## Dynamics of ultrashort pulse propagation through fiber gratings

Optics Express, Vol. 1, Issue 9, pp. 242-249 (1997)

http://dx.doi.org/10.1364/OE.1.000242

Acrobat PDF (201 KB)

### Abstract

By directly integrating the time-domain coupled-mode equations, we can explicitly obtain and examine the backward and forward propagating waves as a function of position and time within fiber grating structures. We apply this numerical procedure to calculate the temporal reflection and transmission response of fiber gratings subject to ultrashort pulse inputs. This allows us to study the dynamics of the ultrashort pulse-grating interaction.

© Optical Society of America

## 1. Introduction

1. See, for example, W. W. Morey, G. A. Ball, and G. Meltz, “Photoinduced Bragg gratings in optical fibers,” Opt. and Photon. News **5**, 8–14 (1994). [CrossRef]

2. See, for example, J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A **11**, 1307–1320 (1994). [CrossRef]

3. D. Taverner, D. J. Richardson, J.-L. Archambault, L. Reekie, P. St. J. Russell, and D. N. Payne, “Experimental investigation of picosecond pulse reflection from fiber gratings,” Opt. Lett. **20**, 282–284 (1995). [CrossRef] [PubMed]

4. See, for example, B.J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. **76**, 1627–1630 (1996). [CrossRef] [PubMed]

8. L. R. Chen, S. D. Benjamin, P. W. E. Smith, and J. E. Sipe, “Ultrashort pulse propagation through fiber gratings: theory and experiment,” presented at *Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals Topical Meeting 1997*, Williamsburg, Virginia, paper BMB2.

7. L. R. Chen, S. D. Benjamin, P. W. E. Smith, and J. E. Sipe, “Ultrashort pulse reflection from fiber gratings: a numerical investigation,” IEEE/OSA J. Lightwave Technol. **15**, 1503–1512 (1997). [CrossRef]

8. L. R. Chen, S. D. Benjamin, P. W. E. Smith, and J. E. Sipe, “Ultrashort pulse propagation through fiber gratings: theory and experiment,” presented at *Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals Topical Meeting 1997*, Williamsburg, Virginia, paper BMB2.

7. L. R. Chen, S. D. Benjamin, P. W. E. Smith, and J. E. Sipe, “Ultrashort pulse reflection from fiber gratings: a numerical investigation,” IEEE/OSA J. Lightwave Technol. **15**, 1503–1512 (1997). [CrossRef]

9. C. M. de Sterke, K. R. Jackson, and B. D. Robert, Nonlinear coupled-mode equations on a finite interval: a numerical procedure,” J. Opt. Soc. Am B **8**, 403–412 (1991). [CrossRef]

## 2. Time-domain coupled-mode equations

2. See, for example, J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A **11**, 1307–1320 (1994). [CrossRef]

7. L. R. Chen, S. D. Benjamin, P. W. E. Smith, and J. E. Sipe, “Ultrashort pulse reflection from fiber gratings: a numerical investigation,” IEEE/OSA J. Lightwave Technol. **15**, 1503–1512 (1997). [CrossRef]

10. L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E , **54**, 2963–2975 (1996). [CrossRef]

*k*is the wavenumber at the design resonant wavelength (with corresponding resonance frequency ω

_{0}_{0}= (

*ck*)/

_{0}*n*and

_{0}*c*is the speed of light in vacuum),

*n*is the effective mode index of the unmodified core, and

_{0}*σ, κ, ϕ*are slowly varying functions of

*z*. The function

*σ*characterizes the space averaged background refractive index increase due to the envelope of the induced index change (DC background envelope),

*κ*is the position dependent coupling coefficient, and

*ϕ*describes the position dependent phase (chirp) of the grating. Writing the electric field within the grating structure as a sum of forward and backward propagating waves

*a*

_{±}(

*z, t*) satisfy the following time-domain coupled-mode equations [10

10. L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E , **54**, 2963–2975 (1996). [CrossRef]

*a*

_{+}(0,

*t*) =

*A*(

*t*) and

*a*

_{-}(

*L,t*) = 0 and initial conditions

*a*

_{±}(

*ζ*,0) = 0 where

*A*(

*t*) is the envelope of the input pulse,

*L*is the length of the grating, and

*ζ*=

*k*(further details regarding the derivation or the significance of the terms in Eq. (3) can be found in [7

_{0}z**15**, 1503–1512 (1997). [CrossRef]

10. L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E , **54**, 2963–2975 (1996). [CrossRef]

9. C. M. de Sterke, K. R. Jackson, and B. D. Robert, Nonlinear coupled-mode equations on a finite interval: a numerical procedure,” J. Opt. Soc. Am B **8**, 403–412 (1991). [CrossRef]

**15**, 1503–1512 (1997). [CrossRef]

## 3. Simulations and discussion

**15**, 1503–1512 (1997). [CrossRef]

### 3.1 Weak uniform grating

**15**, 1503–1512 (1997). [CrossRef]

2. See, for example, J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A **11**, 1307–1320 (1994). [CrossRef]

### 3.2 Weak grating with linear chirp

### 3.3 Very strong uniform grating

11. M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrafast pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E **54**, R1078–R1081 (1996). [CrossRef]

## 4. Summary

## Acknowledgment

## Footnotes

† | J. E. Sipe is with the Department of Physics, University of Toronto, 60 St. George Street, Toronto, Ontario, Canada M5S 1A7. |

## References and links

1. | See, for example, W. W. Morey, G. A. Ball, and G. Meltz, “Photoinduced Bragg gratings in optical fibers,” Opt. and Photon. News |

2. | See, for example, J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A |

3. | D. Taverner, D. J. Richardson, J.-L. Archambault, L. Reekie, P. St. J. Russell, and D. N. Payne, “Experimental investigation of picosecond pulse reflection from fiber gratings,” Opt. Lett. |

4. | See, for example, B.J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. |

5. | K. Rottwitt, M. J. Guy, A. Boskovic, D. U. Noske, J. R. Taylor, and R. Kashyap, “Interaction of uniform phase picosecond pulses with chirped and unchirped photosensitive fiber Bragg gratings,” Electron. Lett. |

6. | L. R. Chen, S. D. Benjamin, P. W. E. Smith, J. E. Sipe, and S. Juma, “Ultrashort pulse propagation in multiple-grating fiber structures,” Opt. Lett. |

7. | L. R. Chen, S. D. Benjamin, P. W. E. Smith, and J. E. Sipe, “Ultrashort pulse reflection from fiber gratings: a numerical investigation,” IEEE/OSA J. Lightwave Technol. |

8. | L. R. Chen, S. D. Benjamin, P. W. E. Smith, and J. E. Sipe, “Ultrashort pulse propagation through fiber gratings: theory and experiment,” presented at |

9. | C. M. de Sterke, K. R. Jackson, and B. D. Robert, Nonlinear coupled-mode equations on a finite interval: a numerical procedure,” J. Opt. Soc. Am B |

10. | L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E , |

11. | M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrafast pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E |

**OCIS Codes**

(050.2770) Diffraction and gratings : Gratings

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

(320.7140) Ultrafast optics : Ultrafast processes in fibers

**ToC Category:**

Research Papers

**History**

Original Manuscript: September 9, 1997

Revised Manuscript: August 21, 1997

Published: October 27, 1997

**Citation**

Lawrence Chen, Seldon Benjamin, Howard Jung, Peter W. E. Smith, and John Sipe, "Dynamics of Ultrashort Pulse Propagation Through Fiber
Gratings," Opt. Express **1**, 242-249 (1997)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-1-9-242

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### References

- See, for example, W. W. Morey, G. A. Ball, and G. Meltz, "Photoinduced Bragg gratings in optical fibers," Opt. and Photon. News 5, 8-14 (1994). [CrossRef]
- See, for example, J. E. Sipe, L. Poladian, and C. M. de Sterke, "Propagation through nonuniform grating structures," J. Opt. Soc. Am. A 11, 1307-1320 (1994). [CrossRef]
- D. Taverner, D. J. Richardson, J.-L. Archambault, L. Reekie, P. St. J. Russell, and D. N. Payne, "Experimental investigation of picosecond pulse reflection from fiber gratings," Opt. Lett. 20, 282-284 (1995). [CrossRef] [PubMed]
- See, for example, B.J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg grating solitons," Phys. Rev. Lett. 76, 1627-1630 (1996). [CrossRef] [PubMed]
- K. Rottwitt, M. J. Guy, A. Boskovic, D. U. Noske, J. R. Taylor, and R. Kashyap, "Interaction of uniform phase picosecond pulses with chirped and unchirped photosensitive fiber Bragg gratings," Electron. Lett. 30, 995-996 (1994). [CrossRef]
- L. R. Chen, S. D. Benjamin, P. W. E. Smith, J. E. Sipe, and S. Juma, "Ultrashort pulse propagation in multiple-grating fiber structures," Opt. Lett. 22, 402-404 (1997). [CrossRef] [PubMed]
- L. R. Chen, S. D. Benjamin, P. W. E. Smith, and J. E. Sipe, "Ultrashort pulse reflection from fiber gratings: a numerical investigation," IEEE/OSA J. Lightwave Technol. 15, 1503-1512 (1997). [CrossRef]
- L. R. Chen, S. D. Benjamin, P. W. E. Smith, and J. E. Sipe, "Ultrashort pulse propagation through fiber gratings: theory and experiment," presented at Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals Topical Meeting 1997, Williamsburg, Virginia, paper BMB2.
- C. M. de Sterke, K. R. Jackson, and B. D. Robert, Nonlinear coupled-mode equations on a finite interval: a numerical procedure," J. Opt. Soc. Am B 8, 403-412 (1991). [CrossRef]
- L. Poladian, "Resonance mode expansions and exact solutions for nonuniform gratings," Phys. Rev. E, 54, 2963-2975 (1996). [CrossRef]
- M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, "Ultrafast pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss," Phys. Rev. E 54, R1078-R1081 (1996). [CrossRef]

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