Abstract

We present two sets of equations to describe nonlinear pulse propagation in a birefringent fiber Bragg grating. The first set uses a coupled-mode formalism to describe light in or near the photonic band gap of the grating. The second set is a pair of coupled nonlinear Schroedinger equations. We use these equations to examine viable switching experiments in the presence of birefringence. We show how the birefringence can both aid and hinder device applications.

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References

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  1. Pochi Yeh, Optical Waves in Layered Media, (Wiley, New York, 1988).
  2. C. M. de Sterke and J. E. Sipe, "Gap Solitons" in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1994).
  3. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug and J. E. Sipe, "Bragg Grating Solitons," Phys. Lett. 76, 1627-30 (1996).
    [CrossRef]
  4. see, e.g., M. J. Steel and C. M. de Sterke, "Schroedinger equation description for cross-phase modulation in grating structures," Phys. Rev. A 49, 5048-55 (1994).
    [CrossRef] [PubMed]
  5. W. Samir, S. J. Garth and C. Pask, "Interplay of grating and nonlinearity in mode-coupling," J. Opt. Soc. Am. B 11, 64-71 (1994).
    [CrossRef]
  6. S. Lee and S. T. Ho, "Optical switching scheme based on the transmission of coupled gap solitons in nonlinear periodic dielectric media," Opt. Lett. 18, 962-64 (1993).
    [CrossRef] [PubMed]
  7. D. Taverner, N. G. R. Broderick, D. J. Richardson, M. Ibsen and R. I. Laming, "All-optical AND gate based on coupled gap-soliton formation in a fiber Bragg grating," Opt. Lett. 23, 259-61 (1998).
    [CrossRef]
  8. Y. Silberberg and Y. Barad, "Rotating vector solitary waves in isotropic fibers," Opt. Lett. 20, 246-48 (1995).
    [CrossRef] [PubMed]
  9. C. M. de Sterke, D. G. Salinas and J. E. Sipe, "Coupled-mode theory for light propagation through deep nonlinear gratings," Phys. Rev. E 54, 1969-89 (1996).
    [CrossRef]
  10. Govind Agrawal, Nonlinear Fiber Optics, (Academic Press, Boston, 1989).
  11. T. Erdogan and V. Mizrahi, "Characterization of UV-induced birefringence in photosensitive Ge-doped silica optical fibers," J. Opt. Soc. Am. B 11, 2100-05 (1994).
    [CrossRef]
  12. C. R. Menyuk, "Stability of solitons in birefringent optical fibers. I: Equal propagation amplitudes," Opt. Lett. 12, 614-16 (1987).
    [CrossRef] [PubMed]

Other

Pochi Yeh, Optical Waves in Layered Media, (Wiley, New York, 1988).

C. M. de Sterke and J. E. Sipe, "Gap Solitons" in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1994).

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug and J. E. Sipe, "Bragg Grating Solitons," Phys. Lett. 76, 1627-30 (1996).
[CrossRef]

see, e.g., M. J. Steel and C. M. de Sterke, "Schroedinger equation description for cross-phase modulation in grating structures," Phys. Rev. A 49, 5048-55 (1994).
[CrossRef] [PubMed]

W. Samir, S. J. Garth and C. Pask, "Interplay of grating and nonlinearity in mode-coupling," J. Opt. Soc. Am. B 11, 64-71 (1994).
[CrossRef]

S. Lee and S. T. Ho, "Optical switching scheme based on the transmission of coupled gap solitons in nonlinear periodic dielectric media," Opt. Lett. 18, 962-64 (1993).
[CrossRef] [PubMed]

D. Taverner, N. G. R. Broderick, D. J. Richardson, M. Ibsen and R. I. Laming, "All-optical AND gate based on coupled gap-soliton formation in a fiber Bragg grating," Opt. Lett. 23, 259-61 (1998).
[CrossRef]

Y. Silberberg and Y. Barad, "Rotating vector solitary waves in isotropic fibers," Opt. Lett. 20, 246-48 (1995).
[CrossRef] [PubMed]

C. M. de Sterke, D. G. Salinas and J. E. Sipe, "Coupled-mode theory for light propagation through deep nonlinear gratings," Phys. Rev. E 54, 1969-89 (1996).
[CrossRef]

Govind Agrawal, Nonlinear Fiber Optics, (Academic Press, Boston, 1989).

T. Erdogan and V. Mizrahi, "Characterization of UV-induced birefringence in photosensitive Ge-doped silica optical fibers," J. Opt. Soc. Am. B 11, 2100-05 (1994).
[CrossRef]

C. R. Menyuk, "Stability of solitons in birefringent optical fibers. I: Equal propagation amplitudes," Opt. Lett. 12, 614-16 (1987).
[CrossRef] [PubMed]

Supplementary Material (5)

» Media 1: MOV (127 KB)     
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» Media 3: MOV (108 KB)     
» Media 4: MOV (85 KB)     
» Media 5: MOV (110 KB)     

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Figures (9)

Figure 1.
Figure 1.

Dispersion relation for a birefringent medium with ny<nx. The y band is shifted up relative to the x band so that an input frequency as shown will be more deeply in the y gap than in the x gap. The dashed lines represent the x polarization dispersion relation in a uniform medium; the x band asymptotically approaches the uniform values as one detunes from the bandgap.

