Deep learning and model predictive control for self-tuning mode-locked lasers

Thomas Baumeister; Steven L. Brunton; J. Nathan Kutz

doi:10.1364/JOSAB.35.000617

Journal of the Optical Society of America B
Vol. 35,
Issue 3,
pp. 617-626
(2018)
•https://doi.org/10.1364/JOSAB.35.000617

Deep learning and model predictive control for self-tuning mode-locked lasers

Thomas Baumeister, Steven L. Brunton, and J. Nathan Kutz

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Thomas Baumeister, Steven L. Brunton, and J. Nathan Kutz, "Deep learning and model predictive control for self-tuning mode-locked lasers," J. Opt. Soc. Am. B 35, 617-626 (2018)

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Related Topics
Table of Contents Category
- Lasers and Laser Optics
Optics & Photonics Topics
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The topics in this list come from the Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Optical systems (120.4820)
- Lasers, fiber (140.3510)
- Mode-locked lasers (140.4050)
- Robotic and machine control (150.5758)

About this Article
History
- Original Manuscript: November 3, 2017
- Revised Manuscript: January 17, 2018
- Manuscript Accepted: January 17, 2018
- Published: February 20, 2018

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Abstract

Self-tuning optical systems are of growing importance in technological applications such as mode-locked fiber lasers. Such self-tuning paradigms require intelligent algorithms capable of inferring approximate models of the underlying physics and discovering appropriate control laws in order to maintain robust performance for a given objective. In this work, we demonstrate the first integration of a deep-learning (DL) architecture with model predictive control (MPC) in order to self-tune a mode-locked fiber laser. Not only can our DL-MPC algorithmic architecture approximate the unknown fiber birefringence, it also builds a dynamical model of the laser and appropriate control law for maintaining robust, high-energy pulses despite a stochastically drifting birefringence. We demonstrate the effectiveness of this method on a fiber laser that is mode-locked by nonlinear polarization rotation. The method advocated can be broadly applied to a variety of optical systems that require robust controllers.

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[Crossref]

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2015 (3)

U. Andral, R. S. Fodil, F. Amrani, F. Billard, E. Hertz, and P. Grelu, “Fiber laser mode locked through an evolutionary algorithm,” Optica 2, 275–278 (2015).
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[Crossref]

2014 (2)

X. Fu, S. L. Brunton, and J. N. Kutz, “Classification of birefringence in mode-locked fiber lasers using machine learning and sparse representation,” Opt. Express 22, 8585–8597 (2014).
[Crossref]

S. L. Brunton, X. Fu, and J. N. Kutz, “Self-tuning fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 20, 464–471 (2014).
[Crossref]

2013 (3)

S. L. Brunton, X. Fu, and J. N. Kutz, “Extremum-seeking control of a mode-locked laser,” IEEE J. Quantum Electron. 49, 852–861 (2013).
[Crossref]

D. Radnatarov, S. Khripunov, S. Kobtsev, A. Ivanenko, and S. Kukarin, “Automatic electronic-controlled mode locking self-start in fibre lasers with non-linear polarisation evolution,” Opt. Express 21, 20626–20631 (2013).
[Crossref]

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[Crossref]

2012 (2)

X. Shen, W. Li, M. Yan, and H. Zeng, “Electronic control of nonlinear-polarization-rotation mode locking in Yb-doped fiber lasers,” Opt. Lett. 37, 3426–3428 (2012).
[Crossref]

E. Ding, W. H. Renninger, F. W. Wise, P. Grelu, E. Shlizerman, and J. N. Kutz, “High-energy passive mode-locking of fiber lasers,” Int. J. Opt. 2012, 1–17 (2012).
[Crossref]

2011 (1)

J. H. Lee, “Model predictive control: Review of the three decades of development,” Int. J. Control Autom. Syst. 9, 415–424 (2011).
[Crossref]

2010 (1)

