Abstract

A detailed analysis of mode-locking is presented in which the nonlinear mode-coupling behavior in a waveguide array, dual-core fiber, and/or fiber array is used to achieve stable and robust passive mode-locking. By using the discrete, nearest-neighbor spatial coupling of these nonlinear mode-coupling devices, low-intensity light can be transferred to the neighboring waveguides and ejected (attenuated) from the laser cavity. In contrast, higher intensity light is self-focused in the launch waveguide and remains largely unaffected. This nonlinear effect, which is a discrete Kerr lens effect, leads to the temporal intensity discrimination required in the laser cavity for mode-locking. Numerical studies of this pulse shaping mechanism show that using current waveguide arrays, fiber-arrays, or dual-core fibers in conjunction with standard optical fiber technology, stable and robust mode-locked soliton-like pulses are produced.

© 2005 Optical Society of America

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References

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  1. H. A. Haus, �??Mode-Locking of Lasers,�?? IEEE J. Sel. Top. Quantum Electron. 6, 1173-1185, (2000).
    [CrossRef]
  2. I. N. Duling III and M. L. Dennis, Compact sources of ultrashort pulses. Cambridge, U.K.: Cambridge University Press, 1995.
    [CrossRef]
  3. K. Tamura, H. A. Haus and E. P. Ippen, �??Self-starting additive pulse mode-locked erbium fiber ring laser,�?? Electron. Lett. 28, 2226-2228, (1992).
    [CrossRef]
  4. H. A. Haus, E. P. Ippen and K. Tamura, �??Additive-pulse mode-locking in fiber lasers,�?? IEEE J. Quantum. Electron. 30, 200-208, (1994).
    [CrossRef]
  5. M. E. Fermann, M. J. Andrejco, Y. Silberberg and M. L. Stock, �??Passive mode-locking by using nonlinear polarization evolution in a polarizing-maintaining erbium-doped fiber laser,�?? Opt. Lett. 29, 447-449, (1993).
  6. D. Y. Tang, W. S. Man and H. Y. Tam, �??Stimulated soliton pulse formation and its mechanism in a passively mode-locked fibre soliton laser,�?? Opt. Commun. 165, 189-194, (1999).
    [CrossRef]
  7. I. N. Duling, �??Subpicosecond all-fiber erbium laser,�?? Electron. Lett. 27, 544-545 (1991).
    [CrossRef]
  8. D. J. Richardson, R. I. Laming, D. N. Payne, V. J. Matsas and M. W. Phillips, �??Self-starting, passively modelocked erbium fiber laser based on the amplifying Sagnac switch,�?? Electron. Lett. 27, 542-544 (1991).
    [CrossRef]
  9. M. L. Dennis and I. N. Duling, �??High repetition rate figure eight laser with extracavity feedback,�?? Electron. Lett. 28, 1894-1896 (1992).
    [CrossRef]
  10. F. O. Ilday, F.W.Wise, and T. Sosnowski, �??High-energy femtosecond stretched-pulse fiber laser with a nonlinear optical loop mirror,�?? Opt. Lett. 27, 1531-1533 (2002).
    [CrossRef]
  11. F. X. K¨artner and U. Keller, �??Stabilization of solitonlike pulses with a slow saturable absorber,�?? Opt. Lett. 20, 16-18 (1995).
    [CrossRef] [PubMed]
  12. B. Collings , S. Tsuda, S. Cundiff, J. N. Kutz, M. Koch, W. Knox, K. Bergman, �??Short cavity Erbium/Ytterbium fiber lasers mode-locked with a saturable Bragg reflector,�?? IEEE J. Sel. Top. Quantum Electron. 3, 1065-1075 (1998).
  13. S. Tsuda, W. H. Knox, E. A. DeSouza, W. J. Jan and J. E. Cunningham, �??Low-loss intracavity AlAs/AlGaAs saturable Bragg reflector for femtosecond mode-locking in solid-state lasers,�?? Opt. Lett. 20, 1406-1408 (1995).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  16. J. N. Kutz, �??Mode-Locking of Fiber Lasers via Nonlinear Mode-Coupling,�?? Dissipative Solitons, Lecture Notes in Physics, Eds. N. N. Akhmediev and A. Ankiewicz, 241-265, Springer-Verlag, Berlin (2005).
    [CrossRef]
  17. J. Proctor and J. N. Kutz, �??Theory and Simulation of Passive Mode-Locking with Waveguide Arrays,�?? Opt. Lett. 13, 2013-2015 (2005).
    [CrossRef]
  18. H. G. Winful and D. T. Walton, �??Passive mode locking through nonlinear coupling in a dual-core fiber laser,�?? Opt. Lett., 17, 1688-1690 (1992).
    [CrossRef] [PubMed]
  19. K. Intrachat and J. N. Kutz, �??Theory and simulation of passive mode-locking dynamics using a long period fiber grating,�?? IEEE J. Quantum. Electron. 39, 1572-1578 (2003).
    [CrossRef]
  20. Y. Oh, S. L. Doty, J. W. Haus, and R. L. Fork, �??Robust operation of a dual-core fiber ring laser,�?? J. Opt. Soc. Am. B 12, 2502-2507 (1995).
