Coupled nonlinear Schrödinger equations with dissipative terms and control of soliton propagation in broadband optical waveguide systems Avner Peleg Racah.
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Coupled nonlinear Schrödinger equations with dissipative terms and control of soliton propagation in
broadband optical waveguide systems
Avner Peleg
Racah Institute of Physics
Workshop on Quantum (and Classical) Physics with Non-Hermitian Operators
The Israel Institute for Advanced StudiesThe Hebrew University of Jerusalem, July 2015
In collaboration with: Q.M. Nguyen, Y. Chung,
D. Chakraborty, J.-H. Jung, T.P. Tran, T.T. Huynh,
M. Chertkov, and I. Gabitov
Outline
• Introduction to broadband (multichannel) optical waveguide transmission.
• The nonlinear Schrödinger (NLS) equation and soliton pulses.
• Perturbative description of single-soliton propagation and of a single two-pulse
collision.
• Effects of dissipative perturbations and crosstalk.
• Coupled-NLS models for soliton propagation.
• N-dimensional Lotka-Volterra (LV) models for amplitude dynamics.
• Stability and bifurcation analysis for the LV models for stabilization and switching.
• Comparison between the coupled-NLS models and the LV models.
• Further stabilization by frequency dependent linear gain-loss
• Conclusions.
Broadband (multichannel) optical waveguide systems and crosstalk
• Broadband (multichannel) transmission is used for
enhancing transmission rates in optical waveguide links.
• In these broadband systems, many pulse sequences
propagate through the same waveguide.
• Pulses from different sequences (frequency channels)
propagate with different group velocities.
=> Collisions between pulses from different channels are
very frequent, and can severely limit transmission quality.
• Interchannel crosstalk – energy exchange in collisions
between pulses from different frequency channels.
• Two main mechanisms: (a) delayed Raman response,
(b) nonlinear loss or gain (cubic or higher order).
• Example: Raman crosstalk in an on-off-keyed optical
fiber transmission system with 101 frequency channels. [AP, Phys. Lett. A 360, 533 (2007)]
Two broadband (multichannel) optical waveguide experiments
• A 109-channel fiber optics system
operating at 10 Gbits/s per channel.
• Dispersion-managed solitons. • Experiment: Mollenauer et al. [Opt.
Lett. (2003)].
• Standard requirement: BER<10-9.
Crosstalk in silicon nanowaveguides
• A 2-channel silicon nanowaveguide transmission system at 10 Gbits/s per channel.
• BER increases with increasing input power.
• Experiment by Okawachi et al. [IEEE Photon. Technol. Lett., (2012)].
Nonlinear Schrödinger equation and solitons
• Hasegawa and Tappert (1973) - pulse propagation in optical waveguides can be described by the nonlinear
Schrödinger (NLS) equation:
0Ψ|Ψ|2κΨdΨi 22t2z
(t,z) – proportional to the envelope of the electric field z – distance along the waveguide, t – timed2 – second-order dispersion coefficientκ – Kerr nonlinearity coefficient In dimensionless form:
•The single soliton solution of the NLS equation in a frequency channel β:
ηβ, yβ, αβ - the soliton amplitude, position, and phase.
•Solitons are stable and shape-preserving => Soliton-based transmission is advantageous compared with other transmission formats.
0Ψ|Ψ|2ΨΨi 22tz
)]β2y(tcosh[η
)z]βi(η)y-(tiexp[iαηz)(t,Ψ
ββ
22βββ
ββ z
a
b
n1
n2
Radial Distance
Ref
. In
dex
n0
a
ab
Soliton collisions
• In an ideal fiber soliton collisions are elastic: the amplitude, frequency, and shape do not change as a result of the collision.
• In real optical fibers this elastic nature of the collisions breaks down due to the presence of perturbations (corrections to the ideal NLSE).
• In this case soliton collisions might lead to: emission of radiation, change in the soliton amplitude and group velocity, corruption of the shape, etc.
Effects of delayed Raman response on single-soliton propagation
• Pulse propagation is described by the perturbed NLSE:
• Use adiabatic perturbation theory for the NLSE soliton
[D.J. Kaup, Phys. Rev. A (1990) and (1991)]. • Look for a solution: , where
is the soliton part with slowly varying parameters, and vrad(t,z) is the radiation part.
