Abstract

We demonstrate an inverse scattering algorithm for reconstructing the structure of lossy fiber Bragg gratings. The algorithm enables us to extract the profiles of the refractive index and the loss coefficient along the grating from the grating transmission spectrum and from the reflection spectra, measured from both sides of the grating. Such an algorithm can be used to develop novel distributed evanescent-wave fiber Bragg sensors that measure the change in both the refractive index and the attenuation coefficient of the medium surrounding the grating. The algorithm can also be used to analyze and to design fiber Bragg gratings written in fiber amplifiers. A novel method to overcome instability problems in extracting the parameters of the lossy grating is introduced. The new method also makes it possible to reduce the spectral resolution needed to accurately extract the grating parameters.

© 2004 Optical Society of America

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  1. A. D. Kersey, “A Review of recent developments in fiber optic sensors technology,” Opt. Fiber Technol. 2, 291–317 (1996).
    [CrossRef]
  2. V. Goloborodko, S. Keren, A. Rosenthal, B. Levit, M. Horowitz, “Measuring temperature profiles in high-power optical components,” Appl. Opt. 42, 2284–2288 (2003).
    [CrossRef] [PubMed]
  3. S. Huang, M. M. Ohn, M. LeBlanc, R. M. Measures, “Continuous arbitrary strain profile measurements with fiber Bragg gratings,” Smart Mater. Struct. 7, 248–256 (1998).
    [CrossRef]
  4. M. Volanthen, H. Geiger, M. J. Cole, J. P. Dakin, “Measurement of arbitrary strain profiles within fibre gratings,” Electron. Lett. 32, 1028–1029 (1996).
    [CrossRef]
  5. P. H. Paul, G. Kychakoff, “Fiber-optic evanescent field absorption sensor,” Appl. Phys. Lett. 51, 12–14 (1987).
    [CrossRef]
  6. H. Tai, H. Tanaka, T. Yoshino, “Fiber-optic evanescent-wave methane-gas sensor using optical absorption for the 3.392-µm line of a He–Ne laser,” Opt. Lett. 12, 437–439 (1987).
    [CrossRef] [PubMed]
  7. V. Ruddy, B. D. MacCraith, J. A. Murphy, “Evanescent wave absorption spectroscopy using multimode fibers,” J. Appl. Phys. 67, 6070–6074 (1990).
    [CrossRef]
  8. G. P. Anderson, J. P. Golden, L. K. Cao, D. Wijesuriya, L. C. Shriver-Lake, F. S. Ligler, “Development of an evanescent wave fiber optic biosensor,” IEEE Eng. Med. Biol. Mag. 13, 358–363 (1994).
    [CrossRef]
  9. A. Messica, A. Greenstein, A. Katzir, “Theory of fiber-optic, evanescent-wave spectroscopy and sensors,” Appl. Opt. 35, 2274–2284 (1996).
    [CrossRef] [PubMed]
  10. S. K. Khijwania, B. D. Gupta, “Fiber optic evanscent field absorption sensor: effect of fiber parameters and geometry of the probe,” Opt. Quantum Electron. 31, 625–636 (1999).
    [CrossRef]
  11. P. K. Choudhury, T. Yoshino, “On the optical sensing means for the study of chemical kinetics,” Meas. Sci. Technol. 13, 1793–1797 (2002).
    [CrossRef]
  12. K. Schroeder, W. Ecke, R. Mueller, R. Willsch, A. Andreev, “A fibre Bragg grating refractometer,” Meas. Sci. Technol. 12, 757–764 (2001).
    [CrossRef]
  13. J. Dubendorfer, R. E. Kunz, G. Jobst, I. Moser, G. Urban, “Integrated optical pH sensor using replicated chirped grating coupler sensor chips,” Sens. Actuators B 50, 210–219 (1998).
    [CrossRef]
  14. S. Keren, M. Horowitz, “Distributed three-dimensional fiber Bragg grating refractometer for biochemical sensing,” Opt. Lett. 28, 2037–2039 (2003).
    [CrossRef] [PubMed]
  15. S. Keren, M. Horowitz, “Interrogation of fiber gratings by use of low-coherence spectral interferometry of noiselike pulses,” Opt. Lett. 26, 328–330 (2001).
    [CrossRef]
  16. R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
    [CrossRef]
  17. J. Skaar, L. Wang, T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” J. Lightwave Technol. 37, 165–173 (2001).
  18. A. Rosenthal, M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 1018–1026 (2003).
    [CrossRef]
  19. A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 45, 312–335 (1985).
    [CrossRef]
  20. J. Frolik, A. E. Yagle, “Forward and inverse scattering for discrete layered lossy and absorbing media,” IEEE Trans. Circuits Syst. 44, 710–722 (1997).
    [CrossRef]
  21. W. H. Loh, R. I. Laming, “1.55 µm phase-shifted distributed feedback fibre laser,” Electron. Lett. 31, 1440–1442 (1995).
    [CrossRef]
  22. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [CrossRef]
  23. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
    [CrossRef]
  24. M. J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform (Society for Applied Mathematics, Philadelphia, Pa., 1981).
  25. S. Keren, A. Rosenthal, M. Horowitz, “Measuring the structure of highly reflecting fiber Bragg gratings,” IEEE Photon. Technol. Lett. 15, 575–577 (2003).
    [CrossRef]
  26. A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962).
  27. J. G. Proakis, D. G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, 3rd ed. (Prentice-Hall International, London, 1996).

