Abstract

A superresolution technique for the measurement of transmission, reflection, and absorption spectra is proposed. An ultrashort laser pulse is propagated in a dispersive element and then periodically phase modulated. The temporal modulation is transformed into periodic spectral modulation, for which the number of harmonics, 2M+1, is determined by the modulation index. The modulated pulse is transmitted through (reflected from) the sample to be tested and measured by a spectrometer. By performing 2M+1 measurements for 2M+1 delays between the dispersed pulse and modulation signal, one can restore the spectral response of the sample with superresolution after simple processing. We numerically demonstrate the measurement of the transmission spectrum of an ultranarrow optical filter with a minimum feature of 0.43 pm by an optical spectrum analyzer with a 10 pm resolution. A twentyfold enhancement of the resolution is achieved in the presence of noise with a level of 0.1%. The advantage of the system is its full reconfigurability.

© 2012 Optical Society of America

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References

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  1. Y. Ozaki, S. Šašić, and J. H. Jiang, “How can we unravel complicated near infrared spectra? Recent progress in spectral analysis methods for resolution enhancement and band assignments in the near infrared region,” J. Near Infrared Spectrosc. 9, 63–95 (2001).
    [CrossRef]
  2. D. K. Buslov, N. A. Nikonenko, N. I. Sushko, and R. G. Zhbankov, “Resolution enhancement in IR spectra of carbohydrates by the deconvolution method and comparison of the results with low-temperature spectra,” Appl. Spectrosc. 54, 1651–1658 (2000).
    [CrossRef]
  3. V. A. Lórenz-Fonfría, J. Villaverde, and E. Padrós, “Fourier deconvolution in non-self-deconvolving conditions. Effective narrowing, signal-to-noise degradation, and curve fitting,” Appl. Spectrosc. 56, 232–242 (2002).
    [CrossRef]
  4. T. Ohara, H. Takara, T. Yamamoto, H. Masuda, T. Morioka, M. Abe, and H. Takahashi, “Over-1000-channel ultradense WDM transmission with supercontinuum multicarrier source,” J. Lightwave Technol. 24, 2311–2317 (2006).
    [CrossRef]
  5. X. Liu, A. Lin, G. Sun, D. S. Moon, D. Hwang, and Y. Chung, “Identical-dual-bandpass sampled fiber Bragg grating and its application to ultranarrow filters,” Appl. Opt. 47, 5637–5643 (2008).
    [CrossRef]
  6. M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
    [CrossRef]
  7. J. Azaña, N. K. Berger, B. Levit, and B. Fischer, “Spectro-temporal imaging of optical pulses with a single time lens,” IEEE Photon. Technol. Lett. 16, 882–884 (2004).
    [CrossRef]
  8. T. Mansuryan, A. Zeytunyan, M. Kalashyan, G. Yesayan, L. Mouradian, F. Louradour, and A. Barthélémy, “Parabolic temporal lensing and spectrotemporal imaging: a femtosecond optical oscilloscope,” J. Opt. Soc. Am. B 25, A101–A110 (2008).
    [CrossRef]
  9. D. J. Erskine, J. Edelstein, W. M. Feuerstein, and B. Welsh, “High-resolution broadband spectroscopy using an externally dispersed interferometer,” Astrophys. J. 592, L103–L106 (2003).
    [CrossRef]
  10. A. Brablec, D. Trunec, and F. Štastný, “Deconvolution of spectral line profiles: solution of the inversion problem,” J. Phys. D 32, 1870–1875 (1999).
    [CrossRef]
  11. G. M. Petrov, “A simple algorithm for spectral line deconvolution,” J. Quant. Spectrosc. Radiat. Transfer 72, 281–287 (2002).
    [CrossRef]
  12. M. Morháč and V. Matoušek, “Complete positive deconvolution of spectrometric data,” Digit. Signal Process. 19, 372–392 (2009).
    [CrossRef]
  13. A. S. Kaminskii, E. L. Kosarev, and E. V. Lavrov, “Using comb-like instrumental functions in high-resolution spectroscopy,” Meas. Sci. Technol. 8, 864–870 (1997).
    [CrossRef]
  14. P. A. Jansson, “Modern constrained nonlinear methods,” in Deconvolution of Images and Spectra, P. A. Jansson, ed. (Academic, 1997), pp. 107–181.
  15. J. L. Harris, “Diffraction and resolving power,” J. Opt. Soc. Am. 54, 931–936 (1964).
    [CrossRef]
  16. W. E. Blass and G. W. Halsey, “Instrumental considerations,” in Deconvolution of Images and Spectra, P. A. Jansson, ed. (Academic, 1997), pp. 200–235.
  17. V. Torres-Company, J. Lancis, and P. Andrés, “Spectral imaging system for scaling the power spectrum of optical waveforms,” Opt. Lett. 32, 2849–2851 (2007).
    [CrossRef]
  18. Y. Okawachi, R. Salem, M. A. Foster, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “High-resolution spectroscopy using a frequency magnifier,” Opt. Express 17, 5691–5697 (2009).
    [CrossRef]
  19. B. Szafraniec, A. Lee, J. Y. Law, W. I. McAlexander, R. D. Pering, T. S. Tan, and D. M. Baney, “Swept coherent optical spectrum analysis,” IEEE Trans. Instrum. Meas. 53, 203–215 (2004).
    [CrossRef]
  20. J. M. Helbert, P. Laforie, and P. Miche, “Conditions of pressure scanning of a Fabry–Perot interferometer over a wide spectrum range,” Appl. Opt. 16, 2119–2126 (1977).
    [CrossRef]
  21. N. K. Berger, “Spectral measurements with superresolution based on periodic modulation of the spectrum,” Appl. Opt. 47, 6535–6542 (2008).
    [CrossRef]
  22. N. K. Berger, “Enhancement of resolution of optical spectrum analysers with thermally tuned sampled fibre Bragg grating,” Electron. Lett. 46, 1457–1458 (2010).
    [CrossRef]
  23. P. Bousquet, Spectroscopy and Its Instrumentation (Hilger, 1971).
  24. N. K. Berger, B. Levit, and B. Fischer, “Measurement of fiber chromatic dispersion using spectral interferometry with modulation of dispersed laser pulses,” Opt. Commun. 283, 3953–3956 (2010).
    [CrossRef]
  25. N. K. Berger, B. Levit, B. Fischer, and J. Azaña, “Picosecond flat-top pulse generation by low-bandwidth electro-optic sinusoidal phase modulation,” Opt. Lett. 33, 125–127 (2008).
    [CrossRef]

