## Classification of birefringence in mode-locked fiber lasers using machine learning and sparse representation |

Optics Express, Vol. 22, Issue 7, pp. 8585-8597 (2014)

http://dx.doi.org/10.1364/OE.22.008585

Acrobat PDF (2819 KB)

### Abstract

It has been observed that changes in the birefringence, which are difficult or impossible to directly measure, can significantly affect mode-locking in a fiber laser. In this work we develop techniques to estimate the effective birefringence by comparing a test measurement of a given objective function against a learned library. In particular, a toroidal search algorithm is applied to the laser cavity for various birefringence values by varying the waveplate and polarizer angles at incommensurate angular frequencies, thus producing a time-series of the objective function. The resulting time series, which is converted to a spectrogram and then dimensionally reduced with a singular value decomposition, is then labelled with the corresponding effective birefringence and concatenated into a library of modes. A sparse search algorithm (*L*_{1}-norm optimization) is then applied to a test measurement in order to classify the birefringence of the fiber laser. Simulations show that the sparse search algorithm performs very well in recognizing cavity birefringence even in the presence of noise and/or noisy measurements. Once classified, the wave plates and polarizers can be adjusted using servo-control motors to the optimal positions obtained from the toroidal search. The result is an efficient, self-tuning laser.

© 2014 Optical Society of America

## 1. Introduction

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

*qualitative*in nature. The underlying and primary reason which has prevented

*quantitative*modeling efforts is the fiber birefringence [2

2. C. D. Poole and R. E. Wagner, “Phenomenological approach to polarization dispersion in long single-mode fibers,” Electron. Lett. **22**, 1029–1030 (1986). [CrossRef]

5. P. K. A. Wai and C. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Light. Tech. **14**, 148–157 (1996). [CrossRef]

6. J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” PNAS **97**, 4541–4550 (2000). [CrossRef] [PubMed]

2. C. D. Poole and R. E. Wagner, “Phenomenological approach to polarization dispersion in long single-mode fibers,” Electron. Lett. **22**, 1029–1030 (1986). [CrossRef]

6. J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” PNAS **97**, 4541–4550 (2000). [CrossRef] [PubMed]

7. J. N. Kutz, “Mode-locked soliton lasers,” SIAM Review **48**, 629–678 (2006). [CrossRef]

8. H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quant. Elec. **6**, 1173–1185 (2000). [CrossRef]

^{−7}index of refraction difference in the two modes, the corresponding beat length

*L*is about 10 meters with variations occurring on lengths of 100 meters, which is often on the same order as the dispersive and/or nonlinear length scales. As a result, the birefringence can have a significant impact on mode-locking dynamics.

_{B}8. H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quant. Elec. **6**, 1173–1185 (2000). [CrossRef]

11. G. Lenz, K. Tamura, H. A. Haus, and E. P. Ippen, “All-solid-state femtosecond source at 1.55 μm,” Opt. Lett. **20**, 1289–1291 (1995). [CrossRef] [PubMed]

12. A. Chong, W. H. Renninger, and F. W. Wise, “Properties of normal-dispersion femtosecond fiber lasers,” J. Opt. Soc. Am. B **25**, 140–148 (2008). [CrossRef]

14. W. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A **77**, 023814 (2008). [CrossRef]

15. F.Ö. Ilday, J. Buckley, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser cavity,” Phys. Rev. Lett. **92**, 213902 (2004). [CrossRef]

17. B. Bale and S. Wabnitz, “Strong spectral filtering for a mode-locked similariton fiber laser,” Opt. Lett. **35**, 2466–2468 (2010). [CrossRef] [PubMed]

18. F. Li, P. K. A. Wai, and J. N. Kutz, “Geometrical description of the onset of multi-pulsing in mode-locked laser cavities,” J. Opt. Soc. Am. B **27**, 2068–2077 (2010). [CrossRef]

20. X. Fu and J. N. Kutz, “High-energy mode-locked fiber lasers using multiple transmission filters and a genetic algorithm,” Opt. Express **21**, 6526–6537 (2013). [CrossRef] [PubMed]

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

22. 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]

23. 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] [PubMed]

## 2. Fiber laser model and objective function

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

*local maximum*of the objective function instead of the desired

*global maximum*. As a result, we propose a data-driven technique which allows us to build a library of time series of the objective function that globally samples the entire parameter space for different birefringence values. Once a sufficiently large library is obtained, the state of fiber laser system can be then characterized by matching the current system behavior with library entries, hence the global optimal performance can be identified, and then subsequently maintained by ESC. In order to perform such a task, an example model laser system is introduced.

### 2.1. Laser Cavity Model

3. C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr media,” IEEE J. Quant. Electron. **25**, 2674–2682 (1989). [CrossRef]

4. C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quant. Electron. **23**, 174–176 (1987). [CrossRef]

*u*and

*v*represent two orthogonally polarized electric field envelopes (the fast- and slow-axis respectively) in the fiber cavity of an optical fiber with birefringence

*K*. The variable

*z*denotes the propagation distance which is normalized by the length of the first fiber section and

*t*is the retarded time normalized by the full-width at half-maximum of the pulse. The parameter

*D*is the averaged group velocity dispersion of the fiber section. It is positive for anomalous dispersion and negative for normal dispersion. The nonlinear coupling parameters A (cross-phase modulation) and B (four-wave mixing) are determined by the material of the optical fiber. For axially symmetric fibers

*A*= 2/3 and

*B*= 1/3. The right hand side of the equations, which are dissipative terms, account for the bandwidth limited gain saturation and attenuation, where the operator

*R*of the dissipative terms is defined as follows: Here

*g*

_{0}and

*e*

_{0}are the nondimensional pumping strength and the saturating energy of the gain. Parameter

*τ*characterizes the bandwidth of the pump, and Γ measures the losses (taken to be distributed) caused by the output coupling and the fiber attenuation.

*α*(

_{k}*k*= 1, 2, 3,

*p*), the Jones matrices are modified so that where

*W*is one of the given Jones matrices and

*R*is the rotation (alignment) matrix: This provides a full characterization of the waveplates and polarizers along with their alignment back to the principal axes of the fiber itself.

### 2.2. Objective Function

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

*O*was introduced which was obtained by dividing the pulse energy

*E*by the spectral kurtosis

*M*

_{4}(fourth-moment) of the wave form: This objective function, which has been shown to be successful for applying adaptive control, is large (optimal) when we have a large amount of energy in a tightly confined temporal wave packet [21

**49**, 852–861 (2013). [CrossRef]

*M*

_{4}measures the spread of the waveform.

*α*

_{3}and polarizer

*α*. The objective function selected is ideal for optimizing pulse energy while simultaneously keeping the mode-locking away from instability boundaries (gray regions) [21

_{p}**49**, 852–861 (2013). [CrossRef]

## 3. Toroidal search and library building

22. 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]

23. 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] [PubMed]

### 3.1. Toroidal Search

*π*, thereby creating a parameter space that is a 4-torus. Multiple NPR sections can be included, let’s say

*N*of them, to further enhance performance [18

18. F. Li, P. K. A. Wai, and J. N. Kutz, “Geometrical description of the onset of multi-pulsing in mode-locked laser cavities,” J. Opt. Soc. Am. B **27**, 2068–2077 (2010). [CrossRef]

20. X. Fu and J. N. Kutz, “High-energy mode-locked fiber lasers using multiple transmission filters and a genetic algorithm,” Opt. Express **21**, 6526–6537 (2013). [CrossRef] [PubMed]

*N*-torus), a toroidal search algorithm is developed. If we want to sample this 4

*N*-dimensional torus, then 4

*N*time series are constructed from: for

*j*∈ [1,...,4

*N*], where

*θ*

_{j0}are initial parameter values,

*ω*are angular frequencies which are incommensurate, i.e. equation doesn’t have integer solution. In other words,

_{i}*ω*is irrational for any

_{j}/ω_{k}*j*,

*k*∈ [1,...,4

*N*], given

*j*≠

*k*. It is easy to prove that under such conditions, [

*θ*

_{1}(

*t*) ...

*θ*

_{4}

*(*

_{N}*t*)] is dense on the torus [24]. Thus using this method, it is guaranteed that one can sample any torus sufficiently well if sampling for a long enough time or using a high enough sample rate. In the following sections of this paper, we take a 2-torus for a single NPR laser as a simple example case. However, the methods mentioned in this paper can be applied to toroidal parameter spaces of any dimensionality. Figure 2 shows how the toroidal sampling works on a 2-torus comprised of parameters

*α*

_{3}and

*α*. Specifically, the resulting time series of the objective function

_{p}*O*is demonstrated as the 2-torus is sampled. In Fig. 2(a), the torus in under-sampled and aliasing of the objective function occurs. However, as the sampling rate is increased, as shown in Fig. 2(e), the objective function is fully constructed and an evaluation of best performance can be ascertained. Indeed, the red dot in Fig. 2(f) shows the optimal global maxima of the laser cavity. The narrow shaded region around this peak performance is highlighted in Fig. 2(h) where an additional evaluation is made of whether the solution is mode-locked or not.

