OSA's Digital Library

Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 17 — Aug. 21, 2006
  • pp: 7610–7616
« Show journal navigation

Perturbative approach to continuum generation in a fiber Bragg grating

P. S. Westbrook and J. W. Nicholson  »View Author Affiliations


Optics Express, Vol. 14, Issue 17, pp. 7610-7616 (2006)
http://dx.doi.org/10.1364/OE.14.007610


View Full Text Article

Acrobat PDF (156 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

Fiber Bragg gratings (FBGs) are periodic modulations of the fiber core refractive index that form a one dimensional photonic band gap over which guided light is strongly reflected. FBGs are also known to have substantial dispersion for transmitted light [1

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]

]. It was suggested as early as 1970 that such photonic bandgap dispersion could affect and enhance nonlinear optical processes [2

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

]. Over the years, numerous nonlinear effects exploiting this dispersion have been studied [3

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

]. Examples include Bragg solitons [4

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]

], optical self-switching and gap solitons [5

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]

], and enhanced harmonic generation [6

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

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

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

]. In all of these processes, the bandwidth of the transmitted light is similar to or less than the grating bandgap. However, grating dispersion is also expected to influence more extreme nonlinear effects such as continuum generation, [9

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

] that can generate optical spectrum far in excess of a Bragg grating photonic bandgap. In particular, recent experiments have shown that when a continuum is generated using femtosecond pulses in the presence of a fiber Bragg grating, spectral peaks, some greater than an order of magnitude above the surrounding spectrum, appear in the transmitted continuum near the grating [10

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

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]

]. These grating enhancements may have significant impact in frequency metrology, where they can coherently amplify an arbitrary portion of a frequency comb, greatly increasing the signal to noise of beat notes between the comb and other frequency-stabilized light sources [12

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]

]. We have simulated the qualitative features of these grating-induced enhancements with a nonlinear Schrödinger equation (NLSE) that included the grating transmission response [10

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]

]. The enhancement was predominantly on one side of the grating bandgap (the long wavelength side) and could be as large as 20dB above the surrounding continuum. While NLSE simulations provide a direct tool to compute the enhancement from a given grating, they provide little physical insight and can be very time consuming due to the necessity for a very fine time/frequency grid to accommodate the disparate grating (~0.1THz) and continuum (>10THz) bandwidths.

2. Perturbative Solution of NLSE

A(ω,z)z=iD(ω)A(ω,z)+βfbg(ω)A(ω,z)
+K(ωω1)A(ω1,z)A(ω2,z)A*(ω1+ω2ω,z)dω1dω2
(1)

Here A is the amplitude of the E-field, D(ω) = β(ω)-β(ω 0)-β 1(ω 0)(ω-ω 0) is the fiber dispersion operator, β and β1 are the fiber propagation constant and its first frequency derivative at ω0, 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

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

]. The grating is included by adding a term to the fiber dispersion operator, δβfbg, which is the difference between the fiber and FBG dispersion operators: δβfbg =Dfbg -D, where Dfbg (ω) = βfbg (ω)-βfbg (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 βfbg. In [10

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]

] βfbg was approximated by an ideal, uniform 1D photonic bandgap. Here, we take βfbg from the FBG transmission coefficient tfpg computed using coupled mode theory: tfbg (ω)=exp(fbg (ω L), where L is the length of the grating. This choice of βfbg 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 δβfbg describes the large dispersion (and low group velocity) of transmitted light. For strong gratings, δβfbg has a maximum value of ~κ=πδnη/λBragg, where δn is the grating index modulation, η is the core-mode overlap, and λBragg is the Bragg wavelength [14

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

]. Strong gratings are defined by κL>>1. The imaginary part of δβfbg 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 δβfbg.

