## Optical bistability and cutoff solitons in photonic bandgap fibers

Optics Express, Vol. 12, Issue 8, pp. 1518-1527 (2004)

http://dx.doi.org/10.1364/OPEX.12.001518

Acrobat PDF (709 KB)

### Abstract

We present detailed theoretical and numerical analysis of certain novel non-linear optical phenomena enabled by photonic bandgap fibers. In particular, we demonstrate the feasibility of optical bistability in an axially modulated nonlinear photonic bandgap fiber through analytical theory and detailed numerical experiments. At 1.55µm carrier wavelength, the in-fiber devices we propose can operate with only a few tens of mW of power, have a nearly instantaneous response and recovery time, and be shorter than 100µm. Furthermore, we predict existence of gap-like solitons (which have thus-far been described only in axially periodic systems) in axially uniform photonic bandgap fibers.

© 2004 Optical Society of America

## 1. Introduction

1. P. Russell, “Photonic crystal fibers,” Science **299**, 358–362 (2003). [CrossRef] [PubMed]

*λ*) in-fiber device that can exhibit optical bistability at power levels close to the ones used in telecommunication networks today. In Section 3, we demonstrate existence of gap-like solitons in axially uniform systems. We conclude in Section 4.

## 2. Optical bistability in axially modulated photonic bandgap fibers

2. Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” J. Lightwave. Technol. **17**, 2039–2041, (1999). [CrossRef]

3. Steven G. Johnson, Mihai Ibanescu, M. Skorobogatiy, Ori Weisberg, Torkel D. Engeness, Marin Soljačić, Steven A. Jacobs, J. D. Joannopoulos, and Yoel Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express **9**, 748–779, (2001). [CrossRef] [PubMed]

4. S.D. Hart, G.R. Maskaly, B. Temelkuran, P.H. Prideaux, J.D. Joannopoulos, and Y. Fink, “External Reflection from Omnidirectional Dielectric Mirror Fibers,” Science **296**, 510–513, (2002). [CrossRef] [PubMed]

5. B. Temelkuran, S.D. Hart, G. Benoit, J.D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO_{2} laser transmission,” Nature **420**, 650–653, (2002). [CrossRef] [PubMed]

*40mW*, whose highly nonlinear input/output power relation is key to many applications [6

6. B.E.A. Saleh and M.C. Teich, *Fundamentals Of Photonics* (John Wiley&Sons, 1991). [CrossRef]

7. Marin Soljačić, Mihai Ibanescu, Steven G. Johnson, Yoel Fink, and J.D. Joannopoulos, “Optimal Bistable Switching in Non-Linear Photonic Crystals,” Phys. Rev. E **66**, 055601(R) (2002). [CrossRef]

8. J.S. Foresi, P.R. Villeneuve, J. Ferrera, E.R. Thoen, G. Steinmeyer, S. Fan, J.D. Joannopoulos, L.C. Kimerling, H.I. Smith, and E.P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature **390**, 143–145, (1997). [CrossRef]

9. Marin Soljačić, Mihai Ibanescu, Steven G. Johnson, J.D. Joannopoulos, and Yoel: Fink“Optical Bistability in Axially Modulated OmniGuide Fibers,” Opt. Lett. **28**, 516–518, (2003). [CrossRef]

*λ*

_{0}, where

*λ*

_{0}is the carrier wavelength in vacuum. The cladding consists of 7 bilayers (periods), each 0.3

*λ*

_{0}thick, 30% of which thickness is the high index (

*n*

_{H}=2.8) material. The inner-most layer, adjacent to the core, is low-index (

*n*

_{L}=1.5). The axial modulation consists of 41 (

*n*

_{SPH}=1.5) spheres tightly filling the core (diameter 0.41

*λ*

_{0}). The periodicity of the axial modulation opens a bandgap for the propagating mode. Our 3D frequency-domain simulations [10

10. Steven G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173–190, (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173. [CrossRef] [PubMed]

*6%*or larger (vs.<

*0.1%*in grated silica fibers).

