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 (L1-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

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References

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2013

2012

2011

2010

2009

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]

2008

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

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]

2006

2004

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]

2000

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

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

1996

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]

1995

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]

1993

1989

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

1987

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

1986

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

Bale, B.

Brunton, S. L.

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]

Buckley, J.

A. Chong, J. Buckley, W. Renninger, F. Wise, “All-normal-dispersion femtosecond fiber laser,” Opt. Express 14, 10095 (2006).
[CrossRef] [PubMed]

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]

Chong, A.

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]

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

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]

Clarkson, W. A.

Ding, E.

Fu, X.

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. 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]

Ganesh, A.

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]

Gordon, J. P.

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

Haus, H. A.

Haus, H.A.

Ilday, F.Ö.

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]

Ippen, E. P.

Ippen, E.P.

Ivanenko, A.

Khripunov, S.

Kobtsev, S.

Kogelnik, H.

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

Kukarin, S.

Kutz, J. N.

Lenz, G.

Li, F.

Li, W.

Ma, Y.

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]

Menyuk, C.

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]

Menyuk, C. R.

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]

Nakazawa, M.

K. Tamura, M. Nakazawa, “Optimizing power extraction in stretched pulse fiber ring lasers,” App. Phys. Lett. 67, 3691–3693 (1995).
[CrossRef]

Needell, D.

D. Needell, J. A. Tropp, “CoSaMP: iterative signal recovery from incomplete and inaccurate samples,” Comm. of the ACM 53, 93–100 (2010).
[CrossRef]

Nelson, L.E.

Nilsson, J.

Poole, C. D.

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

Radnatarov, D.

Renninger, W.

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

A. Chong, J. Buckley, W. Renninger, F. Wise, “All-normal-dispersion femtosecond fiber laser,” Opt. Express 14, 10095 (2006).
[CrossRef] [PubMed]

Renninger, W. H.

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]

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]

Richardson, D. J.

Sastry, S.

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]

Shen, X.

Tamura, K.

Tropp, J. A.

D. Needell, J. A. Tropp, “CoSaMP: iterative signal recovery from incomplete and inaccurate samples,” Comm. of the ACM 53, 93–100 (2010).
[CrossRef]

Wabnitz, S.

Wagner, R. E.

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

Wai, P. K. A.

Wiggins, S.

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2, (Springer2003).

Wise, F.

Wise, F. W.

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]

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

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]

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]

Wright, J.

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]

Yan, M.

Yang, A.

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]

Zeng, H.

App. Phys. Lett.

K. Tamura, M. Nakazawa, “Optimizing power extraction in stretched pulse fiber ring lasers,” App. Phys. Lett. 67, 3691–3693 (1995).
[CrossRef]

Comm. of the ACM

D. Needell, J. A. Tropp, “CoSaMP: iterative signal recovery from incomplete and inaccurate samples,” Comm. of the ACM 53, 93–100 (2010).
[CrossRef]

Electron. Lett.

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

IEEE J. Quant. Electron.

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]

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]

IEEE J. Sel. Top. Quant. Elec.

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

IEEE Trans. Pattern Ana. Mach. Int.

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]

J. Light. Tech.

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. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

Phys. Rev. A

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

Phys. Rev. Lett.

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]

PNAS

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

SIAM Review

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

Other

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2, (Springer2003).

J. N. Kutz, Data-Driven Modeling and Scientific Computation (Oxford2013).

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Figures (8)

Fig. 1
Fig. 1

(A) Schematic of a mode-locked laser cavity which includes a ring fiber with saturable absorber (SA) and gain element. (B) The SA is generated by nonlinear polarization rotation interacting with three waveplates (αj where j = 1, 2, 3) and a polarizer (αp). Incoming polarized light (P) is attenuated by the polarizer if it is not in alignment with the transmitting axis. Thus only a single polarization direction (P+) is transmitted. (C) The fiber itself is subject to stochastic fluctuations in the birefringence, i.e. random rotations of the principal fast- and slow-axes, u and v respectively. Shown is an example portion of fiber where the rotations depend sensitively on bend, twist, anisotropic stress, and/or ambient temperature.

