Abstract

Several methods exist to measure the group delay of a fiber Bragg grating. Here, we compare two such methods, namely the Hilbert transform (HT) of the device transmission spectrum and standard Fourier spectral interferometry. Numerical simulations demonstrate that both methods work not only for ideal, lossless devices but also for ones with realistic absorption. Experimental measurements show that the HT is more straightforward to implement and is significantly less susceptible to phase noise, which can significantly reduce the standard deviation between measurements.

© 2014 Optical Society of America

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  1. F. Ouellette, “Dispersion cancellation using linearly chirped Bragg grating filters in optical waveguides,” Opt. Lett. 12, 847–849 (1987).
    [CrossRef]
  2. N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: theoretical model and design criteria for nearly ideal pulse recompression,” J. Lightwave Technol. 15, 1303–1313 (1997).
    [CrossRef]
  3. 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, 58–65 (1996).
    [CrossRef]
  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]
  5. H. Wen, M. Terrel, S. Fan, and M. Digonnet, “Sensing with slow light in fiber Bragg gratings,” IEEE Sens. J. 12, 156–163 (2012).
    [CrossRef]
  6. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
    [CrossRef]
  7. A. Melloni, M. Chinello, and M. Martinelli, “All-optical switching in phase-shifted fiber Bragg grating,” IEEE Photon. Technol. Lett. 12, 42–44 (2000).
    [CrossRef]
  8. R. W. Boyd, “Material slow light and structural slow light: similarities and differences for nonlinear optics,” J. Opt. Soc. Am. B 28, A38–A44 (2011).
    [CrossRef]
  9. J. T. Mok, M. Ibsen, C. M. de Sterke, and B. J. Eggleton, “Dispersionless slow light with 5-pulse-width delay in fibre Bragg grating,” Electron. Lett. 43, 1418–1419 (2007).
    [CrossRef]
  10. H. Shahoei, M. Li, and J. Yao, “Continuously tunable time delay using an optically pumped linear chirped fiber Bragg grating,” J. Lightwave Technol. 29, 1465–1472 (2011).
    [CrossRef]
  11. A. Gomez-Iglesias, D. O’Brien, L. O’Faolain, A. Miller, and T. Krauss, “Direct measurement of the group index of photonic crystal waveguides via Fourier transform spectral interferometry,” Appl. Phys. Lett. 90, 261107 (2007).
    [CrossRef]
  12. C. Dorrer, N. Belabas, J.-P. Likforman, and M. Joffre, “Spectral resolution and sampling issues in Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B 17, 1795–1802 (2000).
    [CrossRef]
  13. K.-E. Peiponen and E. M. Vartiainen, “Kramers-Kronig relations in optical data inversion,” Phys. Rev. B 44, 8301–8303 (1991).
    [CrossRef]
  14. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).
  15. A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, 1962).
  16. A. Carballar and C. Janer, “Complete fiber Bragg grating characterization using an alternative method based on spectral interferometry and minimum-phase reconstruction algorithms,” J. Lightwave Technol. 30, 2574–2582 (2012).
    [CrossRef]
  17. A. Mecozzi, “Retrieving the full optical response from amplitude data by Hilbert transform,” Opt. Commun. 282, 4183–4187 (2009).
    [CrossRef]
  18. R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk, “Kramers-Kronig relations for an interferometer,” Opt. Commun. 138, 118–126 (1997).
    [CrossRef]
  19. L. Poladian, “Group-delay reconstruction for fiber Bragg gratings in reflection and transmission,” Opt. Lett. 22, 1571–1573 (1997).
    [CrossRef]
  20. M. J. Erro, I. Arnedo, M. A. G. Laso, T. Lopetegi, and M. A. Muriel, “Phase-reconstruction in photonic crystals from S-parameter magnitude in microstrip technology,” Opt. Quantum Electron. 39, 321–331 (2007).
    [CrossRef]
  21. J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11, 1307–1320 (1994).
    [CrossRef]
  22. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
    [CrossRef]
  23. J. Upham, I. De Leon, D. Grobnic, E. Ma, M.-C. N. Dicaire, S. A. Schulz, S. Murugkar, and R. W. Boyd, “Enhancing optical field intensities in Gaussian-profile fiber Bragg gratings,” Opt. Lett. 39, 849–852 (2014).
    [CrossRef]
  24. I. C. M. Littler, T. Grujic, and B. J. Eggleton, “Photothermal effects in fiber Bragg gratings,” Appl. Opt. 45, 4679–4685 (2006).
    [CrossRef]

