Abstract

Laser mode and frequency combs, as lasers, are commonly one-dimensional systems. Here we present a construction of multi-dimensional laser-mode lattices (mode hyper-combs) with unique properties. They are obtained from regular 1-dimensional combs by multi-frequency modulation in active mode-locking (AML). The hyper-comb, with near neighbor mode coupling and noise functioning as temperature, is mapped to interacting magnetic-spins lattices in the spherical-model which is one of the few statistical-mechanics systems soluble in all dimensions. The important result is that such systems have, in d>2 dimensions, a phase-transition to a global phase-ordered mode hyper-comb. It can therefore change the nature of AML lasers by capturing very broad coherent frequency bandwidths and obtaining ultimately short and robust pulses. Additionally, the hyper-combs can serve as a rare physical realization of the spherical-model in any dimension.

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2012 (1)

2010 (5)

Z. Chang, P. B. Corkum, “Attosecond photon sources: the first decade and beyond,” J. Opt. Soc. Am. B 27(11), B9–B17 (2010).
[CrossRef]

P. Popmintchev, M. C. Chen, P. Arpin, M. M. Murnane, H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[CrossRef]

A. Rosen, R. Weill, B. Levit, V. Smulakovsky, A. Bekker, B. Fischer, “Experimental observation of critical phenomena in a laser light system,” Phys. Rev. Lett. 105(1), 013905 (2010).
[CrossRef] [PubMed]

R. Weill, B. Fischer, O. Gat, “Light-mode condensation in actively-mode-locked lasers,” Phys. Rev. Lett. 104(17), 173901 (2010).
[CrossRef] [PubMed]

R. Weill, B. Levit, A. Bekker, O. Gat, B. Fischer, “Laser light condensate: Experimental demonstration of light-mode condensation in actively mode locked laser,” Opt. Express 18(16), 16520–16525 (2010).
[CrossRef] [PubMed]

2007 (1)

H. C. Kapteyn, O. Cohen, I. Christov, M. M. Murnane, “Harnessing attosecond science in the quest for coherent X-rays,” Science 317(5839), 775–778 (2007).
[CrossRef] [PubMed]

2006 (1)

M. Katz, A. Gordon, O. Gat, B. Fischer, “Non-Gibbsian Stochastic light-mode dynamics of passive mode locking,” Phys. Rev. Lett. 97(11), 113902 (2006).
[CrossRef] [PubMed]

2005 (2)

O. Gat, A. Gordon, B. Fischer, “Light-mode locking: a new class of solvable statistical physics systems,” New J. Phys. 7, 151 (2005).
[CrossRef]

R. Weill, A. Rosen, A. Gordon, O. Gat, B. Fischer, “Critical behavior of light in mode-locked lasers,” Phys. Rev. Lett. 95(1), 013903 (2005).
[CrossRef] [PubMed]

2004 (2)

A. Gordon, B. Fischer, “Statistical-mechanics theory of active mode locking with noise,” Opt. Lett. 29(9), 1022–1024 (2004).
[CrossRef] [PubMed]

B. Vodonos, R. Weill, A. Gordon, A. Bekker, V. Smulakovsky, O. Gat, B. Fischer, “Formation and annihilation of laser light pulse quanta in a thermodynamic-like pathway,” Phys. Rev. Lett. 93(15), 153901 (2004).
[CrossRef] [PubMed]

2003 (2)

A. Gordon, B. Vodonos, V. Smulakovski, B. Fischer, “Melting and freezing of light pulses and modes in mode-locked lasers,” Opt. Express 11(25), 3418–3424 (2003).
[CrossRef] [PubMed]

T. R. Schibli, J. W. Kuzucu, J. W. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kaertner, V. Scheuer, G. Angelow, “Toward single-cycle laser systems,” IEEE J. Select. Topics in Quantum. Electron. 9, 990–1001 (2003).

