Abstract

Strongly pumped mode-locked lasers often form pulse bunches. Although several mechanisms of pulse interaction are known, none yields the experimentally observed long-range attraction. Here we demonstrate theoretically and experimentally a new pulse interaction mechanism mediated by the continuum noise floor that is a universal feature in multipulse passively mode-locked lasers. Long-range attraction is facilitated by the depletion of the gain by the pulses, leading to an inhomogeneous noise floor that biases the timing jitter of the pulses and produces an effective interpulse potential with stable pulse bunch configurations. The pulses attract by suppressing electromagnetic fluctuations, as do conductors in the Casimir effect of quantum electrodynamics. This enables manipulation and design of multipulse waveforms to ultimately make them useful for application of mode-locked lasers.

© 2016 Optical Society of America

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References

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2014 (1)

C. Godey, I. V. Balakireva, A. Coillet, and Y. K. Chembo, Phys. Rev. A 89, 063814 (2014).
[Crossref]

2013 (2)

J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, Nat. Photonics 7, 657 (2013).
[Crossref]

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2013).
[Crossref]

2011 (2)

A. M. Weiner, Opt. Commun. 284, 3669 (2011).
[Crossref]

R. Weill, A. Bekker, V. Smulakovsky, B. Fischer, and O. Gat, Phys. Rev. A 83, 043831 (2011).
[Crossref]

2010 (1)

2009 (1)

L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 102, 083901 (2009).
[Crossref]

2007 (1)

R. Weill, B. Vodonos, A. Gordon, O. Gat, and B. Fischer, Phys. Rev. E 76, 031112 (2007).
[Crossref]

2006 (3)

M. Katz, A. Gordon, O. Gat, and B. Fischer, Phys. Rev. Lett. 97, 113902 (2006).
[Crossref]

L. Angelani, C. Conti, G. Ruocco, and F. Zamponi, Phys. Rev. B 74, 104207 (2006).
[Crossref]

J. N. Kutz, SIAM Rev. 48, 629 (2006).
[Crossref]

2005 (2)

O. Gat, A. Gordon, and B. Fischer, New J. Phys. 7, 151 (2005).
[Crossref]

D. Tang, B. Zhao, L. Zhao, and H. Tam, Phys. Rev. E 72, 016616 (2005).
[Crossref]

2004 (3)

B. Vodonos, R. Weill, A. Gordon, A. Bekker, V. Smulakovsky, O. Gat, and B. Fischer, Phys. Rev. Lett. 93, 153901 (2004).
[Crossref]

O. Gat, A. Gordon, and B. Fischer, Phys. Rev. E 70, 046108 (2004).
[Crossref]

A. Picozzi and M. Haelterman, Phys. Rev. Lett. 92, 103901 (2004).
[Crossref]

2003 (1)

2002 (3)

D. Tang, B. Zhao, D. Shen, C. Lu, W. Man, and H. Tam, Phys. Rev. A 66, 033806 (2002).
[Crossref]

P. Grelu, F. Belhache, F. Gutty, and J. M. Soto-Crespo, Opt. Lett. 27, 966 (2002).
[Crossref]

A. Gordon and B. Fischer, Phys. Rev. Lett. 89, 103901 (2002).
[Crossref]

2000 (1)

H. A. Haus, IEEE J. Select. Top. Quantum Electron. 6, 1173 (2000).
[Crossref]

1999 (1)

1998 (2)

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, IEEE J. Quantum Electron. 34, 1749 (1998).
[Crossref]

N. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, J. Opt. Soc. Am. B 15, 515 (1998).
[Crossref]

1997 (1)

1995 (1)

1992 (2)

1990 (1)

Akhmediev, N.

Akhmediev, N. N.

Angelani, L.

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[Crossref]

L. Angelani, C. Conti, G. Ruocco, and F. Zamponi, Phys. Rev. B 74, 104207 (2006).
[Crossref]

Ankiewicz, A.

Balakireva, I. V.

C. Godey, I. V. Balakireva, A. Coillet, and Y. K. Chembo, Phys. Rev. A 89, 063814 (2014).
[Crossref]

Bekker, A.