2010 (2)

N. K. Berger, “Enhancement of resolution of optical spectrum analysers with thermally tuned sampled fibre Bragg grating,” Electron. Lett. 46, 1457–1458 (2010).
[CrossRef]

N. K. Berger, B. Levit, and B. Fischer, “Measurement of fiber chromatic dispersion using spectral interferometry with modulation of dispersed laser pulses,” Opt. Commun. 283, 3953–3956 (2010).
[CrossRef]

2009 (2)

2008 (4)

2007 (1)

2006 (1)

2004 (2)

J. Azaña, N. K. Berger, B. Levit, and B. Fischer, “Spectro-temporal imaging of optical pulses with a single time lens,” IEEE Photon. Technol. Lett. 16, 882–884 (2004).
[CrossRef]

B. Szafraniec, A. Lee, J. Y. Law, W. I. McAlexander, R. D. Pering, T. S. Tan, and D. M. Baney, “Swept coherent optical spectrum analysis,” IEEE Trans. Instrum. Meas. 53, 203–215 (2004).
[CrossRef]

2003 (1)

D. J. Erskine, J. Edelstein, W. M. Feuerstein, and B. Welsh, “High-resolution broadband spectroscopy using an externally dispersed interferometer,” Astrophys. J. 592, L103–L106 (2003).
[CrossRef]

