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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 6 — Mar. 17, 2008
  • pp: 4206–4216
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High-order dispersion in chirped-pulse oscillators

Vladimir L. Kalashnikov, Alma Fernández, and Alexander Apolonski  »View Author Affiliations


Optics Express, Vol. 16, Issue 6, pp. 4206-4216 (2008)
http://dx.doi.org/10.1364/OE.16.004206


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Abstract

The effects of high-order dispersion on a chirped-pulse oscillator operating in the positive dispersion regime were studied both theoretically and experimentally. It was found that odd and negative even high-order dispersions impair the oscillator stability owing to resonance with the dispersion waves, but can broaden the spectrum as in the case of continuum generation in the fibers. Positive fourth-order dispersion enhances the stability and shifts the stability range into negative dispersion. The destabilization mechanism was found to be a parametrical instability which causes noisy mode locking around zero dispersion.

© 2008 Optical Society of America

1. Introduction

Generation of over-100 nJ femtosecond pulses at MHz repetition rates directly from a laser oscillator is of interest for numerous applications, including frequency conversion, gas sensing, metrology, micro-machining etc. One can approach such energy frontiers by stretching an oscillator cavity, which allows energy scalability ∝Tcav (Tcav is the cavity period) without extra amplification [1

1. S.H. Cho, B.E. Bouma, E.P. Ippen, and J.G. Fujimoto, “Low-repetition-rate high-peak-power Kerr-lens mode-locked TiAl2O3 laser with a multiple-pass cavity,” Opt. Lett. 24, 417–419 (1999). [CrossRef]

, 2

2. A. Apolonski, A. Poppe, G. Tempea, C. Spielmann, T. Udem, R. Holzwarth, T. Hänsch, and F. Krausz, “Controlling the phase evolution of few cycle light pulses,” Phys. Rev. Lett. 85, 740–743 (2000). [CrossRef] [PubMed]

]. The main problem is that a long-cavity oscillator has low stability owing to enhanced nonlinear effects at high peak power. To shift the instability threshold to higher energies, one has to stretch the pulse, i.e. decrease its peak power. In the negative dispersion regime (NDR), pulse stretching requires a fair amount of net negative group-delay dispersion (GDD) [3

3. S.H. Cho, F.X. Kärtner, U. Morgner, E.P. Ippen, J.G. Fujimoto, J.E. Cunnighan, and W.H. Knox, “Generation of 90-nJ pulses with a 4-MHz repetition-rate Kerr-lens mode-locked Ti:Al2O3 laser operating with net positive and negative intracavity dispersion,” Opt. Lett. 26, 560–562 (2001). [CrossRef]

, 4

4. A.M. Kowalevicz, A. Tucay Zare, F.X. Kärtner, J.G. Fujimoto, S. Dewald, U. Morgner, V. Scheuer, and G. Angelow, “Generation of 150-nJ pulses from a multiple-pass cavity Kerr-lens mode-locked Ti:Al2O3 oscillator,” Opt. Lett. 28, 1597–1599 (2003). [CrossRef] [PubMed]

]. Since the pulse is chirp-free in the NDR, such stretching is irreversible and requires the use of an extra-cavity nonlinear compression technique to achieve sub-100 fs pulse duration (τ).

An alternative technique providing sub-100 fs pulses with over-100 nJ energies uses an oscillator operating in the net positive dispersion regime (PDR realized in a chirped-pulse oscillator, CPO) [5

5. B. Proctor, E. Westwig, and F. Wise, “Characterization of a Kerr-lens mode-locked Ti:sapphire laser with positive group-velocity dispersion,” Opt. Lett. 18, 1654–1656 (1993). [CrossRef] [PubMed]

]. In that case the pulse has a large chirp Ψ [6

6. J.M. Dudley, S.M. Boussen, D.M.J. Cameron, and J.D. Harvey, “Complete characterization of a self-mode-locked Ti:sapphire laser in the vicinity of zero group-delay dispersion by frequency-resolved optical gating,” Appl. Opt. 38, 3308–3315 (1999). [CrossRef]

] that results in substantial pulse stretching (up to few picoseconds), reduction of its peak power and, thereby, stabilization [7

7. V.L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators: theory and comparison with experiment,” New J. Phys. 7, 217 (2005). [CrossRef]

]. Since Ψ≫1, the pulse spectrum is broad (the spectral width Δ in a mode-locked Ti:sapphire oscillator operating in the PDR can approach 100 nm [9

9. S. Naumov, A. Fernandez, R. Graf, P. Dombi, F. Krausz, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators,” New J. Phys. 7, 216 (2005). [CrossRef]

]), that provides its compressibility down to T≈2/Δ (<40 fs in a Ti:sapphire oscillator [8

8. A. Fernandez, T. Fuji, A. Poppe, A. Fürbach, F. Krausz, and A. Apolonski, “Chirped-pulse oscillators: a route to high-power femtosecond pulses without external amplification,” Opt. Lett. 29, 1366–1368 (2004). [CrossRef] [PubMed]

, 9

9. S. Naumov, A. Fernandez, R. Graf, P. Dombi, F. Krausz, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators,” New J. Phys. 7, 216 (2005). [CrossRef]

]).