2003

V. Goloborodko, S. Keren, A. Rosenthal, B. Levit, M. Horowitz, “Measuring temperature profiles in high-power optical components,” Appl. Opt. 42, 2284–2288 (2003).
[CrossRef] [PubMed]

S. Keren, M. Horowitz, “Distributed three-dimensional fiber Bragg grating refractometer for biochemical sensing,” Opt. Lett. 28, 2037–2039 (2003).
[CrossRef] [PubMed]

A. Rosenthal, M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 1018–1026 (2003).
[CrossRef]

S. Keren, A. Rosenthal, M. Horowitz, “Measuring the structure of highly reflecting fiber Bragg gratings,” IEEE Photon. Technol. Lett. 15, 575–577 (2003).
[CrossRef]

2002

P. K. Choudhury, T. Yoshino, “On the optical sensing means for the study of chemical kinetics,” Meas. Sci. Technol. 13, 1793–1797 (2002).
[CrossRef]

2001

K. Schroeder, W. Ecke, R. Mueller, R. Willsch, A. Andreev, “A fibre Bragg grating refractometer,” Meas. Sci. Technol. 12, 757–764 (2001).
[CrossRef]

S. Keren, M. Horowitz, “Interrogation of fiber gratings by use of low-coherence spectral interferometry of noiselike pulses,” Opt. Lett. 26, 328–330 (2001).
[CrossRef]

J. Skaar, L. Wang, T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” J. Lightwave Technol. 37, 165–173 (2001).

1999

S. K. Khijwania, B. D. Gupta, “Fiber optic evanscent field absorption sensor: effect of fiber parameters and geometry of the probe,” Opt. Quantum Electron. 31, 625–636 (1999).
[CrossRef]

R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

1998

J. Dubendorfer, R. E. Kunz, G. Jobst, I. Moser, G. Urban, “Integrated optical pH sensor using replicated chirped grating coupler sensor chips,” Sens. Actuators B 50, 210–219 (1998).
[CrossRef]

S. Huang, M. M. Ohn, M. LeBlanc, R. M. Measures, “Continuous arbitrary strain profile measurements with fiber Bragg gratings,” Smart Mater. Struct. 7, 248–256 (1998).
[CrossRef]

