Ultrafast electro-optic delay Reservoir
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1 Ultrafast electro-optic delay Reservoir Laurent Larger 1, A. Baylón Fuentes 1, R. Martinenghi 1, M. Jacquot 1, Y.K. Chembo 1, and V.S. Udaltsov 1,2 1 University Bourgogne Franche-Comté, FEMTO-ST institute / Optics Dpt. 15B, Avenue des Montboucons, Besançon, France 2 Institute for Laser Physics, Saint-Petersburg, Russia Dynamical Systems and Brain-Inspired Information Processing 2-3 Nov. 2015, Univ. Bourgogne Franche-Comté, LMB & FEMTO-ST, Besançon, France
2 Outline Introduction Electro-Optic phase delay dynamics RC with EO phase delay dynamics Conclusions Hidden bonus slides Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 1 / 12
3 Background and Motivation Reservoir Computing / Nonlinear Transient Computing Requires a complex dynamical system (e.g. network of neurons) Delay dynamics demonstrated as efficient for photonic hardware RC Photonic setups have potential for high processing speed Complex delay FEMTO-ST Wavelength delay dynamics (patent 96) for chaos communications Electro-optic intensity chaos (IEEE JQE 2001), field experiment in chaos commun. (Nature 2005) EO Phase chaos: State of the art in optical chaos comm., 10 Gb/s & 10 7 BER over 120 km (IEEE JQE 2010) First photonic RC (Opt. Expr. Jan. 2012) First chimera states with electronic (Phys. Rev. Lett. 2013) and photonic (Nat. Comm. 2015) delay systems Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 2 / 12
4 Outline Introduction Electro-Optic phase delay dynamics RC with EO phase delay dynamics Conclusions Hidden bonus slides Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 2 / 12
5 Broadband and low noise EO chaos generator Electro-optic phase delay dynamics Setup, physical principles. DPSK optical modulation Temporally nonlocal non linearity Intrinsically high speed Signal injection by external ΦM Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 3 / 12
6 Broadband and low noise EO chaos generator Electro-optic phase delay dynamics Setup, physical principles. DPSK optical modulation Temporally nonlocal non linearity Intrinsically high speed Signal injection by external ΦM Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 3 / 12
7 Broadband and low noise EO chaos generator Electro-optic phase delay dynamics Setup, physical principles. DPSK optical modulation Temporally nonlocal non linearity Intrinsically high speed Signal injection by external ΦM Chaotic ΦM in the optical spectrum (Lavrov et al., Phys. Rev. E 2009) Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 3 / 12
8 EO phase setup: Modeling RF Bandpass filter, DPSK NL delayed feedback Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 4 / 12
9 EO phase setup: Modeling RF Bandpass filter, DPSK NL delayed feedback Integro-differential (linear bandpass filter) nonlinear delay equation 1 θ t t 0 ϕ(ξ) dξ + ϕ(t) + τ dϕ dt (t) = β [f (t τd )(ϕ ) ] Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 4 / 12
10 EO phase setup: Modeling RF Bandpass filter, DPSK NL delayed feedback Integro-differential (linear bandpass filter) nonlinear delay equation 1 θ t t 0 ϕ(ξ) dξ + ϕ(t) + τ dϕ dt (t) = β [f (t τd )(ϕ ) ] Non linearity via imbalanced interferometer (temporal non locality) Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 4 / 12
11 EO phase setup: Modeling RF Bandpass filter, DPSK NL delayed feedback Integro-differential (linear bandpass filter) nonlinear delay equation 1 θ t t 0 ϕ(ξ) dξ + ϕ(t) + τ dϕ dt (t) = β [f (t τd )(ϕ ) ] Non linearity via imbalanced interferometer (temporal non locality) Standard DPSK demodulator f t (ϕ) = {1 + cos[ϕ(t) ϕ(t δt) + Φ 0 ]} Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 4 / 12
12 EO phase setup: Modeling RF Bandpass filter, DPSK NL delayed feedback Integro-differential (linear bandpass filter) nonlinear delay equation 1 θ t t 0 ϕ(ξ) dξ + ϕ(t) + τ dϕ dt (t) = β [f (t τd )(ϕ ) ] Non linearity via imbalanced interferometer (temporal non locality) Standard DPSK demodulator f t (ϕ) = {1 + cos[ϕ(t) ϕ(t δt) + Φ 0 ]} Generalized multiple wave interferometer f t (ϕ) = F k α k e i[ϕ(t) ϕ(t δtk)+φk] 2 Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 4 / 12
13 EO phase setup: Exp. & Num. behavior Φ-chaos: 4 time-scales (θ τ D δt τ) Temporal bif. diagrams Time traces Spectral bif. diagram Flat chaotic rf spectrum (Lavrov et al., Phys. Rev. E 2009; Weicker et al., Phys.Rev. E 2012). Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 5 / 12
14 Chaos Commun.: Field 10 Gb/s Lavrov et al., IEEE JQE (2010) Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 6 / 12
15 Outline Introduction Electro-Optic phase delay dynamics RC with EO phase delay dynamics Conclusions Hidden bonus slides Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 6 / 12
