Singh
be improved significantly by adjusting the net residual (NRD) of the
system. For SPM-impaired system the optimal NRD can be obtained
by minimizing the output distortion of signal pulse. The NRD of
SPM-impaired dispersion-managed systems can be optimized by a semi
analytical expression obtained with help of perturbation theory. This
method is verified by numerical simulations for many SPM-impaired
systems [21, 22].
3.2. Applications of SPM Phenomenon
Two important applications of SPM concept are in solitons and in
pulse compression.
3.2.1. Solitons
SPM leads to chirping with lower frequencies in the leading edge
and higher frequencies in the trailing edge. On the other hand the
chirping caused by linear dispersion, in the wavelength region above
zero dispersion wavelength, is associated with higher frequencies in
leading edge and lower frequencies in the trailing edge. Both these
effects are opposite. By proper choice of pulse shape (a hyperbolic
secant-shape) and the power carried by the pulse, one effect can
be compensated with the other. In such situation the pulse would
propagate undistorted by mutual compensation of dispersion and SPM.
Such a pulse would broaden neither in the time domain (as in linear
dispersion) nor in frequency domain (as in SPM) and is called soliton
[11, 18]. Since soliton pulse does not broaden during its propagation,
it has tremendous potential for applications in super high bandwidth
optical communication systems.
3.2.2. Pulse Compression
SPM phenomenon can be used in pulse compression. In the wavelength
region where chromatic dispersion is positive, the red-shifted leading
edge of the pulse travels slower and moves toward the center of pulse.
Similarly, the blue shifted trailing edge travels faster, and also moves
toward the center of the pulse. In this situation SPM causes the pulses
to narrow.
Another simple pulse compression scheme is based on filtering selfphase modulation-broadened spectrum [23].
Progress In Electromagnetics Research, PIER 73, 2007
263
3.2.3. Optically Tunable Delays
In ultra-high speed optical communications, the optical /electronic
conversion of information puts limit on transmission data rate.
Therefore, it is desirable to have all-optical components for buffering
and delaying signal pulses. Tunable all-optical delays are important for
application in telecommunication, optical coherence tomography and
optical sampling. There is a novel technique for all-optical delays which
involves spectral broadening via self-phase modulation and wavelength
filtering [24, 25]. Tunable delays of more than 4.2 ns for a 3.5-ps input
pulse is demonstrated by using this technique.
3.2.4. Optical 40 Gb/s 3R Regenerator
Combined effect of self-phase modulation and cross-absorption
modulation is utilized in all optical 3R regenerators [26]. The
performance of such regenerators is experimentally verified for 40 Gb/s
data rate. The introduction of a predistortion block configuration
including a highly non-linear fiber enhances the chromatic dispersion
tolerance.
4. CROSS PHASE MODULATION (CPM)
SPM is the major nonlinear limitation in a single channel system. The
intensity dependence of refractive index leads to another nonlinear
phenomenon known as cross-phase modulation (CPM). When two
or more optical pulses propagate simultaneously, the cross-phase
modulation is always accompanied by SPM and occurs because the
nonlinear refractive index seen by an optical beam depends not only
on the intensity of that beam but also on the intensity of the other
copropagating beams [13]. In fact CPM converts power fluctuations
in a particular wavelength channel to phase fluctuations in other
copropagating channels. The result of CPM may be asymmetric
spectral broadening and distortion of the pulse shape.
The effective refractive index of a nonlinear medium can be
expressed in terms of the input power (P ) and effective core area (Aef f )
as,
P
(26)
nef f = nl + nnl
Aef f
The nonlinear effects depend on ratio of light power to the crosssectional area of the fiber. If the first-order perturbation theory is
applied to investigate how fiber modes are affected by the nonlinear
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Singh and Singh
refractive index, it is found that the mode shape does not change but
the propagation constant becomes power dependent.
kef f = kl + knl P
(27)
where kl is the linear portion of the propagation constant and knl is
nonlinear propagation constant. The phase shift caused by nonlinear
propagation constant in traveling a distance L inside fiber is given as
φnl =
L
(kef f − kl )dz
(28)
0
Using equations (27) and (14) nonlinear phase shift becomes,
φnl = knl Pin Lef f
(29)
When several optical pulses propagate simultaneously the
nonlinear phase shift of first channel φ1nl (say) depends not only on
the power of that channel but also on signal power of other channels.
