Abstract

We demonstrate a trade-off between linewidth and loss-of-lock rate in a mode-locked laser employing active feedback to control the carrier-envelope offset phase difference. In frequency metrology applications, the linewidth translates directly to uncertainty in the measured frequency, whereas the impact of lock loss and recovery on the measured frequency is less well understood. We reduce the dynamics to stochastic differential equations, specifically diffusion processes, and compare the linearized linewidth to the rate of lock loss determined by the mean time to exit, as calculated from large deviation theory.

© 2014 Optical Society of America

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References

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  1. J. Ye, H. Schnatz, and L. Hollberg, IEEE J. Sel. Top. Quantum Electron. 9, 1041 (2003).
    [CrossRef]
  2. A. J. Viterbi, Principles of Coherent Communication, (McGraw-Hill, 1966).
  3. S. T. Cundiff, J. Phys. D 35, R43 (2002).
    [CrossRef]
  4. J. K. Wahlstrand, J. T. Willits, C. R. Menyuk, and S. T. Cundiff, Opt. Express 16, 18624 (2008).
    [CrossRef]
  5. N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, N. D. Lemke, K. Beloy, M. Pizzocaro, C. W. Oates, and A. D. Ludlow, Science 341, 1215 (2013).
    [CrossRef]
  6. S. T. Cundiff, in Dissipative Solitons (Springer, 2005), p. 183206.
  7. G. M. Donovan and W. L. Kath, in Photonic Applications Systems Technologies Conference (2007).
  8. D. Anderson, M. Lisak, and A. Berntson, Pramana 57, 917 (2001).
    [CrossRef]
  9. M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems (Springer, 2012).
  10. B. Z. Bobrovsky and Z. Schuss, SIAM J. Appl. Math. 42, 174 (1982).
    [CrossRef]
  11. M. Heymann and E. Vanden-Eijnden, Commun. Pure Appl. Math. 61, 1052 (2008).
    [CrossRef]

2013 (1)

N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, N. D. Lemke, K. Beloy, M. Pizzocaro, C. W. Oates, and A. D. Ludlow, Science 341, 1215 (2013).
[CrossRef]

2008 (2)

2003 (1)

J. Ye, H. Schnatz, and L. Hollberg, IEEE J. Sel. Top. Quantum Electron. 9, 1041 (2003).
[CrossRef]

2002 (1)

S. T. Cundiff, J. Phys. D 35, R43 (2002).
[CrossRef]

2001 (1)

D. Anderson, M. Lisak, and A. Berntson, Pramana 57, 917 (2001).
[CrossRef]

1982 (1)

B. Z. Bobrovsky and Z. Schuss, SIAM J. Appl. Math. 42, 174 (1982).
[CrossRef]

Anderson, D.

D. Anderson, M. Lisak, and A. Berntson, Pramana 57, 917 (2001).
[CrossRef]

Beloy, K.

N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, N. D. Lemke, K. Beloy, M. Pizzocaro, C. W. Oates, and A. D. Ludlow, Science 341, 1215 (2013).
[CrossRef]

Berntson, A.

D. Anderson, M. Lisak, and A. Berntson, Pramana 57, 917 (2001).
[CrossRef]

Bobrovsky, B. Z.

B. Z. Bobrovsky and Z. Schuss, SIAM J. Appl. Math. 42, 174 (1982).
[CrossRef]

Cundiff, S. T.

J. K. Wahlstrand, J. T. Willits, C. R. Menyuk, and S. T. Cundiff, Opt. Express 16, 18624 (2008).
[CrossRef]

S. T. Cundiff, J. Phys. D 35, R43 (2002).
[CrossRef]

S. T. Cundiff, in Dissipative Solitons (Springer, 2005), p. 183206.

Donovan, G. M.

G. M. Donovan and W. L. Kath, in Photonic Applications Systems Technologies Conference (2007).

Freidlin, M. I.

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems (Springer, 2012).

Heymann, M.

M. Heymann and E. Vanden-Eijnden, Commun. Pure Appl. Math. 61, 1052 (2008).
[CrossRef]

Hinkley, N.

N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, N. D. Lemke, K. Beloy, M. Pizzocaro, C. W. Oates, and A. D. Ludlow, Science 341, 1215 (2013).
[CrossRef]

Hollberg, L.

J. Ye, H. Schnatz, and L. Hollberg, IEEE J. Sel. Top. Quantum Electron. 9, 1041 (2003).
[CrossRef]

Kath, W. L.

G. M. Donovan and W. L. Kath, in Photonic Applications Systems Technologies Conference (2007).

Lemke, N. D.

N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, N. D. Lemke, K. Beloy, M. Pizzocaro, C. W. Oates, and A. D. Ludlow, Science 341, 1215 (2013).
[CrossRef]

Lisak, M.

D. Anderson, M. Lisak, and A. Berntson, Pramana 57, 917 (2001).
[CrossRef]

Ludlow, A. D.

N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, N. D. Lemke, K. Beloy, M. Pizzocaro, C. W. Oates, and A. D. Ludlow, Science 341, 1215 (2013).
[CrossRef]

Menyuk, C. R.

Oates, C. W.