2002 (3)

Th. Udem, R. Holzwarth, T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[CrossRef] [PubMed]

J. Itatani, F. Quéré, G. L. Yudin, M. Y. Ivanov, F. Krausz, P. B. Corkum, “Attosecond streak camera,” Phys. Rev. Lett. 88(17), 173903 (2002).
[CrossRef] [PubMed]

A. Gordon, B. Fischer, “Phase transition theory of many-mode ordering and pulse formation in lasers,” Phys. Rev. Lett. 89(10), 103901 (2002).
[CrossRef] [PubMed]

2001 (1)

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. B. Corkum, U. Heinzmann, M. Drescher, F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001).
[CrossRef] [PubMed]

2000 (2)

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

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–639 (2000).
[CrossRef] [PubMed]

1970 (1)

D. J. Kuizenga, A. E. Siegman, “Modulator frequency detuning effects in the FM-mode-locked laser,” IEEE J. Quantum Electron. QE-6, 803–808 (1970).

1969 (1)

H. E. Stanley, “Exact solution for a linear chain of isotropically interacting classical spins of arbitrary dimensionality,” Phys. Rev. 179(2), 570–577 (1969).
[CrossRef]

1952 (1)

T. H. Berlin, M. Kac, “The spherical model of a ferromagnet,” Phys. Rev. 86(6), 821–835 (1952).
[CrossRef]

Angelow, G.

T. R. Schibli, J. W. Kuzucu, J. W. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kaertner, V. Scheuer, G. Angelow, “Toward single-cycle laser systems,” IEEE J. Select. Topics in Quantum. Electron. 9, 990–1001 (2003).

Arpin, P.

P. Popmintchev, M. C. Chen, P. Arpin, M. M. Murnane, H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[CrossRef]

Bekker, A.

A. Rosen, R. Weill, B. Levit, V. Smulakovsky, A. Bekker, B. Fischer, “Experimental observation of critical phenomena in a laser light system,” Phys. Rev. Lett. 105(1), 013905 (2010).
[CrossRef] [PubMed]

R. Weill, B. Levit, A. Bekker, O. Gat, B. Fischer, “Laser light condensate: Experimental demonstration of light-mode condensation in actively mode locked laser,” Opt. Express 18(16), 16520–16525 (2010).
[CrossRef] [PubMed]

B. Vodonos, R. Weill, A. Gordon, A. Bekker, V. Smulakovsky, O. Gat, B. Fischer, “Formation and annihilation of laser light pulse quanta in a thermodynamic-like pathway,” Phys. Rev. Lett. 93(15), 153901 (2004).
[CrossRef] [PubMed]

Berlin, T. H.

T. H. Berlin, M. Kac, “The spherical model of a ferromagnet,” Phys. Rev. 86(6), 821–835 (1952).
[CrossRef]

Brabec, T.

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. B. Corkum, U. Heinzmann, M. Drescher, F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001).
[CrossRef] [PubMed]

Chang, Z.

Chen, M. C.

P. Popmintchev, M. C. Chen, P. Arpin, M. M. Murnane, H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[CrossRef]

Christov, I.

H. C. Kapteyn, O. Cohen, I. Christov, M. M. Murnane, “Harnessing attosecond science in the quest for coherent X-rays,” Science 317(5839), 775–778 (2007).
[CrossRef] [PubMed]

Cohen, O.

H. C. Kapteyn, O. Cohen, I. Christov, M. M. Murnane, “Harnessing attosecond science in the quest for coherent X-rays,” Science 317(5839), 775–778 (2007).
[CrossRef] [PubMed]

Corkum, P. B.

Z. Chang, P. B. Corkum, “Attosecond photon sources: the first decade and beyond,” J. Opt. Soc. Am. B 27(11), B9–B17 (2010).
[CrossRef]

J. Itatani, F. Quéré, G. L. Yudin, M. Y. Ivanov, F. Krausz, P. B. Corkum, “Attosecond streak camera,” Phys. Rev. Lett. 88(17), 173903 (2002).
[CrossRef] [PubMed]

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. B. Corkum, U. Heinzmann, M. Drescher, F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001).
[CrossRef] [PubMed]

Cundiff, S. T.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–639 (2000).
[CrossRef] [PubMed]

Diddams, S. A.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–639 (2000).
[CrossRef] [PubMed]

Drescher, M.

