Abstract

We present a detailed study of spectral self-imaging phenomena, namely, integer and fractional Talbot effects, observed in the reflection response of chirped sampled fiber Bragg gratings (C-SFBGs). The basic condition for observing spectral self-imaging effects is first derived heuristically, and an intuitive interpretation of the problem based on the notion of multislit interference is also provided. We then present a rigorous analysis of the spectral self-imaging problem in C-SFBGs, including the formal derivation of the conditions for observing the different spectral Talbot effects. This analysis reveals the existence of new effects, in particular, inverse integer and fractional self-images, which are described here for the first time, to our knowledge. Moreover, we also show that the grating physical parameters need to satisfy additional conditions in order for one to be able to observe spectral self-imaging phenomena in C-SFBGs. We also evaluate the impact of deviations from these ideal conditions on the reflection spectrum of a real device. We confirm our theoretical predictions by using numerical simulations, and we report the first experimental observation of fractional spectral Talbot effects in C-SFBGs. Besides their intrinsic physical interest, the results presented constitute the basis for exploiting the spectral self-imaging effect for practical applications, e.g., to optimize the design of SFBGs for applications requiring periodic comb filters with low in-band dispersion.

© 2005 Optical Society of America

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  1. R. Kashyap, Fiber Bragg Gratings (Academic, 1999).
  2. V. Jayaraman, Z. M. Chuang, and L. A. Coldren, "Theory, design, and performance of extended tuning range semiconconductor lasers with sampled gratings," IEEE J. Quantum Electron. 29, 1824-1834 (1993)
    [CrossRef]
  3. B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, "Long periodic superstructure Bragg gratings in optical fibers," Electron. Lett. 30, 1620-1622 (1994).
    [CrossRef]
  4. F. Ouellette, P. A. Krug, T. Stephens, G. Dhosi, and B. Eggleton, "Broadband and WDM dispersion compensation using chirped sampled fiber Bragg gratings," Electron. Lett. 31, 899-901 (1995).
    [CrossRef]
  5. J. Chow, G. Town, B. Eggleton, M. Ibsen, K. Sugden, and I. Bennion, "Multiwavelength generation in an erbium-doped fiber laser using in-fiber comb filters," IEEE Photonics Technol. Lett. 8, 60-62 (1996).
    [CrossRef]
  6. M. Ibsen, M. K. Durkin, M. J. Cole, and R. I. Laming, "Sinc-sampled fiber Bragg gratings for identical multiple wavelength operation," IEEE Photonics Technol. Lett. 10, 842-844 (1998).
    [CrossRef]
  7. W. H. Loh, F. Q. Zhou, and J. J. Pan, "Sampled fiber grating based-dispersion slope compensator," IEEE Photonics Technol. Lett. 11, 1280-1282 (1999).
    [CrossRef]
  8. W. H. Loh, F. Q. Zhou, and J. J. Pan, "Novel designs for sampled grating-based multiplexers-demultiplexers," Opt. Lett. 24, 1457-1459 (1999).
    [CrossRef]
  9. J.-X. Cai, K.-M. Feng, A. E. Willner, V. Grubsky, D. S. Starodubov, and J. Feinberg, "Simultaneous tunable dispersion compensation of many WDM channels using a sampled nonlinearly chirped fiber Bragg grating," IEEE Photon. Technol. Lett. 11, 1455-1457 (1999).
    [CrossRef]
  10. X.-F. Chen, C.-C. Fan, T. Wu, and S.-Z. Xie, "Analytical expression of sampled Bragg gratings with chirp in the sampling period and its application in dispersion management design in a WDM system," IEEE Photon. Technol. Lett. 12, 1013-1015 (2000).
    [CrossRef]
  11. Y. Xie, S. Lee, Z. Pan, J.-X. Cai, A. E. Willner, V. Grubsky, D. S. Starodubov, E. Salik, and J. Feinberg, "Tunable compensation of the dispersion slope mismatch in dispersion-managed systems using a sampled nonlinear chirped FBG," IEEE Photon. Technol. Lett. 12, 1417-1419 (2000).
    [CrossRef]
  12. J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Dammann fiber Bragg gratings and phase-only sampling for high channel counts," IEEE Photon. Technol. Lett. 14, 1309-1311 (2002).
    [CrossRef]
  13. C. Wang, L. R. Chen, and P. W. E. Smith, "Analysis of chirped-sampled and sampled-chirped fiber Bragg gratings," Appl. Opt. 41, 1654-1660 (2002).
    [CrossRef] [PubMed]
  14. A. V. Buryak, K. Y. Kolossovski, and D. Y. Stepanov, "Optimization of refractive index sampling for multichannel fiber Bragg gratings," IEEE J. Quantum Electron. 39, 91-98 (2003).
    [CrossRef]
  15. P. Petropoulos, M. Ibsen, M. N. Zervas, and D. J. Richardson, "Generation of a 40-GHz pulse stream by pulse multiplication with a sampled fiber Bragg grating," Opt. Lett. 25, 521-523 (2000).
    [CrossRef]
  16. C. Wang, J. Azaña, and L. R. Chen, "Spectral Talbot phenomena in one-dimensional photonic bandgap structures," Opt. Lett. 15, 1590-1592 (2004).
    [CrossRef]
  17. J. Azaña and M. A. Muriel, "Temporal self-imaging effects: theory and application for multiplying pulse repetition rates," IEEE J. Sel. Top. Quantum Electron. 7, 728-744 (2001).
    [CrossRef]
  18. B. H. Lee and U.-C. Paek, "Multislit interpretation of cascaded fiber gratings", IEEE J. Quantum Electron. 20, 1750-1761 (2002).
  19. S. Huang, M. Leblanc, M. M. Ohn, and R. M. Measures, "Bragg intragrating structural sensing," Appl. Opt. 34, 5003-5008 (1995).
    [CrossRef] [PubMed]
  20. H. Kogelnik, "Filter response of nonuniform almost-periodic structure," Bell Syst. Tech. J. 55, 109-126 (1976).
    [CrossRef]
  21. A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, 1987).
  22. C. Wang, J. Azaña, and L. R. Chen, "Efficient technique for increasing the channel density in multiwavelength sampled fiber Bragg grating filters," IEEE Photon. Technol. Lett. 16, 1867-1869 (2004).
    [CrossRef]
  23. J. Albert, K. O. Hill, B. Malo, S. Theriault, F. Bilodeau, D. C. Johnson, and L. E. Erickson, "Apodization of the spectral response of fiber Bragg gratings using a phase mask with variable diffraction efficiency," Electron. Lett. 31, 222-223 (1995).
    [CrossRef]

