Abstract

It is shown that light propagation in an apodized fiber Bragg grating with a Kerr nonlinearity approximately obeys a nonlinear Schrödinger-like equation, but with extra terms because the eigenstates of the grating vary with position. It is shown that propagation through such a grating leads to field enhancement, and to a nontrivial phase shift; an approximate expression for the reflectivity is also found.

© 1998 Optical Society of America

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References

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  1. C. M. de Sterke and J. E. Sipe, “Gap solitons,” in Progress in Optics33, E. Wolf, ed. (Elsevier, Amsterdam, 1994). 203–260.
  2. C. M. de Sterke and B. J. Eggleton, “Bragg solitons and the nonlinear Schrödinger equation,” in press Phys. Rev. E .
  3. P. S. Cross and H. Kogelnik, Opt. Lett. 1, 43–45 (1977).
    [Crossref] [PubMed]
  4. B. Malo, D. C. Johnson, F. Bilodeau, J. Albert, and K. O. Hill, Elect. Lett. 31, 223–225 (1995).
    [Crossref]
  5. F. Ouellette, Opt. Lett. 12, 847–849 (1987).
    [Crossref] [PubMed]
  6. B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Bragg grating solitons in the nonlinear Schrödinger limit: theory and experiment,” submitted to J. Opt. Soc. Am. B
  7. L. Poladian, Phys. Rev. E 484758–4767 (1993).
    [Crossref]
  8. D. Marcuse, Theory of dielectric optical waveguides, 2nd Ed. (Academic, San Diego, 1991).

1995 (1)

B. Malo, D. C. Johnson, F. Bilodeau, J. Albert, and K. O. Hill, Elect. Lett. 31, 223–225 (1995).
[Crossref]

1993 (1)

L. Poladian, Phys. Rev. E 484758–4767 (1993).
[Crossref]

1987 (1)

1977 (1)

Albert, J.

B. Malo, D. C. Johnson, F. Bilodeau, J. Albert, and K. O. Hill, Elect. Lett. 31, 223–225 (1995).
[Crossref]

Bilodeau, F.

B. Malo, D. C. Johnson, F. Bilodeau, J. Albert, and K. O. Hill, Elect. Lett. 31, 223–225 (1995).
[Crossref]

Cross, P. S.

de Sterke, C. M.

C. M. de Sterke and J. E. Sipe, “Gap solitons,” in Progress in Optics33, E. Wolf, ed. (Elsevier, Amsterdam, 1994). 203–260.

C. M. de Sterke and B. J. Eggleton, “Bragg solitons and the nonlinear Schrödinger equation,” in press Phys. Rev. E .

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Bragg grating solitons in the nonlinear Schrödinger limit: theory and experiment,” submitted to J. Opt. Soc. Am. B

Eggleton, B. J.

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Bragg grating solitons in the nonlinear Schrödinger limit: theory and experiment,” submitted to J. Opt. Soc. Am. B

C. M. de Sterke and B. J. Eggleton, “Bragg solitons and the nonlinear Schrödinger equation,” in press Phys. Rev. E .

Hill, K. O.

B. Malo, D. C. Johnson, F. Bilodeau, J. Albert, and K. O. Hill, Elect. Lett. 31, 223–225 (1995).
[Crossref]

Johnson, D. C.

B. Malo, D. C. Johnson, F. Bilodeau, J. Albert, and K. O. Hill, Elect. Lett. 31, 223–225 (1995).
[Crossref]

Kogelnik, H.

Malo, B.

B. Malo, D. C. Johnson, F. Bilodeau, J. Albert, and K. O. Hill, Elect. Lett. 31, 223–225 (1995).
[Crossref]

Marcuse, D.

D. Marcuse, Theory of dielectric optical waveguides, 2nd Ed. (Academic, San Diego, 1991).

Ouellette, F.

Poladian, L.

L. Poladian, Phys. Rev. E 484758–4767 (1993).
[Crossref]

Sipe, J. E.

C. M. de Sterke and J. E. Sipe, “Gap solitons,” in Progress in Optics33, E. Wolf, ed. (Elsevier, Amsterdam, 1994). 203–260.

Slusher, R. E.