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. B. Corkum, U. Heinzmann, M. Drescher, F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001).
[CrossRef] [PubMed]

Fischer, B.

B. Fischer, R. Weill, “When does single-mode lasing become a condensation phenomenon?” Opt. Express 20(24), 26704–26713 (2012).
[CrossRef] [PubMed]

R. Weill, B. Fischer, O. Gat, “Light-mode condensation in actively-mode-locked lasers,” Phys. Rev. Lett. 104(17), 173901 (2010).
[CrossRef] [PubMed]

R. Weill, B. Levit, A. Bekker, O. Gat, B. Fischer, “Laser light condensate: Experimental demonstration of light-mode condensation in actively mode locked laser,” Opt. Express 18(16), 16520–16525 (2010).
[CrossRef] [PubMed]

A. Rosen, R. Weill, B. Levit, V. Smulakovsky, A. Bekker, B. Fischer, “Experimental observation of critical phenomena in a laser light system,” Phys. Rev. Lett. 105(1), 013905 (2010).
[CrossRef] [PubMed]

M. Katz, A. Gordon, O. Gat, B. Fischer, “Non-Gibbsian Stochastic light-mode dynamics of passive mode locking,” Phys. Rev. Lett. 97(11), 113902 (2006).
[CrossRef] [PubMed]

O. Gat, A. Gordon, B. Fischer, “Light-mode locking: a new class of solvable statistical physics systems,” New J. Phys. 7, 151 (2005).
[CrossRef]

R. Weill, A. Rosen, A. Gordon, O. Gat, B. Fischer, “Critical behavior of light in mode-locked lasers,” Phys. Rev. Lett. 95(1), 013903 (2005).
[CrossRef] [PubMed]

B. Vodonos, R. Weill, A. Gordon, A. Bekker, V. Smulakovsky, O. Gat, B. Fischer, “Formation and annihilation of laser light pulse quanta in a thermodynamic-like pathway,” Phys. Rev. Lett. 93(15), 153901 (2004).
[CrossRef] [PubMed]

A. Gordon, B. Fischer, “Statistical-mechanics theory of active mode locking with noise,” Opt. Lett. 29(9), 1022–1024 (2004).
[CrossRef] [PubMed]

A. Gordon, B. Vodonos, V. Smulakovski, B. Fischer, “Melting and freezing of light pulses and modes in mode-locked lasers,” Opt. Express 11(25), 3418–3424 (2003).
[CrossRef] [PubMed]

A. Gordon, B. Fischer, “Phase transition theory of many-mode ordering and pulse formation in lasers,” Phys. Rev. Lett. 89(10), 103901 (2002).
[CrossRef] [PubMed]

Fujimoto, J. G.

T. R. Schibli, J. W. Kuzucu, J. W. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kaertner, V. Scheuer, G. Angelow, “Toward single-cycle laser systems,” IEEE J. Select. Topics in Quantum. Electron. 9, 990–1001 (2003).

Gat, O.

R. Weill, B. Levit, A. Bekker, O. Gat, B. Fischer, “Laser light condensate: Experimental demonstration of light-mode condensation in actively mode locked laser,” Opt. Express 18(16), 16520–16525 (2010).
[CrossRef] [PubMed]

R. Weill, B. Fischer, O. Gat, “Light-mode condensation in actively-mode-locked lasers,” Phys. Rev. Lett. 104(17), 173901 (2010).
[CrossRef] [PubMed]

M. Katz, A. Gordon, O. Gat, B. Fischer, “Non-Gibbsian Stochastic light-mode dynamics of passive mode locking,” Phys. Rev. Lett. 97(11), 113902 (2006).
[CrossRef] [PubMed]

O. Gat, A. Gordon, B. Fischer, “Light-mode locking: a new class of solvable statistical physics systems,” New J. Phys. 7, 151 (2005).
[CrossRef]