A vague parallel to such a technique is provided by a similariton fiber oscillator operating in the PDR [10

10. J.M. Dudley, Ch. Finot, D.J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics”, Nature Physics 3, 597 (2007). [CrossRef]

]: substantial pulse stretching prevents instabilities with energy growth and strong chirp provides pulse compressibility down to few optical-cycle duration. As was pointed out in Ref. [11

11. Y. Logvin and H. Anis, “Similariton pulse instability in mode-locked Yb-doped fiber laser in the vicinity of zero cavity dispersion,” Opt. Express 15, 13607–13612 (2007). [CrossRef] [PubMed]

], the similariton fiber oscillator characteristics (such as the spectrum profile, the dependence of pulse duration, spectral width, stability etc. on GDD and energy) are interrelated (on average) with those of a solid-state CPO. Such an interrelation of a solid-state CPO and a similariton laser motivates additional study of the PDR.

Since the spectrum in the PDR is broad enough, the effect of high-order dispersion (HOD) is present and can substantially transform the spectral shape [7

7. V.L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators: theory and comparison with experiment,” New J. Phys. 7, 217 (2005). [CrossRef]

]. Such an effect in the NDR has been the subject of a great deal of speculation. Analysis has shown that the Schrödinger soliton [12

12. G.P. Agrawal, Nonlinear Fiber Optics, 2nd ed., (Academic, San Diego, Calif., 1995).

] developing in the NDR is unstable in the presence of third-order dispersion (TOD) if GDD is close to zero [13

13. P.K.A. Wai, C.R. Menyuk, H.H. Chen, and Y.C. Lee, “Soliton at the zero-group-dispersion wavelength of a single-model fiber,” Opt. Lett. 12, 628–630 (1987). [CrossRef] [PubMed]

, 14

14. P.K.A. Wai, H.H. Chen, and Y.C. Lee, “Radiations by “solitons” at the zero group-dispersion wavelength of single-mode optical fibers,” Phys. Rev. A 41, 426–439 (1990). [CrossRef] [PubMed]

, 15

15. J.N. Elgin, “Soliton propagation in an optical fiber with third-order dispersion,” Opt. Lett. 15, 1409–1410 (1992). [CrossRef]

, 16

16. V.E. Zakharov and E.A. Kuznetsov, “Optical solitons and quasisolitons,” J. Exp. Theor. Phys. 86, 1035–1046 (1998). [CrossRef]

], but can be stabilized in the presence of stimulated Raman scattering and nonlinear dispersion [17

17. M. Santagiustina and E.M. Wright, “Supression of third-order dispersion radiation in solid-state soliton lasers” Opt. Lett. 20, 2267–2269 (1995). [CrossRef] [PubMed]

, 18

18. Zh. Li, L. Li, H. Tian, G. Zhou, and K.H. Spatschek, “Chirped femtosecond solitonlike laser pulse form with self-frequency shift,” Phys. Rev. Lett. 89, 263901 (2002). [CrossRef] [PubMed]

]. In an oscillator, TOD i) shifts the spectrum to the range where GDD is more negative [19

19. M. Santagiustina, “Third-order dispersion radiation in solid-state solitary laser,” J. Opt. Soc. Am. B 14, 1484–1495 (1997). [CrossRef]

], ii) broadens the spectrum owing to dispersion wave radiation [20

20. F.W. Wise, I.A. Walmsley, and C.L. Tang, “Simultaneous formation of solitons and diespersive waves in a femtosecond ring dye laser” Opt. Lett. 13, 129–131 (1988). [CrossRef] [PubMed]

, 21

21. H.A. Haus, J.D. Moores, and L.E. Nelson, “Effect of third-order dispersion on passive mode locking,” Opt. Lett. 18, 51–53 (1993). [CrossRef] [PubMed]

, 22

22. M.L. Dennis and I.N. Duling III, “Third-order dispersion in femtosecond fiber lasers,” Opt. Lett. 19, 1750–1752 (1994). [CrossRef] [PubMed]

, 23

23. A. Rundquist, C. Durfee, Z. Chang, G. Taft, E. Zeek, S. Backus, M.M. Murnane, H.C. Kapteyn, I. Christov, and V. Stoev, “Ultrafast laser and amplifier sources,” Appl. Phys. B 65, 161–174 (1997). [CrossRef]

] the underlying mechanism is analogous to the Cherenkov radiation [24

24. N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted in optical fibers” Phys. Rev. E 51, 2602–2607 (1995).

]), iii) increases the minimum pulse width [22

22. M.L. Dennis and I.N. Duling III, “Third-order dispersion in femtosecond fiber lasers,” Opt. Lett. 19, 1750–1752 (1994). [CrossRef] [PubMed]

, 26

26. Ch. Spielmann, P.F. Curley, T. Brabec, and F. Krausz, “Ultrabroadband femtosecond lasers,” IEEE J. Quantum Electron. 30, 1100–1114 (1994). [CrossRef]

, 27

27. J. Herrmann, V.P. Kalosha, and M. Müller, “Highr-order phase dispersions in femtosecond Kerr-lens mode-locked solid-state lasers: sideband generation and pulse splitting,” Opt. Lett. 22, 236–238 (1997). [CrossRef] [PubMed]

, 28

28. V.P. Kalosha, M. Müller, J. Herrmann, and S. Gatz, “Spatiotemporal model of femtosecond pulse generation in Kerrlens mode-locked solid-state lasers,” J. Opt. Soc. Am. B 15, 535–550 (1998). [CrossRef]

, 29

29. V.L. Kalashnikov, E. Sorokin, S. Naumov, and I.T. Sorokina, “Spectral poperties of the Kerr-lens mode-locked Cr4+:YAG laser,” J. Opt. Soc. Am. B 20, 2084–2092 (2003). [CrossRef]

