Abstract

We derive a perturbative solution to the nonlinear Schrödinger equation to include the effect of a fiber Bragg grating whose bandgap is much smaller than the pulse bandwidth. The grating generates a slow dispersive wave which may be computed from an integral over the unperturbed solution if nonlinear interaction between the grating and unperturbed waves is negligible. Our approach allows rapid estimation of large grating continuum enhancement peaks from a single nonlinear simulation of the waveguide without grating. We apply our method to uniform and sampled gratings, finding good agreement with full nonlinear simulations, and qualitatively reproducing experimental results.

© 2006 Optical Society of America

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  1. B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhosi, Z. Brodzeli, and F. Ouellette, "Dispersion compensation using a fibre grating in transmission," Electron. Lett. 32, 1610 (1996).
    [CrossRef]
  2. N. Bloembergen and A. J. Sievers, "Nonlinear optical properties of periodic laminar structures," Appl. Phys. Lett. 17, 483 (1970).
    [CrossRef]
  3. R. E. Slusher and B. J. Eggleton, ed., Nonlinear Photonic Crystals, (Springer-Verlag, New York, 2003).
  4. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg Grating Solitons," Phys. Rev. Lett. 76,1627 (1996).
    [CrossRef] [PubMed]
  5. D. Taverner, N. G. R. Broderick, D. J. Richardson, R. I. Laming, and M. Ibsen, "Nonlinear self-switching and multiple gap-soliton formation in a fiber Bragg grating," Opt. Lett. 23, 328-330 (1998).
    [CrossRef]
  6. M. J. Steel and C. M. de Sterke, "Second-harmonic generation in second-harmonic fiber Bragg gratings," Appl. Opt. 35, 3211-3222 (1996).
    [CrossRef] [PubMed]
  7. P. P. Markowicz, V. K. S. Hsiao, H. Tiryaki, A. N. Cartwright, P. N. Prasad, K. Dolgaleva, N. N. Lepeshkin, and R. W. Boyd, "Enhancement of third-harmonic generation in a polymer-dispersed liquid-crystal grating,"Appl. Phys. Lett. 87, 51102 (2005).
    [CrossRef]
  8. J. W. Nicholson, P. S. Westbrook, and K. S. Feder, "Localized Enhancement of Supercontinuum Generation using Fiber Bragg Gratings," CLEO 2005, paper CThU2.
  9. R. R. Alfano, ed., The Supercontinuum Laser Source, 2nd ed., (Springer, NewYork, 2006).
    [CrossRef]
  10. P. S. Westbrook, J. W. Nicholson, K. S. Feder, Y. Li, and T. Brown, "Supercontinuum generation in a fiber grating," Appl. Phys. Lett. 85, 4600-4602 (2004).
    [CrossRef]
  11. Y. Li, F. C. Salisbury, Z. Zhu, T. G. Brown, P. S. Westbrook, K. S. Feder, and R. S. Windeler, "Interaction of supercontinuum and Raman solitons with microstructure fiber gratings," Opt. Express 13, 998-1007 (2005).
    [CrossRef] [PubMed]
  12. K. Kim, S. A. Diddams, P. S. Westbrook, J. W. Nicholson, and K. S. Feder, "Improved stabilization of a 1.3 μm femtosecond optical frequency comb by use of a spectrally tailored continuum from a nonlinear fiber grating," Opt. Lett. 31, 277-279 (2006).
    [CrossRef] [PubMed]
  13. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, 1995).
  14. R. Kashyap, Fiber Bragg Gratings, (Academic, San Diego, 1999).
  15. P. S. Westbrook, J. W. Nicholson, K. S. Feder, Y. Li, and T. Brown, "Enhanced supercontinuum generation near fiber Bragg resonances," OFC 2005 paper OThQ4.

2006 (1)

2005 (2)

P. P. Markowicz, V. K. S. Hsiao, H. Tiryaki, A. N. Cartwright, P. N. Prasad, K. Dolgaleva, N. N. Lepeshkin, and R. W. Boyd, "Enhancement of third-harmonic generation in a polymer-dispersed liquid-crystal grating,"Appl. Phys. Lett. 87, 51102 (2005).
[CrossRef]

Y. Li, F. C. Salisbury, Z. Zhu, T. G. Brown, P. S. Westbrook, K. S. Feder, and R. S. Windeler, "Interaction of supercontinuum and Raman solitons with microstructure fiber gratings," Opt. Express 13, 998-1007 (2005).
[CrossRef] [PubMed]

2004 (1)

P. S. Westbrook, J. W. Nicholson, K. S. Feder, Y. Li, and T. Brown, "Supercontinuum generation in a fiber grating," Appl. Phys. Lett. 85, 4600-4602 (2004).
[CrossRef]

1998 (1)

1996 (3)

M. J. Steel and C. M. de Sterke, "Second-harmonic generation in second-harmonic fiber Bragg gratings," Appl. Opt. 35, 3211-3222 (1996).
[CrossRef] [PubMed]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg Grating Solitons," Phys. Rev. Lett. 76,1627 (1996).
[CrossRef] [PubMed]

B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhosi, Z. Brodzeli, and F. Ouellette, "Dispersion compensation using a fibre grating in transmission," Electron. Lett. 32, 1610 (1996).
[CrossRef]

1970 (1)

N. Bloembergen and A. J. Sievers, "Nonlinear optical properties of periodic laminar structures," Appl. Phys. Lett. 17, 483 (1970).
[CrossRef]

Bloembergen, N.

N. Bloembergen and A. J. Sievers, "Nonlinear optical properties of periodic laminar structures," Appl. Phys. Lett. 17, 483 (1970).
[CrossRef]

Boyd, R. W.

P. P. Markowicz, V. K. S. Hsiao, H. Tiryaki, A. N. Cartwright, P. N. Prasad, K. Dolgaleva, N. N. Lepeshkin, and R. W. Boyd, "Enhancement of third-harmonic generation in a polymer-dispersed liquid-crystal grating,"Appl. Phys. Lett. 87, 51102 (2005).
[CrossRef]

Broderick, N. G. R.

Brodzeli, Z.

B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhosi, Z. Brodzeli, and F. Ouellette, "Dispersion compensation using a fibre grating in transmission," Electron. Lett. 32, 1610 (1996).
[CrossRef]

Brown, T.

P. S. Westbrook, J. W. Nicholson, K. S. Feder, Y. Li, and T. Brown, "Supercontinuum generation in a fiber grating," Appl. Phys. Lett. 85, 4600-4602 (2004).
[CrossRef]

Brown, T. G.

Cartwright, A. N.

P. P. Markowicz, V. K. S. Hsiao, H. Tiryaki, A. N. Cartwright, P. N. Prasad, K. Dolgaleva, N. N. Lepeshkin, and R. W. Boyd, "Enhancement of third-harmonic generation in a polymer-dispersed liquid-crystal grating,"Appl. Phys. Lett. 87, 51102 (2005).
[CrossRef]

de Sterke, C. M.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg Grating Solitons," Phys. Rev. Lett. 76,1627 (1996).
[CrossRef] [PubMed]

M. J. Steel and C. M. de Sterke, "Second-harmonic generation in second-harmonic fiber Bragg gratings," Appl. Opt. 35, 3211-3222 (1996).
[CrossRef] [PubMed]

Dhosi, G.

B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhosi, Z. Brodzeli, and F. Ouellette, "Dispersion compensation using a fibre grating in transmission," Electron. Lett. 32, 1610 (1996).
[CrossRef]

Diddams, S. A.

Dolgaleva, K.

