Abstract

Performance analysis of optical isolators based on the nonreciprocal property of optical bistability in tapered and chirped nonlinear Bragg grating (NLBG) has been carried out. The design method for detuning adjustment according to input power level and expected on–off ratio is also provided. The results display that, for tapered NLBG, the broad operation range and greater on–off ratio can be achieved by using larger taper slope; there exists an optimization range of chirp coefficient for chirped NLBG; the operation range boundary is shifted toward lower input power with the increased detuning; and the tapered NLBG is more preferable to obtain the larger on–off ratio and wider operation range simultaneously compared with chirped NLBG.

© 2007 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  20. B. S. Kim, Y. Chung, and J. S. Lee, "An efficient split-step time-domain dynamic modeling of DFB/DBR laser diodes," IEEE J. Quantum Electron. 36, 787-794 (2000).
    [CrossRef]

2006

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, "Dispersionless slow light using gap solitons," Nat. Phys. 2, 775-780 (2006).
[CrossRef]

2005

A. Maitra, C. G. Poulton, J. Wang, J. Leuthold, and W. Freude, "Low switching threshold using nonlinearities in stopband-tapered waveguide Bragg gratings," IEEE J. Quantum Electron. 41, 1303-1308 (2005).
[CrossRef]

J. T. Mok, I. C. M. Littler, E. Tsoy, and B. J. Eggleton, "Soliton compression and pulse-train generation by use of microchip Q-switched pulses in Bragg gratings," Opt. Lett. 30, 2457-2459 (2005).
[CrossRef] [PubMed]

2004

2003

H. Lee and G. P. Agrawal, "Nonlinear switching of optical pulses in fiber Bragg gratings," IEEE J. Quantum Electron. 39, 508-515 (2003).
[CrossRef]

2000

L. Brzozowski and E. H. Sargent, "Optical signal processing using nonlinear distributed feedback structures," IEEE J. Quantum Electron. 36, 550-555 (2000).
[CrossRef]

B. J. Eggleton, G. Lenz, and N. M. Litchinitser, "Optical pulse compression schemes that use nonlinear Bragg gratings," Fiber Integr. Opt. 19, 383-421 (2000).
[CrossRef]

B. S. Kim, Y. Chung, and J. S. Lee, "An efficient split-step time-domain dynamic modeling of DFB/DBR laser diodes," IEEE J. Quantum Electron. 36, 787-794 (2000).
[CrossRef]

1999

1998

1996

J. M. Liu, C. J. Liao, S. H. Liu, and W. C. Xu, "The dynamics of direction-dependent switching in nonlinear chirped gratings," Opt. Commun. 130, 295-301 (1996).
[CrossRef]

1995

S. Radic, N. George, and G. P. Agrawal, "Theory of low-threshold optical switching in nonlinear, phase-shifted, periodic structures," J. Opt. Soc. Am. B 12, 671-680 (1995).
[CrossRef]

S. Radic, N. George, and G. P. Agrawal, "Analysis of nonuniform nonlinear distributed feedback structures: generalized transfer matrix method," IEEE J. Quantum Electron. 31, 1326-1336 (1995).
[CrossRef]

1994

G. P. Agrawal and S. Radic, "Phased-shifted fiber Bragg gratings and their application for wavelength demultiplexing," IEEE Photonics Technol. Lett. 6, 995-997 (1994).
[CrossRef]

S. Radic, N. George, and G. P. Agrawal, "Optical switching in λ/4-shifted nonlinear periodic structures," Opt. Lett. 19, 1789-1791 (1994).
[CrossRef] [PubMed]

1986

1985

1979

H. G. Winful, J. H. Marburger, and E. Garmire, "Theory of bistability in nonlinear distributed feedback structures," Appl. Phys. Lett. 35, 379-381 (1979).
[CrossRef]

Agrawal, G. P.

H. Lee and G. P. Agrawal, "Nonlinear switching of optical pulses in fiber Bragg gratings," IEEE J. Quantum Electron. 39, 508-515 (2003).
[CrossRef]

S. Radic, N. George, and G. P. Agrawal, "Theory of low-threshold optical switching in nonlinear, phase-shifted, periodic structures," J. Opt. Soc. Am. B 12, 671-680 (1995).
[CrossRef]

S. Radic, N. George, and G. P. Agrawal, "Analysis of nonuniform nonlinear distributed feedback structures: generalized transfer matrix method," IEEE J. Quantum Electron. 31, 1326-1336 (1995).
[CrossRef]

G. P. Agrawal and S. Radic, "Phased-shifted fiber Bragg gratings and their application for wavelength demultiplexing," IEEE Photonics Technol. Lett. 6, 995-997 (1994).
[CrossRef]

S. Radic, N. George, and G. P. Agrawal, "Optical switching in λ/4-shifted nonlinear periodic structures," Opt. Lett. 19, 1789-1791 (1994).
[CrossRef] [PubMed]

Broderick, N. G. R.