2002 (2)

2001 (1)

Y. Ozaki, S. Šašić, and J. H. Jiang, “How can we unravel complicated near infrared spectra? Recent progress in spectral analysis methods for resolution enhancement and band assignments in the near infrared region,” J. Near Infrared Spectrosc. 9, 63–95 (2001).
[CrossRef]

2000 (1)

1999 (1)

A. Brablec, D. Trunec, and F. Štastný, “Deconvolution of spectral line profiles: solution of the inversion problem,” J. Phys. D 32, 1870–1875 (1999).
[CrossRef]

1997 (1)

A. S. Kaminskii, E. L. Kosarev, and E. V. Lavrov, “Using comb-like instrumental functions in high-resolution spectroscopy,” Meas. Sci. Technol. 8, 864–870 (1997).
[CrossRef]

1994 (1)

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
[CrossRef]

1977 (1)

1964 (1)

Abe, M.

Andrés, P.

Azaña, J.

N. K. Berger, B. Levit, B. Fischer, and J. Azaña, “Picosecond flat-top pulse generation by low-bandwidth electro-optic sinusoidal phase modulation,” Opt. Lett. 33, 125–127 (2008).
[CrossRef]

J. Azaña, N. K. Berger, B. Levit, and B. Fischer, “Spectro-temporal imaging of optical pulses with a single time lens,” IEEE Photon. Technol. Lett. 16, 882–884 (2004).
[CrossRef]

Baney, D. M.

B. Szafraniec, A. Lee, J. Y. Law, W. I. McAlexander, R. D. Pering, T. S. Tan, and D. M. Baney, “Swept coherent optical spectrum analysis,” IEEE Trans. Instrum. Meas. 53, 203–215 (2004).
[CrossRef]

Banyai, W. C.

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
[CrossRef]

Barthélémy, A.

Berger, N. K.

N. K. Berger, “Enhancement of resolution of optical spectrum analysers with thermally tuned sampled fibre Bragg grating,” Electron. Lett. 46, 1457–1458 (2010).
[CrossRef]

N. K. Berger, B. Levit, and B. Fischer, “Measurement of fiber chromatic dispersion using spectral interferometry with modulation of dispersed laser pulses,” Opt. Commun. 283, 3953–3956 (2010).
[CrossRef]

N. K. Berger, “Spectral measurements with superresolution based on periodic modulation of the spectrum,” Appl. Opt. 47, 6535–6542 (2008).
[CrossRef]

N. K. Berger, B. Levit, B. Fischer, and J. Azaña, “Picosecond flat-top pulse generation by low-bandwidth electro-optic sinusoidal phase modulation,” Opt. Lett. 33, 125–127 (2008).
[CrossRef]

J. Azaña, N. K. Berger, B. Levit, and B. Fischer, “Spectro-temporal imaging of optical pulses with a single time lens,” IEEE Photon. Technol. Lett. 16, 882–884 (2004).
[CrossRef]

Blass, W. E.

W. E. Blass and G. W. Halsey, “Instrumental considerations,” in Deconvolution of Images and Spectra, P. A. Jansson, ed. (Academic, 1997), pp. 200–235.

Bloom, D. M.

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
[CrossRef]

Bousquet, P.

P. Bousquet, Spectroscopy and Its Instrumentation (Hilger, 1971).

Brablec, A.

A. Brablec, D. Trunec, and F. Štastný, “Deconvolution of spectral line profiles: solution of the inversion problem,” J. Phys. D 32, 1870–1875 (1999).
[CrossRef]

Buslov, D. K.

Chung, Y.

Edelstein, J.

D. J. Erskine, J. Edelstein, W. M. Feuerstein, and B. Welsh, “High-resolution broadband spectroscopy using an externally dispersed interferometer,” Astrophys. J. 592, L103–L106 (2003).
[CrossRef]

Erskine, D. J.

