## Perturbative approach to continuum generation in a fiber Bragg grating

Optics Express, Vol. 14, Issue 17, pp. 7610-7616 (2006)

http://dx.doi.org/10.1364/OE.14.007610

Acrobat PDF (156 KB)

### 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

## 1. Introduction

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]

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]

9. R. R. Alfano, ed., *The Supercontinuum Laser Source*, 2^{nd} 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]

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]

## 2. Perturbative Solution of NLSE

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]

*D*(

*ω*) =

*β*(

*ω*)-

*β*(

*ω*

_{0})-

*β*

_{1}(

*ω*

_{0})(

*ω*-

*ω*

_{0}) is the fiber dispersion operator, β and β

_{1}are the fiber propagation constant and its first frequency derivative at ω

_{0}, the center frequency of the pulse, and

*K(ω)*is the Fourier transform of the time dependent nonlinear response function of the third-order susceptibility [13]. The grating is included by adding a term to the fiber dispersion operator, δβ

_{fbg}, which is the difference between the fiber and FBG dispersion operators:

*δβ*

_{fbg}=

*D*

_{fbg}-

*D*, where

*D*

_{fbg}(

*ω*) =

*β*

_{fbg}(

*ω*)-

*β*

_{fbg}(

*ω*

_{0})-

*β*

_{1,fbg}(

*ω*

_{0})(

*ω*-

*ω*

_{0}). The grating response is treated like an atomic resonance, localized to every point within the grating, and characterized by the propagation constant β

_{fbg}. In [10

**85**, 4600–4602 (2004). [CrossRef]

_{fbg}was approximated by an ideal, uniform 1D photonic bandgap. Here, we take β

_{fbg}from the FBG transmission coefficient

*t*

_{fpg}computed using coupled mode theory:

*t*

_{fbg}(

*ω*)=exp(

*iβ*

_{fbg}(

*ω*

*L*), where L is the length of the grating. This choice of β

_{fbg}gives the exact, causal result for linear propagation through the grating, and allows us to estimate the nonlinear response arising in non-uniform gratings. The real part of δβ

_{fbg}describes the large dispersion (and low group velocity) of transmitted light. For strong gratings, δβ

_{fbg}has a maximum value of ~κ=πδnη/λ

_{Bragg}, where δn is the grating index modulation, η is the core-mode overlap, and λ

_{Bragg}is the Bragg wavelength [14]. Strong gratings are defined by κL>>1. The imaginary part of δβ

_{fbg}represents the reduced transmission due to reflection for inband frequencies and is roughly zero outside the bandgap for the apodized gratings that we consider here. Figure 1(a) shows a typical δβ

_{fbg}.

_{0}when δβ

_{fbg}=0. The effect of the grating enters as a perturbation:

*A*≈

*A*

_{0}+

*A*

_{1}. When this is substituted into Eq. (1), and the purely A

_{0}terms are cancelled out, A

_{1}satisfies:

_{1}and A

_{0}. The key to our perturbation approximation is that we neglect nonlinear interaction between the grating-induced wave A

_{1}and the unperturbed continuum wave A

_{0}, keeping only the first two terms on the right hand side. The A

_{1}-A

_{0}interaction is small for two reasons: Firstly, because A

_{1}depends on δβ

_{fbg}, it is non-negligible only near the grating photonic bandgap. Since the grating bandwidth is much smaller than the continuum bandwidth (~1/100), one of the integrations in the A

_{1}-A

_{0}Kerr terms is smaller by roughly the ratio of these bandwidths. Secondly, the grating field amplitude A

_{1}is small because the grating dispersion spreads the grating induced light A

_{1}in the time domain. As shown below (see Fig. 1(d)), these assumptions imply that A

_{1}has a small temporal overlap with A

_{0}. Note that we retain the term δβ

_{fbg}A

_{1}. As previously stated, our approximation requires a narrow band grating resonance. However, this still allows for strong Bragg gratings with δβ

_{fbg}L>>1 as long as their bandwidth is much less than the continuum bandwidth, making the A

_{1}-A

_{0}interaction negligible. In strong gratings, δβ

_{fbg}A

_{1}may be the dominant term in Eq. (2) for frequencies near the bandgap. As a result, A

_{1}can become much larger than A

_{0}, and this requires that the dispersive effect of the grating on A

_{1}be included through δβ

_{fbg}A

_{1}. Note also that even though A

_{1}may become larger than A

_{0}near the bandgap, we may still neglect the higher order nonlinear terms in Eq. (2), since these depend on an integral over A

_{1}, which will still be small because of the small spectral extent of A

_{1}.

_{0}(using the substitution A

_{1}=

*ae*

^{i(D+δβfbg )z}). Substituting this solution into

*A*≈

*A*

_{0}+

*A*

_{1}, the approximate solution to Eq. 1 is then:

_{0}(ω,z) is computed from the NLSE without a grating present, Eq. (3) can then be used to estimate A(ω,L), the full continuum at L generated in the presence of the grating. Equation (3) is exact if there is no Kerr nonlinearity, i.e., A

_{0}propagates linearly. The grating-induced peaks arise only when new frequencies are generated by the Kerr nonlinearity, making A

_{0}increase along z near the Bragg wavelength. In the time domain, these nonlinearly generated frequencies are slowed by the grating dispersion and build up in a dispersive wave that has little interaction with the continuum pulse.