    [CrossRef]
  21. S. Trillo and S. Wabnitz, �??Nonlinear nonreciprocity in a coherent mismatched directional coupler,�?? App. Phys. Lett. 49, 752-754 (1986).
    [CrossRef]
  22. S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, �??Soliton switching in fiber nonlinear directional couplers,�?? Opt. Lett. 13, 672-674 (1988).
    [CrossRef] [PubMed]
  23. S. Trillo and S. Wabnitz, �??Weak-pulse-activated coherent soliton switching in nonlinear couplers,�?? Opt. Lett. 16, 1-3 (1991).
    [CrossRef] [PubMed]
  24. S. R. Friberg, A. M. Weiner, Y. Silberberg, B. G. Sfez, and P. S. Smith, �??Femtosecond switching in a dualcore- fiber nonlinear coupler,�?? Opt. Lett. 13, 904-906 (1988).
    [CrossRef] [PubMed]
  25. S. Lan, E. DelRe, Z. Chen, M. Shih, and M. Segev, �??Directional coupler with soliton-induced waveguides,�?? Opt. Lett. 24, 475-477 (1999).
    [CrossRef]
  26. D. N. Christodoulides and R. I. Joseph, �??Discrete self-focusing in nonlinear arrays of coupled waveguides,�?? Opt. Lett. 13, 794-796 (1988).
    [CrossRef] [PubMed]
  27. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd and J. S. Aitchison, �??Discrete spatial optical solitons in waveguide arrays,�?? Phys. Rev. Lett. 81, 3383-3386 (1998).
    [CrossRef]
  28. A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, �??Discrete selftrapping soliton interactions, and beam steering in nonlinear waveguide arrays,�?? Phys. Rev. E 53, 1172-1189 (1996).
    [CrossRef]
  29. H. S. Eisenberg, R. Morandotti, Y. Silberberg, J. M. Arnold, G. Pennelli, and J. S. Aitchison, �??Optical discrete solitons in waveguide arrays. 1. Soliton formation,�?? J. Opt. Soc. Am. B 19, 2938-1944 (2002).
    [CrossRef]
  30. U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and and F. Lederer, �??Optical discrete solitons in waveguide arrays. 2. Dynamics properties,�?? J. Opt. Soc. Am. B 19, 2637- 2644 (2002).
    [CrossRef]
  31. T. Brabec, Ch. Spielmann, P. F. Curley, and F. Krausz, �??Kerr lens mode locking,�?? Opt. Lett. 17, 1292-1294 (1992).
    [CrossRef] [PubMed]
  32. L. Spinelli, B. Couilland, N. Goldblatt, and D. K. Negus, in Conference on Lasers and Electro-Optics, 10 (1991) OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper CPDP7.
  33. T. P.White, R. C.McPhedran, C. M. de Sterke, M. N. Litchinitser, and B. J. Eggleton, �??Resonance and scattering in microstructured optical fibers,�?? Opt. Lett. 27, 1977-1979 (2002).
    [CrossRef]
  34. T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. T¨unnermann, F. Lederer, �??Nonlinearity and Disorder in Fiber Arrays,�?? Phys. Rev. Lett. 93, 053901 (2004).
    [CrossRef] [PubMed]
  35. S. M. Jensen, �??The nonlinear coherent coupler.�?? IEEE J. Quantum Electron. QE18, 1580-1583 (1982).
    [CrossRef]
  36. M. Asobe, T. Kanamori, and K. Kubodera, �??Applications of highly nonlinear chalcogenide glass fibers in ultrafast all-optical switches,�?? IEEE J. Quantum Electron., 29, 2325-2333 (1993).
    [CrossRef]
  37. M. Asobe, T. Ohara, I. Yokohoma, and T. Kaino, �??Fabrication of Bragg grating in chalcogenide glass fiber using transverse holographic method,�?? Electron. Lett., 32, 1611-1613 (1996).
    [CrossRef]
  38. G. Lenz, J. Zimmerman, T. Katsufuji, M. E. Lines, H. Y. Hwang, S. Sp¨alter, R. E. Slusher, S. -W. Cheong, J. S. Sanghera, and I. D. Aggarwal, �??Large Kerr effect in bulk Se-based chalcogenide glasses,�?? Opt. Lett., 25, 254-256 (2000).
    [CrossRef]
  39. F. Smetkala, C. Qu´emard, L. Leneindre, J. Lucas, A. Barth´el´emy, and C. DeAngelis, �??Chalcogenide glasses with large nonlinear refractive indices,�?? J. Non-Cryst. Solids, 239, 139-142 (1998).
    [CrossRef]
  40. J.M. Harbold, F. ¨O . Ilday, and F.W.Wise, �??Highly Nonlinear Ge-As-Se and Ge-As-S-Se Glasses for All-Optical Switching,�?? IEEE Photonic Technol. Lett. 14, 822-824 (2002).
    [CrossRef]
  41. T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. T¨unnnermann, and F. Lederer, �??Nonlinearity and Disorder in Fiber Arrays,�?? Phys. Rev. Lett. 93, 053901 (2004).
    [CrossRef] [PubMed]
  42. J. N. Kutz, B. Collings, K. Bergman, and W. Knox, �??Stabilized pulse spacing in soliton lasers due to gain deple- tion and recovery,�?? IEEE J. Quantum. Electron. 34, 1749-1757 (1998).
    [CrossRef]