D. Richardson, J. Nilsson, and W. Clarkson, “High power fiber lasers: current status and future perspectives,” J. Opt. Soc. Am. B 27, B63–B92 (2010).
[Crossref]

2009 (4)

S. Mohanty, “Artificial neural network based system identification and model predictive control of a flotation column,” J. Process Control 19, 991–999 (2009).
[Crossref]

H. Peng, J. Wu, G. Inoussa, Q. Deng, and K. Nakano, “Nonlinear system modeling and predictive control using the RBF nets-based quasi-linear ARX model,” Control Eng. Pract. 17, 59–66 (2009).
[Crossref]

C.-C. Tsai, S.-C. Lin, T.-Y. Wang, and F.-J. Teng, “Stochastic model reference predictive temperature control with integral action for an industrial oil-cooling process,” Control Eng. Pract. 17, 302–310 (2009).
[Crossref]

E. Ding and J. N. Kutz, “Operating regimes, split-step modeling, and the Haus master mode-locking model,” J. Opt. Soc. Am. B 26, 2290–2300 (2009).
[Crossref]

2008 (2)

A. Grancharova, J. Kocijan, and T. A. Johansen, “Explicit stochastic predictive control of combustion plants based on gaussian process models,” Automatica 44, 1621–1631 (2008).
[Crossref]

X. Wu, V. Kumar, J. R. Quinlan, J. Ghosh, Q. Yang, H. Motoda, G. J. McLachlan, A. Ng, B. Liu, S. Y. Philip, Z.-H. Zhou, M. Steinbach, D. J. Hand, and D. Steinberg, “Top 10 algorithms in data mining,” Knowl. Inf. Syst. 14, 1–37 (2008).
[Crossref]

2006 (3)

J. N. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48, 629–678 (2006).
[Crossref]

G. E. Hinton, S. Osindero, and Y.-W. Teh, “A fast learning algorithm for deep belief nets,” Neural Comput. 18, 1527–1554 (2006).
[Crossref]

G. Hinton and R. Salakhutdinov, “Reducing the dimensionality of data with neural networks,” Science 313, 504–507 (2006).
[Crossref]

2000 (3)

M. Krstić and H. Wang, “Stability of extremum seeking feedback for general nonlinear dynamic systems,” Automatica 36, 595–601 (2000).
[Crossref]

H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 1173–1185 (2000).
[Crossref]

O. Albert, L. Sherman, G. Mourou, T. Norris, and G. Vdovin, “Smart microscope: an adaptive optics learning system for aberration correction in multiphoton confocal microscopy,” Opt. Lett. 25, 52–54 (2000).
[Crossref]

1997 (1)

S. Hochreiter and S. Jürgem, “Long short-term memory,” Neural Comput. 9, 1735–1780 (1997).
[Crossref]

1992 (1)

H. Weisberg Andersen and M. Kümmel, “Evaluating estimation of gain directionality,” J. Process Control 2, 67–86 (1992).
[Crossref]

1990 (1)

K. Hornik, M. Stinchcombe, and H. White, “Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks,” Neural Netw. 3, 551–560 (1990).
[Crossref]

1989 (3)

G. Cybenko, “Approximation by superpositions of a sigmoidal function,” Math. Control Signals Syst. 2, 303–314 (1989).
[Crossref]

K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Netw. 2, 359–366 (1989).
[Crossref]

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[Crossref]

1941 (1)

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[Crossref]

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Agåker, M.

Al Khawli, T.

Al-Arashi, W. H.

Albert, O.

Amrani, F.

U. Andral, J. Buguet, R. S. Fodil, F. Amrani, F. Billard, E. Hertz, and P. Grelu, “Toward an autosetting mode-locked fiber laser cavity,” J. Opt. Soc. Am. B 33, 825–833 (2016).
[Crossref]

U. Andral, R. S. Fodil, F. Amrani, F. Billard, E. Hertz, and P. Grelu, “Fiber laser mode locked through an evolutionary algorithm,” Optica 2, 275–278 (2015).
[Crossref]

Anandkumar, A.