*n*

_{DEF}=1.9. Tight confinement in the transverse direction is provided by the large band-gap of the OmniGuide cladding, while strong confinement in the axial direction is provided by the large axial bandgap. The cavity couples to the waveguide (axially uniform fiber) through tunneling processes. We model the low-index material to have an instantaneous Kerr non-linearity (the index change is

*δn*(

*r*,

*t*)=

*cn*

_{L}

*ε*

_{0}

*n*

_{2}|

**(**

*E***,**

*r**t*)|

^{2}, where

*n*

_{2}is the Kerr coefficient.)

*/3*and axial length of the mode≈

*6λ*

_{0}). Note that the transverse modal area is 2 orders of magnitude smaller than in typical low index contrast fibers; one could couple such fibers with tapering. The system has a nearly Lorentzian transmission spectrum:

*T*(

*ω*)≡

*P*

_{OUT}(

*ω*)/

*P*

_{IN}(

*ω*)≈

*γW*

^{2}/[(

*γ*

_{R}+

*γW*)

^{2}+(

*ω*-

*ω*

_{RES})

^{2}], where

*P*

_{OUT}and

*P*

_{IN}are the outgoing and incoming powers respectively,

*ω*

_{RES}is the resonant frequency,

*γ*

_{W}is the decay width due to the coupling of the cavity mode to the waveguides, while

*γ*

_{R}is the decay width due to the coupling to the cladding modes [12]. We measure a quality factor

*Q*=

*ω*

_{RES}/[

*2*(

*γ*

_{R}+

*γ*

_{W})]=

*540*, and a peak transmission TP=0.88; from this we obtain that the radiation quality factor

*Q*

_{R}=

*ω*

_{RES}/

*2γ*

_{R}

*=8700. Q*

_{R}is finite due to the coupling of energy to the radiating cladding modes; this coupling is the primary cause of losses in our system, but can be controlled [13

13. S.G. Johnson, S. Fan, A. Mekis, and J.D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. **78**, 3388–3390, (2001). [CrossRef]

*ω*

_{0}=

*ω*

_{RES}-

*3.2*(

*γ*

_{W}+

*γ*

_{R}), and the full-width half-maximum (FWHM) bandwidth of our pulses is

*ω*

_{0}/

*FWHM*=796. The ratio of the transmitted (

*E*

_{OUT}) vs. incoming (

*E*

_{IN}) pulse energy increases sharply as we approach the bistability threshold, and decreases after the threshold is passed, as shown in the upper-left panel of Fig. 2. Transmitted pulse spectra are also shown for a few pulses in upper-right panel of Fig. 2; the non-linear cavity redistributes the energy in the frequency spectrum; if these changes to the spectrum are undesirable, they can be removed by: optimizing the device, using time-integrating non-linearity, or by adding a frequency-dependent filter to the output of the device. Typical output-pulse shapes are shown in the lower panels of Fig. 2. As one can see in the lower-left panel, even without optimizing our system, we still obtain well-behaved shapes of output pulses in the regime where

*E*

_{OUT}

*/E*

_{IN}is maximal. Interestingly enough, the output shape is typically less well behaved as the bandwidth of the pulse becomes narrower with respect to the line-width of the cavity. The reason for this is that during the passage through the system, the pulse observes (self-induced) time-dependent shifts in the resonance position of the cavity, which leave a signature on the pulse itself. It turns out that for shorter-lasting pulses, these signatures get smoothed out. Of course, when the pulse bandwidth becomes larger than the line-width of the cavity, one starts observing distortions for a different reason: in that case, even the linear transmission becomes distorted.

