Interaction between dispersion and SPM
Interactions between dispersion and SPM
The interaction between linear dispersion and the nonlinear SPM effect
results from the different velocities at which different parts of the
spectrally modified pulse travel. For D < 0 the pulse always broadens while
for D > 0, there may exists a balance between D and SPM so the pulse
width stays constant.
Due to D < 0
and SPM
Linear
broadening
Normal dispersion pulse broadens
Due to D > 0
and SPM
Linear
broadening
Anomalous dispersion pulse width constant - soliton
In order to quantify the interaction between dispersion and SPM we
define normalized parameters for the time and the distance
τ=
t
T0
ξ=
z
The field is then
LD
A = (ξ , τ ) = P0 exp( −αz )U (ξ , τ )
Recall that the evolution of the normalized pulse envelope is governed by the
NLSE
∂U
sg n ( β 2 ) ∂ 2U
ex p ( −α z )
⎧+1 for β2 > 0 (normal GVD)
2
+i
=
i
U
U
=
sgn(
β
)
⎨
2
∂z
∂τ 2
LNL
2 LD
⎩−1 for β2 < 0 (anomalous GVD)
Changing variable z → ξ and defining
Recalling:
LD =
T02
β2
dispersion length,
LD
γ P0T02
N =
=
LNL
β2
2
L NL =
1
nonlinear length
γ P0
∂U
s g n ( β 2 ) ∂ 2U
Leads to the NLSE
+i
= iN
2
∂ξ
2
∂τ
2
exp( −α z ) U
2
U
The parameter N governs the relative roles of SPM and GVD.
N<<1
Dispersion dominates
N>>1
SPM dominates
N~1
SPM and GVD play
an equal role
The parameter N has also an important physical meaning in determining
pulse broadening (for D < 0) and the type (order) of the soliton (for D > 0)
The solution of the NLSE for a specific N follows a scaling law:
for N=1 T0=1ps and P0=1mW, the results hold for T0= 0.1 ps and P0= 100mW
In other words, for a specific N, T02P0 = constant.
The combined effects of GVD and SPM
N << 1 – Dispersion dominates. An unchirped pulse broadens linearly
D<0
D>0
The chirp is linear
δ ω (T ) =
(
sgn ( β 2 ) 2 z
1+
(
z
LD
LD
)
2
)
t
T 02
Pulse broadening does not depend on
the sign of D
N >> 1 – SPM dominates.
Nonlinear chirp.
The chirp is nonlinear
2
⎛
L
⎛ t ⎞ ⎞
2 e ff ⎛ t ⎞
δω (t ) =
⎜
⎟ e x p ⎜⎜ − ⎜
⎟ ⎟⎟
T0 L NL ⎝ T0 ⎠
T
⎝ ⎝ 0 ⎠ ⎠
When the dispersion is negligible, N >> 1, SPM modifies the spectrum but
no change is the pulse envelope occurs.
Effect of GVD on pulse evolution in the presence of SPM, N ≈ 1
The nature of the interaction between SPM and dispersion in the
regime where N ≈ 1 depends on the sign of β2.
β2 > 0 normal dispersion
1
intensity
0.8
0.6
0.4
0.2
0
5
4
50
3
z/L
2
d
0
1
0 -50
ti m
p
e in
s
SPM generates red-shifted
frequencies at the leading edge
and blue-shifted frequency at the
trailing edge. In the normal
dispersion regime, red shifted
frequencies travel faster than
blue shifted frequencies. The
pulse is “torn apart” and
consequently, SPM leads to an
enhanced rate of pulse broadening
compared to linear broadening.
The pulse evolution in anomalous dispersion regime (β2 <0 ) regime is
shown on the following figure :
β2 < 0 anomalous dispersion
The dispersion induced chirp and the
SPM induced chirp have opposite signs
and therefore linear broadening and
the SPM induced pulse change
compensate for each other to some
degree. This compensation evolves into
a complete balance at z > 4LD yielding
an envelope which stays constant – a
soliton.
