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.

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References

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  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]
  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. 30, 995-996 (1994).
    [CrossRef]
  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. 22, 402-404 (1997).
    [CrossRef] [PubMed]
  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.
  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]
  10. L. Poladian, "Resonance mode expansions and exact solutions for nonuniform gratings," Phys. Rev. E, 54, 2963-2975 (1996).
    [CrossRef]
  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]

Other

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]

Supplementary Material (3)

» Media 1: MOV (6649 KB)     
» Media 2: MOV (5401 KB)     
» Media 3: MOV (5737 KB)     

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Figures (3)

Fig. 1.
Fig. 1.

A single frame from the movie illustrating the propagation of a 1-ps pulse through a weak uniform grating. The dotted lines indicate the grating boundaries. The backward propagating wave appears in red and the forward propagating wave in blue. The peak of the forward propagating pulse is off the scale since the scale has been expanded to retain as much detail as possible. To run the movie, click on the above figure. [Media 1]

Fig. 2.
Fig. 2.

A single frame from the movie illustrating the propagation of a 1-ps pulse through a weak uniform grating. The dotted lines indicate the grating boundaries. The backward propagating wave appears in red and the forward propagating wave in blue. The peak of the forward propagating pulse is off the scale since the scale has been expanded to retain as much detail as possible. To run the movie, click on the above figure. [Media 2]

Fig. 3.
Fig. 3.

A single frame from the movie illustrating the propagation of a 1-ps pulse through a very strong uniform grating. The dotted lines indicate the grating boundaries. The backward propagating wave appears in red and the forward propagating wave in blue. The peak of the forward propagating pulse is initially off the scale since the scale has been expanded to retain as much detail as possible. To run the movie, click on the above figure. [Media 3]

Tables (1)

Tables Icon

Table 1. Characteristics of gratings used in the numerical simulations.

Equations (4)

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n ( z ) = n 0 + σ ( z ) + 2 κ ( z ) cos [ 2 k 0 z + ϕ ( z ) ]
E z t = a + z t e j ( ω 0 t + δt k 0 z + ϕ ( z ) 2 ) + a z t e j ( ω 0 t + δt + k 0 z ϕ ( z ) 2 )
j ( a + ζ t ζ + n 0 c k 0 a + ζ t ζ ) + ( σ ( ζ ) 2 n 0 2 + n 0 δ c k 0 1 2 ϕ ( ζ ) ) a + ( ζ , t ) + κ ( ζ ) 2 n 0 2 a ζ t = 0
j ( a ζ t ζ n 0 c k 0 a ζ t t ) + ( σ ( ζ ) 2 n 0 2 + n 0 δ c k 0 1 2 ϕ ( ζ ) ) a ζ t + κ ( ζ ) 2 n 0 2 a + ζ t = 0

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