Abstract

We propose a new method for synthesizing piecewise-uniform long-period fiber gratings (LPFGs) by using an extended fundamental matrix model with thermal changes. The proposed synthesis method is then applied to the design of the LPFG tuned by the thermal changes for erbium gain equalization by using the simulated-annealing and steepest-descent optimization techniques. We describe how a piecewise-uniform LPFG can be constructed by utilizing the inverted gain spectrum of erbium-doped fiber amplifiers (EDFAs), from the thermal change parameters’ search process. A sensitivity analysis also is done to study the tolerance of our approach against possible error sources, such as the temperature controller, the fabrication of the LPFG, and the EDFA spectrum, by using Monte Carlo simulations. To verify the validity of the proposed synthesis method experimentally, we manufactured the piecewise-uniform LPFG with thermal changes by using a divided coil heater. We observe that the spectrum designed by the proposed synthesis method is close to the corresponding measured spectrum in the wavelength band of interest. We also compare the performance of the proposed method with traditional approaches, such as Newton-like methods.

© 2006 Optical Society of America

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    [CrossRef]
  4. J. Bae, J. Chun, and S. B. Lee, "Equalization of the nonflat erbium gain spectrum using the multiport lattice filter model," in Optical Fiber Communication Conference (Optical Society of America, 2000), pp. 80-83.
  5. Y. Liu, J. A. R. Williams, L. Zhang, and I. Bennion, "Phase shifted and cascaded long-period fiber gratings," Opt. Commun. 164, 27-31 (1999).
    [CrossRef]
  6. M. Harumoto, M. Shigehara, and H. Suganuma, "Gain-flattening filter using long-period fiber gratings," J. Lightwave Technol. 20, 1027-1033 (2002).
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  7. X. Gu, "Wavelength-division multiplexing isolation fiber filter and light source using cascaded long-period fiber gratings," Opt. Lett. 23, 509-510 (1998).
    [CrossRef]
  8. A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, "Long-period fiber gratings as band-rejection filters," J. Lightwave Technol. 14, 58-64 (1996).
    [CrossRef]
  9. A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, "Fiber grating sensors," J. Lightwave Technol. 15, 1442-1463 (1997).
    [CrossRef]
  10. Y. Han, C. S. Kim, U. C. Paek, and Y. Chung, "Performance enhancement of long period fiber gratings for strain and temperature sensing," IEICE Trans. Electron. E83-C, 1-6 (2000).
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    [CrossRef]
  12. S. Matsumoto, T. Ohira, M. Takabayashi, K. Yoshiara, and T. Sugihara, "Tunable dispersion equalizer with a divided thin-film heater for 40-Gb/s RZ transmission," IEEE Photonics Technol. Lett. 13, 827-829 (2001).
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    [CrossRef]
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    [CrossRef] [PubMed]
  24. J. Bae, S. B. Lee, H. D. Ku, S. H. Kim, and J. Chun, "Fabrication of piecewise-uniform LPFG designed using multiport lattice filter model," in Optical Fiber Communication Conference (Optical Society of America, 2003), pp. 570-571.
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  26. E. Peral, J. Capmany, and J. Marti, "Iterative solution to the Gel'fand-Levitan-Marchenko coupled equations and application to synthesis of fiber gratings," IEEE J. Quantum Electron. 32, 2078-2084 (1996).
    [CrossRef]
  27. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge University, 1992).
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    [CrossRef]
  32. J. Stoer and R. Bulirsch, Introduction to Numerical Analysis (Springer-Verlag, 1980).
  33. X. Shu, L. Zhang, and I. Bennion, "Sensitivity characteristics of long-period fiber gratings," J. Lightwave Technol. 20, 255-266 (2002).
    [CrossRef]
  34. R. Fletcher, Practical Methods of Optimization (Wiley, 1979).
  35. J. A. Rogers, B. J. Eggleton, J. R. Pedrazzani, and T. A. Strasser, "Distributed on-fiber thin-film heaters for Bragg gratings with adjustable chip," Appl. Phys. Lett. 74, 3131-3133 (1999).
    [CrossRef]
  36. B. J. Eggleton, J. A. Rogers, S. Wetbrook, and T. A. Strasser, "Electrically tunable power efficient dispersion compensating fiber Bragg grating," IEEE Photonics Technol. Lett. 11, 854-856 (1999).
    [CrossRef]
  37. J. A. Rogers, B. J. Eggleton, R. J. Jackman, G. R. Kowach, and T. A. Strasser, "Dual on-fiber thin-film heaters for fiber gratings with independently adjustable chirp and wavelength," Opt. Lett. 24, 1328-1330 (1999).
    [CrossRef]
  38. H. S. Kim, S. H. Yun, I. K. Kwang, and B. Y. Kim, "All-fiber acousto-optic tunable notch filter with electronically controllable spectral profile," Opt. Lett. 22, 1476-1478 (1997).
    [CrossRef]

2004 (2)

2003 (1)

J. K. Bae, S. H. Kim, J. H. Kim, J. Bae, S. B. Lee, and J. M. Jeong, "Spectral shape tunable band-rejection filter using a long-period fiber grating with divided coil heaters," IEEE Photonics Technol. Lett. 15, 407-409 (2003).
[CrossRef]

2002 (2)

2001 (4)

S. Matsumoto, T. Ohira, M. Takabayashi, K. Yoshiara, and T. Sugihara, "Tunable dispersion equalizer with a divided thin-film heater for 40-Gb/s RZ transmission," IEEE Photonics Technol. Lett. 13, 827-829 (2001).
[CrossRef]

L. R. Chen, "Phase-shifted long-period fiber gratings by reflective index shifting," Opt. Commun. 200, 187-191 (2001).
[CrossRef]