], iv) decreases the energy [29

29. V.L. Kalashnikov, E. Sorokin, S. Naumov, and I.T. Sorokina, “Spectral poperties of the Kerr-lens mode-locked Cr4+:YAG laser,” J. Opt. Soc. Am. B 20, 2084–2092 (2003). [CrossRef]

], and v) enhances the pulse instability (in particular, as a result of multipulsing and bounded perturbations) [19

19. M. Santagiustina, “Third-order dispersion radiation in solid-state solitary laser,” J. Opt. Soc. Am. B 14, 1484–1495 (1997). [CrossRef]

, 25

25. T. Brabec and S.M.J. Kelly, “Third-order dispersion as limiting factor to mode locking in femtosecond solitary lasers,” Opt. Lett. 18, 2002–2004 (1993). [CrossRef] [PubMed]

, 26

26. Ch. Spielmann, P.F. Curley, T. Brabec, and F. Krausz, “Ultrabroadband femtosecond lasers,” IEEE J. Quantum Electron. 30, 1100–1114 (1994). [CrossRef]

, 27

27. J. Herrmann, V.P. Kalosha, and M. Müller, “Highr-order phase dispersions in femtosecond Kerr-lens mode-locked solid-state lasers: sideband generation and pulse splitting,” Opt. Lett. 22, 236–238 (1997). [CrossRef] [PubMed]

, 28

28. V.P. Kalosha, M. Müller, J. Herrmann, and S. Gatz, “Spatiotemporal model of femtosecond pulse generation in Kerrlens mode-locked solid-state lasers,” J. Opt. Soc. Am. B 15, 535–550 (1998). [CrossRef]

, 29

29. V.L. Kalashnikov, E. Sorokin, S. Naumov, and I.T. Sorokina, “Spectral poperties of the Kerr-lens mode-locked Cr4+:YAG laser,” J. Opt. Soc. Am. B 20, 2084–2092 (2003). [CrossRef]

]. For relatively large negative GDD, TOD can reduce the pulse duration owing to extra broadening of the spectrum [29

29. V.L. Kalashnikov, E. Sorokin, S. Naumov, and I.T. Sorokina, “Spectral poperties of the Kerr-lens mode-locked Cr4+:YAG laser,” J. Opt. Soc. Am. B 20, 2084–2092 (2003). [CrossRef]

, 30

30. Y. Logvin, V.P. Kalosha, and H. Anis, “Third-order dispersion impact on mode-locking regimes of Yb-doped fiber laser with photonic bandgap fiber for dispersion compensation,” Opt. Express 15, 985–991 (2007). [CrossRef] [PubMed]

] and even enhance the stability owing to negative feedback provided by the spectral loss of the side-bands [29

29. V.L. Kalashnikov, E. Sorokin, S. Naumov, and I.T. Sorokina, “Spectral poperties of the Kerr-lens mode-locked Cr4+:YAG laser,” J. Opt. Soc. Am. B 20, 2084–2092 (2003). [CrossRef]

].

The effect of TOD on the chirped solitary pulse developing in the PDR [31

31. H.A. Haus, J.G. Fujimoto, and E.P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B 8, 2068–2076 (1991). [CrossRef]

, 32

32. E. Podivilov and V.L. Kalashnikov, “Heavily-chirped solitary pulses in te normal dispersion region: new solutions of the cubic-quintic complex Ginzburg-Landau equation,” JETP Letters 82, 524–528 (2005). [CrossRef]

, 33

33. V.L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, “Chirped-pulse oscillators: theory and experiment,” Appl. Phys. B 83, 503–510 (2006). [CrossRef]

] is not so well investigated. It is known that TOD causes asymmetry of the spectrum [6

6. J.M. Dudley, S.M. Boussen, D.M.J. Cameron, and J.D. Harvey, “Complete characterization of a self-mode-locked Ti:sapphire laser in the vicinity of zero group-delay dispersion by frequency-resolved optical gating,” Appl. Opt. 38, 3308–3315 (1999). [CrossRef]

, 7

7. V.L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators: theory and comparison with experiment,” New J. Phys. 7, 217 (2005). [CrossRef]

] as well as pulse broadening and leads to higher GDD values being necessary for stable operation in the so-called similariton regime referred to mode-locking in a fiber oscillator [30

30. Y. Logvin, V.P. Kalosha, and H. Anis, “Third-order dispersion impact on mode-locking regimes of Yb-doped fiber laser with photonic bandgap fiber for dispersion compensation,” Opt. Express 15, 985–991 (2007). [CrossRef] [PubMed]

, 11

11. Y. Logvin and H. Anis, “Similariton pulse instability in mode-locked Yb-doped fiber laser in the vicinity of zero cavity dispersion,” Opt. Express 15, 13607–13612 (2007). [CrossRef] [PubMed]

].

The effect of fourth-order dispersion (FOD) in the NDR substantially differs from that of TOD. If FOD is positive (i.e. GDD grows towards the edges of the pulse spectrum), the dispersion waves (the side-bands) radiate when the net GDD approaches zero [24

24. N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted in optical fibers” Phys. Rev. E 51, 2602–2607 (1995).

, 34

34. A. Höök and M. Karlssson, “Ultrashort solitons at the minimum-dispersion wavelength: effect of fourth-order dispersion,” Opt. Lett. 18, 1388–1390 (1993). [CrossRef] [PubMed]

, 35

35. Y.H. Cha, B. Yoo, J. Rhee, and Ch. Kim, “Numerical analysis of sideband generation in femtosecond solid-state lasers with high-order dispersion,” J. Korean Physical Sociaty 44, 859–863 (2004).