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Bragg grating solitons in the nonlinear Schrödinger limit: theory and experiment,” submitted to J. Opt. Soc. Am. B

Elect. Lett. (1)

B. Malo, D. C. Johnson, F. Bilodeau, J. Albert, and K. O. Hill, Elect. Lett. 31, 223–225 (1995).
[Crossref]

Opt. Lett. (2)

Phys. Rev. E (1)

L. Poladian, Phys. Rev. E 484758–4767 (1993).
[Crossref]

Other (4)

D. Marcuse, Theory of dielectric optical waveguides, 2nd Ed. (Academic, San Diego, 1991).

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Bragg grating solitons in the nonlinear Schrödinger limit: theory and experiment,” submitted to J. Opt. Soc. Am. B

C. M. de Sterke and J. E. Sipe, “Gap solitons,” in Progress in Optics33, E. Wolf, ed. (Elsevier, Amsterdam, 1994). 203–260.

C. M. de Sterke and B. J. Eggleton, “Bragg solitons and the nonlinear Schrödinger equation,” in press Phys. Rev. E .

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Figures (3)

Figure 1.
Figure 1.

Argument of A vs position in a tapered grating. Black lines are exact, red lines follow by integrating Eq. (11). In (a) Δ/ṽ = 1.512, in (b) Δ/vį = 1.25.

Figure 2.
Figure 2.

Absolute value of the argument of envelope A at z = L vs the normalized detuning Δ/ṽ for an apodized grating with ṽL = 15. The black line follows from Eq. (11), the dots are exact results, and the red line is analytic approximation (13).

Figure 3.
Figure 3.

Reflectivity vs normalized detuning Δ/κ̃ for two apodized gratings. In (a) κ̃L = 15, in (b) κ̃L = 100. Black lines are exact results, red lines follow from approximation (18).

Equations (19)

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i V ε ± t ± i ε ± z + κ ( z ) ε + Γ ( ε ± 2 + 2 ε 2 ) ε ± = 0 ,
Q = κ γv , Ω ± = ± V κ γ ,
ε = ( μ a ( z 1 , z 2 ; t 1 , t 2 ) φ + ( z 1 , z 2 ) + μ 2 b ( z 1 , z 2 ; t 1 , t 2 ) φ ( z 1 , z 2 )
+ μ 3 c ( z 1 , z 2 ; t 1 , t 2 ) φ ( z 1 , z 2 ) ) e i Ω + t 0 e i Φ ( z 0 ) ,
Φ ( z ) = 0 z 0 Q ( z ) d z ,
i ( a z 1 ± a t 1 ) ± i 2 1 1 ± v v z 1 a + 2 κ 1 ± v b = 0 .
v a z 1 + a t 1 + 1 2 d v d z 1 a = 0 .
a ( z 1 , t 1 ) = f [ t 1 0 z 1 d z v ( z ) ] v ( z ) ,
b = i 2 κ γ 2 a z 1 + i v 4 κ d v d z 1 a .
v a z 2 + a t 2 + 1 2 v d z 2 a + 1 γ b z 1 γ v 2 d v d z 1 b i Γ 2 ( 3 v 2 ) a 2 a = 0
i ( A t + v A z ) + A z z 2 κ γ 3 ( 1 + v 2 ) v 2 κ γ v A z v 4 κ γ v A γ ( 4 v 2 3 ) ( v ) 2 8 κ v 2 A + 3 v 2 2 v Γ A 2 A = 0 .
i v A z v 2 κ γ v A γ ( 4 v 2 3 ) ( v ) 2 8 κ v 2 A = 0 .
κ ( z ) = κ ˜ sin 2 ( π z L ) ( 0 z L ) ,
arg [ A ( L ) ] = π 2 16 ( κ ˜ Δ ) 3 1 κ ˜ L ( 1 + 25 35 ( κ ˜ Δ ) 2 ) ,
ε = [ ( a + φ + + b + φ ) e i Φ + ( a φ + + b φ ) e i Φ ] e i Ω t ,
v a z a t + v 2 a = ( a + t γ + γ 2 v a + + 2 i κ γ v b + ) e 2 i Φ .
A z = A + z γ e 2 i Φ ,
A z = i A + [ v 4 κ γ v + γ ( 4 v 2 3 ) ( v ) 2 8 κ v 2 ] e 2 i Φ
R = 0 L [ v 4 κ γ v + γ ( 4 v 2 3 ) ( v ) 2 8 κ v 2 ] e 2 i Φ d z 2 .

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