R. Weill, A. Rosen, A. Gordon, O. Gat, B. Fischer, “Critical behavior of light in mode-locked lasers,” Phys. Rev. Lett. 95(1), 013903 (2005).
[CrossRef] [PubMed]

B. Vodonos, R. Weill, A. Gordon, A. Bekker, V. Smulakovsky, O. Gat, B. Fischer, “Formation and annihilation of laser light pulse quanta in a thermodynamic-like pathway,” Phys. Rev. Lett. 93(15), 153901 (2004).
[CrossRef] [PubMed]

Gordon, A.

M. Katz, A. Gordon, O. Gat, B. Fischer, “Non-Gibbsian Stochastic light-mode dynamics of passive mode locking,” Phys. Rev. Lett. 97(11), 113902 (2006).
[CrossRef] [PubMed]

O. Gat, A. Gordon, B. Fischer, “Light-mode locking: a new class of solvable statistical physics systems,” New J. Phys. 7, 151 (2005).
[CrossRef]

R. Weill, A. Rosen, A. Gordon, O. Gat, B. Fischer, “Critical behavior of light in mode-locked lasers,” Phys. Rev. Lett. 95(1), 013903 (2005).
[CrossRef] [PubMed]

B. Vodonos, R. Weill, A. Gordon, A. Bekker, V. Smulakovsky, O. Gat, B. Fischer, “Formation and annihilation of laser light pulse quanta in a thermodynamic-like pathway,” Phys. Rev. Lett. 93(15), 153901 (2004).
[CrossRef] [PubMed]

A. Gordon, B. Fischer, “Statistical-mechanics theory of active mode locking with noise,” Opt. Lett. 29(9), 1022–1024 (2004).
[CrossRef] [PubMed]

A. Gordon, B. Vodonos, V. Smulakovski, B. Fischer, “Melting and freezing of light pulses and modes in mode-locked lasers,” Opt. Express 11(25), 3418–3424 (2003).
[CrossRef] [PubMed]

A. Gordon, B. Fischer, “Phase transition theory of many-mode ordering and pulse formation in lasers,” Phys. Rev. Lett. 89(10), 103901 (2002).
[CrossRef] [PubMed]

Hall, J. L.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–639 (2000).
[CrossRef] [PubMed]

Hänsch, T. W.

Th. Udem, R. Holzwarth, T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[CrossRef] [PubMed]

Haus, H. A.

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

Heinzmann, U.

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. B. Corkum, U. Heinzmann, M. Drescher, F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001).
[CrossRef] [PubMed]

Hentschel, M.

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. B. Corkum, U. Heinzmann, M. Drescher, F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001).
[CrossRef] [PubMed]

Holzwarth, R.

Th. Udem, R. Holzwarth, T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[CrossRef] [PubMed]

Ippen, E. P.

T. R. Schibli, J. W. Kuzucu, J. W. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kaertner, V. Scheuer, G. Angelow, “Toward single-cycle laser systems,” IEEE J. Select. Topics in Quantum. Electron. 9, 990–1001 (2003).

Itatani, J.

J. Itatani, F. Quéré, G. L. Yudin, M. Y. Ivanov, F. Krausz, P. B. Corkum, “Attosecond streak camera,” Phys. Rev. Lett. 88(17), 173903 (2002).
[CrossRef] [PubMed]

Ivanov, M. Y.

J. Itatani, F. Quéré, G. L. Yudin, M. Y. Ivanov, F. Krausz, P. B. Corkum, “Attosecond streak camera,” Phys. Rev. Lett. 88(17), 173903 (2002).
[CrossRef] [PubMed]

Jones, D. J.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–639 (2000).
[CrossRef] [PubMed]

Kac, M.

T. H. Berlin, M. Kac, “The spherical model of a ferromagnet,” Phys. Rev. 86(6), 821–835 (1952).
[CrossRef]

Kaertner, F. X.