### 3.2. The Gábor Transform (spectrogram) and Library Building

*g*(

*τ*−

*t*) was introduced with the aim of localizing both time and frequency. The Gábor transform, also known as the short-time Fourier transform is then defined as: where the bar denotes the complex conjugate of the function. Thus the function

*g*(

*τ*−

*t*) acts as a time filter for localizing the signal and its frequency content over a specific window of time, allowing for the construction of a spectrogram. A spectrogram represents a time series (signal) in both the time and spectral domain, as shown in Fig. 3.

## 4. Birefringence classification and recognition

*S*for a large number of possible birefringence values where

_{k}*k*ranges from 1 to

*M*, and for each

*k*, a singular value decomposition (SVD) is applied to the spectrogram [25]: and For each

*k*value, we keep the first

*m*(

*m*<

*n*) modes (low-rank approximation) of

*U*which has the highest energy and store them in the modes library

_{k}*U*such that where the

_{L}*k*-th sub-library

*Ũ*contains the first

_{k}*m*modes of

*U*: Once we have constructed our dimensionally reduced modes library, we can take a measurement of the laser system (the objective function) and compute the spectrogram. Note that the sampling time does not have to be of the same length as the time series collected when the library was built. We perform an SVD reduction on the measured spectrogram and keep the first

_{k}*m*modes as before, as illustrated in Fig. 5. With the most important (dominant) modes from the measurement in hand, we can do an

*L*

_{1}-norm library search, thus promoting sparsity in our solution [25]. In the

*L*

_{1}-norm search, our objective is to find a vector subject to Here we require the number of library modes to be greater than the dimensionality of the frequency domain. Given this condition, this becomes an underdetermined linear system of equations. The

*L*

_{1}-norm minimization produces a sparse vector

*a*, i.e. only a small portion of the elements are non-zero, as shown in Fig. 6. The non-zero elements of vector

*a*act as a classifier (indicator function) for identifying which sub-library the birefringence falls into. Thus if the largest element falls into the

*i*-th sub-library, the recognized birefringence value is equal to

*K*. This sparsity promoting optimization, when used in conjunction with the unique spectrograms, gives a rapid and accurate classification scheme for the fiber birefringence. Thus birefringence recognition can be easily accomplished. Note that our classification scheme essentially uses the

_{i}*L*

_{1}-norm minimization produce as an indicator function for the correct library elements. More sophisticated sparse classification/recognition strategies can be applied if desired [26

26. J. Wright, A. Yang, A. Ganesh, S. Sastry, and Y. Ma, “Robust face recognition via sparse representation,” IEEE Trans. Pattern Ana. Mach. Int. **31**, 210–227 (2009). [CrossRef]

## 5. Classification results

*K*is varied following a gaussian random walk. For each trial, the spectrogram corresponding to the current birefringence is computed and the

*L*

_{1}-norm sparse search is executed. Recognition results and errors are showed in Fig 7. In the figure, the recognition algorithm is tested in two scenarios: (i) well-aligned data given the assumption that the servo motors that control the waveplates and polarizers work without error, and (ii) the mis-aligned data that considers the error in the initial angle of the servo motors. In both of these two scenarios, the sparse search works very well. In the well-aligned case, a birefringence recognition (classification) rate of 98% is achieved while in the mis-aligned case, we get a 88% recognition rate. It should also be noted that even when our recognition algorithm fails to find the correct birefringence value, the error between the true birefringence and the recognized value is very small. Thus, even if we use the mis-classified birefringence, is is likely that the predicted optimal parameters will be near the true optimal parameters and the adaptive controller [21

**49**, 852–861 (2013). [CrossRef]

## 6. Conclusions and outlook

*quantitatively*accurate modeling of fiber lasers for optimizing their performance. Indeed, all other physical parameters in the system, such as the Kerr nonlinearity, dispersion characteristics as a function of wavelength, gain and gain bandwidth, can be fairly well characterized in theoretical models. Thus only the birefringence remains unknown and randomly varying. And unlike optical communications, where statistical averaging methods can be used to quantify its effects statistically, a fixed laser cavity represents a single, and unknown, statistical realization of the birefringence which is highly susceptible and sensitive to environmental factors such as bend, twist, anisotropic stress, and ambient conditions such as temperature. Such a system requires new modeling methods which are based upon state-of-the-art data-driven strategies.

