Abstract

The dynamics of optical pulses in nonuniform fiber gratings is studied analytically and numerically. Our approach is based on the inhomogeneous nonlinear Schrödinger equation, derived from the nonlinear coupled-mode equations. The problem is analyzed in the context of an adiabatic soliton compressor. The dependences of the soliton amplitude and width on the propagation distance are found. We also show the presence of the phase modulation during propagation. The possibility of optimizing the fiber-grating compressor by choice of a proper length to minimize the phase modulation is suggested. Comparison of analytical results with numerical simulations shows qualitatively good agreement.

© 2001 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. V. Mamyshev, S. V. Chernikov, and E. M. Dianov, “Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines,” IEEE J. Quantum Electron. 27, 2347–2355 (1991).
    [CrossRef]
  2. G. Lenz and B. J. Eggleton, “Adiabatic Bragg soliton compression in nonuniform grating structures,” J. Opt. Soc. Am. B 15, 2979–2985 (1998).
    [CrossRef]
  3. 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]
  4. 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]
  5. P. Millar, R. M. De La Rue, T. F. Krauss, J. S. Aitchison, N. G. R. Broderick, and D. J. Richardson, “Nonlinear propagation effects in AlGaAs Bragg grating filter,” Opt. Lett. 24, 685–687 (1999).
    [CrossRef]
  6. See, e.g., J. Lauzon, S. Thibault, J. Martin, and F. Quellette, “Implementation and characterization of fiber Bragg gratings linearly chirped by a temperature gradient,” Opt. Lett. 19, 2027–2029 (1994); P. C. Hill and B. J. Eggleton, “Strain gradient chirp of fibre Bragg gratings,” Electron. Lett. 30, 1172–1174 (1994); J. L. Cruz, A. Diez, M. V. Andres, A. Segura, B. Ortega, and L. Dong, “Fibre Bragg gratings tuned and chirped using magnetic fields,” Electron. Lett. ELLEAK 33, 235–236 (1997).
    [CrossRef] [PubMed]
  7. D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
    [CrossRef] [PubMed]
  8. A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37–42 (1989).
    [CrossRef]
  9. J. E. Sipe and H. G. Winful, “Nonlinear Schrödinger solitons in a periodic structure,” Opt. Lett. 13, 132–133 (1988); C. M. de Sterke and J. E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 38, 5149–5165 (1988).
    [CrossRef] [PubMed]
  10. B. J. Eggleton, G. Lenz, and N. M. Litchinitser, “Optical pulse compression schemes that use nonlinear Bragg gratings,” Fiber Integr. Opt. (to be published).
  11. C. M. de Sterke and J. E. Sipe, “Gap solitons,” in Progress in Optics XXXIII, E. Wolf, ed. (Elsevier, Amsterdam, 1994), Chap. III, pp. 203–260.
  12. C. M. de Sterke, “Wave propagation through nonuniform gratings with slowly varying parameters,” J. Lightwave Technol. 17, 2405–2413 (1999).
    [CrossRef]
  13. E. N. Tsoy and C. M. de Sterke, “Propagation of nonlinear pulses in chirped fiber gratings,” Phys. Rev. E 62, 2882–2890 (2000).
    [CrossRef]
  14. C. M. de Sterke and B. J. Eggleton, “Bragg solitons and the nonlinear Schrödinger equation,” Phys. Rev. E 59, 1267–1269 (1999).
    [CrossRef]
  15. A. I. Maimistov, “Evolution of solitary waves which are approximately solitons of a nonlinear Schrödinger equation,” Zh. Eksp. Teor. Fiz. 104, 3620–3629 (1993)[Sov. Phys. JETP 77, 727–731 (1993)].
  16. D. Anderson, “High transmission rate communication systems using lossy optical fibers,” Opt. Commun. 48, 107–113 (1983).
    [CrossRef]

2000 (1)

E. N. Tsoy and C. M. de Sterke, “Propagation of nonlinear pulses in chirped fiber gratings,” Phys. Rev. E 62, 2882–2890 (2000).
[CrossRef]

1999 (3)

1998 (2)

1996 (1)

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]

1991 (1)

P. V. Mamyshev, S. V. Chernikov, and E. M. Dianov, “Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines,” IEEE J. Quantum Electron. 27, 2347–2355 (1991).
[CrossRef]

1989 (2)

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[CrossRef] [PubMed]

A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37–42 (1989).
[CrossRef]

1983 (1)

D. Anderson, “High transmission rate communication systems using lossy optical fibers,” Opt. Commun. 48, 107–113 (1983).
[CrossRef]

Aceves, A. B.

A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37–42 (1989).
[CrossRef]

Aitchison, J. S.