1997

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

J. Frolik, A. E. Yagle, “Forward and inverse scattering for discrete layered lossy and absorbing media,” IEEE Trans. Circuits Syst. 44, 710–722 (1997).
[CrossRef]

1996

M. Volanthen, H. Geiger, M. J. Cole, J. P. Dakin, “Measurement of arbitrary strain profiles within fibre gratings,” Electron. Lett. 32, 1028–1029 (1996).
[CrossRef]

A. D. Kersey, “A Review of recent developments in fiber optic sensors technology,” Opt. Fiber Technol. 2, 291–317 (1996).
[CrossRef]

A. Messica, A. Greenstein, A. Katzir, “Theory of fiber-optic, evanescent-wave spectroscopy and sensors,” Appl. Opt. 35, 2274–2284 (1996).
[CrossRef] [PubMed]

1995

W. H. Loh, R. I. Laming, “1.55 µm phase-shifted distributed feedback fibre laser,” Electron. Lett. 31, 1440–1442 (1995).
[CrossRef]

1994

G. P. Anderson, J. P. Golden, L. K. Cao, D. Wijesuriya, L. C. Shriver-Lake, F. S. Ligler, “Development of an evanescent wave fiber optic biosensor,” IEEE Eng. Med. Biol. Mag. 13, 358–363 (1994).
[CrossRef]

1990

V. Ruddy, B. D. MacCraith, J. A. Murphy, “Evanescent wave absorption spectroscopy using multimode fibers,” J. Appl. Phys. 67, 6070–6074 (1990).
[CrossRef]

1987

1985

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 45, 312–335 (1985).
[CrossRef]

1969

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Ablowitz, M. J.

M. J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform (Society for Applied Mathematics, Philadelphia, Pa., 1981).

Anderson, G. P.

G. P. Anderson, J. P. Golden, L. K. Cao, D. Wijesuriya, L. C. Shriver-Lake, F. S. Ligler, “Development of an evanescent wave fiber optic biosensor,” IEEE Eng. Med. Biol. Mag. 13, 358–363 (1994).
[CrossRef]

Andreev, A.

K. Schroeder, W. Ecke, R. Mueller, R. Willsch, A. Andreev, “A fibre Bragg grating refractometer,” Meas. Sci. Technol. 12, 757–764 (2001).
[CrossRef]

Bruckstein, A. M.

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 45, 312–335 (1985).
[CrossRef]

Cao, L. K.

G. P. Anderson, J. P. Golden, L. K. Cao, D. Wijesuriya, L. C. Shriver-Lake, F. S. Ligler, “Development of an evanescent wave fiber optic biosensor,” IEEE Eng. Med. Biol. Mag. 13, 358–363 (1994).
[CrossRef]

Choudhury, P. K.

P. K. Choudhury, T. Yoshino, “On the optical sensing means for the study of chemical kinetics,” Meas. Sci. Technol. 13, 1793–1797 (2002).
[CrossRef]

Cole, M. J.

M. Volanthen, H. Geiger, M. J. Cole, J. P. Dakin, “Measurement of arbitrary strain profiles within fibre gratings,” Electron. Lett. 32, 1028–1029 (1996).
[CrossRef]

Dakin, J. P.

M. Volanthen, H. Geiger, M. J. Cole, J. P. Dakin, “Measurement of arbitrary strain profiles within fibre gratings,” Electron. Lett. 32, 1028–1029 (1996).
[CrossRef]

Dubendorfer, J.

J. Dubendorfer, R. E. Kunz, G. Jobst, I. Moser, G. Urban, “Integrated optical pH sensor using replicated chirped grating coupler sensor chips,” Sens. Actuators B 50, 210–219 (1998).
[CrossRef]

Ecke, W.

K. Schroeder, W. Ecke, R. Mueller, R. Willsch, A. Andreev, “A fibre Bragg grating refractometer,” Meas. Sci. Technol. 12, 757–764 (2001).
[CrossRef]

Erdogan, T.