1
intensity
0.8
0.6
0.4
0.2
0
5
4
50
3
z/L
2
d
0
1
0 -50
ti m
p
e in
s
Optical solitons
When SPM compensates completely for the effect of dispersion in the
anomalous dispersion regime, the optical pulse propagates without changing
in its envelope – this is called an optical soliton
The soliton is the solution of the equation :
∂U
i ∂ 2U
−
= iN
∂ξ
2 ∂τ 2
2
U
2
U
Under the assumption of no fiber loss.
N=1 corresponds to a so called 1st order soliton.
The soliton envelope is :
U(ξ,τ ) = sec h(τ )exp(iξ /2)
1
2
sec h(τ ) =
= τ −τ
cosh(τ ) e + e
Propagation of a first order soliton
1.4
intensity
1.2
1
0.8
0.6
0.4
0.2
0
5
4
50
3
z/L
d
2
0
1
0
-50
in
time
ps
U ( 0 , τ ) = s e c h (τ )
The first order soliton can be launched as such or evolve from a sufficiently
intense pulse within 4 LD
The first order soliton is defined by the condition that balances dispersion
and SPM. The key relationship is :
2
PT
0 0
β2
= cons tant =
γ
For a given fiber, γ is fixed. The pulse wavelength determines β2.
The peak power P0 determines the soliton width T0. If the power is large
and there is some bandwidth limitation fixing the minimum T0 , the nonlinear
system sheds the extra energy from portions of the spectrum which are
beyond the spectral region needed to support the soliton of width T0 .
If the power is lower than needed, the soliton modifies its spectrum to
β2
2
support a width corresponding to PT
.
0 0 =
γ
Evolution of a Gaussian pulse with N = 1 : the pulse evolves towards a
fundamental soliton by changing its shape and width
1.4
normalized Power
1.2
1
0.8
0.6
0.4
0.2
0
10
8
10
6
z/Ld
4
0
t/T0
2
0
-10
Higher order solitons
If the initial pulse shape is of the form U(0,τ ) = N sec h(τ ) with N > 1 the pulse
evolves into a so called high order, or Nth order soliton. Also, any pulse shape
having sufficient peak power may evolve into such an Nth order soliton. The
envelope of a high order soliton is not constant but rather it has a periodic
(with ζ ) behavior with a period of Z0 = πLd/2.
Obviously, U(ζ ,τ ) = N sech(τ ) is a proper solution to the NLSE.
The high order soliton has at some points ζ a very complex envelope and its
complete evolution is not intuitive. Therefore, when using solitons as optical
pulses for communication, fundamental solitons are used exclusively.
2nd order soliton
3rd order soliton
4
7
3.5
6
5
intensity
intensity
3
2.5
2
1.5
4
3
1
2
0.5
1
0
0
1
1
0.8
50
0.6
0
50
0.6
0.4
z/ z
0.8
0
0.2
0 -50
in p
e
tim
s
z/
z
0
0.4
0
0.2
0 -50
ti m
n
ei
ps
Effect of Fiber Loss
Since the soliton condition is P0T02 = cons tant a constant pulse envelope
requires a constant P0 which is not possible in practical fibers which have
loss. The loss causes a pulse width broadening so as to maintain the soliton
condition.
The NLSE for N = 1 in the anomalous dispersion regime :
Define Γ = α LD = α
Recall ξ =
T02
β2
z
t
, τ=
LD
T0
∂U i ∂ 2U
1
2
−
− i U U = − ΓU
2
∂ξ 2 ∂τ
2
For small losses, Γ <<1, the term − 1 ΓU serves as a small
2
signal perturbation leading to the following solution:
−Γξ
⎛
⎞
−
1
e
(
)
− Γξ
−Γξ
U (ξ ,τ ) ≈ e sec h(τ e ) exp ⎜ i
⎟
⎜
4Γ ⎟
⎝
⎠
For small losses, the soliton pulse width increases exponentially with
distance as the peak power decreases.