16 NTC operation of the EO phase setup Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 7 / 12
17 NTC operation of the EO phase setup Amplitude parameters Input ΦM amplitude: 1.2π feedback gain: β 0.7 offset phase: Φ 0 2π/5 (nearly parabolic) Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 7 / 12
18 NTC operation of the EO phase setup Amplitude parameters Input ΦM amplitude: 1.2π feedback gain: β 0.7 offset phase: Φ 0 2π/5 (nearly parabolic) Time parameters Loop filter bandwidth: 566 MHz δτ 56.8 ps (AWG limited, 17.6 GS/s) Time delay: τ D ns (a few meters of fiber) internal input sample memory: virtual nodes / input sample, or 1113 virtual nodes / time delay: hidden layers within the delay Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 7 / 12
19 NTC operation of the EO phase setup Amplitude parameters Input ΦM amplitude: 1.2π feedback gain: β 0.7 offset phase: Φ 0 2π/5 (nearly parabolic) Time parameters Loop filter bandwidth: 566 MHz δτ 56.8 ps (AWG limited, 17.6 GS/s) Time delay: τ D ns (a few meters of fiber) internal input sample memory: virtual nodes / input sample, or 1113 virtual nodes / time delay: hidden layers within the delay Unvealing the RNN emulation Delay dynamics as a convolution product: Space-time analogy (Larger et al., Nat. Commun. 2015) Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 7 / 12
20 NTC operation of the EO phase setup ESN model: discrete time & space Input layer: u in k (n) = Q q=1 wi kq u q(n) [ K ] Reservoir dynamics: x k(n) = f NL j=1 wr kj x j(n 1) + u in k (n) Read Out: y m(n) = K k=1 wr mk x k(n) Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 8 / 12
21 NTC operation of the EO phase setup x(t) = ESN model: discrete time & space Input layer: u in k (n) = Q q=1 wi kq u q(n) [ K ] Reservoir dynamics: x k(n) = f NL j=1 wr kj x j(n 1) + u in k (n) Read Out: y m(n) = K k=1 wr mk x k(n) Delay NTC model: discrete time, continuous virtual space t h(t ξ) f NL [x(ξ τ D)] dξ x k(n) = x k(n 1)+ σk σ k τ D h(σ k σ) f NL [x σ(n 1)] dσ Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 8 / 12
![NTC operation of the EO phase setup x(t) = ESN model: discrete time & space Input layer: u in k (n) = Q q=1 wi kq u q(n) [ K ] Reservoir dynamics: x k(n) = f NL j=1 wr kj x](/docs-images/95/123073998/images/22-2.jpg "j(n 1) + u in k (n) Read Out: y m(n) = K k=1 wr mk x k(n) Delay NTC model: discrete time, continuous virtual space t h(t ξ) f NL [x(ξ τ D)] dξ x k(n) = x k(n 1)+ Input layer:")
![u I σ(n) = K k=1 Reservoir dynamics: x k(n) = x k(n 1) + σ k Read Out: y m(n) = K k=1 wr mk x σk [ Q ] q=1 wi kq u q(n) p δτ (σ σ k) σ k τ D h(σ k σ) f NL [x σ(n 1)] dσ [ σ k](/docs-images/95/123073998/images/22-3.jpg "τ D h(σ σ k) f NL xσ(n 1) + ρ u I σ(n 1) ] dσ ( ) [ [ n τ D NL + σk R, with σk R 0; τ D NL Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 8 / 12")
22 NTC operation of the EO phase setup x(t) = ESN model: discrete time & space Input layer: u in k (n) = Q q=1 wi kq u q(n) [ K ] Reservoir dynamics: x k(n) = f NL j=1 wr kj x j(n 1) + u in k (n) Read Out: y m(n) = K k=1 wr mk x k(n) Delay NTC model: discrete time, continuous virtual space t h(t ξ) f NL [x(ξ τ D)] dξ x k(n) = x k(n 1)+ Input layer: u I σ(n) = K k=1 Reservoir dynamics: x k(n) = x k(n 1) + σ k Read Out: y m(n) = K k=1 wr mk x σk [ Q ] q=1 wi kq u q(n) p δτ (σ σ k) σ k τ D h(σ k σ) f NL [x σ(n 1)] dσ [ σ k τ D h(σ σ k) f NL xσ(n 1) + ρ u I σ(n 1) ] dσ ( ) [ [ n τ D NL + σk R, with σk R 0; τ D NL Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 8 / 12
23 Dynamical Processing of Spoken Digits Input pre-processing Lyon Ear Model transformation (Time & Frequency 2D formatting, 60 Samples x 86 Freq.channel) Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 9 / 12
24 Dynamical Processing of Spoken Digits Input pre-processing Lyon Ear Model transformation (Time & Frequency 2D formatting, 60 Samples x 86 Freq.channel) Sparse connection of the 86 Freq. channel to the 371 neurons: random connection matrix Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 9 / 12
25 Dynamical Processing of Spoken Digits Input pre-processing Lyon Ear Model transformation (Time & Frequency 2D formatting, 60 Samples x 86 Freq.channel) Sparse connection of the 86 Freq. channel to the 371 neurons: random connection matrix Reservoir transient response: Time series record for Read-Out post-processing Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 9 / 12
26 Read-Out, Training, and Testing Training of the Read-Out with target output function Learning: optimization of the W matrix, for each different digit Regression problem for A W B: W opt = (A T A λi) 1 A T B Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 10 / 12