Brzozowski, L.

L. Brzozowski and E. H. Sargent, "Optical signal processing using nonlinear distributed feedback structures," IEEE J. Quantum Electron. 36, 550-555 (2000).
[CrossRef]

Chung, Y.

B. S. Kim, Y. Chung, and J. S. Lee, "An efficient split-step time-domain dynamic modeling of DFB/DBR laser diodes," IEEE J. Quantum Electron. 36, 787-794 (2000).
[CrossRef]

de Sterke, C. M.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, "Dispersionless slow light using gap solitons," Nat. Phys. 2, 775-780 (2006).
[CrossRef]

R. E. Slusher, B. J. Eggleton, C. M. de Sterke, and T. A. Strasser,"Nonlinear pulse reflections from chirped fiber gratings," Opt. Express 3, 465-475 (1998).
[CrossRef] [PubMed]

Dobiasch, P.

Eggleton, B. J.

Freude, W.

A. Maitra, C. G. Poulton, J. Wang, J. Leuthold, and W. Freude, "Low switching threshold using nonlinearities in stopband-tapered waveguide Bragg gratings," IEEE J. Quantum Electron. 41, 1303-1308 (2005).
[CrossRef]

Garmire, E.

H. G. Winful, J. H. Marburger, and E. Garmire, "Theory of bistability in nonlinear distributed feedback structures," Appl. Phys. Lett. 35, 379-381 (1979).
[CrossRef]

George, N.

Gibbs, H. M.

Jia, X. H.

Kim, B. S.

B. S. Kim, Y. Chung, and J. S. Lee, "An efficient split-step time-domain dynamic modeling of DFB/DBR laser diodes," IEEE J. Quantum Electron. 36, 787-794 (2000).
[CrossRef]

Lee, H.

H. Lee and G. P. Agrawal, "Nonlinear switching of optical pulses in fiber Bragg gratings," IEEE J. Quantum Electron. 39, 508-515 (2003).
[CrossRef]

Lee, J. S.

B. S. Kim, Y. Chung, and J. S. Lee, "An efficient split-step time-domain dynamic modeling of DFB/DBR laser diodes," IEEE J. Quantum Electron. 36, 787-794 (2000).
[CrossRef]

Lenz, G.

B. J. Eggleton, G. Lenz, and N. M. Litchinitser, "Optical pulse compression schemes that use nonlinear Bragg gratings," Fiber Integr. Opt. 19, 383-421 (2000).
[CrossRef]

G. Lenz and B. J. Eggleton, "Adiabatic Bragg soliton compression in nonuniform fiber gratings," J. Opt. Soc. Am. B 15, 2979-2985 (1999).
[CrossRef]

Leuthold, J.

A. Maitra, C. G. Poulton, J. Wang, J. Leuthold, and W. Freude, "Low switching threshold using nonlinearities in stopband-tapered waveguide Bragg gratings," IEEE J. Quantum Electron. 41, 1303-1308 (2005).
[CrossRef]

Liao, C. J.

J. M. Liu, C. J. Liao, S. H. Liu, and W. C. Xu, "The dynamics of direction-dependent switching in nonlinear chirped gratings," Opt. Commun. 130, 295-301 (1996).
[CrossRef]

Litchinitser, N. M.

B. J. Eggleton, G. Lenz, and N. M. Litchinitser, "Optical pulse compression schemes that use nonlinear Bragg gratings," Fiber Integr. Opt. 19, 383-421 (2000).
[CrossRef]

Littler, I. C. M.

Liu, J. M.

J. M. Liu, C. J. Liao, S. H. Liu, and W. C. Xu, "The dynamics of direction-dependent switching in nonlinear chirped gratings," Opt. Commun. 130, 295-301 (1996).
[CrossRef]

Liu, S. H.