H. Sedghi and A. Anandkumar, “Training input-output recurrent neural networks through spectral methods,” arXiv:1603.00954 (2016).

Andral, U.

U. Andral, J. Buguet, R. S. Fodil, F. Amrani, F. Billard, E. Hertz, and P. Grelu, “Toward an autosetting mode-locked fiber laser cavity,” J. Opt. Soc. Am. B 33, 825–833 (2016).
[Crossref]

U. Andral, R. S. Fodil, F. Amrani, F. Billard, E. Hertz, and P. Grelu, “Fiber laser mode locked through an evolutionary algorithm,” Optica 2, 275–278 (2015).
[Crossref]

Antonoglou, I.

Barillot, T. R.

Beattie, C.

Belforte, D.

G. Overton, A. Nogee, D. Belforte, and C. Holton, “Annual laser market review & forecast: where have all the lasers gone?” Laser Focus World 53, 1–24 (2017).

Bellemare, M. G.

Bengio, Y.

Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521, 436–444 (2015).
[Crossref]

X. Glorot and Y. Bengio, “Understanding the difficulty of training deep feedforward neural networks,” in 13th International Conference on Artificial Intelligence and Statistics, Machine Learning Research, Y. W. Teh and M. Titterington, eds. (PMLR, 2010), vol. 9, pp. 249–256.

I. Goodfellow, Y. Bengio, and A. Courville, Deep Learning (MIT, 2016).

Berrah, N.

Billard, F.

U. Andral, J. Buguet, R. S. Fodil, F. Amrani, F. Billard, E. Hertz, and P. Grelu, “Toward an autosetting mode-locked fiber laser cavity,” J. Opt. Soc. Am. B 33, 825–833 (2016).
[Crossref]

U. Andral, R. S. Fodil, F. Amrani, F. Billard, E. Hertz, and P. Grelu, “Fiber laser mode locked through an evolutionary algorithm,” Optica 2, 275–278 (2015).
[Crossref]

Bostedt, C.

Bozek, J. D.

Brunton, S. L.

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Acta Autom. Sin. (1)

Y.-G. Xi, D.-W. Li, and S. Lin, “Model predictive control — status and challenges,” Acta Autom. Sin. 39, 222–236 (2013).
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Automatica (3)

C. E. Garcia, D. M. Prett, and M. Morari, “Model predictive control: theory and practice—a survey,” Automatica 25, 335–348 (1989).
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M. Krstić and H. Wang, “Stability of extremum seeking feedback for general nonlinear dynamic systems,” Automatica 36, 595–601 (2000).
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Control Eng. Pract. (2)

IEEE J. Quantum Electron. (1)

S. L. Brunton, X. Fu, and J. N. Kutz, “Extremum-seeking control of a mode-locked laser,” IEEE J. Quantum Electron. 49, 852–861 (2013).
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IEEE J. Sel. Top. Quantum Electron. (2)

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H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 1173–1185 (2000).
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IEEE Photon. Technol. Lett. (1)

Int. J. Control Autom. Syst. (1)

J. H. Lee, “Model predictive control: Review of the three decades of development,” Int. J. Control Autom. Syst. 9, 415–424 (2011).
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Int. J. Opt. (1)

E. Ding, W. H. Renninger, F. W. Wise, P. Grelu, E. Shlizerman, and J. N. Kutz, “High-energy passive mode-locking of fiber lasers,” Int. J. Opt. 2012, 1–17 (2012).
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J. Lightwave Technol. (1)

D. Zibar, M. Piels, R. Jones, and C. G. Schäeffer, “Machine learning techniques in optical communication,” J. Lightwave Technol. 34, 1442–1452 (2016).
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J. Opt. Soc. Am. B (3)

E. Ding and J. N. Kutz, “Operating regimes, split-step modeling, and the Haus master mode-locking model,” J. Opt. Soc. Am. B 26, 2290–2300 (2009).
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U. Andral, J. Buguet, R. S. Fodil, F. Amrani, F. Billard, E. Hertz, and P. Grelu, “Toward an autosetting mode-locked fiber laser cavity,” J. Opt. Soc. Am. B 33, 825–833 (2016).
[Crossref]

