## Laser mode hyper-combs |

Optics Express, Vol. 21, Issue 5, pp. 6196-6204 (2013)

http://dx.doi.org/10.1364/OE.21.006196

Acrobat PDF (1031 KB)

### 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.

© 2013 OSA

## 1. Introduction

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

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

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

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

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

4. D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and 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]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

21. R. Weill, B. Levit, A. Bekker, O. Gat, and 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]

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

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

27. 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]

*d*> 2, excluding 2;

*d*in statistical-mechanics calculations is not restricted to integers) there is a second-order phase-transition from disordered spin phase to long-range ordering, due to the high-dimensional connectivity [23

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

27. 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]

## 2. The mode hyper-comb construction

*d*frequencies for

*d*dimensions. The first frequency matches the basic cavity resonance

*l*is the cavity roundtrip length and

*n*is the refractive index), We then add more terms with multiples of the lower frequencies, having altogether:

*n*= 1,2,…) induces coupling (mutual injection) between modes

*i*and

*i ± m*. (We assume cosinusoidal amplitude modulation that produces two sidebands around each mode, but any higher, but finite number of sidebands will not change the basic results.) For

_{n}*d = 1*(with

*i*and

*i ± 1.*A second modulation frequency

*j*and

*n + 1*axis is higher by a factor

*n*.

## 3. The mode hyper-comb and the spherical-model

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

27. 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]

**29**(9), 1022–1024 (2004). [CrossRef] [PubMed]

*A*is the complex modulation amplitude at frequency

_{n}*g(P)*is the total power

*P*dependent slow (compared to one roundtrip) saturable gain, and

*l*is the loss in the cavity. The equation does not include dispersion.

*T*has the role of temperature. Equation (2) can be written as:

**89**(10), 103901 (2002). [CrossRef] [PubMed]

**29**(9), 1022–1024 (2004). [CrossRef] [PubMed]

**29**(9), 1022–1024 (2004). [CrossRef] [PubMed]

**29**(9), 1022–1024 (2004). [CrossRef] [PubMed]

**29**(9), 1022–1024 (2004). [CrossRef] [PubMed]

*T*functioning as the (interaction) energy and temperature. This is the central base that connects the mode system to statistical mechanics [12

**89**(10), 103901 (2002). [CrossRef] [PubMed]

**29**(9), 1022–1024 (2004). [CrossRef] [PubMed]

**29**(9), 1022–1024 (2004). [CrossRef] [PubMed]

*P*that is usually varied, although

_{0}*T*can also be changed by noise injection [13

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

14. B. Vodonos, R. Weill, A. Gordon, A. Bekker, V. Smulakovsky, O. Gat, and 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*.

**86**(6), 821–835 (1952). [CrossRef]

**179**(2), 570–577 (1969). [CrossRef]

*d >2*undergoes a second-order phase-transition. It formally occurs in the thermodynamic limit, but practically finite hyper-comb sizes can be sufficient (discussed below). The average (over magnitude and phase) normalized mode complex amplitude

*A*- the modulation amplitude,

*N*- the overall mode number, and

*T*is the noise strength (“temperature”). The second order transition occurs at

*d>2*, (

*J*is the zero order Bessel function. Equation (7) describes a second-order phase transition at

_{0}**86**(6), 821–835 (1952). [CrossRef]

*d = 3*:

*K*grows, is between all modes, so that

*K*for

**29**(9), 1022–1024 (2004). [CrossRef] [PubMed]

*d = 3*below the phase-transition, the correlation is again decaying to zero with increasing distance, and the correlation length is [26]

## 4. Light pulse generation of the mode hyper-comb

*d>2*has a phase-transition to a phase-ordered mode structure. The high-dimensional mode connectivity overcomes the entropy randomizing force in the free energy, yielding a transition to an ordered mode-phase structure with enhanced resistance against noise, just as it happens in thermodynamics for interacting particles in solids and magnetic spin systems. In the time domain, the robust phase ordered mode-comb then gives ultimately short pulses that can use very broad frequency bandwidths. Below the phase transition or for

*K.*Therefore, the pulse captures the full frequency bandwidth of all

*N*modes, resulting in an ultimate pulse-width

**29**(9), 1022–1024 (2004). [CrossRef] [PubMed]

*d*axis (

*i = d,*in the folding procedure of the hyper-comb construction).

*d*axis mode separation, which is the largest difference in terms of frequency, the closest near-neighbor modes (along the first axis) remain coupled and dominate the pulse rate. Therefore, there is a basic difference between the hyper-comb and 1-dimensional AML with only one high harmonic modulation, say the highest one in the hyper-comb (

*m*) harmonic frequency modulation, however, we have multi-pulse oscillation with a rate

_{d}*m*modes apart, thus having

_{d}*m*interleaved sub-combs (“supermodes” [28]), usually without phase coupling between them [28], resulting in non-optimized pulses.