R. Weill, A. Bekker, V. Smulakovsky, B. Fischer, and O. Gat, Phys. Rev. A 83, 043831 (2011).
[Crossref]

B. Vodonos, R. Weill, A. Gordon, A. Bekker, V. Smulakovsky, O. Gat, and B. Fischer, Phys. Rev. Lett. 93, 153901 (2004).
[Crossref]

Belhache, F.

Bergman, K.

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, IEEE J. Quantum Electron. 34, 1749 (1998).
[Crossref]

Brasch, V.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2013).
[Crossref]

Casimir, H.

H. Casimir, Proceedings of the Koninklijke Nederlandse Akademie Van Wetenschappen (1997), Vol. 100, pp. 61–63 [Reprinted from Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen (1948), Vol 51, pp. 793-795].

Chembo, Y. K.

C. Godey, I. V. Balakireva, A. Coillet, and Y. K. Chembo, Phys. Rev. A 89, 063814 (2014).
[Crossref]

Coen, S.

J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, Nat. Photonics 7, 657 (2013).
[Crossref]

Coillet, A.

C. Godey, I. V. Balakireva, A. Coillet, and Y. K. Chembo, Phys. Rev. A 89, 063814 (2014).
[Crossref]

Collings, B. C.

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, IEEE J. Quantum Electron. 34, 1749 (1998).
[Crossref]

Conti, C.

L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 102, 083901 (2009).
[Crossref]

L. Angelani, C. Conti, G. Ruocco, and F. Zamponi, Phys. Rev. B 74, 104207 (2006).
[Crossref]

Dianov, E. M.

Erkintalo, M.

J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, Nat. Photonics 7, 657 (2013).
[Crossref]

Fischer, B.

R. Weill, A. Bekker, V. Smulakovsky, B. Fischer, and O. Gat, Phys. Rev. A 83, 043831 (2011).
[Crossref]

M. Katz, O. Gat, and B. Fischer, Opt. Lett. 35, 297 (2010).
[Crossref]

R. Weill, B. Vodonos, A. Gordon, O. Gat, and B. Fischer, Phys. Rev. E 76, 031112 (2007).
[Crossref]

M. Katz, A. Gordon, O. Gat, and B. Fischer, Phys. Rev. Lett. 97, 113902 (2006).
[Crossref]

O. Gat, A. Gordon, and B. Fischer, New J. Phys. 7, 151 (2005).
[Crossref]

O. Gat, A. Gordon, and B. Fischer, Phys. Rev. E 70, 046108 (2004).
[Crossref]

B. Vodonos, R. Weill, A. Gordon, A. Bekker, V. Smulakovsky, O. Gat, and B. Fischer, Phys. Rev. Lett. 93, 153901 (2004).
[Crossref]

A. Gordon and B. Fischer, Phys. Rev. Lett. 89, 103901 (2002).
[Crossref]

Folli, V.

L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 102, 083901 (2009).
[Crossref]

Gat, O.

R. Weill, A. Bekker, V. Smulakovsky, B. Fischer, and O. Gat, Phys. Rev. A 83, 043831 (2011).
[Crossref]

M. Katz, O. Gat, and B. Fischer, Opt. Lett. 35, 297 (2010).
[Crossref]

R. Weill, B. Vodonos, A. Gordon, O. Gat, and B. Fischer, Phys. Rev. E 76, 031112 (2007).
[Crossref]

M. Katz, A. Gordon, O. Gat, and B. Fischer, Phys. Rev. Lett. 97, 113902 (2006).
[Crossref]

O. Gat, A. Gordon, and B. Fischer, New J. Phys. 7, 151 (2005).
[Crossref]

B. Vodonos, R. Weill, A. Gordon, A. Bekker, V. Smulakovsky, O. Gat, and B. Fischer, Phys. Rev. Lett. 93, 153901 (2004).
[Crossref]

O. Gat, A. Gordon, and B. Fischer, Phys. Rev. E 70, 046108 (2004).
[Crossref]

Godey, C.

C. Godey, I. V. Balakireva, A. Coillet, and Y. K. Chembo, Phys. Rev. A 89, 063814 (2014).
[Crossref]

Golovchenko, E. A.

Gordon, A.