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[Crossref]

J. Process Control (2)

S. Mohanty, “Artificial neural network based system identification and model predictive control of a flotation column,” J. Process Control 19, 991–999 (2009).
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Knowl. Inf. Syst. (1)

Laser Focus World (1)

G. Overton, A. Nogee, D. Belforte, and C. Holton, “Annual laser market review & forecast: where have all the lasers gone?” Laser Focus World 53, 1–24 (2017).

Math. Control Signals Syst. (1)

G. Cybenko, “Approximation by superpositions of a sigmoidal function,” Math. Control Signals Syst. 2, 303–314 (1989).
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Nat. Commun. (1)

Nature (3)

Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521, 436–444 (2015).
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Neural Comput. (2)

S. Hochreiter and S. Jürgem, “Long short-term memory,” Neural Comput. 9, 1735–1780 (1997).
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G. E. Hinton, S. Osindero, and Y.-W. Teh, “A fast learning algorithm for deep belief nets,” Neural Comput. 18, 1527–1554 (2006).
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K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Netw. 2, 359–366 (1989).
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Opt. Express (3)

Opt. Lett. (2)

X. Shen, W. Li, M. Yan, and H. Zeng, “Electronic control of nonlinear-polarization-rotation mode locking in Yb-doped fiber lasers,” Opt. Lett. 37, 3426–3428 (2012).
[Crossref]

Optica (1)

U. Andral, R. S. Fodil, F. Amrani, F. Billard, E. Hertz, and P. Grelu, “Fiber laser mode locked through an evolutionary algorithm,” Optica 2, 275–278 (2015).
[Crossref]

Prod. Eng. (1)

Sci. Rep. (1)

R. Woodward and E. Kelleher, “Towards ‘smart lasers’: self-optimisation of an ultrafast pulse source using a genetic algorithm,” Sci. Rep. 6, 37616 (2016).

Science (1)

G. Hinton and R. Salakhutdinov, “Reducing the dimensionality of data with neural networks,” Science 313, 504–507 (2006).
[Crossref]

SIAM Rev. (1)

J. N. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48, 629–678 (2006).
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Fig. 1. Schematic of the self-tuning fiber laser. (a) The laser cavity, the optic components, and the laser’s objective function are discussed in Section 3.A. (b) The Variational Autoencoder is discussed in Section 3.B; (c) the Latent Variable Mapping in Section 3.C; and (d) the Model Prediction in Section 3.D.

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Fig. 2. Typical stable mode-locking dynamics where a stable pulse is formed from initial noise in the cavity. The objective function of Eq. (1) is aimed at producing temporally short, high-energy pulses. Thus if the kurtosis is small, then the pulse is tightly confined in time since the kurtosis is in the denominator of the objective function.

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Fig. 3. Schematic of the Deep Learning Controller. The inputs to the controller are sequences of the states of the laser

E

M

and of the control inputs

α

. The Model Prediction is a RNN that first predicts the birefringence

K_{t + 1}

of time step

t + 1

and maps it to good initial control inputs

α_{t + 1}

. Second, the system’s states

v_{t + 1}

are predicted. This is done recurrently to predict

N

time steps in the future. Then, the control inputs are updated such that the objective function is optimized. The optimized control inputs

α_{t + Δ t : t + N Δ t}

are used to regulate the laser system for the next

N

time steps. Once the difference between the prediction and the true output exceeds a certain threshold, the VAE is used to infer

K

and, then,

K - α

mapping maps it to the control input

α

. This inner loop is necessary to stabilize the control system.

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Fig. 4. Comparison of the true birefringence (blue line) and the samples from the two dimensional VAE’s latent space. While the samples from the first dimension seem to capture just random noise, the samples from the second dimension follow the true birefringence with high accuracy.