Fig. 2
Fig. 2

(a, c, e) 2-torus of α3 and αp with sample points shown (dots) for different sample rates (1.25Hz-black, 5Hz-magenta, 20Hz-blue, the global optimum is marked in red). (b, d, f) The time-series of the corresponding objective function with the global optimum again marked in red. (g) Wave forms of the laser output corresponding to different parameter values marked in (h). (h) Zoomed in objective function (red) plot near the global optimum, pulse energy (black) and kurtosis (blue) are also shown (all normalized to the same scale for comparison).

Fig. 3
Fig. 3

Top: objecive function time series sampled at K = 0.17143 (black solid), a Gaussian Gábor window centered at τ = 20 is also shown (red solid). Bottom: corresponding spectrogram obtained using Gábor transform with the Gaussian window shown in top panel.

Fig. 4
Fig. 4

Spectrograms for different birefringence values, various (and unique) temporal dynamics can be observed from the comparison.

Fig. 5
Fig. 5

Top left: Time series of objective function obtained by toroidal search, sampled at 20Hz. Top right: Spectrogram of the time series. Bottom left: Singular values from SVD, the largest 15 singular values (corresponding to SVD modes used in the library) are plotted in red and the rest are plotted in blue. Bottom right: SVD modes correspond to the largest 15 singular values.

Fig. 6
Fig. 6

Top: Barplot of the components of vector a from L1 optimization where sparsity can be observed, i.e. it is mostly comprised of zeros. The indicator function nature of the sparse representation is clearly observed. Bottom: Barplot of components of vector a from L2-norm optimization, showing it does not produce any classification.

Fig. 7
Fig. 7

Left: Recognition results and errors using well-aligned data. A 98% correct birefringence classification is achieved. Right: recognition results and errors using mis-aligned (shifted) data. In this case, an 88% correct birefringence classification is achieved. Note that the blue dots represent the true birefringence labels while the red circles are the classified birefringence. Even if misclassified, the algorithm produces a birefringence that is only slightly off, thus still allowing for a rapid tuning of the laser cavity to the optimal waveplate and polarizer settings.

Fig. 8
Fig. 8

(a) Setup of the proposed mode-locked fiber laser wrapped with servos and machine learning module. (b) Flowchart of training algorithm. (c) Flowchart of execution algorithm. Colored boxes have corresponding pseudo code provided in Table 1.

Tables (1)

Tables Icon

Table 1 Algorithms and pseudo code for training and execution of machine learning module in Fig 8. ( represents built-in MATLAB functions svd and spectrogram. The L-1 norm library search can be implemented using the cvx package with details provide in Section 4, or with the compressive sampling matching pursuit (CoSaMP) [27].)

Equations (19)

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

i u z + D 2 2 u t 2 K u + ( | u | 2 + A | v | 2 ) u + B v 2 u * = i R u ,
i v z + D 2 2 v t 2 K v + ( | v | 2 + A | u | 2 ) v + B u 2 v * = i R v ,
R = 2 g 0 1 + 1 e 0 ( | u | 2 + | v | 2 ) d t ( 1 + τ 2 t 2 ) Γ .
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 k = R ( α k ) W R ( α k ) ,
R ( α k ) = ( cos ( α k ) sin ( α k ) sin ( α k ) cos ( α k ) ) .
O = E M 4 .
θ j ( t ) = ω j t + θ j 0
m ω j + n ω k = 0
g t , ω ( τ ) = e i ω τ g ( τ t ) ,
f ˜ g ( t , ω ) = f ( τ ) g ¯ ( τ t ) e i ω τ d τ ,
S k = U k k V k *
U k = [ u k 1 u k 2 u k n ] .
U L = [ U ˜ 1 U ˜ 2 U ˜ M ]
U ˜ k = [ u k 1 u k 2 u k m ] .
a = arg min a a 1
U L . a u m 1 .

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