_{d}**29**(9), 1022–1024 (2004). [CrossRef] [PubMed]

*m*that gives

*K*the laser lies deeply in the ordered phase (for

## 5. Hyper-comb size

*n*axis is given by

**29**(9), 1022–1024 (2004). [CrossRef] [PubMed]

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

*m,*the modulation can start with sub

*d>2*, we can expect that in 2-dimensional mode-combs, only a few additional layers in the third dimension can be sufficient for getting a phase-transition to the phase-ordered comb.

## 6. Conclusion

## Acknowledgments

## References and links

1. | T. R. Schibli, J. W. Kuzucu, J. W. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kaertner, V. Scheuer, and G. Angelow, “Toward single-cycle laser systems,” IEEE J. Select. Topics in Quantum. Electron. |

2. | F. X. Kärtner, E. P. Ippen, and S. T. Cundiff, “Femtosecond laser development,” in |

3. | Th. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature |

4. | D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science |

5. | M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. B. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature |

6. | Z. Chang and P. B. Corkum, “Attosecond photon sources: the first decade and beyond,” J. Opt. Soc. Am. B |

7. | J. Itatani, F. Quéré, G. L. Yudin, M. Y. Ivanov, F. Krausz, and P. B. Corkum, “Attosecond streak camera,” Phys. Rev. Lett. |

8. | H. C. Kapteyn, O. Cohen, I. Christov, and M. M. Murnane, “Harnessing attosecond science in the quest for coherent X-rays,” Science |

9. | P. Popmintchev, M. C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics |

10. | H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. |

11. | D. J. Kuizenga and A. E. Siegman, “Modulator frequency detuning effects in the FM-mode-locked laser,” IEEE J. Quantum Electron. |

12. | A. Gordon and B. Fischer, “Phase transition theory of many-mode ordering and pulse formation in lasers,” Phys. Rev. Lett. |

13. | A. Gordon, B. Vodonos, V. Smulakovski, and B. Fischer, “Melting and freezing of light pulses and modes in mode-locked lasers,” Opt. Express |

14. | B. Vodonos, R. Weill, A. Gordon, A. Bekker, V. Smulakovsky, O. Gat, and B. Fischer, “Formation and annihilation of laser light pulse quanta in a thermodynamic-like pathway,” Phys. Rev. Lett. |

15. | O. Gat, A. Gordon, and B. Fischer, “Light-mode locking: a new class of solvable statistical physics systems,” New J. Phys. |

16. | R. Weill, A. Rosen, A. Gordon, O. Gat, and B. Fischer, “Critical behavior of light in mode-locked lasers,” Phys. Rev. Lett. |

17. | A. Rosen, R. Weill, B. Levit, V. Smulakovsky, A. Bekker, and B. Fischer, “Experimental observation of critical phenomena in a laser light system,” Phys. Rev. Lett. |

18. | M. Katz, A. Gordon, O. Gat, and B. Fischer, “Non-Gibbsian Stochastic light-mode dynamics of passive mode locking,” Phys. Rev. Lett. |

19. | A. Gordon and B. Fischer, “Statistical-mechanics theory of active mode locking with noise,” Opt. Lett. |

20. | R. Weill, B. Fischer, and O. Gat, “Light-mode condensation in actively-mode-locked lasers,” Phys. Rev. Lett. |

21. | R. Weill, B. Levit, A. Bekker, O. Gat, and B. Fischer, “Laser light condensate: Experimental demonstration of light-mode condensation in actively mode locked laser,” Opt. Express |

22. | B. Fischer and R. Weill, “When does single-mode lasing become a condensation phenomenon?” Opt. Express |

23. | T. H. Berlin and M. Kac, “The spherical model of a ferromagnet,” Phys. Rev. |

24. | G. S. Joyce, “Critical properties of the spherical model,” |

25. | R. J. Baxter, |

26. | H. E. Stanley, |

27. | H. E. Stanley, “Exact solution for a linear chain of isotropically interacting classical spins of arbitrary dimensionality,” Phys. Rev. |

28. | A. Siegman, |

**OCIS Codes**

(000.6590) General : Statistical mechanics

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

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: January 3, 2013

Revised Manuscript: February 10, 2013

Manuscript Accepted: February 10, 2013

Published: March 5, 2013

**Citation**

Alon Schwartz and Baruch Fischer, "Laser mode hyper-combs," Opt. Express **21**, 6196-6204 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-5-6196

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### References

- 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).
- 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).
- Th. Udem, R. Holzwarth, T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002). [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]
- 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]
- 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]
- 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]
- 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. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1173–1185 (2000). [CrossRef]
- 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).
- 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]
- 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]
- 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]
- 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]
- 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]
- A. Gordon, B. Fischer, “Statistical-mechanics theory of active mode locking with noise,” Opt. Lett. 29(9), 1022–1024 (2004). [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]
- B. Fischer, R. Weill, “When does single-mode lasing become a condensation phenomenon?” Opt. Express 20(24), 26704–26713 (2012). [CrossRef] [PubMed]
- T. H. Berlin, M. Kac, “The spherical model of a ferromagnet,” Phys. Rev. 86(6), 821–835 (1952). [CrossRef]
- 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).
- 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]
- A. Siegman, Lasers (University Science Books, 1986).

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