Abstract

An efficient numerical method to solve multimode-coupled equations with two point boundary conditions is improved. Our method [Abrishamian et al., Opt. Fiber Technol. 13, 32–38 (2007)] based on theoretical matrix integration of coupled differential equations and then solving the system of equations by use of initial or final conditions would be straightforward and thus beneficial in comparison with previously used fundamental matrix methods that depend strongly on the initial guess. However, we found that the new analysis depends on how accurately the integrals of the matrix element are calculated. For accuracy in the matrix integration it is required to divide the system of equations into a large number of subsections. Then, the reflectivity calculated is found to be comparable to experimental data reported so far. The present method is highly applicable for simulation of any type of fiber Bragg gratings modulated by long period gratings.

© 2009 Optical Society of America

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References

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  1. K. O. Hill and G. Meltz, “Fiber Bragg grating technology: Fundamentals and overview,” J. Lightwave Technol. 15, 1263-1276 (1997).
    [CrossRef]
  2. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277-1294 (1997).
    [CrossRef]
  3. M. Yamada and K. Sakuda, “Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach,” Appl. Opt. 26, 3474-3478 (1987).
    [CrossRef] [PubMed]
  4. G. J. Liu, Q. Li, G. L. Jin, and B. M. Liang, “Transfer matrix method analysis of apodized grating couplers,” Opt. Commun. 235, 319-324 (2004).
    [CrossRef]
  5. R. Helan, “Comparison of methods for fiber Bragg gratings simulation,” in 29th International Spring Seminar on Electronics Technology ISSE 2006 (2006), pp. 161-166.
    [CrossRef]
  6. Z. Luo, C. Ye, Z. Cai, X. Dai, Y. Kang, and H. Xu, “Numerical analysis and optimization of optical spectral characteristics of fiber Bragg gratings modulated by a transverse acoustic wave,” Appl. Opt. 46, 6959-6965 (2007).
    [CrossRef] [PubMed]
  7. W. F. Liu, P. St. J. Russell, and L. Dong, “100% efficient narrow-band acoustooptic tunable reflector using fiber Bragg grating,” J. Lightwave Technol. 16, 2006-2009 (1998).
    [CrossRef]
  8. P. St. J. Russell and W.-F. Liu, “Acousto optic supperlattice modulation in fiber Bragg gratings,” J. Opt. Soc. Am. A 17, 1421-1429 (2000).
    [CrossRef]
  9. T. A. Birks, P. St. J. Russell, and D. O. Culverhouse, “The acousto-optic effect in single-mode fiber tapers and couplers,” J. Lightwave Technol. 14, 2519-2529 (1996).
    [CrossRef]
  10. D. W. Hang, W. F. Liu, C. W. Wu, and C. C. Yang, “Reflectivity-tunable fiber Bragg grating reflectors,” IEEE Photon. Technol. 12, 176-178 (2000).
    [CrossRef]
  11. H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, 1975), Vol. 7, pp. 13-81.
    [CrossRef]
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    [CrossRef]
  13. L. A. Weller-Brophy, “Analysis of waveguide gratings: Application of Round's method,” J. Opt. Soc. Am. A 2, 863-871(1985).
    [CrossRef]
  14. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University Press, 2002), pp. 710-722.
  15. F. Abrishamian, Y. Nakai, S. Sato, and M. Imai, “An efficient approach for calculating the reflection and transmission spectra of fiber Bragg gratings with acoustically induced microbending,” Opt. Fiber Technol. 13, 32-38 (2007).
    [CrossRef]
  16. F. Abrishamian, S. Sato, and M. Imai, “Numerical analysis of multimode-coupled equations and its application to modulated fiber Bragg gratings,” in Asia Optical Fiber Communication and Optoelectronic Exposition & Conference (AOE) CD-ROM (Optical Society of America, 2008), paper SaE5.
  17. W.-F. Liu, I.-M. Liu, L.-W. Chung, D.-W. Huang, and C. C. Yang, “Acoustic-induced switching of the reflection wavelength in a fiber Bragg grating,” Opt. Lett. 25, 1319-1321(2000).
    [CrossRef]
  18. N.-H. Sun, C.-C. Chou, M.-J. Chang, C.-N. Lin, C.-C. Yang, Y.-W. Kiang, and W.-F. Liu, “Analysis of phase-matching conditions in flexural-wave modulated fiber Bragg grating,” J. Lightwave Technol. 20, 311-315 (2002).
    [CrossRef]
  19. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++: The Art of Scientific Computing (Cambridge University Press, 2002), 2nd ed., pp. 130-133.
    [CrossRef]
  20. T. A. Birks, P. St. J. Russell, and D. O. Culverhouse, “The acousto-optic effect in single-mode fiber tapers and couplers,” J. Lightwave Technol. 14, 2519-2529 (1996).
    [CrossRef]
  21. T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760-1773 (1997).
    [CrossRef]
  22. K. Morishita, “Numerical analysis of pulse broadening in graded index optical fibers,” IEEE Trans. Microwave Theory Tech. 29, 348-352 (1981).
    [CrossRef]
  23. F. Abrishamian, S. Nagai, S. Sato, and M. Imai, “Design theory and experiment of acousto-optic tunable filter by use of flexural waves applied to thin optical fiber,” Opt. Quant. Electron. 40, 665-676 (2008).
    [CrossRef]
  24. T. Lin, Y. Furuumi, M. Imai, Y. Tsuji, and M. Koshiba, “Design theory and experiment of LP01-LP11 mode converter utilizing fused taper fiber coupler,” Trans. IEICE Electron. J82-C-I, 587-595 (1999), in Japanese.

2008 (1)

F. Abrishamian, S. Nagai, S. Sato, and M. Imai, “Design theory and experiment of acousto-optic tunable filter by use of flexural waves applied to thin optical fiber,” Opt. Quant. Electron. 40, 665-676 (2008).
[CrossRef]

2007 (2)

F. Abrishamian, Y. Nakai, S. Sato, and M. Imai, “An efficient approach for calculating the reflection and transmission spectra of fiber Bragg gratings with acoustically induced microbending,” Opt. Fiber Technol. 13, 32-38 (2007).
[CrossRef]

Z. Luo, C. Ye, Z. Cai, X. Dai, Y. Kang, and H. Xu, “Numerical analysis and optimization of optical spectral characteristics of fiber Bragg gratings modulated by a transverse acoustic wave,” Appl. Opt. 46, 6959-6965 (2007).
[CrossRef] [PubMed]

2004 (1)

G. J. Liu, Q. Li, G. L. Jin, and B. M. Liang, “Transfer matrix method analysis of apodized grating couplers,” Opt. Commun. 235, 319-324 (2004).
[CrossRef]

2002 (1)

2000 (3)

1999 (1)

T. Lin, Y. Furuumi, M. Imai, Y. Tsuji, and M. Koshiba, “Design theory and experiment of LP01-LP11 mode converter utilizing fused taper fiber coupler,” Trans. IEICE Electron. J82-C-I, 587-595 (1999), in Japanese.

1998 (1)

1997 (4)

S. M. Norton, T. Erdogan, and G. M. Morris, “Coupled-mode theory of resonant-grating filters,” J. Opt. Soc. Am. A 14, 629-639 (1997).
[CrossRef]

T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760-1773 (1997).
[CrossRef]

K. O. Hill and G. Meltz, “Fiber Bragg grating technology: Fundamentals and overview,” J. Lightwave Technol. 15, 1263-1276 (1997).
[CrossRef]

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277-1294 (1997).
[CrossRef]

1996 (2)

T. A. Birks, P. St. J. Russell, and D. O. Culverhouse, “The acousto-optic effect in single-mode fiber tapers and couplers,” J. Lightwave Technol. 14, 2519-2529 (1996).
[CrossRef]

T. A. Birks, P. St. J. Russell, and D. O. Culverhouse, “The acousto-optic effect in single-mode fiber tapers and couplers,” J. Lightwave Technol. 14, 2519-2529 (1996).
[CrossRef]

1987 (1)

1985 (1)

1981 (1)

K. Morishita, “Numerical analysis of pulse broadening in graded index optical fibers,” IEEE Trans. Microwave Theory Tech. 29, 348-352 (1981).
[CrossRef]

Abrishamian, F.

F. Abrishamian, S. Nagai, S. Sato, and M. Imai, “Design theory and experiment of acousto-optic tunable filter by use of flexural waves applied to thin optical fiber,” Opt. Quant. Electron. 40, 665-676 (2008).
[CrossRef]

F. Abrishamian, Y. Nakai, S. Sato, and M. Imai, “An efficient approach for calculating the reflection and transmission spectra of fiber Bragg gratings with acoustically induced microbending,” Opt. Fiber Technol. 13, 32-38 (2007).
[CrossRef]

F. Abrishamian, S. Sato, and M. Imai, “Numerical analysis of multimode-coupled equations and its application to modulated fiber Bragg gratings,” in Asia Optical Fiber Communication and Optoelectronic Exposition & Conference (AOE) CD-ROM (Optical Society of America, 2008), paper SaE5.

Birks, T. A.

T. A. Birks, P. St. J. Russell, and D. O. Culverhouse, “The acousto-optic effect in single-mode fiber tapers and couplers,” J. Lightwave Technol. 14, 2519-2529 (1996).
[CrossRef]

T. A. Birks, P. St. J. Russell, and D. O. Culverhouse, “The acousto-optic effect in single-mode fiber tapers and couplers,” J. Lightwave Technol. 14, 2519-2529 (1996).
[CrossRef]

Cai, Z.

Chang, M.-J.

Chou, C.-C.

Chung, L.-W.

Culverhouse, D. O.

T. A. Birks, P. St. J. Russell, and D. O. Culverhouse, “The acousto-optic effect in single-mode fiber tapers and couplers,” J. Lightwave Technol. 14, 2519-2529 (1996).
[CrossRef]

T. A. Birks, P. St. J. Russell, and D. O. Culverhouse, “The acousto-optic effect in single-mode fiber tapers and couplers,” J. Lightwave Technol. 14, 2519-2529 (1996).
[CrossRef]

Dai, X.

Dong, L.

Erdogan, T.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++: The Art of Scientific Computing (Cambridge University Press, 2002), 2nd ed., pp. 130-133.
[CrossRef]

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University Press, 2002), pp. 710-722.

Furuumi, Y.

T. Lin, Y. Furuumi, M. Imai, Y. Tsuji, and M. Koshiba, “Design theory and experiment of LP01-LP11 mode converter utilizing fused taper fiber coupler,” Trans. IEICE Electron. J82-C-I, 587-595 (1999), in Japanese.

Hang, D. W.

D. W. Hang, W. F. Liu, C. W. Wu, and C. C. Yang, “Reflectivity-tunable fiber Bragg grating reflectors,” IEEE Photon. Technol. 12, 176-178 (2000).
[CrossRef]

Helan, R.

R. Helan, “Comparison of methods for fiber Bragg gratings simulation,” in 29th International Spring Seminar on Electronics Technology ISSE 2006 (2006), pp. 161-166.
[CrossRef]

Hill, K. O.

K. O. Hill and G. Meltz, “Fiber Bragg grating technology: Fundamentals and overview,” J. Lightwave Technol. 15, 1263-1276 (1997).
[CrossRef]

Huang, D.-W.

Imai, M.

F. Abrishamian, S. Nagai, S. Sato, and M. Imai, “Design theory and experiment of acousto-optic tunable filter by use of flexural waves applied to thin optical fiber,” Opt. Quant. Electron. 40, 665-676 (2008).
[CrossRef]

F. Abrishamian, Y. Nakai, S. Sato, and M. Imai, “An efficient approach for calculating the reflection and transmission spectra of fiber Bragg gratings with acoustically induced microbending,” Opt. Fiber Technol. 13, 32-38 (2007).
[CrossRef]

T. Lin, Y. Furuumi, M. Imai, Y. Tsuji, and M. Koshiba, “Design theory and experiment of LP01-LP11 mode converter utilizing fused taper fiber coupler,” Trans. IEICE Electron. J82-C-I, 587-595 (1999), in Japanese.

F. Abrishamian, S. Sato, and M. Imai, “Numerical analysis of multimode-coupled equations and its application to modulated fiber Bragg gratings,” in Asia Optical Fiber Communication and Optoelectronic Exposition & Conference (AOE) CD-ROM (Optical Society of America, 2008), paper SaE5.

Jin, G. L.

G. J. Liu, Q. Li, G. L. Jin, and B. M. Liang, “Transfer matrix method analysis of apodized grating couplers,” Opt. Commun. 235, 319-324 (2004).
[CrossRef]

Kang, Y.

Kiang, Y.-W.

Kogelnik, H.