Figure 2.
Figure 2.

Operation of gap soliton AND gate device. When the x or y polarization are sent in alone, the output is zero. However, when both are sent in together, there is a significant output, corresponding to 25% of the input power. Both pulses experience significant pulse compression due to their high intensity.

Figure 3.
Figure 3.

Quicktime movie showing the evolution of field intensity within the grating as time increases for the case when the x or y pulse is sent into the system alone. Neither pulse transmits through the grating, but the x pulse gets a bit deeper, as descibed in the text. [Media 1]

Figure 4.
Figure 4.

Quicktime movie showing the evolution of field intensity within the grating as time increases for the case when both pulses are sent into the system together. The combined cross-phase modulation pulls some of the energy from each of the pulses through the grating. The build up of nonlinear compression can be seen. [Media 2]

Figure 5.
Figure 5.

Quicktime movie showing the evolution of field intensity within the grating as time increases for the non-birefringent case. Here we inject two pulses (superimposed). Part of the energy forms into a gap soliton, but the remaining energy is too weak to transmit through the grating. [Media 3]

Figure 6.
Figure 6.

Evolution of the azimuthal angle of the elliptically polarized input pulse with grating length for an isotropic medium. The curve is not quite linear because the injected Gaussian pulse is still evolving into a soliton.

Figure 7.
Figure 7.

Quicktime movie showing the evolution of the elliptically polarized pulse and the profile of its azimuthal angle as grating length increases. As the Gaussian pulse evolves into a soliton, its azimuthal angle increases linearly with grating length, and the azimuthal profile flattens. [Media 4]

Figure 8.
Figure 8.

Quicktime movie showing the evolution of the elliptically polarized pulse, and the profile of its azimuthal angle as the grating’s birefringence increases. The length of the grating is fixed at 4 cm. Eventually the birefringence becomes so high that pulse walk-off destroys the device operation. [Media 5]

Figure 9.
Figure 9.

Numerical integration of the NBCME using an elliptically polarized Gaussian input pulse as discussed in the text. At a birefringence of 10-6 the azimuthal profile is relatively flat, but at 1.6×10-6 the profile is destroyed.

Equations (38)

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ε ( z ) = [ ε x x ( z ) 0 0 ε y y ( z ) ] .
n i 2 ( z ) = ε ii ( z ) ε 0 ,
E ( r , t ) = x ̂ E x ( z , t ) + y ̂ E y ( z , t ) ,
H ( r , t ) = x ̂ H x ( z , t ) + y ̂ H y ( z , t ) ,
H x , y t = ± 1 μ 0 E y , x z ,
E x , y t = ± 1 ε 0 n x , y 2 ( z ) H y , x z .
P i ( z , t ) = ε 0 χ ijkl ( 3 ) E j ( z , t ) E k ( z , t ) E l ( z , t ) ,
n x , y ( z ) = n 0 x , y + δ n cos ( 2 π d z ) .
0 = i X + z + i v x X + t + κ X + α x { X + 2 + 2 X 2 } X +
+ β x { Y + 2 + 2 Y 2 } X + + β x X Y * Y +
+ γ x { X + * Y + 2 + 2 X * Y + Y } e iδt ,
0 = i X z + i v x X t + κ X + + α x { X 2 + 2 X + 2 } X
+ β x { Y + 2 + Y 2 } X + β x X + Y + * Y
+ γ x { X * Y 2 + 2 X + * Y Y + } e iδt ,
0 = i Y z + i v y Y t + κ y Y + + α y { Y 2 + 2 Y + 2 } Y
+ β y { X + 2 + X 2 } Y + β y Y + X + * X
+ γ y { Y * X 2 + 2 Y + * X X + } e iδt .
0 = i Y + z + i v y Y + t + κ y Y + α y { Y + 2 + 2 Y 2 } Y +
+ β y { X + 2 + X 2 } Y + + β y Y X * X +
+ γ y { Y + * X + 2 + 2 Y * X + X } e iδt ,
α i = n 2 A eff k 0 n 0 i
{ α i : β i : γ i } = { 3 : 2 : 1 } .
δ = 2 c k 0 ( 1 n 0 x 1 n 0 y ) .
κ x , y = δ n 2 n 0 x , y π d ,
v x , y = c n 0 x , y .
ω 0 x , y = c n 0 x , y π d ,
0 = i ( t + ω x z ) X + 1 2 ω x 2 X z 2
+ { α spm x X 2 + α cpm x Y 2 } X + α p c x Y 2 X * e iλt ,
0 = i ( t + ω y z ) Y + 1 2 ω y 2 Y z 2
+ { α spm y Y 2 + α cpm y X 2 } Y + α p c y X 2 Y * e iλt .
λ ( k ) = 2 ( ω x ( k ) ω y ( k ) ) ,
α spm i = n 2 A eff k 0 n 0 i 3 ρ i 2 2 ρ i 2 ,
{ α i : β i : γ i } = { 3 : 2 : 1 } .
0 = i U z + 1 2 β 2 2 U t 2 + α 3 { U 2 + 2 V 2 } U ,
0 = i V z + 1 2 β 2 2 V t 2 + α 3 { V 2 + 2 U 2 } V ,
U = 6 η sec h ( 2 η z ) e i η z ,
V = V 0 sec h s ( 2 η z ) e i ς z ,
ς = ( 4 s ) η

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