## 3. Comparison with full NLSE simulations and experiment

_{fbg}L<<1, and we expect a weak enhancement: A

_{1}<<A

_{0}. Equation (3) gives a continuum intensity of:

_{0}terms are slowly varying in frequency, the frequency dependence of the enhancement should follow Re{δβ

_{fbg}(ω)} in the region just outside of the bandgap where Im{δβ

_{fbg}(ω)}=0. The sign and shape of the enhancement will depend on the phase of

*A*

_{0}

*dz*. We did not examine very weak gratings, however weak enhancements (A

_{1}<A

_{0}, or equivalently a peak less than 3dB) roughly proportional to Re{δβ

_{fbg}(ω)} were observed in ref [11

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]

_{fbg}(ω)L<1 for wavelengths a few tenths of a nm from the minimum and maximum values of the grating features.

_{Bragg}=980nm, and |δβ

_{fbg}|L~50 near ω

_{Bragg}. To simplify the computation, the grating response was computed from 967 to 993nm with a very fine wavelength resolution to capture the rapid phase variations near the bandgap. Outside this range, δβ

_{fbg}was small enough that we simply set it to zero, thus avoiding the need to compute the grating response over the entire continuum bandwidth. This step was justified since we are only interested in the continuum within a few nms of the bandgap. This value of δβ

_{fbg}was used for both the approximate computation and, when added to fiber dispersion operator D, for the full NLSE simulations as well. The NLSE simulation used a Gaussian pulse of width 50fs, pulse energy 4nJ, and a fiber with dispersion zero near 1375nm, and dispersion, effective area, and nonlinear coefficient of 7ps/nmkm, 13μm

^{2}, and 10.6 W

^{-1}km

^{-1}respectively, near 1580nm. Figure 1(a) shows Re{δβ

_{fbg}(ω)} (dashed) and Im{δβ

_{fbg}(ω)} (solid). Figure 1(b) shows the continuum computed from the full NLSE simulation (Eq. (1), filled circles plus dotted line), and from the approximate solution (Eq. (3), solid). It also shows the unperturbed continuum without a grating present (dashed). The filled circles are the actual NLSE simulated points, showing that the grid spacing was barely sufficient to resolve the grating features. Substitution of A

_{1}and A

_{0}into Eq. (2) shows that the linear terms dominate the nonlinear terms over ~4nm around the Bragg wavelength, roughly consistent with the range over which there is agreement with the NLSE simulation. The inset of Fig. 1(b) shows the full NLSE simulations with and without grating on a larger wavelength scale. The grating changes the entire continuum to some extent, which can most easily be seen where the continuum is weak near 1100nm. The continuum is plotted on a log scale and little power is near these minima. Such changes are small enough for our approach to remain accurate. For comparison, Fig. 1(c) shows a measured response from a similar grating using the experimental setup of [10

**85**, 4600–4602 (2004). [CrossRef]

**85**, 4600–4602 (2004). [CrossRef]

_{samp}, much larger than the grating period. Such gratings reproduce the unsampled grating transmission response at evenly spaced frequency intervals proportional to 1/Λ

_{samp}. We therefore expect enhancements at multiple wavelengths from such a grating. The grating parameters were: L=3cm; δn=0.002; Λ

_{samp}=375μm; duty cycle 25%; λ

_{Bragg}=1250nm. Figure 2(b) shows that we again had good agreement with the full NLSE within the simulation resolution. Figure 2(c) shows a measured response for a similar grating resonance near 1120nm [15] and also shows multiple enhancement peaks. Full and approximate NLSE simulations at 1120nm were in agreement but gave little enhancement. Clearly, the enhancement peaks for strong gratings depend on the spectral phase of the unperturbed continuum envelope A

_{0}, as was shown for weak gratings above. It is known that continuum generation is very sensitive to pulse and fiber parameters. With a sufficiently precise model of the unperturbed continuum generation we would expect better agreement on the exact Bragg wavelengths at which enhancement peaks are experimentally observed.

## 4. Conclusion

## References and links

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. |

2. | N. Bloembergen and A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. |

3. | R. E. Slusher and B. J. Eggleton, ed., |

4. | B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg Grating Solitons,” Phys. Rev. Lett. |

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. |

6. | M. J. Steel and C. M. de Sterke, “Second-harmonic generation in second-harmonic fiber Bragg gratings,” Appl. Opt. |

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. |

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., |

10. | P. S. Westbrook, J. W. Nicholson, K. S. Feder, Y. Li, and T. Brown, “Supercontinuum generation in a fiber grating,” Appl. Phys. Lett. |

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 |

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. |

13. | G. P. Agrawal, |

14. | R. Kashyap, |

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. |

**OCIS Codes**

(050.2770) Diffraction and gratings : Gratings

(060.2310) Fiber optics and optical communications : Fiber optics

(190.4370) Nonlinear optics : Nonlinear optics, fibers

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: May 18, 2006

Revised Manuscript: June 28, 2006

Manuscript Accepted: June 28, 2006

Published: August 21, 2006

**Citation**

P. S. Westbrook and J. W. Nicholson, "Perturbative approach to continuum generation
in a fiber Bragg grating," Opt. Express **14**, 7610-7616 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-17-7610

Sort: Year | Journal | Reset

### References

- 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]
- N. Bloembergen and A. J. Sievers, "Nonlinear optical properties of periodic laminar structures," Appl. Phys. Lett. 17, 483 (1970). [CrossRef]
- R. E. Slusher and B. J. Eggleton, ed., Nonlinear Photonic Crystals, (Springer-Verlag, New York, 2003).
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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.

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.