N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, N. D. Lemke, K. Beloy, M. Pizzocaro, C. W. Oates, and A. D. Ludlow, Science 341, 1215 (2013).
[CrossRef]

Phillips, N. B.

N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, N. D. Lemke, K. Beloy, M. Pizzocaro, C. W. Oates, and A. D. Ludlow, Science 341, 1215 (2013).
[CrossRef]

Pizzocaro, M.

N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, N. D. Lemke, K. Beloy, M. Pizzocaro, C. W. Oates, and A. D. Ludlow, Science 341, 1215 (2013).
[CrossRef]

Schioppo, M.

N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, N. D. Lemke, K. Beloy, M. Pizzocaro, C. W. Oates, and A. D. Ludlow, Science 341, 1215 (2013).
[CrossRef]

Schnatz, H.

J. Ye, H. Schnatz, and L. Hollberg, IEEE J. Sel. Top. Quantum Electron. 9, 1041 (2003).
[CrossRef]

Schuss, Z.

B. Z. Bobrovsky and Z. Schuss, SIAM J. Appl. Math. 42, 174 (1982).
[CrossRef]

Sherman, J. A.

N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, N. D. Lemke, K. Beloy, M. Pizzocaro, C. W. Oates, and A. D. Ludlow, Science 341, 1215 (2013).
[CrossRef]

Vanden-Eijnden, E.

M. Heymann and E. Vanden-Eijnden, Commun. Pure Appl. Math. 61, 1052 (2008).
[CrossRef]

Viterbi, A. J.

A. J. Viterbi, Principles of Coherent Communication, (McGraw-Hill, 1966).

Wahlstrand, J. K.

Wentzell, A. D.

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems (Springer, 2012).

Willits, J. T.

Ye, J.

J. Ye, H. Schnatz, and L. Hollberg, IEEE J. Sel. Top. Quantum Electron. 9, 1041 (2003).
[CrossRef]

Commun. Pure Appl. Math. (1)

M. Heymann and E. Vanden-Eijnden, Commun. Pure Appl. Math. 61, 1052 (2008).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

J. Ye, H. Schnatz, and L. Hollberg, IEEE J. Sel. Top. Quantum Electron. 9, 1041 (2003).
[CrossRef]

J. Phys. D (1)

S. T. Cundiff, J. Phys. D 35, R43 (2002).
[CrossRef]

Opt. Express (1)

Pramana (1)

D. Anderson, M. Lisak, and A. Berntson, Pramana 57, 917 (2001).
[CrossRef]

Science (1)

N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, N. D. Lemke, K. Beloy, M. Pizzocaro, C. W. Oates, and A. D. Ludlow, Science 341, 1215 (2013).
[CrossRef]

SIAM J. Appl. Math. (1)

B. Z. Bobrovsky and Z. Schuss, SIAM J. Appl. Math. 42, 174 (1982).
[CrossRef]

Other (4)

S. T. Cundiff, in Dissipative Solitons (Springer, 2005), p. 183206.

G. M. Donovan and W. L. Kath, in Photonic Applications Systems Technologies Conference (2007).

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems (Springer, 2012).

A. J. Viterbi, Principles of Coherent Communication, (McGraw-Hill, 1966).

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Figures (3)

Fig. 1.
Fig. 1.

Measures of uncertainty versus control parameter ω. Plotted are phase slip rate (solid) and standard deviations of amplitude A (dashed), frequency Ω (dotted), and timing T (dashed–dotted).

Fig. 2.
Fig. 2.

Most likely exit path corresponding to lowest phase slip rate in Fig. 1.

Fig. 3.
Fig. 3.

Most likely exit path for ω=10.

Equations (17)

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

iuz+122ut2+|u|2u=bcos(ωt)uic1u+ic22ut2+id1|u|2uid2|u|4u+iϵf(t,z),
us(t,z)=A(z)sech[A(z)(tT(z))]eiϕ(z)+itΩ(z),
L[t,z,u(t,z),ut(t,z),uz(t,z)]=Im(uz*u)12|ut|2+12|u|4+bcos(ωt)|u|2
LpjzLp˙j=2Rei[c1us+c2ustt+d1|us|2usd2|us|4us+ϵf(t,z)]us*pjdt.
dU=F(U)dz+ϵσ(U)dW
F=(2c1U1+(43d123c2)U131615d2U152c2U1U2243c2U12U2πbω22U13csch(πω2U1)sin(ωU3)U2)
σ(U)=(U1000U130U3U10π212U13+U32U1),
dϕ=[πωbA3cos(ωT)csch(πω2A)(1+πω2A2coth(πω2A))T+12(A2Ω2)]dz+ϵ(12+π26A)dW4,
A02=516d2[2d1c2+(2d1c2)2965c1d2],
M=(M11000M22M23010),
(43c2A02)2>2πbω3A03csch(πω2A0);
ΣU=ϵ22πds[(isIM)1σ][(isIM)1σ].
σA2=ϵ2A02|M11|,
σϕ2=2A0ϵ26χ(1+π2|M23|4A04),
σT2=2π2ϵ224A03χ(1+M222|M23|),
χ=(M222+2M23+M224+4M222M23+M222+2M23M224+4M222M23).
S=infψ120(ψ˙F(ψ))T(σ(ψ)σ(ψ)T)1(ψ˙F(ψ))dt,

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