• Substitute into the perturbed NLSE to obtain
• Project both sides on the eigenmodes of the linear operator describing small perturbations
about the NLSE soliton. The only effect of delayed Raman response on soliton parameters in
O(εR) is a frequency downshift:
222tz |ψ|ψ-ψ|ψ|2ψψi tR
)zt,(vz)(t,ψz)ψ(t, rads
}y(z)]-[t)(η{cosh
y(z)]}-[t)(iβ)(exp{iα)(ηz)(t,ψs z
zzz
1
1
)(cosh
)tanh(-2i)O(εpart radiation
1-
1
cosh(x)
i2
1
1
)cosh(
)tanh(
1-
1
cosh(x)
ix
1
1
cosh(x)
xtanh(x)-1
342
R
222
x
x
dz
dy
dz
d
dz
dy
x
x
dz
d
dz
d
R
Raman self-frequency shift (Gordon 1986)
15/8/ 4 Rdzd
Effects of delayed Raman response on a single collision
• Consider a single collision between a soliton in the reference channel (β=0) and a soliton in the β channel (Chi & Wen 1989, Malomed 1991, Kumar 1998, Lakoba and Kaup 1999, Chung and AP 2005, Nguyen and AP 2010).
• Assumptions: 1/| β | « 1, εR « 1
• An O(εR) change in the soliton amplitude (Raman-induced crosstalk)
• An O(εR/ β) frequency change (Raman induced cross frequency shift)
• Assuming εR « 1/ | β | « 1 we can neglect effects of O(εR 2) or higher.
0βR(c)0 ηηβsgn2εΔη
||3
ηη8ε(c)0
20βRΔβ
0βR)1(
01 ψηβsgnε
0tRβ(1)
02 ψε4iη
ΔΦ β
independent ofthe magnitude of β
Analysis of a single two-soliton collision (1)
Example – perturbed NLS equation with delayed Raman response:
• Consider a single collision between a soliton in the reference channel (β=0) and a soliton in the β channel.• Assume: 1/|β| « 1, εR « 1 (typical for broadband transmission).
• Look for a two-soliton solution of the perturbed equation in the form
ψ0, ψβ – single-pulse solutions of the perturbed NLS equation in channels 0 and β.
φ0, φβ – collision effects in channels 0 and β.
• Solve, for example, for the pulse in the 0 channel.
• Substitute
• Use resonant approximation (|β|»1), and neglect terms rapidly oscillating in z.
2tR
22tz |ψ|ψ-εψ|ψ|2ψψi
...ψψψ β0β0two
00total0 ψψ
))exp(iχ(xΦz)(t, ))exp(iχ(xΨz)(t,ψ
))exp(iχ(xΦz)(t, ))exp(iχ(xΨz)(t,ψ
ββββββββ
00000000
[Y. Chung and AP, Nonlinearity (2005); Q.M. Nguyen and AP, JOSA B (2010)]
Analysis of a single two-soliton collision (2)
• Expand Φ0 in a perturbation series:
• The equation in O(εR):
• Integrate over z from -∞ to ∞
• So
• On the other hand
fj(x0), j=0, …, 3 – the four localized eigenmodes of the linear operator describing small
perturbations about the NLS soliton.
• Project (ΔΦ(x0),ΔΦ*(x0))T onto the localized eigenmode f0(x0) to obtain the collision-
induced amplitude shift (Raman crosstalk):
...(2)02
(1)02
(0)02
(1)01
(0)010
Ψ|Ψ|βε 02
βR)1(
01z
1
1)(xΨηβsgnε
)(xΔΦ
)(xΔΦ00βR
0*(1)
01
0(1)01
...)(xfΔη)(xfβi)(xfΔyη)(xfΔαiη)(xΔΦ
)(xΔΦ030020010
200000
0(1)*01
0(1)01
[Q.M. Nguyen and AP, JOSA B (2010)]
)(xΨηβsgnε)(xΔΦ 00βR0(1)01
0βR(c)0 ηηβsgn2εΔη
from Raman term
Crosstalk in broadband (multichannel) waveguide systems
• In amplitude-keyed transmission, crosstalk leads to severe transmission degradation due to
the interplay between collision-induced amplitude shifts and amplitude-pattern randomness.
• A method for overcoming crosstalk – encode information
in the phase => phase-shift-keyed (PSK) transmission. • In PSK transmission all time slots are occupied
=> Crosstalk-induced amplitude dynamics is deterministic.
• Is it possible to achieve stable stationary transmission with nonzero amplitudes in all channels?
• Answer this question by obtaining a reduced ODE model for pulse amplitudes.