2014

2012

2011

2009

A. Mecozzi, “Retrieving the full optical response from amplitude data by Hilbert transform,” Opt. Commun. 282, 4183–4187 (2009).
[CrossRef]

2007

A. Gomez-Iglesias, D. O’Brien, L. O’Faolain, A. Miller, and T. Krauss, “Direct measurement of the group index of photonic crystal waveguides via Fourier transform spectral interferometry,” Appl. Phys. Lett. 90, 261107 (2007).
[CrossRef]

J. T. Mok, M. Ibsen, C. M. de Sterke, and B. J. Eggleton, “Dispersionless slow light with 5-pulse-width delay in fibre Bragg grating,” Electron. Lett. 43, 1418–1419 (2007).
[CrossRef]

M. J. Erro, I. Arnedo, M. A. G. Laso, T. Lopetegi, and M. A. Muriel, “Phase-reconstruction in photonic crystals from S-parameter magnitude in microstrip technology,” Opt. Quantum Electron. 39, 321–331 (2007).
[CrossRef]

2006

2000

A. Melloni, M. Chinello, and M. Martinelli, “All-optical switching in phase-shifted fiber Bragg grating,” IEEE Photon. Technol. Lett. 12, 42–44 (2000).
[CrossRef]

C. Dorrer, N. Belabas, J.-P. Likforman, and M. Joffre, “Spectral resolution and sampling issues in Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B 17, 1795–1802 (2000).
[CrossRef]

1997

R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk, “Kramers-Kronig relations for an interferometer,” Opt. Commun. 138, 118–126 (1997).
[CrossRef]

L. Poladian, “Group-delay reconstruction for fiber Bragg gratings in reflection and transmission,” Opt. Lett. 22, 1571–1573 (1997).
[CrossRef]

N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: theoretical model and design criteria for nearly ideal pulse recompression,” J. Lightwave Technol. 15, 1303–1313 (1997).
[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]

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

1996

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, 58–65 (1996).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef]

1994

1991

K.-E. Peiponen and E. M. Vartiainen, “Kramers-Kronig relations in optical data inversion,” Phys. Rev. B 44, 8301–8303 (1991).
[CrossRef]

1987

Arnedo, I.

M. J. Erro, I. Arnedo, M. A. G. Laso, T. Lopetegi, and M. A. Muriel, “Phase-reconstruction in photonic crystals from S-parameter magnitude in microstrip technology,” Opt. Quantum Electron. 39, 321–331 (2007).
[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]

Belabas, N.

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, 58–65 (1996).
[CrossRef]

Boyd, R. W.

Carballar, A.

Chinello, M.

A. Melloni, M. Chinello, and M. Martinelli, “All-optical switching in phase-shifted fiber Bragg grating,” IEEE Photon. Technol. Lett. 12, 42–44 (2000).
[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]

De Leon, I.

de Sterke, C. M.

J. T. Mok, M. Ibsen, C. M. de Sterke, and B. J. Eggleton, “Dispersionless slow light with 5-pulse-width delay in fibre Bragg grating,” Electron. Lett. 43, 1418–1419 (2007).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef]

J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11, 1307–1320 (1994).
[CrossRef]

de Vries, P.

R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk, “Kramers-Kronig relations for an interferometer,” Opt. Commun. 138, 118–126 (1997).
[CrossRef]

Dicaire, M.-C. N.

Digonnet, M.

H. Wen, M. Terrel, S. Fan, and M. Digonnet, “Sensing with slow light in fiber Bragg gratings,” IEEE Sens. J. 12, 156–163 (2012).
[CrossRef]

Dorrer, C.

Eggleton, B. J.