J. Skaar, L. Wang, T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” J. Lightwave Technol. 37, 165–173 (2001).

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

Feced, R.

R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

Frolik, J.

J. Frolik, A. E. Yagle, “Forward and inverse scattering for discrete layered lossy and absorbing media,” IEEE Trans. Circuits Syst. 44, 710–722 (1997).
[CrossRef]

Geiger, H.

M. Volanthen, H. Geiger, M. J. Cole, J. P. Dakin, “Measurement of arbitrary strain profiles within fibre gratings,” Electron. Lett. 32, 1028–1029 (1996).
[CrossRef]

Golden, J. P.

G. P. Anderson, J. P. Golden, L. K. Cao, D. Wijesuriya, L. C. Shriver-Lake, F. S. Ligler, “Development of an evanescent wave fiber optic biosensor,” IEEE Eng. Med. Biol. Mag. 13, 358–363 (1994).
[CrossRef]

Goloborodko, V.

Greenstein, A.

Gupta, B. D.

S. K. Khijwania, B. D. Gupta, “Fiber optic evanscent field absorption sensor: effect of fiber parameters and geometry of the probe,” Opt. Quantum Electron. 31, 625–636 (1999).
[CrossRef]

Horowitz, M.

Huang, S.

S. Huang, M. M. Ohn, M. LeBlanc, R. M. Measures, “Continuous arbitrary strain profile measurements with fiber Bragg gratings,” Smart Mater. Struct. 7, 248–256 (1998).
[CrossRef]

Jobst, G.

J. Dubendorfer, R. E. Kunz, G. Jobst, I. Moser, G. Urban, “Integrated optical pH sensor using replicated chirped grating coupler sensor chips,” Sens. Actuators B 50, 210–219 (1998).
[CrossRef]

Kailath, T.

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 45, 312–335 (1985).
[CrossRef]

Katzir, A.

Keren, S.

Kersey, A. D.

A. D. Kersey, “A Review of recent developments in fiber optic sensors technology,” Opt. Fiber Technol. 2, 291–317 (1996).
[CrossRef]

Khijwania, S. K.

S. K. Khijwania, B. D. Gupta, “Fiber optic evanscent field absorption sensor: effect of fiber parameters and geometry of the probe,” Opt. Quantum Electron. 31, 625–636 (1999).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Kunz, R. E.

J. Dubendorfer, R. E. Kunz, G. Jobst, I. Moser, G. Urban, “Integrated optical pH sensor using replicated chirped grating coupler sensor chips,” Sens. Actuators B 50, 210–219 (1998).
[CrossRef]

Kychakoff, G.

P. H. Paul, G. Kychakoff, “Fiber-optic evanescent field absorption sensor,” Appl. Phys. Lett. 51, 12–14 (1987).
[CrossRef]

Laming, R. I.

W. H. Loh, R. I. Laming, “1.55 µm phase-shifted distributed feedback fibre laser,” Electron. Lett. 31, 1440–1442 (1995).
[CrossRef]

LeBlanc, M.

S. Huang, M. M. Ohn, M. LeBlanc, R. M. Measures, “Continuous arbitrary strain profile measurements with fiber Bragg gratings,” Smart Mater. Struct. 7, 248–256 (1998).
[CrossRef]

Levit, B.

Levy, B. C.

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 45, 312–335 (1985).
[CrossRef]

Ligler, F. S.

G. P. Anderson, J. P. Golden, L. K. Cao, D. Wijesuriya, L. C. Shriver-Lake, F. S. Ligler, “Development of an evanescent wave fiber optic biosensor,” IEEE Eng. Med. Biol. Mag. 13, 358–363 (1994).
[CrossRef]

Loh, W. H.

W. H. Loh, R. I. Laming, “1.55 µm phase-shifted distributed feedback fibre laser,” Electron. Lett. 31, 1440–1442 (1995).
[CrossRef]

MacCraith, B. D.