L. Wang and T. Erdogan, "Layer peeling algorithm for reconstruction of long-period fiber gratings," Electron. Lett. 37, 154-156 (2001).
[CrossRef]

X. Shu, T. Allsop, B. Gwandu, and L. Zhang, "High-temperature sensitivity of long-period grating in B-Ge codoped fiber," IEEE Photonics Technol. Lett. 13, 818-820 (2001).
[CrossRef]

2000 (2)

J. Bae, J. Chun, and S. B. Lee, "Multiport lattice filter model for long-period fiber gratings," Jpn. J. Appl. Phys. Part 1 39, 6576-6577 (2000).
[CrossRef]

Y. Han, C. S. Kim, U. C. Paek, and Y. Chung, "Performance enhancement of long period fiber gratings for strain and temperature sensing," IEICE Trans. Electron. E83-C, 1-6 (2000).

1999 (5)

J. Bae, J. Chun, and S. B. Lee, "Two methods for synthesizing the long period fiber gratings with the inverted erbium gain spectrum," Jpn. J. Appl. Phys. Part 2 38, L819-L822 (1999).
[CrossRef]

Y. Liu, J. A. R. Williams, L. Zhang, and I. Bennion, "Phase shifted and cascaded long-period fiber gratings," Opt. Commun. 164, 27-31 (1999).
[CrossRef]

J. A. Rogers, B. J. Eggleton, J. R. Pedrazzani, and T. A. Strasser, "Distributed on-fiber thin-film heaters for Bragg gratings with adjustable chip," Appl. Phys. Lett. 74, 3131-3133 (1999).
[CrossRef]

B. J. Eggleton, J. A. Rogers, S. Wetbrook, and T. A. Strasser, "Electrically tunable power efficient dispersion compensating fiber Bragg grating," IEEE Photonics Technol. Lett. 11, 854-856 (1999).
[CrossRef]

J. A. Rogers, B. J. Eggleton, R. J. Jackman, G. R. Kowach, and T. A. Strasser, "Dual on-fiber thin-film heaters for fiber gratings with independently adjustable chirp and wavelength," Opt. Lett. 24, 1328-1330 (1999).
[CrossRef]

1998 (1)

1997 (4)

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, "Fiber grating sensors," J. Lightwave Technol. 15, 1442-1463 (1997).
[CrossRef]

T. Erdogan, "Fiber grating spectra," J. Lightwave Technol. 15, 1277-1294 (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]

H. S. Kim, S. H. Yun, I. K. Kwang, and B. Y. Kim, "All-fiber acousto-optic tunable notch filter with electronically controllable spectral profile," Opt. Lett. 22, 1476-1478 (1997).
[CrossRef]

1996 (3)

A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, and P. J. Lemaire, "Long-period fiber-grating-based gain equalizers," Opt. Lett. 21, 336-338 (1996).
[CrossRef] [PubMed]

E. Peral, J. Capmany, and J. Marti, "Iterative solution to the Gel'fand-Levitan-Marchenko coupled equations and application to synthesis of fiber gratings," IEEE J. Quantum Electron. 32, 2078-2084 (1996).
[CrossRef]

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, "Long-period fiber gratings as band-rejection filters," J. Lightwave Technol. 14, 58-64 (1996).
[CrossRef]

1987 (2)

M. Yamada and K. Sakuda, "Analysis of almost-periodic distributed feedback slab waveguides via a funda-mental matrix approach," Appl. Opt. 26, 3474-3478 (1987).
[CrossRef] [PubMed]

A. Corana, M. Marchesi, C. Martini, and S. Ridella, "Minimizing multimodal functions of continuous variables with the "simulated annealing" algorithm," ACM Trans. Math. Softw. 13, 262-280 (1987).
[CrossRef]

1986 (1)

I. O. Bohachevsky, M. E. Johnson, and M. L. Stein, "Generalized simulated annealing for function optimization," Technometrics 28, 209-217 (1986).
[CrossRef]

Allsop, T.

X. Shu, T. Allsop, B. Gwandu, and L. Zhang, "High-temperature sensitivity of long-period grating in B-Ge codoped fiber," IEEE Photonics Technol. Lett. 13, 818-820 (2001).
[CrossRef]

Askins, C. G.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, "Fiber grating sensors," J. Lightwave Technol. 15, 1442-1463 (1997).
[CrossRef]

Bae, J.

J. Bae, J. K. Bae, S. H. Kim, S. B. Lee, and J. Chun, "Analysis for long period fiber gratings using thermal kernel function," Opt. Express 12, 797-810 (2004).
[CrossRef] [PubMed]

J. K. Bae, S. H. Kim, J. H. Kim, J. Bae, S. B. Lee, and J. M. Jeong, "Spectral shape tunable band-rejection filter using a long-period fiber grating with divided coil heaters," IEEE Photonics Technol. Lett. 15, 407-409 (2003).
[CrossRef]

J. Bae, J. Chun, and S. B. Lee, "Multiport lattice filter model for long-period fiber gratings," Jpn. J. Appl. Phys. Part 1 39, 6576-6577 (2000).
[CrossRef]

J. Bae, J. Chun, and S. B. Lee, "Two methods for synthesizing the long period fiber gratings with the inverted erbium gain spectrum," Jpn. J. Appl. Phys. Part 2 38, L819-L822 (1999).
[CrossRef]

J. Bae, J. Chun, and S. B. Lee, "Equalization of the nonflat erbium gain spectrum using the multiport lattice filter model," in Optical Fiber Communication Conference (Optical Society of America, 2000), pp. 80-83.

