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

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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.

© 2018 Optical Society of America

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Figures (7)

Fig. 1.
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.
Fig. 2.
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.
Fig. 3.
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.
Fig. 4.
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.
Fig. 5.
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.
Fig. 6.
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.
Fig. 7.
Fig. 7. Fully connected deep neural network to map the latent variable K to good initial control inputs u .

Equations (21)

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O = E / M ,
i u z + D 2 2 u t 2 K u + ( | u | 2 + A | v | 2 ) u + B v 2 u * = i g ( z ) ( 1 + τ 2 t 2 ) u i Γ u ,
i v z + D 2 2 v t 2 + K v + ( A | u | 2 + | v | 2 ) v + B u 2 v * = i g ( z ) ( 1 + τ 2 t 2 ) v i Γ v .
g ( z ) = 2 g 0 1 + 1 E 0 ( | u | 2 + | v | 2 ) d t ,
W λ 4 = ( e i π / 4 0 0 e i π / 4 ) , W λ 2 = ( i 0 0 i ) , W p = ( 1 0 0 0 ) .
J j = R ( α j ) W R ( α j ) ,
p ( x ) = p ( x | z ) p ( z ) d z = 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 = 1 2 u ^ u 2 2 .
arg max u t + 1 : t + N O t + 1 : t + N = arg max u t + 1 : t + N { E M } t + 1 : t + N .
h l , past = relu ( i W l , past i x t 2 b : t b 1 i + b l , past ) ,
h l , current = relu ( i W l , current i x t b : t i + b l , current ) ,
I current = relu ( i W h l i h l , past i h l , current i + k W l l k I past k + b h l ) ,
h current = relu ( i W current i x t b : t i + b current ) ,
h future = relu ( i W future i u t b + 1 : t + 1 + b future ) ,
h latent = relu ( i W l i I current i + b latent ) ,
K t + 1 = i W K o i h latent i + b K o .
v t + 1 = i W o i h latent i h current i h future i + b o .
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