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Nonlinear interference noise in optical fibers- properties, measurement, and applications Ori Golani What is NLIN? Nonlinear Interference Noise Tx Rx Tx Rx Tx Rx Tx Rx Tx Rx IC IC COI IC IC frequency 2 What is NLIN? Nonlinear

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Nonlinear interference noise in

  • ptical fibers- properties,

measurement, and applications

Ori Golani

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SLIDE 2

Rx Rx Rx Rx Rx Tx Tx Tx Tx Tx

2

COI IC IC IC IC

frequency

Nonlinear Interference Noise

What is NLIN?

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SLIDE 3

3

Nonlinear Interference Noise

What is NLIN?

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4

Channel model

π‘‘π‘œ = π‘π‘œ + π‘₯π‘œ + ෍

𝐽𝐷

෍

β„Ž,𝑙,π‘š

π‘—π›Ώπ‘Œβ„Ž,𝑙,π‘š π‘β„Žπ‘π‘™

βˆ—π‘π‘š

User of interest Interfering users

Received signal Transmitted symbol AWGN Nonlinear interference

π‘π‘œ π‘π‘œ,IC𝑂 π‘π‘œ,IC1

…

π‘‘π‘œ

mux demux

π‘₯π‘œ

Fiber link

βŠ•

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Treating NLIN- interference as noise

π‘‘π‘œ = π‘π‘œ + π‘₯π‘œ + ෍

𝐽𝐷

෍

β„Ž,𝑙,π‘š

π‘—π›Ώπ‘Œβ„Ž,𝑙,π‘š π‘β„Žπ‘π‘™

βˆ—π‘π‘š

= π‘π‘œ + π‘₯π‘œ + π‘€π‘œ

Approximately Gaussian, πœπ‘€

2 = 𝛽𝑄3 Treatment as AWGN

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Stochastic ISI model of NLIN

Total NLIN contribution:

= ෍

π‘š

෍

𝐽𝐷

෍

β„Ž,𝑙

π‘—π›Ώπ‘Œβ„Ž,𝑙,π‘š π‘β„Žπ‘π‘™

βˆ— π‘π‘œβˆ’π‘š = ෍ π‘š

π‘†π‘š

(π‘œ)π‘π‘œβˆ’π‘š

NLIN = ෍

𝐽𝐷

෍

β„Ž,𝑙,π‘š

π‘—π›Ώπ‘Œβ„Ž,𝑙,π‘š π‘β„Žπ‘π‘™

βˆ—π‘π‘š

Sum on unknown ICs

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SLIDE 7

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What is this model good for?

Nonlinear problem Linear Time varying

  • NLIN behaves like a doubly-selective linear channel- we can use tools from RF

communication

  • If the ISI coefficients, π‘†π‘š

(π‘œ), change slow enough, we can track them and mitigate

their effect

Ξ”π‘π‘œ = ෍

π‘š

π‘†π‘š

(π‘œ)π‘π‘œβˆ’π‘š

Ξ”π‘π‘œ = ෍

𝐽𝐷

෍

β„Ž,𝑙,π‘š

π‘—π›Ώπ‘Œβ„Ž,𝑙,π‘š π‘β„Žπ‘π‘™

βˆ—π‘π‘š

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Characterization of time-varying ISI model

Characterizing the statistics of the ISI coefficients π‘†π‘š

(π‘œ)

  • Temporal auto-correlation functions (how do they change over time)
  • Cross-correlation between different elements

Analytical approach- solve a lot of integrals Experimental approach- get the statistics from a transmission experiment

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Characterization: analytical approach

π‘†π‘š

(π‘œ) = ෍ 𝐽𝐷

෍

β„Ž,𝑙

π‘—π›Ώπ‘Œβ„Ž,𝑙,π‘š π‘β„Žπ‘π‘™

βˆ—

ACF Ξ”π‘œ = 𝔽 π‘†π‘š

π‘œ π‘†π‘š π‘œ+Ξ”π‘œ βˆ—

= ෍

𝐽𝐷

෍

β„Ž,𝑙

π‘—π›Ώπ‘Œβ„Ž,𝑙,π‘šπ‘Œβ„Žβ€²+Ξ”π‘œ,𝑙′+Ξ”π‘œ,π‘š

βˆ—

𝔽[π‘β„Žπ‘π‘™

βˆ—π‘β„Žβ€²π‘π‘™β€² βˆ— ]

The channel coefficients are unknown, but we can describe their statistics Autocorrelation function: Surprisingly, we can find these functions analytically (with some numeric integration…)

  • Dependent on: link structure, bandwidth & frequency of ICs, modulation format…
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Characterization: analytical approach

Symbol delay Ξ”π‘œ Symbol delay Ξ”π‘œ 𝔽 π‘†π‘š

(π‘œ)π‘†π‘š π‘œβ€² βˆ— βˆ’ 𝔽 π‘†π‘š 2

5x100km link, 32GBuad 500km link with distributed amp, 32GBuad Dots= SSFM results, lines= model predictions l=0 𝔽 π‘†π‘š

(π‘œ)π‘†π‘š π‘œβ€² βˆ— βˆ’ 𝔽 π‘†π‘š 2

Golani et al, β€œCorrelations and phase noise in NLIN- modelling and system implications,” OFC 2016