J. Bae, S. B. Lee, H. D. Ku, S. H. Kim, and J. Chun, "Fabrication of piecewise-uniform LPFG designed using multiport lattice filter model," in Optical Fiber Communication Conference (Optical Society of America, 2003), pp. 570-571.

Bae, J. K.

J. Bae, J. K. Bae, S. H. Kim, S. B. Lee, and J. Chun, "Analysis for long period fiber gratings using thermal kernel function," Opt. Express 12, 797-810 (2004).
[CrossRef] [PubMed]

J. K. Bae, S. H. Kim, J. H. Kim, J. Bae, S. B. Lee, and J. M. Jeong, "Spectral shape tunable band-rejection filter using a long-period fiber grating with divided coil heaters," IEEE Photonics Technol. Lett. 15, 407-409 (2003).
[CrossRef]

Bennion, I.

X. Shu, L. Zhang, and I. Bennion, "Sensitivity characteristics of long-period fiber gratings," J. Lightwave Technol. 20, 255-266 (2002).
[CrossRef]

Y. Liu, J. A. R. Williams, L. Zhang, and I. Bennion, "Phase shifted and cascaded long-period fiber gratings," Opt. Commun. 164, 27-31 (1999).
[CrossRef]

Bhatia, V.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, "Long-period fiber gratings as band-rejection filters," J. Lightwave Technol. 14, 58-64 (1996).
[CrossRef]

V. Bhatia, D. K. Campbell, and T. D'Alberto, "Standard optical fiber long-period gratings with reduced temperature sensitivity for strain and refractive-index sensing," in Optical Fiber Communication Conference (Optical Society of America, 1997), pp. 346-347.
[CrossRef]

Bohachevsky, I. O.

I. O. Bohachevsky, M. E. Johnson, and M. L. Stein, "Generalized simulated annealing for function optimization," Technometrics 28, 209-217 (1986).
[CrossRef]

Bulirsch, R.

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis (Springer-Verlag, 1980).

Campbell, D. K.

V. Bhatia, D. K. Campbell, and T. D'Alberto, "Standard optical fiber long-period gratings with reduced temperature sensitivity for strain and refractive-index sensing," in Optical Fiber Communication Conference (Optical Society of America, 1997), pp. 346-347.
[CrossRef]

Capmany, J.

E. Peral, J. Capmany, and J. Marti, "Iterative solution to the Gel'fand-Levitan-Marchenko coupled equations and application to synthesis of fiber gratings," IEEE J. Quantum Electron. 32, 2078-2084 (1996).
[CrossRef]

Chen, L. R.

L. R. Chen, "Phase-shifted long-period fiber gratings by reflective index shifting," Opt. Commun. 200, 187-191 (2001).
[CrossRef]

Chesnoy, J.

J. Chesnoy, Undersea Fiber Communication System (Academic, 2002).

Chun, J.

J. Bae, J. K. Bae, S. H. Kim, S. B. Lee, and J. Chun, "Analysis for long period fiber gratings using thermal kernel function," Opt. Express 12, 797-810 (2004).
[CrossRef] [PubMed]

J. Bae, J. Chun, and S. B. Lee, "Multiport lattice filter model for long-period fiber gratings," Jpn. J. Appl. Phys. Part 1 39, 6576-6577 (2000).
[CrossRef]

J. Bae, J. Chun, and S. B. Lee, "Two methods for synthesizing the long period fiber gratings with the inverted erbium gain spectrum," Jpn. J. Appl. Phys. Part 2 38, L819-L822 (1999).
[CrossRef]

J. Bae, J. Chun, and S. B. Lee, "Equalization of the nonflat erbium gain spectrum using the multiport lattice filter model," in Optical Fiber Communication Conference (Optical Society of America, 2000), pp. 80-83.

J. Bae, S. B. Lee, H. D. Ku, S. H. Kim, and J. Chun, "Fabrication of piecewise-uniform LPFG designed using multiport lattice filter model," in Optical Fiber Communication Conference (Optical Society of America, 2003), pp. 570-571.

Chung, Y.

Y. Han, C. S. Kim, U. C. Paek, and Y. Chung, "Performance enhancement of long period fiber gratings for strain and temperature sensing," IEICE Trans. Electron. E83-C, 1-6 (2000).

Corana, A.

A. Corana, M. Marchesi, C. Martini, and S. Ridella, "Minimizing multimodal functions of continuous variables with the "simulated annealing" algorithm," ACM Trans. Math. Softw. 13, 262-280 (1987).
[CrossRef]

D'Alberto, T.

V. Bhatia, D. K. Campbell, and T. D'Alberto, "Standard optical fiber long-period gratings with reduced temperature sensitivity for strain and refractive-index sensing," in Optical Fiber Communication Conference (Optical Society of America, 1997), pp. 346-347.
[CrossRef]

Davis, M. A.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, "Fiber grating sensors," J. Lightwave Technol. 15, 1442-1463 (1997).
[CrossRef]

Eggleton, B. J.

J. A. Rogers, B. J. Eggleton, J. R. Pedrazzani, and T. A. Strasser, "Distributed on-fiber thin-film heaters for Bragg gratings with adjustable chip," Appl. Phys. Lett. 74, 3131-3133 (1999).
[CrossRef]

B. J. Eggleton, J. A. Rogers, S. Wetbrook, and T. A. Strasser, "Electrically tunable power efficient dispersion compensating fiber Bragg grating," IEEE Photonics Technol. Lett. 11, 854-856 (1999).
[CrossRef]

J. A. Rogers, B. J. Eggleton, R. J. Jackman, G. R. Kowach, and T. A. Strasser, "Dual on-fiber thin-film heaters for fiber gratings with independently adjustable chirp and wavelength," Opt. Lett. 24, 1328-1330 (1999).
[CrossRef]

Enomoto, T.