Anderson, D.

D. Anderson, “High transmission rate communication systems using lossy optical fibers,” Opt. Commun. 48, 107–113 (1983).
[CrossRef]

Broderick, N. G. R.

Chernikov, S. V.

P. V. Mamyshev, S. V. Chernikov, and E. M. Dianov, “Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines,” IEEE J. Quantum Electron. 27, 2347–2355 (1991).
[CrossRef]

Christodoulides, D. N.

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[CrossRef] [PubMed]

De La Rue, R. M.

de Sterke, C. M.

C. M. de Sterke, “Wave propagation through nonuniform gratings with slowly varying parameters,” J. Lightwave Technol. 17, 2405–2413 (1999).
[CrossRef]

C. M. de Sterke and B. J. Eggleton, “Bragg solitons and the nonlinear Schrödinger equation,” Phys. Rev. E 59, 1267–1269 (1999).
[CrossRef]

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]

Dianov, E. M.

P. V. Mamyshev, S. V. Chernikov, and E. M. Dianov, “Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines,” IEEE J. Quantum Electron. 27, 2347–2355 (1991).
[CrossRef]

Eggleton, B. J.

C. M. de Sterke and B. J. Eggleton, “Bragg solitons and the nonlinear Schrödinger equation,” Phys. Rev. E 59, 1267–1269 (1999).
[CrossRef]

G. Lenz and B. J. Eggleton, “Adiabatic Bragg soliton compression in nonuniform grating structures,” J. Opt. Soc. Am. B 15, 2979–2985 (1998).
[CrossRef]

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]

Ibsen, M.

Joseph, R. I.

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[CrossRef] [PubMed]

Krauss, T. F.

Krug, P. A.

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]

Laming, R. I.

Lenz, G.

Mamyshev, P. V.

P. V. Mamyshev, S. V. Chernikov, and E. M. Dianov, “Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines,” IEEE J. Quantum Electron. 27, 2347–2355 (1991).
[CrossRef]

Millar, P.

Richardson, D. J.

Sipe, J. E.

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]

Slusher, R. E.

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]

Sterke, C. M. de

E. N. Tsoy and C. M. de Sterke, “Propagation of nonlinear pulses in chirped fiber gratings,” Phys. Rev. E 62, 2882–2890 (2000).
[CrossRef]

Taverner, D.

Tsoy, E. N.

E. N. Tsoy and C. M. de Sterke, “Propagation of nonlinear pulses in chirped fiber gratings,” Phys. Rev. E 62, 2882–2890 (2000).
[CrossRef]

Wabnitz, S.

A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37–42 (1989).
[CrossRef]

IEEE J. Quantum Electron. (1)

P. V. Mamyshev, S. V. Chernikov, and E. M. Dianov, “Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines,” IEEE J. Quantum Electron. 27, 2347–2355 (1991).
[CrossRef]

J. Lightwave Technol. (1)

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

Opt. Commun. (1)

D. Anderson, “High transmission rate communication systems using lossy optical fibers,” Opt. Commun. 48, 107–113 (1983).
[CrossRef]

Opt. Lett. (2)

Phys. Lett. A (1)

A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37–42 (1989).
[CrossRef]

Phys. Rev. E (2)

E. N. Tsoy and C. M. de Sterke, “Propagation of nonlinear pulses in chirped fiber gratings,” Phys. Rev. E 62, 2882–2890 (2000).
[CrossRef]

C. M. de Sterke and B. J. Eggleton, “Bragg solitons and the nonlinear Schrödinger equation,” Phys. Rev. E 59, 1267–1269 (1999).
[CrossRef]

Phys. Rev. Lett. (2)

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]

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[CrossRef] [PubMed]

Other (5)

See, e.g., J. Lauzon, S. Thibault, J. Martin, and F. Quellette, “Implementation and characterization of fiber Bragg gratings linearly chirped by a temperature gradient,” Opt. Lett. 19, 2027–2029 (1994); P. C. Hill and B. J. Eggleton, “Strain gradient chirp of fibre Bragg gratings,” Electron. Lett. 30, 1172–1174 (1994); J. L. Cruz, A. Diez, M. V. Andres, A. Segura, B. Ortega, and L. Dong, “Fibre Bragg gratings tuned and chirped using magnetic fields,” Electron. Lett. ELLEAK 33, 235–236 (1997).
[CrossRef] [PubMed]

A. I. Maimistov, “Evolution of solitary waves which are approximately solitons of a nonlinear Schrödinger equation,” Zh. Eksp. Teor. Fiz. 104, 3620–3629 (1993)[Sov. Phys. JETP 77, 727–731 (1993)].