V. Ruddy, B. D. MacCraith, J. A. Murphy, “Evanescent wave absorption spectroscopy using multimode fibers,” J. Appl. Phys. 67, 6070–6074 (1990).
[CrossRef]

Manolakis, D. G.

J. G. Proakis, D. G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, 3rd ed. (Prentice-Hall International, London, 1996).

Measures, R. M.

S. Huang, M. M. Ohn, M. LeBlanc, R. M. Measures, “Continuous arbitrary strain profile measurements with fiber Bragg gratings,” Smart Mater. Struct. 7, 248–256 (1998).
[CrossRef]

Messica, A.

Moser, I.

J. Dubendorfer, R. E. Kunz, G. Jobst, I. Moser, G. Urban, “Integrated optical pH sensor using replicated chirped grating coupler sensor chips,” Sens. Actuators B 50, 210–219 (1998).
[CrossRef]

Mueller, R.

K. Schroeder, W. Ecke, R. Mueller, R. Willsch, A. Andreev, “A fibre Bragg grating refractometer,” Meas. Sci. Technol. 12, 757–764 (2001).
[CrossRef]

Muriel, M. A.

R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

Murphy, J. A.

V. Ruddy, B. D. MacCraith, J. A. Murphy, “Evanescent wave absorption spectroscopy using multimode fibers,” J. Appl. Phys. 67, 6070–6074 (1990).
[CrossRef]

Ohn, M. M.

S. Huang, M. M. Ohn, M. LeBlanc, R. M. Measures, “Continuous arbitrary strain profile measurements with fiber Bragg gratings,” Smart Mater. Struct. 7, 248–256 (1998).
[CrossRef]

Papoulis, A.

A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962).

Paul, P. H.

P. H. Paul, G. Kychakoff, “Fiber-optic evanescent field absorption sensor,” Appl. Phys. Lett. 51, 12–14 (1987).
[CrossRef]

Proakis, J. G.

J. G. Proakis, D. G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, 3rd ed. (Prentice-Hall International, London, 1996).

Rosenthal, A.

S. Keren, A. Rosenthal, M. Horowitz, “Measuring the structure of highly reflecting fiber Bragg gratings,” IEEE Photon. Technol. Lett. 15, 575–577 (2003).
[CrossRef]

A. Rosenthal, M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 1018–1026 (2003).
[CrossRef]

V. Goloborodko, S. Keren, A. Rosenthal, B. Levit, M. Horowitz, “Measuring temperature profiles in high-power optical components,” Appl. Opt. 42, 2284–2288 (2003).
[CrossRef] [PubMed]

Ruddy, V.

V. Ruddy, B. D. MacCraith, J. A. Murphy, “Evanescent wave absorption spectroscopy using multimode fibers,” J. Appl. Phys. 67, 6070–6074 (1990).
[CrossRef]

Schroeder, K.

K. Schroeder, W. Ecke, R. Mueller, R. Willsch, A. Andreev, “A fibre Bragg grating refractometer,” Meas. Sci. Technol. 12, 757–764 (2001).
[CrossRef]

Segur, H.

M. J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform (Society for Applied Mathematics, Philadelphia, Pa., 1981).

Shriver-Lake, L. C.

G. P. Anderson, J. P. Golden, L. K. Cao, D. Wijesuriya, L. C. Shriver-Lake, F. S. Ligler, “Development of an evanescent wave fiber optic biosensor,” IEEE Eng. Med. Biol. Mag. 13, 358–363 (1994).
[CrossRef]

Skaar, J.

J. Skaar, L. Wang, T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” J. Lightwave Technol. 37, 165–173 (2001).

Tai, H.

Tanaka, H.

Urban, G.

J. Dubendorfer, R. E. Kunz, G. Jobst, I. Moser, G. Urban, “Integrated optical pH sensor using replicated chirped grating coupler sensor chips,” Sens. Actuators B 50, 210–219 (1998).
[CrossRef]

Volanthen, M.