D. J. Erskine, J. Edelstein, W. M. Feuerstein, and B. Welsh, “High-resolution broadband spectroscopy using an externally dispersed interferometer,” Astrophys. J. 592, L103–L106 (2003).
[CrossRef]

Feuerstein, W. M.

D. J. Erskine, J. Edelstein, W. M. Feuerstein, and B. Welsh, “High-resolution broadband spectroscopy using an externally dispersed interferometer,” Astrophys. J. 592, L103–L106 (2003).
[CrossRef]

Fischer, B.

N. K. Berger, B. Levit, and B. Fischer, “Measurement of fiber chromatic dispersion using spectral interferometry with modulation of dispersed laser pulses,” Opt. Commun. 283, 3953–3956 (2010).
[CrossRef]

N. K. Berger, B. Levit, B. Fischer, and J. Azaña, “Picosecond flat-top pulse generation by low-bandwidth electro-optic sinusoidal phase modulation,” Opt. Lett. 33, 125–127 (2008).
[CrossRef]

J. Azaña, N. K. Berger, B. Levit, and B. Fischer, “Spectro-temporal imaging of optical pulses with a single time lens,” IEEE Photon. Technol. Lett. 16, 882–884 (2004).
[CrossRef]

Foster, M. A.

Gaeta, A. L.

Godil, A. A.

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
[CrossRef]

Halsey, G. W.

W. E. Blass and G. W. Halsey, “Instrumental considerations,” in Deconvolution of Images and Spectra, P. A. Jansson, ed. (Academic, 1997), pp. 200–235.

Harris, J. L.

Helbert, J. M.

Hwang, D.

Jansson, P. A.

P. A. Jansson, “Modern constrained nonlinear methods,” in Deconvolution of Images and Spectra, P. A. Jansson, ed. (Academic, 1997), pp. 107–181.

Jiang, J. H.

Y. Ozaki, S. Šašić, and J. H. Jiang, “How can we unravel complicated near infrared spectra? Recent progress in spectral analysis methods for resolution enhancement and band assignments in the near infrared region,” J. Near Infrared Spectrosc. 9, 63–95 (2001).
[CrossRef]

Kalashyan, M.

Kaminskii, A. S.

A. S. Kaminskii, E. L. Kosarev, and E. V. Lavrov, “Using comb-like instrumental functions in high-resolution spectroscopy,” Meas. Sci. Technol. 8, 864–870 (1997).
[CrossRef]

Kauffman, M. T.

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
[CrossRef]

Kosarev, E. L.

A. S. Kaminskii, E. L. Kosarev, and E. V. Lavrov, “Using comb-like instrumental functions in high-resolution spectroscopy,” Meas. Sci. Technol. 8, 864–870 (1997).
[CrossRef]

Laforie, P.

Lancis, J.

Lavrov, E. V.

A. S. Kaminskii, E. L. Kosarev, and E. V. Lavrov, “Using comb-like instrumental functions in high-resolution spectroscopy,” Meas. Sci. Technol. 8, 864–870 (1997).
[CrossRef]

Law, J. Y.

B. Szafraniec, A. Lee, J. Y. Law, W. I. McAlexander, R. D. Pering, T. S. Tan, and D. M. Baney, “Swept coherent optical spectrum analysis,” IEEE Trans. Instrum. Meas. 53, 203–215 (2004).
[CrossRef]

Lee, A.

B. Szafraniec, A. Lee, J. Y. Law, W. I. McAlexander, R. D. Pering, T. S. Tan, and D. M. Baney, “Swept coherent optical spectrum analysis,” IEEE Trans. Instrum. Meas. 53, 203–215 (2004).
[CrossRef]

Levit, B.

N. K. Berger, B. Levit, and B. Fischer, “Measurement of fiber chromatic dispersion using spectral interferometry with modulation of dispersed laser pulses,” Opt. Commun. 283, 3953–3956 (2010).
[CrossRef]

N. K. Berger, B. Levit, B. Fischer, and J. Azaña, “Picosecond flat-top pulse generation by low-bandwidth electro-optic sinusoidal phase modulation,” Opt. Lett. 33, 125–127 (2008).
[CrossRef]

J. Azaña, N. K. Berger, B. Levit, and B. Fischer, “Spectro-temporal imaging of optical pulses with a single time lens,” IEEE Photon. Technol. Lett. 16, 882–884 (2004).
[CrossRef]

Lin, A.

Lipson, M.

Liu, X.

Lórenz-Fonfría, V. A.

Louradour, F.

Mansuryan, T.

Masuda, H.

Matoušek, V.

M. Morháč and V. Matoušek, “Complete positive deconvolution of spectrometric data,” Digit. Signal Process. 19, 372–392 (2009).
[CrossRef]

McAlexander, W. I.

B. Szafraniec, A. Lee, J. Y. Law, W. I. McAlexander, R. D. Pering, T. S. Tan, and D. M. Baney, “Swept coherent optical spectrum analysis,” IEEE Trans. Instrum. Meas. 53, 203–215 (2004).
[CrossRef]

Miche, P.

Moon, D. S.

Morhác, M.

M. Morháč and V. Matoušek, “Complete positive deconvolution of spectrometric data,” Digit. Signal Process. 19, 372–392 (2009).
[CrossRef]

Morioka, T.

Mouradian, L.

Nikonenko, N. A.

Ohara, T.

Okawachi, Y.

Ozaki, Y.

Y. Ozaki, S. Šašić, and J. H. Jiang, “How can we unravel complicated near infrared spectra? Recent progress in spectral analysis methods for resolution enhancement and band assignments in the near infrared region,” J. Near Infrared Spectrosc. 9, 63–95 (2001).
[CrossRef]

Padrós, E.

Pering, R. D.

B. Szafraniec, A. Lee, J. Y. Law, W. I. McAlexander, R. D. Pering, T. S. Tan, and D. M. Baney, “Swept coherent optical spectrum analysis,” IEEE Trans. Instrum. Meas. 53, 203–215 (2004).
[CrossRef]

Petrov, G. M.

G. M. Petrov, “A simple algorithm for spectral line deconvolution,” J. Quant. Spectrosc. Radiat. Transfer 72, 281–287 (2002).
[CrossRef]

Salem, R.

Šašic, S.

Y. Ozaki, S. Šašić, and J. H. Jiang, “How can we unravel complicated near infrared spectra? Recent progress in spectral analysis methods for resolution enhancement and band assignments in the near infrared region,” J. Near Infrared Spectrosc. 9, 63–95 (2001).
[CrossRef]

Štastný, F.

A. Brablec, D. Trunec, and F. Štastný, “Deconvolution of spectral line profiles: solution of the inversion problem,” J. Phys. D 32, 1870–1875 (1999).
[CrossRef]

Sun, G.

Sushko, N. I.

Szafraniec, B.

B. Szafraniec, A. Lee, J. Y. Law, W. I. McAlexander, R. D. Pering, T. S. Tan, and D. M. Baney, “Swept coherent optical spectrum analysis,” IEEE Trans. Instrum. Meas. 53, 203–215 (2004).
[CrossRef]

Takahashi, H.

Takara, H.

Tan, T. S.

B. Szafraniec, A. Lee, J. Y. Law, W. I. McAlexander, R. D. Pering, T. S. Tan, and D. M. Baney, “Swept coherent optical spectrum analysis,” IEEE Trans. Instrum. Meas. 53, 203–215 (2004).
[CrossRef]

Torres-Company, V.

Trunec, D.

A. Brablec, D. Trunec, and F. Štastný, “Deconvolution of spectral line profiles: solution of the inversion problem,” J. Phys. D 32, 1870–1875 (1999).
[CrossRef]

Turner-Foster, A. C.

Villaverde, J.

Welsh, B.