For two channels, φ1nl can be given as,
φ1nl = kef f Lef f (P1 + 2P2 )
(30)
For N -channel transmission system, the shift for ith channel can be
given as [8],
N
φinl
= knl Lef f Pi + 2
n=i
Pn
(31)
The factor 2 in above equation has its origin in the form of
nonlinear susceptibility [5] and indicates that CPM is twice as effective
as SPM for the same amount of power. The first term in above equation
represents the contribution of SPM and second term that of CPM. It
can be observed that CPM is effective only when the interacting signals
superimpose in time.
CPM hinders the system performance through the same
mechanism as SPM: chirping frequency and chromatic dispersion, but
CPM can damage the system performance even more than SPM.
CPM influences the system severely when number of channels is large.
Theoretically, for a 100-channels system, CPM imposes a power limit
of 0.1 mW per channel.
4.1. Thresholds and Management
The CPM-induced phase shift can occur only when two pulses overlap
in time. Due to this overlapping, the intensity-dependent phase shift
Progress In Electromagnetics Research, PIER 73, 2007
265
and consequent chirping is enhanced. Therefore the pulse broadening is
also enhanced, which limits the performance of lightwave systems. The
effects of CPM can be reduced by increasing the wavelength spacing
between individual channels. For increased wavelength spacing, pulse
overlaps for such a short time that CPM effects are virtually negligible.
In fact, owing to fiber dispersion, the propagation constants of these
channels become sufficiently different so that the pulses corresponding
to individual channels walk away from each other. Due to this
pulse walk-off phenomenon the pulses, which were initially temporally
coincident, cease to be so after propagating for some distance and
cannot interact further. Thus, effect of CPM is reduced.
In a WDM system, CPM converts power fluctuations in
a particular wavelength channel to phase fluctuations in other
copropagating channels. This leads to broadening of pulse. It can
be greatly mitigated in WDM systems operating over standard nondispersion shifted single mode fiber [14, 20]. One more advantage of
this kind of fiber is its effective core area, which is typically 80 µm2 .
This large effective area is helpful in reducing nonlinear effects because
knl is inversely proportional to Aef f .
Like SPM, the CPM also depends on interaction length of fiber.
The long interaction length is always helpful in building up this effect
up to a significant level. Keeping inter action length small, one can
reduce this kind of nonlinearity.
4.2. Applications of CPM Phenomenon
Optical switching and pulse compression can be done through the CPM
phenomenon.
4.2.1. Optical Switching
Phase shift, in an optical pulse, due to CPM phenomenon can be
used for optical switching. To take advantage of CPM-induced phase
shift for ultra-fast optical switching many interferometric methods
have been used [5]. Consider a interferometer designed in such a
way that a weak signal pulse, divided equally between its two arms,
experiences identical phase shifts in each arm and is transmitted
through constructive interference. When a pump pulse at different
wavelength is injected into one of the arms, it will change the signal
phase through CPM phenomenon in that arm. If the CPM-induced
phase shift is large (close to π), this much phase shift results in
destructive interference and hence no transmission of signal pulse.
Thus an intense pump pulse can switch the signal pulse.
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4.2.2. Pulse Compression
Like SPM induced frequency chirp, the CPM induced frequency chirp
can also be used for pulse compression. The SPM techniques require
the input pulse to be intense and energetic, but the CPM is able to
compress even weak input pulses because copropagating intense pump
pulse produces the frequency chirp. The CPM induced chirp is affected
by pulse walk-off and depends critically on the initial relative pumpsignal delay. As a result the use of CPM induced pulse compression
requires a careful control of the pump pulse parameters such as its
width, peak power, wavelength and initial delay relative to the signal
pulse.