P. P. Markowicz, V. K. S. Hsiao, H. Tiryaki, A. N. Cartwright, P. N. Prasad, K. Dolgaleva, N. N. Lepeshkin, and R. W. Boyd, "Enhancement of third-harmonic generation in a polymer-dispersed liquid-crystal grating,"Appl. Phys. Lett. 87, 51102 (2005).
[CrossRef]

Eggleton, B. J.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg Grating Solitons," Phys. Rev. Lett. 76,1627 (1996).
[CrossRef] [PubMed]

B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhosi, Z. Brodzeli, and F. Ouellette, "Dispersion compensation using a fibre grating in transmission," Electron. Lett. 32, 1610 (1996).
[CrossRef]

Feder, K. S.

Hsiao, V. K. S.

P. P. Markowicz, V. K. S. Hsiao, H. Tiryaki, A. N. Cartwright, P. N. Prasad, K. Dolgaleva, N. N. Lepeshkin, and R. W. Boyd, "Enhancement of third-harmonic generation in a polymer-dispersed liquid-crystal grating,"Appl. Phys. Lett. 87, 51102 (2005).
[CrossRef]

Ibsen, M.

Kim, K.

Krug, P. A.

B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhosi, Z. Brodzeli, and F. Ouellette, "Dispersion compensation using a fibre grating in transmission," Electron. Lett. 32, 1610 (1996).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg Grating Solitons," Phys. Rev. Lett. 76,1627 (1996).
[CrossRef] [PubMed]

Laming, R. I.

Lepeshkin, N. N.

P. P. Markowicz, V. K. S. Hsiao, H. Tiryaki, A. N. Cartwright, P. N. Prasad, K. Dolgaleva, N. N. Lepeshkin, and R. W. Boyd, "Enhancement of third-harmonic generation in a polymer-dispersed liquid-crystal grating,"Appl. Phys. Lett. 87, 51102 (2005).
[CrossRef]

Li, Y.

Markowicz, P. P.

P. P. Markowicz, V. K. S. Hsiao, H. Tiryaki, A. N. Cartwright, P. N. Prasad, K. Dolgaleva, N. N. Lepeshkin, and R. W. Boyd, "Enhancement of third-harmonic generation in a polymer-dispersed liquid-crystal grating,"Appl. Phys. Lett. 87, 51102 (2005).
[CrossRef]

Nicholson, J. W.

Ouellette, F.

B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhosi, Z. Brodzeli, and F. Ouellette, "Dispersion compensation using a fibre grating in transmission," Electron. Lett. 32, 1610 (1996).
[CrossRef]

Prasad, P. N.

P. P. Markowicz, V. K. S. Hsiao, H. Tiryaki, A. N. Cartwright, P. N. Prasad, K. Dolgaleva, N. N. Lepeshkin, and R. W. Boyd, "Enhancement of third-harmonic generation in a polymer-dispersed liquid-crystal grating,"Appl. Phys. Lett. 87, 51102 (2005).
[CrossRef]

Richardson, D. J.

Salisbury, F. C.

Sievers, A. J.

N. Bloembergen and A. J. Sievers, "Nonlinear optical properties of periodic laminar structures," Appl. Phys. Lett. 17, 483 (1970).
[CrossRef]

Sipe, J. E.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg Grating Solitons," Phys. Rev. Lett. 76,1627 (1996).
[CrossRef] [PubMed]

Slusher, R. E.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg Grating Solitons," Phys. Rev. Lett. 76,1627 (1996).
[CrossRef] [PubMed]

Steel, M. J.

Stephens, T.

B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhosi, Z. Brodzeli, and F. Ouellette, "Dispersion compensation using a fibre grating in transmission," Electron. Lett. 32, 1610 (1996).
[CrossRef]

Taverner, D.

Tiryaki, H.

P. P. Markowicz, V. K. S. Hsiao, H. Tiryaki, A. N. Cartwright, P. N. Prasad, K. Dolgaleva, N. N. Lepeshkin, and R. W. Boyd, "Enhancement of third-harmonic generation in a polymer-dispersed liquid-crystal grating,"Appl. Phys. Lett. 87, 51102 (2005).
[CrossRef]

Westbrook, P. S.

Windeler, R. S.

Zhu, Z.