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π‘‘π‘œ = π‘π‘œ + 𝑗𝑆0

(π‘œ)π‘π‘œ + 𝑗𝑆1 (π‘œ)π‘π‘œβˆ’1 + 𝑗𝑆2 (π‘œ)π‘π‘œβˆ’2 … + π‘₯π‘œ

π‘‘π‘œ βˆ’ π‘π‘œ π‘π‘œβˆ’π‘€ = 𝑗𝑆𝑀

(π‘œ) + π‘ π‘“π‘‘π‘—π‘’π‘£π‘π‘š

Measuring ISI coefficient:

Characterization: experimental approach

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π‘‘π‘œ βˆ’ π‘π‘œ π‘π‘œ Zero-order (phase noise): π‘‘π‘œ βˆ’ π‘π‘œ π‘π‘œβˆ’1 First-order: π‘‘π‘œ βˆ’ π‘π‘œ π‘π‘œβˆ’2 Second-order: π‘‘π‘œ βˆ’ π‘π‘œ π‘π‘œβˆ’3 Third-order:

20x101km link, 7 WDM channels, 40GBuad

The summation is infinite, but the variance of coefficients drops rapidly

Characterization: experimental approach

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Recirculating loop experiment:

  • 64-QAM, dual polarization
  • 40GBaud
  • 101km spans
  • 42.5GHz channel spacing

Experimental setup

Joint work with UCL,

Golani et al, β€œExperimental characterization of the time correlation properties of nonlinear interference noise,” ECOC 2017

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Results- measuring the ACFs

0th order 1st order 2nd order 3rd order

Effect of transmission distance:

*7 WDM channels

Effect of number

  • f ICs:

*2000km link

Can also find cross correlations and other moments from these measurements

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Applications of time-varying ISI model

Simulation and performance estimation

  • β€œVirtual lab” tool- a fast alternative to split-step simulations
  • Predict system performance in the presence of nonlinearity, including

interaction between NLIN and the receiver’s DSP

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Channel model

π‘‘π‘œ = π‘π‘œ + π‘₯π‘œ + ෍

𝐽𝐷

෍

β„Ž,𝑙,π‘š

π‘—π›Ώπ‘Œβ„Ž,𝑙,π‘š π‘β„Žπ‘π‘™

βˆ—π‘π‘š

Channel of interest Interfering channels

Received signal Transmitted symbol AWGN Nonlinear interference

π‘π‘œ π‘π‘œ,IC𝑂 π‘π‘œ,IC1

…

π‘‘π‘œ

mux demux

π‘₯π‘œ

Fiber link

βŠ•

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Simplified channel model

  • O. Golani at Al, β€œModeling the Bit-Error-Rate Performance of Nonlinear Fiber-Optic System,” JLT (2016)
  • O. Golani at Al, β€œCorrelations and phase noise in NLIN- modelling and system implications,” OFC (2016)

ΰ·¨ π‘†π‘œ βŠ— βŠ• βŠ• π‘₯π‘œ π‘π‘œ π‘‘π‘œ

෍

π‘š

ΰ·¨ π‘†π‘œπ‘π‘œ

Elements ΰ·¨ π‘†π‘œ are created artificially If the statistics of the artificial ΰ·¨ π‘†π‘œ are the same as those of π‘†π‘œ, the simplified channel model will behave like the original model

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SLIDE 18

Performed with 11 WDM channels, 500km link Dots= SSFM results, solid lines= model predictions, dashed lines= AWGN model

Lumped Distributed

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A virtual lab for performance assessment

20 40 60 80

Number of channels Bit-error-rate

0.5 dB in Q

Variance based model Virtual Lab

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Applications of time-varying ISI model

Design algorithms for nonlinearity mitigation

  • Use explicit knowledge of the statistics of NLIN to design equalizers tailored

for nonlinearity mitigation

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NLIN mitigation using equalization

Viterbi algorithm Kalman filter Survivor sequences symbol estimations Ϋ§ |ො π‘π‘œ+𝑀 … Ϋ§ |ො π‘π‘œβˆ’π‘€ Ϋ§ |π‘‘π‘œ Ϋ§ |ො π‘π‘œ ΰ·‘ π‘Ίπ‘š

  • O. Golani at Al., β€œKalman-MLSE equalization of nonlinear noise,” OFC (2017)
  • O. Golani at Al., β€œEqualization Methods for NLIN Mitigation,” submitted to JLT

We can use explicit knowledge of ISI statistics to design better equalizers. The equalizer evaluates the ISI coefficients, and attempts to cancel their effect.

Filter uses the statistics of NLIN

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Application of statistical characterization: NLIN mitigation

RLS 1 tap 3 taps 5 taps No adaptive equalizer RLS 1 tap 3 taps 5 taps

5 tap filter requires to measure the ISI coefficients π‘†βˆ’2 … 𝑆2

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Conclusions

  • The time varying-ISI model: a powerful tool to treat fiber nonlinearity
  • Key idea: converting a nonlinear problem into a linear-time varying model
  • Can use this model to import techniques from RF communication to optical

communication

Thanks for listening!