T. R. Schibli, J. W. Kuzucu, J. W. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kaertner, V. Scheuer, G. Angelow, “Toward single-cycle laser systems,” IEEE J. Select. Topics in Quantum. Electron. 9, 990–1001 (2003).

Kapteyn, H. C.

P. Popmintchev, M. C. Chen, P. Arpin, M. M. Murnane, H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[CrossRef]

H. C. Kapteyn, O. Cohen, I. Christov, M. M. Murnane, “Harnessing attosecond science in the quest for coherent X-rays,” Science 317(5839), 775–778 (2007).
[CrossRef] [PubMed]

Katz, M.

M. Katz, A. Gordon, O. Gat, B. Fischer, “Non-Gibbsian Stochastic light-mode dynamics of passive mode locking,” Phys. Rev. Lett. 97(11), 113902 (2006).
[CrossRef] [PubMed]

Kienberger, R.

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. B. Corkum, U. Heinzmann, M. Drescher, F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001).
[CrossRef] [PubMed]

Kim, J. W.

T. R. Schibli, J. W. Kuzucu, J. W. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kaertner, V. Scheuer, G. Angelow, “Toward single-cycle laser systems,” IEEE J. Select. Topics in Quantum. Electron. 9, 990–1001 (2003).

Krausz, F.

J. Itatani, F. Quéré, G. L. Yudin, M. Y. Ivanov, F. Krausz, P. B. Corkum, “Attosecond streak camera,” Phys. Rev. Lett. 88(17), 173903 (2002).
[CrossRef] [PubMed]

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. B. Corkum, U. Heinzmann, M. Drescher, F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001).
[CrossRef] [PubMed]

Kuizenga, D. J.

D. J. Kuizenga, A. E. Siegman, “Modulator frequency detuning effects in the FM-mode-locked laser,” IEEE J. Quantum Electron. QE-6, 803–808 (1970).

Kuzucu, J. W.

T. R. Schibli, J. W. Kuzucu, J. W. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kaertner, V. Scheuer, G. Angelow, “Toward single-cycle laser systems,” IEEE J. Select. Topics in Quantum. Electron. 9, 990–1001 (2003).

Levit, B.

A. Rosen, R. Weill, B. Levit, V. Smulakovsky, A. Bekker, B. Fischer, “Experimental observation of critical phenomena in a laser light system,” Phys. Rev. Lett. 105(1), 013905 (2010).
[CrossRef] [PubMed]

R. Weill, B. Levit, A. Bekker, O. Gat, B. Fischer, “Laser light condensate: Experimental demonstration of light-mode condensation in actively mode locked laser,” Opt. Express 18(16), 16520–16525 (2010).
[CrossRef] [PubMed]

Milosevic, N.

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. B. Corkum, U. Heinzmann, M. Drescher, F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001).
[CrossRef] [PubMed]

Murnane, M. M.

P. Popmintchev, M. C. Chen, P. Arpin, M. M. Murnane, H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[CrossRef]

H. C. Kapteyn, O. Cohen, I. Christov, M. M. Murnane, “Harnessing attosecond science in the quest for coherent X-rays,” Science 317(5839), 775–778 (2007).
[CrossRef] [PubMed]

Popmintchev, P.

P. Popmintchev, M. C. Chen, P. Arpin, M. M. Murnane, H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[CrossRef]

Quéré, F.

J. Itatani, F. Quéré, G. L. Yudin, M. Y. Ivanov, F. Krausz, P. B. Corkum, “Attosecond streak camera,” Phys. Rev. Lett. 88(17), 173903 (2002).
[CrossRef] [PubMed]

Ranka, J. K.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–639 (2000).
[CrossRef] [PubMed]

Reider, G. A.

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. B. Corkum, U. Heinzmann, M. Drescher, F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001).
[CrossRef] [PubMed]

Rosen, A.