M. Volanthen, H. Geiger, M. J. Cole, J. P. Dakin, “Measurement of arbitrary strain profiles within fibre gratings,” Electron. Lett. 32, 1028–1029 (1996).
[CrossRef]

Wang, L.

J. Skaar, L. Wang, T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” J. Lightwave Technol. 37, 165–173 (2001).

Wijesuriya, D.

G. P. Anderson, J. P. Golden, L. K. Cao, D. Wijesuriya, L. C. Shriver-Lake, F. S. Ligler, “Development of an evanescent wave fiber optic biosensor,” IEEE Eng. Med. Biol. Mag. 13, 358–363 (1994).
[CrossRef]

Willsch, R.

K. Schroeder, W. Ecke, R. Mueller, R. Willsch, A. Andreev, “A fibre Bragg grating refractometer,” Meas. Sci. Technol. 12, 757–764 (2001).
[CrossRef]

Yagle, A. E.

J. Frolik, A. E. Yagle, “Forward and inverse scattering for discrete layered lossy and absorbing media,” IEEE Trans. Circuits Syst. 44, 710–722 (1997).
[CrossRef]

Yoshino, T.

P. K. Choudhury, T. Yoshino, “On the optical sensing means for the study of chemical kinetics,” Meas. Sci. Technol. 13, 1793–1797 (2002).
[CrossRef]

H. Tai, H. Tanaka, T. Yoshino, “Fiber-optic evanescent-wave methane-gas sensor using optical absorption for the 3.392-µm line of a He–Ne laser,” Opt. Lett. 12, 437–439 (1987).
[CrossRef] [PubMed]

Zervas, M. N.

R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

P. H. Paul, G. Kychakoff, “Fiber-optic evanescent field absorption sensor,” Appl. Phys. Lett. 51, 12–14 (1987).
[CrossRef]

Bell Syst. Tech. J.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Electron. Lett.

W. H. Loh, R. I. Laming, “1.55 µm phase-shifted distributed feedback fibre laser,” Electron. Lett. 31, 1440–1442 (1995).
[CrossRef]

M. Volanthen, H. Geiger, M. J. Cole, J. P. Dakin, “Measurement of arbitrary strain profiles within fibre gratings,” Electron. Lett. 32, 1028–1029 (1996).
[CrossRef]

IEEE Eng. Med. Biol. Mag.

G. P. Anderson, J. P. Golden, L. K. Cao, D. Wijesuriya, L. C. Shriver-Lake, F. S. Ligler, “Development of an evanescent wave fiber optic biosensor,” IEEE Eng. Med. Biol. Mag. 13, 358–363 (1994).
[CrossRef]

IEEE J. Quantum Electron.

R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

A. Rosenthal, M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 1018–1026 (2003).
[CrossRef]

IEEE Photon. Technol. Lett.

S. Keren, A. Rosenthal, M. Horowitz, “Measuring the structure of highly reflecting fiber Bragg gratings,” IEEE Photon. Technol. Lett. 15, 575–577 (2003).
[CrossRef]

IEEE Trans. Circuits Syst.

J. Frolik, A. E. Yagle, “Forward and inverse scattering for discrete layered lossy and absorbing media,” IEEE Trans. Circuits Syst. 44, 710–722 (1997).
[CrossRef]

J. Appl. Phys.

V. Ruddy, B. D. MacCraith, J. A. Murphy, “Evanescent wave absorption spectroscopy using multimode fibers,” J. Appl. Phys. 67, 6070–6074 (1990).
[CrossRef]

J. Lightwave Technol.

J. Skaar, L. Wang, T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” J. Lightwave Technol. 37, 165–173 (2001).

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

Meas. Sci. Technol.