, 36

36. V.I. Karpman, “Solitons of the fourth order nonlinear Schrödinger equation,” Physics Letters A 193, 355–358 (1994). [CrossRef]

]. These waves take part of the pulse energy (spectral recoil). The energy transfer between the pulse and the dispersive waves is more effective than that for the TOD case [35

35. Y.H. Cha, B. Yoo, J. Rhee, and Ch. Kim, “Numerical analysis of sideband generation in femtosecond solid-state lasers with high-order dispersion,” J. Korean Physical Sociaty 44, 859–863 (2004).

] and can lead to pulse splitting [28

28. V.P. Kalosha, M. Müller, J. Herrmann, and S. Gatz, “Spatiotemporal model of femtosecond pulse generation in Kerrlens mode-locked solid-state lasers,” J. Opt. Soc. Am. B 15, 535–550 (1998). [CrossRef]

]. Nevertheless, soliton propagation without energy transfer to the dispersive waves is possible, if the nonlinearity is saturable [37

37. J. Fujioka and A. Espinosa, “Soliton-like solution of an extended NLS equation existing in resonance with linear dispersive waves,” J. Physical Sociaty of Japan 66, 2601–2607 (1997). [CrossRef]

]. The situation substantially differs if FOD is negative: there is a stable radiationless soliton-like pulse [36

36. V.I. Karpman, “Solitons of the fourth order nonlinear Schrödinger equation,” Physics Letters A 193, 355–358 (1994). [CrossRef]

, 38

38. M. Piché, J.-F. Cormier, and X. Zhu, “Bright optical soliton in the presence of fourth-order dispersion,” Opt. Lett. 21, 845–847 (1996). [CrossRef] [PubMed]

], and a certain minimum FOD provides its stabilization in the vicinity of zero GDD [39

39. I.P. Christov, M.M. Murnane, H.C. Kapteyn, J. Zhou, and Ch.-P. Huang, “Fourth-order dispersion limited solitary pulses,” Opt. Lett.1465–1467 (1994). [CrossRef] [PubMed]

]. In such a regime, FOD can provide additional spectral broadening [23

23. A. Rundquist, C. Durfee, Z. Chang, G. Taft, E. Zeek, S. Backus, M.M. Murnane, H.C. Kapteyn, I. Christov, and V. Stoev, “Ultrafast laser and amplifier sources,” Appl. Phys. B 65, 161–174 (1997). [CrossRef]

].

The FOD effect on the CPO remains largely unexplored. It was found, in particular, that FOD is the source of “M-shaped” spectra [7

7. V.L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators: theory and comparison with experiment,” New J. Phys. 7, 217 (2005). [CrossRef]

], and there are regimes, in which the parts of the spectrum can be located within both the negative and positive GDD regions [23

23. A. Rundquist, C. Durfee, Z. Chang, G. Taft, E. Zeek, S. Backus, M.M. Murnane, H.C. Kapteyn, I. Christov, and V. Stoev, “Ultrafast laser and amplifier sources,” Appl. Phys. B 65, 161–174 (1997). [CrossRef]

]. Also, a soliton with modulated envelope can develop in the presence of FOD when GDD is positive [16

16. V.E. Zakharov and E.A. Kuznetsov, “Optical solitons and quasisolitons,” J. Exp. Theor. Phys. 86, 1035–1046 (1998). [CrossRef]

].

In this paper we present a systematic numerical study of HOD effects in the CPO. We found that HOD plays a crucial role in the oscillator stability and, in particular, show the stabilizing effect of positive FOD as well as the strong destabilizing effect of negative FOD and TOD. We demonstrate that the nature of the instability in the vicinity of zero GDD substantially changes in the presence of HOD: strong irregular oscillation of the pulse occurs instead of CW excitation. As a result, the “chaotic mode locking” appears in the vicinity of zero GDD. Experiments with a Ti:sapphire CPO verify the existence of such a regime. The interconnection between the PDR and noisy mode-locking regimes is not clear, and so they both have to be further studied.

2. Heavily-chirped pulses in the PDR

When an oscillator operates in the PDR, pulses with strong chirp develop [5

5. B. Proctor, E. Westwig, and F. Wise, “Characterization of a Kerr-lens mode-locked Ti:sapphire laser with positive group-velocity dispersion,” Opt. Lett. 18, 1654–1656 (1993). [CrossRef] [PubMed]

, 31

31. H.A. Haus, J.G. Fujimoto, and E.P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B 8, 2068–2076 (1991). [CrossRef]

, 40

40. J.M. Soto-Crespo, N.N. Akhmediev, V.V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E 55, 4783–4796 (1997). [CrossRef]

]. Such a regime can be described on the basis of the distributed generalized complex cubic-quintic nonlinear Ginzburg-Landau model [7

7. V.L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators: theory and comparison with experiment,” New J. Phys. 7, 217 (2005). [CrossRef]

, 32

32. E. Podivilov and V.L. Kalashnikov, “Heavily-chirped solitary pulses in te normal dispersion region: new solutions of the cubic-quintic complex Ginzburg-Landau equation,” JETP Letters 82, 524–528 (2005). [CrossRef]

, 33

33. V.L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, “Chirped-pulse oscillators: theory and experiment,” Appl. Phys. B 83, 503–510 (2006). [CrossRef]

, 40

40. J.M. Soto-Crespo, N.N. Akhmediev, V.V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E 55, 4783–4796 (1997). [CrossRef]

, 41

41. N.N. Akhmediev and A. Ankiewicz, Solitons: nonlinear pulses and beams (Chapman & Hall, London, 1997).

]:

Az=[σA+α2At2k=2Nik+1βkkAtk]+[(κiγ)PκζP2]A.
(1)

Here A(z,t) is the slowly varying field amplitude, P=|A|2 is the power, t is the local time, z the propagation distance, α is the square of the inverse spectral filter bandwidth, and σ is the spectrally independent saturated net gain. The parameters βk describe the kth-order net dispersion (up to Nth order), and γ and κ are the self-phase and self-amplitude modulation coefficients, respectively. Parameter ζ describes saturation of the self-amplitude modulation. In a broadband solid-state oscillator, for example, parameter α is the square of the inverse gain band width multiplied by the saturated gain coefficient; γ is the nonlinear coefficient of the active medium; parameters κ and ζ are defined by the Kerr-lens mode locking mechanism [7

7. V.L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators: theory and comparison with experiment,” New J. Phys. 7, 217 (2005). [CrossRef]

]. The distance z is normalized to the cavity length Lcav, so that all parameters are calculated over Lcav.

It was found [32

32. E. Podivilov and V.L. Kalashnikov, “Heavily-chirped solitary pulses in te normal dispersion region: new solutions of the cubic-quintic complex Ginzburg-Landau equation,” JETP Letters 82, 524–528 (2005). [CrossRef]

, 33

33. V.L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, “Chirped-pulse oscillators: theory and experiment,” Appl. Phys. B 83, 503–510 (2006). [CrossRef]

, 40

40. J.M. Soto-Crespo, N.N. Akhmediev, V.V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E 55, 4783–4796 (1997). [CrossRef]

], that a heavily-chirped pulse can be described as the soliton-like solution of Eq. (1) for N=2. For high-energy pulses the quintic nonlinear term ζ becomes important. Such pulses suffer from the collapsing instability in the framework of the reduced (i.e. cubic nonlinear [31

31. H.A. Haus, J.G. Fujimoto, and E.P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B 8, 2068–2076 (1991). [CrossRef]

, 40

40. J.M. Soto-Crespo, N.N. Akhmediev, V.V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E 55, 4783–4796 (1997). [CrossRef]

]) version of Eq. (1) [42

42. A. Chernykh and S.K. Turitsyn, “Soliton and collapse regimes of pulse generation in passively mode-locking laser systems,” Opt. Lett. 20, 398–400 (1995). [CrossRef] [PubMed]

]. Physically, such an instability means that the Kerr-lensing inside an active medium is not limited by the pump- and laser-mode overlap, which forms the so-called soft aperture [43

43. J. Herrmann, “Theory of Kerr-lens mode locking: role of self-focusing and radially varying gain,” J. Opt. Soc. Am. B 11, 498–512 (1994). [CrossRef]

].

The existence of soliton-like pulses in the PDR can be explained by the combined action of two mechanisms: a pure phase mechanism and a frequency-dissipative one. The first mechanism is the balance of phase contributions from the pulse envelope A(t): β 22 A(t)/∂t 2 and the time-dependent phase ϕ(t): -β 2 A(t)[∂ϕ(t)/∂t]2 (see Eq. (1)). In the cubic nonlinear Ginzburg-Landau equation without HOD, such a balance is possible if the pulse chirp is equal to Ψ 2=2+γP 0/β 2 τ 2 (P 0 is the peak pulse power, β 2>0). However, a pure phase balance is not sufficient for pulse stabilization as the pulse spreads (just as it takes place in the similariton regime [10

10. J.M. Dudley, Ch. Finot, D.J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics”, Nature Physics 3, 597 (2007). [CrossRef]

]), i.e. some dissipative effects are required to form a quasi-soliton [40

40. J.M. Soto-Crespo, N.N. Akhmediev, V.V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E 55, 4783–4796 (1997). [CrossRef]

]. As was pointed out in Ref. [31

31. H.A. Haus, J.G. Fujimoto, and E.P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B 8, 2068–2076 (1991). [CrossRef]

], the pulse lengthening due to GDD can be compensated by its shortening owing to frequency filtering if the pulse is chirped. The chirp results in frequency deviation at the pulse front and tail, and the filter cuts off the high- and low-frequency wings of the pulse, thus shortening it.

Fig. 1. Dependence of the stability threshold on the energy and higher-order dispersion.

The analysis demonstrates that without HOD, the heavily-chirped pulse is stable within the range of its existence, i.e. in the range of σ<0 [33

33. V.L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, “Chirped-pulse oscillators: theory and experiment,” Appl. Phys. B 83, 503–510 (2006). [CrossRef]

]. Negativity of σ, i.e. stability against the CW amplification, can be provided by a certain minimum positive GDD growing with energy. Even in the case of σ<0, the heavily-chirped pulse possesses rich dynamics [44

44. R.J. Deissler and H.R. Brand, “Periodic, quasipereodic, and chaotic localized solutions of the quintic complex Ginzburg-Landau equation,” Phys. Rev. Lett. 72, 478–481 (1994). [CrossRef] [PubMed]

, 45

45. V.L. Kalashnikov and A. Chernykh, “Spectral anomalies and stability of chirped-pulse oscillators,” Phys. Rev. A 75, 033820 (2007). [CrossRef]

], which can be gained by HOD. As a result, the stability conditions can be changed dramatically in the presence of HOD.

