## Nonlinear pulse propagation in birefringent fiber Bragg Gratings

Optics Express, Vol. 3, Issue 11, pp. 418-432 (1998)

http://dx.doi.org/10.1364/OE.3.000418

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### Abstract

We present two sets of equations to describe nonlinear pulse propagation in a birefringent fiber Bragg grating. The first set uses a coupled-mode formalism to describe light in or near the photonic band gap of the grating. The second set is a pair of coupled nonlinear Schroedinger equations. We use these equations to examine viable switching experiments in the presence of birefringence. We show how the birefringence can both aid and hinder device applications.

© Optical Society of America

## 1. Introduction

3. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg Grating Solitons,” Phys. Lett. **76**, 1627–30 (1996). [CrossRef]

4. M. J. Steel and C. M. de Sterke see, e.g., “Schroedinger equation description for cross-phase modulation in grating structures,” Phys. Rev. A **49**, 5048–55 (1994). [CrossRef] [PubMed]

*et al*. that investigated the continuous wave NCME in the presence of birefringence (NBCME) [5

5. W. Samir, S. J. Garth, and C. Pask, “Interplay of grating and nonlinearity in mode-coupling,” J. Opt. Soc. Am. B **11**, 64–71 (1994). [CrossRef]

6. S. Lee and S. T. Ho, “Optical switching scheme based on the transmission of coupled gap solitons in nonlinear periodic dielectric media,” Opt. Lett. **18**, 962–64 (1993). [CrossRef] [PubMed]

7. D. Taverner, N. G. R. Broderick, D. J. Richardson, M. Ibsen, and R. I. Laming, “All-optical AND gate based on coupled gap-soliton formation in a fiber Bragg grating,” Opt. Lett. **23**, 259–61 (1998). [CrossRef]

8. Y. Silberberg and Y. Barad, “Rotating vector solitary waves in isotropic fibers,” Opt. Lett. **20**, 246–48 (1995). [CrossRef] [PubMed]

## 2. Theory

_{0}, we can then define an index of refraction associated with polarization along the x and y axes,

_{0}is the permittivity of free space. We seek transverse fields that depend only on the coordinate z,

*et al*. [9

9. C. M. de Sterke, D. G. Salinas, and J. E. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E **54**, 1969–89 (1996). [CrossRef]

_{0x,y}. The structure of the nonlinear susceptibility χ

_{ijkl}is assumed to be that of an isotropic medium, which limits the validity of our equations to a weakly birefringent medium, but has the advantage of casting our equations in a form familiar from the optical fiber literature. This restriction is not essential, and can be removed by using a nonlinear susceptibility derived for a birefringent medium.

*slowly-varying envelope functions*. Although in heuristic analyses these envelope functions are often taken to modulate plane waves at the carrier frequency, the spirit of our approach is to use the Bloch functions of the grating as a more accurate spatial carrier basis. However, in the weak-grating limit, near the Bragg frequency, our approach reduces to the heuristic approach.

_{+}, X

_{-}, Y

_{+}, Y

_{-}. They represent the forward and backward going amplitudes of the x and y polarizations of the pulse respectively. In the following equations, the fields have been normalized such that their square-modulus represents power.

9. C. M. de Sterke, D. G. Salinas, and J. E. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E **54**, 1969–89 (1996). [CrossRef]

_{x,y}coefficient is

_{2}is the nonlinear index of refraction; k

_{0}is the Bragg wavevector; A

_{eff}is an effective cross-sectional area in the (x,y) plane associated with the problem. The other two coefficients can be determined from α

_{I}because for a weakly birefringent optical fiber they are in the ratio

9. C. M. de Sterke, D. G. Salinas, and J. E. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E **54**, 1969–89 (1996). [CrossRef]

_{i}’=dω

_{i}/dk; the velocity fraction ρ

_{i}=ω

_{i}’/v

_{i}; and the other quantities are defined following (8). The remaining nonlinear coefficients can be determined for a weakly birefringent optical fiber using the ratio

_{i}”=d

^{2}ω

_{i}/dk

^{2}characterize the dispersion relation at the carrier Bloch functions of interest, and result from the grating. From figure 1 it is clear that ω

_{i}’ can take on any value between ±c/n

_{0i}because the dispersion relation approaches that of a uniform medium far from the Bragg frequencies. However, (14) is valid only when the grating dispersion is significantly larger than the underlying, neglected, material dispersion.

_{+}being affected by the intensity of X

_{-}). The second is cross phase modulation between different polarizations (i.e. X

_{+}being affected by the intensity of Y

_{±}). The energy exchange terms are those that couple to the field conjugates, including the α

_{pc}terms in the NBSE and the corresponding terms in the NBCME. If the birefringence is high, then the quickly varying exp(±iδt) term will cause the effect of these energy exchange terms to vanish.

## 3. Optical AND Gate

6. S. Lee and S. T. Ho, “Optical switching scheme based on the transmission of coupled gap solitons in nonlinear periodic dielectric media,” Opt. Lett. **18**, 962–64 (1993). [CrossRef] [PubMed]

7. D. Taverner, N. G. R. Broderick, D. J. Richardson, M. Ibsen, and R. I. Laming, “All-optical AND gate based on coupled gap-soliton formation in a fiber Bragg grating,” Opt. Lett. **23**, 259–61 (1998). [CrossRef]

_{y}<n

_{x}so the y Bragg frequency is slightly higher than the x Bragg frequency. We tune our input pulse near the upper edge of the x gap so that both the x and y polarizations are within their photonic band gap, but the y polarization is more deeply within its gap. If either the x or the y polarizations are sent into the system alone, they are sufficiently deep within their band-gaps that they experience almost 100% reflection. However, when both pulses are present, the cross phase modulation between the polarizations can shift one or both of them out of the gap. Clearly this behavior is that of an AND gate.

_{0x}=1.5; the birefringence is n

_{0x}-n

_{0y}=4×10

^{-6}; the grating index contrast is δn=2.75×10

^{-5}; the nonlinear index is n

_{2}=0.23×10

^{-15}cm

^{2}/W; the pulse width is 2ns; the pulse peak intensity is 10GW/cm

^{2}for each polarization. The long pulse width in this experiment gives a narrow frequency spectrum. A smaller pulse width would broaden the spectrum so that, even if the central pulse frequency was shifted out of the gap, a significant portion of the wings might be left inside.

_{0}. Again the pulses are tuned near the edge of the stop gap so that individually they experience nearly 100% reflection. However, when the two polarizations are injected simultaneously, the isotropic system sees this as being one pulse of power 2P

_{0}. This allows us to cast the familiar nonlinear switching scenarios for gap solitons into a different geometry. We present such a simulation in the following movie (figure 5). In this case the AND gate operates, but the dynamics are different. The initial energy builds into a gap soliton which flushes out of the system, but then the remaining energy is insufficient to form a soliton, so it is reflected.

## 4. Elliptically Polarized Soliton

8. Y. Silberberg and Y. Barad, “Rotating vector solitary waves in isotropic fibers,” Opt. Lett. **20**, 246–48 (1995). [CrossRef] [PubMed]

_{2}(=d

^{2}k/dω

^{2}) by using the appropriate Galilean transformation. We then re-write the equations in a circular basis by letting U = (X+iY)/√2 and V = (X-iY)√2 and find

*i*.

*e*. the V polarization is much weak than the U polarization). The phase accumulation in (20) is represented by ζ which is given by

8. Y. Silberberg and Y. Barad, “Rotating vector solitary waves in isotropic fibers,” Opt. Lett. **20**, 246–48 (1995). [CrossRef] [PubMed]

_{0}=1.5; δn=1.67×10

^{-4}; n

_{2}=0.23×10

^{-15}W/cm

^{2}; pulse width = 85ps; right circularly polarized peak intensity = 9GW/cm

^{2}; left circularly polarized peak intensity = 0.5GW/cm

^{2}. For our system parameters, a right circularly polarized soliton would require a sech profile with a peak intensity of 9GW/cm

^{2}.

*vs*grating length for zero birefringence. This demonstrates that the polarization ellipse does, in fact, rotate upon propagation. The angle is determined by simulating the placement of a linear polarizer at the output and identifying the angle of maximum transmission.

^{-7}and 3×10

^{-5}which encompasses experimentally observed values for the intrinsic birefringence in a fiber Bragg grating. The major effect of the birefringence is to cause walk-off between the x and y polarizations, and hence between the constitutive quantities of the U and V fields (see definitions just before (18)). In some cases the nonlinear interactions can keep the pulses together [12

12. C. R. Menyuk, “Stability of solitons in birefringent optical fibers. I: Equal propagation amplitudes,” Opt. Lett. **12**, 614–16 (1987). [CrossRef] [PubMed]

^{-6}. Past this point the pulse walk-off begins to dominate and the azimuthal profile quickly degenerates. The reader is reminded that the ±π/4 range of azimuthal angle has been converted to a range of (0,π/2) which is why the profile appears to sharply change direction.

^{-6}the azimuthal profile remains relatively flat, but at 1.6×10

^{-6}the profile is destroyed. The effect of the birefringence is more marked with the NBCME because it takes careful account of the reflected waves. As the birefringence increases the two pulses begin to experience both group velocity mismatch, and different amounts of Bragg reflection. These differences combine to destroy the elliptically polarized soliton more quickly than in the NBSE case. Hence, care must be taken in using the NBSE near the gap in place of the more accurate NBCME.

## 5. Conclusions

## 6. Acknowledgements

## References

1. | Pochi Yeh, |

2. | C. M. de Sterke and J. E. Sipe, “Gap Solitons” in |

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

4. | M. J. Steel and C. M. de Sterke see, e.g., “Schroedinger equation description for cross-phase modulation in grating structures,” Phys. Rev. A |

5. | W. Samir, S. J. Garth, and C. Pask, “Interplay of grating and nonlinearity in mode-coupling,” J. Opt. Soc. Am. B |

6. | S. Lee and S. T. Ho, “Optical switching scheme based on the transmission of coupled gap solitons in nonlinear periodic dielectric media,” Opt. Lett. |

7. | D. Taverner, N. G. R. Broderick, D. J. Richardson, M. Ibsen, and R. I. Laming, “All-optical AND gate based on coupled gap-soliton formation in a fiber Bragg grating,” Opt. Lett. |

8. | Y. Silberberg and Y. Barad, “Rotating vector solitary waves in isotropic fibers,” Opt. Lett. |

9. | C. M. de Sterke, D. G. Salinas, and J. E. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E |

10. | Govind Agrawal, |

11. | T. Erdogan and V. Mizrahi, “Characterization of UV-induced birefringence in photosensitive Ge-doped silica optical fibers,” J. Opt. Soc. Am. B |

12. | C. R. Menyuk, “Stability of solitons in birefringent optical fibers. I: Equal propagation amplitudes,” Opt. Lett. |

**OCIS Codes**

(190.4360) Nonlinear optics : Nonlinear optics, devices

(190.4370) Nonlinear optics : Nonlinear optics, fibers

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(230.1150) Optical devices : All-optical devices

(260.1440) Physical optics : Birefringence

**ToC Category:**

Focus Issue: Bragg solitons and nonlinear optics of periodic structures

**History**

Original Manuscript: October 2, 1998

Published: November 23, 1998

**Citation**

Suresh Pereira and John Sipe, "Nonlinear pulse propagation in birefringent fiber Bragg gratings," Opt. Express **3**, 418-432 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-3-11-418

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### References

- Pochi Yeh, Optical Waves in Layered Media, (Wiley, New York, 1988).
- C. M. de Sterke and J. E. Sipe, "Gap Solitons" in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1994).
- B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug and J. E. Sipe, "Bragg Grating Solitons," Phys. Lett. 76, 1627-30 (1996). [CrossRef]
- see, e.g., M. J. Steel and C. M. de Sterke, "Schroedinger equation description for cross-phase modulation in grating structures," Phys. Rev. A 49, 5048-55 (1994). [CrossRef] [PubMed]
- W. Samir, S. J. Garth and C. Pask, "Interplay of grating and nonlinearity in mode-coupling," J. Opt. Soc. Am. B 11, 64-71 (1994). [CrossRef]
- S. Lee and S. T. Ho, "Optical switching scheme based on the transmission of coupled gap solitons in nonlinear periodic dielectric media," Opt. Lett. 18, 962-64 (1993). [CrossRef] [PubMed]
- D. Taverner, N. G. R. Broderick, D. J. Richardson, M. Ibsen and R. I. Laming, "All-optical AND gate based on coupled gap-soliton formation in a fiber Bragg grating," Opt. Lett. 23, 259-61 (1998). [CrossRef]
- Y. Silberberg and Y. Barad, "Rotating vector solitary waves in isotropic fibers," Opt. Lett. 20, 246-48 (1995). [CrossRef] [PubMed]
- C. M. de Sterke, D. G. Salinas and J. E. Sipe, "Coupled-mode theory for light propagation through deep nonlinear gratings," Phys. Rev. E 54, 1969-89 (1996). [CrossRef]
- Govind Agrawal, Nonlinear Fiber Optics, (Academic Press, Boston, 1989).
- T. Erdogan and V. Mizrahi, "Characterization of UV-induced birefringence in photosensitive Ge-doped silica optical fibers," J. Opt. Soc. Am. B 11, 2100-05 (1994). [CrossRef]
- C. R. Menyuk, "Stability of solitons in birefringent optical fibers. I: Equal propagation amplitudes," Opt. Lett. 12, 614-16 (1987). [CrossRef] [PubMed]

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