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