A. Rosen, R. Weill, B. Levit, V. Smulakovsky, A. Bekker, B. Fischer, “Experimental observation of critical phenomena in a laser light system,” Phys. Rev. Lett. 105(1), 013905 (2010).
[CrossRef] [PubMed]

R. Weill, A. Rosen, A. Gordon, O. Gat, B. Fischer, “Critical behavior of light in mode-locked lasers,” Phys. Rev. Lett. 95(1), 013903 (2005).
[CrossRef] [PubMed]

Scheuer, V.

T. R. Schibli, J. W. Kuzucu, J. W. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kaertner, V. Scheuer, G. Angelow, “Toward single-cycle laser systems,” IEEE J. Select. Topics in Quantum. Electron. 9, 990–1001 (2003).

Schibli, T. R.

T. R. Schibli, J. W. Kuzucu, J. W. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kaertner, V. Scheuer, G. Angelow, “Toward single-cycle laser systems,” IEEE J. Select. Topics in Quantum. Electron. 9, 990–1001 (2003).

Siegman, A. E.

D. J. Kuizenga, A. E. Siegman, “Modulator frequency detuning effects in the FM-mode-locked laser,” IEEE J. Quantum Electron. QE-6, 803–808 (1970).

Smulakovski, V.

Smulakovsky, V.

A. Rosen, R. Weill, B. Levit, V. Smulakovsky, A. Bekker, B. Fischer, “Experimental observation of critical phenomena in a laser light system,” Phys. Rev. Lett. 105(1), 013905 (2010).
[CrossRef] [PubMed]

B. Vodonos, R. Weill, A. Gordon, A. Bekker, V. Smulakovsky, O. Gat, B. Fischer, “Formation and annihilation of laser light pulse quanta in a thermodynamic-like pathway,” Phys. Rev. Lett. 93(15), 153901 (2004).
[CrossRef] [PubMed]

Spielmann, Ch.

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. B. Corkum, U. Heinzmann, M. Drescher, F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001).
[CrossRef] [PubMed]

Stanley, H. E.

H. E. Stanley, “Exact solution for a linear chain of isotropically interacting classical spins of arbitrary dimensionality,” Phys. Rev. 179(2), 570–577 (1969).
[CrossRef]

Stentz, A.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–639 (2000).
[CrossRef] [PubMed]

Udem, Th.

Th. Udem, R. Holzwarth, T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[CrossRef] [PubMed]

Vodonos, B.

B. Vodonos, R. Weill, A. Gordon, A. Bekker, V. Smulakovsky, O. Gat, B. Fischer, “Formation and annihilation of laser light pulse quanta in a thermodynamic-like pathway,” Phys. Rev. Lett. 93(15), 153901 (2004).
[CrossRef] [PubMed]

A. Gordon, B. Vodonos, V. Smulakovski, B. Fischer, “Melting and freezing of light pulses and modes in mode-locked lasers,” Opt. Express 11(25), 3418–3424 (2003).
[CrossRef] [PubMed]

Weill, R.

B. Fischer, R. Weill, “When does single-mode lasing become a condensation phenomenon?” Opt. Express 20(24), 26704–26713 (2012).
[CrossRef] [PubMed]

R. Weill, B. Levit, A. Bekker, O. Gat, B. Fischer, “Laser light condensate: Experimental demonstration of light-mode condensation in actively mode locked laser,” Opt. Express 18(16), 16520–16525 (2010).
[CrossRef] [PubMed]

R. Weill, B. Fischer, O. Gat, “Light-mode condensation in actively-mode-locked lasers,” Phys. Rev. Lett. 104(17), 173901 (2010).
[CrossRef] [PubMed]

A. Rosen, R. Weill, B. Levit, V. Smulakovsky, A. Bekker, B. Fischer, “Experimental observation of critical phenomena in a laser light system,” Phys. Rev. Lett. 105(1), 013905 (2010).
[CrossRef] [PubMed]

R. Weill, A. Rosen, A. Gordon, O. Gat, B. Fischer, “Critical behavior of light in mode-locked lasers,” Phys. Rev. Lett. 95(1), 013903 (2005).
[CrossRef] [PubMed]

B. Vodonos, R. Weill, A. Gordon, A. Bekker, V. Smulakovsky, O. Gat, B. Fischer, “Formation and annihilation of laser light pulse quanta in a thermodynamic-like pathway,” Phys. Rev. Lett. 93(15), 153901 (2004).
[CrossRef] [PubMed]

Windeler, R. S.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–639 (2000).
[CrossRef] [PubMed]

Yudin, G. L.