Appl. Opt. (1)

Appl. Phys. Lett. (3)

P. P. Markowicz, V. K. S. Hsiao, H. Tiryaki, A. N. Cartwright, P. N. Prasad, K. Dolgaleva, N. N. Lepeshkin, and R. W. Boyd, "Enhancement of third-harmonic generation in a polymer-dispersed liquid-crystal grating,"Appl. Phys. Lett. 87, 51102 (2005).
[CrossRef]

N. Bloembergen and A. J. Sievers, "Nonlinear optical properties of periodic laminar structures," Appl. Phys. Lett. 17, 483 (1970).
[CrossRef]

P. S. Westbrook, J. W. Nicholson, K. S. Feder, Y. Li, and T. Brown, "Supercontinuum generation in a fiber grating," Appl. Phys. Lett. 85, 4600-4602 (2004).
[CrossRef]

Electron. Lett. (1)

B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhosi, Z. Brodzeli, and F. Ouellette, "Dispersion compensation using a fibre grating in transmission," Electron. Lett. 32, 1610 (1996).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg Grating Solitons," Phys. Rev. Lett. 76,1627 (1996).
[CrossRef] [PubMed]

Other (6)

R. E. Slusher and B. J. Eggleton, ed., Nonlinear Photonic Crystals, (Springer-Verlag, New York, 2003).

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

R. Kashyap, Fiber Bragg Gratings, (Academic, San Diego, 1999).

P. S. Westbrook, J. W. Nicholson, K. S. Feder, Y. Li, and T. Brown, "Enhanced supercontinuum generation near fiber Bragg resonances," OFC 2005 paper OThQ4.

J. W. Nicholson, P. S. Westbrook, and K. S. Feder, "Localized Enhancement of Supercontinuum Generation using Fiber Bragg Gratings," CLEO 2005, paper CThU2.

R. R. Alfano, ed., The Supercontinuum Laser Source, 2nd ed., (Springer, NewYork, 2006).
[CrossRef]

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Figures (2)

Fig. 1.
Fig. 1.

Continuum generation with and without grating. (a) Im(δβfbg), solid. Re(δβfbg), dashed. (b) Continuum intensity near Bragg resonance. Solid: Approximation (Eq. (3)). Filled circles plus dotted line: Full NLSE grating simulation (Eq. (1)). Dashed: Unperturbed solution (δβfbg=0). Inset: Full NLSE simulations with (solid) and without (dotted) grating. (c) Measured enhancement with grating and continuum parameters similar to (b). Inset: full continuum. (d) Time dependence of continuum pulses on a logarithmic time scale: Approximation (solid), full NLSE (dotted), no grating (dashed, indicated with arrow).

Fig. 2.
Fig. 2.

Continuum generation in a sampled grating. All lines as in Fig. 1. (a) Re(δβfbg) and Im(δβfbg). (b) Simulations. (c) Experiment with similar grating.

Equations (6)

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A ( ω , z ) z = iD ( ω ) A ( ω , z ) + β fbg ( ω ) A ( ω , z )
+ K ( ω ω 1 ) A ( ω 1 , z ) A ( ω 2 , z ) A * ( ω 1 + ω 2 ω , z ) d ω 1 d ω 2
A 1 ( ω , z ) z = β fbg ( ω ) A 0 ( ω , z ) + i ( D ( ω ) + δ β fbg ( ω ) ) A 1 ( ω , z )
+ K ( ω ω 1 ) A 0 ( ω 1 , z ) A 0 ( ω 2 , z ) A 1 * ( ω 1 + ω 2 ω , z ) d ω 1 d ω 2 +
A ( ω , L ) A 0 ( ω , L ) + β fbg ( ω ) 0 L A 0 ( ω , z ) e i ( D ( ω ) + δ β fbg ( ω ) ) ( z L ) dz
A ( ω , L ) 2 A 0 ( ω , L ) 2 + 2 Re { β fbg ( ω ) A 0 ( ω , L ) * 0 L A 0 ( ω , z ) dz } .

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