4.2.3. Pulse Retiming
In an anomalous-dispersion polarization-maintained fiber ultra-fast
optical pulses can be retimed by utilizing cross-phase modulation
phenomenon. With help of this phenomenon spectral, temporal and
spatial properties of ultra-short pulses can be controlled [27, 28].
5. FOUR-WAVE MIXING (FWM)
The origin of FWM process lies in the nonlinear response of bound
electrons of a material to an applied optical field. In fact, the
polarization induced in the medium contains not only linear terms but
also the nonlinear terms. The magnitude of these terms is governed
by the nonlinear susceptibilities of different orders. The FWM process
originates from third order nonlinear susceptibility (χ(3) ). If three
optical fields with carrier frequencies ω1 , ω2 and ω3 , copropagate inside
the fiber simultaneously, (χ(3) ) generates a fourth field with frequency
ω4 , which is related to other frequencies by a relation, ω4 = ω1 ±ω2 ±ω3 .
In quantum-mechanical context, FWM occurs when photons from
one or more waves are annihilated and new photons are created at
different frequencies such that net energy and momentum are conserved
during the interaction.
SPM and CPM are significant mainly for high bit rate systems,
but the FWM effect is independent of the bit rate and is critically
dependant on the channel spacing and fiber dispersion. Decreasing
the channel spacing increases the four-wave mixing effect and so does
decreasing the dispersion.
In order to understand the FWM effect [6], consider a WDM
signal, which is sum of n monochromatic plane waves. The electric
Progress In Electromagnetics Research, PIER 73, 2007
267
field of such signal can be written as
n
E=
Ep cos(ωp t − kp z)
(32)
p=1
Then the nonlinear polarization is given by
Pnl = ε0 χ(3) E 3
(33)
For this case Pnl takes the form as
n
n
n
(3)
Ep cos(ωp t−kp z)Eq cos(ωq t−tq z)Er cos(ωr t−kr z)
Pnl = ε0 χ
p=1 q=1 r=1
(34)
Expansion of above expression gives,
n
3
Ep Eq Ep cos(ωp t − tp z)
Pnl = ε0 χ(3)
Ep2 + 2
4
p=1
q=p
n
1
Ep3 cos(3ωp t − 3kp z)
+ ε0 χ(3)
4
p=1
n
3
+ ε0 χ(3)
Ep2 Eq cos{(2ωp − ωq )t − (2kp − kq )z}
4
p=1 q=p
n
3
+ ε0 χ(3)
Ep2 Eq cos{(2ωp + ωq )t − (2kp + kq )z}
4
p=1 q=1
n
6
+ ε0 χ(3)
Ep Eq Er cos{(2ωp +ωq + ωr )t−(kp +kq +kr )z}
4
p=1 q>p r>q
+ cos{(ωp + ωq + ωr )t − (kp + kq + kr )z}
+ cos{(ωp − ωq + ωr )t − (kp − kq + kr )z}
+ cos{(ωp − ωq − ωr )t − (kp − kq − kr )z}
(35)
The first terms in above equation represents the effect of SPM
and CPM. Second, third and fourth terms can be neglected because
of phase mismatch. The reason behind this phase mismatch is
that, in real fibers k(3ω) = 3k(ω) so any difference like (3ω −
3k) is called as phase mismatch. The phase mismatch can also
be understood as the mismatch in phase between different signals
traveling within the fiber at different group velocities. All these
waves can be neglected because they contribute little. The last term
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represents phenomenon of four-wave mixing. It is this term, which
tells that three EM waves propagating in a fiber generate new waves
[16] with frequencies (ωp ± ωq ± ωr ). Four-wave mixing (FWM) is
analogous to intermodulation distortion in electrical systems. The
last term of polarization expression tells that FWM comes from
frequency combinations like (ωp + ωq − ωr ). In compact form all these
combinations can be written as
ωpqr = ωp + ωq − ωr
(2ω1-ω2)
with p, q = r
ω1
ω2
(36)
(2ω2-ω1)
Frequency (ω)
Figure 9. Showing mixing of two waves.