M. Shigehara, T. Enomoto, S. Ishikawa, M. Harumoto, and H. Kanamori, "Wavelength tunable long-period fiber grating," in APCC/OECC '99 Fifth Asia-Pacific Conference and Fourth Optoelectronics and Communications Conference (Institute of Electrical and Electronics Engineers, 1999), pp. 1610-1611.
[CrossRef]

Erdogan, T.

L. Wang and T. Erdogan, "Layer peeling algorithm for reconstruction of long-period fiber gratings," Electron. Lett. 37, 154-156 (2001).
[CrossRef]

T. Erdogan, "Fiber grating spectra," J. Lightwave Technol. 15, 1277-1294 (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]

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, "Long-period fiber gratings as band-rejection filters," J. Lightwave Technol. 14, 58-64 (1996).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge University, 1992).

Fletcher, R.

R. Fletcher, Practical Methods of Optimization (Wiley, 1979).

Friebele, E. J.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, "Fiber grating sensors," J. Lightwave Technol. 15, 1442-1463 (1997).
[CrossRef]

Gu, X.

Gwandu, B.

X. Shu, T. Allsop, B. Gwandu, and L. Zhang, "High-temperature sensitivity of long-period grating in B-Ge codoped fiber," IEEE Photonics Technol. Lett. 13, 818-820 (2001).
[CrossRef]

Han, Y.

Y. Han, S. H. Kim, and S. B. Lee, "Flexibly tunable multichannel filter and bandpass filter based on long-period fiber gratings," Opt. Express 12, 1902-1907 (2004).
[CrossRef] [PubMed]

Y. Han, C. S. Kim, U. C. Paek, and Y. Chung, "Performance enhancement of long period fiber gratings for strain and temperature sensing," IEICE Trans. Electron. E83-C, 1-6 (2000).

Harumoto, M.

M. Harumoto, M. Shigehara, and H. Suganuma, "Gain-flattening filter using long-period fiber gratings," J. Lightwave Technol. 20, 1027-1033 (2002).
[CrossRef]

M. Shigehara, T. Enomoto, S. Ishikawa, M. Harumoto, and H. Kanamori, "Wavelength tunable long-period fiber grating," in APCC/OECC '99 Fifth Asia-Pacific Conference and Fourth Optoelectronics and Communications Conference (Institute of Electrical and Electronics Engineers, 1999), pp. 1610-1611.
[CrossRef]

Ishikawa, S.

M. Shigehara, T. Enomoto, S. Ishikawa, M. Harumoto, and H. Kanamori, "Wavelength tunable long-period fiber grating," in APCC/OECC '99 Fifth Asia-Pacific Conference and Fourth Optoelectronics and Communications Conference (Institute of Electrical and Electronics Engineers, 1999), pp. 1610-1611.
[CrossRef]

Jackman, R. J.

Jeong, J. M.

J. K. Bae, S. H. Kim, J. H. Kim, J. Bae, S. B. Lee, and J. M. Jeong, "Spectral shape tunable band-rejection filter using a long-period fiber grating with divided coil heaters," IEEE Photonics Technol. Lett. 15, 407-409 (2003).
[CrossRef]

Johnson, M. E.

I. O. Bohachevsky, M. E. Johnson, and M. L. Stein, "Generalized simulated annealing for function optimization," Technometrics 28, 209-217 (1986).
[CrossRef]

Judkins, J. B.

A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, and P. J. Lemaire, "Long-period fiber-grating-based gain equalizers," Opt. Lett. 21, 336-338 (1996).
[CrossRef] [PubMed]

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, "Long-period fiber gratings as band-rejection filters," J. Lightwave Technol. 14, 58-64 (1996).
[CrossRef]

Kalli, K.

A. Othonos and K. Kalli, Fiber Bragg Gratings—Fundamentals and Applications in Telecommunications and Sensing (Artech House, 1999).

Kanamori, H.

M. Shigehara, T. Enomoto, S. Ishikawa, M. Harumoto, and H. Kanamori, "Wavelength tunable long-period fiber grating," in APCC/OECC '99 Fifth Asia-Pacific Conference and Fourth Optoelectronics and Communications Conference (Institute of Electrical and Electronics Engineers, 1999), pp. 1610-1611.
[CrossRef]

Kersey, A. D.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, "Fiber grating sensors," J. Lightwave Technol. 15, 1442-1463 (1997).
[CrossRef]

Kim, B. Y.

Kim, C. S.

Y. Han, C. S. Kim, U. C. Paek, and Y. Chung, "Performance enhancement of long period fiber gratings for strain and temperature sensing," IEICE Trans. Electron. E83-C, 1-6 (2000).

Kim, H. S.

Kim, J. H.

J. K. Bae, S. H. Kim, J. H. Kim, J. Bae, S. B. Lee, and J. M. Jeong, "Spectral shape tunable band-rejection filter using a long-period fiber grating with divided coil heaters," IEEE Photonics Technol. Lett. 15, 407-409 (2003).
[CrossRef]

Kim, S. H.

Y. Han, S. H. Kim, and S. B. Lee, "Flexibly tunable multichannel filter and bandpass filter based on long-period fiber gratings," Opt. Express 12, 1902-1907 (2004).
[CrossRef] [PubMed]

J. Bae, J. K. Bae, S. H. Kim, S. B. Lee, and J. Chun, "Analysis for long period fiber gratings using thermal kernel function," Opt. Express 12, 797-810 (2004).
[CrossRef] [PubMed]

J. K. Bae, S. H. Kim, J. H. Kim, J. Bae, S. B. Lee, and J. M. Jeong, "Spectral shape tunable band-rejection filter using a long-period fiber grating with divided coil heaters," IEEE Photonics Technol. Lett. 15, 407-409 (2003).
[CrossRef]

J. Bae, S. B. Lee, H. D. Ku, S. H. Kim, and J. Chun, "Fabrication of piecewise-uniform LPFG designed using multiport lattice filter model," in Optical Fiber Communication Conference (Optical Society of America, 2003), pp. 570-571.