27 Read-Out, Training, and Testing Training of the Read-Out with target output function Learning: optimization of the W matrix, for each different digit Regression problem for A W B: W opt = (A T A λi) 1 A T B Testing with training-defined Read-Out Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 10 / 12
28 Read-Out, Training, and Testing Training of the Read-Out with target output function Learning: optimization of the W matrix, for each different digit Regression problem for A W B: W opt = (A T A λi) 1 A T B Testing with training-defined Read-Out Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 10 / 12
29 Read-Out, Training, and Testing Training of the Read-Out with target output function Learning: optimization of the W matrix, for each different digit Regression problem for A W B: W opt = (A T A λi) 1 A T B Testing with training-defined Read-Out Test result: State of the art (close to 0% Word Error Rate) With Telecom Bandwidth setup: record speed recognition, 1M word/s Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 10 / 12
30 Experiments, practical details Optimal nonlinear operating point Non linearity: Next to an extrema Gain: slightly before the Hopf bifurcation point Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 11 / 12
31 Experiments, practical details Optimal nonlinear operating point Non linearity: Next to an extrema Gain: slightly before the Hopf bifurcation point Asynchronous Write-In and Read-Out More than 10 times improvement in the WER Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 11 / 12
32 Outline Introduction Electro-Optic phase delay dynamics RC with EO phase delay dynamics Conclusions Hidden bonus slides Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 11 / 12
33 Conclusion, and perspectives Ultrafast Photonic RC with EO phase delay dynamics Million Words per second recognition rate achieved Fine modeling supporting rigorously the ESN analogy of delay systems Concept of hidden layers in delay Reservoirs Found beneficial asynchrony between Write-In and Read-Out Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 12 / 12
34 Conclusion, and perspectives Ultrafast Photonic RC with EO phase delay dynamics Million Words per second recognition rate achieved Fine modeling supporting rigorously the ESN analogy of delay systems Concept of hidden layers in delay Reservoirs Found beneficial asynchrony between Write-In and Read-Out Future work On-line processing experiments Memory functionalities through chimera-like delay dynamics Explore the origine for beneficial Write-In vs. Read-Out asynchrony Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 12 / 12
35 Thank you for attention Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 12 / 12
36 Outline Introduction Electro-Optic phase delay dynamics RC with EO phase delay dynamics Conclusions Hidden bonus slides Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 12 / 12
37 A chaotic rainbow... A tentative artistic implementation of the wavelength chaos setup Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 12 / 12
38 A chaotic rainbow... A tentative artistic implementation of the wavelength chaos setup Delay dynamics on the color sliced by an AOTF from the rainbow of a SC white light source Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 12 / 12
39 A chaotic rainbow... A tentative artistic implementation of the wavelength chaos setup Delay dynamics on the color sliced by an AOTF from the rainbow of a SC white light source Friendly science demo (many diffracted rainbows with a chaotically moving dark line; human eye time scale compatible) Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 12 / 12
40 A chaotic rainbow... A tentative artistic implementation of the wavelength chaos setup Delay dynamics on the color sliced by an AOTF from the rainbow of a SC white light source Friendly science demo (many diffracted rainbows with a chaotically moving dark line; human eye time scale compatible) Easily transportable experiment (no optical table required) Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 12 / 12
41 A chaotic rainbow... A tentative artistic implementation of the wavelength chaos setup Delay dynamics on the color sliced by an AOTF from the rainbow of a SC white light source Friendly science demo (many diffracted rainbows with a chaotically moving dark line; human eye time scale compatible) Easily transportable experiment (no optical table required) Setup mimicking the shape of our new FEMTO-ST building in Besançon. 1 st exhibition at the (local) 2015 Science Fair last week end in Besançon, France. Dyn. Syst. and Brain-Insp. Inform. Process., 2-3 Nov. 2015, Besançon, France 12 / 12
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