(a) Analyze perturbation effects on a single collision.
(b) Use (a) and collision-rate calculations to obtain the reduced ODE model.
(c) Analyze stability of equilibrium points of reduced ODE model.
(d) Compare predictions of ODE model with numerical solution of corresponding
coupled-NLS model.
(e) Study role of high-order effects and find ways to control them.
A Lotka-Volterra model for Raman-induced amplitude dynamics (1)
• A broadband fiber optics system with 2N+1 channels and frequency difference
Δβ between adjacent channels.
• The amplitude shift of a jth-channel soliton due to a collision with a kth-channel
soliton:
εR – Raman coefficient [εR=0.006/τ0, τ0 - pulse width in picoseconds]
ηj, ηk – initial amplitudes; βj, βk – initial frequencies.
f(|j-k|) – a constant describing the strength of the Raman interaction.
• Assumptions:
(1) εR « 1/ | β | « 1;
(2) Deterministic pulse sequences;
(3) Sequences are infinitely long (long-haul transmission) or are subject to periodic
temporal boundary conditions (closed fiber loop experiments).
• gj – net linear gain-loss in jth channel.
• Δzc – inter-collision distance for collisions between solitons from adjacent channels.
• Take into account amplitude shift due to: (a) single-pulse propagation, (b) collisions.
kjRj η)η-|)sgn(k-jf(|2εΔη jk
[Q.M. Nguyen and AP, Optics Communications (2010)]
A Lotka-Volterra model for Raman-induced amplitude dynamics (2)
• The change in the jth-channel soliton’s amplitude in the interval (z,z+Δzc):
• Taking the continuum limit
• A predator-prey model with 2N+1 species!
• Determine gj values by looking for an equilibrium state with equal nonzero amplitudes
in all channels:
The gain required for maintaining an equilibrium state with equal amplitudes is not “flat”
(constant) with respect to frequency.
• Model takes the form:
NjNkjfjkT
gdz
d N
Nkk
Rjj
j
- |)(|)(4
NjNkjfjkT
gN
Nk
Rj
- |)(|)(4
NjNkjfjkTdz
d N
Nkkj
Rj
- )|)((|)(4
NjNjNT
g Rtriang
j
- )12(
4 :example
[Q.M. Nguyen and AP, Optics Communications (2010)]
NjN- (z)(z)η|)ηk-jj)f(|-(k2ε(z)Δzηg(z)η)Δz(zηN
-NkkjRcjjjcj
Equilibrium states of the Lotka-Volterra model and their stability • Equilibrium states with non-zero amplitudes are determined by
• “Trivial” equilibrium state: ηj(eq)=η for -N≤j≤N.
• For an odd # of channels there are infinitely many steady states => infinitely many
possibilities for stationary transmission (with unequal amplitudes).
• Show stability by constructing Lyapunov functions for the model:
• VL satisfies: (a) dVL /dz=0 (along trajectories of the model); (b) ;
(c) for any with positive amplitudes;
=> Equilibrium state is stable for any initial condition.
• Stability is robust – it is independent of the details of the approximation for the Raman
interaction [the exact values of the f(|j-k|) coefficients].
NjNkjfjk eqk
N
Nk
- 0)(|)(|)( )(
N
N
jeq
jeq
jeq
jjL
jV )/ln()( )()()(
0)( )( eqLV
0)(
LV )(eq
)(eq
Example: numerical solution of the LV model with 3 channels
• Channels: j=0 (middle), j=50 (highest), and j=-50 (lowest).
• Equations for amplitude dynamics
• Equilibrium states:
With equal amplitudes: η50=η0=η-50=1.
With unequal amplitudes: η50=η-50= (3-η0)/2
=> a line segment of equilibrium states.
• For input amplitude values that are
off the equilibrium states dynamics is
oscillatory => Stable transmission.
)23(4
50
)(4
50
)23(4
50
0505050
505000
5005050
Tdz
d
Tdz
d
Tdz
d
R
R
R
Phase portrait for η50, η0 and η-50
Pulse amplitudes vs propagation distance
Comparison with full-scale coupled-NLS simulations
• The LV model neglects high-order effects, such as radiation emission, which can be
important at large distances, and can lead to the breakdown of the LV model’s description.
• Need to compare the LV model’s predictions with simulations with the full NLS model.
• The dynamics involves a large number of fast collisions.
=> Amplitude measurements are difficult to perform with a single NLS model.