Kogelnik, H.

H. Kogelnik, "Theory of Optical Waveguides," in Guided-Wave Optoelectronics, T.Tamir, ed. (Springer-Verlag,1990).
[CrossRef]

Koo, K. P.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, "Fiber grating sensors," J. Lightwave Technol. 15, 1442-1463 (1997).
[CrossRef]

Kowach, G. R.

Ku, H. D.

J. Bae, S. B. Lee, H. D. Ku, S. H. Kim, and J. Chun, "Fabrication of piecewise-uniform LPFG designed using multiport lattice filter model," in Optical Fiber Communication Conference (Optical Society of America, 2003), pp. 570-571.

Kwang, I. K.

LeBlanc, M.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, "Fiber grating sensors," J. Lightwave Technol. 15, 1442-1463 (1997).
[CrossRef]

Lee, S. B.

Y. Han, S. H. Kim, and S. B. Lee, "Flexibly tunable multichannel filter and bandpass filter based on long-period fiber gratings," Opt. Express 12, 1902-1907 (2004).
[CrossRef] [PubMed]

J. Bae, J. K. Bae, S. H. Kim, S. B. Lee, and J. Chun, "Analysis for long period fiber gratings using thermal kernel function," Opt. Express 12, 797-810 (2004).
[CrossRef] [PubMed]

J. K. Bae, S. H. Kim, J. H. Kim, J. Bae, S. B. Lee, and J. M. Jeong, "Spectral shape tunable band-rejection filter using a long-period fiber grating with divided coil heaters," IEEE Photonics Technol. Lett. 15, 407-409 (2003).
[CrossRef]

J. Bae, J. Chun, and S. B. Lee, "Multiport lattice filter model for long-period fiber gratings," Jpn. J. Appl. Phys. Part 1 39, 6576-6577 (2000).
[CrossRef]

J. Bae, J. Chun, and S. B. Lee, "Two methods for synthesizing the long period fiber gratings with the inverted erbium gain spectrum," Jpn. J. Appl. Phys. Part 2 38, L819-L822 (1999).
[CrossRef]

J. Bae, J. Chun, and S. B. Lee, "Equalization of the nonflat erbium gain spectrum using the multiport lattice filter model," in Optical Fiber Communication Conference (Optical Society of America, 2000), pp. 80-83.

J. Bae, S. B. Lee, H. D. Ku, S. H. Kim, and J. Chun, "Fabrication of piecewise-uniform LPFG designed using multiport lattice filter model," in Optical Fiber Communication Conference (Optical Society of America, 2003), pp. 570-571.

Lemaire, P. J.

A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, and P. J. Lemaire, "Long-period fiber-grating-based gain equalizers," Opt. Lett. 21, 336-338 (1996).
[CrossRef] [PubMed]

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, "Long-period fiber gratings as band-rejection filters," J. Lightwave Technol. 14, 58-64 (1996).
[CrossRef]

Liu, Y.

Y. Liu, J. A. R. Williams, L. Zhang, and I. Bennion, "Phase shifted and cascaded long-period fiber gratings," Opt. Commun. 164, 27-31 (1999).
[CrossRef]

Madsen, C. K.

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis, A Signal Processing Approach (Wiley,1999).

Marchesi, M.

A. Corana, M. Marchesi, C. Martini, and S. Ridella, "Minimizing multimodal functions of continuous variables with the "simulated annealing" algorithm," ACM Trans. Math. Softw. 13, 262-280 (1987).
[CrossRef]

Marti, J.

E. Peral, J. Capmany, and J. Marti, "Iterative solution to the Gel'fand-Levitan-Marchenko coupled equations and application to synthesis of fiber gratings," IEEE J. Quantum Electron. 32, 2078-2084 (1996).
[CrossRef]

Martini, C.

A. Corana, M. Marchesi, C. Martini, and S. Ridella, "Minimizing multimodal functions of continuous variables with the "simulated annealing" algorithm," ACM Trans. Math. Softw. 13, 262-280 (1987).
[CrossRef]

Matsumoto, S.

S. Matsumoto, T. Ohira, M. Takabayashi, K. Yoshiara, and T. Sugihara, "Tunable dispersion equalizer with a divided thin-film heater for 40-Gb/s RZ transmission," IEEE Photonics Technol. Lett. 13, 827-829 (2001).
[CrossRef]

Ohira, T.

S. Matsumoto, T. Ohira, M. Takabayashi, K. Yoshiara, and T. Sugihara, "Tunable dispersion equalizer with a divided thin-film heater for 40-Gb/s RZ transmission," IEEE Photonics Technol. Lett. 13, 827-829 (2001).
[CrossRef]

Othonos, A.

A. Othonos and K. Kalli, Fiber Bragg Gratings—Fundamentals and Applications in Telecommunications and Sensing (Artech House, 1999).

Paek, U. C.

Y. Han, C. S. Kim, U. C. Paek, and Y. Chung, "Performance enhancement of long period fiber gratings for strain and temperature sensing," IEICE Trans. Electron. E83-C, 1-6 (2000).

Patrick, H. J.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, "Fiber grating sensors," J. Lightwave Technol. 15, 1442-1463 (1997).
[CrossRef]

Pedrazzani, J. R.