App. Phys. Lett.

S. Trillo and S. Wabnitz, �??Nonlinear nonreciprocity in a coherent mismatched directional coupler,�?? App. Phys. Lett. 49, 752-754 (1986).
[CrossRef]

Dissipative Solitons

J. N. Kutz, �??Mode-Locking of Fiber Lasers via Nonlinear Mode-Coupling,�?? Dissipative Solitons, Lecture Notes in Physics, Eds. N. N. Akhmediev and A. Ankiewicz, 241-265, Springer-Verlag, Berlin (2005).
[CrossRef]

Electron. Lett.

K. Tamura, H. A. Haus and E. P. Ippen, �??Self-starting additive pulse mode-locked erbium fiber ring laser,�?? Electron. Lett. 28, 2226-2228, (1992).
[CrossRef]

I. N. Duling, �??Subpicosecond all-fiber erbium laser,�?? Electron. Lett. 27, 544-545 (1991).
[CrossRef]

D. J. Richardson, R. I. Laming, D. N. Payne, V. J. Matsas and M. W. Phillips, �??Self-starting, passively modelocked erbium fiber laser based on the amplifying Sagnac switch,�?? Electron. Lett. 27, 542-544 (1991).
[CrossRef]

M. L. Dennis and I. N. Duling, �??High repetition rate figure eight laser with extracavity feedback,�?? Electron. Lett. 28, 1894-1896 (1992).
[CrossRef]

M. Asobe, T. Ohara, I. Yokohoma, and T. Kaino, �??Fabrication of Bragg grating in chalcogenide glass fiber using transverse holographic method,�?? Electron. Lett., 32, 1611-1613 (1996).
[CrossRef]

IEEE J. Quantum Electron.

S. M. Jensen, �??The nonlinear coherent coupler.�?? IEEE J. Quantum Electron. QE18, 1580-1583 (1982).
[CrossRef]

M. Asobe, T. Kanamori, and K. Kubodera, �??Applications of highly nonlinear chalcogenide glass fibers in ultrafast all-optical switches,�?? IEEE J. Quantum Electron., 29, 2325-2333 (1993).
[CrossRef]

IEEE J. Quantum. Electron.