In our perturbative approach we first compute the unperturbed nonlinear solution A0 when δβfbg=0. The effect of the grating enters as a perturbation: AA 0+A 1. When this is substituted into Eq. (1), and the purely A0 terms are cancelled out, A1 satisfies:

A1(ω,z)z=βfbg(ω)A0(ω,z)+i(D(ω)+δβfbg(ω))A1(ω,z)
+K(ωω1)A0(ω1,z)A0(ω2,z)A1*(ω1+ω2ω,z)dω1dω2+
(2)

Here we have shown only one of the many nonlinear terms involving A1 and A0. The key to our perturbation approximation is that we neglect nonlinear interaction between the grating-induced wave A1 and the unperturbed continuum wave A0, keeping only the first two terms on the right hand side. The A1-A0 interaction is small for two reasons: Firstly, because A1 depends on δβfbg, 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 A1-A0 Kerr terms is smaller by roughly the ratio of these bandwidths. Secondly, the grating field amplitude A1 is small because the grating dispersion spreads the grating induced light A1 in the time domain. As shown below (see Fig. 1(d)), these assumptions imply that A1 has a small temporal overlap with A0. Note that we retain the term δβfbgA1. As previously stated, our approximation requires a narrow band grating resonance. However, this still allows for strong Bragg gratings with δβfbgL>>1 as long as their bandwidth is much less than the continuum bandwidth, making the A1-A0 interaction negligible. In strong gratings, δβfbgA1 may be the dominant term in Eq. (2) for frequencies near the bandgap. As a result, A1 can become much larger than A0, and this requires that the dispersive effect of the grating on A1 be included through δβfbgA1. Note also that even though A1 may become larger than A0 near the bandgap, we may still neglect the higher order nonlinear terms in Eq. (2), since these depend on an integral over A1, which will still be small because of the small spectral extent of A1.

With the nonlinear interaction eliminated, the solution to Eq. (2) may be written as an integral over A0 (using the substitution A1 = ae i(D+δβfbg )z). Substituting this solution into AA 0+A 1, the approximate solution to Eq. 1 is then:

A(ω,L)A0(ω,L)+βfbg(ω)0LA0(ω,z)ei(D(ω)+δβfbg(ω))(zL)dz
(3)

Once A0 (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., A0 propagates linearly. The grating-induced peaks arise only when new frequencies are generated by the Kerr nonlinearity, making A0 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

We now apply Eq. (3) to continuum generation in specific FBGs. In the case of a very weak grating, δβfbgL<<1, and we expect a weak enhancement: A1<<A0. Equation (3) gives a continuum intensity of:

A(ω,L)2A0(ω,L)2+2Re{βfbg(ω)A0(ω,L)*0LA0(ω,z)dz}.
(4)

Since the A0 terms are slowly varying in frequency, the frequency dependence of the enhancement should follow Re{δβfbg(ω)} in the region just outside of the bandgap where Im{δβfbg(ω)}=0. The sign and shape of the enhancement will depend on the phase of iA0*0L A 0 dz. We did not examine very weak gratings, however weak enhancements (A1<A0, or equivalently a peak less than 3dB) roughly proportional to Re{δβfbg(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]

]. These gratings had κL>1, but δβfbg(ω)L<1 for wavelengths a few tenths of a nm from the minimum and maximum values of the grating features.

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).

To test the applicability of our technique with a more complex grating, we simulated a sampled grating, whose index modulation is switched on and off with period Λsamp, much larger than the grating period. Such gratings reproduce the unsampled grating transmission response at evenly spaced frequency intervals proportional to 1/Λsamp. We therefore expect enhancements at multiple wavelengths from such a grating. The grating parameters were: L=3cm; δn=0.002; Λsamp=375μm; duty cycle 25%; λBragg=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

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.

] 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 A0, 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.

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.

4. Conclusion

We have shown that the large spectral peaks observed near a Bragg resonance when continuum is generated in the Bragg grating can be described by treating the grating as a perturbation to the nonlinear propagation. The continuum generation in the fiber without grating is computed once as a function of z in the fiber, and then combined with the grating transmission response to arrive at the grating-modified continuum without further solution of the NLSE. Thus, the response near any grating may be computed from an integral over a single computation of the full nonlinear propagation, enabling rapid (~100x faster) simulation of the change in continuum generation near a Bragg resonance. Such grating enhancements have shown great potential to improve signal to noise ratio in frequency metrology applications using fiber continuum combs.

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. 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.

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:  Author  |  Year  |  Journal  |  Reset  

References

  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.

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.

Figures

Fig. 1. Fig. 2.
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited