Abstract

We extend the approach recently applied for achieving spatial and temporal superresolution to spectral measurements with superresolution. The light passes previously through a filter with a periodic spectral response. Owing to the spectral modulation, the parts of the spectrum Fourier transform are shifted and transmitted through the passband of the Fourier transform of the spectrometer instrumental function. Thus, all the parts of the spectrum Fourier transform are recorded by the spectrometer and then restored by a special procedure. An inverse Fourier transform gives the spectrum restored with superresolution. We numerically demonstrate more than tenfold enhancement of the resolution by using a sampled fiber Bragg grating for spectral modulation.

© 2008 Optical Society of America

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References

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  1. Z. Zalevsky and D. Mendlovic, Optical Superresolution (Springer-Verlag, 2004).
  2. R. Heintzmann, T. M. Jovin, and C. Cremer, “Saturated patterned excitation microscopy--a concept for optical resolution improvement,” J. Opt. Soc. Am. A 19, 1599-1609 (2002).
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  3. M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. U.S.A. 102, 13081-13086 (2005).
    [CrossRef] [PubMed]
  4. N. K. Berger and B. Fischer, “Unified approach to short optical pulse recording and spatial imaging with subwavelength resolution,” Opt. Commun. 274, 50-58 (2007).
    [CrossRef]
  5. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
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    [CrossRef]
  7. J. L. Harris, “Diffraction and resolving power,” J. Opt. Soc. Am. 54, 931-936 (1964).
    [CrossRef]
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    [CrossRef]
  9. P. Bousquet, Spectroscopy and its Instrumentation (Hilger, 1971).
  10. P. A. Jansson, “Modern constrained nonlinear methods,” in Deconvolution of Images and Spectra, P. A. Jansson, ed. (Academic, 1997), pp. 107-172.
  11. A. Brablec, D. Trunec, and F. Šťastný, “Deconvolution of spectral line profiles: solution of the inversion problem,” J. Phys. D 32, 1870-1875 (1999).
    [CrossRef]
  12. S. Mijovic and M. Vuceljic, “Comparison between direct and inverse approaches in problems of recovering the true profile of a spectral line,” J. Quant. Spectrosc. Radiat. Transfer 77, 79-86 (2003).
    [CrossRef]
  13. G. M. Petrov, “A simple algorithm for spectral line deconvolution,” J. Quant. Spectrosc. Radiat. Transfer 72, 281-287 (2002).
    [CrossRef]
  14. L. Xu, H. Yang, K. Chen, Q. Tan, Q. He, and G. Jin, “Resolution enhancement by combination of subpixel and deconvolution in miniature spectrometers,” Appl. Opt. 46, 3210-3214 (2007).
    [CrossRef] [PubMed]
  15. 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]
  16. B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, “Long periodic superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1620-1622 (1994).
    [CrossRef]
  17. S. Osawa, N. Wada, K. Kitayama, and W. Chujo, “Arbitrary-shaped optical pulse train synthesis using weight/phase-programmable 32-taped delay line waveguide filter,” Electron. Lett. 37, 1356-1357 (2001).
    [CrossRef]
  18. N. K. Berger, B. Levit, A. Bekker, and B. Fischer, “Compression of periodic optical pulses using temporal fractional Talbot effect,” IEEE Photon. Technol. Lett. 16, 1855-1857(2004).
    [CrossRef]

2007

N. K. Berger and B. Fischer, “Unified approach to short optical pulse recording and spatial imaging with subwavelength resolution,” Opt. Commun. 274, 50-58 (2007).
[CrossRef]

L. Xu, H. Yang, K. Chen, Q. Tan, Q. He, and G. Jin, “Resolution enhancement by combination of subpixel and deconvolution in miniature spectrometers,” Appl. Opt. 46, 3210-3214 (2007).
[CrossRef] [PubMed]

2005

M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. U.S.A. 102, 13081-13086 (2005).
[CrossRef] [PubMed]

2004

N. K. Berger, B. Levit, A. Bekker, and B. Fischer, “Compression of periodic optical pulses using temporal fractional Talbot effect,” IEEE Photon. Technol. Lett. 16, 1855-1857(2004).
[CrossRef]

2003

S. Mijovic and M. Vuceljic, “Comparison between direct and inverse approaches in problems of recovering the true profile of a spectral line,” J. Quant. Spectrosc. Radiat. Transfer 77, 79-86 (2003).
[CrossRef]

2002

2001

S. Osawa, N. Wada, K. Kitayama, and W. Chujo, “Arbitrary-shaped optical pulse train synthesis using weight/phase-programmable 32-taped delay line waveguide filter,” Electron. Lett. 37, 1356-1357 (2001).
[CrossRef]

1999

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

1997

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

B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, “Long periodic superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1620-1622 (1994).
[CrossRef]

1990

E. L. Kosarev, “Shannon's superresolution limit for signal recovery,” Inverse Probl. 6, 55-76 (1990).
[CrossRef]

1968

1964

Bekker, A.