2004

C. Wang, J. Azaña, and L. R. Chen, "Spectral Talbot phenomena in one-dimensional photonic bandgap structures," Opt. Lett. 15, 1590-1592 (2004).
[CrossRef]

C. Wang, J. Azaña, and L. R. Chen, "Efficient technique for increasing the channel density in multiwavelength sampled fiber Bragg grating filters," IEEE Photon. Technol. Lett. 16, 1867-1869 (2004).
[CrossRef]

2003

A. V. Buryak, K. Y. Kolossovski, and D. Y. Stepanov, "Optimization of refractive index sampling for multichannel fiber Bragg gratings," IEEE J. Quantum Electron. 39, 91-98 (2003).
[CrossRef]

2002

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Dammann fiber Bragg gratings and phase-only sampling for high channel counts," IEEE Photon. Technol. Lett. 14, 1309-1311 (2002).
[CrossRef]

B. H. Lee and U.-C. Paek, "Multislit interpretation of cascaded fiber gratings", IEEE J. Quantum Electron. 20, 1750-1761 (2002).

C. Wang, L. R. Chen, and P. W. E. Smith, "Analysis of chirped-sampled and sampled-chirped fiber Bragg gratings," Appl. Opt. 41, 1654-1660 (2002).
[CrossRef] [PubMed]

2001

J. Azaña and M. A. Muriel, "Temporal self-imaging effects: theory and application for multiplying pulse repetition rates," IEEE J. Sel. Top. Quantum Electron. 7, 728-744 (2001).
[CrossRef]

2000

X.-F. Chen, C.-C. Fan, T. Wu, and S.-Z. Xie, "Analytical expression of sampled Bragg gratings with chirp in the sampling period and its application in dispersion management design in a WDM system," IEEE Photon. Technol. Lett. 12, 1013-1015 (2000).
[CrossRef]

Y. Xie, S. Lee, Z. Pan, J.-X. Cai, A. E. Willner, V. Grubsky, D. S. Starodubov, E. Salik, and J. Feinberg, "Tunable compensation of the dispersion slope mismatch in dispersion-managed systems using a sampled nonlinear chirped FBG," IEEE Photon. Technol. Lett. 12, 1417-1419 (2000).
[CrossRef]

P. Petropoulos, M. Ibsen, M. N. Zervas, and D. J. Richardson, "Generation of a 40-GHz pulse stream by pulse multiplication with a sampled fiber Bragg grating," Opt. Lett. 25, 521-523 (2000).
[CrossRef]

1999

W. H. Loh, F. Q. Zhou, and J. J. Pan, "Novel designs for sampled grating-based multiplexers-demultiplexers," Opt. Lett. 24, 1457-1459 (1999).
[CrossRef]

W. H. Loh, F. Q. Zhou, and J. J. Pan, "Sampled fiber grating based-dispersion slope compensator," IEEE Photonics Technol. Lett. 11, 1280-1282 (1999).
[CrossRef]

J.-X. Cai, K.-M. Feng, A. E. Willner, V. Grubsky, D. S. Starodubov, and J. Feinberg, "Simultaneous tunable dispersion compensation of many WDM channels using a sampled nonlinearly chirped fiber Bragg grating," IEEE Photon. Technol. Lett. 11, 1455-1457 (1999).
[CrossRef]

1998

M. Ibsen, M. K. Durkin, M. J. Cole, and R. I. Laming, "Sinc-sampled fiber Bragg gratings for identical multiple wavelength operation," IEEE Photonics Technol. Lett. 10, 842-844 (1998).
[CrossRef]

1996

J. Chow, G. Town, B. Eggleton, M. Ibsen, K. Sugden, and I. Bennion, "Multiwavelength generation in an erbium-doped fiber laser using in-fiber comb filters," IEEE Photonics Technol. Lett. 8, 60-62 (1996).
[CrossRef]

1995

F. Ouellette, P. A. Krug, T. Stephens, G. Dhosi, and B. Eggleton, "Broadband and WDM dispersion compensation using chirped sampled fiber Bragg gratings," Electron. Lett. 31, 899-901 (1995).
[CrossRef]

J. Albert, K. O. Hill, B. Malo, S. Theriault, F. Bilodeau, D. C. Johnson, and L. E. Erickson, "Apodization of the spectral response of fiber Bragg gratings using a phase mask with variable diffraction efficiency," Electron. Lett. 31, 222-223 (1995).
[CrossRef]

S. Huang, M. Leblanc, M. M. Ohn, and R. M. Measures, "Bragg intragrating structural sensing," Appl. Opt. 34, 5003-5008 (1995).
[CrossRef] [PubMed]

1994

B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, "Long periodic superstructure Bragg gratings in optical fibers," Electron. Lett. 30, 1620-1622 (1994).
[CrossRef]

1993

V. Jayaraman, Z. M. Chuang, and L. A. Coldren, "Theory, design, and performance of extended tuning range semiconconductor lasers with sampled gratings," IEEE J. Quantum Electron. 29, 1824-1834 (1993)
[CrossRef]

1976

H. Kogelnik, "Filter response of nonuniform almost-periodic structure," Bell Syst. Tech. J. 55, 109-126 (1976).
[CrossRef]

Albert, J.