J. N. Kutz, B. Collings, K. Bergman, and W. Knox, �??Stabilized pulse spacing in soliton lasers due to gain deple- tion and recovery,�?? IEEE J. Quantum. Electron. 34, 1749-1757 (1998).
[CrossRef]

H. A. Haus, E. P. Ippen and K. Tamura, �??Additive-pulse mode-locking in fiber lasers,�?? IEEE J. Quantum. Electron. 30, 200-208, (1994).
[CrossRef]

H. A. Haus, �??A theory of forced mode locking,�?? IEEE J. Quantum. Electron. 11, 323-330 (1975).
[CrossRef]

K. Intrachat and J. N. Kutz, �??Theory and simulation of passive mode-locking dynamics using a long period fiber grating,�?? IEEE J. Quantum. Electron. 39, 1572-1578 (2003).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

H. A. Haus, �??Mode-Locking of Lasers,�?? IEEE J. Sel. Top. Quantum Electron. 6, 1173-1185, (2000).
[CrossRef]

B. Collings , S. Tsuda, S. Cundiff, J. N. Kutz, M. Koch, W. Knox, K. Bergman, �??Short cavity Erbium/Ytterbium fiber lasers mode-locked with a saturable Bragg reflector,�?? IEEE J. Sel. Top. Quantum Electron. 3, 1065-1075 (1998).

IEEE Photonic Technol. Lett.

J.M. Harbold, F. ¨O . Ilday, and F.W.Wise, �??Highly Nonlinear Ge-As-Se and Ge-As-S-Se Glasses for All-Optical Switching,�?? IEEE Photonic Technol. Lett. 14, 822-824 (2002).
[CrossRef]

J. Non-Cryst. Solids

F. Smetkala, C. Qu´emard, L. Leneindre, J. Lucas, A. Barth´el´emy, and C. DeAngelis, �??Chalcogenide glasses with large nonlinear refractive indices,�?? J. Non-Cryst. Solids, 239, 139-142 (1998).
[CrossRef]

J. Opt. Soc.

Y. Oh, S. L. Doty, J. W. Haus, and R. L. Fork, �??Robust operation of a dual-core fiber ring laser,�?? J. Opt. Soc. Am. B 12, 2502-2507 (1995).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

D. Y. Tang, W. S. Man and H. Y. Tam, �??Stimulated soliton pulse formation and its mechanism in a passively mode-locked fibre soliton laser,�?? Opt. Commun. 165, 189-194, (1999).
[CrossRef]

Opt. Lett.

J. Proctor and J. N. Kutz, �??Theory and Simulation of Passive Mode-Locking with Waveguide Arrays,�?? Opt. Lett. 13, 2013-2015 (2005).
[CrossRef]

M. E. Fermann, M. J. Andrejco, Y. Silberberg and M. L. Stock, �??Passive mode-locking by using nonlinear polarization evolution in a polarizing-maintaining erbium-doped fiber laser,�?? Opt. Lett. 29, 447-449, (1993).

G. Lenz, J. Zimmerman, T. Katsufuji, M. E. Lines, H. Y. Hwang, S. Sp¨alter, R. E. Slusher, S. -W. Cheong, J. S. Sanghera, and I. D. Aggarwal, �??Large Kerr effect in bulk Se-based chalcogenide glasses,�?? Opt. Lett., 25, 254-256 (2000).
[CrossRef]

S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, �??Soliton switching in fiber nonlinear directional couplers,�?? Opt. Lett. 13, 672-674 (1988).
[CrossRef] [PubMed]

D. N. Christodoulides and R. I. Joseph, �??Discrete self-focusing in nonlinear arrays of coupled waveguides,�?? Opt. Lett. 13, 794-796 (1988).
[CrossRef] [PubMed]

S. R. Friberg, A. M. Weiner, Y. Silberberg, B. G. Sfez, and P. S. Smith, �??Femtosecond switching in a dualcore- fiber nonlinear coupler,�?? Opt. Lett. 13, 904-906 (1988).
[CrossRef] [PubMed]

S. Trillo and S. Wabnitz, �??Weak-pulse-activated coherent soliton switching in nonlinear couplers,�?? Opt. Lett. 16, 1-3 (1991).
[CrossRef] [PubMed]

T. Brabec, Ch. Spielmann, P. F. Curley, and F. Krausz, �??Kerr lens mode locking,�?? Opt. Lett. 17, 1292-1294 (1992).
[CrossRef] [PubMed]

H. G. Winful and D. T. Walton, �??Passive mode locking through nonlinear coupling in a dual-core fiber laser,�?? Opt. Lett., 17, 1688-1690 (1992).
[CrossRef] [PubMed]