Figure 9 shows a simple example of mixing of two waves at
frequency ω1 and ω2 . When these waves mixed up, they generate
sidebands at (2ω1 − ω2 ) and (2ω2 − ω1 ). Similarly, three copropagating
waves will create nine new optical sideband waves at frequencies given
by equation (36). These sidebands travel along with original waves
and will grow at the expense of signal-strength depletion. In general
for N -wavelengths launched into fiber, the number of generated mixed
products M is,
M = N 2 /2 · (N − 1)
The efficiency FWM depends on fiber dispersion and the channel
spacing. Since the dispersion varies with wavelength, the signal waves
and the generated waves have different group velocities. This destroys
the phase matching of interacting waves and lowers the efficiency of
power transfer to newly generated frequencies. The higher the group
velocity mismatch and wider the channel spacing, the lower the fourwave mixing. This is shown in Figure 10. The curves show the
frequency-spacing range over which the FWM process is efficient for
Progress In Electromagnetics Research, PIER 73, 2007
269
100
Efficiency
(%)
80
(With 1-ps/nm.km dispersion in
1550 nm window)
DSF
60
40
20
SMF
(16-ps/nm-km
disperion in
1550nm window)
0
50
100
Channel separation (GHz)
Figure 10. Efficiency of four wave mixing with respect to channel
separation.
two dispersion values. It is clear that in conventional SMFs, frequencies
with separations less than 20 GHz will mix efficiently. But for DSFs,
FWM efficiencies are greater than 20% for separation upto 50 GHz.
5.1. Thresholds and Management
Four-wave mixing process results in power transfer from one channel
to other.
This phenomenon results in power depletion of the
channel, which degrades the performance of that channel (i.e., BER
is increased). In order to achieve original BER, some additional power
is required which is termed as power penalty. Since, FWM itself is
interchannel crosstalk it induces interference of information from one
channel with another channel. This interference again degrades the
system performance. To reduce this degradation, channel spacing must
be increased. This increases the group velocity mismatch between
channels and hence FWM penalty is reduced.
Four-wave mixing presents a severe problem in WDM systems
using dispersion-shifted fibers (DSF) [12]. Penalty due to FWM can
be reduced if a little chromatic dispersion is present in the fiber.
Due to chromatic dispersion, different interacting waves travel with
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Singh and Singh
different group velocities. This results in reduced efficiency of FWM
and hence penalty. A non-zero dispersion-shifted fiber is used for this
purpose. The FWM imposes limitations on the maximum transmit
power per channel. This limitation for system operating over standard
single-mode fiber (SMF) and dispersion-shifted fiber (DSF) is shown
in Figure 11.
100
SMF
10
Maximum
transmit
power per
channel
(mW)
1.0
DSF
0.1
0.01
1000
100
10000
Distance (km)
Figure 11. Maximum transmitted power per channel versus distance
imposed by FWM.
Like other nonlinear effects limitations of FWM on a communication system depend on the effective area (Aef f ), effective fiber length
(Lef f ) and, of course, on the intensity of transmitted signal. Using
NZ-DSF of large effective area and small effective length with reduced
transmitted signal power results in reduction of penalty due to FWM
process.
FWM produces severe limitations on performance of WDM alloptical networks. The number of FWM components increases with the
increase in number of users. If these generated wavelengths coincide
with the original signal wavelength, then it results in interference
causing degradation in signal-to-noise ratio (SNR). This effect can
be reduced by using modified repeated unequally spaced channel
allocation [29].
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271
5.2. Applications of FWM Process
Two important applications of FWM are squeezing and wavelength
conversion. These are described below.