J. Albert, K. O. Hill, B. Malo, S. Theriault, F. Bilodeau, D. C. Johnson, and L. E. Erickson, "Apodization of the spectral response of fiber Bragg gratings using a phase mask with variable diffraction efficiency," Electron. Lett. 31, 222-223 (1995).
[CrossRef]

Azaña, J.

C. Wang, J. Azaña, and L. R. Chen, "Spectral Talbot phenomena in one-dimensional photonic bandgap structures," Opt. Lett. 15, 1590-1592 (2004).
[CrossRef]

C. Wang, J. Azaña, and L. R. Chen, "Efficient technique for increasing the channel density in multiwavelength sampled fiber Bragg grating filters," IEEE Photon. Technol. Lett. 16, 1867-1869 (2004).
[CrossRef]

J. Azaña and M. A. Muriel, "Temporal self-imaging effects: theory and application for multiplying pulse repetition rates," IEEE J. Sel. Top. Quantum Electron. 7, 728-744 (2001).
[CrossRef]

Bennion, I.

J. Chow, G. Town, B. Eggleton, M. Ibsen, K. Sugden, and I. Bennion, "Multiwavelength generation in an erbium-doped fiber laser using in-fiber comb filters," IEEE Photonics Technol. Lett. 8, 60-62 (1996).
[CrossRef]

Bilodeau, F.

J. Albert, K. O. Hill, B. Malo, S. Theriault, F. Bilodeau, D. C. Johnson, and L. E. Erickson, "Apodization of the spectral response of fiber Bragg gratings using a phase mask with variable diffraction efficiency," Electron. Lett. 31, 222-223 (1995).
[CrossRef]

Buryak, A. V.

A. V. Buryak, K. Y. Kolossovski, and D. Y. Stepanov, "Optimization of refractive index sampling for multichannel fiber Bragg gratings," IEEE J. Quantum Electron. 39, 91-98 (2003).
[CrossRef]

Cai, J.-X.

Y. Xie, S. Lee, Z. Pan, J.-X. Cai, A. E. Willner, V. Grubsky, D. S. Starodubov, E. Salik, and J. Feinberg, "Tunable compensation of the dispersion slope mismatch in dispersion-managed systems using a sampled nonlinear chirped FBG," IEEE Photon. Technol. Lett. 12, 1417-1419 (2000).
[CrossRef]

J.-X. Cai, K.-M. Feng, A. E. Willner, V. Grubsky, D. S. Starodubov, and J. Feinberg, "Simultaneous tunable dispersion compensation of many WDM channels using a sampled nonlinearly chirped fiber Bragg grating," IEEE Photon. Technol. Lett. 11, 1455-1457 (1999).
[CrossRef]

Chen, L. R.

C. Wang, J. Azaña, and L. R. Chen, "Spectral Talbot phenomena in one-dimensional photonic bandgap structures," Opt. Lett. 15, 1590-1592 (2004).
[CrossRef]

C. Wang, J. Azaña, and L. R. Chen, "Efficient technique for increasing the channel density in multiwavelength sampled fiber Bragg grating filters," IEEE Photon. Technol. Lett. 16, 1867-1869 (2004).
[CrossRef]

C. Wang, L. R. Chen, and P. W. E. Smith, "Analysis of chirped-sampled and sampled-chirped fiber Bragg gratings," Appl. Opt. 41, 1654-1660 (2002).
[CrossRef] [PubMed]

Chen, X.-F.

X.-F. Chen, C.-C. Fan, T. Wu, and S.-Z. Xie, "Analytical expression of sampled Bragg gratings with chirp in the sampling period and its application in dispersion management design in a WDM system," IEEE Photon. Technol. Lett. 12, 1013-1015 (2000).
[CrossRef]

Chow, J.

J. Chow, G. Town, B. Eggleton, M. Ibsen, K. Sugden, and I. Bennion, "Multiwavelength generation in an erbium-doped fiber laser using in-fiber comb filters," IEEE Photonics Technol. Lett. 8, 60-62 (1996).
[CrossRef]

Chuang, Z. M.

V. Jayaraman, Z. M. Chuang, and L. A. Coldren, "Theory, design, and performance of extended tuning range semiconconductor lasers with sampled gratings," IEEE J. Quantum Electron. 29, 1824-1834 (1993)
[CrossRef]

Coldren, L. A.

V. Jayaraman, Z. M. Chuang, and L. A. Coldren, "Theory, design, and performance of extended tuning range semiconconductor lasers with sampled gratings," IEEE J. Quantum Electron. 29, 1824-1834 (1993)
[CrossRef]

Cole, M. J.

M. Ibsen, M. K. Durkin, M. J. Cole, and R. I. Laming, "Sinc-sampled fiber Bragg gratings for identical multiple wavelength operation," IEEE Photonics Technol. Lett. 10, 842-844 (1998).
[CrossRef]

Dhosi, G.

F. Ouellette, P. A. Krug, T. Stephens, G. Dhosi, and B. Eggleton, "Broadband and WDM dispersion compensation using chirped sampled fiber Bragg gratings," Electron. Lett. 31, 899-901 (1995).
[CrossRef]

Durkin, M. K.