T. P.White, R. C.McPhedran, C. M. de Sterke, M. N. Litchinitser, and B. J. Eggleton, �??Resonance and scattering in microstructured optical fibers,�?? Opt. Lett. 27, 1977-1979 (2002).
[CrossRef]

F. X. K¨artner and U. Keller, �??Stabilization of solitonlike pulses with a slow saturable absorber,�?? Opt. Lett. 20, 16-18 (1995).
[CrossRef] [PubMed]

S. Tsuda, W. H. Knox, E. A. DeSouza, W. J. Jan and J. E. Cunningham, �??Low-loss intracavity AlAs/AlGaAs saturable Bragg reflector for femtosecond mode-locking in solid-state lasers,�?? Opt. Lett. 20, 1406-1408 (1995).
[CrossRef] [PubMed]

S. Lan, E. DelRe, Z. Chen, M. Shih, and M. Segev, �??Directional coupler with soliton-induced waveguides,�?? Opt. Lett. 24, 475-477 (1999).
[CrossRef]

F. O. Ilday, F.W.Wise, and T. Sosnowski, �??High-energy femtosecond stretched-pulse fiber laser with a nonlinear optical loop mirror,�?? Opt. Lett. 27, 1531-1533 (2002).
[CrossRef]

OSA Technical Digest Series

L. Spinelli, B. Couilland, N. Goldblatt, and D. K. Negus, in Conference on Lasers and Electro-Optics, 10 (1991) OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper CPDP7.

Phys. Rev. E

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, �??Discrete selftrapping soliton interactions, and beam steering in nonlinear waveguide arrays,�?? Phys. Rev. E 53, 1172-1189 (1996).
[CrossRef]

Phys. Rev. Lett.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. T¨unnermann, F. Lederer, �??Nonlinearity and Disorder in Fiber Arrays,�?? Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef] [PubMed]

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. T¨unnnermann, and F. Lederer, �??Nonlinearity and Disorder in Fiber Arrays,�?? Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd and J. S. Aitchison, �??Discrete spatial optical solitons in waveguide arrays,�?? Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Other

I. N. Duling III and M. L. Dennis, Compact sources of ultrashort pulses. Cambridge, U.K.: Cambridge University Press, 1995.
[CrossRef]

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Figures (13)

Fig. 1.
Fig. 1.

Possible laser cavity configurations which include nonlinear mode-coupling (NLMC) as the mode-locking element. The fiber coupling to the NLMC is illustrated in Fig. 2. In addition to the basic setup, polarization controllers, isolators, and other stabilization mechanisms may be useful or required for successful operation.

Fig. 2.
Fig. 2.

Schematic of butt-coupling implementation of NLMC element in the laser cavity configurations of Fig. 1. Three NLCM are depicted: (a) a waveguide array, (b) a dual-core fiber, and (c) a fiber array. In addition to the basic butt-coupling, index matching materials and tappering to account for core-size mismatch may be required to improve performance. Note that the figures are not drawn to scale.

Fig. 3.
Fig. 3.

The classic representation of spatial diffraction and confinement of electromagnetic energy in a waveguide array considered by Peschel et al. [30]. In the top figure, the intensity is not strong enough to produce self-focusing and confinement in the center waveguide, whereas the bottom figure shows the self-focusing due to the NLMC. The effect of this spatial focusing on a temporal pulse is shown in Fig. 4. Note that light was launched in the center waveguide with initial amplitude A 0(0) = 1 (top) and A 0(0) = 3 (bottom)

Fig. 4.
Fig. 4.

Temporal pulse shaping in the center waveguide via passage through the waveguide array in Fig. 3. The dotted lines show the input waveform (e.g. a hyperbolic secant pulse) while the solid lines show the output. For low intensities (a), the energy in the center waveguide diffracts to the neighboring waveguides as shown in Fig. 3(top). As the intensity is increased, the pulse is temporally compressed due to the resonant coupling of low intensity light to the other waveguides. The temporal reshaping is responsible for the mode-locking.

Fig. 5.
Fig. 5.

Representation of the spatial diffraction and confinement of electromagnetic energy in a dual-core fiber. In the top figure, the intensity is not strong enough to produce self-focusing and confinement in the launch waveguide, whereas the bottom figure shows the self-focusing due to the NLMC. The effect of this spatial focusing on a temporal pulse is shown in Fig. 6. Note that light was launched in the the primary waveguide with initial amplitude A 0(0) = 1 (top) and A 0(0) = 3 (bottom)

Fig. 6.
Fig. 6.