J. T. Mok, M. Ibsen, C. M. de Sterke, and B. J. Eggleton, “Dispersionless slow light with 5-pulse-width delay in fibre Bragg grating,” Electron. Lett. 43, 1418–1419 (2007).
[CrossRef]

I. C. M. Littler, T. Grujic, and B. J. Eggleton, “Photothermal effects in fiber Bragg gratings,” Appl. Opt. 45, 4679–4685 (2006).
[CrossRef]

N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: theoretical model and design criteria for nearly ideal pulse recompression,” J. Lightwave Technol. 15, 1303–1313 (1997).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef]

Erdogan, T.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (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, 58–65 (1996).
[CrossRef]

Erro, M. J.

M. J. Erro, I. Arnedo, M. A. G. Laso, T. Lopetegi, and M. A. Muriel, “Phase-reconstruction in photonic crystals from S-parameter magnitude in microstrip technology,” Opt. Quantum Electron. 39, 321–331 (2007).
[CrossRef]

Fan, S.

H. Wen, M. Terrel, S. Fan, and M. Digonnet, “Sensing with slow light in fiber Bragg gratings,” IEEE Sens. J. 12, 156–163 (2012).
[CrossRef]

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]

Gomez-Iglesias, A.

A. Gomez-Iglesias, D. O’Brien, L. O’Faolain, A. Miller, and T. Krauss, “Direct measurement of the group index of photonic crystal waveguides via Fourier transform spectral interferometry,” Appl. Phys. Lett. 90, 261107 (2007).
[CrossRef]

Grobnic, D.

Grujic, T.

Ibsen, M.

J. T. Mok, M. Ibsen, C. M. de Sterke, and B. J. Eggleton, “Dispersionless slow light with 5-pulse-width delay in fibre Bragg grating,” Electron. Lett. 43, 1418–1419 (2007).
[CrossRef]

Janer, C.

Joffre, M.

Judkins, J. B.

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, 58–65 (1996).
[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]

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]

Kop, R. H. J.

R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk, “Kramers-Kronig relations for an interferometer,” Opt. Commun. 138, 118–126 (1997).
[CrossRef]

Krauss, T.

A. Gomez-Iglesias, D. O’Brien, L. O’Faolain, A. Miller, and T. Krauss, “Direct measurement of the group index of photonic crystal waveguides via Fourier transform spectral interferometry,” Appl. Phys. Lett. 90, 261107 (2007).
[CrossRef]

Krug, P. A.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef]

Lagendijk, A.

R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk, “Kramers-Kronig relations for an interferometer,” Opt. Commun. 138, 118–126 (1997).
[CrossRef]

Laso, M. A. G.

M. J. Erro, I. Arnedo, M. A. G. Laso, T. Lopetegi, and M. A. Muriel, “Phase-reconstruction in photonic crystals from S-parameter magnitude in microstrip technology,” Opt. Quantum Electron. 39, 321–331 (2007).
[CrossRef]

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]

Lemaire, P. J.

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, 58–65 (1996).
[CrossRef]

Li, M.

Likforman, J.-P.

Litchinitser, N. M.

N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: theoretical model and design criteria for nearly ideal pulse recompression,” J. Lightwave Technol. 15, 1303–1313 (1997).
[CrossRef]

Littler, I. C. M.

Lopetegi, T.

M. J. Erro, I. Arnedo, M. A. G. Laso, T. Lopetegi, and M. A. Muriel, “Phase-reconstruction in photonic crystals from S-parameter magnitude in microstrip technology,” Opt. Quantum Electron. 39, 321–331 (2007).
[CrossRef]

Ma, E.

Martinelli, M.

A. Melloni, M. Chinello, and M. Martinelli, “All-optical switching in phase-shifted fiber Bragg grating,” IEEE Photon. Technol. Lett. 12, 42–44 (2000).
[CrossRef]

Mecozzi, A.

A. Mecozzi, “Retrieving the full optical response from amplitude data by Hilbert transform,” Opt. Commun. 282, 4183–4187 (2009).
[CrossRef]

Melloni, A.

A. Melloni, M. Chinello, and M. Martinelli, “All-optical switching in phase-shifted fiber Bragg grating,” IEEE Photon. Technol. Lett. 12, 42–44 (2000).
[CrossRef]

Miller, A.

A. Gomez-Iglesias, D. O’Brien, L. O’Faolain, A. Miller, and T. Krauss, “Direct measurement of the group index of photonic crystal waveguides via Fourier transform spectral interferometry,” Appl. Phys. Lett. 90, 261107 (2007).
[CrossRef]

Mok, J. T.