*L*

_{1}-norm optimization routine. Accuracies as high as 98% are achieved, thus suggesting the algorithm is highly promising for application purposes. And even when birefringence is misclassified, the results are only off by a small percentage, thus suggesting that cavity tuning can still be effective and efficient. Although the algorithm was demonstrated on an underlying theoretical model, the method can be integrated directly into an experimental laser cavity design, i.e. the advocated method does not rely on an underlying model of the laser dynamics.

22. 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]

23. 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] [PubMed]

## Acknowledgments

## References and links

1. | D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives,” J. Opt. Soc. Am. B |

2. | C. D. Poole and R. E. Wagner, “Phenomenological approach to polarization dispersion in long single-mode fibers,” Electron. Lett. |

3. | C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr media,” IEEE J. Quant. Electron. |

4. | C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quant. Electron. |

5. | P. K. A. Wai and C. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Light. Tech. |

6. | J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” PNAS |

7. | J. N. Kutz, “Mode-locked soliton lasers,” SIAM Review |

8. | H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quant. Elec. |

9. | K. Tamura, E.P. Ippen, H.A. Haus, and L.E. Nelson, “77-fs Pulse generation from a stretched-pulse mode-locked all-fiber ring laser,” Opt. Lett. |

10. | K. Tamura and M. Nakazawa, “Optimizing power extraction in stretched pulse fiber ring lasers,” App. Phys. Lett. |

11. | G. Lenz, K. Tamura, H. A. Haus, and E. P. Ippen, “All-solid-state femtosecond source at 1.55 μm,” Opt. Lett. |

12. | A. Chong, W. H. Renninger, and F. W. Wise, “Properties of normal-dispersion femtosecond fiber lasers,” J. Opt. Soc. Am. B |

13. | A. Chong, J. Buckley, W. Renninger, and F. Wise, “All-normal-dispersion femtosecond fiber laser,” Opt. Express |

14. | W. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A |

15. | F.Ö. Ilday, J. Buckley, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser cavity,” Phys. Rev. Lett. |

16. | W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A |

17. | B. Bale and S. Wabnitz, “Strong spectral filtering for a mode-locked similariton fiber laser,” Opt. Lett. |

18. | F. Li, P. K. A. Wai, and J. N. Kutz, “Geometrical description of the onset of multi-pulsing in mode-locked laser cavities,” J. Opt. Soc. Am. B |

19. | F. Li, E. Ding, J. N. Kutz, and P. K. A. Wai, “Dual transmission filters for enhanced energy in mode-locked fiber lasers,” Opt. Express |

20. | X. Fu and J. N. Kutz, “High-energy mode-locked fiber lasers using multiple transmission filters and a genetic algorithm,” Opt. Express |

21. | S. L. Brunton, X. Fu, and J. N. Kutz, “Extremum-seeking control of a mode-locked laser,” IEEE J. Quant. Electron. |

22. | X. Shen, W. Li, M. Yan, and H. Zeng, “Electronic control of nonlinear-polarization-rotation mode locking in Yb-doped fiber lasers,” Opt. Lett. |

23. | 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 |

24. | S. Wiggins, |

25. | J. N. Kutz, |

26. | J. Wright, A. Yang, A. Ganesh, S. Sastry, and Y. Ma, “Robust face recognition via sparse representation,” IEEE Trans. Pattern Ana. Mach. Int. |

27. | D. Needell and J. A. Tropp, “CoSaMP: iterative signal recovery from incomplete and inaccurate samples,” Comm. of the ACM |

**OCIS Codes**

(140.3510) Lasers and laser optics : Lasers, fiber

(140.4050) Lasers and laser optics : Mode-locked lasers

(320.7090) Ultrafast optics : Ultrafast lasers

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: December 18, 2013

Revised Manuscript: March 20, 2014

Manuscript Accepted: March 21, 2014

Published: April 3, 2014

**Citation**

Xing Fu, Steven L. Brunton, and J. Nathan Kutz, "Classification of birefringence in mode-locked fiber lasers using machine learning and sparse representation," Opt. Express **22**, 8585-8597 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-7-8585