D. J. Erskine, J. Edelstein, W. M. Feuerstein, and B. Welsh, “High-resolution broadband spectroscopy using an externally dispersed interferometer,” Astrophys. J. 592, L103–L106 (2003).
[CrossRef]

Yamamoto, T.

Yesayan, G.

Zeytunyan, A.

Zhbankov, R. G.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
[CrossRef]

Appl. Spectrosc. (2)

Astrophys. J. (1)

D. J. Erskine, J. Edelstein, W. M. Feuerstein, and B. Welsh, “High-resolution broadband spectroscopy using an externally dispersed interferometer,” Astrophys. J. 592, L103–L106 (2003).
[CrossRef]

Digit. Signal Process. (1)

M. Morháč and V. Matoušek, “Complete positive deconvolution of spectrometric data,” Digit. Signal Process. 19, 372–392 (2009).
[CrossRef]

Electron. Lett. (1)

N. K. Berger, “Enhancement of resolution of optical spectrum analysers with thermally tuned sampled fibre Bragg grating,” Electron. Lett. 46, 1457–1458 (2010).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

J. Azaña, N. K. Berger, B. Levit, and B. Fischer, “Spectro-temporal imaging of optical pulses with a single time lens,” IEEE Photon. Technol. Lett. 16, 882–884 (2004).
[CrossRef]

IEEE Trans. Instrum. Meas. (1)

B. Szafraniec, A. Lee, J. Y. Law, W. I. McAlexander, R. D. Pering, T. S. Tan, and D. M. Baney, “Swept coherent optical spectrum analysis,” IEEE Trans. Instrum. Meas. 53, 203–215 (2004).
[CrossRef]

J. Lightwave Technol. (1)

J. Near Infrared Spectrosc. (1)

Y. Ozaki, S. Šašić, and J. H. Jiang, “How can we unravel complicated near infrared spectra? Recent progress in spectral analysis methods for resolution enhancement and band assignments in the near infrared region,” J. Near Infrared Spectrosc. 9, 63–95 (2001).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys. D (1)

A. Brablec, D. Trunec, and F. Štastný, “Deconvolution of spectral line profiles: solution of the inversion problem,” J. Phys. D 32, 1870–1875 (1999).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

G. M. Petrov, “A simple algorithm for spectral line deconvolution,” J. Quant. Spectrosc. Radiat. Transfer 72, 281–287 (2002).
[CrossRef]

Meas. Sci. Technol. (1)

A. S. Kaminskii, E. L. Kosarev, and E. V. Lavrov, “Using comb-like instrumental functions in high-resolution spectroscopy,” Meas. Sci. Technol. 8, 864–870 (1997).
[CrossRef]

Opt. Commun. (1)

N. K. Berger, B. Levit, and B. Fischer, “Measurement of fiber chromatic dispersion using spectral interferometry with modulation of dispersed laser pulses,” Opt. Commun. 283, 3953–3956 (2010).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Other (3)

P. Bousquet, Spectroscopy and Its Instrumentation (Hilger, 1971).

P. A. Jansson, “Modern constrained nonlinear methods,” in Deconvolution of Images and Spectra, P. A. Jansson, ed. (Academic, 1997), pp. 107–181.

W. E. Blass and G. W. Halsey, “Instrumental considerations,” in Deconvolution of Images and Spectra, P. A. Jansson, ed. (Academic, 1997), pp. 200–235.

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Figures (15)

Fig. 1.
Fig. 1.

Calculated transmission spectrum of the dual-bandpass SFBG proposed in [5]. The SFBG parameters are given in Section 5.

Fig. 2.
Fig. 2.

Magnified left transmission peak of Fig. 1 (solid curve) and this peak as it would be measured by an OSA with a resolution of 10 pm (dashed curve).

Fig. 3.
Fig. 3.

Left half of the spectrum shown in Fig. 1 as it would be measured by an OSA with a resolution of 10 pm.

Fig. 4.
Fig. 4.