3. Stability of the CPO in the presence of HOD

To study the effect of HOD on the CPO stability, we solve Eq. (1) numerically by the means of the split-step Fourier method [12

12. G.P. Agrawal, Nonlinear Fiber Optics, 2nd ed., (Academic, San Diego, Calif., 1995).

]. The parameters used correspond to those in Ref. [33

33. V.L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, “Chirped-pulse oscillators: theory and experiment,” Appl. Phys. B 83, 503–510 (2006). [CrossRef]

], i.e. a Ti:sapphire CPO: α=1.1 fs2, κ=0.04γ, ζ=0.6γ, γ=4.55MW-1; the local time step equals to 1 fs (215 points); the longitudinal step is Δz=10-3 Lcav, and the simulation interval exceeds 104 Lcav. The energy dependence of the net gain parameter σ is linearized in the vicinity of 0: σ(E)dσdEE=E*E*(EE*1) , where E* corresponds to the energy stored inside an oscillator in the CW regime. Parameter δdσdEE=E*E*=0.05 defines the response of the active medium to the pulse energy E and can be expressed through the gain and loss coefficients only [33

33. V.L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, “Chirped-pulse oscillators: theory and experiment,” Appl. Phys. B 83, 503–510 (2006). [CrossRef]

]. The initial conditions are: a=a 0 sech[(t-t′)/θ], θ=200 fs, a 2 0=2.5×10-3/γ, and σ=-δ. The seeding pulse is located in the center of the time window at t′. In the absence of HOD, the numerical solution converges to the analytical one of Ref. [33

33. V.L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, “Chirped-pulse oscillators: theory and experiment,” Appl. Phys. B 83, 503–510 (2006). [CrossRef]

] during less than 1000 round-trips. Variations of the initial a 0 and θ affect only the convergence time.

Fig. 2. The pulse peak power evolution (a), spectral (b) and temporal (c) profiles (at z=104 Lcav) of the strongly perturbed pulse, E*=288 nJ, β 2=90 fs2, β 3=-300 fs2.

The simulations demonstrate that, as in the case without HOD, there exists some minimum GDD (β 2) for a given energy (E*) and a given HOD (βk, k>2), which provides stability for the CPO. We varied the HOD terms separately, i.e. only a single βk term was nonzero for k>2. The corresponding dependences of the threshold GDD (β 2), which provides the CPO stability, are shown in Fig. 1.

Odd HOD destabilizes the pulse so that a large positive GDD (larger β 2) is required for pulse stabilization (in Fig. 1 only one case with β 3≠0 is shown). As in the case without HOD, the stabilization threshold monotonically increases with the energy. The source of destabilization in the presence of odd HOD is the excitation of dispersive waves, which is caused by resonance with the CW perturbation. Since the spectrum of the chirped pulse is sufficiently broad (and it broadens as β 2 decreases), the resonance frequency (defined from the dispersion relation [12

12. G.P. Agrawal, Nonlinear Fiber Optics, 2nd ed., (Academic, San Diego, Calif., 1995).

, 41

41. N.N. Akhmediev and A. Ankiewicz, Solitons: nonlinear pulses and beams (Chapman & Hall, London, 1997).

]) shifts into the pulse spectrum. The appearing resonance with the dispersive wave causes strong parametric instability and pulse auto-oscillations (Fig. 2(a)) [6

6. J.M. Dudley, S.M. Boussen, D.M.J. Cameron, and J.D. Harvey, “Complete characterization of a self-mode-locked Ti:sapphire laser in the vicinity of zero group-delay dispersion by frequency-resolved optical gating,” Appl. Opt. 38, 3308–3315 (1999). [CrossRef]

, 11

11. Y. Logvin and H. Anis, “Similariton pulse instability in mode-locked Yb-doped fiber laser in the vicinity of zero cavity dispersion,” Opt. Express 15, 13607–13612 (2007). [CrossRef] [PubMed]

]. As a result of this instability, the fragmentation of spectral and temporal envelopes occurs (Figs. 2(b,c)). Hence, the destabilization does not have the form of a CW excitation (i.e. σ>0) but looks like pulse fission. An additional source of destabilization can be enhancement of the asymmetric internal perturbation modes analyzed in Ref. [45

45. V.L. Kalashnikov and A. Chernykh, “Spectral anomalies and stability of chirped-pulse oscillators,” Phys. Rev. A 75, 033820 (2007). [CrossRef]

].

The effect of FOD has to be considered separately. The sign of the β 4 term is substantial in this case. When β 4<0, i.e FOD decreases the total dispersion toward the edges of the spectrum, the pulse is unstable within a wide range of net dispersion (see Fig. 1). Such destabilization is understandable on the basis of the hypothesis about the role of resonance between a solitary pulse and linear waves. Negative FOD results in two resonances of the solitary pulse with the linear waves [12

12. G.P. Agrawal, Nonlinear Fiber Optics, 2nd ed., (Academic, San Diego, Calif., 1995).

, 33

33. V.L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, “Chirped-pulse oscillators: theory and experiment,” Appl. Phys. B 83, 503–510 (2006). [CrossRef]

, 41

41. N.N. Akhmediev and A. Ankiewicz, Solitons: nonlinear pulses and beams (Chapman & Hall, London, 1997).

]: -β 2(Δ/2)2 =β 2 ω 2+β 3 ω 3+β 4 ω 4+…, where Δ/2 is the frequency of the spectral truncation measured from the carrier frequency. Since the pulse cannot be stabilized by the spectral self-shift, as it happens in the case of pure TOD, the destabilization is stronger for β 4<0. Again, the destabilization looks like the strong auto-oscillations of the pulse peak power (like those in Fig. 2(a)) and the fragmentation of the pulse envelope. The fragmentation is similar to that illustrated by Fig. 2, but it is approximately symmetric in both the temporal and spectral domains owing to the symmetry of the two dispersive resonances.

