Rare events in dispersion-managed nonlinear lightwave systems Elaine Spiller Jinglai Li, Gino Biondini, and William Kath State University of New York at Buffalo Department of Mathematics Nonlinear Physics: Theory and Experiment, IV - 27 June, 2006
Outline dispersion management (DM) -ligtwave communication and mode-locked lasers noise in DM systems dispersion managed nonlinear Schrödinger equation (DMNLS) -stationary solutions, map strength symmetries on DMNLS - conservation laws - modes of linearized DMNLS importance sampled Monte-Carlo (IS-MC) for DMNLS 1
Dispersion management - alternating dispersion + nonlinearity iu z + 1 2 d(z)u tt+ u 2 u=0 Dispersion management: periodic, alternating dispersion d(z)=d + for0 z z +, local dispersion, d(z) d 1 average dispersion distance, z d(z)=d forz + <z z a z a =z + +z : dispersion map period Dispersion map: specific parameter choice d 2 Model for Long distance optical communication systesm Mode-locked lasers 2
Noise in dispersion-managed lightwave systems Signal amplification by stimulated emission is always accompanied by spontaneous emission of incoherent photons noise nonlinear transmission can cause resulting statistics to differ greatly from the statistics of the source noise Performance demands require that errors be extremely rare events long distance communication - bit-error ratios (BER) of10 10 or less mode-locked lasers - frequencies accurate to10 18 Goal: estimate performance of these systems Problems: Monte-Carlo (MC) alone cannot capture such rare events Perturbation theory cannot precisely predict probabilities of errors Solution: Importance sampling MC guided by perturbation theory 3
Review: Perturbation theory for NLS solitons (D(z) = 1) u s =Asech(A[t T Ωz])exp(i[Ωt+ 1 2 (A2 Ω 2 )z+φ]) Invariances of NLS imply that it cannot resist parameter fluctuations Build up of parameter changes large pulse distortion Linear modes link perturbations to parameter changes. Consideru=u s +v, where v= Au A + Ωu Ω + Tu T + φu φ +R(t,z) (u K are modes of linearized NLS corresponding to parameterk) Calculating variances at output is not enough to predict rare events Also, varying dispersion kills some symmetries lose linear modes With varying dispersion, how can we recover symmetries? 4
The dispersion-managed nonlinear Schrödinger equation (DMNLS) Multiple scales expansion in powers ofz a d(z)= d+d(z/z a )/z a Leading order: zero-mean, large oscillations in dispersion Fourier domain: û (0) (ω,z;z a )=Û(ω,z)e i(ω2 /2) D(z/z a )dz/z a Fast phase determined by the zero-mean oscillations of d(z) - responsible for pulse breathing (compression/expansion cycle) Û(ω, z) = slowly evolving Fourier amplitude (i.e., core pulse shape) DMNLS:i Û ω z 1 2 dω 2 Û ω + Û ω+ω Û ω+ω Û ω+ω +ω r(ω ω )dω dω =0 5
The dispersion-managed NLS equation Integration kernel: average nonlinearity, mitigated by DM Piecewise-constant two-step dispersion maps:r(x)= 1 sinsx (2π) 2 sx s= d(z) d 1 /4=reduced map strength -measures the strength of dispersion variations local dispersion, d(ζ) d 1 D 1 = d 1 -- d a average dispersion, d a 0 1/2 1 θ/2 (1-- θ)/2 θ distance, ζ = z/z a D 2 = d 2 -- d a 1-- θ d 2 r depends ons, but independent of map details (D 1,2 &θ)! 6
Special solutions: time -independent DMNLS DMNLS in the temporal domain: i U t z +1 2 d 2 U t t 2 + U t+t U t+t U t+t +t R(t,t ;s)dt dt =0 Stationary solutions: U s (t,z)=f(t)e iλ2 z/2 Û s (ω,z)=ˆf(ω)e iλ2 z/2 λ = soliton eigenvalue. Traveling waves can be generated via Galilean invariance. ˆf(ω) solves a nonlinear integral equation: ˆf ω = 2 λ 2 + dω 2 dω dω ˆf ω+ω ˆf ω+ω ˆf ω+ω +ω r(ω ω ;s). Find ˆf(ω) numerically with a modified Neumann iteration 7
Stationary solutions of DMNLS: DM-solitons Profile of the stationary pulse as a function of the map strength,s: s 1(weak DM): solutions are close to NLS solitons s O(1) (strong DM): solutions have Gaussian core exponentially decaying oscillating tails DM-solitons not widely used in communication systems, but a good starting point to study arbitrary pulse shapes DMNLS recovers symmetries lost with dispersion management 8
Symmetries of DMNLS equation Some conservation laws same as NLS phase invariance:u ue iε d u 2 dt=0 dz translation (timing shift) invariance:u u(t ε,z) d Im(u u dz t )dt=0 Galilean invariance:u u(t εz,z)e iεt iε2 z/2 d t u 2 dt= Im(u u )dt (modified conservation law) dz t New conservation law for scale invariance (via Noether s theorem): scale invariance: u au(at,a 2 z) R s (t,t ) a 2 R a2 s(at,at ) (a=1+ε) d dz Ê [(u u t u u t ) is(u u s u u s )]dt = Ê (H[u]+ 1 2 u t 2 )dt+ ÊÊÊ t t (u tu t+t u t+t u t+t +t )R s(t,t )dt dt dt 9
