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.

© 1997 Optical Society of America

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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]

1997 (2)

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]

1996 (3)

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]

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]

1995 (1)

1994 (3)

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]

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]

1991 (1)

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]

Archambault, J.-L.

Ball, G. A.

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]

Bendickson, J. M.

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]

Benjamin, S. D.

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, 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 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.

Bloemer, M. J.

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]

Boskovic, A.

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]

Bowden, C. M

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]

Chen, L. R.

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, 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 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.

de Sterke, C. M.

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]

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]

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]

Dowling, J. P.

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]

Eggleton, B.J.

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]

Flynn, R. J.

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]

Fork, R. L.

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]

Guy, M. J.

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]

Jackson, K. R.

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]

Juma, S.

Kashyap, R.

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]

Krug, P. A.

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]

Leavitt, R. P.

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]

Ledbetter, H. S.

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]

Meltz, G.

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]

Morey, W. W.

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]

Noske, D. U.

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]

Payne, D. N.

Poladian, L.

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

Reekie, L.

Reinhardt, S. B.

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]

Richardson, D. J.

Robert, B. D.

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]

Rottwitt, K.

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]

Russell, P. St. J.

Scalora, M.

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]

Sipe, J. E.

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, J. E. Sipe, and S. Juma, “Ultrashort pulse propagation in multiple-grating fiber structures,” Opt. Lett. 22, 402–404 (1997).
[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]

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]

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.

Slusher, R. E.

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]

Smith, P. W. E.

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, 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 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.

Taverner, D.

Taylor, J. R.

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]

Tocci, M. D.

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]

Electron. Lett. (1)

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]

IEEE/OSA J. Lightwave Technol. (1)

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]

J. Opt. Soc. Am B (1)

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]

J. Opt. Soc. Am. A (1)

Opt. and Photon. News (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]

Opt. Lett. (2)

Phys. Rev. E (2)

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]

Phys. Rev. Lett. (1)

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]

Other (1)

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.

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)

Equations on this page are rendered with MathJax. Learn more.

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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