J. A. Rogers, B. J. Eggleton, J. R. Pedrazzani, and T. A. Strasser, "Distributed on-fiber thin-film heaters for Bragg gratings with adjustable chip," Appl. Phys. Lett. 74, 3131-3133 (1999).
[CrossRef]

A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, and P. J. Lemaire, "Long-period fiber-grating-based gain equalizers," Opt. Lett. 21, 336-338 (1996).
[CrossRef] [PubMed]

Peral, E.

E. Peral, J. Capmany, and J. Marti, "Iterative solution to the Gel'fand-Levitan-Marchenko coupled equations and application to synthesis of fiber gratings," IEEE J. Quantum Electron. 32, 2078-2084 (1996).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge University, 1992).

Putnam, M. A.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, "Fiber grating sensors," J. Lightwave Technol. 15, 1442-1463 (1997).
[CrossRef]

Ridella, S.

A. Corana, M. Marchesi, C. Martini, and S. Ridella, "Minimizing multimodal functions of continuous variables with the "simulated annealing" algorithm," ACM Trans. Math. Softw. 13, 262-280 (1987).
[CrossRef]

Rogers, J. A.

J. A. Rogers, B. J. Eggleton, J. R. Pedrazzani, and T. A. Strasser, "Distributed on-fiber thin-film heaters for Bragg gratings with adjustable chip," Appl. Phys. Lett. 74, 3131-3133 (1999).
[CrossRef]

J. A. Rogers, B. J. Eggleton, R. J. Jackman, G. R. Kowach, and T. A. Strasser, "Dual on-fiber thin-film heaters for fiber gratings with independently adjustable chirp and wavelength," Opt. Lett. 24, 1328-1330 (1999).
[CrossRef]

B. J. Eggleton, J. A. Rogers, S. Wetbrook, and T. A. Strasser, "Electrically tunable power efficient dispersion compensating fiber Bragg grating," IEEE Photonics Technol. Lett. 11, 854-856 (1999).
[CrossRef]

Sakuda, K.

Shigehara, M.

M. Harumoto, M. Shigehara, and H. Suganuma, "Gain-flattening filter using long-period fiber gratings," J. Lightwave Technol. 20, 1027-1033 (2002).
[CrossRef]

M. Shigehara, T. Enomoto, S. Ishikawa, M. Harumoto, and H. Kanamori, "Wavelength tunable long-period fiber grating," in APCC/OECC '99 Fifth Asia-Pacific Conference and Fourth Optoelectronics and Communications Conference (Institute of Electrical and Electronics Engineers, 1999), pp. 1610-1611.
[CrossRef]

Shu, X.

X. Shu, L. Zhang, and I. Bennion, "Sensitivity characteristics of long-period fiber gratings," J. Lightwave Technol. 20, 255-266 (2002).
[CrossRef]

X. Shu, T. Allsop, B. Gwandu, and L. Zhang, "High-temperature sensitivity of long-period grating in B-Ge codoped fiber," IEEE Photonics Technol. Lett. 13, 818-820 (2001).
[CrossRef]

Sipe, J. E.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, "Long-period fiber gratings as band-rejection filters," J. Lightwave Technol. 14, 58-64 (1996).
[CrossRef]

Stein, M. L.

I. O. Bohachevsky, M. E. Johnson, and M. L. Stein, "Generalized simulated annealing for function optimization," Technometrics 28, 209-217 (1986).
[CrossRef]

Stoer, J.

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis (Springer-Verlag, 1980).

Strasser, T. A.

B. J. Eggleton, J. A. Rogers, S. Wetbrook, and T. A. Strasser, "Electrically tunable power efficient dispersion compensating fiber Bragg grating," IEEE Photonics Technol. Lett. 11, 854-856 (1999).
[CrossRef]

J. A. Rogers, B. J. Eggleton, R. J. Jackman, G. R. Kowach, and T. A. Strasser, "Dual on-fiber thin-film heaters for fiber gratings with independently adjustable chirp and wavelength," Opt. Lett. 24, 1328-1330 (1999).
[CrossRef]

J. A. Rogers, B. J. Eggleton, J. R. Pedrazzani, and T. A. Strasser, "Distributed on-fiber thin-film heaters for Bragg gratings with adjustable chip," Appl. Phys. Lett. 74, 3131-3133 (1999).
[CrossRef]

Suganuma, H.

Sugihara, T.

S. Matsumoto, T. Ohira, M. Takabayashi, K. Yoshiara, and T. Sugihara, "Tunable dispersion equalizer with a divided thin-film heater for 40-Gb/s RZ transmission," IEEE Photonics Technol. Lett. 13, 827-829 (2001).
[CrossRef]

Takabayashi, M.

S. Matsumoto, T. Ohira, M. Takabayashi, K. Yoshiara, and T. Sugihara, "Tunable dispersion equalizer with a divided thin-film heater for 40-Gb/s RZ transmission," IEEE Photonics Technol. Lett. 13, 827-829 (2001).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge University, 1992).

Vengsarkar, A. M.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, "Long-period fiber gratings as band-rejection filters," J. Lightwave Technol. 14, 58-64 (1996).
[CrossRef]

A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, and P. J. Lemaire, "Long-period fiber-grating-based gain equalizers," Opt. Lett. 21, 336-338 (1996).
[CrossRef] [PubMed]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge University, 1992).

Wang, L.

L. Wang and T. Erdogan, "Layer peeling algorithm for reconstruction of long-period fiber gratings," Electron. Lett. 37, 154-156 (2001).
[CrossRef]

Wetbrook, S.