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Fig. 5. Performance of the Deep Learning Control despite significant sinusoidal change in birefringence over time. Without control, the objective function plummets and results in failure of the fiber laser to mode lock. With DL-MPC, the system remains at a high-performance mode-locked state.

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Fig. 6. Same as Fig. 5 but with random changes in birefringence. The DL-MPC again stabilizes the objective function of the system at a high level.

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Fig. 7. Fully connected deep neural network to map the latent variable

K

to good initial control inputs

u

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Equations (21)

Equations on this page are rendered with MathJax. Learn more.

O = E / M,

i \frac{\partial u}{\partial z} + \frac{D}{2} \frac{\partial^{2} u}{\partial t^{2}} - K u + ({| u |}^{2} + A {| v |}^{2}) u + B v^{2} u^{*} = i g (z) (1 + τ \frac{\partial^{2}}{\partial t^{2}}) u - i Γ u,

i \frac{\partial v}{\partial z} + \frac{D}{2} \frac{\partial^{2} v}{\partial t^{2}} + K v + (A {| u |}^{2} + {| v |}^{2}) v + B u^{2} v^{*} = i g (z) (1 + τ \frac{\partial^{2}}{\partial t^{2}}) v - i Γ v .

g (z) = \frac{2 g_{0}}{1 + \frac{1}{E_{0}} \int_{- \infty}^{\infty} ({| u |}^{2} + {| v |}^{2}) d t},

W_{\frac{λ}{4}} = (\begin{matrix} e^{- i π / 4} & 0 \\ 0 & e^{i π / 4} \end{matrix}), W_{\frac{λ}{2}} = (\begin{matrix} - i & 0 \\ 0 & i \end{matrix}), W_{p} = (\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}) .

J_{j} = R (α_{j}) W R (- α_{j}),

p (x) = \int_{- \infty}^{\infty} p (x | z) p (z) d z = \frac{p (x | z; θ) p (z)}{p (z | x; θ)},

K L (q (z | x; ϕ) | | p (z | x; θ)) = E L B O (ϕ, θ) + \log p (x),

ELBO (ϕ, θ) = E_{q} [\log p (z | x; θ)] - E_{q} [\log q (z | x; ϕ)] .

q^{*} (z | x; ϕ) = argmin KL (q (z | x; ϕ) | | p (z | x; θ)) .

{ELBO}_{i} (ϕ, θ) = E_{q} [\log p (x_{i} | z; θ)] - K L (q (z | x_{i}; ϕ) | | p (z)) .

L = \frac{1}{2} {‖ \hat{u} - u ‖}_{2}^{2} .

\underset{u_{t + 1 : t + N}}{\arg \max} O_{t + 1 : t + N} = \underset{u_{t + 1 : t + N}}{\arg \max} {\frac{E}{M}}_{t + 1 : t + N} .

h_{l, past} = relu (\sum_{i} W_{l, past}^{i} x_{t - 2 b : t - b - 1}^{i} + b_{l, past}),

h_{l, current} = relu (\sum_{i} W_{l, current}^{i} x_{t - b : t}^{i} + b_{l, current}),

I_{current} = relu (\sum_{i} W_{h l}^{i} h_{l, past}^{i} h_{l, current}^{i} + \sum_{k} W_{l l}^{k} I_{past}^{k} + b_{h l}),

h_{current} = relu (\sum_{i} W_{current}^{i} x_{t - b : t}^{i} + b_{current}),

h_{future} = relu (\sum_{i} W_{future}^{i} u_{t - b + 1 : t + 1} + b_{future}),

h_{latent} = relu (\sum_{i} W_{l}^{i} I_{current}^{i} + b_{latent}),

K_{t + 1} = \sum_{i} W_{K o}^{i} h_{latent}^{i} + b_{K o} .

v_{t + 1} = \sum_{i} W_{o}^{i} h_{latent}^{i} h_{current}^{i} h_{future}^{i} + b_{o} .