DMNLS: from symmetries to linear modes Consider the Lagrange densityl, and a perturbed solutionu =u+εv Euler-Lagrange J[u]= ÊÊ Ldzdt Noether Ý DMNLS Ý conservation laws O(ε) ============ O(ε) ============ O(ε) ============ K[u,v]= ÊÊ Kdzdt Ý linearized DMNLS Ý Euler-Lagrange Noether perturbed conservation laws, linear modes One mode (or generalized mode) corresponding to each symmetry Note: J[u] DMNLS = ÊÊÊ L DMNLS dzdtds 10
Modes of linearized DMNLS equation DMNLS linearized about a solution, u v z =L dmnls(v) =i 2 v t 2 i 2 λ2 v+2i +i ÊÊ ÊÊ v t+t u t+t u t+t +t R(t t ;s)dt dt u t+t u t+t v t+t +t R(t t ;s)dt dt modes ofl dmnls f T = u t f φ =iu f Ω =itu f λ =?? neutral modes (same as for NLS) L dmnls (f T )=0 L dmnls (f φ )=0 generalized mode L dmnls (f Ω )=f T 11
Amplitude, λ, mode of linearized DMNLS f λ =(u+t u t +2s u s )/λ amplitude mode 1 0.8 0.6 0.4 0.2 0 15 10 mapstrength, s 5 0 10 5 0 5 10 time also a generalized model dmnls (f λ )=λf φ s 1(weak DM): similar to amplitude mode of linearized NLS Recall, perturbation theory alone is not enough importance sampling 12
Importance sampling: a very simple example Experiment: 100 coin flips Question: What is the probability of 70 or more heads? Answer:2.4 10 13 But, how do we simulate this directly? Solution: Use an unfair coin! Optimal: Use a coin that gives heads 70% of the time. Correct for biased coin by using likelihood ratio: If on a flip one gets heads, multiply by 0.5/0.7 If on a flip one gets tails, multiply by 0.5/0.3 This corrects statistics: get results for a fair coin But, 10 orders of magnitude simulation speedup 13
IS application: calculating rare events of DM systems goals: predict error rates and understand why errors occur recall: error rates in DM systems must be small<10 10 know: how noise changes DM soliton parameters linear modes idea: use linear modes to bias simulations toward rare events of interest but: Crucial to understand most likely noise realizations that lead to desired events, need to be careful 14
Noise-induced parameter changes Consider a noise-induced perturbation, v, to the solution u v= Tf T + Ωf Ω + φf φ + λf λ +R v will cause the following parameter change K= Ê vf K dt Ê fk f K dt wherek {T,Ω,φ,λ} Note, adjoint modes are related byl dmnls (v)=il dmnls (iv) f T =if Ω, f Ω =if T, f φ =if λ, f λ = if φ Note, DM-soliton solution does not need explicit parameter dependence to isolate perturbation-induced parameters changes 15
Optimal biasing for DMNLS equation (at one amplifier) Bias noise with function of timeb(t) X(t) white, Gaussian i.i.d.,v(t)=x(t)+b(t) Maximize probability of hitting, on average, a desired parameter change, K (K {T,Ω,φ,λ}) Solution: b(t)= K Ê f K f K dt Ê fk 2 dt f K (t) Optimal biasing proportional to adjoint modes 16
Importance-sampled Monte-Carlo for DM systems Pulse without and with biasing by the DMNLSλmode 1 2 2 1.5 1 + 0.5 0 = 1.5 1 0.5 0.5 0.5 0 0 0 50 100 150 200 250 300 1 0 50 100 150 200 250 300 0 50 100 150 200 250 300 1 2 2 1.5 1 + 0.5 0 = 1.5 1 0.5 0.5 0.5 0 0 0 50 100 150 200 250 300 1 0 50 100 150 200 250 300 0 50 100 150 200 250 300 run full simulation at an amplifier, calculate linear modes & bias noise to produce (desired) larger than normal parameter changes update likelihood ratio 17
Samples following targeted path 1.6 energy, a.u. 1.4 1.2 1 0.8 0.6 Targeted output energy 0.4 0 500 1000 1500 2000 2500 3000 3500 4000 distance, km correct statistics with likelihood ratio unbiased statistics of rare events 18
IS-MC results for frequency (left) and energy (right) 10 0 10 0 10 5 probability 10 10 10 15 probability 10 5 10 10 10 20 10 25 0.2 0.1 0 0.1 0.2 Ω at output 10 15 0.4 0.6 0.8 1 1.2 1.4 1.6 output energy (normalized by back to back energy) black dots - 100,000/50,000 MC samples (full NLS with varying dispersion) black curve - Gaussian fit to MC simulations pink curve - 75,000/42,000 IS-MC samples (DMNLS) right simulation params: (s=2, λ=1.5, σ=.03, z a =100km,z final =2,000km) left simulation params: (s=4, λ=2, σ=.06, z a =100km,z final =4,000km) 19
Rare events in DMNLS systems: Conclusions and projects Conclusions identified most likely way to achieve rare events -adjoint modes of linearized DMNLS identified linear mode associate with λ -Noether s theorem relates invariances, conservation laws, and linear modes demonstrated effectiveness IS-MC to find rare events -test cases: frequency jitter, amplitude jitter next: pdfs for phase 20