T1 ( z ) = T0 exp( Γ ξ ) ≡ T0 exp(α z )
70
Perturbation
Γ=0.035
60
N=1
T1/T0
50
40
Dispersion only
30
20
Exact
10
0
0
10
20
30
40
50
z/LD
60
70
80
90
100
For small losses, the soliton broadens adiabatically but eventually the power
gets sufficiently low that the nonlinearities are negligible and the soliton is
lost. In order to avoid this loss, the system has to recover the soliton and this
is done in one of two ways, periodic amplification or continuous dispersion
modification.
Periodic amplification : loss managed soliton
G
TX
LA
G
LA
G
LA
The idea is to periodically amplify the soliton so as to not allow complete
distortion. The amplifier spacing LA is chosen to be much smaller than LD
In order to ensure that the soliton width does not broaden too much.
The gain G compensates exactly the fiber losses over LA Km.
A fundamental soliton can be created and maintained if its path average peak
power equals the soliton peak power, P0 that would have been used in a
hypothetical lossless fiber.
av
Ppeak
1
=
LA
β2
∫0 Ppeak ( z )dz = P0 = γ T02
LA
The soliton power decrease exponentially along amplification span :
P paeavk ( z ) = P p ea k (0 ) ex p ( − α z )
The path average peak soliton power over one amplifier span is :
Ppeak
1
=
LA
=
LA
∫
0
1
Ppeak ( z )dz =
LA
LA
∫P
peak
0
Ppeak (0) 1 − exp(−α LA )
LA
α
(0) exp(−α z )dz
=
Leff
LA
Ppeak (0)
The condition
β2
P ( z ) = P0 = 2
γ T0
av
peak
Since G=exp(αLA)
Ppeak (0)
P0
=
yields
α LA
Ppeak (0) = P0
1 − exp(−α LA )
ln(G ) G ln(G )
=
= f LM
1
G −1
1−
G
fLM is the power enhancement factor in loss managed soliton systems.
In a lossless fiber, the input pulse that becomes a first fundamental
soliton obeys the condition
P0 =
β2
γ T02
For a lossy fiber, the input pulse should obey the condition
Ppeak (0) = f LM P0
Evolution of loss managed soliton over 5000 km (LA=50 km,
α=0.2 dB/km, β2=-2.5 ps2/km)
LD=200km
LD=20km
When the pulse width determines to a dispersion length of 200 km, the
soliton is quite well preserved even after 5000 km since the condition LA<<LD
is reasonably satisfied. However, when the dispersion length is reduced to 20
km, the soliton is unstable and unable to sustain itself due to an excess of
dispersive wave emission.
A second way to maintain the soliton is to adiabatically modify the constant
β2 / γ by continuously changing β2
This is achieved in a so called dispersion decreasing fiber (DDF) where the
dispersion changes continuously.
In order to fulfill the condition N2(z)=1 in the presence of loss
γ P( z )T 2 ( z )
and maintain T(z)=T0
N ( z) =
β2 ( z)
2
γ P ( 0 ) exp(−α z )T ( z )
γ P(0)T0
∀z , N ( z ) =
= N 2 (0) =
⇒ β 2 ( z ) = β 2 (0) exp(−α z )
β2 ( z)
β 2 (0)
2
⇒ D( z ) = D (0) exp(−α z )
β 2 ( z ) = β 2 (0 ) ex p (−α z )
The dispersion has an exponentially decreasing profile similar to the fiber
loss profile. If the soliton peak power P0 decreases exponentially with z,
the requirement N=1 can be fulfilled at every point along the fiber if β2
decreases similarly.
The fiber with such a profile is called Dispersion Decreasing Fiber (DDF)
but it is not used in commercial systems because of its limited length (20
to 40 km for a dispersion change by a factor 10).
Since DDF are not available commercially, fiber loss is commonly
compensated by periodic amplifications.