J. E. Sipe and H. G. Winful, “Nonlinear Schrödinger solitons in a periodic structure,” Opt. Lett. 13, 132–133 (1988); C. M. de Sterke and J. E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 38, 5149–5165 (1988).
[CrossRef] [PubMed]

B. J. Eggleton, G. Lenz, and N. M. Litchinitser, “Optical pulse compression schemes that use nonlinear Bragg gratings,” Fiber Integr. Opt. (to be published).

C. M. de Sterke and J. E. Sipe, “Gap solitons,” in Progress in Optics XXXIII, E. Wolf, ed. (Elsevier, Amsterdam, 1994), Chap. III, pp. 203–260.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Band diagram, κ(z), of the tapered grating with apodization at the ends.

Fig. 2
Fig. 2

Evolution of the soliton parameters for different initial peak intensities, Pin=0.8Pfund (open circles, 1), Pfund (closed circles, 2), 1.2Pfund (open triangles, 3), and σ=1.2. Symbols are the numerical simulation of Eqs. (1). Solid and dashed curves are the solution of Eqs. (10)–(14) and of Eq. (5), respectively. (a) Intensity P+=|E+|2, (b) the FWHM, (c) the linear phase coefficient, -σ+c1/τ0, and (d) the phase chirp coefficient, c2/τ02.

Fig. 3
Fig. 3

Evolution of the soliton parameters for different values of the slope α=-0.0005 (open circles, 1), -0.001 (closed circles, 2), and -0.0015 (open triangles, 3). The detuning σ=1.2, and Pin=Pfund=0.00943. Symbols are the numerical simulation of Eqs. (1). Solid and dashed curves are the solution of Eqs. (10)–(14) and of Eq. (5), respectively. (a) Intensity P+=|E+|2, (b) the FWHM, (c) the linear phase coefficient, -σ+c1/τ0, and (d) the phase chirp coefficient, c2/τ02.

Fig. 4
Fig. 4

Dependence of the FWHM of the transmitted pulse on detuning. Open circles and triangles are numerical simulation of Eqs. (1) and of Eqs. (10)–(14), respectively. The thin solid curve is a numerical simulation of Eq. (5). The thick solid curve is the Wentzel–Kramers–Brillouin solution (21). The dashed curve is the dependence (20), τp=τmin. The dash-dotted line is Eq. (24) but without the v0/v(L) factor.

Equations (32)

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

iV E±t±i E±z+κ(z)E+Γ(|E±|2+2|E|2)E±=0,
Ω±=±Vκ2+Q2.
iv az+iV at+12κγ3v2V2 2at2+Γ(3-v2)2|a|2a
=-i2va-12κγvV κκγ2+(2+v2)vv at-14κv vκκγ+γ(3-2v2)v22v-vγa,
E+E-a2 1+v-1-vexpi0zQ(z)dz-Ω+t.
iψy+12ψττ+|ψ|2ψ=-i0ψ-1ψτ,
0=as,yas+vy2v,
1=12γv2Vκτ0 κyκγ2+(2+v2)vyv.
y=0z dzzs,τ=1τ0 t-1V 0z dzv,ψ=aas,
zs(z)=τ02V2κγ3v3,
as(z)=2/[vVτ0γ3κΓ(3-v2)].
ψ(y, τ)=A(y)sech(x)exp(iφ),
x=(τ-τc)/τd,φ=ϕ+c1τdx+c2τd2x2.
ddy(τdA2)=-21c1+vy2v+as,yasτdA2,
dτcdy=c1-π231c2τd2,
dc1dy=-213τd2(1+π2c22τd4),
dc2dy=21π2τd4-c22-A2π2τd2,
dτddy=2c2τd.
Einc(t)=E0 sech(t/τ0)exp(-iσVt),
A(0)=E0/[as(0)v(0)],τd(0)=1,
τc(0)=c1(0)=c2(0)=0.
τdA2τd0A02=as02v0as2v,
d2τddy2=4π2 1τd3-A02 as02v0as2v 1τd2,
τminas2vA02as02v0=1A02 v0γ02(3-v02)vγ2(3-v2).
τd=τmin{1+β cos[Φ(y)+ϑ0]},
A02-1=β cos ϑ0,
2c20-A02 dτmin(0)dy=-2πA02β sin ϑ0.
Φ(v)=-[κz(v)τ02V2]-1v0v[γv2τmin2(v)]-1dv,
dτmindy=-κz(v)τ02V2γv2 dτmindv,
1τd(z=L)[vγ2(3-v2)]z=Lv0γ02(3-v02)
|α|αthκ0/[π2zs(0)].
0=ατ02V2 γ3v(v4+3)2(3-v2),1=-ατ0Vκv.

Metrics