H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, 1975), Vol. 7, pp. 13-81.
[CrossRef]

Koshiba, M.

T. Lin, Y. Furuumi, M. Imai, Y. Tsuji, and M. Koshiba, “Design theory and experiment of LP01-LP11 mode converter utilizing fused taper fiber coupler,” Trans. IEICE Electron. J82-C-I, 587-595 (1999), in Japanese.

Li, Q.

G. J. Liu, Q. Li, G. L. Jin, and B. M. Liang, “Transfer matrix method analysis of apodized grating couplers,” Opt. Commun. 235, 319-324 (2004).
[CrossRef]

Liang, B. M.

G. J. Liu, Q. Li, G. L. Jin, and B. M. Liang, “Transfer matrix method analysis of apodized grating couplers,” Opt. Commun. 235, 319-324 (2004).
[CrossRef]

Lin, C.-N.

Lin, T.

T. Lin, Y. Furuumi, M. Imai, Y. Tsuji, and M. Koshiba, “Design theory and experiment of LP01-LP11 mode converter utilizing fused taper fiber coupler,” Trans. IEICE Electron. J82-C-I, 587-595 (1999), in Japanese.

Liu, G. J.

G. J. Liu, Q. Li, G. L. Jin, and B. M. Liang, “Transfer matrix method analysis of apodized grating couplers,” Opt. Commun. 235, 319-324 (2004).
[CrossRef]

Liu, I.-M.

Liu, W. F.

D. W. Hang, W. F. Liu, C. W. Wu, and C. C. Yang, “Reflectivity-tunable fiber Bragg grating reflectors,” IEEE Photon. Technol. 12, 176-178 (2000).
[CrossRef]

W. F. Liu, P. St. J. Russell, and L. Dong, “100% efficient narrow-band acoustooptic tunable reflector using fiber Bragg grating,” J. Lightwave Technol. 16, 2006-2009 (1998).
[CrossRef]

Liu, W.-F.

Luo, Z.

Meltz, G.

K. O. Hill and G. Meltz, “Fiber Bragg grating technology: Fundamentals and overview,” J. Lightwave Technol. 15, 1263-1276 (1997).
[CrossRef]

Morishita, K.

K. Morishita, “Numerical analysis of pulse broadening in graded index optical fibers,” IEEE Trans. Microwave Theory Tech. 29, 348-352 (1981).
[CrossRef]

Morris, G. M.

Nagai, S.

F. Abrishamian, S. Nagai, S. Sato, and M. Imai, “Design theory and experiment of acousto-optic tunable filter by use of flexural waves applied to thin optical fiber,” Opt. Quant. Electron. 40, 665-676 (2008).
[CrossRef]

Nakai, Y.

F. Abrishamian, Y. Nakai, S. Sato, and M. Imai, “An efficient approach for calculating the reflection and transmission spectra of fiber Bragg gratings with acoustically induced microbending,” Opt. Fiber Technol. 13, 32-38 (2007).
[CrossRef]

Norton, S. M.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++: The Art of Scientific Computing (Cambridge University Press, 2002), 2nd ed., pp. 130-133.
[CrossRef]

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University Press, 2002), pp. 710-722.

Russell, P. St. J.

P. St. J. Russell and W.-F. Liu, “Acousto optic supperlattice modulation in fiber Bragg gratings,” J. Opt. Soc. Am. A 17, 1421-1429 (2000).
[CrossRef]

W. F. Liu, P. St. J. Russell, and L. Dong, “100% efficient narrow-band acoustooptic tunable reflector using fiber Bragg grating,” J. Lightwave Technol. 16, 2006-2009 (1998).
[CrossRef]

T. A. Birks, P. St. J. Russell, and D. O. Culverhouse, “The acousto-optic effect in single-mode fiber tapers and couplers,” J. Lightwave Technol. 14, 2519-2529 (1996).
[CrossRef]

T. A. Birks, P. St. J. Russell, and D. O. Culverhouse, “The acousto-optic effect in single-mode fiber tapers and couplers,” J. Lightwave Technol. 14, 2519-2529 (1996).
[CrossRef]

Sakuda, K.

Sato, S.