• We therefore work with the following equivalent coupled-NLS model:
ψj – envelope of the electric field of the jth sequence
gj – linear gain-loss coefficient for the jth sequence
• We numerically solve the coupled-NLS model with periodic boundary conditions and an
initial condition consisting of 2N+1 periodic soliton sequences:
*kjtk
2ktj
N
-NkjkR
2jtjR
jjj2
kjkj2
jj2tjz
ψψψ|ψ|ψδ1ε|ψ|ψε
2/ψigψ|ψ|δ14ψ|ψ|2ψψi
N
Nk
NjN-
|)(|)(4
N
Nk
Rj kjfjk
Tg
[AP, Q.M. Nguyen and T.P. Tran, arXiv:1501.06300]
NjN- kT)](0)(tcosh[η
]kT)-(0)(tβexp[i(0)η(t,0)ψ
J
Jk j
jjj
Comparison with coupled-NLS simulations: 2 channels
• A 2-channel system with εR=0.0024,
T=18, Δβ=40, and 5 solitons in each sequence.
• Good agreement between the LV
model predictions and the coupled-NLS
simulations up to a distance z=2500.
• Frequency difference also oscillates
due to coupling to amplitude dynamics.
The oscillations are captured by the
following perturbed predator-prey model:
• The shapes of the solitons are retained
up to z=2500, but at larger distances, soliton
shapes degrade, due to radiation emission.
[AP, Q.M. Nguyen, T.P. Tran, arXiv:1501.06300]
41
42
21
221111
121222
15
8
4
4
R
R
R
dz
d
Tg
dz
d
Tg
dz
d
Comparison with coupled-NLS simulations: 4 channels
• A 4-channel system with εR=0.0018,
T=20, Δβ=15, and 2 solitons in each sequence.
• Good agreement between the LV model
and the coupled-NLS simulations up to a z=800.
• The shapes of the solitons are retained up
to z=800. At larger distances, soliton shapes
degrade, due to radiation emission.
• In a four-channel system with Δβ=15,
radiative sidebands for the jth sequence develop at frequencies βk(z) of the other soliton sequences.
• These sidebands can be suppressed by
increasing Δβ or by using a fiber coupler with frequency dependent linear gain-loss.
[AP, Q.M. Nguyen, T.P. Tran, arXiv:1501.06300]
Further stabilization by frequency dependent linear gain-loss
• The Fourier transform of the soliton part of ψj(t,z):
• Since |βk-βj|»1, the are well-separated
and this can be used to suppress radiation emission.
• Use frequency-dependent linear gain-loss.
For example, in a nonlinear N-waveguide coupler, we can choose:
where gL<0.
• The coupled-NLS model
• Stable propagation extended to z=5000. No generation of radiation sidebands.
[Q.M. Nguyen, AP, and T.P. Tran, arXiv:1501.06300]
(z)](z)]/[2ηβπ[ωcosh
)]kTωcos(21[(z)eη)π/2(z),ω(ψ̂
jj
1(z)yωi-(z)θij
2/1j
jj
J
k
z),ω(ψ̂ j
2/)0(βor 2/)0(β if
2/)0(β2/)0(β if g)ω(
jj
jjj
j WWg
WWg
L
*kjtk
2ktj
N
-NkjkR
2jtjR
jj1-
j2
k
N
Nkjkj
2jj
2tjz
ψψψ|ψ|ψδ1ε|ψ|ψε
2/))ω(ψ̂)ω(g(iFψ|ψ|δ14ψ|ψ|2ψψi
Transmission switching in the presence of nonlinear gain-lossExample: A Ginzburg-Landau gain-loss profile in waveguide lasers
• Consider a nonlinear waveguide with weak linear loss, cubic gain, and quintic loss, i.e.,
with a Ginzburg-Landau gain-loss profile.
• Perturbed NLS equation:
• Amplitude shift in a single two-soliton collision in the presence of quintic loss:
• The LV model for crosstalk-induced amplitude dynamics in a two-channel system:
• The corresponding coupled-NLS model:
ψ|ψ|iε-ψ|ψ|iεψ-iεψ|ψ|2ψψi 45
231
22tz
|β|)/η(2ηηη4εΔη 2β
20β05
(2s)0 Amplitude shift is quartic
in soliton amplitudes
0ε /εε κ1,2k 1,2j
]3η)η(2η[ηT
8-η)-(η
T
8κ)η(η
15
16-)η(η
3
4κη4ε
dz
dη
553
32k
2jkk
44j
22jj5
j
)/6/215/43/(ε4g /
ψ|ψ||ψ|6iεψ|ψ|3iεψ|ψ|iε
ψ|ψ|iε2ψ|ψ|iε2/ψigψ|ψ|4ψ|ψ|2ψψi
235j53
j2
j2
k5j4
k5j4
j5
j2
k3j2
j3jjj2
kj2
jj2tjz
TT
[AP and Y. Chung, Phys. Rev. A (2012)]
Amplitude dynamics with a Ginzburg-Landau gain-loss profile
• Stable propagation for a wide range of ε5 values, including
ε5=0.5, i.e., outside of the perturbative regime.