M. Ibsen, M. K. Durkin, M. J. Cole, and R. I. Laming, "Sinc-sampled fiber Bragg gratings for identical multiple wavelength operation," IEEE Photonics Technol. Lett. 10, 842-844 (1998).
[CrossRef]

Eggleton, B.

J. Chow, G. Town, B. Eggleton, M. Ibsen, K. Sugden, and I. Bennion, "Multiwavelength generation in an erbium-doped fiber laser using in-fiber comb filters," IEEE Photonics Technol. Lett. 8, 60-62 (1996).
[CrossRef]

F. Ouellette, P. A. Krug, T. Stephens, G. Dhosi, and B. Eggleton, "Broadband and WDM dispersion compensation using chirped sampled fiber Bragg gratings," Electron. Lett. 31, 899-901 (1995).
[CrossRef]

Eggleton, B. J.

B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, "Long periodic superstructure Bragg gratings in optical fibers," Electron. Lett. 30, 1620-1622 (1994).
[CrossRef]

Erickson, L. E.

J. Albert, K. O. Hill, B. Malo, S. Theriault, F. Bilodeau, D. C. Johnson, and L. E. Erickson, "Apodization of the spectral response of fiber Bragg gratings using a phase mask with variable diffraction efficiency," Electron. Lett. 31, 222-223 (1995).
[CrossRef]

Fan, C.-C.

X.-F. Chen, C.-C. Fan, T. Wu, and S.-Z. Xie, "Analytical expression of sampled Bragg gratings with chirp in the sampling period and its application in dispersion management design in a WDM system," IEEE Photon. Technol. Lett. 12, 1013-1015 (2000).
[CrossRef]

Feinberg, J.

Y. Xie, S. Lee, Z. Pan, J.-X. Cai, A. E. Willner, V. Grubsky, D. S. Starodubov, E. Salik, and J. Feinberg, "Tunable compensation of the dispersion slope mismatch in dispersion-managed systems using a sampled nonlinear chirped FBG," IEEE Photon. Technol. Lett. 12, 1417-1419 (2000).
[CrossRef]

J.-X. Cai, K.-M. Feng, A. E. Willner, V. Grubsky, D. S. Starodubov, and J. Feinberg, "Simultaneous tunable dispersion compensation of many WDM channels using a sampled nonlinearly chirped fiber Bragg grating," IEEE Photon. Technol. Lett. 11, 1455-1457 (1999).
[CrossRef]

Feng, K.-M.

J.-X. Cai, K.-M. Feng, A. E. Willner, V. Grubsky, D. S. Starodubov, and J. Feinberg, "Simultaneous tunable dispersion compensation of many WDM channels using a sampled nonlinearly chirped fiber Bragg grating," IEEE Photon. Technol. Lett. 11, 1455-1457 (1999).
[CrossRef]

Grubsky, V.

Y. Xie, S. Lee, Z. Pan, J.-X. Cai, A. E. Willner, V. Grubsky, D. S. Starodubov, E. Salik, and J. Feinberg, "Tunable compensation of the dispersion slope mismatch in dispersion-managed systems using a sampled nonlinear chirped FBG," IEEE Photon. Technol. Lett. 12, 1417-1419 (2000).
[CrossRef]

J.-X. Cai, K.-M. Feng, A. E. Willner, V. Grubsky, D. S. Starodubov, and J. Feinberg, "Simultaneous tunable dispersion compensation of many WDM channels using a sampled nonlinearly chirped fiber Bragg grating," IEEE Photon. Technol. Lett. 11, 1455-1457 (1999).
[CrossRef]

Hill, K. O.

J. Albert, K. O. Hill, B. Malo, S. Theriault, F. Bilodeau, D. C. Johnson, and L. E. Erickson, "Apodization of the spectral response of fiber Bragg gratings using a phase mask with variable diffraction efficiency," Electron. Lett. 31, 222-223 (1995).
[CrossRef]

Huang, S.

Ibsen, M.

P. Petropoulos, M. Ibsen, M. N. Zervas, and D. J. Richardson, "Generation of a 40-GHz pulse stream by pulse multiplication with a sampled fiber Bragg grating," Opt. Lett. 25, 521-523 (2000).
[CrossRef]

M. Ibsen, M. K. Durkin, M. J. Cole, and R. I. Laming, "Sinc-sampled fiber Bragg gratings for identical multiple wavelength operation," IEEE Photonics Technol. Lett. 10, 842-844 (1998).
[CrossRef]

J. Chow, G. Town, B. Eggleton, M. Ibsen, K. Sugden, and I. Bennion, "Multiwavelength generation in an erbium-doped fiber laser using in-fiber comb filters," IEEE Photonics Technol. Lett. 8, 60-62 (1996).
[CrossRef]

Jayaraman, V.

V. Jayaraman, Z. M. Chuang, and L. A. Coldren, "Theory, design, and performance of extended tuning range semiconconductor lasers with sampled gratings," IEEE J. Quantum Electron. 29, 1824-1834 (1993)
[CrossRef]

Johnson, D. C.

J. Albert, K. O. Hill, B. Malo, S. Theriault, F. Bilodeau, D. C. Johnson, and L. E. Erickson, "Apodization of the spectral response of fiber Bragg gratings using a phase mask with variable diffraction efficiency," Electron. Lett. 31, 222-223 (1995).
[CrossRef]

Kashyap, R.

R. Kashyap, Fiber Bragg Gratings (Academic, 1999).

Kogelnik, H.

H. Kogelnik, "Filter response of nonuniform almost-periodic structure," Bell Syst. Tech. J. 55, 109-126 (1976).
[CrossRef]

Kolossovski, K. Y.