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### References

- D. J. Richardson, J. Nilsson, W. A. Clarkson, “High power fiber lasers: current status and future perspectives,” J. Opt. Soc. Am. B 27, B63–B92 (2010). [CrossRef]
- C. D. Poole, R. E. Wagner, “Phenomenological approach to polarization dispersion in long single-mode fibers,” Electron. Lett. 22, 1029–1030 (1986). [CrossRef]
- C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr media,” IEEE J. Quant. Electron. 25, 2674–2682 (1989). [CrossRef]
- C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quant. Electron. 23, 174–176 (1987). [CrossRef]
- P. K. A. Wai, C. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Light. Tech. 14, 148–157 (1996). [CrossRef]
- J. P. Gordon, H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” PNAS 97, 4541–4550 (2000). [CrossRef] [PubMed]
- J. N. Kutz, “Mode-locked soliton lasers,” SIAM Review 48, 629–678 (2006). [CrossRef]
- H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quant. Elec. 6, 1173–1185 (2000). [CrossRef]
- K. Tamura, E.P. Ippen, H.A. Haus, L.E. Nelson, “77-fs Pulse generation from a stretched-pulse mode-locked all-fiber ring laser,” Opt. Lett. 18, 1080–1082 (1993). [CrossRef] [PubMed]
- K. Tamura, M. Nakazawa, “Optimizing power extraction in stretched pulse fiber ring lasers,” App. Phys. Lett. 67, 3691–3693 (1995). [CrossRef]
- G. Lenz, K. Tamura, H. A. Haus, E. P. Ippen, “All-solid-state femtosecond source at 1.55 μm,” Opt. Lett. 20, 1289–1291 (1995). [CrossRef] [PubMed]
- A. Chong, W. H. Renninger, F. W. Wise, “Properties of normal-dispersion femtosecond fiber lasers,” J. Opt. Soc. Am. B 25, 140–148 (2008). [CrossRef]
- A. Chong, J. Buckley, W. Renninger, F. Wise, “All-normal-dispersion femtosecond fiber laser,” Opt. Express 14, 10095 (2006). [CrossRef] [PubMed]
- W. Renninger, A. Chong, F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008). [CrossRef]
- F.Ö. Ilday, J. Buckley, F. W. Wise, “Self-similar evolution of parabolic pulses in a laser cavity,” Phys. Rev. Lett. 92, 213902 (2004). [CrossRef]
- W. H. Renninger, A. Chong, F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A 82, 021805 (2010). [CrossRef]
- B. Bale, S. Wabnitz, “Strong spectral filtering for a mode-locked similariton fiber laser,” Opt. Lett. 35, 2466–2468 (2010). [CrossRef] [PubMed]
- F. Li, P. K. A. Wai, J. N. Kutz, “Geometrical description of the onset of multi-pulsing in mode-locked laser cavities,” J. Opt. Soc. Am. B 27, 2068–2077 (2010). [CrossRef]
- F. Li, E. Ding, J. N. Kutz, P. K. A. Wai, “Dual transmission filters for enhanced energy in mode-locked fiber lasers,” Opt. Express 19, 23408–23419 (2011). [CrossRef] [PubMed]
- X. Fu, J. N. Kutz, “High-energy mode-locked fiber lasers using multiple transmission filters and a genetic algorithm,” Opt. Express 21, 6526–6537 (2013). [CrossRef] [PubMed]
- S. L. Brunton, X. Fu, J. N. Kutz, “Extremum-seeking control of a mode-locked laser,” IEEE J. Quant. Electron. 49, 852–861 (2013). [CrossRef]
- X. Shen, W. Li, M. Yan, H. Zeng, “Electronic control of nonlinear-polarization-rotation mode locking in Yb-doped fiber lasers,” Opt. Lett. 37, 3426–3428 (2012). [CrossRef]
- D. Radnatarov, S. Khripunov, S. Kobtsev, A. Ivanenko, S. Kukarin, “Automatic electronic-controlled mode locking self-start in fibre lasers with non-linear polarisation evolution,” Opt. Express 21, 20626–20631 (2013). [CrossRef] [PubMed]
- S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2, (Springer2003).
- J. N. Kutz, Data-Driven Modeling and Scientific Computation (Oxford2013).
- J. Wright, A. Yang, A. Ganesh, S. Sastry, Y. Ma, “Robust face recognition via sparse representation,” IEEE Trans. Pattern Ana. Mach. Int. 31, 210–227 (2009). [CrossRef]
- D. Needell, J. A. Tropp, “CoSaMP: iterative signal recovery from incomplete and inaccurate samples,” Comm. of the ACM 53, 93–100 (2010). [CrossRef]

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