F. Abrishamian, S. Nagai, S. Sato, and M. Imai, “Design theory and experiment of acousto-optic tunable filter by use of flexural waves applied to thin optical fiber,” Opt. Quant. Electron. 40, 665-676 (2008).
[CrossRef]

F. Abrishamian, Y. Nakai, S. Sato, and M. Imai, “An efficient approach for calculating the reflection and transmission spectra of fiber Bragg gratings with acoustically induced microbending,” Opt. Fiber Technol. 13, 32-38 (2007).
[CrossRef]

F. Abrishamian, S. Sato, and M. Imai, “Numerical analysis of multimode-coupled equations and its application to modulated fiber Bragg gratings,” in Asia Optical Fiber Communication and Optoelectronic Exposition & Conference (AOE) CD-ROM (Optical Society of America, 2008), paper SaE5.

Sun, N.-H.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University Press, 2002), pp. 710-722.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++: The Art of Scientific Computing (Cambridge University Press, 2002), 2nd ed., pp. 130-133.
[CrossRef]

Tsuji, Y.

T. Lin, Y. Furuumi, M. Imai, Y. Tsuji, and M. Koshiba, “Design theory and experiment of LP01-LP11 mode converter utilizing fused taper fiber coupler,” Trans. IEICE Electron. J82-C-I, 587-595 (1999), in Japanese.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University Press, 2002), pp. 710-722.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++: The Art of Scientific Computing (Cambridge University Press, 2002), 2nd ed., pp. 130-133.
[CrossRef]

Weller-Brophy, L. A.

Wu, C. W.

D. W. Hang, W. F. Liu, C. W. Wu, and C. C. Yang, “Reflectivity-tunable fiber Bragg grating reflectors,” IEEE Photon. Technol. 12, 176-178 (2000).
[CrossRef]

Xu, H.

Yamada, M.

Yang, C. C.

D. W. Hang, W. F. Liu, C. W. Wu, and C. C. Yang, “Reflectivity-tunable fiber Bragg grating reflectors,” IEEE Photon. Technol. 12, 176-178 (2000).
[CrossRef]

W.-F. Liu, I.-M. Liu, L.-W. Chung, D.-W. Huang, and C. C. Yang, “Acoustic-induced switching of the reflection wavelength in a fiber Bragg grating,” Opt. Lett. 25, 1319-1321(2000).
[CrossRef]

Yang, C.-C.

Ye, C.

Appl. Opt. (2)

IEEE Photon. Technol. (1)

D. W. Hang, W. F. Liu, C. W. Wu, and C. C. Yang, “Reflectivity-tunable fiber Bragg grating reflectors,” IEEE Photon. Technol. 12, 176-178 (2000).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

K. Morishita, “Numerical analysis of pulse broadening in graded index optical fibers,” IEEE Trans. Microwave Theory Tech. 29, 348-352 (1981).
[CrossRef]

J. Lightwave Technol. (6)

T. A. Birks, P. St. J. Russell, and D. O. Culverhouse, “The acousto-optic effect in single-mode fiber tapers and couplers,” J. Lightwave Technol. 14, 2519-2529 (1996).
[CrossRef]

W. F. Liu, P. St. J. Russell, and L. Dong, “100% efficient narrow-band acoustooptic tunable reflector using fiber Bragg grating,” J. Lightwave Technol. 16, 2006-2009 (1998).
[CrossRef]

K. O. Hill and G. Meltz, “Fiber Bragg grating technology: Fundamentals and overview,” J. Lightwave Technol. 15, 1263-1276 (1997).
[CrossRef]

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277-1294 (1997).
[CrossRef]

T. A. Birks, P. St. J. Russell, and D. O. Culverhouse, “The acousto-optic effect in single-mode fiber tapers and couplers,” J. Lightwave Technol. 14, 2519-2529 (1996).
[CrossRef]

N.-H. Sun, C.-C. Chou, M.-J. Chang, C.-N. Lin, C.-C. Yang, Y.-W. Kiang, and W.-F. Liu, “Analysis of phase-matching conditions in flexural-wave modulated fiber Bragg grating,” J. Lightwave Technol. 20, 311-315 (2002).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

G. J. Liu, Q. Li, G. L. Jin, and B. M. Liang, “Transfer matrix method analysis of apodized grating couplers,” Opt. Commun. 235, 319-324 (2004).
[CrossRef]

Opt. Fiber Technol. (1)

F. Abrishamian, Y. Nakai, S. Sato, and M. Imai, “An efficient approach for calculating the reflection and transmission spectra of fiber Bragg gratings with acoustically induced microbending,” Opt. Fiber Technol. 13, 32-38 (2007).
[CrossRef]