A. V. Buryak, K. Y. Kolossovski, and D. Y. Stepanov, "Optimization of refractive index sampling for multichannel fiber Bragg gratings," IEEE J. Quantum Electron. 39, 91-98 (2003).
[CrossRef]

Krug, P. A.

F. Ouellette, P. A. Krug, T. Stephens, G. Dhosi, and B. Eggleton, "Broadband and WDM dispersion compensation using chirped sampled fiber Bragg gratings," Electron. Lett. 31, 899-901 (1995).
[CrossRef]

B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, "Long periodic superstructure Bragg gratings in optical fibers," Electron. Lett. 30, 1620-1622 (1994).
[CrossRef]

Laming, R. I.

M. Ibsen, M. K. Durkin, M. J. Cole, and R. I. Laming, "Sinc-sampled fiber Bragg gratings for identical multiple wavelength operation," IEEE Photonics Technol. Lett. 10, 842-844 (1998).
[CrossRef]

Leblanc, M.

Lee, B. H.

B. H. Lee and U.-C. Paek, "Multislit interpretation of cascaded fiber gratings", IEEE J. Quantum Electron. 20, 1750-1761 (2002).

Lee, S.

Y. Xie, S. Lee, Z. Pan, J.-X. Cai, A. E. Willner, V. Grubsky, D. S. Starodubov, E. Salik, and J. Feinberg, "Tunable compensation of the dispersion slope mismatch in dispersion-managed systems using a sampled nonlinear chirped FBG," IEEE Photon. Technol. Lett. 12, 1417-1419 (2000).
[CrossRef]

Li, H.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Dammann fiber Bragg gratings and phase-only sampling for high channel counts," IEEE Photon. Technol. Lett. 14, 1309-1311 (2002).
[CrossRef]

Li, Y.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Dammann fiber Bragg gratings and phase-only sampling for high channel counts," IEEE Photon. Technol. Lett. 14, 1309-1311 (2002).
[CrossRef]

Loh, W. H.

W. H. Loh, F. Q. Zhou, and J. J. Pan, "Novel designs for sampled grating-based multiplexers-demultiplexers," Opt. Lett. 24, 1457-1459 (1999).
[CrossRef]

W. H. Loh, F. Q. Zhou, and J. J. Pan, "Sampled fiber grating based-dispersion slope compensator," IEEE Photonics Technol. Lett. 11, 1280-1282 (1999).
[CrossRef]

Malo, B.

J. Albert, K. O. Hill, B. Malo, S. Theriault, F. Bilodeau, D. C. Johnson, and L. E. Erickson, "Apodization of the spectral response of fiber Bragg gratings using a phase mask with variable diffraction efficiency," Electron. Lett. 31, 222-223 (1995).
[CrossRef]

Measures, R. M.

Muriel, M. A.

J. Azaña and M. A. Muriel, "Temporal self-imaging effects: theory and application for multiplying pulse repetition rates," IEEE J. Sel. Top. Quantum Electron. 7, 728-744 (2001).
[CrossRef]

Ohn, M. M.

Ouellette, F.

F. Ouellette, P. A. Krug, T. Stephens, G. Dhosi, and B. Eggleton, "Broadband and WDM dispersion compensation using chirped sampled fiber Bragg gratings," Electron. Lett. 31, 899-901 (1995).
[CrossRef]

B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, "Long periodic superstructure Bragg gratings in optical fibers," Electron. Lett. 30, 1620-1622 (1994).
[CrossRef]

Paek, U.-C.

B. H. Lee and U.-C. Paek, "Multislit interpretation of cascaded fiber gratings", IEEE J. Quantum Electron. 20, 1750-1761 (2002).

Pan, J. J.

W. H. Loh, F. Q. Zhou, and J. J. Pan, "Sampled fiber grating based-dispersion slope compensator," IEEE Photonics Technol. Lett. 11, 1280-1282 (1999).
[CrossRef]

W. H. Loh, F. Q. Zhou, and J. J. Pan, "Novel designs for sampled grating-based multiplexers-demultiplexers," Opt. Lett. 24, 1457-1459 (1999).
[CrossRef]

Pan, Z.

Y. Xie, S. Lee, Z. Pan, J.-X. Cai, A. E. Willner, V. Grubsky, D. S. Starodubov, E. Salik, and J. Feinberg, "Tunable compensation of the dispersion slope mismatch in dispersion-managed systems using a sampled nonlinear chirped FBG," IEEE Photon. Technol. Lett. 12, 1417-1419 (2000).
[CrossRef]

Papoulis, A.

A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, 1987).

Petropoulos, P.

Poladian, L.

B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, "Long periodic superstructure Bragg gratings in optical fibers," Electron. Lett. 30, 1620-1622 (1994).
[CrossRef]

Popelek, J.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Dammann fiber Bragg gratings and phase-only sampling for high channel counts," IEEE Photon. Technol. Lett. 14, 1309-1311 (2002).
[CrossRef]

Richardson, D. J.

Rothenberg, J. E.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Dammann fiber Bragg gratings and phase-only sampling for high channel counts," IEEE Photon. Technol. Lett. 14, 1309-1311 (2002).
[CrossRef]

Salik, E.

Y. Xie, S. Lee, Z. Pan, J.-X. Cai, A. E. Willner, V. Grubsky, D. S. Starodubov, E. Salik, and J. Feinberg, "Tunable compensation of the dispersion slope mismatch in dispersion-managed systems using a sampled nonlinear chirped FBG," IEEE Photon. Technol. Lett. 12, 1417-1419 (2000).
[CrossRef]

Sheng, Y.

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[CrossRef]

Smith, P. W.

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Y. Xie, S. Lee, Z. Pan, J.-X. Cai, A. E. Willner, V. Grubsky, D. S. Starodubov, E. Salik, and J. Feinberg, "Tunable compensation of the dispersion slope mismatch in dispersion-managed systems using a sampled nonlinear chirped FBG," IEEE Photon. Technol. Lett. 12, 1417-1419 (2000).
[CrossRef]

J.-X. Cai, K.-M. Feng, A. E. Willner, V. Grubsky, D. S. Starodubov, and J. Feinberg, "Simultaneous tunable dispersion compensation of many WDM channels using a sampled nonlinearly chirped fiber Bragg grating," IEEE Photon. Technol. Lett. 11, 1455-1457 (1999).
[CrossRef]

Stepanov, D. Y.

A. V. Buryak, K. Y. Kolossovski, and D. Y. Stepanov, "Optimization of refractive index sampling for multichannel fiber Bragg gratings," IEEE J. Quantum Electron. 39, 91-98 (2003).
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J. Albert, K. O. Hill, B. Malo, S. Theriault, F. Bilodeau, D. C. Johnson, and L. E. Erickson, "Apodization of the spectral response of fiber Bragg gratings using a phase mask with variable diffraction efficiency," Electron. Lett. 31, 222-223 (1995).
[CrossRef]

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J. Chow, G. Town, B. Eggleton, M. Ibsen, K. Sugden, and I. Bennion, "Multiwavelength generation in an erbium-doped fiber laser using in-fiber comb filters," IEEE Photonics Technol. Lett. 8, 60-62 (1996).
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C. Wang, J. Azaña, and L. R. Chen, "Spectral Talbot phenomena in one-dimensional photonic bandgap structures," Opt. Lett. 15, 1590-1592 (2004).
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C. Wang, J. Azaña, and L. R. Chen, "Efficient technique for increasing the channel density in multiwavelength sampled fiber Bragg grating filters," IEEE Photon. Technol. Lett. 16, 1867-1869 (2004).
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C. Wang, L. R. Chen, and P. W. E. Smith, "Analysis of chirped-sampled and sampled-chirped fiber Bragg gratings," Appl. Opt. 41, 1654-1660 (2002).
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J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Dammann fiber Bragg gratings and phase-only sampling for high channel counts," IEEE Photon. Technol. Lett. 14, 1309-1311 (2002).
[CrossRef]

Wilcox, R. B.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Dammann fiber Bragg gratings and phase-only sampling for high channel counts," IEEE Photon. Technol. Lett. 14, 1309-1311 (2002).
[CrossRef]

Willner, A. E.

Y. Xie, S. Lee, Z. Pan, J.-X. Cai, A. E. Willner, V. Grubsky, D. S. Starodubov, E. Salik, and J. Feinberg, "Tunable compensation of the dispersion slope mismatch in dispersion-managed systems using a sampled nonlinear chirped FBG," IEEE Photon. Technol. Lett. 12, 1417-1419 (2000).
[CrossRef]

J.-X. Cai, K.-M. Feng, A. E. Willner, V. Grubsky, D. S. Starodubov, and J. Feinberg, "Simultaneous tunable dispersion compensation of many WDM channels using a sampled nonlinearly chirped fiber Bragg grating," IEEE Photon. Technol. Lett. 11, 1455-1457 (1999).
[CrossRef]

Wu, T.

X.-F. Chen, C.-C. Fan, T. Wu, and S.-Z. Xie, "Analytical expression of sampled Bragg gratings with chirp in the sampling period and its application in dispersion management design in a WDM system," IEEE Photon. Technol. Lett. 12, 1013-1015 (2000).
[CrossRef]

Xie, S.-Z.

X.-F. Chen, C.-C. Fan, T. Wu, and S.-Z. Xie, "Analytical expression of sampled Bragg gratings with chirp in the sampling period and its application in dispersion management design in a WDM system," IEEE Photon. Technol. Lett. 12, 1013-1015 (2000).
[CrossRef]

Xie, Y.

Y. Xie, S. Lee, Z. Pan, J.-X. Cai, A. E. Willner, V. Grubsky, D. S. Starodubov, E. Salik, and J. Feinberg, "Tunable compensation of the dispersion slope mismatch in dispersion-managed systems using a sampled nonlinear chirped FBG," IEEE Photon. Technol. Lett. 12, 1417-1419 (2000).
[CrossRef]

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Zhou, F. Q.

W. H. Loh, F. Q. Zhou, and J. J. Pan, "Sampled fiber grating based-dispersion slope compensator," IEEE Photonics Technol. Lett. 11, 1280-1282 (1999).
[CrossRef]

W. H. Loh, F. Q. Zhou, and J. J. Pan, "Novel designs for sampled grating-based multiplexers-demultiplexers," Opt. Lett. 24, 1457-1459 (1999).
[CrossRef]

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J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Dammann fiber Bragg gratings and phase-only sampling for high channel counts," IEEE Photon. Technol. Lett. 14, 1309-1311 (2002).
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[CrossRef]

F. Ouellette, P. A. Krug, T. Stephens, G. Dhosi, and B. Eggleton, "Broadband and WDM dispersion compensation using chirped sampled fiber Bragg gratings," Electron. Lett. 31, 899-901 (1995).
[CrossRef]

J. Albert, K. O. Hill, B. Malo, S. Theriault, F. Bilodeau, D. C. Johnson, and L. E. Erickson, "Apodization of the spectral response of fiber Bragg gratings using a phase mask with variable diffraction efficiency," Electron. Lett. 31, 222-223 (1995).
[CrossRef]

IEEE J. Quantum Electron.

