Abstract

An analytical expression for calculating the position of the reflection-peak wavelength of a chirped sampled fiber Bragg grating (C-SFBG) is obtained for what is believed to be the first time. Using Fourier theory, the chirped sampling function of the C-SFBG is expanded, and an equivalent local Bragg period is then obtained to derive the expression of the peak wavelength. The calculated results based on the expression are in excellent agreement with the numerical reflection spectra obtained by the conventional transfer-matrix method.

© 2008 Optical Society of America

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References

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  1. M. Ibsen, M. K. Durkin, M. J. Cole, and R. I. Laming, “Sinc-sampled fiber Bragg gratings for identical multiple wavelength operation,” IEEE Photon. Technol. Lett. 10, 842-844(1998).
    [CrossRef]
  2. 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]
  3. N. Yusuke and Y. Shinji, “Densification of sampled fiber Bragg gratings using multiple phase shift (MPS) technique,” J. Lightwave Technol. 23, 1808-1817 (2005).
    [CrossRef]
  4. W. H. Loh, F. Q. Zhou, and J. J. Pan, “Sampled fiber grating based-dispersion slope compensator,” IEEE Photon. Technol. Lett. 11, 1280-1282 (1999).
    [CrossRef]
  5. 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. 11, 899-901 (1995).
    [CrossRef]
  6. X.-F. Chen, Y. Luo, 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]
  7. M. S. Kumar and A. Bekal, “Performance evaluation of SSFBG based optical CDMA systems employing golden sequences,” Opt. Fiber Technol. 11, 56-68 (2005).
    [CrossRef]
  8. X. H. Zou, W. Pan, B. Luo, W. L. Zhang, and M. Y. Wang, “Accurate analytical expression for reflection-peak wavelengths of sampled Bragg grating,” IEEE Photon. Technol. Lett. 18, 529-531 (2006).
    [CrossRef]
  9. C. H. 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]

2006 (1)

X. H. Zou, W. Pan, B. Luo, W. L. Zhang, and M. Y. Wang, “Accurate analytical expression for reflection-peak wavelengths of sampled Bragg grating,” IEEE Photon. Technol. Lett. 18, 529-531 (2006).
[CrossRef]

2005 (2)

M. S. Kumar and A. Bekal, “Performance evaluation of SSFBG based optical CDMA systems employing golden sequences,” Opt. Fiber Technol. 11, 56-68 (2005).
[CrossRef]

N. Yusuke and Y. Shinji, “Densification of sampled fiber Bragg gratings using multiple phase shift (MPS) technique,” J. Lightwave Technol. 23, 1808-1817 (2005).
[CrossRef]

2004 (1)

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]

2002 (1)

2000 (1)

X.-F. Chen, Y. Luo, 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]

1999 (1)

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

1998 (1)

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

1995 (1)

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. 11, 899-901 (1995).
[CrossRef]

Appl. Opt. (1)

Electron. Lett. (1)

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. 11, 899-901 (1995).
[CrossRef]

IEEE Photon. Technol. Lett. (5)

X.-F. Chen, Y. Luo, 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]

X. H. Zou, W. Pan, B. Luo, W. L. Zhang, and M. Y. Wang, “Accurate analytical expression for reflection-peak wavelengths of sampled Bragg grating,” IEEE Photon. Technol. Lett. 18, 529-531 (2006).
[CrossRef]

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

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

J. Lightwave Technol. (1)

Opt. Fiber Technol. (1)

M. S. Kumar and A. Bekal, “Performance evaluation of SSFBG based optical CDMA systems employing golden sequences,” Opt. Fiber Technol. 11, 56-68 (2005).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Configurations of sampled fiber Bragg gratings: (a) Uniform sampled fiber Bragg grating (U-SFBG), (b) chirped sampled fiber Bragg grating (C-FBG), where b is the sampling period and a is the grating length in one period, and ( b a ) is the nongrating length.

Fig. 2
Fig. 2

(a) Wavelength shift of the corresponding reflection peaks between a U-SFBG ( c s = 0 ) and C-SFBG with different chirp coefficient c s . In the calculation, b 0 = 1 mm (the first period) and a = 0.08 mm (a is fixed for all of the sampling period) are assumed, and three different c s are used: c s = 0 (U-SFBG), c s = 70 × 10 4 , and c s = 1.5 × 10 3 . (b) Enlarged details of the wavelength shift over three channels of Fig. 2(a).

Fig. 3
Fig. 3

Comparison of the RPWs results obtained with analytical and numerical methods for three different chirp coefficients: c s = 0 (U-SFBG), c s = 1.5 × 10 3 , and c s = 3 × 10 3 . Lines, numerical; symbols, analytical.

Fig. 4
Fig. 4

Relative deviations between the analytical and the numerical results for different Fourier orders with different chirp coefficients. c s = 0 (squares), c s = 1.5 × 10 3 (circles), and c s = 3 × 10 3 (triangles).

Fig. 5
Fig. 5

Comparison between the analytical and numerical results with a chirp coefficient of c s = 1.5 × 10 3 for different sampling periods, b 0 = 1 mm (squares), b 0 = 2 mm (circles), and b 0 = 3 mm (triangles). All numerical results are plotted with lines.

Fig. 6
Fig. 6

Wavelength shift of the corresponding reflection peaks with different total number of sampling periods N ( = 10 , 25, and 40). In the calculation, b 0 = 1 mm (the first period), a = 0.08 mm (a is fixed for all the sampling period), and c s = 1.5 × 10 3 .

Fig. 7
Fig. 7

Comparison of the RPWs results obtained with analytical and numerical methods for three different total number of sampling periods N ( = 10 , 25, and 40). The chirp coefficient c s = 3 × 10 3 , and the other parameters are the same as those used in Fig. 6. Numerical results (num), lines; analytical results (ana), symbols.

Tables (1)

Tables Icon

Table 1 Calculated RPWs of Three Different Chirp Coefficients c s Based on Analytical Expression and the T-Matrix Method

Equations (9)

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b ( i ) = b 0 [ 1 + ( i - 1 ) c s ] ,
Δ n ( z ) = Δ n 0 s ( z ) Re [ exp ( j 2 π Λ 0 ) ] ,
s ( z ) = rect ( z a ) * m = - + δ { z - m b 0 [ 1 + ( m - 1 ) 2 c s ] } ,
s ( z ) = k = - + F k exp ( 2 π j k z / p ) ,
Δ n ( z ) = Δ n 0 k = - + F k exp [ 2 π j ( 1 Λ ( k ) ) z ] ,
Λ ( k ) = p Λ 0 p - k Λ 0 .
λ max ( k ) = 2 n 0 Λ ( k ) .
λ max ( k ) = 2 n 0 · b 0 Λ 0 [ 2 + ( N 1 ) c s ] b 0 [ 2 + ( N - 1 ) c s ] - 2 k Λ 0 .
Δ λ ( k ) = 2 n 0 [ k Λ 0 2 ( b 0 - p ) ( b 0 - k Λ 0 ) ( p - k Λ 0 ) ] ,

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