R. Weill, B. Vodonos, A. Gordon, O. Gat, and B. Fischer, Phys. Rev. E 76, 031112 (2007).
[Crossref]

M. Katz, A. Gordon, O. Gat, and B. Fischer, Phys. Rev. Lett. 97, 113902 (2006).
[Crossref]

O. Gat, A. Gordon, and B. Fischer, New J. Phys. 7, 151 (2005).
[Crossref]

O. Gat, A. Gordon, and B. Fischer, Phys. Rev. E 70, 046108 (2004).
[Crossref]

B. Vodonos, R. Weill, A. Gordon, A. Bekker, V. Smulakovsky, O. Gat, and B. Fischer, Phys. Rev. Lett. 93, 153901 (2004).
[Crossref]

A. Gordon and B. Fischer, Phys. Rev. Lett. 89, 103901 (2002).
[Crossref]

Gordon, J. P.

Gorodetsky, M. L.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2013).
[Crossref]

Gray, S.

Grelu, P.

Grudinin, A. B.

Gutty, F.

Haelterman, M.

A. Picozzi and M. Haelterman, Phys. Rev. Lett. 92, 103901 (2004).
[Crossref]

Haus, H. A.

H. A. Haus, IEEE J. Select. Top. Quantum Electron. 6, 1173 (2000).
[Crossref]

Herr, T.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2013).
[Crossref]

Jagadish, C.

Jang, J. K.

J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, Nat. Photonics 7, 657 (2013).
[Crossref]

Jost, J. D.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2013).
[Crossref]

Katz, M.

M. Katz, O. Gat, and B. Fischer, Opt. Lett. 35, 297 (2010).
[Crossref]

M. Katz, A. Gordon, O. Gat, and B. Fischer, Phys. Rev. Lett. 97, 113902 (2006).
[Crossref]

Kelly, S.

S. Kelly, Electron. Lett. 28, 806 (1992).
[Crossref]

Kippenberg, T. J.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2013).
[Crossref]

Knox, W. H.

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, IEEE J. Quantum Electron. 34, 1749 (1998).
[Crossref]

Kondratiev, N. M.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2013).
[Crossref]

Kutz, J. N.

J. N. Kutz, SIAM Rev. 48, 629 (2006).
[Crossref]

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, IEEE J. Quantum Electron. 34, 1749 (1998).
[Crossref]

Lederer, M. J.

Leuzzi, L.

L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 102, 083901 (2009).
[Crossref]

Lu, C.

D. Tang, B. Zhao, D. Shen, C. Lu, W. Man, and H. Tam, Phys. Rev. A 66, 033806 (2002).
[Crossref]

Luchnikov, A. V.

Luther-Davies, B.

Man, W.

D. Tang, B. Zhao, D. Shen, C. Lu, W. Man, and H. Tam, Phys. Rev. A 66, 033806 (2002).
[Crossref]

Menyuk, C. R.

Murdoch, S. G.

J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, Nat. Photonics 7, 657 (2013).
[Crossref]

Picozzi, A.

A. Picozzi and M. Haelterman, Phys. Rev. Lett. 92, 103901 (2004).
[Crossref]

Pilipetskii, A. N.

Risken, H.

H. Risken, The Fokker-Planck Equation (Springer, 1989).

Ruocco, G.

L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 102, 083901 (2009).
[Crossref]

L. Angelani, C. Conti, G. Ruocco, and F. Zamponi, Phys. Rev. B 74, 104207 (2006).
[Crossref]

Shen, D.

D. Tang, B. Zhao, D. Shen, C. Lu, W. Man, and H. Tam, Phys. Rev. A 66, 033806 (2002).
[Crossref]

Smulakovsky, V.

R. Weill, A. Bekker, V. Smulakovsky, B. Fischer, and O. Gat, Phys. Rev. A 83, 043831 (2011).
[Crossref]

B. Vodonos, R. Weill, A. Gordon, A. Bekker, V. Smulakovsky, O. Gat, and B. Fischer, Phys. Rev. Lett. 93, 153901 (2004).
[Crossref]

Soto-Crespo, J. M.

Starodumov, A. N.

Tam, H.

D. Tang, B. Zhao, L. Zhao, and H. Tam, Phys. Rev. E 72, 016616 (2005).
[Crossref]

D. Tang, B. Zhao, D. Shen, C. Lu, W. Man, and H. Tam, Phys. Rev. A 66, 033806 (2002).
[Crossref]

Tan, H. H.

Tang, D.

D. Tang, B. Zhao, L. Zhao, and H. Tam, Phys. Rev. E 72, 016616 (2005).
[Crossref]

D. Tang, B. Zhao, D. Shen, C. Lu, W. Man, and H. Tam, Phys. Rev. A 66, 033806 (2002).
[Crossref]

Vodonos, B.

R. Weill, B. Vodonos, A. Gordon, O. Gat, and B. Fischer, Phys. Rev. E 76, 031112 (2007).
[Crossref]

B. Vodonos, R. Weill, A. Gordon, A. Bekker, V. Smulakovsky, O. Gat, and B. Fischer, Phys. Rev. Lett. 93, 153901 (2004).
[Crossref]

Wang, C. Y.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2013).
[Crossref]

Weill, R.

R. Weill, A. Bekker, V. Smulakovsky, B. Fischer, and O. Gat, Phys. Rev. A 83, 043831 (2011).
[Crossref]

R. Weill, B. Vodonos, A. Gordon, O. Gat, and B. Fischer, Phys. Rev. E 76, 031112 (2007).
[Crossref]

B. Vodonos, R. Weill, A. Gordon, A. Bekker, V. Smulakovsky, O. Gat, and B. Fischer, Phys. Rev. Lett. 93, 153901 (2004).
[Crossref]

Weiner, A. M.

A. M. Weiner, Opt. Commun. 284, 3669 (2011).
[Crossref]

Zamponi, F.

L. Angelani, C. Conti, G. Ruocco, and F. Zamponi, Phys. Rev. B 74, 104207 (2006).
[Crossref]

Zhao, B.

D. Tang, B. Zhao, L. Zhao, and H. Tam, Phys. Rev. E 72, 016616 (2005).
[Crossref]

D. Tang, B. Zhao, D. Shen, C. Lu, W. Man, and H. Tam, Phys. Rev. A 66, 033806 (2002).
[Crossref]

Zhao, L.

D. Tang, B. Zhao, L. Zhao, and H. Tam, Phys. Rev. E 72, 016616 (2005).
[Crossref]

Electron. Lett. (1)

S. Kelly, Electron. Lett. 28, 806 (1992).
[Crossref]

IEEE J. Quantum Electron. (1)

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, IEEE J. Quantum Electron. 34, 1749 (1998).
[Crossref]

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

H. A. Haus, IEEE J. Select. Top. Quantum Electron. 6, 1173 (2000).
[Crossref]

J. Opt. Soc. Am. B (4)

Nat. Photonics (2)

J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, Nat. Photonics 7, 657 (2013).
[Crossref]

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2013).
[Crossref]

New J. Phys. (1)

O. Gat, A. Gordon, and B. Fischer, New J. Phys. 7, 151 (2005).
[Crossref]

Opt. Commun. (1)

A. M. Weiner, Opt. Commun. 284, 3669 (2011).
[Crossref]

Opt. Lett. (5)

Phys. Rev. A (3)

C. Godey, I. V. Balakireva, A. Coillet, and Y. K. Chembo, Phys. Rev. A 89, 063814 (2014).
[Crossref]

D. Tang, B. Zhao, D. Shen, C. Lu, W. Man, and H. Tam, Phys. Rev. A 66, 033806 (2002).
[Crossref]

R. Weill, A. Bekker, V. Smulakovsky, B. Fischer, and O. Gat, Phys. Rev. A 83, 043831 (2011).
[Crossref]

Phys. Rev. B (1)

L. Angelani, C. Conti, G. Ruocco, and F. Zamponi, Phys. Rev. B 74, 104207 (2006).
[Crossref]

Phys. Rev. E (3)

O. Gat, A. Gordon, and B. Fischer, Phys. Rev. E 70, 046108 (2004).
[Crossref]

D. Tang, B. Zhao, L. Zhao, and H. Tam, Phys. Rev. E 72, 016616 (2005).
[Crossref]

R. Weill, B. Vodonos, A. Gordon, O. Gat, and B. Fischer, Phys. Rev. E 76, 031112 (2007).
[Crossref]

Phys. Rev. Lett. (5)

B. Vodonos, R. Weill, A. Gordon, A. Bekker, V. Smulakovsky, O. Gat, and B. Fischer, Phys. Rev. Lett. 93, 153901 (2004).
[Crossref]

A. Picozzi and M. Haelterman, Phys. Rev. Lett. 92, 103901 (2004).
[Crossref]

L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 102, 083901 (2009).
[Crossref]

M. Katz, A. Gordon, O. Gat, and B. Fischer, Phys. Rev. Lett. 97, 113902 (2006).
[Crossref]

A. Gordon and B. Fischer, Phys. Rev. Lett. 89, 103901 (2002).
[Crossref]

SIAM Rev. (1)

J. N. Kutz, SIAM Rev. 48, 629 (2006).
[Crossref]

Other (2)

H. Risken, The Fokker-Planck Equation (Springer, 1989).

H. Casimir, Proceedings of the Koninklijke Nederlandse Akademie Van Wetenschappen (1997), Vol. 100, pp. 61–63 [Reprinted from Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen (1948), Vol 51, pp. 793-795].

Supplementary Material (1)

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» Supplement 1: PDF (921 KB)      Supplemental Document

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Figures (4)

Fig. 1.
Fig. 1. Schematic of the pulse bunching process. (a) A single pulse configuration (b) becomes unstable to the formation of a second pulse at a random cavity position, (c) which is then attracted to the second pulse by noise interaction (d) until it reaches a steady state  determined by pedestal overlap. The pulse, pedestal, and continuum waveforms are shown in blue, green, and brown, respectively. The (negative) net gain profile for the given separations is shown in violet. The pulse and pedestal widths are 0.001 and 0.01 round-trip time, respectively, and they saturate 70% and 20%, respectively, of the maximum gain. The pulse waveform is clipped at high powers for visibility.
Fig. 2.
Fig. 2. Inhomogeneous continuum and its dependence on pumping. Experimental measurement (left) and theoretical calculation (right) of the continuum power in a multipulse PML for several pumping levels. The laser was injected with 0.6  mW of amplified spontaneous emission. The continuum power was measured by averaging output power over several hundred round trips; since the continuum is the only nonnegligible component of the waveform in more than 99% of the cavity, one can identify the output waveform with the continuum waveform, except near the sharp peak in the middle. Experimental graphs are ordered by increasing pumping from 12 to 45  mW and theoretical graphs by increasing mean gain g0 from bottom to top, showing the increase of continuum power and its variation with increased pumping.
Fig. 3.
Fig. 3. Pulse motion and evolution of the continuum. A similar measurement to the one shown in Fig. 2 with a single pumping value, showing (a)–(c) three consecutive transient states and (d) the steady state. The experiment shows how the continuum power profile changes as the pulses approach each other to form the bunch, seen as a single peak in panel (d) due to the limited resolution of the oscilloscope.
Fig. 4.
Fig. 4. Two-pulse noise-mediated interaction effective potential. The timing separation diffusion coefficient Ds times 12 (normalized by the single pulse diffusion coefficient D0) as a function of the separation ts (normalized by the cavity round-trip time tR). The black (red) curve shows the interaction potential for pulses that deplete 70% of the net gain without (with) pedestals, whose width is 0.01 of the round-trip time, that deplete an additional 20% of the net gain. The latter displays a stable fixed point near ts=0.024tR. Inset: zoom on the small separation region that includes the minimum induced by the pedestal, overlaid with the pedestal waveform (green).

Equations (6)

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zψ=i(t2ψ+2|ψ|2ψ)+G[ψ]L[ψ]+Γ(z,t),
tnz=12Dntn.
tsz=12Dsts,with  Ds=D1+D2.
Ds(ts)=12D0(κ(Δp,Δb,Δb)+κ(1Δp,Δb,Δb)+κ(Δp1,Δb,Δb)+κ(Δp,Δb,Δb)),
κ(Δ1,Δ2,Δ3)=(1Δ1tstR+Δ2Δ3(ets/tb+e(tstR)/tb))1/2.
ts(eq)=tblog(ΔbΔptRtb)

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