A. V. Buryak, K. Y. Kolossovski, and D. Y. Stepanov, "Optimization of refractive index sampling for multichannel fiber Bragg gratings," IEEE J. Quantum Electron. 39, 91-98 (2003).
[CrossRef]

V. Jayaraman, Z. M. Chuang, and L. A. Coldren, "Theory, design, and performance of extended tuning range semiconconductor lasers with sampled gratings," IEEE J. Quantum Electron. 29, 1824-1834 (1993)
[CrossRef]

B. H. Lee and U.-C. Paek, "Multislit interpretation of cascaded fiber gratings", IEEE J. Quantum Electron. 20, 1750-1761 (2002).

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J. Azaña and M. A. Muriel, "Temporal self-imaging effects: theory and application for multiplying pulse repetition rates," IEEE J. Sel. Top. Quantum Electron. 7, 728-744 (2001).
[CrossRef]

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J.-X. Cai, K.-M. Feng, A. E. Willner, V. Grubsky, D. S. Starodubov, and J. Feinberg, "Simultaneous tunable dispersion compensation of many WDM channels using a sampled nonlinearly chirped fiber Bragg grating," IEEE Photon. Technol. Lett. 11, 1455-1457 (1999).
[CrossRef]

X.-F. Chen, C.-C. Fan, T. Wu, and S.-Z. Xie, "Analytical expression of sampled Bragg gratings with chirp in the sampling period and its application in dispersion management design in a WDM system," IEEE Photon. Technol. Lett. 12, 1013-1015 (2000).
[CrossRef]

Y. Xie, S. Lee, Z. Pan, J.-X. Cai, A. E. Willner, V. Grubsky, D. S. Starodubov, E. Salik, and J. Feinberg, "Tunable compensation of the dispersion slope mismatch in dispersion-managed systems using a sampled nonlinear chirped FBG," IEEE Photon. Technol. Lett. 12, 1417-1419 (2000).
[CrossRef]

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Dammann fiber Bragg gratings and phase-only sampling for high channel counts," IEEE Photon. Technol. Lett. 14, 1309-1311 (2002).
[CrossRef]

C. Wang, J. Azaña, and L. R. Chen, "Efficient technique for increasing the channel density in multiwavelength sampled fiber Bragg grating filters," IEEE Photon. Technol. Lett. 16, 1867-1869 (2004).
[CrossRef]

IEEE Photonics Technol. Lett.

J. Chow, G. Town, B. Eggleton, M. Ibsen, K. Sugden, and I. Bennion, "Multiwavelength generation in an erbium-doped fiber laser using in-fiber comb filters," IEEE Photonics Technol. Lett. 8, 60-62 (1996).
[CrossRef]

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[CrossRef]

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[CrossRef]

Opt. Lett.

Other

R. Kashyap, Fiber Bragg Gratings (Academic, 1999).

A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, 1987).

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Figures (10)

Fig. 1
Fig. 1

Illustration of SFBG.

Fig. 2
Fig. 2

Decomposition of (a) a U-SFBG into a sequence of single uniform gratings and (b) a C-SFBG into a sequence of single chirped gratings.

Fig. 3
Fig. 3

Reflection coefficient (including amplitude and phase) from the first three gratings of a U-SFBG: (a) first grating, (b) second grating, and (c) third grating; see text for grating parameters.

Fig. 4
Fig. 4

Reflectivity from the U-SFBG in Fig. 3, calculated by using coherent summation of the sequence of decomposed gratings.

Fig. 5
Fig. 5

Amplitudes of reflection coefficients from a sequence of decomposed gratings (the first four gratings shown in the graph) from a C-SFBG, with the parameters given in the text (same parameters as those in Fig. 3 and C Λ 1.361822 × 10 7 ).

Fig. 6
Fig. 6

Result of coherent summation of reflection coefficients from a sequence of decomposed gratings for the C-SFBG in Fig. 5.

Fig. 7
Fig. 7

Representation of the signals involved in the problems of temporal and spectral self-imaging illustrating the time–frequency equivalence between the two problems. The labels corresponding to the temporal (SPECTRAL) self-imaging problem are in lower (UPPER) case letters.

Fig. 8
Fig. 8

Reflectivity (top) and group delay (bottom) corresponding to the reflection bands indicated by arrows for a C-SFBG with varying grating chirp coefficients (fixed to observe integer spectral self-imaging effects): (a) m = 1 , s = 0 (U-SFBG); (b) m = 1 , s = 1 ( C Λ = 2.723645 × 10 7 ) ; and (c) m = 1 , s = 2 ( C Λ = 5.4472895 × 10 7 ) . Note that (b) corresponds to the case of inverse integer spectral self-imaging, whereas (c) corresponds to the case of direct integer spectral self-imaging.

Fig. 9
Fig. 9

Reflectivity (top) and group delay (bottom) corresponding to the reflection bands indicated by arrows for a C-SFBG with varying grating chirp coefficients (fixed to observe fractional spectral self-imaging effects): (a) m = 3 , s = 1 ( C Λ = 0.907881 × 10 7 ) ; (b) m = 3 , s = 2 ( C Λ = 1.815763 × 10 7 ) ; and (c) m = 3 , s = 4 ( C Λ = 3.6315267 × 10 7 ) . Note that (a) corresponds to the case of inverse fractional spectral self-imaging, whereas (b) and (c) correspond to the case of direct fractional spectral self-imaging.

Fig. 10
Fig. 10

(a) Comparing the measured reflectivity of a U-SFBG and a C-SFBG having the same sampling parameters. (b) Measured group delay of the C-SFBG over the entire wavelength range and a zoom over a narrower span clearly show that the in-band group delay is approximately constant, though there is a group-delay variation from band to band (the overall group-delay response follows that of the seed grating). The grating parameters are given in the text.