B. J. Eggleton, J. A. Rogers, S. Wetbrook, and T. A. Strasser, "Electrically tunable power efficient dispersion compensating fiber Bragg grating," IEEE Photonics Technol. Lett. 11, 854-856 (1999).
[CrossRef]

Williams, J. A. R.

Y. Liu, J. A. R. Williams, L. Zhang, and I. Bennion, "Phase shifted and cascaded long-period fiber gratings," Opt. Commun. 164, 27-31 (1999).
[CrossRef]

Yamada, M.

Yoshiara, K.

S. Matsumoto, T. Ohira, M. Takabayashi, K. Yoshiara, and T. Sugihara, "Tunable dispersion equalizer with a divided thin-film heater for 40-Gb/s RZ transmission," IEEE Photonics Technol. Lett. 13, 827-829 (2001).
[CrossRef]

Yun, S. H.

Zhang, L.

X. Shu, L. Zhang, and I. Bennion, "Sensitivity characteristics of long-period fiber gratings," J. Lightwave Technol. 20, 255-266 (2002).
[CrossRef]

X. Shu, T. Allsop, B. Gwandu, and L. Zhang, "High-temperature sensitivity of long-period grating in B-Ge codoped fiber," IEEE Photonics Technol. Lett. 13, 818-820 (2001).
[CrossRef]

Y. Liu, J. A. R. Williams, L. Zhang, and I. Bennion, "Phase shifted and cascaded long-period fiber gratings," Opt. Commun. 164, 27-31 (1999).
[CrossRef]

Zhao, J. H.

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis, A Signal Processing Approach (Wiley,1999).

ACM Trans. Math. Softw. (1)

A. Corana, M. Marchesi, C. Martini, and S. Ridella, "Minimizing multimodal functions of continuous variables with the "simulated annealing" algorithm," ACM Trans. Math. Softw. 13, 262-280 (1987).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

J. A. Rogers, B. J. Eggleton, J. R. Pedrazzani, and T. A. Strasser, "Distributed on-fiber thin-film heaters for Bragg gratings with adjustable chip," Appl. Phys. Lett. 74, 3131-3133 (1999).
[CrossRef]

Electron. Lett. (1)

L. Wang and T. Erdogan, "Layer peeling algorithm for reconstruction of long-period fiber gratings," Electron. Lett. 37, 154-156 (2001).
[CrossRef]

IEEE J. Quantum Electron. (1)

E. Peral, J. Capmany, and J. Marti, "Iterative solution to the Gel'fand-Levitan-Marchenko coupled equations and application to synthesis of fiber gratings," IEEE J. Quantum Electron. 32, 2078-2084 (1996).
[CrossRef]

IEEE Photonics Technol. Lett. (4)

X. Shu, T. Allsop, B. Gwandu, and L. Zhang, "High-temperature sensitivity of long-period grating in B-Ge codoped fiber," IEEE Photonics Technol. Lett. 13, 818-820 (2001).
[CrossRef]

S. Matsumoto, T. Ohira, M. Takabayashi, K. Yoshiara, and T. Sugihara, "Tunable dispersion equalizer with a divided thin-film heater for 40-Gb/s RZ transmission," IEEE Photonics Technol. Lett. 13, 827-829 (2001).
[CrossRef]

J. K. Bae, S. H. Kim, J. H. Kim, J. Bae, S. B. Lee, and J. M. Jeong, "Spectral shape tunable band-rejection filter using a long-period fiber grating with divided coil heaters," IEEE Photonics Technol. Lett. 15, 407-409 (2003).
[CrossRef]

B. J. Eggleton, J. A. Rogers, S. Wetbrook, and T. A. Strasser, "Electrically tunable power efficient dispersion compensating fiber Bragg grating," IEEE Photonics Technol. Lett. 11, 854-856 (1999).
[CrossRef]

IEICE Trans. Electron. (1)

Y. Han, C. S. Kim, U. C. Paek, and Y. Chung, "Performance enhancement of long period fiber gratings for strain and temperature sensing," IEICE Trans. Electron. E83-C, 1-6 (2000).

J. Lightwave Technol. (5)

M. Harumoto, M. Shigehara, and H. Suganuma, "Gain-flattening filter using long-period fiber gratings," J. Lightwave Technol. 20, 1027-1033 (2002).
[CrossRef]

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, "Long-period fiber gratings as band-rejection filters," J. Lightwave Technol. 14, 58-64 (1996).
[CrossRef]

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, "Fiber grating sensors," J. Lightwave Technol. 15, 1442-1463 (1997).
[CrossRef]

X. Shu, L. Zhang, and I. Bennion, "Sensitivity characteristics of long-period fiber gratings," J. Lightwave Technol. 20, 255-266 (2002).
[CrossRef]

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

J. Opt. Soc. Am. A (1)

Jpn. J. Appl. Phys. Part 1 (1)

J. Bae, J. Chun, and S. B. Lee, "Multiport lattice filter model for long-period fiber gratings," Jpn. J. Appl. Phys. Part 1 39, 6576-6577 (2000).
[CrossRef]

Jpn. J. Appl. Phys. Part 2 (1)

J. Bae, J. Chun, and S. B. Lee, "Two methods for synthesizing the long period fiber gratings with the inverted erbium gain spectrum," Jpn. J. Appl. Phys. Part 2 38, L819-L822 (1999).
[CrossRef]

Opt. Commun. (2)

L. R. Chen, "Phase-shifted long-period fiber gratings by reflective index shifting," Opt. Commun. 200, 187-191 (2001).
[CrossRef]

Y. Liu, J. A. R. Williams, L. Zhang, and I. Bennion, "Phase shifted and cascaded long-period fiber gratings," Opt. Commun. 164, 27-31 (1999).
[CrossRef]

Opt. Express (2)

Opt. Lett. (4)

Technometrics (1)

I. O. Bohachevsky, M. E. Johnson, and M. L. Stein, "Generalized simulated annealing for function optimization," Technometrics 28, 209-217 (1986).
[CrossRef]

Other (11)

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis, A Signal Processing Approach (Wiley,1999).