*κ*(derived in detail elsewhere [7

7. Marin Soljačić, Mihai Ibanescu, Steven G. Johnson, Yoel Fink, and J.D. Joannopoulos, “Optimal Bistable Switching in Non-Linear Photonic Crystals,” Phys. Rev. E **66**, 055601(R) (2002). [CrossRef]

*n*(

*) is the unperturbed index of refraction,*

**r***(*

**E***,*

**r***t*)=[

*(*

**E***)*

**r***exp*(

*iωt*)+

**(**

*E***)*

**r***exp*(-

*iωt*)]/

*2*is the electric field,

*n*

_{2}(

*) is the local Kerr coefficient,*

**r***VOL*of integration is over the extent of the mode, and

*d*is the dimensionality of the system in question. As can be seen from Eq.(1),

*κ*is dimensionless and scale-invariant. It is determined by the degree of spatial confinement of the field in the non-linear material. To an excellent approximation, it is independent of

*n*

_{2}, the peak electric field amplitude,

*Q*, and small deviations in

*ω*

_{0}. Let us denote by

*P*

_{IN}and

*P*

_{OUT}respectively. Using a lorentzian transmission spectrum in the linear case and perturbation theory for small

*δn*(

*), we obtain:*

**r***δ*=(

*ω*

_{RES}-

*ω*

_{0})/(

*γ*

_{W}+

*γ*

_{R}),

*T*

_{P}=[

*γ*

_{W}+

*γ*

_{R})

^{2}] is the peak transmission, and

*P*

_{0}is a “characteristic power” of this cavity given by:

*κ*=

*0.020*from a single non-linear computation; together with the knowledge of

*Q*and

*ω*

_{RES}, we obtain

*δ*=

*3.2*which we plot, as a solid line in Fig. 3. We compare our analytical theory with numerical experiments in which we launch smoothly turned-on CW signals into the cavity. To observe the upper hysteresis branch we launch large-amplitude and wide-width gaussian pulses that decay into smaller steady state CW values. The small discrepancy between our analytical theory and the numerical experiments in Fig. 3 is mostly attributable to the fact that the linear-regime transmission curve is not a perfect Lorentzian; we operate fairly close to the edge of the axial band-gap, and the linear transmission shape looks more like a “tilted” Lorentizan.

*Q*s would be prohibitively long, our analytical model allows us to predict the behavior of a high-Q device. According to Eq. (3), the power requirements drop with 1/

*Q*

^{2}.

*Q*can be increased by adding more spheres to the “walls” of the cavity. For

*Q*of

*6000*(which is still compatible with

*10Gbit/sec*signals), the non-linear index changes are <

*0.001*(which is still compatible with nearly instantaneous non-linear materials, and below the damage threshold for many materials, including many Chalcogenide glasses [14

14. G. Lenz, J. Zimmerman, T. Katsufuji, M. E. Lines, H. Y. Hwang, S. Spälter, 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]

*34mW*(assuming

*n*

_{2}=

*1.5*10*

^{-17}

*m*

^{2}

*/W*, a value typical for the Chalcogenide glasses [14

14. G. Lenz, J. Zimmerman, T. Katsufuji, M. E. Lines, H. Y. Hwang, S. Spälter, 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]

15. Jeffrey M. Harbold, F. Ömer Ilday, Frank W. Wise, and Bruce G. Aitken, “Highly Nonlinear Ge-As-Se and Ge-As-S-Se Glasses for All-Optical Switching,” IEEE Photon. Technol. Lett. **14**, 822–824, (2002). [CrossRef]

*5mW*single-channel peak power levels used in telecommunications. The power can be further decreased by reducing the modal volume and/or by using materials with larger Kerr coefficient.

16. S. Coen and M. Haelterman, “Competition between modulational instability and switching in optical bistability,” Opt. Lett. **24**, 80–82, (1999). [CrossRef]

17. Stojan Radić, Nicholas George, and Govind P. Agrawal, “Theory of low-threshold optical switching in nonlinear phase-shifted periodic structures,” J. Opt. Soc. Am. B **12**, 671–680, (1995). [CrossRef]

*δn*(

*) induced in the system is much larger than in other systems with much larger modal volumes [17*

**r**17. Stojan Radić, Nicholas George, and Govind P. Agrawal, “Theory of low-threshold optical switching in nonlinear phase-shifted periodic structures,” J. Opt. Soc. Am. B **12**, 671–680, (1995). [CrossRef]

*κ*=

*0.020*is fairly large. (For comparison, if one had a system in which all the energy of the mode were contained uniformly inside a volume (

*λ*

_{0}/

*2n*

_{L})