J. Itatani, F. Quéré, G. L. Yudin, M. Y. Ivanov, F. Krausz, P. B. Corkum, “Attosecond streak camera,” Phys. Rev. Lett. 88(17), 173903 (2002).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

D. J. Kuizenga, A. E. Siegman, “Modulator frequency detuning effects in the FM-mode-locked laser,” IEEE J. Quantum Electron. QE-6, 803–808 (1970).

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

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

IEEE J. Select. Topics in Quantum. Electron. (1)

T. R. Schibli, J. W. Kuzucu, J. W. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kaertner, V. Scheuer, G. Angelow, “Toward single-cycle laser systems,” IEEE J. Select. Topics in Quantum. Electron. 9, 990–1001 (2003).

J. Opt. Soc. Am. B (1)

Nat. Photonics (1)

P. Popmintchev, M. C. Chen, P. Arpin, M. M. Murnane, H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[CrossRef]

Nature (2)

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. B. Corkum, U. Heinzmann, M. Drescher, F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001).
[CrossRef] [PubMed]

Th. Udem, R. Holzwarth, T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[CrossRef] [PubMed]

New J. Phys. (1)

O. Gat, A. Gordon, B. Fischer, “Light-mode locking: a new class of solvable statistical physics systems,” New J. Phys. 7, 151 (2005).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. (2)

H. E. Stanley, “Exact solution for a linear chain of isotropically interacting classical spins of arbitrary dimensionality,” Phys. Rev. 179(2), 570–577 (1969).
[CrossRef]

T. H. Berlin, M. Kac, “The spherical model of a ferromagnet,” Phys. Rev. 86(6), 821–835 (1952).
[CrossRef]

Phys. Rev. Lett. (7)

R. Weill, B. Fischer, O. Gat, “Light-mode condensation in actively-mode-locked lasers,” Phys. Rev. Lett. 104(17), 173901 (2010).
[CrossRef] [PubMed]

B. Vodonos, R. Weill, A. Gordon, A. Bekker, V. Smulakovsky, O. Gat, B. Fischer, “Formation and annihilation of laser light pulse quanta in a thermodynamic-like pathway,” Phys. Rev. Lett. 93(15), 153901 (2004).
[CrossRef] [PubMed]

A. Gordon, B. Fischer, “Phase transition theory of many-mode ordering and pulse formation in lasers,” Phys. Rev. Lett. 89(10), 103901 (2002).
[CrossRef] [PubMed]

R. Weill, A. Rosen, A. Gordon, O. Gat, B. Fischer, “Critical behavior of light in mode-locked lasers,” Phys. Rev. Lett. 95(1), 013903 (2005).
[CrossRef] [PubMed]

A. Rosen, R. Weill, B. Levit, V. Smulakovsky, A. Bekker, B. Fischer, “Experimental observation of critical phenomena in a laser light system,” Phys. Rev. Lett. 105(1), 013905 (2010).
[CrossRef] [PubMed]

M. Katz, A. Gordon, O. Gat, B. Fischer, “Non-Gibbsian Stochastic light-mode dynamics of passive mode locking,” Phys. Rev. Lett. 97(11), 113902 (2006).
[CrossRef] [PubMed]

J. Itatani, F. Quéré, G. L. Yudin, M. Y. Ivanov, F. Krausz, P. B. Corkum, “Attosecond streak camera,” Phys. Rev. Lett. 88(17), 173903 (2002).
[CrossRef] [PubMed]

Science (2)

H. C. Kapteyn, O. Cohen, I. Christov, M. M. Murnane, “Harnessing attosecond science in the quest for coherent X-rays,” Science 317(5839), 775–778 (2007).
[CrossRef] [PubMed]

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–639 (2000).
[CrossRef] [PubMed]

Other (5)

F. X. Kärtner, E. P. Ippen, and S. T. Cundiff, “Femtosecond laser development,” in Femtosecond Optical Frequency Comb: Principle, Operation and Applications, J. Ye and S.T. Cundiff, eds., (Springer Verlag, 2005).