J. T. Mok, M. Ibsen, C. M. de Sterke, and B. J. Eggleton, “Dispersionless slow light with 5-pulse-width delay in fibre Bragg grating,” Electron. Lett. 43, 1418–1419 (2007).
[CrossRef]

Muriel, M. A.

M. J. Erro, I. Arnedo, M. A. G. Laso, T. Lopetegi, and M. A. Muriel, “Phase-reconstruction in photonic crystals from S-parameter magnitude in microstrip technology,” Opt. Quantum Electron. 39, 321–331 (2007).
[CrossRef]

Murugkar, S.

O’Brien, D.

A. Gomez-Iglesias, D. O’Brien, L. O’Faolain, A. Miller, and T. Krauss, “Direct measurement of the group index of photonic crystal waveguides via Fourier transform spectral interferometry,” Appl. Phys. Lett. 90, 261107 (2007).
[CrossRef]

O’Faolain, L.

A. Gomez-Iglesias, D. O’Brien, L. O’Faolain, A. Miller, and T. Krauss, “Direct measurement of the group index of photonic crystal waveguides via Fourier transform spectral interferometry,” Appl. Phys. Lett. 90, 261107 (2007).
[CrossRef]

Ouellette, F.

Papoulis, A.

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

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]

Patterson, D. B.

N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: theoretical model and design criteria for nearly ideal pulse recompression,” J. Lightwave Technol. 15, 1303–1313 (1997).
[CrossRef]

Peiponen, K.-E.

K.-E. Peiponen and E. M. Vartiainen, “Kramers-Kronig relations in optical data inversion,” Phys. Rev. B 44, 8301–8303 (1991).
[CrossRef]

Poladian, L.

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]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

Schulz, S. A.

Shahoei, H.

Sipe, J. E.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (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, 58–65 (1996).
[CrossRef]

J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11, 1307–1320 (1994).
[CrossRef]

Slusher, R. E.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef]

Sprik, R.

R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk, “Kramers-Kronig relations for an interferometer,” Opt. Commun. 138, 118–126 (1997).
[CrossRef]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

Terrel, M.

H. Wen, M. Terrel, S. Fan, and M. Digonnet, “Sensing with slow light in fiber Bragg gratings,” IEEE Sens. J. 12, 156–163 (2012).
[CrossRef]

Upham, J.

Vartiainen, E. M.

K.-E. Peiponen and E. M. Vartiainen, “Kramers-Kronig relations in optical data inversion,” Phys. Rev. B 44, 8301–8303 (1991).
[CrossRef]

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, 58–65 (1996).
[CrossRef]

Wen, H.

H. Wen, M. Terrel, S. Fan, and M. Digonnet, “Sensing with slow light in fiber Bragg gratings,” IEEE Sens. J. 12, 156–163 (2012).
[CrossRef]

Yao, J.

Appl. Opt.

Appl. Phys. Lett.

A. Gomez-Iglesias, D. O’Brien, L. O’Faolain, A. Miller, and T. Krauss, “Direct measurement of the group index of photonic crystal waveguides via Fourier transform spectral interferometry,” Appl. Phys. Lett. 90, 261107 (2007).
[CrossRef]

Electron. Lett.

J. T. Mok, M. Ibsen, C. M. de Sterke, and B. J. Eggleton, “Dispersionless slow light with 5-pulse-width delay in fibre Bragg grating,” Electron. Lett. 43, 1418–1419 (2007).
[CrossRef]

IEEE Photon. Technol. Lett.

A. Melloni, M. Chinello, and M. Martinelli, “All-optical switching in phase-shifted fiber Bragg grating,” IEEE Photon. Technol. Lett. 12, 42–44 (2000).
[CrossRef]

IEEE Sens. J.