P. K. Choudhury, T. Yoshino, “On the optical sensing means for the study of chemical kinetics,” Meas. Sci. Technol. 13, 1793–1797 (2002).
[CrossRef]

K. Schroeder, W. Ecke, R. Mueller, R. Willsch, A. Andreev, “A fibre Bragg grating refractometer,” Meas. Sci. Technol. 12, 757–764 (2001).
[CrossRef]

Opt. Fiber Technol.

A. D. Kersey, “A Review of recent developments in fiber optic sensors technology,” Opt. Fiber Technol. 2, 291–317 (1996).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

S. K. Khijwania, B. D. Gupta, “Fiber optic evanscent field absorption sensor: effect of fiber parameters and geometry of the probe,” Opt. Quantum Electron. 31, 625–636 (1999).
[CrossRef]

Sens. Actuators B

J. Dubendorfer, R. E. Kunz, G. Jobst, I. Moser, G. Urban, “Integrated optical pH sensor using replicated chirped grating coupler sensor chips,” Sens. Actuators B 50, 210–219 (1998).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Appl. Math.

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 45, 312–335 (1985).
[CrossRef]

Smart Mater. Struct.

S. Huang, M. M. Ohn, M. LeBlanc, R. M. Measures, “Continuous arbitrary strain profile measurements with fiber Bragg gratings,” Smart Mater. Struct. 7, 248–256 (1998).
[CrossRef]

Other

M. J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform (Society for Applied Mathematics, Philadelphia, Pa., 1981).

A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962).

J. G. Proakis, D. G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, 3rd ed. (Prentice-Hall International, London, 1996).

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Figures (4)

Fig. 1
Fig. 1

Reconstruction of a grating with a chirped Gaussian coupling coefficient, q(z)=600 exp[-105(z-L/2)2(2.5+20i)] m-1 and with a sinusoidal loss profile α=70[1-cos(10πz/L)] m-1 written in the region [0, L=1 cm]. The reconstructed parameters (solid curves) are compared with the original parameters (dashed curves). The reflection spectra, obtained from both sides of the grating, and the transmission spectrum were sampled over a bandwidth of 10 nm with a spectral resolution of 0.02 nm.

Fig. 2
Fig. 2

Reconstruction of a grating with a coupling coefficient, |q|=30 m-1, a Gaussian phase derivative, d/dz[arg(q)]=120 exp[-(z-L/2)2/1.6×10-5] m-1, and a Gaussian loss coefficient α(z)=100 exp[-(z-L/2)2/1.6×10-5] m-1, written in the region [0, L=2 cm]. The reconstructed parameters (solid curves) are compared with the original parameters (dashed curves). The reflection spectra, obtained from both sides of the grating, and the transmission spectrum were sampled over a bandwidth of 10 nm with a spectral resolution of 0.005 nm.

Fig. 3
Fig. 3

(a) Fourier transform of the forward reflection function of an unstable conjugate system and (b) the forward impulse response, calculated with the method described in Section 3. The coupling-coefficient profile of the grating was equal to q(z)=250[1+0.9 cos(2πz/L)] m-1, and the loss profile was given by α(z)=250[1-cos(2πz/L)] m-1. The grating was written in the region [0, L=4 mm]. The Fourier transform of the grating reflection spectrum gives a wrong result since it corresponds to a noncausal function.

Fig. 4
Fig. 4

Reconstruction of a grating with an unstable conjugate system, analyzed in Fig. 3. The grating was written in the region [0, L=4 mm] and had a coupling-coefficient profile of q(z)=250[1+0.9 cos(2πz/L)] m-1 and a loss profile α(z)=250[1-cos(2πz/L)] m-1. Since the conjugate system is unstable, the impulse-response function was calculated with the method described in Section 3. The reconstructed parameters (solid curves) are compared with the original parameters (dashed curves).