Fourier transform, H(t), of the 1.25 GHz (10 pm) Gaussian instrumental function of an OSA. The dashed lines bound the chosen interval of the “passband,” Δtpas=1.11ns.

Fig. 5.
Fig. 5.

Fourier transforms, H(t), of the instrumental function of an OSA from Fig. 4, and Ffilt(t) of the left half of the SFBG transmission spectrum, Sfilt(ω), shown in Fig. 1, multiplied by Slas(ω)Sap(ω). Ffilt(t) is shifted relative to H(t) by the third harmonic of the spectral modulation by 3Δtpas.

Fig. 6.
Fig. 6.

Experimental setup proposed for spectral measurements with superresolution.

Fig. 7.
Fig. 7.

Three periods of the quasi-periodic spectrum, Smod(ω), of the modulated stretched laser pulse. The spectrum period is Λω/(2π)=0.899GHz.

Fig. 8.
Fig. 8.

Absolute values of the Fourier coefficients of the spectrum shown in Fig. 7.

Fig. 9.
Fig. 9.

Spectrum of the modulated stretched laser pulse, transmitted through the SFBG that would be measured by an OSA with the 1.25 GHz (10 pm) instrumental function after the 17th shift of the modulation voltage by T/27=0.926ps.

Fig. 10.
Fig. 10.

Fourier transform of the “measured” spectrum shown in Fig. 9.

Fig. 11.
Fig. 11.

Restored spectrum Fourier transform, Ffilt(t), composed of the parts Ffilt,s(t) calculated according to Eq. (11).

Fig. 12.
Fig. 12.

Restored transmission spectrum, Sfilt(ω), of the SFBG calculated as the inverse Fourier transform of the function Ffilt(t) shown in Fig. 11, divided by the Gaussian apodization function Sap(ω) and laser pulse spectrum Slas(ω).

Fig. 13.
Fig. 13.

Magnified central transmission peak of the restored SFBG spectrum shown in Fig. 12 (solid curve) and the same (left) peak of the original SFBG spectrum presented in Fig. 1 (dashed curve).

Fig. 14.
Fig. 14.

Transmission spectrum, Sfilt(ω), of the SFBG restored in the presence of the noise of the “measured” spectra with a level of 0.1%.

Fig. 15.
Fig. 15.

Restored parts (a) Ffilt,9(t) and (b) Ffilt,11(t) corresponding to the modulation harmonic orders 9 and 11, respectively, for a noise level of 0.5% (solid curves) and in the absence of noise (dashed curves).

Equations (17)

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

Smeas(ω)=I(ωΩ)Sin(Ω)dΩ,
Fmeas(t)=H(t)Fin(t),
2M+1=ΔtF/Δtpas,
SRC=ΔtF/ΔtH.
fspec(ω)=s=MMBsexp(isΔtpasω),
Λω=2π/Δtpas.
Sin(ω)=Sfilt(ω)s=MMBsexp(isΔtpasω),
Sfilt(ω)=Sfilt(ω)Slas(ω).
Sin,l(ω)=Sfilt(ω)s=MMBsexp{isΔtpas[ωlΛω/(2M+1)]}.
Fmeas,l(t)=H(t)s=MMBsexp[i2πsl/(2M+1)]Ffilt(tsΔtpas).
Ffilt,s(t)={1/[(2M+1)H(t)Bs]}l=02MFmeas,l(t)exp[i2πsl/(2M+1)].
ftemp(t)=n=NNcnexp(inωmt),
Smod(ω)Slas(ω)s=MMBsexp(isβ2Lωmω),Bs=k=NNck+sck*exp{i[(k+s)2k2]β2Lωm2/2},|k+s|N,
Sin(ω)=Smod(ω)Sfilt(ω)=Sfilt(ω)Slas(ω)s=MMBsexp[isβ2Lωmω].
β2Lωm=Δtpas.
Δτl=lT/(2M+1)
Δωl=lΛω/(2M+1).

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