^{3},

*κ*would be≈0.15.) Second, a large quality factor can be achieved at the same time as the large

*κ*. Increasing

*Q*while keeping other parameters fixed decreases the power requirements as 1/

*Q*

^{2}[18

18. J.E. Heebner and R. Boyd, “Enhanced all-optical switching by use of a nonlinear fiber ring resonator,” Opt. Lett. **24**, 847–849, (1999). [CrossRef]

19. A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. **14**, 483–485, (2002). [CrossRef]

*Q*appears because increasing

*Q*for a fixed

*P*

_{IN}leads to a larger peak electric field inside the cavity, due to the longer energy accumulation. The second factor of

*Q*comes from the fact that the resonance peak width is ~

*1/Q*, thereby reducing the required frequency (index) shift by

*1/Q*as well.

## 3. Cutoff solitons in axially uniform photonic bandgap fibers

21. H.G. Winful, J.H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. **35**, 379–381, (1979). [CrossRef]

25. C. Martijn de Sterke and J.E. Sipe, “Switching dynamics of finite periodic nonlinear media: A numerical study,” Phys. Rev. A **42**, 2858–2869, (1990). [CrossRef]

26. B.J. Eggleton, C. Martijn de Sterke, and R.E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B **14**, 2980–2993, (1997). [CrossRef]

27. 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]

28. S. Janz, J. He, Z.R. Wasilewski, and M. Cada, “Low threshold optical bistable switching in an asymmetric ¼-shifted distributed-feedback heterostructure,” Appl. Phys. Lett. **67**, 1051–1053, (1995). [CrossRef]

2. Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” J. Lightwave. Technol. **17**, 2039–2041, (1999). [CrossRef]

3. Steven G. Johnson, Mihai Ibanescu, M. Skorobogatiy, Ori Weisberg, Torkel D. Engeness, Marin Soljačić, Steven A. Jacobs, J. D. Joannopoulos, and Yoel Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express **9**, 748–779, (2001). [CrossRef] [PubMed]

5. B. Temelkuran, S.D. Hart, G. Benoit, J.D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO_{2} laser transmission,” Nature **420**, 650–653, (2002). [CrossRef] [PubMed]

30. R.F. Gregan, B.J. Mangan, J.C. Knight, T.A. Birks, P.St.J. Russell, P.J. Roberts, and D.C. Allan, “Single-Mode Photonic Band Gap Guidance of Light in Air,” Science **285**, 1537–1539, (1999). [CrossRef]

*k*=0: tuning the input frequency below the cutoff provides for an effective axial spectral-gap. This is particularly true, since there are minimal radiation losses (we operate far from the light line), resulting in a distributed feedback mechanism similar to a Bragg reflector. Also, stationary solutions with very low group velocities (even zero) come in naturally at

*k*=0 [31]. We term these solitons “cutoff solitons” in order to distinguish them from all other gap solitons which have thus far only been described for axially

*periodic*structures.

*k*=0 and the absence of a complete spectral gap. In Fig. 4(b) we plot the guided mode dispersion relation in the linear (low-intensity) limit, as calculated by the finite-difference time-domain (FDTD) method. Any small change in linear refractive index will result in an almost constant frequency shift of the dispersion relations.

*ω*<

*ω*

_{c}there are no available guided states in the core. Assume an input port consisting of a similar fiber with a higher-index core

*n′*and cutoff frequency

*ω′*

_{c}<

*ω*

_{c}. Low-intensity incident guided waves with

*ω′*

_{c}<

*ω*<

*ω*

_{c}will decay exponentially in the axial direction and result in a strong reflection. This is similar to waves incident onto a Bragg reflector and is in contrast to

*all*other axially uniform fibers. At high input power we observe a wide range of nonlinear phenomena, such as bistability and self-pulsing, similar to those found in nonlinear axially periodic gratings. We study these phenomena in the limit of small nonlinearities (which is the experimentally correct limit).

*ω′*

_{c}<

*ω*

_{c}. For example, enlarging the core area would have the desired effect. For simplicity we just use a higher-index core of

*n′*=1.9. In Fig. 5 we plot the fiber’s nonlinear response for two nonlinear-core lengths,

*L*=5

*a*and

*L*=8

*a*. A CW excitation with

*ω′*

_{c}<

*ω*<

*ω*

_{c}is the input for both cases. Fields and flux are monitored at the output while perfectly-matched-layer absorbing boundary conditions are used to simulate perfect absorption at the edges of the computational cell.

*k*=0 in our case. We create a 1D model where the dispersion relations are fit to quadratic forms that include the nonlinear shift δω:

*(*

**F***x*,

*y*) and a slowly-varying amplitude

*A*(

*z*,

*t*),

*γ*is calculated using the FDTD method as

*γ*≡0.02·

*n*

_{0}·

*n*

_{2}|

_{max}at

*ω*

_{0}=

*ω*

_{c}. It is a weakly increasing function of frequency, but for simplicity we assume it constant. Other parameters used are

*ω*

_{c}=0.26215(2

*πc/a*) [

*ω′*

_{c}=0.244(2

*πc/a*)] and

*α*=0.564(

*ac*/2

*π*) [

*α′*=0.463(

*ac*/2

*π*)].

*linear*transmission coefficient for the

*L*=5

*a*system. This is plotted in Fig. 6(a) along with the full-2D FDTD data. Excellent agreement is found above cutoff. Note that the 1D model does show structure below cutoff, because it ignores the small coupling with the cladding modes (gray areas in Fig. 4(b)). This coupling could be further suppressed by reducing the index contrast between the different segments.

*γ*, neglecting second order corrections in

*γ*, and to nonzero contributions from cladding modes. In Fig. 6(b) we plot the intensity along the nonlinear core for the

*L*=5

*a*system at two different points: at the peak of the upper transmission branch where a cutoff soliton has been excited, and at the lower branch where the wave decays exponentially; the 1D model captures all the essential features of the 2D system’s nonlinear response.

*A*. In frequency domain, the equation for

*Ã*(

*z*,

*ω*-

*ω*

_{c}) is

*∂*

^{2}

*Ã*/

*∂Z*

^{2}+

*k*

^{2}

*Ã*=0, where we expand around the cutoff frequency

*ω*

_{c}, and

*k*contains the nonlinear index change (first order terms vanish at cutoff). Using the dispersion relations of Eq. (1),

*k*

^{2}=(

*ω*-

*ω*

_{c}-

*δω*)/

*α*, and transforming back into the time domain we arrive at the nonlinear Schrödinger equation

^{14}(NLSE):

23. C. Martijn de Sterke and J.E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A **38**, 5149–5165, (1988). [CrossRef]

*any*arbitrary group velocity, even zero. Any nonlinear system with similar dispersion relations is thus well described by Eq. (6). Such systems include metallic waveguides, multilayer stacks, PBG fibers, and photonic crystal linear-defect waveguides.

## 4. Conclusion

## Acknowledgments

## References and links

1. | P. Russell, “Photonic crystal fibers,” Science |

2. | Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” J. Lightwave. Technol. |

3. | Steven G. Johnson, Mihai Ibanescu, M. Skorobogatiy, Ori Weisberg, Torkel D. Engeness, Marin Soljačić, Steven A. Jacobs, J. D. Joannopoulos, and Yoel Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express |

4. | S.D. Hart, G.R. Maskaly, B. Temelkuran, P.H. Prideaux, J.D. Joannopoulos, and Y. Fink, “External Reflection from Omnidirectional Dielectric Mirror Fibers,” Science |

5. | B. Temelkuran, S.D. Hart, G. Benoit, J.D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO |

6. | B.E.A. Saleh and M.C. Teich, |

7. | Marin Soljačić, Mihai Ibanescu, Steven G. Johnson, Yoel Fink, and J.D. Joannopoulos, “Optimal Bistable Switching in Non-Linear Photonic Crystals,” Phys. Rev. E |

8. | J.S. Foresi, P.R. Villeneuve, J. Ferrera, E.R. Thoen, G. Steinmeyer, S. Fan, J.D. Joannopoulos, L.C. Kimerling, H.I. Smith, and E.P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature |

9. | Marin Soljačić, Mihai Ibanescu, Steven G. Johnson, J.D. Joannopoulos, and Yoel: Fink“Optical Bistability in Axially Modulated OmniGuide Fibers,” Opt. Lett. |

10. | Steven G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

11. | For a review, see A. Taflove, |

12. | H.A. Haus, |

13. | S.G. Johnson, S. Fan, A. Mekis, and J.D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. |

14. | G. Lenz, J. Zimmerman, T. Katsufuji, M. E. Lines, H. Y. Hwang, S. Spälter, R. E. Slusher, S.-W. Cheong, J. S. Sanghera, and I. D. Aggarwal, “Large Kerr effect in bulk Se-based chalcogenide glasses,” Opt. Lett. |

15. | Jeffrey M. Harbold, F. Ömer Ilday, Frank W. Wise, and Bruce G. Aitken, “Highly Nonlinear Ge-As-Se and Ge-As-S-Se Glasses for All-Optical Switching,” IEEE Photon. Technol. Lett. |

16. | S. Coen and M. Haelterman, “Competition between modulational instability and switching in optical bistability,” Opt. Lett. |

17. | Stojan Radić, Nicholas George, and Govind P. Agrawal, “Theory of low-threshold optical switching in nonlinear phase-shifted periodic structures,” J. Opt. Soc. Am. B |

18. | J.E. Heebner and R. Boyd, “Enhanced all-optical switching by use of a nonlinear fiber ring resonator,” Opt. Lett. |

19. | A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. |

20. | C. Kerbage and B.J. Eggleton, “Microstructured Optical Fibers,” Optics&Photonics News 38–42, (September 2002). |

21. | H.G. Winful, J.H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. |

22. | Wei Chen and D.L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. |

23. | C. Martijn de Sterke and J.E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A |

24. | D.N. Christodoulides and R.I Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. |

25. | C. Martijn de Sterke and J.E. Sipe, “Switching dynamics of finite periodic nonlinear media: A numerical study,” Phys. Rev. A |

26. | B.J. Eggleton, C. Martijn de Sterke, and R.E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B |

27. | 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. |

28. | S. Janz, J. He, Z.R. Wasilewski, and M. Cada, “Low threshold optical bistable switching in an asymmetric ¼-shifted distributed-feedback heterostructure,” Appl. Phys. Lett. |

29. | Elefterios Lidorikis, Marin Soljacic, Mihai Ibanescu, Yoel Fink, and J.D. Joannopoulos, “Gap solitons and optical switching in axially uniform systems,” Opt. Lett., in press. |

30. | R.F. Gregan, B.J. Mangan, J.C. Knight, T.A. Birks, P.St.J. Russell, P.J. Roberts, and D.C. Allan, “Single-Mode Photonic Band Gap Guidance of Light in Air,” Science |

31. | This special kind of modal cutoff is different from the ones found in many simple waveguiding systems which appear very close to the light line ( |

32. | E. Lidorikis, K. Busch, Qiming Li, C.T. Chan, and C.M. Soukoulis, “Optical nonlinear response of a single nonlinear dielectric layer sandwiched between two linear dielectric structures,” Phys. Rev. B |

33. | G. P. Agrawal, |

**OCIS Codes**

(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons

(190.1450) Nonlinear optics : Bistability

(190.4370) Nonlinear optics : Nonlinear optics, fibers

(230.4320) Optical devices : Nonlinear optical devices

**ToC Category:**

Focus Issue: Photonic crystals and holey fibers

**History**

Original Manuscript: February 19, 2004

Revised Manuscript: March 24, 2004

Manuscript Accepted: March 24, 2004

**Citation**

Marin Soljačić, Elefterios Lidorikis, Mihai Ibanescu, Steven G. Johnson, J.D. Joannopoulos, and Yoel Fink, "Optical bistability and cutoff solitons in photonic bandgap fibers," Opt. Express **12**, 1518-1527 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-8-1518

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### References

- P. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003). [CrossRef] [PubMed]
- Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” J. Lightwave. Technol. 17, 2039–2041, (1999). [CrossRef]
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