• On-off (off-on) transmission switching:
turning off (on) transmission of one of the soliton sequences,
using bifurcations of the equilibrium state with equal
amplitudes in both channels. • Example: use the saddle-node bifurcation of (1,1) at
κc=(8T-15)/(5T-15) to turn off (on) transmission of sequence 2.
• In on-off switching, κ is increased from κi<κc to κf>κc, such
that (1,1) becomes unstable, while (ηs,0) is stable. As a result, η2
and η1 tend to 0 and ηs after the switching.
0 200 400 600 800 1000 1200
1.00
1.05
1.10
1.15
1.20
1.25
1.30
1
z
CNLS 5=0.01
CNLS 5=0.06
CNLS 5=0.1
LV 5=0.01
LV 5=0.06
LV 5=0.1
• In off-on switching, κ is decreased from κi>κc to κf<κc, such that (1,1)
becomes stable. As a result, both η2
and η1 tend to 1 after the switching.
on-off switching
off-on switching
[D. Chakraborty, AP, and J.-H. Jung, Phys. Rev. A (2013)]
Conclusions
• We developed a general framework for transmission control in broadband (multichannel) soliton-based optical waveguide systems.
• Using single-collision analysis and collision-rate calculations, we showed that amplitude dynamics in an N-channel waveguide system can be described by N-dimensional Lotka-Volterra (LV) models, where the form of the LV model depends on the physical perturbation.
• Stability and bifurcation analysis of the steady states of the LV models is used to develop methods for achieving robust transmission stabilization and switching for the main nonlinear dissipative processes, including delayed Raman response and nonlinear loss and gain.
• The method can find applications in a variety of waveguide systems, including fiber optics communication systems, data transfer on computer processors, and multiwavelength waveguide lasers.
Main Publications
Crosstalk-induced dynamics in broadband waveguide systems
• Q.M. Nguyen and AP, Opt. Commun. 283, 3500 (2010).
• AP, Q.M. Nguyen, and Y. Chung, Phys. Rev. A 82, 053830 (2010).
• AP and Y. Chung, Phys. Rev. A 85, 063828 (2012).
• D. Chakraborty, AP, and J.-H. Jung, Phys. Rev. A 88, 023845 (2013).
• Q.M. Nguyen, AP, and T.P. Tran, Phys. Rev. A 91, 013839 (2015).
• AP, Q.M. Nguyen, and T.P. Tran, submitted, arXiv:1501.06300.
• AP, Q.M. Nguyen, and T.T. Huynh, submitted, arXiv:1506.01124.
Single-collision analysis
• AP, M. Chertkov, and I. Gabitov, Phys. Rev. E 68, 026605 (2003).
• J. Soneson and AP, Physica D 195, 123 (2004).
• Y. Chung and AP, Nonlinearity 18, 1555 (2005).
• Q.M. Nguyen and AP, J. Opt. Soc. Am. B 27, 1985 (2010).
• AP, Q.M. Nguyen, and P. Glenn, Phys. Rev. E 89, 043201 (2014).
Back of the envelope derivation of the NLSE
E(t,z) – the envelope of the electric field
Taylor expansion of the wave number
c.c]t)e[E(z,2
1t)e(z, t)ωzi(k 00 slow varying envelope
approximation: 1/(ω0τ0) «1
22
200000
2 |E||E|
k)ω)(ω(ω'k'
2
1)ω)(ω(ωk'k)|E|,k(
ω
:Eon operating and i ωω and i- kk Replacing t0z0
0E|E|k)(E'k'2
1 -Eik'Ei 2
|E|2ttz 2
zk'tz/vtt
/cnωκ :tynonlineariKerr
'k'd :dispersionorder -second
|E|n)ω(nck/ωn :index refractive
g
20
2
220
0E2|E|2E2t2d -Ezi
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