Equations (40)

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neff(z)=navg+n0(z)+iη(z)+n1(z)sin2πΛz+θ(z),
E(x, y, z, t)=[a1(z)exp(-iβz)+a2(z)exp(iβz)]e(x, y)exp(-iωt),
da1(z)dz=-ia1(z)[σ(z)+iα(z)]+a2(z)κ(z)×exp(-iθ(z)+2ikz),
da2(z)dz=ia2(z)[σ(z)+iα(z)]+a1(k)κ(z)×exp(iθ(z)-2ikz),
σ(z)=ωcn0(x, y, z)|e(x, y)|2dxdy|e(x, y)|2dxdy,
α(z)=ωcη(x, y, z)|e(x, y)|2dxdy|e(x, y)|2dxdy,
κ(z)=ωcn1(x, y, z)|e(x, y)|2dxdy|e(x, y)|2dxdy.
v1(k, z)=a1(k, z)×exp-ikz-0zα(ζ)dζ+i0zσ(ζ)dζ,
v2(k, z)=a2(k, z)×exp+ikz+0zα(ζ)dζ-i0zσ(ζ)dζ,
dV(k, z)dz=-ikq1(z)q2(z)ikV(k, z),
q1(z)=q(z)exp-20zα(ζ)dζ,
q2(z)=q*(z)exp+20zα(ζ)dζ.
q(z)=κ(z)exp-iθ(z)+2i0zσ(ζ)dζ.
Φ(k, z=0)=10,Ψ(k, z=L)=01,
Φ¯(k, z=0)=01,Ψ¯(k, z=L)=10.
Ψ(k, z)=a(k)Φ¯(k, z)+b(k)Φ(k, z),
Ψ¯(k, z)=a¯(k)Φ(k, z)+b¯(k)Φ¯(k, z).
b(k)b¯(k)-a(k)a¯(k)=-1.
dU˜(k, z)dz=-ikq2(z)q1(z)ikU˜(k, z),
r˜f(k)=-rb(-k)[t(-k)]2-rf(-k)rb(-k).
rmf(k)rf(k),
tm(k)exp+ikL+0Lα(ζ)dζ-i0Lσ(ζ)dζt(k),
rmb(k)exp+2ikL+20Lα(ζ)dζ-2i0Lσ(ζ)dζ
rb(k).
h˜f(τ)=12π-+iμ+iμr˜f(κ)exp(-iκτ)dκ,
Ψ¯(κ, z)=10exp[-iκ(z-L)]+z-LL-zK¯(τ, z)exp(iκτ)dτ,0zL.
a¯(τ)b¯(τ)=δ(τ-L)0+K¯(τ, 0).
a¯(-k-iμ)b¯(-k-iμ)=exp[-i(k+iμ)L]0+-LLK¯(τ, 0)exp[-i(k+iμ)τ]dτ,
u1(k, (n+1)Δz)u2(k, (n+1)Δz)=Tn(k)u1(k, nΔz)u2(k, nΔz),
Tn=(1-q1,nq2,nΔz2)-1/2×exp(-ikΔz)00exp(ikΔz)1q1,nΔzq2,nΔz1,
T˜n=(1-q1,nq2,nΔz2)-1/2exp(-ikΔz)00exp(ikΔz)×1q2,nΔzq1,nΔz1.
rf(k, z)=u1(k, z)/u2(k, z),
r˜f(k, z)=u˜1(k, z)/u˜2(k, z).
rn+1f(k)=exp(-2ikΔz)rnf(k)+q1,n1+rnf(k)q2,n,
r˜n+1f(k)=exp(-2ikΔz)r˜nf(k)+q2,n1+r˜nf(k)q1,n.
q1,n=-1MΔzm=1Mrnf(km).
q2,n=-1MΔzm=1Mr˜nf(km).
r˜n+1f(k+iμ)=exp(-2i(k+iμ)Δz)r˜nf(k+iμ)+q2,n1+r˜nf(k+iμ)q1,n
q2,n=-1MΔzm=1Mr˜f(km+iμ)
f(τ)=12π-r˜f(k)exp(-ikτ)dk=j=1Ncj exp(-iκjτ)forτ<0,

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