Opt. Lett. (1)

Opt. Quant. Electron. (1)

F. Abrishamian, S. Nagai, S. Sato, and M. Imai, “Design theory and experiment of acousto-optic tunable filter by use of flexural waves applied to thin optical fiber,” Opt. Quant. Electron. 40, 665-676 (2008).
[CrossRef]

Trans. IEICE Electron. (1)

T. Lin, Y. Furuumi, M. Imai, Y. Tsuji, and M. Koshiba, “Design theory and experiment of LP01-LP11 mode converter utilizing fused taper fiber coupler,” Trans. IEICE Electron. J82-C-I, 587-595 (1999), in Japanese.

Other (5)

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++: The Art of Scientific Computing (Cambridge University Press, 2002), 2nd ed., pp. 130-133.
[CrossRef]

F. Abrishamian, S. Sato, and M. Imai, “Numerical analysis of multimode-coupled equations and its application to modulated fiber Bragg gratings,” in Asia Optical Fiber Communication and Optoelectronic Exposition & Conference (AOE) CD-ROM (Optical Society of America, 2008), paper SaE5.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University Press, 2002), pp. 710-722.

R. Helan, “Comparison of methods for fiber Bragg gratings simulation,” in 29th International Spring Seminar on Electronics Technology ISSE 2006 (2006), pp. 161-166.
[CrossRef]

H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, 1975), Vol. 7, pp. 13-81.
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Schematic of a FBG modulated by acoustically induced LPG.

Fig. 2
Fig. 2

Dependence of the calculated maximum spectra of the propagating core mode at λ B , λ S 1 , and λ S 2 on the number of intervals for n = 10 100 in the case of κ S 1 L = 0.9 and κ S 2 L = 0.3 .

Fig. 3
Fig. 3

(a) Calculated reflection spectra of the propagating core mode through the acoustically modulated FBG for three different values of n = 30 , 50, and 100. (b) Calculated reflection spectra for n = 200 obtained by the present modeling technique. (c) Experimentally observed reflection spectra of core mode reported by Yang’s group [17].

Fig. 4
Fig. 4

Calculated reflection spectra using the improved technique of dividing the interval into n = 100 for (a)  κ S 1 L = 1.2 and κ S 2 L = 0.9 , (b)  κ S 1 L = 1.1 and κ S 2 L = 0.5 .

Equations (23)

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

d y 1 d z = p 11 ( z ) y 1 ( z ) + p 12 ( z ) y 2 ( z ) + ... + p 1 N ( z ) y N ( z ) , d y 2 d z = p 21 ( z ) y 1 ( z ) + p 22 ( z ) y 2 ( z ) + ... + p 2 N ( z ) y N ( z ) , d y N d z = p N 1 ( z ) y 1 ( z ) + p N 2 ( z ) y 2 ( z ) + ... + p N N ( z ) y N ( z ) ,
( y 1 ( z ) y 2 ( z ) : y N ( z ) ) = ( y 1 ( z = 0 ) y 2 ( z = 0 ) : y N ( z = 0 ) ) . exp [ P N × N ( z ) d z ] .
R N × N = exp ( 0 z = L P ( z ) d z ) = V N × N 1 . ( exp ( λ 1 ) 0 .. 0 0 exp ( λ 2 ) .. 0 : : : : 0 0 0 0 exp ( λ N ) ) . V N × N ,
n 2 n eff L λ B ,
[ A co ( z ) B co ( z ) A cl 1 ( z ) B cl 1 ( z ) A cl 2 ( z ) B cl 2 ( z ) ] = [ A co ( 0 ) B co ( 0 ) A cl 1 ( 0 ) B cl 1 ( 0 ) A cl 2 ( 0 ) B cl 2 ( 0 ) ] . exp ( 0 z P ( z ) d z ) ,
P 6 × 6 = [ { p 11 = 0 , p 12 = i κ co exp ( i 2 δ co z ) , p 13 = i κ S 1 exp ( i 2 δ S 1 z ) , p 14 = i κ cl 1 exp ( i 2 δ cl 1 z ) , p 15 = i κ S 2 exp ( i 2 δ S 2 z ) , p 16 = i κ cl 2 exp ( i 2 δ cl 2 z ) } , { p 21 = i κ co exp ( i 2 δ co z ) , p 22 = 0 , p 23 = i κ cl 1 exp ( i 2 δ cl 1 z ) , p 24 = i κ S 1 exp ( i 2 δ S 1 z ) , p 25 = i κ cl 2 exp ( i 2 δ cl 2 z ) , p 26 = i κ S 2 exp ( i 2 δ S 2 z ) } , { p 31 = i κ S 1 exp ( i 2 δ S 1 z ) , p 32 = i κ cl 1 exp ( i 2 δ cl 1 z ) , p 33 = 0 , p 34 = 0 , p 35 = 0 , p 36 = 0 } , { p 41 = i κ cl 1 exp ( i 2 δ cl 1 z ) , p 42 = i κ S 1 exp ( i 2 δ S 1 z ) , p 43 = 0 , p 44 = 0 , p 45 = 0 , p 46 = 0 } , { p 51 = i κ S 2 exp ( i 2 δ S 2 z ) , p 52 = i κ cl 2 exp ( i 2 δ cl 2 z ) , p 53 = 0 , p 54 = 0 , p 55 = 0 , p 56 = 0 } , { p 61 = i κ cl 2 exp ( i 2 δ cl 2 z ) , p 62 = i κ S 2 exp ( i 2 δ S 2 z ) , p 63 = 0 , p 64 = 0 , p 65 = 0 , p 66 = 0 } ] .
δ co = 1 2 ( 2 β co 2 π Λ B ) ,
δ cl 1 = 1 2 ( β co + β cl 1 2 π Λ B ) ,
δ cl 2 = 1 2 ( β co + β cl 2 2 π Λ B ) ,
δ S 1 = 1 2 ( β co β cl 1 2 π Λ S 1 ) ,
δ S 2 = 1 2 ( β co β cl 2 2 π Λ S 2 ) ,
κ v μ ( z ) = k 2 × z 0 0 2 π d ϕ 0 r d r Δ ε ( r , z ) E v t ( r , ϕ ) E μ t * ( r , ϕ ) ,
κ clm = 1 z 0 λ ( Δ n core n core 2 a A norm - co A norm - clad ) × 1 u 1 2 u 2 2 ( u 1 J m ( a u 2 ) J m + 1 ( a u 1 ) u 2 J m ( a u 1 ) J m + 1 ( a u 2 ) ) ,
u 1 = k n core 2 n 01 2 ,
u 2 = k n core 2 n m 1 2 .
( 0 a r ( A norm - clad × J m ( u 2 r ) ) 2 d r ) + ( a b r ( B J m ( u 4 r ) + C N m ( u 4 r ) ) 2 d r ) + ( b r ( D K m ( u 5 r ) ) 2 d r ) = 1 ,
( 0 a r ( A norm - core × J m ( u 1 r ) ) 2 d r ) + ( a r ( A norm - core × J m ( u 1 a ) K m ( u 3 a ) K m ( u 3 r ) ) 2 d r ) = 1 ,
D = J m ( u 4 b ) K m ( u 5 b ) × B + N m ( u 4 b ) K m ( u 5 b ) × C ,
B = A norm - clad ( ( p 11 q 22 ) ( q 12 p 21 ) ( q 11 q 22 ) ( q 12 q 21 ) ) ,
C = A norm - clad ( ( p 11 q 21 ) ( q 11 p 21 ) ( q 12 q 21 ) ( q 11 q 22 ) ) ,
p 11 = J m ( a u 1 ) , p 12 = N m ( a u 1 ) , p 21 = u m ( J m 1 ( a u 1 ) J m + 1 ( a u 1 ) ) 2 β , p 22 = u 1 ( N m 1 ( a u 1 ) N m + 1 ( a u 1 ) ) 2 β ,
q 11 = J m ( a u 4 ) , q 12 = N m ( a u 4 ) , q 21 = u 4 ( J m 1 ( a u 4 ) J m + 1 ( a u 4 ) ) 2 β , q 22 = u 4 ( N m 1 ( a u 4 ) N m + 1 ( a u 4 ) ) 2 β ,
β = k n m 1 , u 2 = k n 01 2 n clad 2 , u 3 = k n core 2 n m 1 2 , u 4 = k n clad 2 n m 1 2 , u 5 = k n m 1 2 n air 2 .

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