N. K. Berger, B. Levit, A. Bekker, and B. Fischer, “Compression of periodic optical pulses using temporal fractional Talbot effect,” IEEE Photon. Technol. Lett. 16, 1855-1857(2004).
[CrossRef]

Berger, N. K.

N. K. Berger and B. Fischer, “Unified approach to short optical pulse recording and spatial imaging with subwavelength resolution,” Opt. Commun. 274, 50-58 (2007).
[CrossRef]

N. K. Berger, B. Levit, A. Bekker, and B. Fischer, “Compression of periodic optical pulses using temporal fractional Talbot effect,” IEEE Photon. Technol. Lett. 16, 1855-1857(2004).
[CrossRef]

Bousquet, P.

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

Brablec, A.

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

Chen, K.

Chujo, W.

S. Osawa, N. Wada, K. Kitayama, and W. Chujo, “Arbitrary-shaped optical pulse train synthesis using weight/phase-programmable 32-taped delay line waveguide filter,” Electron. Lett. 37, 1356-1357 (2001).
[CrossRef]

Cremer, C.

Eggleton, B. J.

B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, “Long periodic superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1620-1622 (1994).
[CrossRef]

Fischer, B.

N. K. Berger and B. Fischer, “Unified approach to short optical pulse recording and spatial imaging with subwavelength resolution,” Opt. Commun. 274, 50-58 (2007).
[CrossRef]

N. K. Berger, B. Levit, A. Bekker, and B. Fischer, “Compression of periodic optical pulses using temporal fractional Talbot effect,” IEEE Photon. Technol. Lett. 16, 1855-1857(2004).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Gustafsson, M. G. L.

M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. U.S.A. 102, 13081-13086 (2005).
[CrossRef] [PubMed]

Harris, J. L.

Harris, R. W.

He, Q.

Heintzmann, R.

Jansson, P. A.

P. A. Jansson, “Modern constrained nonlinear methods,” in Deconvolution of Images and Spectra, P. A. Jansson, ed. (Academic, 1997), pp. 107-172.

Jin, G.

Jovin, T. 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]

Kitayama, K.

S. Osawa, N. Wada, K. Kitayama, and W. Chujo, “Arbitrary-shaped optical pulse train synthesis using weight/phase-programmable 32-taped delay line waveguide filter,” Electron. Lett. 37, 1356-1357 (2001).
[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]

E. L. Kosarev, “Shannon's superresolution limit for signal recovery,” Inverse Probl. 6, 55-76 (1990).
[CrossRef]

Krug, P. A.

B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, “Long periodic superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1620-1622 (1994).
[CrossRef]

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]

Levit, B.

N. K. Berger, B. Levit, A. Bekker, and B. Fischer, “Compression of periodic optical pulses using temporal fractional Talbot effect,” IEEE Photon. Technol. Lett. 16, 1855-1857(2004).
[CrossRef]

Mendlovic, D.

Z. Zalevsky and D. Mendlovic, Optical Superresolution (Springer-Verlag, 2004).

Mijovic, S.

S. Mijovic and M. Vuceljic, “Comparison between direct and inverse approaches in problems of recovering the true profile of a spectral line,” J. Quant. Spectrosc. Radiat. Transfer 77, 79-86 (2003).
[CrossRef]

Osawa, S.

S. Osawa, N. Wada, K. Kitayama, and W. Chujo, “Arbitrary-shaped optical pulse train synthesis using weight/phase-programmable 32-taped delay line waveguide filter,” Electron. Lett. 37, 1356-1357 (2001).
[CrossRef]

Ouellette, F.

B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, “Long periodic superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1620-1622 (1994).
[CrossRef]

Petrov, G. M.

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

Poladian, L.

B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, “Long periodic superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1620-1622 (1994).
[CrossRef]

Rushforth, C. K.

Štastný, F.

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

Tan, Q.

Trunec, D.

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

Vuceljic, M.

S. Mijovic and M. Vuceljic, “Comparison between direct and inverse approaches in problems of recovering the true profile of a spectral line,” J. Quant. Spectrosc. Radiat. Transfer 77, 79-86 (2003).
[CrossRef]

Wada, N.

S. Osawa, N. Wada, K. Kitayama, and W. Chujo, “Arbitrary-shaped optical pulse train synthesis using weight/phase-programmable 32-taped delay line waveguide filter,” Electron. Lett. 37, 1356-1357 (2001).
[CrossRef]

Xu, L.

Yang, H.

Zalevsky, Z.

Z. Zalevsky and D. Mendlovic, Optical Superresolution (Springer-Verlag, 2004).

Appl. Opt.

Electron. Lett.

B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, “Long periodic superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1620-1622 (1994).
[CrossRef]

S. Osawa, N. Wada, K. Kitayama, and W. Chujo, “Arbitrary-shaped optical pulse train synthesis using weight/phase-programmable 32-taped delay line waveguide filter,” Electron. Lett. 37, 1356-1357 (2001).
[CrossRef]

IEEE Photon. Technol. Lett.

N. K. Berger, B. Levit, A. Bekker, and B. Fischer, “Compression of periodic optical pulses using temporal fractional Talbot effect,” IEEE Photon. Technol. Lett. 16, 1855-1857(2004).
[CrossRef]

Inverse Probl.

E. L. Kosarev, “Shannon's superresolution limit for signal recovery,” Inverse Probl. 6, 55-76 (1990).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. D

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

J. Quant. Spectrosc. Radiat. Transfer

S. Mijovic and M. Vuceljic, “Comparison between direct and inverse approaches in problems of recovering the true profile of a spectral line,” J. Quant. Spectrosc. Radiat. Transfer 77, 79-86 (2003).
[CrossRef]

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

Meas. Sci. Technol.

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.

N. K. Berger and B. Fischer, “Unified approach to short optical pulse recording and spatial imaging with subwavelength resolution,” Opt. Commun. 274, 50-58 (2007).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A.

M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. U.S.A. 102, 13081-13086 (2005).
[CrossRef] [PubMed]

Other

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

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. 107-172.

Z. Zalevsky and D. Mendlovic, Optical Superresolution (Springer-Verlag, 2004).

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Figures (11)

Fig. 1
Fig. 1

1, Original spectrum; 2, instrumental function of a spectrometer; 3, spectrum that could be measured without superresolution.

Fig. 2
Fig. 2

0, Fourier transform of the original spectrum; 1, its replica shifted by τ by the first-order modulation harmonic; 2, Fourier transform of the instrumental function, H ( t ) . The dashed lines bound the interval ( τ / 2 , τ / 2 ) .

Fig. 3
Fig. 3

Shifted parts U ( t n τ ) of the spectrum Fourier transform within the interval ( τ / 2 , τ / 2 ) . Curves 0 and 1 are the portions of curves 0 and 1 shown in Fig. 2, respectively, lying between the dashed lines.

Fig. 4
Fig. 4

Filters with periodic spectral responses for spectral modulation: (a) sampled FBG consisting of equally spaced uniform FBGs, (b) waveguide (or fiber) array with the equal delay differences between the waveguides (fibers), (c) Fabry–Perot inter ferometer. M 1 , M 2 : mirrors.

Fig. 5
Fig. 5

Reflectivities of the sampled FBG shown in Fig. 4a for two cases: 1, all of the reflection phases of the individual gratings are equal to zero; 2, only the reflection phase of the fifth grating is nonzero and equal to 0.2 rad .

Fig. 6
Fig. 6

Fourier coefficients of the spectral responses shown in Fig. 5.

Fig. 7
Fig. 7

Results of the spectral modulation of the original spectrum by the filter with the spectral responses shown in Fig. 5.

Fig. 8
Fig. 8

Results of the “measurement” of the modulated spectra shown in Fig. 7 by a spectrometer with a resolution of 0.02 nm .

Fig. 9
Fig. 9

Fourier transforms of the measured modulated spectra shown in Fig. 8.

Fig. 10
Fig. 10

Original (dotted curve) and restored (solid curve) spectra in the absence of noise. The width of the spectrum dip is 200 MHz ( 1.6 pm ) .

Fig. 11
Fig. 11

Original (dotted curve) and restored (solid curve) spectra with a noise of 0.1%. The width of the spectrum dip is 125 MHz ( 1 pm ) . The parameters of the sampled FBG are identical to those for Fig. 10.

Equations (9)

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K ( Ω ω ) u ( ω ) d ω = v ( Ω ) ,
H ( t ) U ( t ) = V ( t ) ,
( 2 N + 1 ) τ T ,
R ( ω ) = n = N N c n exp ( i n τ ω ) , τ = 2 π / Λ ω ,
u mod ( ω ) = R ( ω ) u ( ω ) = u ( ω ) n = N N c n exp ( i n τ ω ) .
U mod ( t ) = n = N N c n U ( t n τ ) .
H ( t ) U mod ( t ) = V mod ( t ) ,
H ( t ) n = N N c n U ( t n τ ) = V mod ( t ) .
H ( k Δ t ) n = - N N c n ( m ) U n ( k Δ t ) = V mod ( m ) ( k Δ t ) ,

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