Equations (35)

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n ( z ) = n av + Δ n peak f ( z ) cos [ 2 π Λ 0 z + φ ( z ) ] , L 2 z L 2 ,
f ( z ) = a ( z ) q = + g ( z q P ) ,
Κ ( z ) = Κ 0 + C Κ z ,
D ν = T 2 m ,
Δ λ λ B 2 ( 2 n av P ) ,
D ν , grating ( λ B c ) 2 1 C Λ ,
D ν , grating = m Δ ν 2 ,
C Λ 1 m Λ 0 2 P 2 .
r = i r i exp ( i ϕ i ) ,
δ λ = 2 n av C Λ P .
h ̑ ( t ) { f ( z ) exp [ j φ ( z ) ] } z = c 2 n av t ,
h ̑ ( t ) a ( t ) exp ( j ϕ t 2 t 2 ) q = + g ( t q T R ) ,
ϕ t = ( c 2 n av ) 2 C κ ,
T R = 2 n av c P .
H ̑ δ , uniform ( ω ) = I { h ̑ δ , uniform ( t ) } q = + A ( ω q ω R ) ,
h ̑ δ ( t ) a ( t ) exp ( j ϕ t 2 t 2 ) q = + δ ( t q T R ) = a ( t ) q = + exp ( j q 2 ϕ t 2 T R 2 ) δ ( t q T R ) .
ϕ t T R 2 = 2 π s m ,
H ̑ δ ( ω ) q = + exp ( j ϑ q ) A ( ω q ω R m γ ω R 2 m ) .
C Κ P 2 = 2 π s m
C Λ P 2 = s m Λ 0 2 .
H ̑ ( ω ) exp [ j ( 1 2 ϕ t ) ω 2 ] ( g ( t = ω ϕ t ) { exp [ j ( 1 2 ϕ t ) ω 2 ] q = + exp ( j ϑ q ) A ( ω q ω R m γ ω R 2 m ) } ) .
H ̑ δ ( ω ) q = + exp ( j ϑ q ) A ( ω q ω R m γ ω R 2 m )
P 1 m s P 2 L
N p P 1 m s P ,
H ̑ ( ω ) exp [ j ( 1 2 ϕ t ) ω 2 ] { A ( ω ) [ G ( ω ) q = + δ ( ω q ω R ) ] } = exp [ j ( 1 2 ϕ t ) ω 2 ] [ G ( ω ) q = + A ( ω q ω R ) ] ,
H ̑ ( ω ) exp [ j ( 1 2 ϕ t ) ω 2 ] [ G ( ω ) H ̑ δ , uniform ( ω ) ] .
H ̑ ( ω ) exp [ j ( 1 2 ϕ t ) ω 2 ] + G ( Ω ) H ̑ δ ( Ω ) exp [ j ( 1 2 ϕ t ) Ω 2 ] exp [ j ( 1 ϕ t ) ω Ω ] d Ω = exp [ j ( 1 2 ϕ t ) ω 2 ] { g ( t = ω ϕ t ) + H ̑ δ , uniform ( Ω ) exp [ j ( 1 2 ϕ t ) Ω 2 ] exp [ j ( 1 ϕ t ) ω Ω ] d Ω } .
+ H ̑ δ , uniform ( Ω ) exp [ j ( 1 2 ϕ t ) Ω 2 ] exp [ j ( 1 ϕ t ) ω Ω ] d Ω = exp [ j ( 1 2 ϕ t ) ω 2 ] + H ̑ δ , uniform ( Ω ) exp [ j ( 1 2 ϕ t ) ( ω Ω ) 2 ] d Ω = exp [ j ( 1 2 ϕ t ) ω 2 ] J { h ̑ δ , uniform ( t ) exp [ j ( ϕ t 2 ) t 2 ] } = exp [ j ( 1 2 ϕ t ) ω 2 ] J { h ̑ δ ( t ) } = exp [ j ( 1 2 ϕ t ) ω 2 ] H ̑ δ ( ω ) .
H ̑ ( ω ) exp [ j ( 1 2 ϕ t ) ω 2 ] ( g ( t = ω ϕ t ) { exp [ j ( 1 2 ϕ t ) ω 2 ] q = + exp ( j ϑ q ) A ( ω q ω R m γ ω R 2 m ) } ) .
h ̑ ( t ) exp [ j ( ϕ t 2 ) t 2 ] { G ( ω = ϕ t t ) q = + exp ( j ϑ q ) a ( t τ g , q ) exp [ j ω R ( q m + γ 2 m ) ( t τ g , q ) ] } ,
h ̑ ( t ) exp [ j ( ϕ t 2 ) t 2 ] q = + G ( ω = ϕ t τ g , q ) exp ( j ϑ q ) a ( t τ g , q ) exp [ j ω R ( q m + γ 2 m ) ( t τ g , q ) ] ,
H ̑ ( ω ) exp [ j ( 1 2 ϕ t ) ω 2 ] q = + G ( ω = ϕ t τ g , q ) exp [ j ( ω τ g , q ϑ q ) ] A ( ω q ω R m γ ω R 2 m ) = q = + G [ ω = ( q ω R m ) + ( γ ω R 2 m ) ] exp ( j ϑ q ) A ( ω q ω R m γ ω R 2 m ) = G ( ω ) q = + exp ( j ϑ q ) A ( ω q ω R m γ ω R 2 m ) = G ( ω ) H ̑ δ ( ω ) ,
T 1 ϕ t Δ ω 0 ,
2 π s m T 1 T R 2 Δ ω 0 ,
P 1 m s P 2 L .

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