A. Siegman, Lasers (University Science Books, 1986).

G. S. Joyce, “Critical properties of the spherical model,” 2, Chapter 10, 375, in Phasetransitions and Critical Phenomena, C. Domb and M.S. Green, eds. (Academic Press, 1972).

R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic, 1989).

H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford University, 1971).

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Figures (3)

Fig. 1
Fig. 1

Construction of a 2-dimensional mode hyper-comb: The small arrows at the lattice points denote the modes phasor, where the directions indicate the phases. (The comb size in the figure serves only for illustration). (a) 1-dimensional mode frequency-comb: AML modulation with two frequencies, Ω 1 = m 1 Ω 0 , Ω 2 = m 2 Ω 0 with m 1 =1 and m 2 =7 , that induce coupling Jn between modes. (b) 2-dimensionsl mode-lattice constructed by shifting mode segments (here each of seven modes) from the linear mode chain in (a), one above the other. Each mode is coupled to its four nearest-neighbor modes. Note the “broken bonds” between segments denoted by X, that can be neglected for large hyper-mode sizes.

Fig. 2
Fig. 2

Construction of a d-dimensional mode hyper-comb: (The comb size in the figure serves only for illustration.) (a) 1-dimensionsl mode system with three modulation frequencies, Ω n = m n Ω 0 that induce coupling Jn between modes j and j ± mn. In this figure m 1 =1, m 2 =7, m 3 =35 , and generally m 3 is a multiple of m 2 . (c) 3-dimensionsl mode-hyper-comb (with the above three modulation frequencies), constructed by further shifting the 2-dimensional mode matrices of Fig. 1(b). Each mode is coupled to its six nearest-neighbor modes. This procedure can be proceeded to higher dimensions.

Fig. 3
Fig. 3

Triangular (hexagonal) mode-lattice: (a) 1-dimensionsl mode-comb with three frequencies, Ω 1 = Ω 0 , Ω 2 = m 2 Ω 0 , and Ω 3 = m 3 Ω 0 =( m 2 +1) Ω 0 , and mode coupling Jn. (In this figure m 1 =1, m 2 =7, m 3 =8 ). (b) Reconstruction to a 2-dimensionsl triangular (hexagonal) mode-lattice. In the reconstruction procedure each additional segment that is placed as a new row is horizontally shifted by half a period. It reflects the coupling of each mode in the former row to two adjacent modes in the above row, besides the coupling to nearest neighbor modes in the same row. Experimentally, those couplings can be different in strength and sign.

Equations (8)

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ν ijk... ν i'j'k'... =[(ii') m 1 +(jj') m 2 +(kk') m 3 +...] Ω 0 /2π
a ijk... τ = n=1 d A n 2 i,j,k... a i±1,j±1,k±1... +[g(P)l] a ijk... + Γ ijk... (t,τ) ,
a ijk... /τ=H/ a * i±1,j±1,k±1... + Γ ijk... (t,τ) a * ijk... /τ=H/ a i±1,j±1,k±1... + Γ * ijk... (t,τ),
H= H int g 0 ln( P sat +P)+lP,
H int = n=1 d A n 2 i,j,k... a ijk... a * i±1,j±1,k±1... .
ρ( a ijk... ,... a ijk... )δ(P P 0 )exp( H int /2T).
< a ˜ ijk >={ ( 1 K c /K ) 1/2 forK K c 0forK< K c ,
< a ijk >={ ( P 0 /N ) 1/2 ( 1 P c / P 0 ) 1/2 for P 0 P c 0for P 0 < P c .

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