Temporal pulse shaping in the launch waveguide via passage through the dual-core fiber. The dotted lines show the input waveform (e.g. a hyperbolic secant pulse) while the solid lines show the output. For low intensities (a), the energy in the launch waveguide couples to the neighboring core as shown in Fig. 5(top). As the intensity is increased, the pulse is temporally compressed due to the resonant coupling of low intensity light to the neighboring core. The temporal reshaping is responsible for the mode-locking.

Fig. 7.
Fig. 7.

Schematic of the fiber array configuration. Here the coupling is to the nearest neighbors. In a hexagonal configuration [41], the evanescent coupling is then dominated by six neighboring core modes.

Fig. 8.
Fig. 8.

Intensity dependent spatial diffraction and confinement of electromagnetic energy in a fiber array. In the top figures (a), the intensity is not strong enough to produce self-focusing and confinement in the launch waveguide. Specifically, the electromagnetic field which is initially launched with A 0,0 = 1 ((a)-left) quickly diffracts energy to its neighboring waveguides as it propagates over 2 cm ((a)-middle) and 4 cm ((a)-right). For sufficiently high intensities, A 0,0 = 3, the self-focusing confines the energy to the launch waveguide over the propagation distances of 0, 2, and 4 cm ((b)-left, middle, and right respectively).

Fig. 9.
Fig. 9.

Temporal pulse shaping in the launch waveguide via passage through a fiber array. The dotted lines show the input waveform (e.g. a hyperbolic secant pulse) while the solid lines show the output. For low intensities (a), the energy in the launch waveguide couples to the neighboring fiber cores as shown in Fig. 8(a). As the intensity is increased, the self-focusing begins to dominate and the pulse is temporally compressed due to the resonant coupling of low intensity light to the neighboring cores. The temporal reshaping is responsible for the mode-locking.

Fig. 10.
Fig. 10.

Stable mode-locking using a waveguide array for (a) a fixed gain model g(Z) = g 0 = 0.263 and (b) a saturable gain model of Eq. Eq. (2) with g 0 = 0.7. The mode-locking is robust to the specific gain model, cavity parameter changes, and cavity perturbations. Here is is assumed that a 20% coupling loss occurs at the input and output of the waveguide array.

Fig. 11.
Fig. 11.

Output temporal response of the waveguide array under stable mode-locked operation. The center panel depicts the input (dashed line) and output (solid line) in the center waveguide A 0. The temporal output of the two nearest waveguides, A ±1 and A ±2, are also depicted. The central waveguide retains 94% of the incoming light while the neighboring waveguides contain 1.5% (A ±1) and 2.7% (A ±2). Note the characteristic shape of the temporal energy in the neighboring waveguides due to the nonlinear mode-coupling.

Fig. 12.
Fig. 12.

Multi-pulse per round trip mode-locked operation for g(Z) = g 0 = 0.475 and 40% loss before and after waveguide array. On average, there are 3–4 stabilized pulses in the cavity for the given gain.

Fig. 13.
Fig. 13.

Stable mode-locking using (a) a dual-core fiber with g 0 = 0.39 and (b) using a fiber array with g 0 = 0.5. The mode-locking is robust to cavity parameter changes and cavity perturbations. Here is is assumed that a 20% coupling loss occurs at the input and output of the dual-core fiber and fiber array.

Equations (8)

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

i Q Z + 1 2 2 Q T 2 + Q 2 Q + iγQ ig ( Z ) ( 1 + τ 2 T 2 ) Q = 0 ,
g ( Z ) = 2 g 0 1 + Q 2 e 0 ,
i dA n + C ( A n 1 + A n + 1 ) + β A n 2 A n = 0 ,
i dA 1 + CA 2 + β A 1 2 A 1 = 0
i dA 2 + CA 1 + β A 2 2 A 2 = 0
i dA m , n + β A m , n 2 A m , n + C m + 1 , n A m + 1 , n
+ C m 1 , n A m 1 , n + C m , n + 1 A m , n + 1 + C m , n 1 A m , n 1
+ C m + 1 , n + 1 A m + 1 , n + 1 + C m 1 , n 1 A m 1 , n 1 = 0

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