H. Wen, M. Terrel, S. Fan, and M. Digonnet, “Sensing with slow light in fiber Bragg gratings,” IEEE Sens. J. 12, 156–163 (2012).
[CrossRef]

J. Lightwave Technol.

N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: theoretical model and design criteria for nearly ideal pulse recompression,” J. Lightwave Technol. 15, 1303–1313 (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, 58–65 (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]

H. Shahoei, M. Li, and J. Yao, “Continuously tunable time delay using an optically pumped linear chirped fiber Bragg grating,” J. Lightwave Technol. 29, 1465–1472 (2011).
[CrossRef]

A. Carballar and C. Janer, “Complete fiber Bragg grating characterization using an alternative method based on spectral interferometry and minimum-phase reconstruction algorithms,” J. Lightwave Technol. 30, 2574–2582 (2012).
[CrossRef]

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

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

A. Mecozzi, “Retrieving the full optical response from amplitude data by Hilbert transform,” Opt. Commun. 282, 4183–4187 (2009).
[CrossRef]

R. H. J. Kop, P. de Vries, R. Sprik, and A. Lagendijk, “Kramers-Kronig relations for an interferometer,” Opt. Commun. 138, 118–126 (1997).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

M. J. Erro, I. Arnedo, M. A. G. Laso, T. Lopetegi, and M. A. Muriel, “Phase-reconstruction in photonic crystals from S-parameter magnitude in microstrip technology,” Opt. Quantum Electron. 39, 321–331 (2007).
[CrossRef]

Phys. Rev. B

K.-E. Peiponen and E. M. Vartiainen, “Kramers-Kronig relations in optical data inversion,” Phys. Rev. B 44, 8301–8303 (1991).
[CrossRef]

Phys. Rev. Lett.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef]

Other

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

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

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Figures (6)

Fig. 1.
Fig. 1.

Experimental transmittance (dashed red) and reflectance (solid blue) curves of a rectified Gaussian FBG. The narrow peaks are the result of a resonant cavity inside the photonic bandgap. Inset: bandgap diagram of the same device along the fiber axis. The shaded area represents the bandgap, where propagation of light is forbidden. At short wavelengths, the propagating region is bounded by two forbidden regions, thus creating a cavity where light resonates.

Fig. 2.
Fig. 2.

Sketch of a Mach–Zehnder interferometer, suitable for the FTSI method. The optical input is split into two arms, one short reference arm and a longer arm containing the sample under investigation, where τF is the time delay associated with the total length difference between the two arms at frequencies far from the bandgap.

Fig. 3.
Fig. 3.

Numerical group delay spectra for a lossless rectified Gaussian FBG. The different lines are hard to distinguish as the HT and FTSI derived group delays agree well with the actual group delay.

Fig. 4.
Fig. 4.

Same case as Fig. 3, in the presence of absorption losses. Even with loss, both the HT and FTSI derived group delays still agree well with the true value.

Fig. 5.
Fig. 5.

Group delay spectra obtained experimentally with FTSI (blue) and HT (red) methods. The lines represent the average group delay values, while the shaded regions represent the associated standard deviation. Top: overview of the whole region of interest. The HT method has a large uncertainty between the peaks, as there is no transmission: only random noise, with random phase, is detected. The FTSI method has less noise in that region because there is still transmission from the reference arm. However, this is not a concern since we are only interested in the peak values. Bottom images: zoomed-in spectra of the group delay peaks. The average values are in good agreement, and the standard deviation for the HT method is consistently lower than the FTSI standard deviation, indicating a cleaner measurement.

Fig. 6.
Fig. 6.

Standard deviation of the group delay peaks, normalized to the peak height. The FTSI method (blue crosses) has large uncertainties, ranging from 25% to 185%. The HT method’s (red dots) uncertainties are much better, at most 10% of the peak height.

Equations (10)

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τg(ω)=dϕ(ω)dω.
[I[G(ω0)]]=1π+I[G(ω)]ωω0dω=R[G(ω0)],
[I[G(ω)]]=F1[isgn(t)·F[I[G(ω)]]],
t(ω)=T(ω)eiϕ(ω),
lnt(ω)=lnT(ω)2+iϕ(ω).
ϕ(ω0)=12πω1ω2ln[T(ω)]ωω0dωω1τ1,
I(ω)=S(ω)+R(ω)+2S(ω)R(ω)cos(ϕ(ω)ωτF).
ϕ(ω)=arg[F1(W(t)·F(I(ω)))]+ωτF,
τg(ω)=ddω{arg[F1(W(t)·F(I(ω)))]}τF+τ1,
τg(ω)=ddω{F1(isgn(t)·F(ln(T(ω))2))}+τ1,

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