Fig. 3. Pulse spectrum (a), temporal profile (b) and enlarged temporal profile (c) at the stability border (z=104 Lcav). E=144 nJ, β 4=4000 fs4. The red curve is the GDD profile β 2+6β 4 ω 2 [7].

FOD of opposite sign (β 4>0) forbids resonance with the linear waves within the range β 2>0. An important feature of the regime with β 4>0 is that it is very stable in the vicinity of β 2=0. The parameter β 2 that stabilizes the regime decreases with positive FOD (Fig. 1) and can “penetrate” even into the negative dispersion range. Within the range β 2<0, two resonances with the linear waves are possible if β 4>0. As a result, the destabilization arises. Therefore, there exists a certain maximum positive FOD providing a stable pulse within the widest range of parameter β 2. Higher values of FOD destabilize the pulse. It should be noted that for sufficiently large positive FOD the dependence of the stabilization threshold on energy exists only for comparatively low energies (Fig. 1). When the positive FOD and energy values are not too large, the destabilization mechanism is the CW amplification (σ>0). Growth of both energy and positive FOD results in a change of the destabilization mechanism, which is pulse splitting.

4. Temporal and spectral shapes of the chirped-pulse in the presence of HOD

The chirped pulse under the influence of HOD substantially differs from the usual chirped solitary pulse. If β 4>0, the spectral components are pushed out of the spectral range, where β 2 is close to zero (Fig. 3(a)). Such spectra are widely-observable in the experiment (Fig. 4). A strong asymmetry of the dispersion curve relative to its minimum resulted from a contribution of odd-order HODs can cause the spectrum asymmetry.

Fig. 4. Broadest experimental spectrum for the 10.7 MHz Ti:sapphire oscillator (blue curve) and net GDD (black curve). β 2≈55 fs2, β 3≈126 fs3, β 4≈4500 fs4.
Fig. 5. Dependence of the spectral Δ (black curve) and temporal T (red curve) widths on β 4 at the stability threshold. E=144 nJ, z=104 Lcav.

In contrast to the chirped pulse without HOD [32

32. E. Podivilov and V.L. Kalashnikov, “Heavily-chirped solitary pulses in te normal dispersion region: new solutions of the cubic-quintic complex Ginzburg-Landau equation,” JETP Letters 82, 524–528 (2005). [CrossRef]

], the pulse is Λ-shaped (Fig. 3(b)) with the profile peak (Fig. 3(c)) if β 4>0. The effect resembles optical wave breaking [12

12. G.P. Agrawal, Nonlinear Fiber Optics, 2nd ed., (Academic, San Diego, Calif., 1995).

] and will be analyzed elsewhere.

For the variable parameter β 4, the variations of the temporal and spectral pulse widths are shown in Fig. 5. The minimum pulse and maximum spectral widths correspond to β 2≈0. This is caused by shift of the spectral components to the edges of the spectrum if β 4∝0. The stability of the regime is provided by the both spectral splitting and shifting out of the zero GDD point. As a result, spectral dissipation grows and narrows the pulse spectrum. The pulse parameter variations with FOD are much smaller if β 4>0.

In combination with the generation of dispersive waves, the odd HOD terms allow the spectral shape to be controlled. In particular, the spectrum can be extra-broadened (see Fig. 6, where the full spectral widths are ≈0.53 fs-1 (black curve) vs. ≈0.36 fs-1 for β 5=0 (red curve)). One can see that the fifth-order dispersion provides the zero net-GDD at ω≈0.15 fs-1. As a result, the intensive dispersive wave develops around the zero-GDD as well as in the negative dispersion range. The mechanism of broadening is similar to that in fibers, where a spectral continuum appears [46

46. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006). [CrossRef]

]. Such extra broadening is possible in a similariton fiber oscillator operating in the PDR [30

30. Y. Logvin, V.P. Kalosha, and H. Anis, “Third-order dispersion impact on mode-locking regimes of Yb-doped fiber laser with photonic bandgap fiber for dispersion compensation,” Opt. Express 15, 985–991 (2007). [CrossRef] [PubMed]

] and in a solid-state oscillator operating in the NDR [29

29. V.L. Kalashnikov, E. Sorokin, S. Naumov, and I.T. Sorokina, “Spectral poperties of the Kerr-lens mode-locked Cr4+:YAG laser,” J. Opt. Soc. Am. B 20, 2084–2092 (2003). [CrossRef]

], as well.

Fig. 6. Spectrum at z=104 Lcav for E=144 nJ, β 2=31 fs2, β 5 =-103 fs5 (black) and 0 fs5 (red). The blue curve is the GDD profile.

5. Chaotic mode-locking in the vicinity of zero GDD

As it was described above, positive FOD stabilizes the chirped pulse in the vicinity of zero GDD against CW amplification. Stable operation at β 2≈0 (but not over the entire dispersion range) was reported in Ref. [6

6. J.M. Dudley, S.M. Boussen, D.M.J. Cameron, and J.D. Harvey, “Complete characterization of a self-mode-locked Ti:sapphire laser in the vicinity of zero group-delay dispersion by frequency-resolved optical gating,” Appl. Opt. 38, 3308–3315 (1999). [CrossRef]

].