R. Fletcher, Practical Methods of Optimization (Wiley, 1979).

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis (Springer-Verlag, 1980).

J. Bae, S. B. Lee, H. D. Ku, S. H. Kim, and J. Chun, "Fabrication of piecewise-uniform LPFG designed using multiport lattice filter model," in Optical Fiber Communication Conference (Optical Society of America, 2003), pp. 570-571.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge University, 1992).

J. Bae, J. Chun, and S. B. Lee, "Equalization of the nonflat erbium gain spectrum using the multiport lattice filter model," in Optical Fiber Communication Conference (Optical Society of America, 2000), pp. 80-83.

A. Othonos and K. Kalli, Fiber Bragg Gratings—Fundamentals and Applications in Telecommunications and Sensing (Artech House, 1999).

J. Chesnoy, Undersea Fiber Communication System (Academic, 2002).

H. Kogelnik, "Theory of Optical Waveguides," in Guided-Wave Optoelectronics, T.Tamir, ed. (Springer-Verlag,1990).
[CrossRef]

M. Shigehara, T. Enomoto, S. Ishikawa, M. Harumoto, and H. Kanamori, "Wavelength tunable long-period fiber grating," in APCC/OECC '99 Fifth Asia-Pacific Conference and Fourth Optoelectronics and Communications Conference (Institute of Electrical and Electronics Engineers, 1999), pp. 1610-1611.
[CrossRef]

V. Bhatia, D. K. Campbell, and T. D'Alberto, "Standard optical fiber long-period gratings with reduced temperature sensitivity for strain and refractive-index sensing," in Optical Fiber Communication Conference (Optical Society of America, 1997), pp. 346-347.
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Block diagram of the extended fundamental matrix model for LPFGs with thermal changes.

Fig. 2
Fig. 2

(a) Transmission spectra with thermal changes; (b) X and δ ( T 1 = 24.9 ° C ) .

Fig. 3
Fig. 3

(a) The a ( T 1 ) with thermal changes. [For the detailed display, we subtracted 0.9999 from a ( T 1 ) .] (b) Attained coefficients for a ( T 1 ) . (c) The b ( T 1 ) with thermal changes. (d) Attained coefficients for b ( T 1 ) . (e) The c ( T 1 ) with thermal changes. (f) Attained coefficients for c ( T 1 ) .

Fig. 4
Fig. 4

Flow chart for the proposed synthesis method.

Fig. 5
Fig. 5

(a) Convergence behavior of the proposed optimization method: SA, simulated annealing; SD, steepest decent. (b) The V ( Θ ) value. (c) Convergence behavior of BFGS and SD. (d) Cost of BFGS. (e) Cost of SD.

Fig. 6
Fig. 6

(a) X at 24.9 ° C for Δ n = 0.00019 ; (b) δ at 24.9 ° C ; (c) attained transmission spectrum curve; (d) corresponding thermal changes (attained periods are Λ 1 = 433.56 μ m and Λ 2 = 446.9 μ m ).

Fig. 7
Fig. 7

(a) Transmission spectra calculated by the Monte Carlo simulations; (b) histogram for the maximum power variation.

Fig. 8
Fig. 8

(a) Calculated and attained transmission spectrum curves; (b) corresponding thermal changes with 15 sections; (c) X and δ at 24.9 ° C ( Δ n = 0.000197 ) .

Fig. 9
Fig. 9

(a) Monte Carlo simulations for ϵ X and ϵ δ ; (b) histogram of the maximum power variation for the initial spectrum; (c) histogram of the maximum power variation for the fitted spectrum.

Equations (15)

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

[ A i B i ] = F i [ A i 1 B i 1 ] ,
F i = [ cos ( Υ i L i ) + j δ T i Υ i sin ( Υ i L i ) j κ T i Υ i sin ( Υ i L i ) j κ T i Υ i sin ( Υ i L i ) cos ( Υ i L i ) j δ T i Υ i sin ( Υ i L i ) ] ,
Υ i = def [ ( κ T i ) 2 + ( δ T i ) 2 ] 1 2
κ T i 1 λ [ a ( T i ) X + b ( T i ) ] ,
δ T i δ + c ( T i ) .
a ( T i ) a 0 T i m 1 + a 1 T i m 1 1 + + a m 1 1 T i 1 + a m 1 ,
b ( T i ) b 0 T i m 2 + b 1 T i m 2 1 + + b m 2 1 T i 1 + b m 2 ,
c ( T i ) c 0 T i m 3 + c 1 T i m 3 1 + + c m 3 1 T i 1 + c m 3 .
n co = 1.45248 , n cl = 1.44532 , n air = 1 ,
r co = 2.815 μ m , r cl = 62.51 μ m ,
[ A M B M ] = F [ A 0 B 0 ] ,
F F M F M 1 F 1 [ F 11 F 12 F 21 F 22 ] .
V ( Θ ) = p = 1 P w p ( Y ( λ p , Θ ) Y p d ) 2 ,
V ( Θ ) θ k = 2 p = 1 P w p ( Y ( λ p , Θ ) Y p d ) Y ( λ p , Θ ) θ k ,
Y ( λ p , Θ ) θ k = 1 Y ( λ p , Θ ) Re [ Y * ( λ p , Θ ) Y ( λ p , Θ ) θ k ] .

Metrics