J. M. Liu, C. J. Liao, S. H. Liu, and W. C. Xu, "The dynamics of direction-dependent switching in nonlinear chirped gratings," Opt. Commun. 130, 295-301 (1996).
[CrossRef]

Logvin, Y. A.

Maitra, A.

A. Maitra, C. G. Poulton, J. Wang, J. Leuthold, and W. Freude, "Low switching threshold using nonlinearities in stopband-tapered waveguide Bragg gratings," IEEE J. Quantum Electron. 41, 1303-1308 (2005).
[CrossRef]

Marburger, J. H.

H. G. Winful, J. H. Marburger, and E. Garmire, "Theory of bistability in nonlinear distributed feedback structures," Appl. Phys. Lett. 35, 379-381 (1979).
[CrossRef]

Marquis, F.

Meystre, P.

Mok, J. T.

Peyghambarian, N.

Poulton, C. G.

A. Maitra, C. G. Poulton, J. Wang, J. Leuthold, and W. Freude, "Low switching threshold using nonlinearities in stopband-tapered waveguide Bragg gratings," IEEE J. Quantum Electron. 41, 1303-1308 (2005).
[CrossRef]

Radic, S.

S. Radic, N. George, and G. P. Agrawal, "Analysis of nonuniform nonlinear distributed feedback structures: generalized transfer matrix method," IEEE J. Quantum Electron. 31, 1326-1336 (1995).
[CrossRef]

S. Radic, N. George, and G. P. Agrawal, "Theory of low-threshold optical switching in nonlinear, phase-shifted, periodic structures," J. Opt. Soc. Am. B 12, 671-680 (1995).
[CrossRef]

G. P. Agrawal and S. Radic, "Phased-shifted fiber Bragg gratings and their application for wavelength demultiplexing," IEEE Photonics Technol. Lett. 6, 995-997 (1994).
[CrossRef]

S. Radic, N. George, and G. P. Agrawal, "Optical switching in λ/4-shifted nonlinear periodic structures," Opt. Lett. 19, 1789-1791 (1994).
[CrossRef] [PubMed]

Richardson, D. J.

Sargent, E. H.

L. Brzozowski and E. H. Sargent, "Optical signal processing using nonlinear distributed feedback structures," IEEE J. Quantum Electron. 36, 550-555 (2000).
[CrossRef]

Slusher, R. E.

Strasser, T. A.

Taverner, D.

Tsoy, E.

Volkov, V. M.

Wang, J.

A. Maitra, C. G. Poulton, J. Wang, J. Leuthold, and W. Freude, "Low switching threshold using nonlinearities in stopband-tapered waveguide Bragg gratings," IEEE J. Quantum Electron. 41, 1303-1308 (2005).
[CrossRef]

Winful, H. G.

H. G. Winful, J. H. Marburger, and E. Garmire, "Theory of bistability in nonlinear distributed feedback structures," Appl. Phys. Lett. 35, 379-381 (1979).
[CrossRef]

Wright, E. M.

Wu, Z. M.

Xia, G. Q.

Xu, W. C.

J. M. Liu, C. J. Liao, S. H. Liu, and W. C. Xu, "The dynamics of direction-dependent switching in nonlinear chirped gratings," Opt. Commun. 130, 295-301 (1996).
[CrossRef]

Appl. Phys. Lett.

H. G. Winful, J. H. Marburger, and E. Garmire, "Theory of bistability in nonlinear distributed feedback structures," Appl. Phys. Lett. 35, 379-381 (1979).
[CrossRef]

Fiber Integr. Opt.

B. J. Eggleton, G. Lenz, and N. M. Litchinitser, "Optical pulse compression schemes that use nonlinear Bragg gratings," Fiber Integr. Opt. 19, 383-421 (2000).
[CrossRef]

IEEE J. Quantum Electron.

B. S. Kim, Y. Chung, and J. S. Lee, "An efficient split-step time-domain dynamic modeling of DFB/DBR laser diodes," IEEE J. Quantum Electron. 36, 787-794 (2000).
[CrossRef]

H. Lee and G. P. Agrawal, "Nonlinear switching of optical pulses in fiber Bragg gratings," IEEE J. Quantum Electron. 39, 508-515 (2003).
[CrossRef]

A. Maitra, C. G. Poulton, J. Wang, J. Leuthold, and W. Freude, "Low switching threshold using nonlinearities in stopband-tapered waveguide Bragg gratings," IEEE J. Quantum Electron. 41, 1303-1308 (2005).
[CrossRef]

L. Brzozowski and E. H. Sargent, "Optical signal processing using nonlinear distributed feedback structures," IEEE J. Quantum Electron. 36, 550-555 (2000).
[CrossRef]

S. Radic, N. George, and G. P. Agrawal, "Analysis of nonuniform nonlinear distributed feedback structures: generalized transfer matrix method," IEEE J. Quantum Electron. 31, 1326-1336 (1995).
[CrossRef]

IEEE Photonics Technol. Lett.

G. P. Agrawal and S. Radic, "Phased-shifted fiber Bragg gratings and their application for wavelength demultiplexing," IEEE Photonics Technol. Lett. 6, 995-997 (1994).
[CrossRef]

J. Opt. Soc. Am. B

Nat. Phys.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, "Dispersionless slow light using gap solitons," Nat. Phys. 2, 775-780 (2006).
[CrossRef]

Opt. Commun.

J. M. Liu, C. J. Liao, S. H. Liu, and W. C. Xu, "The dynamics of direction-dependent switching in nonlinear chirped gratings," Opt. Commun. 130, 295-301 (1996).
[CrossRef]

Opt. Express

Opt. Lett.

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Figures (4)

Fig. 1
Fig. 1

(a) Input–output characteristic curves for LT-NLBG, where Δ k = 30 % , δ L = 3 ; (b) contour diagram of on–off ratio of LT-NLBG with the variation of input power and Δ k , where δ L = 3 ; (c) contour diagram of on–off ratio of LT-NLBG with the variation of input power and Δ k , where δ L = 4 ; (d) contour diagram of on–off ratio of LT-NLBG with the variation of input power and detuning, where Δ k = 30 % .

Fig. 2
Fig. 2

(a) Input–output characteristic curves for LC-NLBG, where C = 3 , δ L = 3 ; (b) contour diagram of on–off ratio of LC-NLBG with the variation of input power and C , where δ L = 3 ; (c) contour diagram of on–off ratio of LC-NLBG with the variation of input power and C when δ L = 4 ; (d) contour diagram of on–off ratio of LC-NLBG with the variation of input power and detuning, where C = 3 .

Fig. 3
Fig. 3

Input and output pulse shapes for LT-NLBG, where Δ k = 30 % , δ L = 3 ; the normalized input peak power P in P c is (a) 0.65 and (b) 0.75.

Fig. 4
Fig. 4

Input and output pulse shapes for LC-NLBG, where C = 3 , δ L = 3 ; the normalized input peak power P in P c is (a) 0.5 and (b) 0.6.

Equations (14)

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

n ( z ) = n 0 + n 1 ( z ) cos [ 2 π Λ z + ϕ ( z ) ] + n 2 E ( z ) 2 ,
E = A f exp [ i ( β 0 z ϖ t ) ] + A b exp [ i ( β 0 z + ϖ t ) ] ,
A f z + 1 v g A f t = i [ δ A f + Γ ( A f 2 + 2 A b 2 ) A f + κ A b ] ,
A b z 1 v g A b t = i [ δ A b + Γ ( A b 2 + 2 A f 2 ) A b + κ * A f ] ,
δ = β β 0 = n 0 ϖ c β 0 , Γ = 2 π n 2 λ 0 ,
κ ( z ) = π n 1 ( z ) η λ 0 exp [ i ϕ ( z ) ] ,
κ ( z ) = κ 0 [ 1 + Δ κ z L ] ,
Δ κ = κ ( L ) κ ( 0 ) κ ( 0 ) ,
β 0 = β 0 + C L 2 ( z L 2 )
z = 0 : A f ( 0 , t ) = A i ( 0 , t ) , A r ( 0 , t ) = A b ( 0 , t ) ,
z = L : A b ( L , t ) = 0 , A t ( L , t ) = A f ( L , t ) ,
[ A f , i + 1 ( t + Δ t ) A b , i ( t + Δ t ) ] = T P T C [ A f , i ( t ) A b , i + 1 ( t ) ] ,
T C = [ sec h ( k Δ z ) i tanh ( k Δ z ) i tanh ( k Δ z ) sec h ( k Δ z ) ] ,
T p = [ exp { i Δ z [ δ + Γ ( A f , j ( t ) 2 + 2 A b , j ( t ) 2 ) ] } 0 0 exp { i Δ z [ δ + Γ ( A b , j + 1 ( t ) 2 + 2 A f , j + 1 ( t ) 2 ) ] } ] .

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