Our experiments have demonstrated that an oscillator in the vicinity of zero GDD can suffer from auto-oscillation instability (so-called chaotic mode-locking [44

44. R.J. Deissler and H.R. Brand, “Periodic, quasipereodic, and chaotic localized solutions of the quintic complex Ginzburg-Landau equation,” Phys. Rev. Lett. 72, 478–481 (1994). [CrossRef] [PubMed]

, 45

45. V.L. Kalashnikov and A. Chernykh, “Spectral anomalies and stability of chirped-pulse oscillators,” Phys. Rev. A 75, 033820 (2007). [CrossRef]

]; Fig. 7(a,c)). Even in the NDR, the chaotic mode-locked spectrum can mimic that in the PDR (Fig. 7(b)): it is truncated but not too abruptly. Smoothing of the spectrum edges also takes place in the PDR when β 2 approaches zero, thereby violating the condition αβ 2 [32

32. E. Podivilov and V.L. Kalashnikov, “Heavily-chirped solitary pulses in te normal dispersion region: new solutions of the cubic-quintic complex Ginzburg-Landau equation,” JETP Letters 82, 524–528 (2005). [CrossRef]

, 33

33. V.L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, “Chirped-pulse oscillators: theory and experiment,” Appl. Phys. B 83, 503–510 (2006). [CrossRef]

]. Hence, these regimes cannot be unambiguously spectrally distinguished.

The simulations on the basis of Eq. (1) reproduce the chaotic mode locking in the presence of positive FOD. The standard deviation of the peak pulse power set (that is i=1N(PiP¯)2(N1)P¯ , where is the average peak power, Pi is the peak power at i th-round- trip; N=104) at the stability threshold as a function of β 4 is shown in Fig. 8. The pulse auto-oscillations grow with β 4. Nevertheless, the pulse is not destroyed and there is no CW amplification in spite of the strong chaotic auto-oscillations of the pulse peak power.

Such an instability substantially differs from multipulsing or CW amplification when σ crosses the zero-level [47

47. V. L. Kalashnikov, E. Sorokin, and I.T. Sorokina, “Multipulse operation and limits of the Kerr-lens mode locking stability,” IEEE J. Quantum Electron. 39, 323–336 (2003). [CrossRef]

], and is similar to the parametrical instability causing the picosecond supercontinuum generation in fibers [46

46. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006). [CrossRef]

].

Fig. 7. The pulse train (a), generated spectrum (b), and autocorrelation trace (c) realized in the Ti:sapphire chaotic mode-locked oscillator.
Fig. 8. Relative standard deviation of the peak power evolution in dependence on β 4 at the stability threshold. E=144 nJ.
Fig. 9. Relative standard deviation of the peak power evolution as a function of β 2. E=144 nJ, β 4=4000 fs4.

6. Conclusion

Effects of HOD on CPO characteristics have been studied both theoretically and experimentally. It is found that odd HOD destabilizes the pulse so that larger positive GDD is required for pulse stabilization. The mechanism of the pulse destabilization is the parametric instability caused by resonance with dispersive waves when the resonance frequencies reach the heavily-chirped pulse spectrum.

The effect of FOD depends on the sign of β 4: negative FOD strongly destabilizes the CPO, while positive FOD enhances the pulse stability, suppresses its energy dependence, and allows stable operation for both positive and negative GDD values. Growth of FOD results in the pulse broadening and its spectrum narrowing, while for positive FOD this dependence is weaker.

The chaotic mode-locking regime was also studied. It was found that in the presence of positive FOD low GDD leads to irregular oscillations of the pulse peak power. Such auto-oscillations grow with FOD. Thus, there is a range in the vicinity of zero GDD where the chirped pulse exists, but its parameters oscillate. Such a range can spread into the negative GDD range.

The main results were obtained by studying a solid-state CPO, where a chirped solitary pulse develops. However, a solitary pulse is a particular case of a similariton. Therefore, the effects of HOD in a similariton fiber laser can be similar, so that the main conclusions of this paper can be valid for fiber oscillators, as well.

Acknowledgements

This work was supported in part by the DFG Cluster of Excellence “Munich Centre for Advanced Photonics” (www.munich-photonics.de). Author V.L.K. acknowledges the support from Max Planck Institute for Quantum Optics (MPQ), Garching, Germany and the Austrian National Science Fund (Fonds zur Forderung der wissenschaftlichen Forschung, FWF), projects P20293 and P17973.

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Y. Logvin and H. Anis, “Similariton pulse instability in mode-locked Yb-doped fiber laser in the vicinity of zero cavity dispersion,” Opt. Express 15, 13607–13612 (2007). [CrossRef] [PubMed]

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45.

V.L. Kalashnikov and A. Chernykh, “Spectral anomalies and stability of chirped-pulse oscillators,” Phys. Rev. A 75, 033820 (2007). [CrossRef]

46.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006). [CrossRef]

47.

V. L. Kalashnikov, E. Sorokin, and I.T. Sorokina, “Multipulse operation and limits of the Kerr-lens mode locking stability,” IEEE J. Quantum Electron. 39, 323–336 (2003). [CrossRef]

OCIS Codes
(140.4050) Lasers and laser optics : Mode-locked lasers
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: January 22, 2008
Revised Manuscript: March 7, 2008
Manuscript Accepted: March 8, 2008
Published: March 12, 2008

Citation
Vladimir L. Kalashnikov, Alma Fernández, and Alexander Apolonski, "High-order dispersion in chirped-pulse oscillators," Opt. Express 16, 4206-4216 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-6-4206


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References

  1. S. H. Cho, B. E. Bouma, E. P. Ippen, and J. G. Fujimoto, "Low-repetition-rate high-peak-power Kerr-lens mode-locked TiAl2O3 laser with a multiple-pass cavity," Opt. Lett. 24, 417-419 (1999). [CrossRef]
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