5.2.1. Squeezing
The FWM process can be used to reduce quantum noise through
a phenomenon called squeezing. In fact squeezing is a process of
generating the special states of an electromagnetic field for which
noise fluctuations, in some frequency range, are reduced below the
quantum-noise level [5]. FWM can be used for squeezing as noise
components at the signal and idler frequencies are coupled through
the fiber nonlinearity.
Physically, squeezing can be understood as deamplification of
signal and idler waves for certain values of relative phase between
the two waves. Photons of random phases are generated due to
spontaneous emission at the signal and idler frequencies. Four-wave
mixing process increases or decreases the number of signal-idler photon
pairs depending on their relative phases. Noise is reduced below the
quantum-noise level when the phase of the local oscillator is adjusted
to match the relative phase corresponding to the photon pair,whose
number was reduced as a result of FWM process.
5.2.2. Wavelength Conversion
Four-wave-mixing phenomenon can be used effectively for wavelength
conversion too. The function of wavelength converter is to transform
information from one wavelength to another. A phenomenological
method of wavelength conversion is shown in Figure 12. When a data
input (λ1 ) and a probe signal (λ2 ) are injected into a nonlinear medium,
due to mixing process a new signal (λ3 ) is generated in association with
All signals at different
wavelengths
Data signal
(λ 1)
Nonlinear
Medium
Probe signal
(λ 2)
Filter
Converted
signal ( λ 3)
Figure 12. Phenomenological description of wavelength conversion
through FWM process.
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Singh and Singh
other signal wavelengths such that;
1
2
1
=
−
λ3
λ1 λ2
or
ω3 = 2ω1 − ω2
where ω is angular frequency.
The wavelength conversion is an important component in alloptical networks, since the wavelength of incoming signal may already
be in use by another information channel residing on the destined
outgoing path. Converting the incoming signal to new wavelength
will allow both information channels to traverse the same fiber
simultaneously.
Four-wave-mixing based wavelength conversion at 1.55 µm in
a 2.2 m long dispersion-shifted lead-silicate holy fiber has been
investigated [30]. It is shown that highly efficient and broadband
wavelength conversion, covering the entire C band, can be achieved
for such fibers at reasonable optical pump power.
Table 2. Comparison of nonlinear refractive effects.
Nonlinear
Phenomenon
CPM
SPM
Characteristics
FWM
1. Bit-rate
Dependent
Dependent
Independent
2. Origin
Nonlinear
susceptibility
χ (3)
Nonlinear
susceptibility
χ (3)
Nonlinear
susceptibility
χ (3)
3. Effects of χ (3)
Phase shift due to Phase shift is alone
pulse itself only due to copropagating
signals
Symmetrical
May be a symmetrical
New waves are
generated
5. Energy transfer
between m edium
and optical pulse
No
No
6. Channel spacing
No effect
4. Shape of
broadening
No
Increases on
decreasing the spacing
__
Increases on
decreasing the
spacing
Progress In Electromagnetics Research, PIER 73, 2007
273
6. COMPARISON OF DIFFERENT NONLINEAR
EFFECTS
Different nonlinear effects based on Kerr-effect are compared in
Table 2. The parameters taken are bit-rate, origin, effects of thirdorder susceptibility, shape of broadening, energy transfer between
medium and optical pulse and effect of channel spacing.
7. CONCLUSION
Nonlinear effects such as SPM, CPM, and FWM are discussed. These
effects degrade the performance of fiber optic systems. Impact of
SPM is negligible if power per channel is below 19.6 mW. FWM has
severe effects in WDM systems, which uses dispersion-shifted fiber. If
some dispersion is their, then effect of FWM is reduced. That is why
non-zero dispersion-shifted fibers are normally used in WDM systems.
Though these effects degrade nature, they are also useful for many
applications such as SPM in solitons and pulse compression, CPM in
optical switching, and FWM in squeezing and wavelength conversion.
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